Properties

Label 361.2.e.g
Level $361$
Weight $2$
Character orbit 361.e
Analytic conductor $2.883$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 361 = 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 361.e (of order \(9\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.88259951297\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/361\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\zeta_{18}^{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
−0.766044 + 0.642788i
0.939693 0.342020i
−0.173648 + 0.984808i
−0.173648 0.984808i
0.939693 + 0.342020i
−0.766044 0.642788i
0.152704 + 0.866025i 0.407604 0.342020i 1.15270 0.419550i 2.37939 + 0.866025i 0.358441 + 0.300767i 0.939693 1.62760i 1.41875 + 2.45734i −0.471782 + 2.67561i −0.386659 + 2.19285i
54.1 −1.03209 0.866025i 2.70574 0.984808i −0.0320889 0.181985i 0.152704 0.866025i −3.64543 1.32683i −0.173648 0.300767i −1.47178 + 2.54920i 4.05303 3.40090i −0.907604 + 0.761570i
62.1 2.37939 0.866025i −0.113341 + 0.642788i 3.37939 2.83564i −1.03209 0.866025i 0.286989 + 1.62760i −0.766044 1.32683i 3.05303 5.28801i 2.41875 + 0.880352i −3.20574 1.16679i
99.1 2.37939 + 0.866025i −0.113341 0.642788i 3.37939 + 2.83564i −1.03209 + 0.866025i 0.286989 1.62760i −0.766044 + 1.32683i 3.05303 + 5.28801i 2.41875 0.880352i −3.20574 + 1.16679i
234.1 −1.03209 + 0.866025i 2.70574 + 0.984808i −0.0320889 + 0.181985i 0.152704 + 0.866025i −3.64543 + 1.32683i −0.173648 + 0.300767i −1.47178 2.54920i 4.05303 + 3.40090i −0.907604 0.761570i
245.1 0.152704 0.866025i 0.407604 + 0.342020i 1.15270 + 0.419550i 2.37939 0.866025i 0.358441 0.300767i 0.939693 + 1.62760i 1.41875 2.45734i −0.471782 2.67561i −0.386659 2.19285i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 245.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 361.2.e.g 6
19.b odd 2 1 361.2.e.a 6
19.c even 3 1 19.2.e.a 6
19.c even 3 1 361.2.e.f 6
19.d odd 6 1 361.2.e.b 6
19.d odd 6 1 361.2.e.h 6
19.e even 9 1 19.2.e.a 6
19.e even 9 1 361.2.a.g 3
19.e even 9 2 361.2.c.i 6
19.e even 9 1 361.2.e.f 6
19.e even 9 1 inner 361.2.e.g 6
19.f odd 18 1 361.2.a.h 3
19.f odd 18 2 361.2.c.h 6
19.f odd 18 1 361.2.e.a 6
19.f odd 18 1 361.2.e.b 6
19.f odd 18 1 361.2.e.h 6
57.h odd 6 1 171.2.u.c 6
57.j even 18 1 3249.2.a.s 3
57.l odd 18 1 171.2.u.c 6
57.l odd 18 1 3249.2.a.z 3
76.g odd 6 1 304.2.u.b 6
76.k even 18 1 5776.2.a.bi 3
76.l odd 18 1 304.2.u.b 6
76.l odd 18 1 5776.2.a.br 3
95.i even 6 1 475.2.l.a 6
95.m odd 12 2 475.2.u.a 12
95.o odd 18 1 9025.2.a.x 3
95.p even 18 1 475.2.l.a 6
95.p even 18 1 9025.2.a.bd 3
95.q odd 36 2 475.2.u.a 12
133.g even 3 1 931.2.v.b 6
133.h even 3 1 931.2.x.a 6
133.k odd 6 1 931.2.v.a 6
133.m odd 6 1 931.2.w.a 6
133.t odd 6 1 931.2.x.b 6
133.u even 9 1 931.2.x.a 6
133.w even 9 1 931.2.v.b 6
133.x odd 18 1 931.2.x.b 6
133.y odd 18 1 931.2.w.a 6
133.z odd 18 1 931.2.v.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.e.a 6 19.c even 3 1
19.2.e.a 6 19.e even 9 1
171.2.u.c 6 57.h odd 6 1
171.2.u.c 6 57.l odd 18 1
304.2.u.b 6 76.g odd 6 1
304.2.u.b 6 76.l odd 18 1
361.2.a.g 3 19.e even 9 1
361.2.a.h 3 19.f odd 18 1
361.2.c.h 6 19.f odd 18 2
361.2.c.i 6 19.e even 9 2
361.2.e.a 6 19.b odd 2 1
361.2.e.a 6 19.f odd 18 1
361.2.e.b 6 19.d odd 6 1
361.2.e.b 6 19.f odd 18 1
361.2.e.f 6 19.c even 3 1
361.2.e.f 6 19.e even 9 1
361.2.e.g 6 1.a even 1 1 trivial
361.2.e.g 6 19.e even 9 1 inner
361.2.e.h 6 19.d odd 6 1
361.2.e.h 6 19.f odd 18 1
475.2.l.a 6 95.i even 6 1
475.2.l.a 6 95.p even 18 1
475.2.u.a 12 95.m odd 12 2
475.2.u.a 12 95.q odd 36 2
931.2.v.a 6 133.k odd 6 1
931.2.v.a 6 133.z odd 18 1
931.2.v.b 6 133.g even 3 1
931.2.v.b 6 133.w even 9 1
931.2.w.a 6 133.m odd 6 1
931.2.w.a 6 133.y odd 18 1
931.2.x.a 6 133.h even 3 1
931.2.x.a 6 133.u even 9 1
931.2.x.b 6 133.t odd 6 1
931.2.x.b 6 133.x odd 18 1
3249.2.a.s 3 57.j even 18 1
3249.2.a.z 3 57.l odd 18 1
5776.2.a.bi 3 76.k even 18 1
5776.2.a.br 3 76.l odd 18 1
9025.2.a.x 3 95.o odd 18 1
9025.2.a.bd 3 95.p even 18 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(361, [\chi])\):

\( T_{2}^{6} - 3T_{2}^{5} + 3T_{2}^{3} + 9T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{3}^{6} - 6T_{3}^{5} + 12T_{3}^{4} - 8T_{3}^{3} + 6T_{3}^{2} - 3T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 3 T^{5} + 3 T^{3} + 9 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{6} - 6 T^{5} + 12 T^{4} - 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} - 3 T^{5} + 3 T^{3} + 9 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{6} + 9 T^{4} - 18 T^{3} + 81 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{6} + 12 T^{5} + 69 T^{4} + \cdots + 1369 \) Copy content Toggle raw display
$17$ \( T^{6} + 6 T^{5} + 9 T^{4} + 3 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 12 T^{5} + 72 T^{4} + \cdots + 576 \) Copy content Toggle raw display
$29$ \( T^{6} + 3 T^{5} - 9 T^{4} + \cdots + 12321 \) Copy content Toggle raw display
$31$ \( T^{6} - 9 T^{5} + 75 T^{4} + \cdots + 2809 \) Copy content Toggle raw display
$37$ \( (T^{3} - 21 T - 17)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 15 T^{5} + 198 T^{4} + \cdots + 12321 \) Copy content Toggle raw display
$43$ \( T^{6} + 21 T^{5} + 204 T^{4} + \cdots + 26569 \) Copy content Toggle raw display
$47$ \( T^{6} + 12 T^{5} + 63 T^{4} + 51 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$53$ \( T^{6} - 15 T^{5} + 90 T^{4} + \cdots + 2601 \) Copy content Toggle raw display
$59$ \( T^{6} + 6 T^{5} + 108 T^{4} + \cdots + 71289 \) Copy content Toggle raw display
$61$ \( T^{6} + 12 T^{5} + 132 T^{4} + \cdots + 32761 \) Copy content Toggle raw display
$67$ \( T^{6} - 24 T^{5} + 456 T^{4} + \cdots + 179776 \) Copy content Toggle raw display
$71$ \( T^{6} - 12 T^{5} - 672 T^{3} + \cdots + 788544 \) Copy content Toggle raw display
$73$ \( T^{6} + 12 T^{5} + 96 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$79$ \( T^{6} - 15 T^{5} + 87 T^{4} + \cdots + 654481 \) Copy content Toggle raw display
$83$ \( T^{6} + 189 T^{4} + 918 T^{3} + \cdots + 210681 \) Copy content Toggle raw display
$89$ \( T^{6} - 15 T^{5} + 36 T^{4} + \cdots + 3249 \) Copy content Toggle raw display
$97$ \( T^{6} - 54 T^{4} + 413 T^{3} + \cdots + 16129 \) Copy content Toggle raw display
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