Properties

Label 3744.2.g.e
Level $3744$
Weight $2$
Character orbit 3744.g
Analytic conductor $29.896$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3744.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.8959905168\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 2 x^{15} + 2 x^{14} - 4 x^{13} + 9 x^{12} - 10 x^{11} + 2 x^{10} - 8 x^{9} + 28 x^{8} - 16 x^{7} + 8 x^{6} - 80 x^{5} + 144 x^{4} - 128 x^{3} + 128 x^{2} - 256 x + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: no (minimal twist has level 312)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{12} q^{5} -\beta_{5} q^{7} +O(q^{10})\) \( q + \beta_{12} q^{5} -\beta_{5} q^{7} + \beta_{10} q^{11} -\beta_{6} q^{13} + ( -1 + \beta_{2} ) q^{17} + ( \beta_{1} - \beta_{6} ) q^{19} + ( -1 + \beta_{3} + \beta_{5} + \beta_{8} ) q^{23} + ( -2 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{11} ) q^{25} + ( \beta_{1} - \beta_{12} - \beta_{14} ) q^{29} + \beta_{11} q^{31} + ( -\beta_{6} - \beta_{7} - \beta_{12} - \beta_{14} ) q^{35} + ( \beta_{6} - \beta_{10} + \beta_{14} ) q^{37} + ( 2 + \beta_{8} - \beta_{13} ) q^{41} + ( 2 \beta_{6} - \beta_{10} + \beta_{14} - \beta_{15} ) q^{43} + ( 2 - \beta_{3} - \beta_{5} ) q^{47} + ( 3 + \beta_{2} - \beta_{3} - \beta_{8} + \beta_{11} ) q^{49} + ( \beta_{1} - \beta_{7} + \beta_{9} - \beta_{10} + \beta_{12} - \beta_{15} ) q^{53} + ( -2 - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{8} + \beta_{11} ) q^{55} + ( -\beta_{7} - \beta_{9} + \beta_{10} - 2 \beta_{12} ) q^{59} + ( -\beta_{6} + \beta_{7} - \beta_{10} + \beta_{12} + \beta_{15} ) q^{61} -\beta_{3} q^{65} + ( -\beta_{1} - 3 \beta_{6} - \beta_{7} + \beta_{9} ) q^{67} + ( -2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{11} ) q^{71} + ( -3 + 2 \beta_{3} + \beta_{4} + \beta_{8} - 2 \beta_{13} ) q^{73} + ( -2 \beta_{1} - 2 \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} + \beta_{14} - \beta_{15} ) q^{77} + ( 1 - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{8} - \beta_{11} + \beta_{13} ) q^{79} + ( 3 \beta_{6} - 2 \beta_{9} + 2 \beta_{12} + \beta_{14} ) q^{83} + ( -\beta_{6} + \beta_{10} - 2 \beta_{12} + \beta_{14} ) q^{85} + ( 3 + 2 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{11} - \beta_{13} ) q^{89} + \beta_{9} q^{91} + ( -1 - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{8} + 2 \beta_{13} ) q^{95} + ( 3 - \beta_{4} - 2 \beta_{5} + \beta_{8} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{7} + O(q^{10}) \) \( 16 q - 4 q^{7} - 16 q^{17} - 8 q^{23} - 32 q^{25} + 4 q^{31} + 36 q^{41} + 24 q^{47} + 48 q^{49} - 24 q^{55} - 4 q^{65} - 32 q^{73} + 60 q^{89} - 24 q^{95} + 40 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 2 x^{15} + 2 x^{14} - 4 x^{13} + 9 x^{12} - 10 x^{11} + 2 x^{10} - 8 x^{9} + 28 x^{8} - 16 x^{7} + 8 x^{6} - 80 x^{5} + 144 x^{4} - 128 x^{3} + 128 x^{2} - 256 x + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -6 \nu^{15} + 23 \nu^{14} - 18 \nu^{13} + 50 \nu^{12} - 50 \nu^{11} + 7 \nu^{10} - 26 \nu^{9} + 42 \nu^{8} - 112 \nu^{7} + 124 \nu^{6} - 224 \nu^{5} + 1096 \nu^{4} - 624 \nu^{3} + 400 \nu^{2} - 1856 \nu + 384 \)\()/448\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{15} - 8 \nu^{14} + 2 \nu^{13} + 25 \nu^{11} - 16 \nu^{10} - 14 \nu^{9} - 4 \nu^{8} + 68 \nu^{7} - 24 \nu^{6} - 40 \nu^{5} - 192 \nu^{4} + 400 \nu^{3} + 32 \nu^{2} + 320 \nu - 448 \)\()/64\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{15} + 10 \nu^{14} - 2 \nu^{13} + 12 \nu^{12} - 51 \nu^{11} + 18 \nu^{10} + 30 \nu^{9} + 56 \nu^{8} - 124 \nu^{7} - 64 \nu^{6} + 24 \nu^{5} + 528 \nu^{4} - 528 \nu^{3} - 832 \nu + 1280 \)\()/128\)
\(\beta_{4}\)\(=\)\((\)\( -5 \nu^{15} + 22 \nu^{14} - 6 \nu^{13} + 20 \nu^{12} - 117 \nu^{11} + 46 \nu^{10} + 58 \nu^{9} + 136 \nu^{8} - 260 \nu^{7} - 64 \nu^{6} + 104 \nu^{5} + 1072 \nu^{4} - 1264 \nu^{3} - 256 \nu^{2} - 1856 \nu + 2816 \)\()/128\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{15} - 8 \nu^{14} + 2 \nu^{13} - 8 \nu^{12} + 43 \nu^{11} - 16 \nu^{10} - 30 \nu^{9} - 52 \nu^{8} + 108 \nu^{7} + 24 \nu^{6} - 56 \nu^{5} - 416 \nu^{4} + 496 \nu^{3} + 32 \nu^{2} + 512 \nu - 1024 \)\()/64\)
\(\beta_{6}\)\(=\)\((\)\( -25 \nu^{15} + 13 \nu^{14} + 16 \nu^{13} + 94 \nu^{12} - 101 \nu^{11} - 91 \nu^{10} - 8 \nu^{9} + 322 \nu^{8} - 140 \nu^{7} - 356 \nu^{6} - 616 \nu^{5} + 1720 \nu^{4} + 256 \nu^{3} + 528 \nu^{2} - 2880 \nu + 256 \)\()/448\)
\(\beta_{7}\)\(=\)\((\)\( 39 \nu^{15} - 34 \nu^{14} - 2 \nu^{13} - 164 \nu^{12} + 199 \nu^{11} + 70 \nu^{10} + 78 \nu^{9} - 448 \nu^{8} + 252 \nu^{7} + 384 \nu^{6} + 840 \nu^{5} - 2672 \nu^{4} + 80 \nu^{3} - 1984 \nu^{2} + 6016 \nu - 1152 \)\()/448\)
\(\beta_{8}\)\(=\)\((\)\( 13 \nu^{15} - 38 \nu^{14} + 6 \nu^{13} - 20 \nu^{12} + 157 \nu^{11} - 62 \nu^{10} - 122 \nu^{9} - 168 \nu^{8} + 452 \nu^{7} + 64 \nu^{6} - 168 \nu^{5} - 1584 \nu^{4} + 1776 \nu^{3} + 256 \nu^{2} + 2112 \nu - 3456 \)\()/128\)
\(\beta_{9}\)\(=\)\((\)\( 23 \nu^{15} - 10 \nu^{14} - 8 \nu^{13} - 96 \nu^{12} + 103 \nu^{11} + 70 \nu^{10} + 32 \nu^{9} - 364 \nu^{8} + 140 \nu^{7} + 416 \nu^{6} + 616 \nu^{5} - 1728 \nu^{4} - 16 \nu^{3} - 544 \nu^{2} + 3232 \nu - 1472 \)\()/224\)
\(\beta_{10}\)\(=\)\((\)\( 107 \nu^{15} - 38 \nu^{14} - 134 \nu^{13} - 404 \nu^{12} + 411 \nu^{11} + 434 \nu^{10} - 38 \nu^{9} - 1568 \nu^{8} + 588 \nu^{7} + 2096 \nu^{6} + 2184 \nu^{5} - 7216 \nu^{4} - 2032 \nu^{3} - 320 \nu^{2} + 12864 \nu - 1024 \)\()/896\)
\(\beta_{11}\)\(=\)\((\)\( -4 \nu^{15} + 13 \nu^{14} - 4 \nu^{13} + 10 \nu^{12} - 56 \nu^{11} + 29 \nu^{10} + 28 \nu^{9} + 58 \nu^{8} - 152 \nu^{7} - 4 \nu^{6} + 32 \nu^{5} + 536 \nu^{4} - 656 \nu^{3} + 16 \nu^{2} - 736 \nu + 1280 \)\()/32\)
\(\beta_{12}\)\(=\)\((\)\( -117 \nu^{15} + 74 \nu^{14} + 34 \nu^{13} + 492 \nu^{12} - 485 \nu^{11} - 350 \nu^{10} - 94 \nu^{9} + 1568 \nu^{8} - 532 \nu^{7} - 1712 \nu^{6} - 3192 \nu^{5} + 8016 \nu^{4} + 656 \nu^{3} + 3264 \nu^{2} - 14912 \nu + 2560 \)\()/896\)
\(\beta_{13}\)\(=\)\((\)\( 7 \nu^{15} - 23 \nu^{14} + 2 \nu^{13} - 14 \nu^{12} + 107 \nu^{11} - 39 \nu^{10} - 78 \nu^{9} - 122 \nu^{8} + 308 \nu^{7} + 60 \nu^{6} - 152 \nu^{5} - 1016 \nu^{4} + 1184 \nu^{3} + 304 \nu^{2} + 1312 \nu - 2464 \)\()/32\)
\(\beta_{14}\)\(=\)\((\)\( 199 \nu^{15} - 64 \nu^{14} - 194 \nu^{13} - 816 \nu^{12} + 767 \nu^{11} + 952 \nu^{10} + 174 \nu^{9} - 3052 \nu^{8} + 588 \nu^{7} + 4264 \nu^{6} + 4872 \nu^{5} - 13792 \nu^{4} - 5456 \nu^{3} - 1824 \nu^{2} + 24896 \nu - 1536 \)\()/896\)
\(\beta_{15}\)\(=\)\((\)\( -104 \nu^{15} + 65 \nu^{14} + 66 \nu^{13} + 470 \nu^{12} - 428 \nu^{11} - 399 \nu^{10} - 166 \nu^{9} + 1526 \nu^{8} - 392 \nu^{7} - 1612 \nu^{6} - 3024 \nu^{5} + 7704 \nu^{4} + 1616 \nu^{3} + 3312 \nu^{2} - 15744 \nu + 384 \)\()/448\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{14} + \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + 1\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{15} - \beta_{12} + \beta_{10} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{1}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{14} + \beta_{11} + \beta_{10} + 3 \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} + 3 \beta_{5} + \beta_{4} + 4 \beta_{2} + 3\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{14} + \beta_{13} - 3 \beta_{12} + \beta_{11} - \beta_{8} + \beta_{7} + 3 \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} + 2 \beta_{1} - 3\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-3 \beta_{14} - 4 \beta_{12} - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - 5 \beta_{7} - 17 \beta_{6} - 7 \beta_{5} - 3 \beta_{4} - 4 \beta_{3} + 4 \beta_{1} - 1\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(\beta_{15} + 2 \beta_{14} - \beta_{12} - \beta_{10} + 2 \beta_{9} + 8 \beta_{6} - \beta_{5} + 3 \beta_{4} - 7 \beta_{3} + 2 \beta_{2} + \beta_{1} + 14\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-7 \beta_{14} + 8 \beta_{13} + 3 \beta_{11} - \beta_{10} + 9 \beta_{9} - 7 \beta_{8} + 5 \beta_{7} - 9 \beta_{6} - 15 \beta_{5} + 7 \beta_{4} - 8 \beta_{3} - 4 \beta_{2} + 29\)\()/8\)
\(\nu^{8}\)\(=\)\((\)\(8 \beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} + 6 \beta_{10} - 2 \beta_{9} - \beta_{8} + 5 \beta_{7} - 21 \beta_{6} - 5 \beta_{5} - 2 \beta_{4} - 13 \beta_{3} + 3 \beta_{2} - 6 \beta_{1} - 11\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(8 \beta_{15} + 15 \beta_{14} + 28 \beta_{12} - 11 \beta_{11} - 3 \beta_{10} + 21 \beta_{9} - 11 \beta_{8} + 9 \beta_{7} + 5 \beta_{6} - 5 \beta_{5} + 7 \beta_{4} + 12 \beta_{3} + 16 \beta_{2} - 12 \beta_{1} + 5\)\()/8\)
\(\nu^{10}\)\(=\)\((\)\(-5 \beta_{15} + 8 \beta_{13} + 5 \beta_{12} + 14 \beta_{11} - 21 \beta_{10} + 8 \beta_{9} - 6 \beta_{8} + 2 \beta_{7} - 26 \beta_{6} + 7 \beta_{5} + 7 \beta_{4} + 3 \beta_{3} + 14 \beta_{2} + 3 \beta_{1} - 44\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(16 \beta_{15} + 11 \beta_{14} - 8 \beta_{12} - 7 \beta_{11} - 27 \beta_{10} - 45 \beta_{9} - 21 \beta_{8} + 15 \beta_{7} - 139 \beta_{6} - 13 \beta_{5} - 35 \beta_{4} - 4 \beta_{2} + 32 \beta_{1} + 39\)\()/8\)
\(\nu^{12}\)\(=\)\((\)\(-8 \beta_{15} + 11 \beta_{14} - 11 \beta_{13} + 45 \beta_{12} - 3 \beta_{11} - 28 \beta_{10} + 20 \beta_{9} + 31 \beta_{8} - 7 \beta_{7} - 53 \beta_{6} - 9 \beta_{5} - 8 \beta_{4} + 9 \beta_{3} + \beta_{2} + 22 \beta_{1} + 49\)\()/4\)
\(\nu^{13}\)\(=\)\((\)\(32 \beta_{15} + 21 \beta_{14} + 16 \beta_{13} - 68 \beta_{12} + 39 \beta_{11} - 89 \beta_{10} + 39 \beta_{9} + 39 \beta_{8} + 35 \beta_{7} + 55 \beta_{6} - 31 \beta_{5} - 11 \beta_{4} + 28 \beta_{3} - 48 \beta_{2} - 28 \beta_{1} + 295\)\()/8\)
\(\nu^{14}\)\(=\)\((\)\(41 \beta_{15} - 42 \beta_{14} - 8 \beta_{13} + 23 \beta_{12} + 12 \beta_{11} + 51 \beta_{10} + 38 \beta_{9} - 4 \beta_{8} + 28 \beta_{7} - 196 \beta_{6} + 3 \beta_{5} - 49 \beta_{4} - 15 \beta_{3} - 30 \beta_{2} - 15 \beta_{1} + 194\)\()/4\)
\(\nu^{15}\)\(=\)\((\)\(192 \beta_{15} + \beta_{14} - 72 \beta_{13} - 48 \beta_{12} + 11 \beta_{11} + 199 \beta_{10} + 113 \beta_{9} + 161 \beta_{8} + 157 \beta_{7} + 47 \beta_{6} + 41 \beta_{5} - 49 \beta_{4} + 232 \beta_{3} + 76 \beta_{2} - 64 \beta_{1} - 315\)\()/8\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(2017\) \(2081\) \(2341\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1873.1
−1.32561 0.492712i
−0.879244 1.10767i
1.27276 + 0.616518i
0.791485 1.17199i
−0.654116 + 1.25385i
0.802079 + 1.16476i
−0.414573 1.35208i
1.40722 + 0.140463i
1.40722 0.140463i
−0.414573 + 1.35208i
0.802079 1.16476i
−0.654116 1.25385i
0.791485 + 1.17199i
1.27276 0.616518i
−0.879244 + 1.10767i
−1.32561 + 0.492712i
0 0 0 4.33571i 0 2.30442 0 0 0
1873.2 0 0 0 3.31390i 0 −4.17825 0 0 0
1873.3 0 0 0 3.29521i 0 −2.97802 0 0 0
1873.4 0 0 0 3.05343i 0 0.397397 0 0 0
1873.5 0 0 0 1.87654i 0 −0.584696 0 0 0
1873.6 0 0 0 1.47174i 0 2.93973 0 0 0
1873.7 0 0 0 0.550135i 0 −4.37841 0 0 0
1873.8 0 0 0 0.218531i 0 4.47783 0 0 0
1873.9 0 0 0 0.218531i 0 4.47783 0 0 0
1873.10 0 0 0 0.550135i 0 −4.37841 0 0 0
1873.11 0 0 0 1.47174i 0 2.93973 0 0 0
1873.12 0 0 0 1.87654i 0 −0.584696 0 0 0
1873.13 0 0 0 3.05343i 0 0.397397 0 0 0
1873.14 0 0 0 3.29521i 0 −2.97802 0 0 0
1873.15 0 0 0 3.31390i 0 −4.17825 0 0 0
1873.16 0 0 0 4.33571i 0 2.30442 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1873.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3744.2.g.e 16
3.b odd 2 1 1248.2.g.b 16
4.b odd 2 1 936.2.g.e 16
8.b even 2 1 inner 3744.2.g.e 16
8.d odd 2 1 936.2.g.e 16
12.b even 2 1 312.2.g.b 16
24.f even 2 1 312.2.g.b 16
24.h odd 2 1 1248.2.g.b 16
48.i odd 4 1 9984.2.a.bs 8
48.i odd 4 1 9984.2.a.bv 8
48.k even 4 1 9984.2.a.bt 8
48.k even 4 1 9984.2.a.bu 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.g.b 16 12.b even 2 1
312.2.g.b 16 24.f even 2 1
936.2.g.e 16 4.b odd 2 1
936.2.g.e 16 8.d odd 2 1
1248.2.g.b 16 3.b odd 2 1
1248.2.g.b 16 24.h odd 2 1
3744.2.g.e 16 1.a even 1 1 trivial
3744.2.g.e 16 8.b even 2 1 inner
9984.2.a.bs 8 48.i odd 4 1
9984.2.a.bt 8 48.k even 4 1
9984.2.a.bu 8 48.k even 4 1
9984.2.a.bv 8 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{16} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(3744, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( T^{16} \)
$5$ \( 2304 + 58368 T^{2} + 221248 T^{4} + 197888 T^{6} + 73152 T^{8} + 13152 T^{10} + 1220 T^{12} + 56 T^{14} + T^{16} \)
$7$ \( ( 384 - 384 T - 1696 T^{2} + 416 T^{3} + 444 T^{4} - 60 T^{5} - 38 T^{6} + 2 T^{7} + T^{8} )^{2} \)
$11$ \( 1679616 + 20772864 T^{2} + 52308480 T^{4} + 17463808 T^{6} + 2420448 T^{8} + 171776 T^{10} + 6560 T^{12} + 128 T^{14} + T^{16} \)
$13$ \( ( 1 + T^{2} )^{8} \)
$17$ \( ( -64256 - 4864 T + 37056 T^{2} + 14720 T^{3} - 720 T^{5} - 68 T^{6} + 8 T^{7} + T^{8} )^{2} \)
$19$ \( 22429696 + 5707161600 T^{2} + 2373474304 T^{4} + 343948032 T^{6} + 24400016 T^{8} + 943712 T^{10} + 20204 T^{12} + 224 T^{14} + T^{16} \)
$23$ \( ( -166912 - 153600 T + 3584 T^{2} + 24448 T^{3} + 3536 T^{4} - 624 T^{5} - 120 T^{6} + 4 T^{7} + T^{8} )^{2} \)
$29$ \( 4671995904 + 6215335936 T^{2} + 2247888896 T^{4} + 355358720 T^{6} + 28005376 T^{8} + 1138304 T^{10} + 23952 T^{12} + 248 T^{14} + T^{16} \)
$31$ \( ( 110464 + 1920 T - 45408 T^{2} - 2496 T^{3} + 3804 T^{4} + 140 T^{5} - 110 T^{6} - 2 T^{7} + T^{8} )^{2} \)
$37$ \( 157351936 + 1165492224 T^{2} + 1151582208 T^{4} + 262275072 T^{6} + 24667648 T^{8} + 1103872 T^{10} + 24512 T^{12} + 256 T^{14} + T^{16} \)
$41$ \( ( -139792 - 14928 T + 77912 T^{2} - 15400 T^{3} - 4008 T^{4} + 1116 T^{5} + 14 T^{6} - 18 T^{7} + T^{8} )^{2} \)
$43$ \( 561468473344 + 373267726336 T^{2} + 56422666240 T^{4} + 3908044800 T^{6} + 149857280 T^{8} + 3392384 T^{10} + 45200 T^{12} + 328 T^{14} + T^{16} \)
$47$ \( ( 48 + 1600 T + 2016 T^{2} - 2960 T^{3} - 480 T^{4} + 368 T^{5} - 12 T^{7} + T^{8} )^{2} \)
$53$ \( 462422016 + 47784656896 T^{2} + 45540622336 T^{4} + 8906956800 T^{6} + 425390336 T^{8} + 8962304 T^{10} + 95136 T^{12} + 496 T^{14} + T^{16} \)
$59$ \( 276642337024 + 855183167488 T^{2} + 264285707776 T^{4} + 21081421824 T^{6} + 716910560 T^{8} + 12457856 T^{10} + 115808 T^{12} + 544 T^{14} + T^{16} \)
$61$ \( 329127100416 + 215901306880 T^{2} + 48761671680 T^{4} + 5171939328 T^{6} + 278176768 T^{8} + 7458944 T^{10} + 94736 T^{12} + 520 T^{14} + T^{16} \)
$67$ \( 409763856384 + 333682647040 T^{2} + 94434490368 T^{4} + 11087418624 T^{6} + 509533712 T^{8} + 10552416 T^{10} + 107500 T^{12} + 528 T^{14} + T^{16} \)
$71$ \( ( 21431088 + 3108384 T - 1886592 T^{2} - 112384 T^{3} + 49664 T^{4} + 600 T^{5} - 408 T^{6} + T^{8} )^{2} \)
$73$ \( ( 3326208 - 7414784 T - 618496 T^{2} + 392960 T^{3} + 22240 T^{4} - 5312 T^{5} - 320 T^{6} + 16 T^{7} + T^{8} )^{2} \)
$79$ \( ( 2166016 - 1229824 T - 1101568 T^{2} + 53760 T^{3} + 35360 T^{4} - 320 T^{5} - 336 T^{6} + T^{8} )^{2} \)
$83$ \( 6293836597504 + 3852663542784 T^{2} + 621437020160 T^{4} + 35690143488 T^{6} + 996178016 T^{8} + 15118656 T^{10} + 127424 T^{12} + 560 T^{14} + T^{16} \)
$89$ \( ( 37581936 - 20801808 T + 3470840 T^{2} + 43112 T^{3} - 69736 T^{4} + 5916 T^{5} + 86 T^{6} - 30 T^{7} + T^{8} )^{2} \)
$97$ \( ( -637696 - 549632 T + 564736 T^{2} - 91328 T^{3} - 15584 T^{4} + 4272 T^{5} - 128 T^{6} - 20 T^{7} + T^{8} )^{2} \)
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