Properties

Label 180.3.c.b
Level $180$
Weight $3$
Character orbit 180.c
Analytic conductor $4.905$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 180.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.90464475849\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.85100625.1
Defining polynomial: \(x^{8} - x^{7} - 2 x^{6} + x^{5} + 3 x^{4} + 2 x^{3} - 8 x^{2} - 8 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{5} q^{2} + ( 1 + \beta_{3} + \beta_{4} + \beta_{5} ) q^{4} + \beta_{3} q^{5} + ( -1 + \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} ) q^{7} + ( 2 - \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{8} +O(q^{10})\) \( q -\beta_{5} q^{2} + ( 1 + \beta_{3} + \beta_{4} + \beta_{5} ) q^{4} + \beta_{3} q^{5} + ( -1 + \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} ) q^{7} + ( 2 - \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{8} + ( 1 - \beta_{3} + \beta_{4} - \beta_{6} ) q^{10} + ( -1 - \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{11} + ( 3 + \beta_{1} + \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 3 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{13} + ( 4 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{14} + ( 8 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 4 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{16} + ( -2 + 6 \beta_{3} - \beta_{4} + 5 \beta_{5} + \beta_{6} - \beta_{7} ) q^{17} + ( -3 - \beta_{1} + 3 \beta_{2} + 3 \beta_{5} - 5 \beta_{6} - 2 \beta_{7} ) q^{19} + ( 6 - \beta_{1} + \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} ) q^{20} + ( 8 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{22} + ( -5 + 5 \beta_{1} + \beta_{2} + 4 \beta_{4} + 9 \beta_{5} - 7 \beta_{6} - 2 \beta_{7} ) q^{23} + 5 q^{25} + ( 10 + 4 \beta_{1} - 6 \beta_{3} + 10 \beta_{4} + 2 \beta_{6} - 4 \beta_{7} ) q^{26} + ( 6 + 6 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{28} + ( -10 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 10 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} ) q^{29} + ( 5 + 11 \beta_{1} - \beta_{2} + 10 \beta_{4} + 5 \beta_{5} + 5 \beta_{6} ) q^{31} + ( 10 - \beta_{1} + 3 \beta_{3} + 5 \beta_{4} - 4 \beta_{6} - 6 \beta_{7} ) q^{32} + ( -14 - 10 \beta_{3} + 2 \beta_{4} - 6 \beta_{6} ) q^{34} + ( 2 + 4 \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{35} + ( -19 - \beta_{1} - \beta_{2} + 14 \beta_{3} + 2 \beta_{4} + 13 \beta_{5} + \beta_{6} - 4 \beta_{7} ) q^{37} + ( 8 - 8 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} - 4 \beta_{4} - 8 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{38} + ( 2 + 3 \beta_{1} + \beta_{2} + \beta_{3} + 5 \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{40} + ( 4 + 2 \beta_{1} + 2 \beta_{2} - 8 \beta_{4} - 6 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{41} + ( -7 - 15 \beta_{1} + \beta_{2} - 7 \beta_{4} + 7 \beta_{7} ) q^{43} + ( -14 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{4} - 8 \beta_{5} + 6 \beta_{6} + 2 \beta_{7} ) q^{44} + ( 24 - 8 \beta_{1} - 4 \beta_{2} - 16 \beta_{3} - 8 \beta_{4} - 16 \beta_{5} + 8 \beta_{6} + 8 \beta_{7} ) q^{46} + ( -7 - 9 \beta_{1} - \beta_{2} - 12 \beta_{4} - 5 \beta_{5} - 9 \beta_{6} - 2 \beta_{7} ) q^{47} + ( -15 - 8 \beta_{3} - 4 \beta_{4} + 20 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} ) q^{49} -5 \beta_{5} q^{50} + ( -24 - 14 \beta_{1} - 4 \beta_{2} + 18 \beta_{3} - 12 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{52} + ( -46 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 8 \beta_{4} + 6 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} ) q^{53} + ( 3 - 5 \beta_{1} + \beta_{2} - 3 \beta_{4} - 6 \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{55} + ( -16 + 4 \beta_{1} - 10 \beta_{2} + 8 \beta_{4} - 2 \beta_{6} + 4 \beta_{7} ) q^{56} + ( -22 + 8 \beta_{1} - 18 \beta_{3} + 10 \beta_{4} + 12 \beta_{5} + 6 \beta_{6} - 8 \beta_{7} ) q^{58} + ( -1 - \beta_{1} + \beta_{2} + 5 \beta_{4} + 6 \beta_{5} + 4 \beta_{6} + 5 \beta_{7} ) q^{59} + ( -16 - 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 12 \beta_{4} - 14 \beta_{5} - 6 \beta_{6} ) q^{61} + ( -8 \beta_{1} - 12 \beta_{2} - 4 \beta_{4} - 4 \beta_{6} - 4 \beta_{7} ) q^{62} + ( -16 - 11 \beta_{1} + \beta_{2} + 5 \beta_{3} - 3 \beta_{4} - 12 \beta_{5} - 5 \beta_{6} + 10 \beta_{7} ) q^{64} + ( 6 - 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 7 \beta_{4} + 11 \beta_{5} - \beta_{6} - 5 \beta_{7} ) q^{65} + ( -3 - 3 \beta_{1} + \beta_{2} - 11 \beta_{4} - 8 \beta_{5} - 12 \beta_{6} - 9 \beta_{7} ) q^{67} + ( 24 - 2 \beta_{1} + 6 \beta_{3} - 12 \beta_{4} + 2 \beta_{5} + 14 \beta_{6} + 6 \beta_{7} ) q^{68} + ( -8 + 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{70} + ( -2 - 18 \beta_{1} + 2 \beta_{2} - 10 \beta_{4} - 8 \beta_{5} + 4 \beta_{6} + 6 \beta_{7} ) q^{71} + ( -20 + 2 \beta_{1} + 2 \beta_{2} - 12 \beta_{3} - 4 \beta_{4} - 26 \beta_{5} - 2 \beta_{6} + 8 \beta_{7} ) q^{73} + ( -34 - 4 \beta_{1} - 18 \beta_{3} - 2 \beta_{4} + 12 \beta_{5} - 18 \beta_{6} + 4 \beta_{7} ) q^{74} + ( 12 \beta_{2} + 16 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} - 12 \beta_{7} ) q^{76} + ( 38 - 2 \beta_{1} - 2 \beta_{2} + 24 \beta_{3} + 10 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} ) q^{77} + ( 21 + 7 \beta_{1} - 13 \beta_{2} - 4 \beta_{4} - 25 \beta_{5} + 23 \beta_{6} + 2 \beta_{7} ) q^{79} + ( 6 - 9 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{80} + ( 16 + 8 \beta_{1} - 8 \beta_{3} + 16 \beta_{4} + 2 \beta_{5} + 8 \beta_{6} - 8 \beta_{7} ) q^{82} + ( 15 + 23 \beta_{1} - 5 \beta_{2} + 23 \beta_{4} + 8 \beta_{5} + 20 \beta_{6} + 5 \beta_{7} ) q^{83} + ( 17 + \beta_{1} + \beta_{2} + 4 \beta_{3} - 6 \beta_{4} + 7 \beta_{5} + 3 \beta_{6} ) q^{85} + ( 28 + 12 \beta_{1} + 16 \beta_{2} - 8 \beta_{4} - 8 \beta_{6} - 8 \beta_{7} ) q^{86} + ( -8 \beta_{1} + 6 \beta_{2} + 12 \beta_{3} + 4 \beta_{4} + 32 \beta_{5} - 10 \beta_{6} - 12 \beta_{7} ) q^{88} + ( -12 - 2 \beta_{1} - 2 \beta_{2} - 44 \beta_{3} + 8 \beta_{4} + 6 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} ) q^{89} + ( -18 + 2 \beta_{1} - 2 \beta_{2} + 18 \beta_{5} - 18 \beta_{6} ) q^{91} + ( -4 + 24 \beta_{1} + 4 \beta_{2} + 36 \beta_{3} + 8 \beta_{6} - 12 \beta_{7} ) q^{92} + ( 12 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} + 8 \beta_{4} - 4 \beta_{5} + 12 \beta_{6} + 12 \beta_{7} ) q^{94} + ( 4 - 2 \beta_{1} + 6 \beta_{2} - 3 \beta_{4} - 7 \beta_{5} - 3 \beta_{6} - 7 \beta_{7} ) q^{95} + ( 64 - 2 \beta_{1} - 2 \beta_{2} - 24 \beta_{3} + 14 \beta_{4} - 24 \beta_{5} - 8 \beta_{6} + 2 \beta_{7} ) q^{97} + ( -88 - 8 \beta_{3} - 24 \beta_{4} + 7 \beta_{5} + 8 \beta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{2} + 10q^{4} + 20q^{8} + O(q^{10}) \) \( 8q - 4q^{2} + 10q^{4} + 20q^{8} + 10q^{10} + 16q^{13} + 20q^{14} + 34q^{16} + 40q^{20} + 68q^{22} + 40q^{25} + 36q^{26} + 28q^{28} - 64q^{29} + 76q^{32} - 92q^{34} - 112q^{37} + 40q^{38} - 10q^{40} + 16q^{41} - 172q^{44} + 152q^{46} - 56q^{49} - 20q^{50} - 128q^{52} - 352q^{53} - 116q^{56} - 204q^{58} - 176q^{61} + 56q^{62} - 110q^{64} + 80q^{65} + 184q^{68} - 60q^{70} - 240q^{73} - 132q^{74} - 24q^{76} + 288q^{77} + 80q^{80} + 40q^{82} + 160q^{85} + 200q^{86} + 140q^{88} - 80q^{89} - 144q^{92} - 96q^{94} + 432q^{97} - 660q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - 2 x^{6} + x^{5} + 3 x^{4} + 2 x^{3} - 8 x^{2} - 8 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{7} + 3 \nu^{6} + 4 \nu^{5} + 7 \nu^{4} - 17 \nu^{3} - 8 \nu^{2} + 24 \nu + 8 \)\()/16\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} - 3 \nu^{6} + 4 \nu^{5} + \nu^{4} + \nu^{3} - 8 \nu^{2} + 8 \nu + 8 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{7} - \nu^{6} - 4 \nu^{5} - 5 \nu^{4} - 5 \nu^{3} + 16 \nu^{2} - 8 \nu - 24 \)\()/16\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} - \nu^{6} - 2 \nu^{5} + \nu^{4} + 3 \nu^{3} + 2 \nu^{2} - 8 \nu - 8 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( -5 \nu^{7} - \nu^{6} + 4 \nu^{5} + 3 \nu^{4} - 5 \nu^{3} - 24 \nu^{2} + 24 \nu + 56 \)\()/16\)
\(\beta_{6}\)\(=\)\((\)\( 7 \nu^{7} + 3 \nu^{6} - 12 \nu^{5} - 9 \nu^{4} + 15 \nu^{3} + 32 \nu^{2} - 8 \nu - 88 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( -15 \nu^{7} - 11 \nu^{6} + 20 \nu^{5} + 25 \nu^{4} - 23 \nu^{3} - 48 \nu^{2} + 40 \nu + 168 \)\()/16\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + 2 \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{2} + \beta_{1} + 1\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(3 \beta_{7} + 2 \beta_{6} - 6 \beta_{5} - 3 \beta_{4} - \beta_{2} - \beta_{1} + 7\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + 2 \beta_{6} - 2 \beta_{5} - \beta_{4} - 8 \beta_{3} + \beta_{2} - 3 \beta_{1} + 5\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{7} + 2 \beta_{6} - 6 \beta_{5} + 5 \beta_{4} - 8 \beta_{3} - \beta_{2} + 7 \beta_{1} + 7\)\()/8\)
\(\nu^{5}\)\(=\)\((\)\(\beta_{7} - 2 \beta_{6} - 18 \beta_{5} - 9 \beta_{4} - 8 \beta_{3} + 5 \beta_{2} - 3 \beta_{1} - 3\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(-5 \beta_{7} - 2 \beta_{6} - 6 \beta_{5} - 3 \beta_{4} - 16 \beta_{3} - 5 \beta_{2} + 7 \beta_{1} + 23\)\()/8\)
\(\nu^{7}\)\(=\)\((\)\(-7 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 7 \beta_{4} + 13 \beta_{2} + 13 \beta_{1} + 53\)\()/8\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(91\) \(101\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
1.04064 + 0.957636i
1.04064 0.957636i
−0.600040 + 1.28061i
−0.600040 1.28061i
1.40906 0.120653i
1.40906 + 0.120653i
−1.34966 0.422403i
−1.34966 + 0.422403i
−1.87477 0.696577i 0 3.02956 + 2.61185i 2.23607 0 5.46770i −3.86039 7.00695i 0 −4.19212 1.55759i
91.2 −1.87477 + 0.696577i 0 3.02956 2.61185i 2.23607 0 5.46770i −3.86039 + 7.00695i 0 −4.19212 + 1.55759i
91.3 −1.67986 1.08539i 0 1.64388 + 3.64660i −2.23607 0 0.596540i 1.19648 7.91002i 0 3.75629 + 2.42700i
91.4 −1.67986 + 1.08539i 0 1.64388 3.64660i −2.23607 0 0.596540i 1.19648 + 7.91002i 0 3.75629 2.42700i
91.5 −0.438172 1.95141i 0 −3.61601 + 1.71011i −2.23607 0 6.33166i 4.92155 + 6.30701i 0 0.979781 + 4.36349i
91.6 −0.438172 + 1.95141i 0 −3.61601 1.71011i −2.23607 0 6.33166i 4.92155 6.30701i 0 0.979781 4.36349i
91.7 1.99281 0.169449i 0 3.94257 0.675358i 2.23607 0 12.3959i 7.74236 2.01392i 0 4.45606 0.378899i
91.8 1.99281 + 0.169449i 0 3.94257 + 0.675358i 2.23607 0 12.3959i 7.74236 + 2.01392i 0 4.45606 + 0.378899i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.3.c.b 8
3.b odd 2 1 60.3.c.a 8
4.b odd 2 1 inner 180.3.c.b 8
5.b even 2 1 900.3.c.u 8
5.c odd 4 2 900.3.f.f 16
8.b even 2 1 2880.3.e.j 8
8.d odd 2 1 2880.3.e.j 8
12.b even 2 1 60.3.c.a 8
15.d odd 2 1 300.3.c.d 8
15.e even 4 2 300.3.f.b 16
20.d odd 2 1 900.3.c.u 8
20.e even 4 2 900.3.f.f 16
24.f even 2 1 960.3.e.c 8
24.h odd 2 1 960.3.e.c 8
60.h even 2 1 300.3.c.d 8
60.l odd 4 2 300.3.f.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.c.a 8 3.b odd 2 1
60.3.c.a 8 12.b even 2 1
180.3.c.b 8 1.a even 1 1 trivial
180.3.c.b 8 4.b odd 2 1 inner
300.3.c.d 8 15.d odd 2 1
300.3.c.d 8 60.h even 2 1
300.3.f.b 16 15.e even 4 2
300.3.f.b 16 60.l odd 4 2
900.3.c.u 8 5.b even 2 1
900.3.c.u 8 20.d odd 2 1
900.3.f.f 16 5.c odd 4 2
900.3.f.f 16 20.e even 4 2
960.3.e.c 8 24.f even 2 1
960.3.e.c 8 24.h odd 2 1
2880.3.e.j 8 8.b even 2 1
2880.3.e.j 8 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 224 T_{7}^{6} + 12032 T_{7}^{4} + 188416 T_{7}^{2} + 65536 \) acting on \(S_{3}^{\mathrm{new}}(180, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 256 + 256 T + 48 T^{2} - 64 T^{3} - 52 T^{4} - 16 T^{5} + 3 T^{6} + 4 T^{7} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( -5 + T^{2} )^{4} \)
$7$ \( 65536 + 188416 T^{2} + 12032 T^{4} + 224 T^{6} + T^{8} \)
$11$ \( ( 10496 + 208 T^{2} + T^{4} )^{2} \)
$13$ \( ( -12464 + 5792 T - 472 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$17$ \( ( -8816 - 3840 T - 424 T^{2} + T^{4} )^{2} \)
$19$ \( 6544162816 + 173686784 T^{2} + 925952 T^{4} + 1696 T^{6} + T^{8} \)
$23$ \( 101419319296 + 1884176384 T^{2} + 4397312 T^{4} + 3616 T^{6} + T^{8} \)
$29$ \( ( 1334416 - 34688 T - 2152 T^{2} + 32 T^{3} + T^{4} )^{2} \)
$31$ \( 59895709696 + 2731491328 T^{2} + 7432448 T^{4} + 5408 T^{6} + T^{8} \)
$37$ \( ( -244784 - 55136 T - 1528 T^{2} + 56 T^{3} + T^{4} )^{2} \)
$41$ \( ( 87184 + 7264 T - 1800 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$43$ \( 33624411406336 + 62108155904 T^{2} + 40259072 T^{4} + 10816 T^{6} + T^{8} \)
$47$ \( 1056981385216 + 9701752832 T^{2} + 15726848 T^{4} + 8032 T^{6} + T^{8} \)
$53$ \( ( -478064 + 161344 T + 9752 T^{2} + 176 T^{3} + T^{4} )^{2} \)
$59$ \( 173909016576 + 2174459904 T^{2} + 6273792 T^{4} + 4896 T^{6} + T^{8} \)
$61$ \( ( -2142704 - 273568 T - 2536 T^{2} + 88 T^{3} + T^{4} )^{2} \)
$67$ \( 281086590976 + 15044755456 T^{2} + 57554432 T^{4} + 16064 T^{6} + T^{8} \)
$71$ \( 16079971680256 + 101402017792 T^{2} + 64237568 T^{4} + 13952 T^{6} + T^{8} \)
$73$ \( ( 4962064 - 325920 T - 1576 T^{2} + 120 T^{3} + T^{4} )^{2} \)
$79$ \( 3198642669223936 + 2420601929728 T^{2} + 550899968 T^{4} + 41888 T^{6} + T^{8} \)
$83$ \( 4284940379815936 + 2381453017088 T^{2} + 464465408 T^{4} + 36928 T^{6} + T^{8} \)
$89$ \( ( 70652944 - 757600 T - 20584 T^{2} + 40 T^{3} + T^{4} )^{2} \)
$97$ \( ( -59281776 + 1154592 T + 5880 T^{2} - 216 T^{3} + T^{4} )^{2} \)
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