Properties

Label 403.2.a.c
Level 403
Weight 2
Character orbit 403.a
Self dual yes
Analytic conductor 3.218
Analytic rank 0
Dimension 7
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 403.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.21797120146\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( 1 - \beta_{1} ) q^{3} + ( \beta_{1} + \beta_{4} + \beta_{6} ) q^{4} + ( 2 - \beta_{4} ) q^{5} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{6} + ( 1 - \beta_{6} ) q^{7} + ( \beta_{4} + \beta_{5} + \beta_{6} ) q^{8} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{3} q^{2} + ( 1 - \beta_{1} ) q^{3} + ( \beta_{1} + \beta_{4} + \beta_{6} ) q^{4} + ( 2 - \beta_{4} ) q^{5} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{6} + ( 1 - \beta_{6} ) q^{7} + ( \beta_{4} + \beta_{5} + \beta_{6} ) q^{8} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{9} + ( -\beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{10} + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{11} + ( -1 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{12} - q^{13} + ( -\beta_{1} - \beta_{6} ) q^{14} + ( 2 - 3 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{15} + ( 1 - \beta_{1} + \beta_{4} + 2 \beta_{5} ) q^{16} + ( 2 \beta_{1} + \beta_{4} ) q^{17} + ( -1 - \beta_{1} - \beta_{5} - \beta_{6} ) q^{18} + ( 1 - \beta_{1} - \beta_{2} - \beta_{6} ) q^{19} + ( -2 + 3 \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{20} + ( -1 + \beta_{3} - \beta_{5} ) q^{21} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{22} + ( 3 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{6} ) q^{23} + ( 1 - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{24} + ( 2 + \beta_{3} - 3 \beta_{4} + \beta_{6} ) q^{25} -\beta_{3} q^{26} + ( 3 - \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{6} ) q^{27} + ( -2 + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{6} ) q^{28} + ( -2 + 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{29} + ( -1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{30} + q^{31} + ( 2 + \beta_{1} - 2 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{32} + ( \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{33} + ( -2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{34} + ( 1 - \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{35} + ( -3 + 2 \beta_{1} - \beta_{5} - 2 \beta_{6} ) q^{36} + ( 4 + \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{37} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{38} + ( -1 + \beta_{1} ) q^{39} + ( -\beta_{1} - 2 \beta_{4} - \beta_{5} ) q^{40} + ( 2 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{41} + ( 1 + \beta_{1} + \beta_{2} - \beta_{5} ) q^{42} + ( -2 + \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{6} ) q^{43} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{44} + ( 4 - 4 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{45} + ( -4 + 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{46} + ( -4 + 2 \beta_{1} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{47} + ( 2 + \beta_{2} - 2 \beta_{3} + \beta_{4} + 3 \beta_{5} + 2 \beta_{6} ) q^{48} + ( -1 - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{6} ) q^{49} + ( 2 + 2 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 5 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} ) q^{50} + ( -6 + \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} ) q^{51} + ( -\beta_{1} - \beta_{4} - \beta_{6} ) q^{52} + ( 5 - \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{53} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{54} + ( 1 + 2 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{55} + ( -3 - \beta_{3} + \beta_{6} ) q^{56} + ( 2 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{57} + ( 2 - 4 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{58} + ( -1 + \beta_{1} + 3 \beta_{2} - 5 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} ) q^{59} + ( -7 + \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} ) q^{60} + ( 1 - 2 \beta_{2} + \beta_{4} + \beta_{5} ) q^{61} + \beta_{3} q^{62} + ( -3 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{63} + ( -5 - \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{6} ) q^{64} + ( -2 + \beta_{4} ) q^{65} + ( -3 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{66} + ( -4 + \beta_{2} - \beta_{3} - \beta_{6} ) q^{67} + ( 4 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} ) q^{68} + ( 4 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{69} + ( -1 - 3 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{70} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} ) q^{71} + ( 1 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{6} ) q^{72} + ( -2 + \beta_{1} + 4 \beta_{2} + \beta_{4} + 2 \beta_{6} ) q^{73} + ( -1 - \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{74} + ( 4 - 5 \beta_{1} + \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{75} + ( -1 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{6} ) q^{76} + ( 1 - 2 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{77} + ( -\beta_{1} - \beta_{2} - \beta_{4} ) q^{78} + ( -2 + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{79} + ( 3 - 5 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{80} + ( 3 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{6} ) q^{81} + ( -5 + \beta_{1} + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} + 4 \beta_{5} ) q^{82} + ( 1 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{83} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{6} ) q^{84} + ( -3 + 6 \beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{85} + ( -4 + \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{86} + ( -10 + 3 \beta_{1} - \beta_{2} + 7 \beta_{3} + \beta_{4} - 4 \beta_{5} - 2 \beta_{6} ) q^{87} + ( -2 + \beta_{1} - 2 \beta_{4} - \beta_{5} ) q^{88} + ( 6 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{89} + ( -5 - 2 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{90} + ( -1 + \beta_{6} ) q^{91} + ( -4 - \beta_{1} - \beta_{2} - 4 \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{6} ) q^{92} + ( 1 - \beta_{1} ) q^{93} + ( \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} ) q^{94} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{95} + ( -2 - \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{96} + ( 1 + \beta_{1} - \beta_{2} + 3 \beta_{3} - 4 \beta_{4} - 5 \beta_{5} - 2 \beta_{6} ) q^{97} + ( 3 - \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{98} + ( -5 + \beta_{2} + 3 \beta_{4} - \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 2q^{2} + 5q^{3} + 8q^{4} + 11q^{5} + 6q^{6} + 4q^{7} + 3q^{8} + 4q^{9} + O(q^{10}) \) \( 7q + 2q^{2} + 5q^{3} + 8q^{4} + 11q^{5} + 6q^{6} + 4q^{7} + 3q^{8} + 4q^{9} + 8q^{11} - 3q^{12} - 7q^{13} - 5q^{14} + 2q^{16} + 7q^{17} - 9q^{18} + q^{19} - 2q^{21} + 6q^{22} + 6q^{23} + 5q^{24} + 10q^{25} - 2q^{26} + 11q^{27} - 11q^{28} - 2q^{29} - 3q^{30} + 7q^{31} + 18q^{32} + 6q^{33} - 4q^{34} - q^{35} - 20q^{36} + 28q^{37} - 8q^{38} - 5q^{39} - 5q^{40} + 3q^{41} + 13q^{42} - q^{43} + 12q^{44} + 9q^{45} - 37q^{46} - q^{47} + 11q^{48} - 19q^{49} + 21q^{50} - 30q^{51} - 8q^{52} + 29q^{53} + 2q^{54} + 19q^{55} - 20q^{56} + 11q^{57} + 3q^{58} + 3q^{59} - 43q^{60} + 5q^{61} + 2q^{62} + q^{63} - 29q^{64} - 11q^{65} - 29q^{66} - 32q^{67} + 38q^{68} + 17q^{69} - 23q^{70} + 5q^{71} - 17q^{72} + q^{73} - 4q^{74} - 7q^{75} - 12q^{76} - 5q^{77} - 6q^{78} - 15q^{79} - 11q^{80} + 3q^{81} - 36q^{82} + 17q^{83} + 2q^{84} - q^{85} - 23q^{86} - 42q^{87} - 15q^{88} + 26q^{89} - 40q^{90} - 4q^{91} - 24q^{92} + 5q^{93} + 18q^{94} - 21q^{95} - 4q^{96} + 11q^{97} + 6q^{98} - 28q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 2 x^{6} - 9 x^{5} + 12 x^{4} + 22 x^{3} - 18 x^{2} - 13 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{5} - \nu^{4} - 9 \nu^{3} + 2 \nu^{2} + 15 \nu + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{5} - \nu^{4} - 9 \nu^{3} + \nu^{2} + 16 \nu + 4 \)
\(\beta_{4}\)\(=\)\( -\nu^{6} + \nu^{5} + 9 \nu^{4} - \nu^{3} - 17 \nu^{2} - 4 \nu + 3 \)
\(\beta_{5}\)\(=\)\( -\nu^{6} + \nu^{5} + 10 \nu^{4} - 4 \nu^{3} - 21 \nu^{2} + 7 \nu + 3 \)
\(\beta_{6}\)\(=\)\( 2 \nu^{6} - \nu^{5} - 19 \nu^{4} - 6 \nu^{3} + 33 \nu^{2} + 19 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{6} + 2 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 7 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(3 \beta_{6} + \beta_{5} + 5 \beta_{4} - 13 \beta_{3} + 10 \beta_{2} + 14 \beta_{1} + 15\)
\(\nu^{5}\)\(=\)\(12 \beta_{6} + \beta_{5} + 23 \beta_{4} - 38 \beta_{3} + 27 \beta_{2} + 60 \beta_{1} + 17\)
\(\nu^{6}\)\(=\)\(38 \beta_{6} + 10 \beta_{5} + 65 \beta_{4} - 135 \beta_{3} + 98 \beta_{2} + 158 \beta_{1} + 103\)

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
3.17535
−0.434125
−2.01774
1.56423
−1.70914
1.51046
−0.0890306
−2.10645 −2.17535 2.43713 2.39413 4.58226 1.34409 −0.920801 1.73214 −5.04311
1.2 −2.07211 1.43412 2.29366 0.0880198 −2.97167 0.184197 −0.608494 −0.943287 −0.182387
1.3 −0.299542 3.01774 −1.91027 3.67569 −0.903940 −0.783156 1.17129 6.10678 −1.10102
1.4 0.406064 −0.564229 −1.83511 2.08199 −0.229113 4.31735 −1.55730 −2.68165 0.845422
1.5 1.39140 2.70914 −0.0639949 −0.456939 3.76951 1.81179 −2.87185 4.33945 −0.635787
1.6 2.09092 −0.510463 2.37194 4.43975 −1.06734 −2.30124 0.777706 −2.73943 9.28317
1.7 2.58972 1.08903 4.70665 −1.22264 2.82028 −0.573038 7.00945 −1.81401 −3.16629
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

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 403.2.a.c 7
3.b odd 2 1 3627.2.a.n 7
4.b odd 2 1 6448.2.a.ba 7
13.b even 2 1 5239.2.a.h 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.a.c 7 1.a even 1 1 trivial
3627.2.a.n 7 3.b odd 2 1
5239.2.a.h 7 13.b even 2 1
6448.2.a.ba 7 4.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(13\) \(1\)
\(31\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - 2 T_{2}^{6} - 9 T_{2}^{5} + 17 T_{2}^{4} + 20 T_{2}^{3} - 37 T_{2}^{2} + T_{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(403))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 5 T^{2} - 7 T^{3} + 14 T^{4} - 21 T^{5} + 41 T^{6} - 56 T^{7} + 82 T^{8} - 84 T^{9} + 112 T^{10} - 112 T^{11} + 160 T^{12} - 128 T^{13} + 128 T^{14} \)
$3$ \( 1 - 5 T + 21 T^{2} - 62 T^{3} + 164 T^{4} - 360 T^{5} + 735 T^{6} - 1306 T^{7} + 2205 T^{8} - 3240 T^{9} + 4428 T^{10} - 5022 T^{11} + 5103 T^{12} - 3645 T^{13} + 2187 T^{14} \)
$5$ \( 1 - 11 T + 73 T^{2} - 357 T^{3} + 1400 T^{4} - 4585 T^{5} + 12789 T^{6} - 30754 T^{7} + 63945 T^{8} - 114625 T^{9} + 175000 T^{10} - 223125 T^{11} + 228125 T^{12} - 171875 T^{13} + 78125 T^{14} \)
$7$ \( 1 - 4 T + 42 T^{2} - 144 T^{3} + 791 T^{4} - 2291 T^{5} + 8715 T^{6} - 20704 T^{7} + 61005 T^{8} - 112259 T^{9} + 271313 T^{10} - 345744 T^{11} + 705894 T^{12} - 470596 T^{13} + 823543 T^{14} \)
$11$ \( 1 - 8 T + 74 T^{2} - 429 T^{3} + 2386 T^{4} - 10471 T^{5} + 43268 T^{6} - 147557 T^{7} + 475948 T^{8} - 1266991 T^{9} + 3175766 T^{10} - 6280989 T^{11} + 11917774 T^{12} - 14172488 T^{13} + 19487171 T^{14} \)
$13$ \( ( 1 + T )^{7} \)
$17$ \( 1 - 7 T + 99 T^{2} - 487 T^{3} + 3936 T^{4} - 14909 T^{5} + 92493 T^{6} - 294386 T^{7} + 1572381 T^{8} - 4308701 T^{9} + 19337568 T^{10} - 40674727 T^{11} + 140565843 T^{12} - 168962983 T^{13} + 410338673 T^{14} \)
$19$ \( 1 - T + 96 T^{2} - 116 T^{3} + 4435 T^{4} - 5215 T^{5} + 127343 T^{6} - 128176 T^{7} + 2419517 T^{8} - 1882615 T^{9} + 30419665 T^{10} - 15117236 T^{11} + 237705504 T^{12} - 47045881 T^{13} + 893871739 T^{14} \)
$23$ \( 1 - 6 T + 85 T^{2} - 436 T^{3} + 3598 T^{4} - 15182 T^{5} + 100343 T^{6} - 386900 T^{7} + 2307889 T^{8} - 8031278 T^{9} + 43776866 T^{10} - 122010676 T^{11} + 547089155 T^{12} - 888215334 T^{13} + 3404825447 T^{14} \)
$29$ \( 1 + 2 T + 95 T^{2} + 341 T^{3} + 4545 T^{4} + 22874 T^{5} + 158418 T^{6} + 847016 T^{7} + 4594122 T^{8} + 19237034 T^{9} + 110848005 T^{10} + 241182821 T^{11} + 1948559155 T^{12} + 1189646642 T^{13} + 17249876309 T^{14} \)
$31$ \( ( 1 - T )^{7} \)
$37$ \( 1 - 28 T + 543 T^{2} - 7259 T^{3} + 79366 T^{4} - 702049 T^{5} + 5368229 T^{6} - 34835635 T^{7} + 198624473 T^{8} - 961105081 T^{9} + 4020125998 T^{10} - 13604534699 T^{11} + 37653768651 T^{12} - 71840339452 T^{13} + 94931877133 T^{14} \)
$41$ \( 1 - 3 T + 154 T^{2} - 384 T^{3} + 12246 T^{4} - 27106 T^{5} + 677120 T^{6} - 1349900 T^{7} + 27761920 T^{8} - 45565186 T^{9} + 844006566 T^{10} - 1085092224 T^{11} + 17841854954 T^{12} - 14250312723 T^{13} + 194754273881 T^{14} \)
$43$ \( 1 + T + 182 T^{2} + 446 T^{3} + 15284 T^{4} + 56610 T^{5} + 839368 T^{6} + 3380702 T^{7} + 36092824 T^{8} + 104671890 T^{9} + 1215184988 T^{10} + 1524785246 T^{11} + 26755536626 T^{12} + 6321363049 T^{13} + 271818611107 T^{14} \)
$47$ \( 1 + T + 110 T^{2} + 628 T^{3} + 8078 T^{4} + 51096 T^{5} + 603016 T^{6} + 2275594 T^{7} + 28341752 T^{8} + 112871064 T^{9} + 838682194 T^{10} + 3064439668 T^{11} + 25227950770 T^{12} + 10779215329 T^{13} + 506623120463 T^{14} \)
$53$ \( 1 - 29 T + 591 T^{2} - 8451 T^{3} + 101858 T^{4} - 1016019 T^{5} + 9011455 T^{6} - 69125430 T^{7} + 477607115 T^{8} - 2853997371 T^{9} + 15164313466 T^{10} - 66682454931 T^{11} + 247153536363 T^{12} - 642766472741 T^{13} + 1174711139837 T^{14} \)
$59$ \( 1 - 3 T + 105 T^{2} - 407 T^{3} + 5420 T^{4} - 49707 T^{5} + 249413 T^{6} - 4514256 T^{7} + 14715367 T^{8} - 173030067 T^{9} + 1113154180 T^{10} - 4931765927 T^{11} + 75067051395 T^{12} - 126541600923 T^{13} + 2488651484819 T^{14} \)
$61$ \( 1 - 5 T + 315 T^{2} - 1450 T^{3} + 48188 T^{4} - 191734 T^{5} + 4491051 T^{6} - 14924570 T^{7} + 273954111 T^{8} - 713442214 T^{9} + 10937760428 T^{10} - 20076469450 T^{11} + 266047834815 T^{12} - 257601871805 T^{13} + 3142742836021 T^{14} \)
$67$ \( 1 + 32 T + 865 T^{2} + 15244 T^{3} + 234085 T^{4} + 2802431 T^{5} + 29745354 T^{6} + 257912236 T^{7} + 1992938718 T^{8} + 12580112759 T^{9} + 70404106855 T^{10} + 307183688524 T^{11} + 1167858217555 T^{12} + 2894668229408 T^{13} + 6060711605323 T^{14} \)
$71$ \( 1 - 5 T + 232 T^{2} - 1756 T^{3} + 29821 T^{4} - 244839 T^{5} + 2745945 T^{6} - 21305042 T^{7} + 194962095 T^{8} - 1234233399 T^{9} + 10673263931 T^{10} - 44622911836 T^{11} + 418581209432 T^{12} - 640501419605 T^{13} + 9095120158391 T^{14} \)
$73$ \( 1 - T + 155 T^{2} + 1727 T^{3} + 10322 T^{4} + 214941 T^{5} + 2026687 T^{6} + 10857629 T^{7} + 147948151 T^{8} + 1145420589 T^{9} + 4015433474 T^{10} + 49043762207 T^{11} + 321326096915 T^{12} - 151334226289 T^{13} + 11047398519097 T^{14} \)
$79$ \( 1 + 15 T + 496 T^{2} + 6030 T^{3} + 109559 T^{4} + 1086719 T^{5} + 13935127 T^{6} + 111068090 T^{7} + 1100875033 T^{8} + 6782213279 T^{9} + 54016859801 T^{10} + 234868988430 T^{11} + 1526219973904 T^{12} + 3646311832815 T^{13} + 19203908986159 T^{14} \)
$83$ \( 1 - 17 T + 452 T^{2} - 5040 T^{3} + 77092 T^{4} - 623788 T^{5} + 7666358 T^{6} - 53524394 T^{7} + 636307714 T^{8} - 4297275532 T^{9} + 44080203404 T^{10} - 239189937840 T^{11} + 1780446370636 T^{12} - 5557986347273 T^{13} + 27136050989627 T^{14} \)
$89$ \( 1 - 26 T + 730 T^{2} - 11864 T^{3} + 194429 T^{4} - 2319046 T^{5} + 27909218 T^{6} - 261497659 T^{7} + 2483920402 T^{8} - 18369163366 T^{9} + 137066417701 T^{10} - 744373947224 T^{11} + 4076363397770 T^{12} - 12921513564986 T^{13} + 44231334895529 T^{14} \)
$97$ \( 1 - 11 T + 339 T^{2} - 1956 T^{3} + 40349 T^{4} - 18067 T^{5} + 2498592 T^{6} + 13581260 T^{7} + 242363424 T^{8} - 169992403 T^{9} + 36825442877 T^{10} - 173163273636 T^{11} + 2911108347123 T^{12} - 9162692054219 T^{13} + 80798284478113 T^{14} \)
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