Properties

Label 8001.2.a.o
Level $8001$
Weight $2$
Character orbit 8001.a
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - 4 x^{12} - 10 x^{11} + 53 x^{10} + 19 x^{9} - 242 x^{8} + 61 x^{7} + 467 x^{6} - 211 x^{5} - 372 x^{4} + 146 x^{3} + 116 x^{2} - 12 x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 2667)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -1 + \beta_{10} ) q^{5} + q^{7} + ( -1 - \beta_{1} - \beta_{6} + \beta_{7} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -1 + \beta_{10} ) q^{5} + q^{7} + ( -1 - \beta_{1} - \beta_{6} + \beta_{7} ) q^{8} + ( \beta_{1} - \beta_{11} ) q^{10} + \beta_{8} q^{11} + ( 2 + \beta_{4} - \beta_{7} - \beta_{12} ) q^{13} -\beta_{1} q^{14} + ( 1 + \beta_{1} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{12} ) q^{16} + ( -1 - \beta_{4} + \beta_{5} + \beta_{9} ) q^{17} + ( -\beta_{4} - \beta_{8} + \beta_{12} ) q^{19} + ( -3 - \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} ) q^{20} + ( -\beta_{3} - \beta_{4} + \beta_{5} + \beta_{9} + \beta_{12} ) q^{22} + ( 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{23} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{7} - 3 \beta_{10} ) q^{25} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{12} ) q^{26} + ( 1 + \beta_{2} ) q^{28} + ( -2 + \beta_{1} - \beta_{4} + \beta_{6} + \beta_{10} - \beta_{11} ) q^{29} + ( -1 - \beta_{1} + \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} ) q^{31} + ( -2 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} ) q^{32} + ( -1 + 3 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{12} ) q^{34} + ( -1 + \beta_{10} ) q^{35} + ( 1 + \beta_{4} + \beta_{11} + \beta_{12} ) q^{37} + ( 1 + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{9} - \beta_{12} ) q^{38} + ( 4 + 5 \beta_{1} + \beta_{3} + 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} + 3 \beta_{11} ) q^{40} + ( -1 + \beta_{4} + \beta_{6} ) q^{41} + ( -2 - \beta_{4} - 2 \beta_{5} - \beta_{9} - \beta_{12} ) q^{43} + ( 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} ) q^{44} + ( -4 - \beta_{1} - 2 \beta_{3} - 2 \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{46} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} - 2 \beta_{9} - \beta_{10} - \beta_{12} ) q^{47} + q^{49} + ( -1 + \beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{50} + ( 2 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{6} - \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} ) q^{52} + ( -3 - \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{53} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} ) q^{55} + ( -1 - \beta_{1} - \beta_{6} + \beta_{7} ) q^{56} + ( -4 - \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} ) q^{58} + ( -3 + \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{8} - 3 \beta_{10} + \beta_{11} + \beta_{12} ) q^{59} + ( 3 + \beta_{4} + \beta_{6} + \beta_{8} - \beta_{12} ) q^{61} + ( 3 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - \beta_{8} - \beta_{9} - 4 \beta_{10} + 2 \beta_{11} - \beta_{12} ) q^{62} + ( 3 + 4 \beta_{1} - \beta_{2} + 5 \beta_{3} + 2 \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} - 4 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} ) q^{64} + ( -3 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} + \beta_{12} ) q^{65} + ( -2 - \beta_{1} - \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} + 2 \beta_{12} ) q^{67} + ( -3 + 3 \beta_{1} - 4 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{10} + \beta_{11} + \beta_{12} ) q^{68} + ( \beta_{1} - \beta_{11} ) q^{70} + ( -1 + \beta_{1} + \beta_{3} - \beta_{5} - \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} ) q^{71} + ( 1 + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{73} + ( 3 + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{10} - 2 \beta_{12} ) q^{74} + ( -1 - 2 \beta_{1} - 3 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + 3 \beta_{12} ) q^{76} + \beta_{8} q^{77} + ( -4 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{9} + \beta_{12} ) q^{79} + ( -4 - 5 \beta_{1} - \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} + \beta_{10} - 2 \beta_{11} + 2 \beta_{12} ) q^{80} + ( 1 + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{9} - \beta_{12} ) q^{82} + ( -2 - 4 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{11} + \beta_{12} ) q^{83} + ( 1 - \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} ) q^{85} + ( 1 + \beta_{1} + 3 \beta_{5} + \beta_{7} - 2 \beta_{8} - 2 \beta_{10} + \beta_{11} + 2 \beta_{12} ) q^{86} + ( -3 + \beta_{1} - 2 \beta_{2} - \beta_{5} + \beta_{7} - \beta_{8} - \beta_{11} + 2 \beta_{12} ) q^{88} + ( -5 - 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} - \beta_{6} + 2 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} ) q^{89} + ( 2 + \beta_{4} - \beta_{7} - \beta_{12} ) q^{91} + ( 2 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - \beta_{7} - 2 \beta_{10} - \beta_{12} ) q^{92} + ( -3 + \beta_{2} - 3 \beta_{3} - 2 \beta_{6} - \beta_{8} + \beta_{9} + 3 \beta_{10} - \beta_{11} + 2 \beta_{12} ) q^{94} + ( 1 + 2 \beta_{3} + \beta_{7} + \beta_{8} - \beta_{10} - 2 \beta_{12} ) q^{95} + ( -1 - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{10} - 2 \beta_{11} + \beta_{12} ) q^{97} -\beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 4q^{2} + 10q^{4} - 12q^{5} + 13q^{7} - 9q^{8} + O(q^{10}) \) \( 13q - 4q^{2} + 10q^{4} - 12q^{5} + 13q^{7} - 9q^{8} + 6q^{10} - 3q^{11} + 21q^{13} - 4q^{14} + 8q^{16} - 17q^{17} + 5q^{19} - 29q^{20} + q^{22} - 4q^{23} + q^{25} - 22q^{26} + 10q^{28} - 21q^{29} - 7q^{31} - 12q^{32} + 2q^{34} - 12q^{35} + 7q^{37} + 9q^{38} + 29q^{40} - 21q^{41} - 9q^{43} + 2q^{44} - 28q^{46} - 23q^{47} + 13q^{49} - 15q^{50} + 15q^{52} - 31q^{53} - 8q^{55} - 9q^{56} - 25q^{58} - 28q^{59} + 29q^{61} + 3q^{62} + 9q^{64} - 30q^{65} - 18q^{67} - 34q^{68} + 6q^{70} - 10q^{71} + 24q^{73} + 19q^{74} - 3q^{77} - 28q^{79} - 26q^{80} + 18q^{82} - 26q^{83} + 20q^{85} + 2q^{86} - 17q^{88} - 44q^{89} + 21q^{91} - 6q^{92} - 9q^{94} + 2q^{95} + 17q^{97} - 4q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13} - 4 x^{12} - 10 x^{11} + 53 x^{10} + 19 x^{9} - 242 x^{8} + 61 x^{7} + 467 x^{6} - 211 x^{5} - 372 x^{4} + 146 x^{3} + 116 x^{2} - 12 x - 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{12} - 2 \nu^{11} - 18 \nu^{10} + 29 \nu^{9} + 125 \nu^{8} - 144 \nu^{7} - 411 \nu^{6} + 285 \nu^{5} + 615 \nu^{4} - 190 \nu^{3} - 334 \nu^{2} - 12 \nu + 16 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{12} - 2 \nu^{11} - 15 \nu^{10} + 26 \nu^{9} + 84 \nu^{8} - 114 \nu^{7} - 224 \nu^{6} + 198 \nu^{5} + 287 \nu^{4} - 109 \nu^{3} - 150 \nu^{2} - 10 \nu + 10 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{12} + 2 \nu^{11} - 22 \nu^{10} - 27 \nu^{9} + 169 \nu^{8} + 120 \nu^{7} - 571 \nu^{6} - 203 \nu^{5} + 871 \nu^{4} + 118 \nu^{3} - 502 \nu^{2} - 28 \nu + 48 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{12} - 8 \nu^{11} - 40 \nu^{10} + 105 \nu^{9} + 187 \nu^{8} - 470 \nu^{7} - 387 \nu^{6} + 863 \nu^{5} + 363 \nu^{4} - 586 \nu^{3} - 154 \nu^{2} + 72 \nu + 8 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( 3 \nu^{12} - 8 \nu^{11} - 40 \nu^{10} + 105 \nu^{9} + 187 \nu^{8} - 470 \nu^{7} - 387 \nu^{6} + 863 \nu^{5} + 363 \nu^{4} - 590 \nu^{3} - 154 \nu^{2} + 92 \nu + 12 \)\()/4\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{12} + 4 \nu^{11} + 8 \nu^{10} - 51 \nu^{9} + 9 \nu^{8} + 220 \nu^{7} - 193 \nu^{6} - 387 \nu^{5} + 455 \nu^{4} + 246 \nu^{3} - 302 \nu^{2} - 44 \nu + 28 \)\()/2\)
\(\beta_{9}\)\(=\)\((\)\( -2 \nu^{12} + 5 \nu^{11} + 27 \nu^{10} - 65 \nu^{9} - 128 \nu^{8} + 285 \nu^{7} + 269 \nu^{6} - 498 \nu^{5} - 265 \nu^{4} + 298 \nu^{3} + 138 \nu^{2} - 30 \nu - 16 \)\()/2\)
\(\beta_{10}\)\(=\)\((\)\( 3 \nu^{12} - 8 \nu^{11} - 44 \nu^{10} + 109 \nu^{9} + 243 \nu^{8} - 510 \nu^{7} - 655 \nu^{6} + 975 \nu^{5} + 883 \nu^{4} - 670 \nu^{3} - 518 \nu^{2} + 60 \nu + 48 \)\()/4\)
\(\beta_{11}\)\(=\)\((\)\( 2 \nu^{12} - 7 \nu^{11} - 25 \nu^{10} + 93 \nu^{9} + 108 \nu^{8} - 419 \nu^{7} - 213 \nu^{6} + 758 \nu^{5} + 223 \nu^{4} - 478 \nu^{3} - 144 \nu^{2} + 42 \nu + 12 \)\()/2\)
\(\beta_{12}\)\(=\)\((\)\( 3 \nu^{12} - 7 \nu^{11} - 42 \nu^{10} + 91 \nu^{9} + 212 \nu^{8} - 399 \nu^{7} - 493 \nu^{6} + 696 \nu^{5} + 550 \nu^{4} - 405 \nu^{3} - 276 \nu^{2} + 12 \nu + 18 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{7} + \beta_{6} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{12} - \beta_{9} - \beta_{7} + \beta_{6} + \beta_{4} + 6 \beta_{2} + \beta_{1} + 15\)
\(\nu^{5}\)\(=\)\(-\beta_{12} + \beta_{11} - 2 \beta_{10} - \beta_{9} - 8 \beta_{7} + 9 \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} + 30 \beta_{1} + 10\)
\(\nu^{6}\)\(=\)\(-12 \beta_{12} + 2 \beta_{11} - 4 \beta_{10} - 11 \beta_{9} - \beta_{8} - 10 \beta_{7} + 11 \beta_{6} + 2 \beta_{5} + 10 \beta_{4} + 5 \beta_{3} + 35 \beta_{2} + 14 \beta_{1} + 89\)
\(\nu^{7}\)\(=\)\(-16 \beta_{12} + 14 \beta_{11} - 27 \beta_{10} - 15 \beta_{9} - 2 \beta_{8} - 58 \beta_{7} + 68 \beta_{6} + 14 \beta_{5} - 8 \beta_{4} + 29 \beta_{3} + 193 \beta_{1} + 86\)
\(\nu^{8}\)\(=\)\(-109 \beta_{12} + 31 \beta_{11} - 59 \beta_{10} - 95 \beta_{9} - 14 \beta_{8} - 84 \beta_{7} + 98 \beta_{6} + 31 \beta_{5} + 77 \beta_{4} + 72 \beta_{3} + 207 \beta_{2} + 141 \beta_{1} + 571\)
\(\nu^{9}\)\(=\)\(-174 \beta_{12} + 139 \beta_{11} - 262 \beta_{10} - 156 \beta_{9} - 31 \beta_{8} - 414 \beta_{7} + 495 \beta_{6} + 140 \beta_{5} - 44 \beta_{4} + 293 \beta_{3} + 4 \beta_{2} + 1284 \beta_{1} + 694\)
\(\nu^{10}\)\(=\)\(-894 \beta_{12} + 327 \beta_{11} - 607 \beta_{10} - 757 \beta_{9} - 140 \beta_{8} - 673 \beta_{7} + 812 \beta_{6} + 328 \beta_{5} + 546 \beta_{4} + 732 \beta_{3} + 1246 \beta_{2} + 1252 \beta_{1} + 3811\)
\(\nu^{11}\)\(=\)\(-1613 \beta_{12} + 1210 \beta_{11} - 2248 \beta_{10} - 1404 \beta_{9} - 328 \beta_{8} - 2952 \beta_{7} + 3573 \beta_{6} + 1226 \beta_{5} - 175 \beta_{4} + 2571 \beta_{3} + 83 \beta_{2} + 8714 \beta_{1} + 5431\)
\(\nu^{12}\)\(=\)\(-6983 \beta_{12} + 2953 \beta_{11} - 5411 \beta_{10} - 5816 \beta_{9} - 1226 \beta_{8} - 5269 \beta_{7} + 6480 \beta_{6} + 2974 \beta_{5} + 3757 \beta_{4} + 6486 \beta_{3} + 7632 \beta_{2} + 10446 \beta_{1} + 26023\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.70878
2.29288
2.28328
1.45503
1.27342
1.18273
0.307326
−0.305711
−0.423652
−0.910949
−1.58856
−1.82297
−2.45160
−2.70878 0 5.33747 −2.87857 0 1.00000 −9.04047 0 7.79741
1.2 −2.29288 0 3.25728 0.132682 0 1.00000 −2.88278 0 −0.304223
1.3 −2.28328 0 3.21337 −2.58645 0 1.00000 −2.77047 0 5.90560
1.4 −1.45503 0 0.117115 0.420019 0 1.00000 2.73966 0 −0.611141
1.5 −1.27342 0 −0.378389 2.08575 0 1.00000 3.02870 0 −2.65604
1.6 −1.18273 0 −0.601156 −4.39766 0 1.00000 3.07646 0 5.20123
1.7 −0.307326 0 −1.90555 0.988649 0 1.00000 1.20028 0 −0.303838
1.8 0.305711 0 −1.90654 0.276458 0 1.00000 −1.19427 0 0.0845162
1.9 0.423652 0 −1.82052 −2.66196 0 1.00000 −1.61857 0 −1.12774
1.10 0.910949 0 −1.17017 2.06751 0 1.00000 −2.88787 0 1.88339
1.11 1.58856 0 0.523519 −3.57671 0 1.00000 −2.34548 0 −5.68181
1.12 1.82297 0 1.32323 −0.630622 0 1.00000 −1.23374 0 −1.14961
1.13 2.45160 0 4.01034 −1.23909 0 1.00000 4.92856 0 −3.03775
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(127\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8001.2.a.o 13
3.b odd 2 1 2667.2.a.l 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.2.a.l 13 3.b odd 2 1
8001.2.a.o 13 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\(T_{2}^{13} + \cdots\)
\(T_{5}^{13} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 8 - 12 T - 116 T^{2} + 146 T^{3} + 372 T^{4} - 211 T^{5} - 467 T^{6} + 61 T^{7} + 242 T^{8} + 19 T^{9} - 53 T^{10} - 10 T^{11} + 4 T^{12} + T^{13} \)
$3$ \( T^{13} \)
$5$ \( 16 - 184 T + 412 T^{2} + 784 T^{3} - 1990 T^{4} - 1590 T^{5} + 1877 T^{6} + 1492 T^{7} - 340 T^{8} - 457 T^{9} - 50 T^{10} + 39 T^{11} + 12 T^{12} + T^{13} \)
$7$ \( ( -1 + T )^{13} \)
$11$ \( -284672 + 416256 T + 67968 T^{2} - 316480 T^{3} + 42736 T^{4} + 95504 T^{5} - 19182 T^{6} - 14937 T^{7} + 2786 T^{8} + 1309 T^{9} - 165 T^{10} - 60 T^{11} + 3 T^{12} + T^{13} \)
$13$ \( -22336 + 9856 T + 209968 T^{2} - 325744 T^{3} + 24820 T^{4} + 220232 T^{5} - 137647 T^{6} + 12055 T^{7} + 13359 T^{8} - 4095 T^{9} + 70 T^{10} + 134 T^{11} - 21 T^{12} + T^{13} \)
$17$ \( -27656 - 250028 T - 673830 T^{2} - 736685 T^{3} - 207396 T^{4} + 235029 T^{5} + 205190 T^{6} + 40886 T^{7} - 13339 T^{8} - 7140 T^{9} - 930 T^{10} + 37 T^{11} + 17 T^{12} + T^{13} \)
$19$ \( -16768 + 79664 T + 9416 T^{2} - 140220 T^{3} - 15696 T^{4} + 84330 T^{5} + 15862 T^{6} - 19207 T^{7} - 3852 T^{8} + 1810 T^{9} + 276 T^{10} - 74 T^{11} - 5 T^{12} + T^{13} \)
$23$ \( -975616 + 3957760 T - 3346576 T^{2} - 1838416 T^{3} + 1965244 T^{4} + 558624 T^{5} - 307907 T^{6} - 74663 T^{7} + 19181 T^{8} + 4454 T^{9} - 491 T^{10} - 115 T^{11} + 4 T^{12} + T^{13} \)
$29$ \( -92930644 + 43800648 T + 87025943 T^{2} - 6872293 T^{3} - 27149845 T^{4} - 5957137 T^{5} + 1323463 T^{6} + 587571 T^{7} + 23498 T^{8} - 14998 T^{9} - 1984 T^{10} + 44 T^{11} + 21 T^{12} + T^{13} \)
$31$ \( 15003392 + 27575680 T - 8831520 T^{2} - 23377632 T^{3} + 5782108 T^{4} + 4685826 T^{5} - 1062279 T^{6} - 338267 T^{7} + 59182 T^{8} + 11829 T^{9} - 1167 T^{10} - 186 T^{11} + 7 T^{12} + T^{13} \)
$37$ \( 1844308 - 2520564 T - 2797989 T^{2} + 6202263 T^{3} - 2760296 T^{4} - 457511 T^{5} + 543356 T^{6} - 44017 T^{7} - 33647 T^{8} + 4936 T^{9} + 803 T^{10} - 132 T^{11} - 7 T^{12} + T^{13} \)
$41$ \( -184216 + 506788 T + 872770 T^{2} - 324213 T^{3} - 818469 T^{4} - 131168 T^{5} + 203826 T^{6} + 75412 T^{7} - 9630 T^{8} - 8063 T^{9} - 930 T^{10} + 83 T^{11} + 21 T^{12} + T^{13} \)
$43$ \( -28880896 + 79970304 T - 53081280 T^{2} - 18626704 T^{3} + 20945376 T^{4} + 1711048 T^{5} - 2519908 T^{6} - 194909 T^{7} + 116274 T^{8} + 11477 T^{9} - 1815 T^{10} - 196 T^{11} + 9 T^{12} + T^{13} \)
$47$ \( 9569152 + 45208960 T + 74002368 T^{2} + 43253040 T^{3} - 4227512 T^{4} - 11766384 T^{5} - 1935456 T^{6} + 688079 T^{7} + 137823 T^{8} - 11700 T^{9} - 3117 T^{10} - 4 T^{11} + 23 T^{12} + T^{13} \)
$53$ \( -405812 - 5163096 T - 22048207 T^{2} - 31846170 T^{3} + 2248510 T^{4} + 15128899 T^{5} + 6199162 T^{6} + 649825 T^{7} - 126259 T^{8} - 34534 T^{9} - 1645 T^{10} + 247 T^{11} + 31 T^{12} + T^{13} \)
$59$ \( -43041792704 + 5684383104 T + 15367230460 T^{2} + 2104708704 T^{3} - 993805642 T^{4} - 247246098 T^{5} + 9346767 T^{6} + 7360873 T^{7} + 457733 T^{8} - 59180 T^{9} - 7471 T^{10} - 19 T^{11} + 28 T^{12} + T^{13} \)
$61$ \( -20477872 + 57784256 T - 35272060 T^{2} - 15765288 T^{3} + 15146442 T^{4} - 328114 T^{5} - 1797497 T^{6} + 321179 T^{7} + 52428 T^{8} - 18647 T^{9} + 789 T^{10} + 230 T^{11} - 29 T^{12} + T^{13} \)
$67$ \( 49278934912 + 11659593856 T - 15678737768 T^{2} - 903605236 T^{3} + 1378938004 T^{4} + 34642310 T^{5} - 52670484 T^{6} - 1399265 T^{7} + 942709 T^{8} + 37150 T^{9} - 7164 T^{10} - 356 T^{11} + 18 T^{12} + T^{13} \)
$71$ \( 15605244416 - 6886633664 T - 5211217608 T^{2} + 1070295788 T^{3} + 720069700 T^{4} + 1286914 T^{5} - 30201104 T^{6} - 1753207 T^{7} + 527775 T^{8} + 42920 T^{9} - 3947 T^{10} - 367 T^{11} + 10 T^{12} + T^{13} \)
$73$ \( 169328048 - 709932416 T + 1180513532 T^{2} - 990795136 T^{3} + 432767258 T^{4} - 82835182 T^{5} - 2330721 T^{6} + 3354895 T^{7} - 343315 T^{8} - 26544 T^{9} + 5637 T^{10} - 79 T^{11} - 24 T^{12} + T^{13} \)
$79$ \( -18990976 + 20270576 T + 143966140 T^{2} - 113818875 T^{3} - 112006527 T^{4} + 1247310 T^{5} + 14248384 T^{6} + 2596057 T^{7} - 164103 T^{8} - 86339 T^{9} - 7530 T^{10} - 2 T^{11} + 28 T^{12} + T^{13} \)
$83$ \( -28884464384 - 15434947392 T + 6700193008 T^{2} + 4113741344 T^{3} - 74445240 T^{4} - 269371256 T^{5} - 21633149 T^{6} + 6100399 T^{7} + 837800 T^{8} - 33468 T^{9} - 9026 T^{10} - 167 T^{11} + 26 T^{12} + T^{13} \)
$89$ \( -180656384 + 703704320 T - 334174176 T^{2} - 623502000 T^{3} - 112738444 T^{4} + 49885698 T^{5} + 17299201 T^{6} + 382063 T^{7} - 461845 T^{8} - 57109 T^{9} + 1033 T^{10} + 632 T^{11} + 44 T^{12} + T^{13} \)
$97$ \( -416205928 + 3821954752 T - 5890557708 T^{2} + 3036212017 T^{3} - 395441756 T^{4} - 110532829 T^{5} + 29387674 T^{6} + 640402 T^{7} - 613355 T^{8} + 17944 T^{9} + 5358 T^{10} - 263 T^{11} - 17 T^{12} + T^{13} \)
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