Properties

Label 126.2.e.c
Level $126$
Weight $2$
Character orbit 126.e
Analytic conductor $1.006$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 126.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.00611506547\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
Defining polynomial: \(x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - \nu^{4} + 5 \nu^{3} + \nu^{2} + 6 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 5 \nu^{3} + 2 \nu^{2} - 3 \nu \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 6 \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{5} - 5 \nu^{4} + 19 \nu^{3} - 22 \nu^{2} + 33 \nu - 9 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + \beta_{1} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(2 \beta_{5} + 3 \beta_{4} - 5 \beta_{3} - 3 \beta_{2} - 6 \beta_{1} + 6\)
\(\nu^{5}\)\(=\)\(-3 \beta_{5} - 2 \beta_{4} - 11 \beta_{3} - 6 \beta_{2} + 8 \beta_{1} + 7\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\)
\(\chi(n)\) \(-1 - \beta_{4}\) \(-1 - \beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
0.500000 0.224437i
0.500000 2.05195i
0.500000 + 1.41036i
0.500000 + 0.224437i
0.500000 + 2.05195i
0.500000 1.41036i
−1.00000 −1.64400 0.545231i 1.00000 −0.794182 + 1.37556i 1.64400 + 0.545231i 1.23855 + 2.33795i −1.00000 2.40545 + 1.79272i 0.794182 1.37556i
25.2 −1.00000 0.933463 1.45899i 1.00000 −0.296790 + 0.514055i −0.933463 + 1.45899i 2.32383 1.26483i −1.00000 −1.25729 2.72382i 0.296790 0.514055i
25.3 −1.00000 1.71053 + 0.272169i 1.00000 1.59097 2.75564i −1.71053 0.272169i −2.56238 + 0.658939i −1.00000 2.85185 + 0.931107i −1.59097 + 2.75564i
121.1 −1.00000 −1.64400 + 0.545231i 1.00000 −0.794182 1.37556i 1.64400 0.545231i 1.23855 2.33795i −1.00000 2.40545 1.79272i 0.794182 + 1.37556i
121.2 −1.00000 0.933463 + 1.45899i 1.00000 −0.296790 0.514055i −0.933463 1.45899i 2.32383 + 1.26483i −1.00000 −1.25729 + 2.72382i 0.296790 + 0.514055i
121.3 −1.00000 1.71053 0.272169i 1.00000 1.59097 + 2.75564i −1.71053 + 0.272169i −2.56238 0.658939i −1.00000 2.85185 0.931107i −1.59097 2.75564i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.2.e.c 6
3.b odd 2 1 378.2.e.d 6
4.b odd 2 1 1008.2.q.g 6
7.b odd 2 1 882.2.e.o 6
7.c even 3 1 126.2.h.d yes 6
7.c even 3 1 882.2.f.n 6
7.d odd 6 1 882.2.f.o 6
7.d odd 6 1 882.2.h.p 6
9.c even 3 1 126.2.h.d yes 6
9.c even 3 1 1134.2.g.m 6
9.d odd 6 1 378.2.h.c 6
9.d odd 6 1 1134.2.g.l 6
12.b even 2 1 3024.2.q.g 6
21.c even 2 1 2646.2.e.p 6
21.g even 6 1 2646.2.f.m 6
21.g even 6 1 2646.2.h.o 6
21.h odd 6 1 378.2.h.c 6
21.h odd 6 1 2646.2.f.l 6
28.g odd 6 1 1008.2.t.h 6
36.f odd 6 1 1008.2.t.h 6
36.h even 6 1 3024.2.t.h 6
63.g even 3 1 882.2.f.n 6
63.g even 3 1 1134.2.g.m 6
63.h even 3 1 inner 126.2.e.c 6
63.h even 3 1 7938.2.a.bv 3
63.i even 6 1 2646.2.e.p 6
63.i even 6 1 7938.2.a.bz 3
63.j odd 6 1 378.2.e.d 6
63.j odd 6 1 7938.2.a.ca 3
63.k odd 6 1 882.2.f.o 6
63.l odd 6 1 882.2.h.p 6
63.n odd 6 1 1134.2.g.l 6
63.n odd 6 1 2646.2.f.l 6
63.o even 6 1 2646.2.h.o 6
63.s even 6 1 2646.2.f.m 6
63.t odd 6 1 882.2.e.o 6
63.t odd 6 1 7938.2.a.bw 3
84.n even 6 1 3024.2.t.h 6
252.u odd 6 1 1008.2.q.g 6
252.bb even 6 1 3024.2.q.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.c 6 1.a even 1 1 trivial
126.2.e.c 6 63.h even 3 1 inner
126.2.h.d yes 6 7.c even 3 1
126.2.h.d yes 6 9.c even 3 1
378.2.e.d 6 3.b odd 2 1
378.2.e.d 6 63.j odd 6 1
378.2.h.c 6 9.d odd 6 1
378.2.h.c 6 21.h odd 6 1
882.2.e.o 6 7.b odd 2 1
882.2.e.o 6 63.t odd 6 1
882.2.f.n 6 7.c even 3 1
882.2.f.n 6 63.g even 3 1
882.2.f.o 6 7.d odd 6 1
882.2.f.o 6 63.k odd 6 1
882.2.h.p 6 7.d odd 6 1
882.2.h.p 6 63.l odd 6 1
1008.2.q.g 6 4.b odd 2 1
1008.2.q.g 6 252.u odd 6 1
1008.2.t.h 6 28.g odd 6 1
1008.2.t.h 6 36.f odd 6 1
1134.2.g.l 6 9.d odd 6 1
1134.2.g.l 6 63.n odd 6 1
1134.2.g.m 6 9.c even 3 1
1134.2.g.m 6 63.g even 3 1
2646.2.e.p 6 21.c even 2 1
2646.2.e.p 6 63.i even 6 1
2646.2.f.l 6 21.h odd 6 1
2646.2.f.l 6 63.n odd 6 1
2646.2.f.m 6 21.g even 6 1
2646.2.f.m 6 63.s even 6 1
2646.2.h.o 6 21.g even 6 1
2646.2.h.o 6 63.o even 6 1
3024.2.q.g 6 12.b even 2 1
3024.2.q.g 6 252.bb even 6 1
3024.2.t.h 6 36.h even 6 1
3024.2.t.h 6 84.n even 6 1
7938.2.a.bv 3 63.h even 3 1
7938.2.a.bw 3 63.t odd 6 1
7938.2.a.bz 3 63.i even 6 1
7938.2.a.ca 3 63.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - T_{5}^{5} + 7 T_{5}^{4} + 12 T_{5}^{3} + 33 T_{5}^{2} + 18 T_{5} + 9 \) acting on \(S_{2}^{\mathrm{new}}(126, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{6} \)
$3$ \( 1 - 2 T - 2 T^{2} + 9 T^{3} - 6 T^{4} - 18 T^{5} + 27 T^{6} \)
$5$ \( 1 - T - 8 T^{2} + 17 T^{3} + 23 T^{4} - 52 T^{5} - 11 T^{6} - 260 T^{7} + 575 T^{8} + 2125 T^{9} - 5000 T^{10} - 3125 T^{11} + 15625 T^{12} \)
$7$ \( 1 - 2 T - 4 T^{2} + 31 T^{3} - 28 T^{4} - 98 T^{5} + 343 T^{6} \)
$11$ \( 1 + T - 26 T^{2} - 23 T^{3} + 407 T^{4} + 202 T^{5} - 4853 T^{6} + 2222 T^{7} + 49247 T^{8} - 30613 T^{9} - 380666 T^{10} + 161051 T^{11} + 1771561 T^{12} \)
$13$ \( 1 - 8 T + 24 T^{2} - 42 T^{3} - 32 T^{4} + 1408 T^{5} - 7901 T^{6} + 18304 T^{7} - 5408 T^{8} - 92274 T^{9} + 685464 T^{10} - 2970344 T^{11} + 4826809 T^{12} \)
$17$ \( 1 + 4 T - 23 T^{2} - 68 T^{3} + 410 T^{4} + 220 T^{5} - 8111 T^{6} + 3740 T^{7} + 118490 T^{8} - 334084 T^{9} - 1920983 T^{10} + 5679428 T^{11} + 24137569 T^{12} \)
$19$ \( 1 + 3 T - 12 T^{2} - 67 T^{3} - 153 T^{4} + 54 T^{5} + 6315 T^{6} + 1026 T^{7} - 55233 T^{8} - 459553 T^{9} - 1563852 T^{10} + 7428297 T^{11} + 47045881 T^{12} \)
$23$ \( 1 + 7 T - 32 T^{2} - 83 T^{3} + 2423 T^{4} + 3946 T^{5} - 46865 T^{6} + 90758 T^{7} + 1281767 T^{8} - 1009861 T^{9} - 8954912 T^{10} + 45054401 T^{11} + 148035889 T^{12} \)
$29$ \( 1 + 5 T + 4 T^{2} + 251 T^{3} + 197 T^{4} - 3418 T^{5} + 20293 T^{6} - 99122 T^{7} + 165677 T^{8} + 6121639 T^{9} + 2829124 T^{10} + 102555745 T^{11} + 594823321 T^{12} \)
$31$ \( ( 1 + 20 T + 214 T^{2} + 1441 T^{3} + 6634 T^{4} + 19220 T^{5} + 29791 T^{6} )^{2} \)
$37$ \( ( 1 - 11 T + 37 T^{2} )^{3}( 1 + 10 T + 37 T^{2} )^{3} \)
$41$ \( 1 - 90 T^{2} + 18 T^{3} + 4410 T^{4} - 810 T^{5} - 194177 T^{6} - 33210 T^{7} + 7413210 T^{8} + 1240578 T^{9} - 254318490 T^{10} + 4750104241 T^{12} \)
$43$ \( ( 1 - 12 T - 6 T^{2} + 547 T^{3} - 258 T^{4} - 22188 T^{5} + 79507 T^{6} )( 1 + 18 T + 198 T^{2} + 1519 T^{3} + 8514 T^{4} + 33282 T^{5} + 79507 T^{6} ) \)
$47$ \( ( 1 - 9 T + 87 T^{2} - 657 T^{3} + 4089 T^{4} - 19881 T^{5} + 103823 T^{6} )^{2} \)
$53$ \( 1 - 15 T - 33 T^{3} + 13635 T^{4} - 60360 T^{5} - 225155 T^{6} - 3199080 T^{7} + 38300715 T^{8} - 4912941 T^{9} - 6272932395 T^{11} + 22164361129 T^{12} \)
$59$ \( ( 1 - 14 T + 216 T^{2} - 1589 T^{3} + 12744 T^{4} - 48734 T^{5} + 205379 T^{6} )^{2} \)
$61$ \( ( 1 + 8 T + 178 T^{2} + 883 T^{3} + 10858 T^{4} + 29768 T^{5} + 226981 T^{6} )^{2} \)
$67$ \( ( 1 + T + 89 T^{2} - 77 T^{3} + 5963 T^{4} + 4489 T^{5} + 300763 T^{6} )^{2} \)
$71$ \( ( 1 - 7 T + 15 T^{2} + 599 T^{3} + 1065 T^{4} - 35287 T^{5} + 357911 T^{6} )^{2} \)
$73$ \( 1 - 19 T + 134 T^{2} - 27 T^{3} - 5759 T^{4} + 41986 T^{5} - 314903 T^{6} + 3064978 T^{7} - 30689711 T^{8} - 10503459 T^{9} + 3805364294 T^{10} - 39388360267 T^{11} + 151334226289 T^{12} \)
$79$ \( ( 1 + 5 T + 163 T^{2} + 469 T^{3} + 12877 T^{4} + 31205 T^{5} + 493039 T^{6} )^{2} \)
$83$ \( 1 - 2 T - 182 T^{2} - 2 T^{3} + 18788 T^{4} + 13564 T^{5} - 1721225 T^{6} + 1125812 T^{7} + 129430532 T^{8} - 1143574 T^{9} - 8637414422 T^{10} - 7878081286 T^{11} + 326940373369 T^{12} \)
$89$ \( 1 + 9 T - 144 T^{2} - 1197 T^{3} + 16101 T^{4} + 73314 T^{5} - 1141967 T^{6} + 6524946 T^{7} + 127536021 T^{8} - 843847893 T^{9} - 9034882704 T^{10} + 50256535041 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 - 28 T + 281 T^{2} - 2724 T^{3} + 45178 T^{4} - 388196 T^{5} + 2169217 T^{6} - 37655012 T^{7} + 425079802 T^{8} - 2486121252 T^{9} + 24876727961 T^{10} - 240445527196 T^{11} + 832972004929 T^{12} \)
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