Properties

Label 931.2.v.b
Level $931$
Weight $2$
Character orbit 931.v
Analytic conductor $7.434$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.v (of order \(9\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.43407242818\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/931\mathbb{Z}\right)^\times\).

\(n\) \(248\) \(344\)
\(\chi(n)\) \(-1 + \zeta_{18}^{3}\) \(-\zeta_{18}^{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1
0.939693 + 0.342020i
−0.173648 + 0.984808i
0.939693 0.342020i
−0.766044 + 0.642788i
−0.173648 0.984808i
−0.766044 0.642788i
−0.439693 2.49362i −0.113341 0.642788i −4.14543 + 1.50881i 1.26604 + 0.460802i −1.55303 + 0.565258i 0 3.05303 + 5.28801i 2.41875 0.880352i 0.592396 3.35965i
214.1 0.673648 + 0.565258i 0.407604 + 0.342020i −0.213011 1.20805i −0.439693 + 2.49362i 0.0812519 + 0.460802i 0 1.41875 2.45734i −0.471782 2.67561i −1.70574 + 1.43128i
263.1 −0.439693 + 2.49362i −0.113341 + 0.642788i −4.14543 1.50881i 1.26604 0.460802i −1.55303 0.565258i 0 3.05303 5.28801i 2.41875 + 0.880352i 0.592396 + 3.35965i
275.1 1.26604 + 0.460802i 2.70574 + 0.984808i −0.141559 0.118782i 0.673648 0.565258i 2.97178 + 2.49362i 0 −1.47178 2.54920i 4.05303 + 3.40090i 1.11334 0.405223i
422.1 0.673648 0.565258i 0.407604 0.342020i −0.213011 + 1.20805i −0.439693 2.49362i 0.0812519 0.460802i 0 1.41875 + 2.45734i −0.471782 + 2.67561i −1.70574 1.43128i
606.1 1.26604 0.460802i 2.70574 0.984808i −0.141559 + 0.118782i 0.673648 + 0.565258i 2.97178 2.49362i 0 −1.47178 + 2.54920i 4.05303 3.40090i 1.11334 + 0.405223i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 606.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
133.u even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.v.b 6
7.b odd 2 1 931.2.v.a 6
7.c even 3 1 19.2.e.a 6
7.c even 3 1 931.2.x.a 6
7.d odd 6 1 931.2.w.a 6
7.d odd 6 1 931.2.x.b 6
19.e even 9 1 931.2.x.a 6
21.h odd 6 1 171.2.u.c 6
28.g odd 6 1 304.2.u.b 6
35.j even 6 1 475.2.l.a 6
35.l odd 12 2 475.2.u.a 12
133.g even 3 1 361.2.e.g 6
133.h even 3 1 361.2.e.f 6
133.j odd 6 1 361.2.e.b 6
133.n odd 6 1 361.2.e.a 6
133.r odd 6 1 361.2.e.h 6
133.u even 9 1 361.2.a.g 3
133.u even 9 2 361.2.c.i 6
133.u even 9 1 inner 931.2.v.b 6
133.w even 9 1 19.2.e.a 6
133.w even 9 1 361.2.e.f 6
133.w even 9 1 361.2.e.g 6
133.x odd 18 1 931.2.v.a 6
133.y odd 18 1 931.2.x.b 6
133.z odd 18 1 931.2.w.a 6
133.bd odd 18 1 361.2.a.h 3
133.bd odd 18 2 361.2.c.h 6
133.be odd 18 1 361.2.e.a 6
133.be odd 18 1 361.2.e.b 6
133.be odd 18 1 361.2.e.h 6
399.bv even 18 1 3249.2.a.s 3
399.ca odd 18 1 171.2.u.c 6
399.ch odd 18 1 3249.2.a.z 3
532.bt odd 18 1 304.2.u.b 6
532.ce even 18 1 5776.2.a.bi 3
532.cf odd 18 1 5776.2.a.br 3
665.cl odd 18 1 9025.2.a.x 3
665.db even 18 1 9025.2.a.bd 3
665.dc even 18 1 475.2.l.a 6
665.dq odd 36 2 475.2.u.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.e.a 6 7.c even 3 1
19.2.e.a 6 133.w even 9 1
171.2.u.c 6 21.h odd 6 1
171.2.u.c 6 399.ca odd 18 1
304.2.u.b 6 28.g odd 6 1
304.2.u.b 6 532.bt odd 18 1
361.2.a.g 3 133.u even 9 1
361.2.a.h 3 133.bd odd 18 1
361.2.c.h 6 133.bd odd 18 2
361.2.c.i 6 133.u even 9 2
361.2.e.a 6 133.n odd 6 1
361.2.e.a 6 133.be odd 18 1
361.2.e.b 6 133.j odd 6 1
361.2.e.b 6 133.be odd 18 1
361.2.e.f 6 133.h even 3 1
361.2.e.f 6 133.w even 9 1
361.2.e.g 6 133.g even 3 1
361.2.e.g 6 133.w even 9 1
361.2.e.h 6 133.r odd 6 1
361.2.e.h 6 133.be odd 18 1
475.2.l.a 6 35.j even 6 1
475.2.l.a 6 665.dc even 18 1
475.2.u.a 12 35.l odd 12 2
475.2.u.a 12 665.dq odd 36 2
931.2.v.a 6 7.b odd 2 1
931.2.v.a 6 133.x odd 18 1
931.2.v.b 6 1.a even 1 1 trivial
931.2.v.b 6 133.u even 9 1 inner
931.2.w.a 6 7.d odd 6 1
931.2.w.a 6 133.z odd 18 1
931.2.x.a 6 7.c even 3 1
931.2.x.a 6 19.e even 9 1
931.2.x.b 6 7.d odd 6 1
931.2.x.b 6 133.y odd 18 1
3249.2.a.s 3 399.bv even 18 1
3249.2.a.z 3 399.ch odd 18 1
5776.2.a.bi 3 532.ce even 18 1
5776.2.a.br 3 532.cf odd 18 1
9025.2.a.x 3 665.cl odd 18 1
9025.2.a.bd 3 665.db even 18 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(931, [\chi])\):

\( T_{2}^{6} - 3 T_{2}^{5} + 9 T_{2}^{4} - 24 T_{2}^{3} + 36 T_{2}^{2} - 27 T_{2} + 9 \)
\( T_{3}^{6} - 6 T_{3}^{5} + 12 T_{3}^{4} - 8 T_{3}^{3} + 6 T_{3}^{2} - 3 T_{3} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 - 27 T + 36 T^{2} - 24 T^{3} + 9 T^{4} - 3 T^{5} + T^{6} \)
$3$ \( 1 - 3 T + 6 T^{2} - 8 T^{3} + 12 T^{4} - 6 T^{5} + T^{6} \)
$5$ \( 9 - 27 T + 36 T^{2} - 24 T^{3} + 9 T^{4} - 3 T^{5} + T^{6} \)
$7$ \( T^{6} \)
$11$ \( 81 - 81 T + 81 T^{2} - 18 T^{3} + 9 T^{4} + T^{6} \)
$13$ \( 1369 + 222 T - 114 T^{2} + 26 T^{3} + 24 T^{4} + 3 T^{5} + T^{6} \)
$17$ \( 9 - 27 T + 36 T^{2} + 3 T^{3} + 9 T^{4} + 6 T^{5} + T^{6} \)
$19$ \( 6859 - 2166 T - 228 T^{2} + 169 T^{3} - 12 T^{4} - 6 T^{5} + T^{6} \)
$23$ \( 576 + 864 T + 576 T^{2} + 240 T^{3} + 72 T^{4} + 12 T^{5} + T^{6} \)
$29$ \( 12321 - 1998 T - 477 T^{2} - 57 T^{3} + 36 T^{4} + 3 T^{5} + T^{6} \)
$31$ \( ( -53 + 6 T + 9 T^{2} + T^{3} )^{2} \)
$37$ \( 289 - 357 T + 441 T^{2} - 34 T^{3} + 21 T^{4} + T^{6} \)
$41$ \( 12321 - 8991 T + 3411 T^{2} - 672 T^{3} + 162 T^{4} - 21 T^{5} + T^{6} \)
$43$ \( 26569 + 5379 T - 663 T^{2} + 8 T^{3} + 60 T^{4} + 3 T^{5} + T^{6} \)
$47$ \( 9 + 27 T + 36 T^{2} + 51 T^{3} + 63 T^{4} + 12 T^{5} + T^{6} \)
$53$ \( 2601 + 3213 T + 1764 T^{2} + 537 T^{3} + 99 T^{4} + 12 T^{5} + T^{6} \)
$59$ \( 71289 - 14418 T - 1800 T^{2} + 57 T^{3} + 108 T^{4} + 6 T^{5} + T^{6} \)
$61$ \( 32761 + 23892 T + 7500 T^{2} + 1259 T^{3} + 132 T^{4} + 12 T^{5} + T^{6} \)
$67$ \( 179776 + 40704 T + 13296 T^{2} + 1448 T^{3} - 48 T^{4} - 6 T^{5} + T^{6} \)
$71$ \( 788544 - 31968 T + 8352 T^{2} - 1536 T^{3} - 36 T^{4} + 6 T^{5} + T^{6} \)
$73$ \( 4096 + 6144 T + 3072 T^{2} + 512 T^{3} + 96 T^{4} + 12 T^{5} + T^{6} \)
$79$ \( 654481 - 259689 T + 44382 T^{2} - 4528 T^{3} + 366 T^{4} - 24 T^{5} + T^{6} \)
$83$ \( 210681 + 86751 T + 35721 T^{2} + 918 T^{3} + 189 T^{4} + T^{6} \)
$89$ \( 3249 + 2052 T + 1035 T^{2} + 213 T^{3} + 36 T^{4} - 15 T^{5} + T^{6} \)
$97$ \( 16129 - 20574 T + 9522 T^{2} - 1855 T^{3} + 234 T^{4} - 18 T^{5} + T^{6} \)
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