Properties

Label 403.2.a.b
Level 403
Weight 2
Character orbit 403.a
Self dual yes
Analytic conductor 3.218
Analytic rank 1
Dimension 6
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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.5748973.1
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 6 x^{4} + 4 x^{3} + 8 x^{2} - 4 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 3 \nu + 2 \)
\(\beta_{4}\)\(=\)\( \nu^{5} - \nu^{4} - 5 \nu^{3} + 3 \nu^{2} + 4 \nu - 2 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - 6 \nu^{3} - 2 \nu^{2} + 6 \nu + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 3 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{5} - \beta_{4} + \beta_{3} + 6 \beta_{2} + \beta_{1} + 6\)
\(\nu^{5}\)\(=\)\(\beta_{5} + 6 \beta_{3} + 8 \beta_{2} + 12 \beta_{1} + 2\)

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
−0.187851
1.31089
−1.85232
0.699790
2.35805
−1.32857
−2.61388 −1.18785 4.83234 1.45573 3.10489 −1.87247 −7.40339 −1.58901 −3.80510
1.2 −1.94648 0.310892 1.78879 −3.27007 −0.605146 3.06335 0.411120 −2.90335 6.36512
1.3 −0.917109 −2.85232 −1.15891 −0.732299 2.61588 4.57774 2.89707 5.13571 0.671598
1.4 0.482827 −0.300210 −1.76688 0.848172 −0.144950 −1.74674 −1.81875 −2.90987 0.409520
1.5 0.543189 1.35805 −1.70495 −3.27954 0.737678 0.540055 −2.01249 −1.15570 −1.78141
1.6 2.45145 −2.32857 4.00960 −4.02200 −5.70836 −4.56194 4.92644 2.42222 −9.85973
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.b 6
3.b odd 2 1 3627.2.a.m 6
4.b odd 2 1 6448.2.a.y 6
13.b even 2 1 5239.2.a.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.a.b 6 1.a even 1 1 trivial
3627.2.a.m 6 3.b odd 2 1
5239.2.a.g 6 13.b even 2 1
6448.2.a.y 6 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 \( T_{2}^{6} + 2 T_{2}^{5} - 7 T_{2}^{4} - 13 T_{2}^{3} + 6 T_{2}^{2} + 7 T_{2} - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(403))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 5 T^{2} + 7 T^{3} + 10 T^{4} + 9 T^{5} + 13 T^{6} + 18 T^{7} + 40 T^{8} + 56 T^{9} + 80 T^{10} + 64 T^{11} + 64 T^{12} \)
$3$ \( 1 + 5 T + 22 T^{2} + 65 T^{3} + 172 T^{4} + 361 T^{5} + 691 T^{6} + 1083 T^{7} + 1548 T^{8} + 1755 T^{9} + 1782 T^{10} + 1215 T^{11} + 729 T^{12} \)
$5$ \( 1 + 9 T + 50 T^{2} + 206 T^{3} + 700 T^{4} + 1979 T^{5} + 4789 T^{6} + 9895 T^{7} + 17500 T^{8} + 25750 T^{9} + 31250 T^{10} + 28125 T^{11} + 15625 T^{12} \)
$7$ \( 1 + 13 T^{2} - 6 T^{3} + 98 T^{4} - 5 T^{5} + 671 T^{6} - 35 T^{7} + 4802 T^{8} - 2058 T^{9} + 31213 T^{10} + 117649 T^{12} \)
$11$ \( 1 + 5 T + 48 T^{2} + 226 T^{3} + 1170 T^{4} + 4357 T^{5} + 16795 T^{6} + 47927 T^{7} + 141570 T^{8} + 300806 T^{9} + 702768 T^{10} + 805255 T^{11} + 1771561 T^{12} \)
$13$ \( ( 1 - T )^{6} \)
$17$ \( 1 + 23 T + 296 T^{2} + 2664 T^{3} + 18400 T^{4} + 101871 T^{5} + 462517 T^{6} + 1731807 T^{7} + 5317600 T^{8} + 13088232 T^{9} + 24722216 T^{10} + 32656711 T^{11} + 24137569 T^{12} \)
$19$ \( 1 - 7 T + 37 T^{2} - 205 T^{3} + 878 T^{4} - 4092 T^{5} + 23147 T^{6} - 77748 T^{7} + 316958 T^{8} - 1406095 T^{9} + 4821877 T^{10} - 17332693 T^{11} + 47045881 T^{12} \)
$23$ \( 1 + 18 T + 230 T^{2} + 2072 T^{3} + 15500 T^{4} + 94612 T^{5} + 496265 T^{6} + 2176076 T^{7} + 8199500 T^{8} + 25210024 T^{9} + 64363430 T^{10} + 115854174 T^{11} + 148035889 T^{12} \)
$29$ \( 1 + 18 T + 256 T^{2} + 2385 T^{3} + 19683 T^{4} + 127703 T^{5} + 761621 T^{6} + 3703387 T^{7} + 16553403 T^{8} + 58167765 T^{9} + 181063936 T^{10} + 369200682 T^{11} + 594823321 T^{12} \)
$31$ \( ( 1 - T )^{6} \)
$37$ \( 1 + 13 T + 167 T^{2} + 1395 T^{3} + 10520 T^{4} + 68826 T^{5} + 424911 T^{6} + 2546562 T^{7} + 14401880 T^{8} + 70660935 T^{9} + 312984887 T^{10} + 901471441 T^{11} + 2565726409 T^{12} \)
$41$ \( 1 + 5 T + 107 T^{2} + 139 T^{3} + 4339 T^{4} - 11907 T^{5} + 124273 T^{6} - 488187 T^{7} + 7293859 T^{8} + 9580019 T^{9} + 302356427 T^{10} + 579281005 T^{11} + 4750104241 T^{12} \)
$43$ \( 1 + 7 T + 157 T^{2} + 799 T^{3} + 10895 T^{4} + 39897 T^{5} + 515997 T^{6} + 1715571 T^{7} + 20144855 T^{8} + 63526093 T^{9} + 536751757 T^{10} + 1029059101 T^{11} + 6321363049 T^{12} \)
$47$ \( 1 + 9 T + 187 T^{2} + 1615 T^{3} + 17941 T^{4} + 128543 T^{5} + 1062573 T^{6} + 6041521 T^{7} + 39631669 T^{8} + 167674145 T^{9} + 912500347 T^{10} + 2064105063 T^{11} + 10779215329 T^{12} \)
$53$ \( 1 + 31 T + 652 T^{2} + 9636 T^{3} + 114432 T^{4} + 1093949 T^{5} + 8760977 T^{6} + 57979297 T^{7} + 321439488 T^{8} + 1434578772 T^{9} + 5144593612 T^{10} + 12964060283 T^{11} + 22164361129 T^{12} \)
$59$ \( 1 + T + 218 T^{2} + 366 T^{3} + 24064 T^{4} + 38337 T^{5} + 1731235 T^{6} + 2261883 T^{7} + 83766784 T^{8} + 75168714 T^{9} + 2641584698 T^{10} + 714924299 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 + 15 T + 272 T^{2} + 2577 T^{3} + 24700 T^{4} + 180103 T^{5} + 1436915 T^{6} + 10986283 T^{7} + 91908700 T^{8} + 584930037 T^{9} + 3766068752 T^{10} + 12668944515 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 + 28 T + 566 T^{2} + 7958 T^{3} + 98051 T^{4} + 987149 T^{5} + 8880619 T^{6} + 66138983 T^{7} + 440150939 T^{8} + 2393471954 T^{9} + 11405534486 T^{10} + 37803502996 T^{11} + 90458382169 T^{12} \)
$71$ \( 1 - T + 147 T^{2} + 509 T^{3} + 7580 T^{4} + 107146 T^{5} + 310755 T^{6} + 7607366 T^{7} + 38210780 T^{8} + 182176699 T^{9} + 3735517107 T^{10} - 1804229351 T^{11} + 128100283921 T^{12} \)
$73$ \( 1 + 20 T + 526 T^{2} + 7007 T^{3} + 103261 T^{4} + 998865 T^{5} + 10249753 T^{6} + 72917145 T^{7} + 550277869 T^{8} + 2725842119 T^{9} + 14937474766 T^{10} + 41461431860 T^{11} + 151334226289 T^{12} \)
$79$ \( 1 + 15 T + 391 T^{2} + 4527 T^{3} + 69134 T^{4} + 634850 T^{5} + 7043949 T^{6} + 50153150 T^{7} + 431465294 T^{8} + 2231987553 T^{9} + 15229481671 T^{10} + 46155845985 T^{11} + 243087455521 T^{12} \)
$83$ \( 1 - T + 425 T^{2} - 363 T^{3} + 80383 T^{4} - 56955 T^{5} + 8627425 T^{6} - 4727265 T^{7} + 553758487 T^{8} - 207558681 T^{9} + 20169786425 T^{10} - 3939040643 T^{11} + 326940373369 T^{12} \)
$89$ \( 1 + T + 282 T^{2} - 601 T^{3} + 33142 T^{4} - 183391 T^{5} + 2838089 T^{6} - 16321799 T^{7} + 262517782 T^{8} - 423686369 T^{9} + 17693311962 T^{10} + 5584059449 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 + 5 T + 330 T^{2} + 1813 T^{3} + 59935 T^{4} + 290002 T^{5} + 7241445 T^{6} + 28130194 T^{7} + 563928415 T^{8} + 1654676149 T^{9} + 29214662730 T^{10} + 42936701285 T^{11} + 832972004929 T^{12} \)
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