Properties

Label 1800.2.d.t
Level 1800
Weight 2
Character orbit 1800.d
Analytic conductor 14.373
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1800.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.3730723638\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.214798336.3
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 600)
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + ( -1 - \beta_{1} + \beta_{3} + \beta_{6} ) q^{4} + ( \beta_{1} - \beta_{3} + \beta_{5} + \beta_{7} ) q^{7} + ( 1 + \beta_{3} + \beta_{5} + \beta_{7} ) q^{8} +O(q^{10})\) \( q + \beta_{6} q^{2} + ( -1 - \beta_{1} + \beta_{3} + \beta_{6} ) q^{4} + ( \beta_{1} - \beta_{3} + \beta_{5} + \beta_{7} ) q^{7} + ( 1 + \beta_{3} + \beta_{5} + \beta_{7} ) q^{8} + ( -\beta_{1} - \beta_{2} - \beta_{6} ) q^{11} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{13} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{14} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{16} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{6} - 2 \beta_{7} ) q^{17} + ( -\beta_{1} - \beta_{2} - 2 \beta_{4} - 3 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{19} + ( 3 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{22} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{23} + ( -\beta_{2} + 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{7} ) q^{26} + ( 3 - \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{28} + ( 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + 4 \beta_{5} + \beta_{6} - \beta_{7} ) q^{29} + ( 3 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{31} + ( -2 \beta_{1} - 4 \beta_{5} + 2 \beta_{6} ) q^{32} + ( -1 + \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{34} + ( -2 \beta_{2} + 2 \beta_{3} + 2 \beta_{6} + 2 \beta_{7} ) q^{37} + ( -3 - \beta_{1} + 3 \beta_{2} + \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{38} + ( 2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - 3 \beta_{6} - \beta_{7} ) q^{41} + ( -1 - 4 \beta_{2} + 2 \beta_{3} + \beta_{4} + 3 \beta_{6} + 3 \beta_{7} ) q^{43} + ( -2 - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{6} ) q^{44} + ( 3 + \beta_{1} - \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{46} + ( 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{47} + ( -6 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{49} + ( 4 - 4 \beta_{2} + \beta_{4} - 3 \beta_{5} ) q^{52} + ( 2 + 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{6} ) q^{53} + ( -1 + 2 \beta_{1} - \beta_{3} + 5 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{56} + ( -1 + \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{58} + ( -4 \beta_{1} + 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{59} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + 5 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{61} + ( -3 + 3 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} - 5 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{62} + ( 2 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 4 \beta_{5} + 2 \beta_{6} ) q^{64} + ( 1 - 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - 3 \beta_{4} + 3 \beta_{6} + \beta_{7} ) q^{67} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{7} ) q^{68} + ( 6 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{6} ) q^{71} + ( -2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{73} + ( -2 \beta_{2} - 4 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{74} + ( -2 - 6 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 7 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} ) q^{76} + ( 4 + \beta_{1} + 5 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 3 \beta_{6} - 2 \beta_{7} ) q^{77} + ( 2 + 6 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{6} - 4 \beta_{7} ) q^{79} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 3 \beta_{4} - 5 \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{82} + ( 2 - 2 \beta_{1} + 6 \beta_{2} - 2 \beta_{4} - 2 \beta_{7} ) q^{83} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - 7 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{86} + ( -2 - 4 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{7} ) q^{88} + ( 4 + 4 \beta_{1} - 2 \beta_{4} - 6 \beta_{6} - 2 \beta_{7} ) q^{89} + ( 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + 5 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{91} + ( -4 - 6 \beta_{2} - 4 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{92} + ( 2 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{94} + ( 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{97} + ( 4 - 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - 8 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{2} - 4q^{4} + 8q^{8} + O(q^{10}) \) \( 8q + 2q^{2} - 4q^{4} + 8q^{8} - 6q^{14} + 8q^{16} + 20q^{22} + 2q^{26} + 24q^{28} + 8q^{31} + 12q^{32} - 12q^{34} - 14q^{38} - 8q^{43} - 12q^{44} + 20q^{46} + 24q^{52} - 8q^{53} - 8q^{56} - 20q^{58} - 26q^{62} + 32q^{64} + 24q^{67} - 36q^{68} + 40q^{71} + 8q^{74} - 20q^{76} + 24q^{77} + 16q^{79} - 16q^{82} + 32q^{83} + 18q^{86} - 8q^{88} - 28q^{92} + 4q^{94} + 40q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - 2 x^{5} + 9 x^{4} - 4 x^{3} - 16 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{7} + 2 \nu^{4} - 5 \nu^{3} - 6 \nu^{2} + 4 \nu + 8 \)\()/8\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} - 2 \nu^{4} + 5 \nu^{3} - 2 \nu^{2} + 4 \nu - 8 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 2 \nu^{4} - \nu^{3} - 2 \nu^{2} - 4 \nu + 4 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( -3 \nu^{7} + 2 \nu^{6} + 4 \nu^{5} + 6 \nu^{4} - 11 \nu^{3} - 8 \nu^{2} + 4 \nu + 24 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( -5 \nu^{7} + 2 \nu^{6} + 4 \nu^{5} + 18 \nu^{4} - 21 \nu^{3} - 12 \nu^{2} - 20 \nu + 56 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( -7 \nu^{7} + 4 \nu^{6} + 8 \nu^{5} + 22 \nu^{4} - 35 \nu^{3} - 22 \nu^{2} - 20 \nu + 88 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( 15 \nu^{7} - 10 \nu^{6} - 12 \nu^{5} - 46 \nu^{4} + 71 \nu^{3} + 32 \nu^{2} + 44 \nu - 168 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_{1} + 1\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_{1} + 3\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-3 \beta_{6} + 5 \beta_{5} + \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + \beta_{1} - 1\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(2 \beta_{7} + 3 \beta_{6} + \beta_{5} + 3 \beta_{4} - \beta_{3} - 2 \beta_{2} - 5 \beta_{1} - 3\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-4 \beta_{7} - 3 \beta_{6} - 5 \beta_{5} - \beta_{4} + \beta_{3} + 10 \beta_{2} - 3 \beta_{1} - 1\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(\beta_{6} + 3 \beta_{5} - 5 \beta_{4} - 9 \beta_{3} + 6 \beta_{2} + \beta_{1} - 3\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1549.1
−0.565036 1.29643i
−0.565036 + 1.29643i
1.23291 + 0.692769i
1.23291 0.692769i
1.41216 + 0.0762223i
1.41216 0.0762223i
−1.08003 0.912978i
−1.08003 + 0.912978i
−0.796431 1.16863i 0 −0.731395 + 1.86147i 0 0 4.72294i 2.75787 0.627801i 0 0
1549.2 −0.796431 + 1.16863i 0 −0.731395 1.86147i 0 0 4.72294i 2.75787 + 0.627801i 0 0
1549.3 −0.192769 1.40101i 0 −1.92568 + 0.540143i 0 0 0.0802864i 1.12796 + 2.59378i 0 0
1549.4 −0.192769 + 1.40101i 0 −1.92568 0.540143i 0 0 0.0802864i 1.12796 2.59378i 0 0
1549.5 0.576222 1.29150i 0 −1.33594 1.48838i 0 0 1.97676i −2.69204 + 0.867721i 0 0
1549.6 0.576222 + 1.29150i 0 −1.33594 + 1.48838i 0 0 1.97676i −2.69204 0.867721i 0 0
1549.7 1.41298 0.0591148i 0 1.99301 0.167056i 0 0 1.33411i 2.80620 0.353863i 0 0
1549.8 1.41298 + 0.0591148i 0 1.99301 + 0.167056i 0 0 1.33411i 2.80620 + 0.353863i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1549.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1800.2.d.t 8
3.b odd 2 1 600.2.d.g 8
4.b odd 2 1 7200.2.d.t 8
5.b even 2 1 1800.2.d.s 8
5.c odd 4 1 1800.2.k.q 8
5.c odd 4 1 1800.2.k.t 8
8.b even 2 1 1800.2.d.s 8
8.d odd 2 1 7200.2.d.s 8
12.b even 2 1 2400.2.d.h 8
15.d odd 2 1 600.2.d.h 8
15.e even 4 1 600.2.k.d 8
15.e even 4 1 600.2.k.e yes 8
20.d odd 2 1 7200.2.d.s 8
20.e even 4 1 7200.2.k.r 8
20.e even 4 1 7200.2.k.s 8
24.f even 2 1 2400.2.d.g 8
24.h odd 2 1 600.2.d.h 8
40.e odd 2 1 7200.2.d.t 8
40.f even 2 1 inner 1800.2.d.t 8
40.i odd 4 1 1800.2.k.q 8
40.i odd 4 1 1800.2.k.t 8
40.k even 4 1 7200.2.k.r 8
40.k even 4 1 7200.2.k.s 8
60.h even 2 1 2400.2.d.g 8
60.l odd 4 1 2400.2.k.d 8
60.l odd 4 1 2400.2.k.e 8
120.i odd 2 1 600.2.d.g 8
120.m even 2 1 2400.2.d.h 8
120.q odd 4 1 2400.2.k.d 8
120.q odd 4 1 2400.2.k.e 8
120.w even 4 1 600.2.k.d 8
120.w even 4 1 600.2.k.e yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.d.g 8 3.b odd 2 1
600.2.d.g 8 120.i odd 2 1
600.2.d.h 8 15.d odd 2 1
600.2.d.h 8 24.h odd 2 1
600.2.k.d 8 15.e even 4 1
600.2.k.d 8 120.w even 4 1
600.2.k.e yes 8 15.e even 4 1
600.2.k.e yes 8 120.w even 4 1
1800.2.d.s 8 5.b even 2 1
1800.2.d.s 8 8.b even 2 1
1800.2.d.t 8 1.a even 1 1 trivial
1800.2.d.t 8 40.f even 2 1 inner
1800.2.k.q 8 5.c odd 4 1
1800.2.k.q 8 40.i odd 4 1
1800.2.k.t 8 5.c odd 4 1
1800.2.k.t 8 40.i odd 4 1
2400.2.d.g 8 24.f even 2 1
2400.2.d.g 8 60.h even 2 1
2400.2.d.h 8 12.b even 2 1
2400.2.d.h 8 120.m even 2 1
2400.2.k.d 8 60.l odd 4 1
2400.2.k.d 8 120.q odd 4 1
2400.2.k.e 8 60.l odd 4 1
2400.2.k.e 8 120.q odd 4 1
7200.2.d.s 8 8.d odd 2 1
7200.2.d.s 8 20.d odd 2 1
7200.2.d.t 8 4.b odd 2 1
7200.2.d.t 8 40.e odd 2 1
7200.2.k.r 8 20.e even 4 1
7200.2.k.r 8 40.k even 4 1
7200.2.k.s 8 20.e even 4 1
7200.2.k.s 8 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1800, [\chi])\):

\( T_{7}^{8} + 28 T_{7}^{6} + 134 T_{7}^{4} + 156 T_{7}^{2} + 1 \)
\( T_{11}^{8} + 32 T_{11}^{6} + 336 T_{11}^{4} + 1344 T_{11}^{2} + 1600 \)
\( T_{13}^{4} - 22 T_{13}^{2} + 32 T_{13} + 9 \)
\( T_{37}^{4} - 64 T_{37}^{2} + 128 T_{37} + 64 \)
\( T_{41}^{4} - 64 T_{41}^{2} - 56 T_{41} + 328 \)
\( T_{53}^{4} + 4 T_{53}^{3} - 120 T_{53}^{2} - 680 T_{53} - 152 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 4 T^{2} - 8 T^{3} + 10 T^{4} - 16 T^{5} + 16 T^{6} - 16 T^{7} + 16 T^{8} \)
$3$ 1
$5$ 1
$7$ \( 1 - 28 T^{2} + 330 T^{4} - 2224 T^{6} + 13203 T^{8} - 108976 T^{10} + 792330 T^{12} - 3294172 T^{14} + 5764801 T^{16} \)
$11$ \( 1 - 56 T^{2} + 1612 T^{4} - 29896 T^{6} + 388998 T^{8} - 3617416 T^{10} + 23601292 T^{12} - 99207416 T^{14} + 214358881 T^{16} \)
$13$ \( ( 1 + 30 T^{2} + 32 T^{3} + 451 T^{4} + 416 T^{5} + 5070 T^{6} + 28561 T^{8} )^{2} \)
$17$ \( 1 - 56 T^{2} + 1484 T^{4} - 24968 T^{6} + 374950 T^{8} - 7215752 T^{10} + 123945164 T^{12} - 1351703864 T^{14} + 6975757441 T^{16} \)
$19$ \( 1 - 36 T^{2} + 1546 T^{4} - 35120 T^{6} + 832243 T^{8} - 12678320 T^{10} + 201476266 T^{12} - 1693651716 T^{14} + 16983563041 T^{16} \)
$23$ \( 1 - 56 T^{2} + 1324 T^{4} - 36616 T^{6} + 1094310 T^{8} - 19369864 T^{10} + 370509484 T^{12} - 8290009784 T^{14} + 78310985281 T^{16} \)
$29$ \( 1 - 88 T^{2} + 4780 T^{4} - 171048 T^{6} + 5385990 T^{8} - 143851368 T^{10} + 3380803180 T^{12} - 52344452248 T^{14} + 500246412961 T^{16} \)
$31$ \( ( 1 - 4 T + 54 T^{2} - 168 T^{3} + 2099 T^{4} - 5208 T^{5} + 51894 T^{6} - 119164 T^{7} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 + 84 T^{2} + 128 T^{3} + 3542 T^{4} + 4736 T^{5} + 114996 T^{6} + 1874161 T^{8} )^{2} \)
$41$ \( ( 1 + 100 T^{2} - 56 T^{3} + 5166 T^{4} - 2296 T^{5} + 168100 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 + 4 T + 58 T^{2} + 336 T^{3} + 3379 T^{4} + 14448 T^{5} + 107242 T^{6} + 318028 T^{7} + 3418801 T^{8} )^{2} \)
$47$ \( 1 - 232 T^{2} + 26076 T^{4} - 1910872 T^{6} + 102863686 T^{8} - 4221116248 T^{10} + 127242561756 T^{12} - 2500777956328 T^{14} + 23811286661761 T^{16} \)
$53$ \( ( 1 + 4 T + 92 T^{2} - 44 T^{3} + 3982 T^{4} - 2332 T^{5} + 258428 T^{6} + 595508 T^{7} + 7890481 T^{8} )^{2} \)
$59$ \( 1 - 40 T^{2} + 2364 T^{4} - 112984 T^{6} + 20250598 T^{8} - 393297304 T^{10} + 28645441404 T^{12} - 1687221345640 T^{14} + 146830437604321 T^{16} \)
$61$ \( 1 - 252 T^{2} + 32650 T^{4} - 2942672 T^{6} + 202734451 T^{8} - 10949682512 T^{10} + 452066708650 T^{12} - 12983134338972 T^{14} + 191707312997281 T^{16} \)
$67$ \( ( 1 - 12 T + 154 T^{2} - 1520 T^{3} + 16755 T^{4} - 101840 T^{5} + 691306 T^{6} - 3609156 T^{7} + 20151121 T^{8} )^{2} \)
$71$ \( ( 1 - 20 T + 380 T^{2} - 4188 T^{3} + 43342 T^{4} - 297348 T^{5} + 1915580 T^{6} - 7158220 T^{7} + 25411681 T^{8} )^{2} \)
$73$ \( 1 - 184 T^{2} + 15196 T^{4} - 1235336 T^{6} + 104948486 T^{8} - 6583105544 T^{10} + 431539670236 T^{12} - 27845497637176 T^{14} + 806460091894081 T^{16} \)
$79$ \( ( 1 - 8 T + 132 T^{2} - 1032 T^{3} + 16454 T^{4} - 81528 T^{5} + 823812 T^{6} - 3944312 T^{7} + 38950081 T^{8} )^{2} \)
$83$ \( ( 1 - 16 T + 276 T^{2} - 2576 T^{3} + 30070 T^{4} - 213808 T^{5} + 1901364 T^{6} - 9148592 T^{7} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 + 132 T^{2} + 64 T^{3} + 18534 T^{4} + 5696 T^{5} + 1045572 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( 1 - 356 T^{2} + 80362 T^{4} - 11927760 T^{6} + 1353370099 T^{8} - 112228293840 T^{10} + 7114390079722 T^{12} - 296538033754724 T^{14} + 7837433594376961 T^{16} \)
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