Properties

Label 3525.2.a.bi
Level $3525$
Weight $2$
Character orbit 3525.a
Self dual yes
Analytic conductor $28.147$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.1472667125\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - 3 x^{12} - 17 x^{11} + 51 x^{10} + 106 x^{9} - 316 x^{8} - 288 x^{7} + 852 x^{6} + 309 x^{5} - 923 x^{4} - 107 x^{3} + 293 x^{2} + 12 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 705)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + q^{3} + ( 1 + \beta_{2} ) q^{4} + \beta_{1} q^{6} + \beta_{10} q^{7} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{8} + q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + q^{3} + ( 1 + \beta_{2} ) q^{4} + \beta_{1} q^{6} + \beta_{10} q^{7} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{8} + q^{9} + ( 1 - \beta_{12} ) q^{11} + ( 1 + \beta_{2} ) q^{12} + ( -1 + \beta_{5} ) q^{13} + ( -\beta_{1} + \beta_{4} - \beta_{7} + \beta_{12} ) q^{14} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{16} + ( 1 + \beta_{7} ) q^{17} + \beta_{1} q^{18} + ( 2 - \beta_{3} - \beta_{5} - \beta_{11} ) q^{19} + \beta_{10} q^{21} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} - 2 \beta_{10} + 2 \beta_{11} ) q^{22} + ( \beta_{1} - \beta_{4} - \beta_{9} ) q^{23} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{24} + ( -\beta_{1} - \beta_{4} + 2 \beta_{6} - \beta_{10} ) q^{26} + q^{27} + ( -1 + \beta_{2} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{28} + ( 1 - \beta_{3} - \beta_{6} + \beta_{10} ) q^{29} + ( 2 + \beta_{1} - \beta_{4} + \beta_{11} ) q^{31} + ( 4 + 2 \beta_{2} + \beta_{4} + \beta_{9} + \beta_{10} ) q^{32} + ( 1 - \beta_{12} ) q^{33} + ( \beta_{1} + \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{10} ) q^{34} + ( 1 + \beta_{2} ) q^{36} + ( \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{6} - \beta_{9} + \beta_{10} ) q^{37} + ( 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{10} - \beta_{11} + \beta_{12} ) q^{38} + ( -1 + \beta_{5} ) q^{39} + ( 2 - \beta_{2} - \beta_{3} + \beta_{8} + \beta_{9} ) q^{41} + ( -\beta_{1} + \beta_{4} - \beta_{7} + \beta_{12} ) q^{42} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{43} + ( 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{11} - 2 \beta_{12} ) q^{44} + ( \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{46} + q^{47} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{48} + ( 1 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{12} ) q^{49} + ( 1 + \beta_{7} ) q^{51} + ( -3 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{12} ) q^{52} + ( 1 + \beta_{2} + \beta_{3} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{53} + \beta_{1} q^{54} + ( -2 \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{7} - \beta_{9} + \beta_{10} - 2 \beta_{11} + 2 \beta_{12} ) q^{56} + ( 2 - \beta_{3} - \beta_{5} - \beta_{11} ) q^{57} + ( 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{8} + \beta_{12} ) q^{58} + ( 3 - 2 \beta_{1} + \beta_{2} + \beta_{5} + \beta_{9} + \beta_{10} ) q^{59} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{8} + \beta_{9} - \beta_{12} ) q^{61} + ( 1 + 3 \beta_{1} - \beta_{4} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} ) q^{62} + \beta_{10} q^{63} + ( 1 + 4 \beta_{1} - \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} ) q^{64} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} - 2 \beta_{10} + 2 \beta_{11} ) q^{66} + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} + 2 \beta_{9} - \beta_{12} ) q^{67} + ( 3 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{12} ) q^{68} + ( \beta_{1} - \beta_{4} - \beta_{9} ) q^{69} + ( 2 - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{6} + \beta_{7} - \beta_{9} + \beta_{10} + \beta_{12} ) q^{71} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{72} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{12} ) q^{73} + ( 3 - 2 \beta_{1} - \beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} ) q^{74} + ( 6 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + 3 \beta_{10} - \beta_{11} + 2 \beta_{12} ) q^{76} + ( 1 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} + 2 \beta_{10} + 2 \beta_{12} ) q^{77} + ( -\beta_{1} - \beta_{4} + 2 \beta_{6} - \beta_{10} ) q^{78} + ( 1 + \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{79} + q^{81} + ( 2 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{82} + ( 1 - \beta_{2} - \beta_{5} + 2 \beta_{6} - \beta_{7} - 2 \beta_{10} + \beta_{12} ) q^{83} + ( -1 + \beta_{2} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{84} + ( 4 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{86} + ( 1 - \beta_{3} - \beta_{6} + \beta_{10} ) q^{87} + ( -4 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{6} + 2 \beta_{7} + \beta_{8} - 3 \beta_{9} - 5 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} ) q^{88} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{89} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - 3 \beta_{10} ) q^{91} + ( -1 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{10} - \beta_{12} ) q^{92} + ( 2 + \beta_{1} - \beta_{4} + \beta_{11} ) q^{93} + \beta_{1} q^{94} + ( 4 + 2 \beta_{2} + \beta_{4} + \beta_{9} + \beta_{10} ) q^{96} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} ) q^{97} + ( -4 + 4 \beta_{1} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} ) q^{98} + ( 1 - \beta_{12} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 3q^{2} + 13q^{3} + 17q^{4} + 3q^{6} - 4q^{7} + 15q^{8} + 13q^{9} + O(q^{10}) \) \( 13q + 3q^{2} + 13q^{3} + 17q^{4} + 3q^{6} - 4q^{7} + 15q^{8} + 13q^{9} + 16q^{11} + 17q^{12} - 8q^{13} - 4q^{14} + 29q^{16} + 12q^{17} + 3q^{18} + 28q^{19} - 4q^{21} + 6q^{23} + 15q^{24} + 4q^{26} + 13q^{27} - 20q^{28} + 12q^{29} + 26q^{31} + 53q^{32} + 16q^{33} + 8q^{34} + 17q^{36} - 4q^{37} + 2q^{38} - 8q^{39} + 24q^{41} - 4q^{42} - 6q^{43} + 4q^{44} + 16q^{46} + 13q^{47} + 29q^{48} + 21q^{49} + 12q^{51} - 32q^{52} + 6q^{53} + 3q^{54} + 28q^{57} - 4q^{58} + 34q^{59} + 24q^{61} + 30q^{62} - 4q^{63} + 13q^{64} - 24q^{67} + 44q^{68} + 6q^{69} + 20q^{71} + 15q^{72} - 6q^{73} + 20q^{74} + 66q^{76} - 2q^{77} + 4q^{78} + 6q^{79} + 13q^{81} + 20q^{82} + 14q^{83} - 20q^{84} + 48q^{86} + 12q^{87} - 22q^{88} + 36q^{89} + 4q^{91} + 4q^{92} + 26q^{93} + 3q^{94} + 53q^{96} - 32q^{97} - 39q^{98} + 16q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13} - 3 x^{12} - 17 x^{11} + 51 x^{10} + 106 x^{9} - 316 x^{8} - 288 x^{7} + 852 x^{6} + 309 x^{5} - 923 x^{4} - 107 x^{3} + 293 x^{2} + 12 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 5 \nu + 2 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - \nu^{3} - 6 \nu^{2} + 4 \nu + 3 \)
\(\beta_{5}\)\(=\)\((\)\( \nu^{9} - 2 \nu^{8} - 13 \nu^{7} + 22 \nu^{6} + 55 \nu^{5} - 72 \nu^{4} - 79 \nu^{3} + 68 \nu^{2} + 32 \nu - 6 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{12} - 2 \nu^{11} - 15 \nu^{10} + 28 \nu^{9} + 78 \nu^{8} - 142 \nu^{7} - 158 \nu^{6} + 326 \nu^{5} + 99 \nu^{4} - 336 \nu^{3} - 7 \nu^{2} + 134 \nu + 10 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{11} + \nu^{10} + 17 \nu^{9} - 12 \nu^{8} - 105 \nu^{7} + 46 \nu^{6} + 281 \nu^{5} - 62 \nu^{4} - 308 \nu^{3} + 13 \nu^{2} + 96 \nu + 4 \)\()/2\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{12} + 2 \nu^{11} + 19 \nu^{10} - 36 \nu^{9} - 134 \nu^{8} + 238 \nu^{7} + 430 \nu^{6} - 694 \nu^{5} - 627 \nu^{4} + 824 \nu^{3} + 379 \nu^{2} - 294 \nu - 50 \)\()/8\)
\(\beta_{9}\)\(=\)\((\)\( -\nu^{12} + 2 \nu^{11} + 17 \nu^{10} - 32 \nu^{9} - 104 \nu^{8} + 186 \nu^{7} + 268 \nu^{6} - 466 \nu^{5} - 257 \nu^{4} + 440 \nu^{3} + 63 \nu^{2} - 98 \nu - 2 \)\()/4\)
\(\beta_{10}\)\(=\)\((\)\( \nu^{12} - 2 \nu^{11} - 17 \nu^{10} + 32 \nu^{9} + 104 \nu^{8} - 186 \nu^{7} - 268 \nu^{6} + 470 \nu^{5} + 253 \nu^{4} - 468 \nu^{3} - 47 \nu^{2} + 130 \nu - 2 \)\()/4\)
\(\beta_{11}\)\(=\)\((\)\( \nu^{12} - 4 \nu^{11} - 17 \nu^{10} + 68 \nu^{9} + 110 \nu^{8} - 418 \nu^{7} - 326 \nu^{6} + 1106 \nu^{5} + 403 \nu^{4} - 1182 \nu^{3} - 169 \nu^{2} + 390 \nu + 6 \)\()/8\)
\(\beta_{12}\)\(=\)\((\)\( \nu^{12} - 2 \nu^{11} - 17 \nu^{10} + 32 \nu^{9} + 106 \nu^{8} - 190 \nu^{7} - 290 \nu^{6} + 506 \nu^{5} + 327 \nu^{4} - 552 \nu^{3} - 113 \nu^{2} + 166 \nu - 2 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{4} + \beta_{3} + 7 \beta_{2} + \beta_{1} + 16\)
\(\nu^{5}\)\(=\)\(\beta_{10} + \beta_{9} + \beta_{4} + 8 \beta_{3} + 10 \beta_{2} + 28 \beta_{1} + 12\)
\(\nu^{6}\)\(=\)\(\beta_{12} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{6} + 10 \beta_{4} + 10 \beta_{3} + 46 \beta_{2} + 14 \beta_{1} + 97\)
\(\nu^{7}\)\(=\)\(\beta_{12} - 2 \beta_{11} + 13 \beta_{10} + 11 \beta_{9} + \beta_{8} + \beta_{7} - 3 \beta_{6} - \beta_{5} + 14 \beta_{4} + 53 \beta_{3} + 80 \beta_{2} + 167 \beta_{1} + 111\)
\(\nu^{8}\)\(=\)\(15 \beta_{12} - 4 \beta_{11} + 17 \beta_{10} + 15 \beta_{9} + 13 \beta_{8} + 2 \beta_{7} - 17 \beta_{6} - 2 \beta_{5} + 83 \beta_{4} + 77 \beta_{3} + 302 \beta_{2} + 139 \beta_{1} + 622\)
\(\nu^{9}\)\(=\)\(21 \beta_{12} - 34 \beta_{11} + 126 \beta_{10} + 96 \beta_{9} + 17 \beta_{8} + 17 \beta_{7} - 51 \beta_{6} - 15 \beta_{5} + 145 \beta_{4} + 334 \beta_{3} + 597 \beta_{2} + 1036 \beta_{1} + 926\)
\(\nu^{10}\)\(=\)\(160 \beta_{12} - 76 \beta_{11} + 202 \beta_{10} + 162 \beta_{9} + 126 \beta_{8} + 38 \beta_{7} - 198 \beta_{6} - 34 \beta_{5} + 660 \beta_{4} + 540 \beta_{3} + 2003 \beta_{2} + 1196 \beta_{1} + 4125\)
\(\nu^{11}\)\(=\)\(278 \beta_{12} - 396 \beta_{11} + 1102 \beta_{10} + 786 \beta_{9} + 200 \beta_{8} + 196 \beta_{7} - 592 \beta_{6} - 160 \beta_{5} + 1338 \beta_{4} + 2067 \beta_{3} + 4325 \beta_{2} + 6611 \beta_{1} + 7325\)
\(\nu^{12}\)\(=\)\(1498 \beta_{12} - 952 \beta_{11} + 2058 \beta_{10} + 1538 \beta_{9} + 1100 \beta_{8} + 472 \beta_{7} - 1976 \beta_{6} - 396 \beta_{5} + 5185 \beta_{4} + 3611 \beta_{3} + 13441 \beta_{2} + 9557 \beta_{1} + 28020\)

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
−2.32477
−2.24873
−2.12231
−1.22883
−0.697808
−0.105426
0.0655745
0.747203
1.32490
1.67335
2.57687
2.59761
2.74237
−2.32477 1.00000 3.40453 0 −2.32477 −4.29166 −3.26521 1.00000 0
1.2 −2.24873 1.00000 3.05678 0 −2.24873 −1.38491 −2.37640 1.00000 0
1.3 −2.12231 1.00000 2.50418 0 −2.12231 4.63433 −1.07002 1.00000 0
1.4 −1.22883 1.00000 −0.489976 0 −1.22883 −1.54483 3.05976 1.00000 0
1.5 −0.697808 1.00000 −1.51306 0 −0.697808 3.46613 2.45144 1.00000 0
1.6 −0.105426 1.00000 −1.98889 0 −0.105426 −3.91364 0.420531 1.00000 0
1.7 0.0655745 1.00000 −1.99570 0 0.0655745 1.54896 −0.262016 1.00000 0
1.8 0.747203 1.00000 −1.44169 0 0.747203 0.654660 −2.57164 1.00000 0
1.9 1.32490 1.00000 −0.244653 0 1.32490 1.72082 −2.97393 1.00000 0
1.10 1.67335 1.00000 0.800100 0 1.67335 −2.75180 −2.00785 1.00000 0
1.11 2.57687 1.00000 4.64024 0 2.57687 1.51741 6.80354 1.00000 0
1.12 2.59761 1.00000 4.74755 0 2.59761 −4.83808 7.13706 1.00000 0
1.13 2.74237 1.00000 5.52059 0 2.74237 1.18262 9.65475 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(47\) \(-1\)

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 3525.2.a.bi 13
5.b even 2 1 3525.2.a.bh 13
5.c odd 4 2 705.2.c.c 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
705.2.c.c 26 5.c odd 4 2
3525.2.a.bh 13 5.b even 2 1
3525.2.a.bi 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3525))\):

\(T_{2}^{13} - \cdots\)
\(T_{7}^{13} + \cdots\)
\(T_{11}^{13} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + 12 T + 293 T^{2} - 107 T^{3} - 923 T^{4} + 309 T^{5} + 852 T^{6} - 288 T^{7} - 316 T^{8} + 106 T^{9} + 51 T^{10} - 17 T^{11} - 3 T^{12} + T^{13} \)
$3$ \( ( -1 + T )^{13} \)
$5$ \( T^{13} \)
$7$ \( -24064 + 56192 T + 1792 T^{2} - 72464 T^{3} + 22104 T^{4} + 33677 T^{5} - 12580 T^{6} - 7272 T^{7} + 2388 T^{8} + 842 T^{9} - 172 T^{10} - 48 T^{11} + 4 T^{12} + T^{13} \)
$11$ \( -2451456 - 6953472 T - 505856 T^{2} + 4668608 T^{3} - 98752 T^{4} - 1014460 T^{5} + 104072 T^{6} + 94552 T^{7} - 15976 T^{8} - 3548 T^{9} + 892 T^{10} + 17 T^{11} - 16 T^{12} + T^{13} \)
$13$ \( -892224 - 2715936 T - 954080 T^{2} + 1698016 T^{3} + 840412 T^{4} - 295298 T^{5} - 190888 T^{6} + 12789 T^{7} + 17462 T^{8} + 771 T^{9} - 662 T^{10} - 65 T^{11} + 8 T^{12} + T^{13} \)
$17$ \( -60736 - 121504 T + 895200 T^{2} - 130688 T^{3} - 1161588 T^{4} - 164530 T^{5} + 260040 T^{6} + 21977 T^{7} - 23986 T^{8} - 205 T^{9} + 954 T^{10} - 49 T^{11} - 12 T^{12} + T^{13} \)
$19$ \( -899200 - 2726880 T - 616000 T^{2} + 3390552 T^{3} + 1681560 T^{4} - 742456 T^{5} - 338742 T^{6} + 99255 T^{7} + 22906 T^{8} - 7923 T^{9} - 144 T^{10} + 245 T^{11} - 28 T^{12} + T^{13} \)
$23$ \( -60416 + 1701760 T - 69056 T^{2} - 2467504 T^{3} - 154772 T^{4} + 962277 T^{5} + 125830 T^{6} - 119328 T^{7} - 15422 T^{8} + 5618 T^{9} + 546 T^{10} - 120 T^{11} - 6 T^{12} + T^{13} \)
$29$ \( 17120 + 42920 T - 14008 T^{2} - 109156 T^{3} - 63390 T^{4} + 41837 T^{5} + 38906 T^{6} - 3988 T^{7} - 7382 T^{8} + 122 T^{9} + 542 T^{10} - 24 T^{11} - 12 T^{12} + T^{13} \)
$31$ \( 17920 + 121792 T + 57152 T^{2} - 537984 T^{3} + 85872 T^{4} + 429192 T^{5} - 245120 T^{6} + 852 T^{7} + 29504 T^{8} - 6752 T^{9} - 116 T^{10} + 212 T^{11} - 26 T^{12} + T^{13} \)
$37$ \( 389395456 - 990185728 T - 853411968 T^{2} - 7905600 T^{3} + 108466992 T^{4} + 12371284 T^{5} - 5264848 T^{6} - 814904 T^{7} + 117876 T^{8} + 21416 T^{9} - 1184 T^{10} - 247 T^{11} + 4 T^{12} + T^{13} \)
$41$ \( 58496 - 1073632 T + 2372256 T^{2} - 675784 T^{3} - 2924832 T^{4} + 3805552 T^{5} - 1983698 T^{6} + 438337 T^{7} + 1264 T^{8} - 16709 T^{9} + 1986 T^{10} + 79 T^{11} - 24 T^{12} + T^{13} \)
$43$ \( 308908544 + 208028928 T - 198561792 T^{2} - 77406336 T^{3} + 36547072 T^{4} + 10249248 T^{5} - 2571040 T^{6} - 596176 T^{7} + 81008 T^{8} + 16416 T^{9} - 1152 T^{10} - 210 T^{11} + 6 T^{12} + T^{13} \)
$47$ \( ( -1 + T )^{13} \)
$53$ \( 25854912 - 65768736 T + 35673568 T^{2} + 19537120 T^{3} - 18044580 T^{4} + 1196574 T^{5} + 1568552 T^{6} - 227755 T^{7} - 55868 T^{8} + 9733 T^{9} + 922 T^{10} - 167 T^{11} - 6 T^{12} + T^{13} \)
$59$ \( -178880 + 1839840 T - 2652128 T^{2} - 1780928 T^{3} + 3530484 T^{4} + 111654 T^{5} - 934736 T^{6} + 111057 T^{7} + 73712 T^{8} - 16669 T^{9} - 282 T^{10} + 359 T^{11} - 34 T^{12} + T^{13} \)
$61$ \( 27648112896 + 287734272 T - 12942005728 T^{2} + 1856748176 T^{3} + 1544438816 T^{4} - 479031528 T^{5} + 17277482 T^{6} + 9035681 T^{7} - 1001276 T^{8} - 33321 T^{9} + 9098 T^{10} - 193 T^{11} - 24 T^{12} + T^{13} \)
$67$ \( -2592738688 + 1533792704 T + 10175277696 T^{2} + 5629868416 T^{3} + 485480512 T^{4} - 263977096 T^{5} - 45824024 T^{6} + 3697388 T^{7} + 1015112 T^{8} - 4212 T^{9} - 8652 T^{10} - 232 T^{11} + 24 T^{12} + T^{13} \)
$71$ \( -9039415936 + 17507763008 T - 12575574688 T^{2} + 3718046672 T^{3} - 98906840 T^{4} - 177853748 T^{5} + 27120946 T^{6} + 2167695 T^{7} - 659258 T^{8} + 4137 T^{9} + 6108 T^{10} - 219 T^{11} - 20 T^{12} + T^{13} \)
$73$ \( -594804992 + 1141024640 T + 122391168 T^{2} - 805047424 T^{3} + 55790448 T^{4} + 102344184 T^{5} - 7850224 T^{6} - 4364972 T^{7} + 240232 T^{8} + 69584 T^{9} - 2256 T^{10} - 456 T^{11} + 6 T^{12} + T^{13} \)
$79$ \( 953661440 - 71421440 T - 715824896 T^{2} - 82899008 T^{3} + 157579776 T^{4} + 42185316 T^{5} - 5191360 T^{6} - 2290088 T^{7} + 3736 T^{8} + 42488 T^{9} + 960 T^{10} - 335 T^{11} - 6 T^{12} + T^{13} \)
$83$ \( 35253854208 - 39110252544 T + 11405691904 T^{2} + 1446557184 T^{3} - 1075276672 T^{4} + 40694432 T^{5} + 36190272 T^{6} - 2842688 T^{7} - 589856 T^{8} + 51736 T^{9} + 4696 T^{10} - 378 T^{11} - 14 T^{12} + T^{13} \)
$89$ \( -461035624960 + 4082174720 T + 182296593024 T^{2} + 27432461376 T^{3} - 10146710240 T^{4} - 1063681584 T^{5} + 240465304 T^{6} + 13592796 T^{7} - 2832024 T^{8} - 52568 T^{9} + 16228 T^{10} - 142 T^{11} - 36 T^{12} + T^{13} \)
$97$ \( -27544367104 - 25975038976 T + 16186004992 T^{2} + 7723190016 T^{3} - 457742144 T^{4} - 439306800 T^{5} - 16610432 T^{6} + 9210320 T^{7} + 776120 T^{8} - 68240 T^{9} - 9252 T^{10} + T^{11} + 32 T^{12} + T^{13} \)
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