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

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

$p$ $F_p(T)$
$2$ \( 27 + 63 T + 40 T^{2} - 21 T^{3} - 4 T^{4} + 3 T^{5} + T^{6} \)
$3$ \( 2187 + 567 T - 194 T^{2} - 63 T^{3} + 20 T^{4} + 9 T^{5} + T^{6} \)
$5$ \( 15876 + 5544 T + 2188 T^{2} + 164 T^{3} + 48 T^{4} + 2 T^{5} + T^{6} \)
$7$ \( ( -94 - 78 T + T^{3} )^{2} \)
$11$ \( ( -3 - 83 T - 13 T^{2} + T^{3} )^{2} \)
$13$ \( 355008 - 28896 T - 9536 T^{2} + 840 T^{3} + 272 T^{4} - 30 T^{5} + T^{6} \)
$17$ \( 5817744 + 1360368 T + 216792 T^{2} + 18864 T^{3} + 1200 T^{4} + 42 T^{5} + T^{6} \)
$19$ \( 47045881 - 3258025 T + 370386 T^{2} - 16967 T^{3} + 1026 T^{4} - 25 T^{5} + T^{6} \)
$23$ \( 381924 - 67980 T + 17044 T^{2} - 356 T^{3} + 174 T^{4} - 8 T^{5} + T^{6} \)
$29$ \( 54289548 - 18887760 T + 2241448 T^{2} - 17760 T^{3} - 1432 T^{4} + 12 T^{5} + T^{6} \)
$31$ \( 2642351052 + 6412156 T^{2} + 4544 T^{4} + T^{6} \)
$37$ \( 38988 + 1967760 T^{2} + 3024 T^{4} + T^{6} \)
$41$ \( 498533643 - 62920971 T + 1834996 T^{2} + 102501 T^{3} - 304 T^{4} - 63 T^{5} + T^{6} \)
$43$ \( 981944896 + 28703776 T + 1904480 T^{2} + 31528 T^{3} + 2072 T^{4} + 34 T^{5} + T^{6} \)
$47$ \( 94361796 + 1612524 T + 590968 T^{2} - 29056 T^{3} + 3198 T^{4} - 58 T^{5} + T^{6} \)
$53$ \( 48771072 - 9870336 T + 714240 T^{2} - 9792 T^{3} - 768 T^{4} + 12 T^{5} + T^{6} \)
$59$ \( 1787690763 + 54558585 T - 3033392 T^{2} - 109515 T^{3} + 6458 T^{4} + 147 T^{5} + T^{6} \)
$61$ \( 661415524 + 71701784 T + 6281300 T^{2} + 213140 T^{3} + 6152 T^{4} - 58 T^{5} + T^{6} \)
$67$ \( 17294403 + 12353145 T + 2458624 T^{2} - 344715 T^{3} + 15182 T^{4} - 201 T^{5} + T^{6} \)
$71$ \( 1259712 + 1562976 T + 712512 T^{2} + 82008 T^{3} + 4272 T^{4} + 102 T^{5} + T^{6} \)
$73$ \( 340734681 - 59530275 T + 10529838 T^{2} - 14343 T^{3} + 3274 T^{4} - 7 T^{5} + T^{6} \)
$79$ \( 44231363328 - 2436251136 T + 44729344 T^{2} - 6688 T^{4} + T^{6} \)
$83$ \( ( 397611 - 5585 T - 73 T^{2} + T^{3} )^{2} \)
$89$ \( 84829321008 - 7954451424 T + 260737056 T^{2} - 1135296 T^{3} - 14040 T^{4} + 72 T^{5} + T^{6} \)
$97$ \( 21248211843 + 3958586883 T + 247598380 T^{2} + 329259 T^{3} - 15532 T^{4} - 21 T^{5} + T^{6} \)
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