# Gravitational search for near Earth black holes or other compact dark
objects

Tomoyo Namigata<EMAIL_ADDRESS>Center for Exploration of Energy and Matter
and Department of Physics, Indiana University, Bloomington, IN 47405, USA  C.
J. Horowitz<EMAIL_ADDRESS>Center for Exploration of Energy and Matter
and Department of Physics, Indiana University, Bloomington, IN 47405, USA  R.
Widmer-Schnidrig<EMAIL_ADDRESS>Black Forest Observatory (BFO),
Heubach 206, D-77709 Wolfach, Germany  Institute of Geodesy, University of
Stuttgart, Stuttgart, Germany

###### Abstract

Primordial black holes, with masses comparable to asteroids, are an attractive
possibility for dark matter. In addition, other forms of dark matter could
form compact dark objects (CDO). We search for small tidal accelerations from
low mass black holes or CDOs orbiting near the Earth, and find none. Using
about 10 years of data from the superconducting gravimeters in the Black
Forest Observatory in South-Western Germany and at Djougou, Northern Benin in
Western Africa we set an upper limit on the maximum mass of any dark object
orbiting the Earth as a function of orbital radius. For semi-major axis less
than two earth radii we exclude all black holes or CDOs with masses larger
than $6.7\times 10^{13}$ kg. Lower mass primordial black holes may be strongly
constrained by Hawking radiation. We conclude that near Earth black holes are
extremely unlikely.

Primordial black holes (PBH), made by density fluctuations in the early
universe, are an attractive candidate for dark matter Green and Kavanagh
(2021); Carr and Kühnel (2020). These PBH could have a large range of possible
masses depending on the scale of the density fluctuations. However, dark
matter made of PBH with masses $M_{PBH}<10^{14}$ kg is constrained by Hawking
radiation Coogan et al. (2021) while $M_{PBH}>10^{19}$ kg is constrained by
microlensing Niikura et al. (2019); Alcock et al. (1995, 2000); Tisserand et
al. (2007); Paczynski (1986). This leaves a very interesting open window of
PBH masses between $10^{14}$ and $10^{19}$ kg that is largely unconstrained by
present observations. For example, De Luca et al. explain NANOgrav
observations Arzoumanian et al. (2020) with gravitational waves from the
formation of PBH with masses in this range De Luca et al. (2021).

There could be black holes in the solar system. Indeed, Scholtz and Unwin
speculate that Planet 9, a possible outer solar system object of several Earth
masses, is a PBH Scholtz and Unwin (2020); Siraj and Loeb (2020). If dark
matter is composed of PBH, we expect $\approx
10^{3}(10^{14}{\rm\,kg}/M_{PBH})$ PBH to be present in the solar system (of
volume $\approx 10^{6}$ AU3) at any given time. It is possible that one of
these objects could approach Earth or even be captured into orbit through a
three body interaction.

More generally, it is possible that Earth may have one or more “dark moon” in
addition to Luna. These objects could be composed of PBH or other forms of
dark matter that allows them to escape detection via electromagnetic
observations. In addition to PBH, other forms of dark matter may concentrate
into compact dark objects (CDO). These objects are assumed to have small non-
gravitational interactions with normal matter. Some possibilities or names for
CDO include Boson Stars Liebling and Palenzuela (2017), Dark Blobs Grabowska
et al. (2018), asymmetric dark matter nuggets Gresham et al. (2018), Exotic
Compact Objects Giudice et al. (2016), Ultra Compact Mini Halos (UCMH)
Bringmann et al. (2012) made for example of axions Yang et al. (2017), and
Macros Jacobs et al. (2015).

The lifetime of near Earth orbits for conventional objects is short because of
air resistance. Dark objects, be they PBH or CDO, may have unique long lived
near Earth orbits because of their small non-gravitational interactions. In
this letter, we directly search for small gravitational accelerations (tides)
from a PBH or CDO, with a $10^{14}-10^{19}$ kg mass, in a near Earth orbit.

In previous work we searched for gravitational waves from CDO merging with
neutron stars Horowitz and Reddy (2019) or orbiting inside the sun Horowitz et
al. (2019). We also searched for small gravitational accelerations from CDO
moving inside the Earth Horowitz and Widmer-Schnidrig (2020), see also Hu et
al. (2020); Figueroa et al. (2021); Tino (2021). In the present letter we
significantly extend ref. Horowitz and Widmer-Schnidrig (2020) to search for
CDO or PBH orbiting partially or completely outside the Earth with a range of
longer orbital periods.

One might observe electromagnetic radiation as material accretes onto a black
hole, see for example Siraj and Loeb (2020). Even if there is little
accretion, one can still search for a dark object’s gravity by accurately
tracking spacecraft, see for example Witten (2020); Lawrence and Rogoszinski
(2020), or using a gravimeter on Earth.

Gravimeters Goodkind (1999) measure the local acceleration due to gravity and
can detect small tides from an orbiting CDO or PBH. Sensitive superconducting
gravimeters have been deployed at several locations around the world Voigt et
al. (2016). They are used to observe a wide range of geophysical phenomena
including Chandler wobble, solid Earth tides, post glacial rebound, seismic
free oscillations and hydrological processes (Hinderer et al., 2007). In
addition to geophysics, they have been used to search for a dependence of
gravity on a hypothetical preferred reference frame Will and Nordtvedt (1972);
Warburton and Goodkind (1976), or the violation of Lorentz invariance Flowers
et al. (2017); Shao et al. (2018), as the Earth translates or rotates. In
addition, gravimeters have been used to search for oscillations of the Earth
excited by gravitational waves Coughlin and Harms (2014).

An object orbiting through or around the Earth with coordinate ${\bf r}(t)$
will have an acceleration,

$\frac{d^{2}{\bf r}(t)}{dt^{2}}=-G\frac{M_{enc}(r)}{r^{3}}{\bf r}\,.$ (1)

Here $G$ is Newton’s constant and $M_{enc}(r)$ is the enclosed mass of the
Earth that is interior to $r$. This reduces to the mass of the Earth $M_{E}$
for $r>R_{E}$ the radius of the Earth, see ref. Horowitz and Widmer-Schnidrig
(2020) for details. We numerically integrate Eq. 1 using the Velocity Verlet
algorithm with a time step $\Delta t=1$ s, typically for a total time of
$2^{26}$ s. This simple procedure works for orbits inside, partially inside,
and outside the Earth. We assume the Earth is spherically symmetric.
Explicitly including the Earth’s quadrupole moment does not significantly
change results for excluded masses, see below. We also neglect small
perturbations from the moon or the sun. These are expected to be unimportant
for $r$ near the Earth.

A gravimeter located at ${\bf R}(t)$ will feel a time-dependent acceleration
$a(t)$ that is the difference of the gravitational acceleration from the
compact object minus the acceleration of Earth towards the object,

$a(t)=GM_{D}\Bigl{\\{}\frac{M_{enc}(r)}{M_{E}\,r^{3}}{\bf r}-\frac{{\bf
r}-{\bf R}}{|{\bf r}-{\bf R}|^{3}}\Bigr{\\}}\cdot\hat{\bf R}.$ (2)

Here $M_{D}$ is the mass of the dark object, either $M_{PBH}$ or $M_{CDO}$,
and $\hat{\bf R}={\bf R}/R$. Note that the gravimeter only measures the
vertical component of ${\bf a}$. We choose a coordinate system where the
gravimeter is located at ${\bf R}(t)=R_{E}(\cos\theta_{l}\cos\omega
t,\cos\theta_{l}\sin\omega t,\sin\theta_{l})$. Here $\theta_{l}$ is the
latitude of the gravimeter (48.33∘ N for the Black Forest Observatory) and
$\omega=2\pi/{\rm day}$. We fast Fourier transform (FFT) $a(t)$ from Eq. 2
(including a Hanning taper) using the numerically integrated orbit ${\bf
r}(t)$. This yields the signal amplitude $A(f)$ for frequency $f$ that will be
compared to the FFT of gravimeter data. We will be interested in $f$ from
$\approx 10^{-6}$ to $2\times 10^{-4}$ Hz depending on the orbit.

We now analyze gravimeter data. Data from super-conducting gravimeters (SGs),
deployed at various locations around the world, has been archived by the
Global Geodynamics Project (GGP, 1997-2015) and by the International
Geodynamics and Earth Tide Service (IGETS, 2015-) Voigt et al. (2016); Boy
(2016). We focus on the SG at the Black Forest Observatory (BFO at 48.33∘N,
8.33∘E) in South Western Germany because of its low noise. We also consider
the SG at Djougou, Northern Benin (DJ at 9.74∘N, 1.61∘E) in West Africa
because of its location near the equator.

We analyze 11.1 years of BFO gravimeter data from Dec. 1, 2009 to Jan. 23,
2021 BFO (1971) and 7.9 years of DJ data from Jan. 1, 2011 to Nov. 30, 2018
Boy et al. (2017). We analyze this data as in Ref. Horowitz and Widmer-
Schnidrig (2020), but now focusing on the lower frequency band $\approx
10^{-6}$ to $2\times 10^{-4}$ Hz. We carefully handle artifacts in the data:
times when the instrument behaved non-linearly due to saturation from large
quakes, operator interference or other malfunctions. We subtract from the data
a synthetic tidal model for the stations that includes the effect of ocean
loading (Wenzel, 1996). In addition we partially correct for accelerations due
to atmospheric mass fluctuations above the gravimeter based on the barometric
pressure recorded near the gravimeter, see Ref. Horowitz and Widmer-Schnidrig
(2020) for details.

In Figs. 1 and 2 we show Fourier amplitude spectra of the pressure corrected
and Hanning tapered gravity residuals for BFO and DJ, respectively. These
spectra are normalized such that a pure time domain signal
$h(t)=A_{0}\sin(2\pi f_{0}t)$ of amplitude $A_{0}$ yields $A(f_{0})=A_{0}$.
The average noise and standard deviation for a number of frequency bins is
shown along with a conservative upper limit that is 10$\sigma$ above the
average noise. A signal from a PBH above this limit can confidently be ruled
out 111If a dark object signal happened to have exactly the same frequency as
one of the lunar or solar tidal lines then the object could “hide” in the
large tidal background. This provides a small exception.. We fit these upper
limit noise curves with

$A_{\rm BFO}(f)\approx 1.25\times 10^{-4}\Bigl{(}\frac{1\ \rm
Hz}{f}\Bigr{)}^{1.1833}\ {\rm pm/s^{2}}$ (3)

for the BFO station and

$A_{\rm DJ}(f)\approx 4.71\times 10^{-3}\Bigl{(}\frac{1\ \rm
Hz}{f}\Bigr{)}^{0.909}\ {\rm pm/s^{2}}$ (4)

for the DJ station. These fits are valid for $10^{-6}$ Hz $<f<2\times 10^{-4}$
Hz.

Figure 1:  Fourier amplitude spectra of gravity residuals for the Black Forest
Observatory gravimeter versus frequency $f$. Shown in black are the mean and
standard deviation $\sigma$. The red line is a conservative upper limit of
mean plus 10$\sigma$ and the dashed blue line is the simple fit to this limit
given by Eq. 3. The three peaks in the upper limit at frequencies above
$10^{-5}$ Hz are solar and lunar diurnal, semidiurnal, and terdiurnal residual
tides. Figure 2:  Fourier amplitude spectra for the Djougou gravimeter as in
Fig. 1. The dashed blue line is the fit to the upper limit in Eq. 4.

We compare Fourier amplitude spectra $A(f)$ for a given orbit to the
gravimeter upper limit noise curves in Eqs. 3, 4 to deduce excluded masses
$M_{ex}$. The excluded mass is the smallest object mass $M_{D}$ in Eq. 2 so
that $A(f)$ just equals the noise limit curve for one harmonic frequency $f$.
Any object with $M_{D}>M_{ex}$ can be ruled out by the gravimeter data. Table
1 presents excluded masses for some selected orbits as shown in Figs. 3,4. The
final excluded mass for a given orbit is the minimum of the excluded mass
based on BFO and DJ observations.

Figure 3: Fourier amplitude $A(f)$ vs frequency $f$ for equatorial (inclination $\theta_{I}=0$) orbits observed with the DJ gravimeter. Eccentricity $\epsilon=0$ orbits with semimajor axis $a=3R_{E}$ (thick black curve) and $a=6R_{E}$ (blue curve) are shown in addition to an elliptical orbit with $a=6.05R_{E}$ and eccentricity $\epsilon=0.1$ (dotted red curve). In each case the object mass is equal to the appropriate excluded mass see Table 1. The dashed blue curve shows the DJ noise limit from Eq. 4. Figure 4:  Fourier amplitude $A(f)$ vs frequency $f$ for circular orbits of radius $a=3R_{E}$ as observed by the BFO gravimeter. The thick black lines are for inclination $\theta_{I}=0^{\circ}$ and the thin red lines are for $\theta_{I}=45^{\circ}$. In each case the object mass is equal to the appropriate excluded mass see Table 1. The blue dashed curve shows the BFO noise limit from Eq. 3.  $a$ ($R_{E}$) |  $\theta_{I}$ (∘) | $\epsilon$ | Gravimeter |  $M_{ex}$ (kg)
---|---|---|---|---
3 | 0 | 0 | DJ | $4.0\times 10^{14}$
6 | 0 | 0 | DJ | $3.7\times 10^{16}$
6.05 | 0 | 0.1 | DJ | $2.7\times 10^{16}$
3 | 0 | 0 | BFO | $5.0\times 10^{14}$
3 | 45 | 0 | BFO | $3.0\times 10^{14}$
Table 1:  Excluded mass $M_{ex}$ for example orbits with semimajor axis $a$,
inclination $\theta_{I}$, and eccentricity $\epsilon$ as observed with the
indicated gravimeter.

Figure 3 shows $A(f)$ for DJ observations of equatorial orbits (inclination
$\theta_{I}=0$). As the semimajor axis $a$ of the orbit increases, $A(f)$
shifts to lower frequencies, where the gravimeter noise is larger. This shift
is because of the larger orbital period. Elliptical orbits (with nonzero
eccentricity $\epsilon$) have higher harmonics with larger amplitudes compared
to $A(f)$ for a circular orbit. As a result $M_{ex}$ for $\epsilon>0$ is
expected to be lower than $M_{ex}$ for a circular orbit (with the same $a$).
This can be seen in another way. Elliptical orbits come closer to the
gravimeter than do circular orbits, with the same $a$. This increases the
gravity signal of the compact object and leads to a smaller $M_{ex}$.

Figure 4 shows $A(f)$ for BFO observations of circular orbits. As the
inclination of the orbit $\theta_{I}$ increases from 0 to 45∘, $M_{ex}$
decreases. In general, inclined orbits will come closer to the BFO gravimeter,
at latitude 48.33∘ N, than do equatorial orbits. This leads to larger $A(f)$,
for a given $M_{D}$, and a smaller excluded mass $M_{ex}$. Furthermore,
inclined orbits will have more harmonics in $A(f)$ involving sums and
differences of the orbital frequency and multiples of the Earth’s rotational
frequency. This is also illustrated in Fig. 4.

Figure 5:  Excluded mass $M_{ex}$ versus $a$ for equatorial orbits
$\theta_{I}=0$. Black circles are for BFO, and red squares DJ, observations of
circular orbits. Elliptical orbits observed with DJ are shown for
eccentricities $\epsilon=0.25$ downward red triangles, $0.5$ diamonds, and
$0.75$ upward triangles. Figure 6:  Excluded mass $M_{ex}$ versus $a$ for
circular orbits. The red squares show DJ observations for $\theta_{I}=0$ while
the black circles show BFO observations of inclined orbits with
$\theta_{I}\geq 30^{\circ}$. The dashed black line is $M_{ex}\approx 2.1\times
10^{12}(a/R_{E})^{5}$ kg, see Eq. 5.

Figure 5 shows excluded mass $M_{ex}$ versus $a$ for equatorial orbits
$\theta_{I}=0$. Here the DJ gravimeter dataset provides tighter constraints
than the dataset from BFO. Furthermore both gravimeters are much more
sensitive for elliptical orbits than for circular orbits.

In general $M_{ex}$ increases smoothly with $a$. However, there are resonances
where the orbital period over the Earth’s rotational period is equal to the
ratio of small integers. Resonances can introduce structure in $M_{ex}$ versus
$a$. There is a large peak in $M_{ex}$ for equatorial orbits with $a\approx
7R_{E}$. This corresponds to nearly geostationary orbits. Here objects move
very slowly with respect to the gravimeter. As a result the gravimeter is
almost blind to their presence. This blind spot is only present for nearly
equatorial circular orbits and largely vanishes by $\theta_{I}=15^{\circ}$.

Excluded masses for $a<R_{E}$ in Fig. 5 correspond to objects moving inside
the Earth as discussed in ref. Horowitz and Widmer-Schnidrig (2020). For these
objects there is a second blind spot at the center of the Earth $a=0$. Here
the object is also at rest with respect to the gravimeter.

Figure 6 shows $M_{ex}$ for circular orbits with a range of inclinations. An
approximate fit to an upper limit for all of the $M_{ex}$ in Fig. 6 is

$M_{ex}\approx 2.1\times 10^{12}\bigl{(}\frac{a}{R_{E}}\bigr{)}^{5}{\ \rm
kg}\,.$ (5)

This is shown as a dashed line in Fig. 6. Furthermore since $M_{ex}$ is
smaller for elliptical orbits, Eq. 5 also serves as a conservative upper limit
for elliptical orbits. Therefore Eq. 5 provides an upper limit to $M_{ex}$
that is valid for orbits of any eccentricity or inclination. It is our primary
result. For example, at $a=2R_{E}$ we have $M_{ex}=6.7\times 10^{13}$ kg. A
spherical object of this mass and terrestrial densities has a radius of a few
kilometers while a black hole of this mass has a Schwarzschild radius of 100
fm.

The $a^{5}$ dependence of $M_{ex}$ in Eq. 5 can be understood as follows.
First, the tidal force falls as $\approx 1/a^{3}$. Second, the orbital
frequency $f$ decreases as $1/a^{3/2}$ (Kepler’s third law). Finally, the
noise limit in Eqs. 3 and 4 increases as $f$ decreases. Therefore the ratio of
signal to noise decreases faster than $1/a^{3}$ so that the excluded mass
grows faster than $a^{3}$.

Our search is limited by the background in the gravimeter signal from
atmospheric fluctuations. It should be possible to somewhat increase the
sensitivity with a coherent search involving gravimeters at several locations.
However, this would likely involve a great increase in complexity because one
would need to analyze gravimeter data separately for every compact object
orbit.

Searches around other solar system bodies are also possible. The seismometer
on the InSight lander on Mars has sensitivity as a gravimeter. Unfortunately,
this instrument is exposed to some wind and thermal noise Pou et al. (2021).
Observations on the Moon could largely avoid atmospheric fluctuations. Indeed
a gravimeter was deployed during Apollo 17 Giganti et al. (1977); Kawamura et
al. (2015) but it did not function well.

We searched for periodic gravimeter signals from small tidal interactions of
dark objects and found none. We rule out all near Earth objects orbiting with
semimajor axis $a$ and mass $M>M_{ex}(a)$, see Eq. 5. We find
$M_{ex}(2R_{E})\approx 6.7\times 10^{13}$ kg and rule out all near Earth black
holes orbiting with $a\leq 2R_{E}$ and $M>6.7\times 10^{13}$ kg. Furthermore,
dark matter made of primordial black holes of mass $M_{PBH}<10^{14}$ kg is
significantly constrained by the lack of observation of Hawking radiation
Coogan et al. (2021). Therefore a near Earth black hole may be extremely
unlikely.

###### Acknowledgements.

We acknowledge helpful discussions with Dmitry Budker, Matt Caplan, Rafael
Lang, Rees McNally, Cole Miller, Maria Alessandra Papa, Tanya Zelevinsky and
Walter Zürn. CJH is supported in part by US Department of Energy grants DE-
FG02-87ER40365 and DE-SC0018083. We gratefully acknowledge the work of the
operators of the superconducting gravimeters: Jacques Hinderer and Frédéric
Little from EOST, Strasbourg for the Djougou station and Thomas Forbriger and
Peter Duffner from the Karlsruhe Institute of Technology for the Black Forest
Observatory. We thank the data centers for archiving and freely distributing
the gravimetric data and Thomas Forbriger for porting the ETERNA software
package for tidal predictions to UNIX (Forbriger, 2018) and Jean-Paul Boy from
EOST Starsbourg for publishing his tidal analyses results.

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  * Pou et al. (2021) L. Pou, F. Nimmo, P. Lognonné, D. Mimoun, R. F. Garcia, B. Pinot, A. Rivoldini, D. Banfield, and W. B. Banerdt, Earth and Space Science 8, e2021EA001669 (2021), e2021EA001669 2021EA001669, URL https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2021EA001669.
  * Giganti et al. (1977) J. J. Giganti, J. V. Larson, J. P. Richard, R. L. Tobias, and J. Weber, Lunar Surface Gravimeter Experiment Final Report, Univ. of Maryland Department of Physics and Astronomy, College Park, Md. (1977).
  * Kawamura et al. (2015) T. Kawamura, N. Kobayashi, S. Tanaka, and P. Lognonné, Journal of Geophysical Research: Planets 120, 343 (2015).
  * Forbriger (2018) T. Forbriger (2018), URL https://git.scc.kit.edu/ToE/Eterna.


\cF(Z\bar{W}\hat{Y}\bar{X}ZX) & = \left(d^4-6 d^2+3\right) c_{0,2}+\left(-6 d^4+34 d^2-19\right) c_{1,1}+\left(-6 d^4+34 d^2-19\right) c_{1,2}\\ & \quad +\left(-6 d^4+34 d^2-19\right) c_{1,3} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Y}\bar{X}ZW) & = \left(d^4-5 d^2+2\right) c_{0,2}+\left(-3 d^4+17 d^2-9\right) c_{1,1}+\left(-3 d^4+17 d^2-9\right) c_{1,2}\\ & \quad +\left(-3 d^4+17 d^2-9\right) c_{1,3} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Y}\bar{X}YX) & = \left(4 d^4-23 d^2+14\right) c_{0,1}+\left(4 d^4-23 d^2+14\right) c_{0,2}+\left(4 d^4-23 d^2+13\right) c_{0,3}\\ & \quad +\left(-9 d^4+51 d^2-28\right) c_{1,1}+\left(-9 d^4+51 d^2-28\right) c_{1,2}+\left(-7 d^4+40 d^2-23\right) c_{1,3} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Y}\bar{X}YW) & = \left(-d^4+6 d^2-3\right) c_{0,2}+\left(d^4-6 d^2+5\right) c_{0,4}+\left(-3 d^4+17 d^2-10\right) c_{1,1}\\ & \quad +\left(-3 d^4+17 d^2-10\right) c_{1,2}+\left(-3 d^4+17 d^2-9\right) c_{1,3} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Y}\bar{W}ZX) & = \left(-d^4+5 d^2-1\right) c_{0,2}+\left(-2 d^4+11 d^2-4\right) c_{1,2} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Y}\bar{W}ZW) & = \left(-2 d^4+11 d^2-3\right) c_{0,2}+\left(-3 d^4+16 d^2-5\right) c_{1,2} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Y}\bar{W}YX) & = \left(2 d^4-11 d^2+6\right) c_{0,1}+\left(2 d^4-11 d^2+6\right) c_{0,2}+\left(2 d^4-11 d^2+6\right) c_{0,3}\\ & \quad +\left(-d^4+6 d^2-3\right) c_{1,2} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Y}\bar{W}YW) & = \left(-d^4+5 d^2-2\right) c_{0,2}+\left(d^4-6 d^2+3\right) c_{0,4}+\left(-d^4+5 d^2-2\right) c_{1,2} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Y}\bar{g}YX) & = \left(d^4-6 d^2+4\right) c_{0,1}+\left(d^4-6 d^2+4\right) c_{0,2}+\left(2 d^4-11 d^2+6\right) c_{0,3}\\ & \quad +\left(d^4-6 d^2+4\right) c_{1,1} \displaybreak[1] \\
\cF(Z\bar{W}\hat{Y}\bar{g}YW) & = \left(-d^4+5 d^2-2\right) c_{0,2}+\left(d^4-6 d^2+3\right) c_{1,1}+c_{0,4} \displaybreak[1] \\
\cF(Y\bar{X}\hat{1}\bar{X}1X) & = \left(d^4-6 d^2+3\right) c_{0,1}+\left(-d^4+6 d^2-4\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{X}\hat{1}\bar{X}ZX) & = \left(-4 d^4+23 d^2-14\right) c_{1,1}-c_{0,1}-c_{0,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{1}\bar{X}ZW) & = \left(-3 d^4+17 d^2-10\right) c_{1,1}+c_{0,3} \displaybreak[1] \\
\cF(Y\bar{X}\hat{1}\bar{X}YX) & = \left(-2 d^4+12 d^2-8\right) c_{0,1}+\left(-d^4+6 d^2-4\right) c_{0,2}+\left(-7 d^4+40 d^2-24\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{X}\hat{1}\bar{X}YW) & = \left(-4 d^4+23 d^2-14\right) c_{1,1}-c_{0,1}-c_{0,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{1}\bar{X}Yg) & = \left(d^4-6 d^2+3\right) c_{0,1}+\left(-d^4+6 d^2-4\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Z}\bar{X}1X) & = \left(-2 d^4+12 d^2-7\right) c_{0,1}+\left(-3 d^4+17 d^2-10\right) c_{1,1}+\left(-d^4+6 d^2-4\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Z}\bar{X}ZX) & = \left(3 d^4-17 d^2+10\right) c_{0,1}+\left(3 d^4-17 d^2+10\right) c_{0,2}+\left(-d^4+6 d^2-4\right) c_{0,4}\\ & \quad +\left(-9 d^4+51 d^2-29\right) c_{1,1}+\left(-4 d^4+23 d^2-14\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Z}\bar{X}ZW) & = \left(-3 d^4+17 d^2-10\right) c_{0,3}+\left(-6 d^4+34 d^2-19\right) c_{1,1}-c_{0,2}\\ & \quad +\left(-3 d^4+17 d^2-10\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Z}\bar{X}YX) & = \left(15 d^4-85 d^2+48\right) c_{0,1}+\left(6 d^4-34 d^2+19\right) c_{0,2}+\left(-13 d^4+74 d^2-43\right) c_{1,1}\\ & \quad +\left(-7 d^4+40 d^2-24\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Z}\bar{X}YW) & = \left(4 d^4-23 d^2+13\right) c_{0,1}+\left(3 d^4-17 d^2+9\right) c_{0,2}+\left(-7 d^4+40 d^2-23\right) c_{1,1}\\ & \quad +\left(-4 d^4+23 d^2-14\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Z}\bar{X}Yg) & = \left(-3 d^4+17 d^2-10\right) c_{0,1}+\left(-d^4+6 d^2-4\right) c_{1,1}+\left(-d^4+6 d^2-4\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Z}\bar{W}ZX) & = \left(2 d^4-11 d^2+6\right) c_{0,1}+\left(2 d^4-11 d^2+6\right) c_{0,2}+\left(-2 d^4+11 d^2-5\right) c_{0,4}\\ & \quad +\left(d^4-6 d^2+4\right) c_{1,2}+\left(4 d^4-23 d^2+14\right) c_{1,3} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Z}\bar{W}ZW) & = \left(-2 d^4+11 d^2-4\right) c_{0,2}+\left(-2 d^4+11 d^2-6\right) c_{0,3}+\left(5 d^2-d^4\right) c_{1,2}\\ & \quad +\left(3 d^4-17 d^2+10\right) c_{1,3} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Z}\bar{W}YX) & = \left(6 d^4-35 d^2+20\right) c_{0,1}+\left(2 d^4-12 d^2+7\right) c_{0,2}+\left(d^4-6 d^2+4\right) c_{1,2}\\ & \quad +\left(6 d^4-34 d^2+19\right) c_{1,3} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Z}\bar{W}YW) & = \left(2 d^4-11 d^2+6\right) c_{0,1}+\left(d^4-5 d^2+3\right) c_{0,2}+c_{1,2}\\ & \quad +\left(3 d^4-17 d^2+10\right) c_{1,3} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Z}\bar{W}Yg) & = \left(-d^4+7 d^2-4\right) c_{0,1}+\left(d^4-6 d^2+3\right) c_{1,3} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{X}1X) & = \left(d^4-6 d^2+4\right) c_{0,1}+\left(4 d^4-23 d^2+14\right) c_{1,1}+\left(d^4-6 d^2+4\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{X}ZX) & = \left(-3 d^4+17 d^2-9\right) c_{0,1}+\left(-3 d^4+17 d^2-9\right) c_{0,2}+\left(d^4-6 d^2+4\right) c_{0,4}\\ & \quad +\left(13 d^4-74 d^2+43\right) c_{1,1}+\left(4 d^4-23 d^2+14\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{X}ZW) & = \left(3 d^4-17 d^2+9\right) c_{0,3}+\left(9 d^4-51 d^2+29\right) c_{1,1}+c_{0,2}\\ & \quad +\left(3 d^4-17 d^2+10\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{X}YX) & = \left(-13 d^4+73 d^2-40\right) c_{0,1}+\left(-5 d^4+28 d^2-15\right) c_{0,2}+\left(20 d^4-114 d^2+67\right) c_{1,1}\\ & \quad +\left(7 d^4-40 d^2+24\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{X}YW) & = \left(-4 d^4+23 d^2-12\right) c_{0,1}+\left(-3 d^4+17 d^2-8\right) c_{0,2}+\left(11 d^4-63 d^2+37\right) c_{1,1}\\ & \quad +\left(4 d^4-23 d^2+14\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{X}Yg) & = \left(2 d^4-11 d^2+7\right) c_{0,1}+\left(2 d^4-12 d^2+8\right) c_{1,1}+\left(d^4-6 d^2+4\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{W}ZX) & = \left(-2 d^4+11 d^2-5\right) c_{0,1}+\left(-2 d^4+11 d^2-5\right) c_{0,2}+\left(d^4-6 d^2+3\right) c_{0,4}\\ & \quad +\left(d^4-6 d^2+5\right) c_{1,1}+\left(d^4-6 d^2+5\right) c_{1,2}+\left(-d^4+6 d^2-4\right) c_{1,4} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{W}ZW) & = \left(d^4-5 d^2+2\right) c_{0,2}+\left(2 d^4-11 d^2+5\right) c_{0,3}+\left(d^4-6 d^2+4\right) c_{1,1}\\ & \quad +\left(d^4-6 d^2+4\right) c_{1,2}+\left(d^4-5 d^2\right) c_{1,4} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{W}YX) & = \left(-4 d^4+22 d^2-13\right) c_{0,1}+\left(-d^4+6 d^2-4\right) c_{0,2}+\left(3 d^4-17 d^2+9\right) c_{1,1}\\ & \quad +\left(3 d^4-17 d^2+9\right) c_{1,2}+\left(-d^4+6 d^2-4\right) c_{1,4} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{W}YW) & = \left(-d^4+7 d^2-4\right) c_{0,1}+\left(d^4-6 d^2+4\right) c_{1,1}-c_{0,2}\\ & \quad +\left(d^4-6 d^2+4\right) c_{1,2}-c_{1,4} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{W}Yg) & = \left(d^4-6 d^2+3\right) c_{0,1}+\left(d^4-5 d^2+2\right) c_{1,1}+\left(d^4-5 d^2+2\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{g}YX) & = \left(-3 d^4+17 d^2-9\right) c_{0,1}+\left(-d^4+6 d^2-3\right) c_{0,2}+\left(-3 d^4+17 d^2-9\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{g}YW) & = \left(-d^4+6 d^2-3\right) c_{0,1}+\left(-d^4+5 d^2-2\right) c_{0,2}+\left(-2 d^4+11 d^2-6\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{X}\hat{Y}\bar{g}Yg) & = \left(1-d^2\right) c_{0,1}+\left(-d^4+6 d^2-3\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Z}\bar{X}1X) & = \left(4 d^4-23 d^2+13\right) c_{0,1}+\left(-3 d^4+17 d^2-9\right) c_{1,2}+\left(6 d^4-34 d^2+19\right) c_{1,4} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Z}\bar{X}ZX) & = \left(-6 d^4+34 d^2-19\right) c_{0,1}+\left(-6 d^4+34 d^2-19\right) c_{0,2}+\left(3 d^4-17 d^2+10\right) c_{0,4}\\ & \quad +\left(-4 d^4+23 d^2-14\right) c_{1,2}+\left(13 d^4-74 d^2+42\right) c_{1,4} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Z}\bar{X}ZW) & = \left(d^4-6 d^2+4\right) c_{0,2}+\left(6 d^4-34 d^2+19\right) c_{0,3}+\left(-3 d^4+17 d^2-9\right) c_{1,2}\\ & \quad +\left(9 d^4-51 d^2+28\right) c_{1,4} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Z}\bar{X}YX) & = \left(-25 d^4+142 d^2-81\right) c_{0,1}+\left(-9 d^4+51 d^2-29\right) c_{0,2}+\left(-6 d^4+34 d^2-19\right) c_{1,2}\\ & \quad +\left(19 d^4-108 d^2+61\right) c_{1,4} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Z}\bar{X}YW) & = \left(-7 d^4+40 d^2-23\right) c_{0,1}+\left(-4 d^4+23 d^2-13\right) c_{0,2}+\left(-3 d^4+17 d^2-9\right) c_{1,2}\\ & \quad +\left(10 d^4-57 d^2+33\right) c_{1,4} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Z}\bar{X}Yg) & = \left(5 d^4-28 d^2+16\right) c_{0,1}+\left(3 d^4-17 d^2+9\right) c_{1,4} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Z}\bar{W}ZX) & = \left(-3 d^4+17 d^2-10\right) c_{0,1}+\left(-3 d^4+17 d^2-10\right) c_{0,2}+\left(3 d^4-17 d^2+9\right) c_{0,4}\\ & \quad +\left(d^4-6 d^2+4\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Z}\bar{W}ZW) & = \left(2 d^4-11 d^2+5\right) c_{0,2}+\left(3 d^4-17 d^2+10\right) c_{0,3}+\left(2 d^4-11 d^2+5\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Z}\bar{W}YX) & = \left(-10 d^4+57 d^2-33\right) c_{0,1}+\left(-3 d^4+17 d^2-10\right) c_{0,2}+\left(d^4-6 d^2+5\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Z}\bar{W}YW) & = \left(-3 d^4+17 d^2-10\right) c_{0,1}+\left(-d^4+6 d^2-4\right) c_{0,2}+\left(d^4-6 d^2+4\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Z}\bar{W}Yg) & = \left(2 d^4-11 d^2+6\right) c_{0,1}+c_{1,2} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{X}1X) & = \left(-4 d^4+23 d^2-13\right) c_{0,1}+\left(-6 d^4+34 d^2-19\right) c_{1,1}+\left(-6 d^4+34 d^2-19\right) c_{1,2}\\ & \quad +\left(-6 d^4+34 d^2-19\right) c_{1,3} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{X}ZX) & = \left(7 d^4-40 d^2+23\right) c_{0,1}+\left(7 d^4-40 d^2+23\right) c_{0,2}+\left(-4 d^4+23 d^2-14\right) c_{0,4}\\ & \quad +\left(-13 d^4+74 d^2-42\right) c_{1,1}+\left(-13 d^4+74 d^2-42\right) c_{1,2}+\left(-10 d^4+57 d^2-33\right) c_{1,3} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{X}ZW) & = \left(-d^4+6 d^2-5\right) c_{0,2}+\left(-7 d^4+40 d^2-23\right) c_{0,3}+\left(-9 d^4+51 d^2-28\right) c_{1,1}\\ & \quad +\left(-9 d^4+51 d^2-28\right) c_{1,2}+\left(-6 d^4+34 d^2-19\right) c_{1,3} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{X}YX) & = \left(30 d^4-171 d^2+98\right) c_{0,1}+\left(10 d^4-57 d^2+33\right) c_{0,2}+\left(-19 d^4+108 d^2-61\right) c_{1,1}\\ & \quad +\left(-19 d^4+108 d^2-61\right) c_{1,2}+\left(-13 d^4+74 d^2-42\right) c_{1,3} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{X}YW) & = \left(9 d^4-51 d^2+29\right) c_{0,1}+\left(4 d^4-23 d^2+14\right) c_{0,2}+\left(-10 d^4+57 d^2-33\right) c_{1,1}\\ & \quad +\left(-10 d^4+57 d^2-33\right) c_{1,2}+\left(-6 d^4+34 d^2-19\right) c_{1,3} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{X}Yg) & = \left(-6 d^4+34 d^2-19\right) c_{0,1}+\left(-3 d^4+17 d^2-9\right) c_{1,1}+\left(-3 d^4+17 d^2-9\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{W}ZX) & = \left(3 d^4-17 d^2+10\right) c_{0,1}+\left(3 d^4-17 d^2+10\right) c_{0,2}+\left(-3 d^4+17 d^2-9\right) c_{0,4}\\ & \quad +\left(-d^4+6 d^2-4\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{W}ZW) & = \left(-2 d^4+11 d^2-5\right) c_{0,2}+\left(-3 d^4+17 d^2-10\right) c_{0,3}+\left(-2 d^4+11 d^2-5\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{W}YX) & = \left(10 d^4-57 d^2+33\right) c_{0,1}+\left(3 d^4-17 d^2+10\right) c_{0,2}+\left(-d^4+6 d^2-5\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{W}YW) & = \left(3 d^4-17 d^2+10\right) c_{0,1}+\left(d^4-6 d^2+4\right) c_{0,2}+\left(-d^4+6 d^2-4\right) c_{1,2} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{W}Yg) & = \left(-2 d^4+11 d^2-6\right) c_{0,1}-c_{1,2} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{g}YX) & = \left(5 d^4-29 d^2+17\right) c_{0,1}+\left(d^4-6 d^2+4\right) c_{0,2}+\left(3 d^4-17 d^2+10\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{g}YW) & = \left(2 d^4-11 d^2+6\right) c_{0,1}+\left(3 d^4-17 d^2+9\right) c_{1,1}+c_{0,2} \displaybreak[1] \\
\cF(Y\bar{W}\hat{Y}\bar{g}Yg) & = \left(-d^4+6 d^2-3\right) c_{0,1}+\left(d^4-6 d^2+4\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{X}1X) & = \left(d^4-6 d^2+3\right) c_{0,1}+\left(-d^4+6 d^2-4\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{X}ZX) & = \left(-d^4+6 d^2-4\right) c_{0,1}+\left(-d^4+6 d^2-4\right) c_{0,2}+\left(2 d^4-11 d^2+5\right) c_{0,4}\\ & \quad +\left(-3 d^4+17 d^2-10\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{X}ZW) & = \left(2 d^4-11 d^2+4\right) c_{0,2}+\left(d^4-6 d^2+4\right) c_{0,3}+\left(-3 d^4+17 d^2-9\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{X}YX) & = \left(-4 d^4+23 d^2-14\right) c_{0,1}+\left(-d^4+6 d^2-4\right) c_{0,2}+\left(-4 d^4+23 d^2-14\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{X}YW) & = \left(-d^4+6 d^2-4\right) c_{0,1}+\left(-3 d^4+17 d^2-10\right) c_{1,1}-c_{0,2} \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{X}Yg) & = \left(d^4-6 d^2+3\right) c_{0,1}+\left(-d^4+6 d^2-4\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{W}ZX) & = -c_{0,1}-c_{0,2}+c_{0,4} -c_{1,1} \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{W}ZW) & = \left(-d^4+5 d^2-1\right) c_{0,2}+c_{0,3}-c_{1,1} \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{W}YX) & = \left(-2 d^4+10 d^2-5\right) c_{0,1}+\left(-d^4+5 d^2-2\right) c_{0,2}+\left(-d^4+6 d^2-3\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{W}YW) & = \left(d^2-1\right) c_{0,1}+\left(-d^4+6 d^2-2\right) c_{0,2}-c_{1,1} \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{W}Yg) & = d^2 c_{0,1}+\left(-d^4+5 d^2-2\right) c_{1,1} \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{g}YX) & = 0 \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{g}YW) & = 0 \displaybreak[1] \\
\cF(Y\bar{g}\hat{Y}\bar{g}Yg) & = 0 \displaybreak[1] \\
\end{align*}

§.§ Flat generators for $\cB$

Only the low weight eigenspace with eigenvalue $1$ in $\cG(\Gamma(\cB))_{3,+}$ contains any flat vectors, and these are spanned by the element $T$ below. (As usual, $d^2$ is the index of $\cA$ and $\cB$, the largest real root of $\lambda^3-6\lambda^2+5\lambda -1=0$, approximately $5.04892$.)

\begin{align*}
% generated by the command 'ToLaTeXString[GPA4Matrix[T], "T"]' in the Mathematica notebook code/connections-and-flat-elements.nb available with the arXiv sources of this article.
T_{1,f} & = \left(
\begin{array}{cc}
0 & 0 \\
0 & 0
\end{array}
\right) \displaybreak[1]\\
T_{1,B} & = \left(
\begin{array}{c}
-5 d^4+34 d^2-20
\end{array}
\right) \displaybreak[1]\\
T_{1,F} & = \left(
\begin{array}{c}
2 d^4-15 d^2+8
\end{array}
\right) \displaybreak[1]\\
T_{A,f} & = \left(
\begin{array}{cccc}
0 & 0 & 9 d^4-50 d^2+29 & -5 d^4+27 d^2-13 \\
0 & 0 & -9 d^4+50 d^2-29 & 5 d^4-27 d^2+13 \\
4 d^4-16 d^2+9 & 3 d^4-19 d^2+12 & d^4-4 d^2+4 & -d^4+4 d^2+3 \\
-4 d^4+16 d^2-9 & -3 d^4+19 d^2-12 & 8 d^4-46 d^2+25 & -4 d^4+23 d^2-16
\end{array}
\right) \displaybreak[1]\\
T_{A,B} & = \left(
\begin{array}{cccc}
-9 d^4+50 d^2-29 & -3 d^4+19 d^2-12 & -6 d^4+31 d^2-17 & -4 d^4+23 d^2-16 \\
d^4-4 d^2+4 & 4 d^4-23 d^2+16 & 8 d^4-46 d^2+25 & -6 d^4+31 d^2-10 \\
-d^4+4 d^2-4 & -4 d^4+23 d^2-16 & -8 d^4+46 d^2-25 & 6 d^4-31 d^2+10 \\
8 d^4-46 d^2+25 & -d^4+4 d^2-4 & -2 d^4+15 d^2-8 & 10 d^4-54 d^2+26
\end{array}
\right) \displaybreak[1]\\
T_{A,F} & = \left(
\begin{array}{ccccc}
5 d^4-27 d^2+13 & -4 d^4+23 d^2-16 & 9 d^4-50 d^2+22 & 6 d^4-31 d^2+17 & -d^4+4 d^2-4 \\
-d^4+4 d^2+3 & -6 d^4+31 d^2-10 & -10 d^4+54 d^2-12 & -2 d^4+8 d^2-1 & -9 d^4+50 d^2-22 \\
-4 d^4+23 d^2-16 & 10 d^4-54 d^2+26 & d^4-4 d^2-10 & -4 d^4+23 d^2-16 & 10 d^4-54 d^2+26 \\
6 d^4-31 d^2+17 & -d^4+4 d^2-4 & 9 d^4-50 d^2+22 & 5 d^4-27 d^2+13 & -4 d^4+23 d^2-16 \\
-2 d^4+8 d^2-1 & -9 d^4+50 d^2-22 & -10 d^4+54 d^2-12 & -d^4+4 d^2+3 & -6 d^4+31 d^2-10
\end{array}
\right) \displaybreak[1]\\
T_{A,z} & = \left(
\begin{array}{c}
2 d^4-15 d^2+8
\end{array}
\right) \displaybreak[1]\\
T_{A,D} & = \left(
\begin{array}{cc}
-5 d^4+27 d^2-13 & 7 d^4-35 d^2+21 \\
7 d^4-35 d^2+14 & 2 d^4-8 d^2+1
\end{array}
\right) \displaybreak[1]\\
T_{C,f} & = \left(
\begin{array}{c}
2 d^4-15 d^2+8
\end{array}
\right) \displaybreak[1]\\
T_{C,B} & = \left(
\begin{array}{ccc}
8 d^4-46 d^2+25 & 9 d^4-50 d^2+29 & 3 d^4-19 d^2+12 \\
-8 d^4+46 d^2-25 & -9 d^4+50 d^2-29 & -3 d^4+19 d^2-12 \\
8 d^4-46 d^2+25 & 9 d^4-50 d^2+29 & 3 d^4-19 d^2+12
\end{array}
\right) \displaybreak[1]\\
T_{C,F} & = \left(
\begin{array}{cc}
2 d^4-8 d^2+1 & 7 d^4-35 d^2+14 \\
7 d^4-35 d^2+21 & -5 d^4+27 d^2-13
\end{array}
\right) \displaybreak[1]\\
T_{C,D} & = \left(
\begin{array}{ccc}
-3 d^4+19 d^2-12 & -d^4+4 d^2-4 & -3 d^4+19 d^2-12 \\
3 d^4-19 d^2+12 & d^4-4 d^2+4 & 3 d^4-19 d^2+12 \\
-3 d^4+19 d^2-12 & -d^4+4 d^2-4 & -3 d^4+19 d^2-12
\end{array}
\right) \displaybreak[1]\\
T_{G,f} & = \left(
\begin{array}{c}
-5 d^4+34 d^2-20
\end{array}
\right) \displaybreak[1]\\
T_{G,B} & = \left(
\begin{array}{c}
2 d^4-15 d^2+8
\end{array}
\right) \displaybreak[1]\\
T_{G,F} & = \left(
\begin{array}{cccc}
-4 d^4+23 d^2-16 & -3 d^4+19 d^2-12 & 8 d^4-46 d^2+25 & -4 d^4+16 d^2-9 \\
5 d^4-27 d^2+13 & 0 & -9 d^4+50 d^2-29 & 0 \\
-d^4+4 d^2+3 & 3 d^4-19 d^2+12 & d^4-4 d^2+4 & 4 d^4-16 d^2+9 \\
-5 d^4+27 d^2-13 & 0 & 9 d^4-50 d^2+29 & 0
\end{array}
\right) \displaybreak[1]\\
T_{G,z} & = \left(
\begin{array}{cc}
0 & 0 \\
0 & 0
\end{array}
\right) \displaybreak[1]\\
T_{G,D} & = \left(
\begin{array}{c}
2 d^4-15 d^2+8
\end{array}
\right) \displaybreak[1]\\
T_{E,f} & = \left(
\begin{array}{c}
2 d^4-15 d^2+8
\end{array}
\right) \displaybreak[1]\\
T_{E,B} & = \left(
\begin{array}{cc}
-12 d^4+69 d^2-41 & -21 d^4+133 d^2-77 \\
7 & 9 d^4-50 d^2+29
\end{array}
\right) \displaybreak[1]\\
T_{E,F} & = \left(
\begin{array}{cccc}
10 d^4-54 d^2+26 & 8 d^4-46 d^2+25 & -d^4+4 d^2-4 & -2 d^4+15 d^2-8 \\
-4 d^4+23 d^2-16 & -9 d^4+50 d^2-29 & -3 d^4+19 d^2-12 & -6 d^4+31 d^2-17 \\
-6 d^4+31 d^2-10 & d^4-4 d^2+4 & 4 d^4-23 d^2+16 & 8 d^4-46 d^2+25 \\
6 d^4-31 d^2+10 & -d^4+4 d^2-4 & -4 d^4+23 d^2-16 & -8 d^4+46 d^2-25
\end{array}
\right) \displaybreak[1]\\
T_{E,z} & = \left(
\begin{array}{c}
-5 d^4+34 d^2-20
\end{array}
\right) \displaybreak[1]\\
T_{E,D} & = \left(
\begin{array}{ccc}
8 d^4-46 d^2+25 & 3 d^4-19 d^2+12 & 9 d^4-50 d^2+29 \\
8 d^4-46 d^2+25 & 3 d^4-19 d^2+12 & 9 d^4-50 d^2+29 \\
-8 d^4+46 d^2-25 & -3 d^4+19 d^2-12 & -9 d^4+50 d^2-29
\end{array}
\right) \displaybreak[1]\\
\end{align*}