Properties

Label 19.3.d.a
Level 19
Weight 3
Character orbit 19.d
Analytic conductor 0.518
Analytic rank 0
Dimension 6
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 19 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 19.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.517712502285\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.6967728.1
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{5} ) q^{2} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{3} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{4} + ( -3 + \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{5} + ( 4 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{6} + ( -1 - 3 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{7} + ( 3 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + \beta_{4} + \beta_{5} ) q^{8} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{5} ) q^{2} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{3} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{4} + ( -3 + \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{5} + ( 4 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{6} + ( -1 - 3 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{7} + ( 3 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + \beta_{4} + \beta_{5} ) q^{8} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{9} + ( -12 - 3 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} - \beta_{4} ) q^{10} + ( 6 + 5 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{11} + ( 4 + 2 \beta_{1} + \beta_{2} + 14 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{12} + ( 4 + 2 \beta_{3} - 4 \beta_{4} ) q^{13} + ( 3 - \beta_{1} - 2 \beta_{2} - 7 \beta_{3} - 4 \beta_{5} ) q^{14} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} + 10 \beta_{4} ) q^{15} + ( 7 + 4 \beta_{1} + 3 \beta_{3} + 8 \beta_{4} + 4 \beta_{5} ) q^{16} + ( -16 - 15 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{17} + ( -10 - 6 \beta_{3} - 7 \beta_{4} - 7 \beta_{5} ) q^{18} + ( 14 - 5 \beta_{1} - 3 \beta_{2} + 13 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} ) q^{19} + ( 17 - 3 \beta_{2} - 5 \beta_{4} + 5 \beta_{5} ) q^{20} + ( -7 - \beta_{1} - 2 \beta_{2} + 14 \beta_{3} + 7 \beta_{5} ) q^{21} + ( -4 - \beta_{1} - 2 \beta_{2} + 5 \beta_{3} + \beta_{5} ) q^{22} + ( -4 + \beta_{1} + \beta_{2} - 5 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} ) q^{23} + ( -26 - 5 \beta_{1} - 26 \beta_{3} ) q^{24} + ( 2 + 8 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{25} + ( -22 + 8 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{26} + ( 2 - 10 \beta_{1} - 5 \beta_{2} + 8 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{27} + ( 7 \beta_{1} + 7 \beta_{2} - 13 \beta_{3} ) q^{28} + ( 9 \beta_{1} - 9 \beta_{2} + 13 \beta_{4} ) q^{29} + ( 45 - 17 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{30} + ( 17 - 2 \beta_{1} - \beta_{2} + 42 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} ) q^{31} + ( 16 - 4 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} + 3 \beta_{4} ) q^{32} + ( 17 + 6 \beta_{1} + 12 \beta_{2} - 15 \beta_{3} + 2 \beta_{5} ) q^{33} + ( -10 - 2 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} - 15 \beta_{4} ) q^{34} + ( -26 - 14 \beta_{1} - 17 \beta_{3} - 18 \beta_{4} - 9 \beta_{5} ) q^{35} + ( -14 - 6 \beta_{1} - 15 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{36} + ( -3 + 2 \beta_{1} + \beta_{2} - 24 \beta_{3} + 9 \beta_{4} + 9 \beta_{5} ) q^{37} + ( 18 + 18 \beta_{1} + 7 \beta_{2} + 33 \beta_{3} + 8 \beta_{4} + \beta_{5} ) q^{38} + ( -4 + 10 \beta_{2} + 6 \beta_{4} - 6 \beta_{5} ) q^{39} + ( -23 + \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 19 \beta_{5} ) q^{40} + ( 3 - 10 \beta_{1} - 20 \beta_{2} - 4 \beta_{3} - \beta_{5} ) q^{41} + ( 26 + 17 \beta_{1} + 17 \beta_{2} + 38 \beta_{3} + 13 \beta_{4} + 26 \beta_{5} ) q^{42} + ( -22 + 8 \beta_{1} - 18 \beta_{3} - 8 \beta_{4} - 4 \beta_{5} ) q^{43} + ( -8 - 15 \beta_{1} - 15 \beta_{2} + 24 \beta_{3} - 4 \beta_{4} - 8 \beta_{5} ) q^{44} + ( -4 + 2 \beta_{2} - 10 \beta_{4} + 10 \beta_{5} ) q^{45} + ( -15 - 10 \beta_{1} - 5 \beta_{2} - 22 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} ) q^{46} + ( 12 - \beta_{1} - \beta_{2} - 13 \beta_{3} + 6 \beta_{4} + 12 \beta_{5} ) q^{47} + ( -46 + \beta_{1} - \beta_{2} - 23 \beta_{3} - 19 \beta_{4} ) q^{48} + ( 5 + 6 \beta_{2} + 7 \beta_{4} - 7 \beta_{5} ) q^{49} + ( 9 - 12 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} + 12 \beta_{4} + 12 \beta_{5} ) q^{50} + ( 34 + 14 \beta_{1} - 14 \beta_{2} + 17 \beta_{3} + 21 \beta_{4} ) q^{51} + ( 22 - 12 \beta_{1} - 24 \beta_{2} - 10 \beta_{3} + 12 \beta_{5} ) q^{52} + ( -8 - 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} - 12 \beta_{4} ) q^{53} + ( 4 + 11 \beta_{1} + 5 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{54} + ( 1 + 17 \beta_{1} - 7 \beta_{3} + 16 \beta_{4} + 8 \beta_{5} ) q^{55} + ( 31 - 6 \beta_{1} - 3 \beta_{2} + 42 \beta_{3} + 10 \beta_{4} + 10 \beta_{5} ) q^{56} + ( -31 - 12 \beta_{1} + 8 \beta_{2} - 3 \beta_{3} - 37 \beta_{4} - 26 \beta_{5} ) q^{57} + ( 29 + \beta_{2} + 9 \beta_{4} - 9 \beta_{5} ) q^{58} + ( 2 + 11 \beta_{1} + 22 \beta_{2} + 20 \beta_{3} + 22 \beta_{5} ) q^{59} + ( -35 + 17 \beta_{1} + 34 \beta_{2} + 11 \beta_{3} - 24 \beta_{5} ) q^{60} + ( -30 - 13 \beta_{1} - 13 \beta_{2} - 30 \beta_{3} - 15 \beta_{4} - 30 \beta_{5} ) q^{61} + ( 3 - 5 \beta_{1} - 17 \beta_{3} + 40 \beta_{4} + 20 \beta_{5} ) q^{62} + ( -24 - 8 \beta_{1} - 8 \beta_{2} - 38 \beta_{3} - 12 \beta_{4} - 24 \beta_{5} ) q^{63} + ( 27 - 2 \beta_{2} - 12 \beta_{4} + 12 \beta_{5} ) q^{64} + ( 18 + 28 \beta_{1} + 14 \beta_{2} + 40 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{65} + ( -18 - 14 \beta_{1} - 14 \beta_{2} - 8 \beta_{3} - 9 \beta_{4} - 18 \beta_{5} ) q^{66} + ( 48 - 13 \beta_{1} + 13 \beta_{2} + 24 \beta_{3} - 8 \beta_{4} ) q^{67} + ( -6 + 24 \beta_{2} - 3 \beta_{4} + 3 \beta_{5} ) q^{68} + ( 25 + 6 \beta_{1} + 3 \beta_{2} + 12 \beta_{3} + 19 \beta_{4} + 19 \beta_{5} ) q^{69} + ( -62 - 4 \beta_{1} + 4 \beta_{2} - 31 \beta_{3} - 31 \beta_{4} ) q^{70} + ( -10 + 4 \beta_{1} + 8 \beta_{2} + 10 \beta_{3} ) q^{71} + ( 46 + 8 \beta_{1} - 8 \beta_{2} + 23 \beta_{3} + 7 \beta_{4} ) q^{72} + ( 25 - 2 \beta_{1} + 14 \beta_{3} + 22 \beta_{4} + 11 \beta_{5} ) q^{73} + ( 31 + 15 \beta_{1} + 42 \beta_{3} - 22 \beta_{4} - 11 \beta_{5} ) q^{74} + ( -24 - 6 \beta_{1} - 3 \beta_{2} - 60 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} ) q^{75} + ( -30 - 11 \beta_{1} - 18 \beta_{2} - 36 \beta_{3} + 31 \beta_{4} + 11 \beta_{5} ) q^{76} + ( -73 - 31 \beta_{2} + 5 \beta_{4} - 5 \beta_{5} ) q^{77} + ( 48 - 22 \beta_{1} - 44 \beta_{2} - 40 \beta_{3} + 8 \beta_{5} ) q^{78} + ( 26 + 10 \beta_{1} + 20 \beta_{2} + 8 \beta_{3} + 34 \beta_{5} ) q^{79} + ( -30 - 29 \beta_{1} - 29 \beta_{2} - 50 \beta_{3} - 15 \beta_{4} - 30 \beta_{5} ) q^{80} + ( 81 + 4 \beta_{1} + 82 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{81} + ( -28 + 28 \beta_{1} + 28 \beta_{2} + 25 \beta_{3} - 14 \beta_{4} - 28 \beta_{5} ) q^{82} + ( 12 - 37 \beta_{2} + 13 \beta_{4} - 13 \beta_{5} ) q^{83} + ( 19 + 26 \beta_{1} + 13 \beta_{2} - 16 \beta_{3} + 27 \beta_{4} + 27 \beta_{5} ) q^{84} + ( 32 - 6 \beta_{1} - 6 \beta_{2} + 48 \beta_{3} + 16 \beta_{4} + 32 \beta_{5} ) q^{85} + ( -56 - 16 \beta_{1} + 16 \beta_{2} - 28 \beta_{3} - 10 \beta_{4} ) q^{86} + ( -99 - 13 \beta_{2} - 8 \beta_{4} + 8 \beta_{5} ) q^{87} + ( -20 + 22 \beta_{1} + 11 \beta_{2} - 50 \beta_{3} + 5 \beta_{4} + 5 \beta_{5} ) q^{88} + ( 6 - 24 \beta_{1} + 24 \beta_{2} + 3 \beta_{3} + 9 \beta_{4} ) q^{89} + ( -32 + 18 \beta_{1} + 36 \beta_{2} + 48 \beta_{3} + 16 \beta_{5} ) q^{90} + ( -52 + 2 \beta_{1} - 2 \beta_{2} - 26 \beta_{3} - 4 \beta_{4} ) q^{91} + ( 7 + 3 \beta_{1} + 15 \beta_{3} - 16 \beta_{4} - 8 \beta_{5} ) q^{92} + ( -61 - 67 \beta_{1} - 46 \beta_{3} - 30 \beta_{4} - 15 \beta_{5} ) q^{93} + ( 17 + 26 \beta_{1} + 13 \beta_{2} + 62 \beta_{3} - 14 \beta_{4} - 14 \beta_{5} ) q^{94} + ( 67 + 10 \beta_{1} - 13 \beta_{2} - 26 \beta_{3} - 4 \beta_{4} - 10 \beta_{5} ) q^{95} + ( 28 + 21 \beta_{2} - 22 \beta_{4} + 22 \beta_{5} ) q^{96} + ( -29 - 2 \beta_{1} - 4 \beta_{2} - 17 \beta_{3} - 46 \beta_{5} ) q^{97} + ( 35 - 20 \beta_{1} - 40 \beta_{2} - 41 \beta_{3} - 6 \beta_{5} ) q^{98} + ( 10 + 2 \beta_{1} + 2 \beta_{2} + 23 \beta_{3} + 5 \beta_{4} + 10 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 3q^{2} - 9q^{3} + 5q^{4} - 2q^{5} + q^{6} + 14q^{9} + O(q^{10}) \) \( 6q - 3q^{2} - 9q^{3} + 5q^{4} - 2q^{5} + q^{6} + 14q^{9} - 60q^{10} + 26q^{11} + 30q^{13} + 54q^{14} - 18q^{15} + q^{16} - 42q^{17} + 25q^{19} + 108q^{20} - 102q^{21} - 39q^{22} + 8q^{23} - 83q^{24} - 17q^{25} - 148q^{26} + 32q^{28} - 12q^{29} + 304q^{30} + 51q^{32} + 123q^{33} - 6q^{34} - 38q^{35} - 54q^{36} - 14q^{38} - 44q^{39} - 96q^{40} + 63q^{41} - 92q^{42} - 34q^{43} - 69q^{44} - 28q^{45} + 58q^{47} - 147q^{48} + 18q^{49} + 132q^{51} + 162q^{52} - 12q^{53} + 29q^{54} - 28q^{55} - 16q^{57} + 172q^{58} - 147q^{59} - 222q^{60} + 58q^{61} - 116q^{62} + 86q^{63} + 166q^{64} + 11q^{66} + 201q^{67} - 84q^{68} - 198q^{70} - 102q^{71} + 210q^{72} + 7q^{73} + 174q^{74} - 173q^{76} - 376q^{77} + 450q^{78} + 134q^{80} + 253q^{81} - 145q^{82} + 146q^{83} - 90q^{85} - 270q^{86} - 568q^{87} - 72q^{89} - 438q^{90} - 216q^{91} + 72q^{92} - 160q^{93} + 558q^{95} + 126q^{96} + 21q^{97} + 411q^{98} - 56q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} + 8 x^{4} + 5 x^{3} + 50 x^{2} - 7 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - 8 \nu^{4} + 64 \nu^{3} - 50 \nu^{2} + 7 \nu - 56 \)\()/393\)
\(\beta_{3}\)\(=\)\((\)\( 56 \nu^{5} - 55 \nu^{4} + 440 \nu^{3} + 344 \nu^{2} + 2750 \nu - 385 \)\()/393\)
\(\beta_{4}\)\(=\)\((\)\( 70 \nu^{5} - 36 \nu^{4} + 550 \nu^{3} + 561 \nu^{2} + 3634 \nu + 534 \)\()/393\)
\(\beta_{5}\)\(=\)\((\)\( 77 \nu^{5} - 92 \nu^{4} + 605 \nu^{3} + 211 \nu^{2} + 3683 \nu - 1430 \)\()/393\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-2 \beta_{5} - \beta_{4} + 4 \beta_{3} - 2\)
\(\nu^{3}\)\(=\)\(-\beta_{5} + \beta_{4} + 7 \beta_{2} - 4\)
\(\nu^{4}\)\(=\)\(8 \beta_{5} + 16 \beta_{4} - 31 \beta_{3} - 6 \beta_{1} - 23\)
\(\nu^{5}\)\(=\)\(28 \beta_{5} + 14 \beta_{4} - 48 \beta_{3} - 55 \beta_{2} - 55 \beta_{1} + 28\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
1.56632 2.71294i
−1.13654 + 1.96854i
0.0702177 0.121621i
1.56632 + 2.71294i
−1.13654 1.96854i
0.0702177 + 0.121621i
−2.90671 1.67819i −3.29225 1.90078i 3.63264 + 6.29191i 3.47303 6.01546i 6.37974 + 11.0500i −1.22892 10.9595i 2.72593 + 4.72145i −20.1902 + 11.6568i
8.2 −0.583430 0.336844i 2.49304 + 1.43936i −1.77307 3.07105i −1.55311 + 2.69006i −0.969676 1.67953i −8.15294 5.08374i −0.356503 0.617481i 1.81226 1.04631i
8.3 1.99014 + 1.14901i −3.70079 2.13665i 0.640435 + 1.10927i −2.91992 + 5.05745i −4.91006 8.50447i 9.38186 6.24860i 4.63057 + 8.02039i −11.6221 + 6.71002i
12.1 −2.90671 + 1.67819i −3.29225 + 1.90078i 3.63264 6.29191i 3.47303 + 6.01546i 6.37974 11.0500i −1.22892 10.9595i 2.72593 4.72145i −20.1902 11.6568i
12.2 −0.583430 + 0.336844i 2.49304 1.43936i −1.77307 + 3.07105i −1.55311 2.69006i −0.969676 + 1.67953i −8.15294 5.08374i −0.356503 + 0.617481i 1.81226 + 1.04631i
12.3 1.99014 1.14901i −3.70079 + 2.13665i 0.640435 1.10927i −2.91992 5.05745i −4.91006 + 8.50447i 9.38186 6.24860i 4.63057 8.02039i −11.6221 6.71002i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 12.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 19.3.d.a 6
3.b odd 2 1 171.3.p.d 6
4.b odd 2 1 304.3.r.b 6
19.b odd 2 1 361.3.d.c 6
19.c even 3 1 361.3.b.b 6
19.c even 3 1 361.3.d.c 6
19.d odd 6 1 inner 19.3.d.a 6
19.d odd 6 1 361.3.b.b 6
19.e even 9 3 361.3.f.h 18
19.e even 9 3 361.3.f.i 18
19.f odd 18 3 361.3.f.h 18
19.f odd 18 3 361.3.f.i 18
57.f even 6 1 171.3.p.d 6
76.f even 6 1 304.3.r.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.d.a 6 1.a even 1 1 trivial
19.3.d.a 6 19.d odd 6 1 inner
171.3.p.d 6 3.b odd 2 1
171.3.p.d 6 57.f even 6 1
304.3.r.b 6 4.b odd 2 1
304.3.r.b 6 76.f even 6 1
361.3.b.b 6 19.c even 3 1
361.3.b.b 6 19.d odd 6 1
361.3.d.c 6 19.b odd 2 1
361.3.d.c 6 19.c even 3 1
361.3.f.h 18 19.e even 9 3
361.3.f.h 18 19.f odd 18 3
361.3.f.i 18 19.e even 9 3
361.3.f.i 18 19.f odd 18 3

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(19, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 8 T^{2} + 15 T^{3} + 20 T^{4} + 27 T^{5} - T^{6} + 108 T^{7} + 320 T^{8} + 960 T^{9} + 2048 T^{10} + 3072 T^{11} + 4096 T^{12} \)
$3$ \( 1 + 9 T + 47 T^{2} + 180 T^{3} + 463 T^{4} + 891 T^{5} + 2142 T^{6} + 8019 T^{7} + 37503 T^{8} + 131220 T^{9} + 308367 T^{10} + 531441 T^{11} + 531441 T^{12} \)
$5$ \( 1 + 2 T - 27 T^{2} + 114 T^{3} + 238 T^{4} - 2656 T^{5} + 5401 T^{6} - 66400 T^{7} + 148750 T^{8} + 1781250 T^{9} - 10546875 T^{10} + 19531250 T^{11} + 244140625 T^{12} \)
$7$ \( ( 1 + 69 T^{2} - 94 T^{3} + 3381 T^{4} + 117649 T^{6} )^{2} \)
$11$ \( ( 1 - 13 T + 280 T^{2} - 3149 T^{3} + 33880 T^{4} - 190333 T^{5} + 1771561 T^{6} )^{2} \)
$13$ \( 1 - 30 T + 779 T^{2} - 14370 T^{3} + 239570 T^{4} - 3505902 T^{5} + 48205499 T^{6} - 592497438 T^{7} + 6842358770 T^{8} - 69361245330 T^{9} + 635454231659 T^{10} - 4135754755470 T^{11} + 23298085122481 T^{12} \)
$17$ \( 1 + 42 T + 333 T^{2} + 6726 T^{3} + 513306 T^{4} + 6959454 T^{5} + 21390509 T^{6} + 2011282206 T^{7} + 42871830426 T^{8} + 162349289094 T^{9} + 2322927227853 T^{10} + 84671743818858 T^{11} + 582622237229761 T^{12} \)
$19$ \( 1 - 25 T + 1026 T^{2} - 16967 T^{3} + 370386 T^{4} - 3258025 T^{5} + 47045881 T^{6} \)
$23$ \( 1 - 8 T - 1413 T^{2} + 3876 T^{3} + 1337428 T^{4} - 1325942 T^{5} - 814307171 T^{6} - 701423318 T^{7} + 374267188948 T^{8} + 573787105764 T^{9} - 110653422202053 T^{10} - 331412089709192 T^{11} + 21914624432020321 T^{12} \)
$29$ \( 1 + 12 T + 1091 T^{2} + 12516 T^{3} + 342470 T^{4} + 25875306 T^{5} + 53097851 T^{6} + 21761132346 T^{7} + 242222524070 T^{8} + 7444808685636 T^{9} + 545768836540451 T^{10} + 5048486799602412 T^{11} + 353814783205469041 T^{12} \)
$31$ \( 1 - 1222 T^{2} + 2797835 T^{4} - 2253009856 T^{6} + 2583859377035 T^{8} - 1042232847752902 T^{10} + 787662783788549761 T^{12} \)
$37$ \( 1 - 5190 T^{2} + 13520751 T^{4} - 22697438888 T^{6} + 25340064214911 T^{8} - 18229768365849990 T^{10} + 6582952005840035281 T^{12} \)
$41$ \( 1 - 63 T + 4739 T^{2} - 215208 T^{3} + 9562553 T^{4} - 413399385 T^{5} + 16368671414 T^{6} - 694924366185 T^{7} + 27021489327833 T^{8} - 1022260433497128 T^{9} + 37840560660804419 T^{10} - 845627536539601263 T^{11} + 22563490300366186081 T^{12} \)
$43$ \( 1 + 34 T - 3475 T^{2} - 31338 T^{3} + 9673978 T^{4} - 28877782 T^{5} - 22052754731 T^{6} - 53395018918 T^{7} + 33073405660378 T^{8} - 198098875229562 T^{9} - 40616495964663475 T^{10} + 734790398651664466 T^{11} + 39959630797262576401 T^{12} \)
$47$ \( 1 - 58 T - 3429 T^{2} + 99066 T^{3} + 16840372 T^{4} - 217034296 T^{5} - 36013746767 T^{6} - 479428759864 T^{7} + 82175643281332 T^{8} + 1067853745782714 T^{9} - 81648901963178469 T^{10} - 3050749669678142842 T^{11} + \)\(11\!\cdots\!41\)\( T^{12} \)
$53$ \( 1 + 12 T + 7659 T^{2} + 91332 T^{3} + 36866070 T^{4} + 510480060 T^{5} + 120117765391 T^{6} + 1433938488540 T^{7} + 290891024879670 T^{8} + 2024315430633828 T^{9} + 476846968860613899 T^{10} + 2098649644386156588 T^{11} + \)\(49\!\cdots\!41\)\( T^{12} \)
$59$ \( 1 + 147 T + 16901 T^{2} + 1425606 T^{3} + 106851335 T^{4} + 7433603271 T^{5} + 456697800146 T^{6} + 25876372986351 T^{7} + 1294756199526935 T^{8} + 60132821841811446 T^{9} + 2481581225950629221 T^{10} + 75134162735194285947 T^{11} + \)\(17\!\cdots\!81\)\( T^{12} \)
$61$ \( 1 - 58 T - 5011 T^{2} + 428958 T^{3} + 12450718 T^{4} - 1018447028 T^{5} + 4642755289 T^{6} - 3789641391188 T^{7} + 172390661763838 T^{8} + 22100076745145838 T^{9} - 960645345429375091 T^{10} - 41373888876447190858 T^{11} + \)\(26\!\cdots\!21\)\( T^{12} \)
$67$ \( 1 - 201 T + 28649 T^{2} - 3051582 T^{3} + 282765251 T^{4} - 23237371929 T^{5} + 1659244019666 T^{6} - 104312562589281 T^{7} + 5698036787496371 T^{8} - 276041170776041358 T^{9} + 11633432894320208009 T^{10} - \)\(36\!\cdots\!49\)\( T^{11} + \)\(81\!\cdots\!61\)\( T^{12} \)
$71$ \( 1 + 102 T + 19395 T^{2} + 1624554 T^{3} + 208411794 T^{4} + 13817724054 T^{5} + 1288263668227 T^{6} + 69655146956214 T^{7} + 5296094025765714 T^{8} + 208105828644996234 T^{9} + 12524389738511534595 T^{10} + \)\(33\!\cdots\!02\)\( T^{11} + \)\(16\!\cdots\!41\)\( T^{12} \)
$73$ \( 1 - 7 T - 12713 T^{2} + 22960 T^{3} + 94466917 T^{4} + 36786071 T^{5} - 563497579498 T^{6} + 196032972359 T^{7} + 2682694275492997 T^{8} + 3474633835595440 T^{9} - 10252527148249451753 T^{10} - 30083380807924903543 T^{11} + \)\(22\!\cdots\!21\)\( T^{12} \)
$79$ \( 1 + 12035 T^{2} + 69730790 T^{4} + 4384013520 T^{5} + 462010537511 T^{6} + 27360628378320 T^{7} + 2716019918693990 T^{8} + 18258404527225461635 T^{10} + \)\(59\!\cdots\!41\)\( T^{12} \)
$83$ \( ( 1 - 73 T + 15082 T^{2} - 608183 T^{3} + 103899898 T^{4} - 3464457433 T^{5} + 326940373369 T^{6} )^{2} \)
$89$ \( 1 + 72 T + 9723 T^{2} + 575640 T^{3} + 40073838 T^{4} + 8635069188 T^{5} + 306038573143 T^{6} + 68398383038148 T^{7} + 2514322401590958 T^{8} + 286082310328790040 T^{9} + 38275452957841333563 T^{10} + \)\(22\!\cdots\!72\)\( T^{11} + \)\(24\!\cdots\!21\)\( T^{12} \)
$97$ \( 1 - 21 T + 12695 T^{2} - 263508 T^{3} + 43921757 T^{4} - 6267660039 T^{5} - 57670086154 T^{6} - 58972413306951 T^{7} + 3888361567466717 T^{8} - 219494787074830932 T^{9} + 99496219480615519895 T^{10} - \)\(15\!\cdots\!29\)\( T^{11} + \)\(69\!\cdots\!41\)\( T^{12} \)
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