Properties

Label 361.2.e.h
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} + 1) q^{2} + (\zeta_{18}^{5} - \zeta_{18}^{3} - \zeta_{18} + 1) q^{3} + ( - \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + \zeta_{18} + 1) q^{4} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{2} - 1) q^{5} + (2 \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{3} - 2 \zeta_{18}^{2}) q^{6} + (\zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{2}) q^{7} + (\zeta_{18}^{5} - 3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + \zeta_{18}) q^{8} + (2 \zeta_{18}^{4} - 3 \zeta_{18}^{3} - 3 \zeta_{18} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} + 1) q^{2} + (\zeta_{18}^{5} - \zeta_{18}^{3} - \zeta_{18} + 1) q^{3} + ( - \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + \zeta_{18} + 1) q^{4} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{2} - 1) q^{5} + (2 \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{3} - 2 \zeta_{18}^{2}) q^{6} + (\zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{2}) q^{7} + (\zeta_{18}^{5} - 3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + \zeta_{18}) q^{8} + (2 \zeta_{18}^{4} - 3 \zeta_{18}^{3} - 3 \zeta_{18} + 2) q^{9} + ( - \zeta_{18}^{4} - \zeta_{18}^{3} - \zeta_{18} - 1) q^{10} + (2 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{2} + 2 \zeta_{18}) q^{11} + (\zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18} - 1) q^{12} + (\zeta_{18}^{4} - 3 \zeta_{18}^{3} + 2 \zeta_{18} + 2) q^{13} + (\zeta_{18}^{5} + 2 \zeta_{18}^{4} + \zeta_{18}^{3}) q^{14} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} - 1) q^{15} + ( - 3 \zeta_{18}^{2} - \zeta_{18} - 3) q^{16} + ( - \zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} + 2 \zeta_{18} + 1) q^{17} + (4 \zeta_{18}^{5} - 4 \zeta_{18}^{2} - 4 \zeta_{18} + 1) q^{18} + (\zeta_{18}^{5} + 2 \zeta_{18}^{4} - 3 \zeta_{18}^{2} - 3 \zeta_{18} - 1) q^{20} + ( - \zeta_{18}^{4} + \zeta_{18}) q^{21} + 3 \zeta_{18} q^{22} + (2 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}) q^{23} + (4 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - \zeta_{18}^{2} + 1) q^{24} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} + \zeta_{18} + 1) q^{25} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} - 5 \zeta_{18}^{3} + \zeta_{18} + 5) q^{26} + (3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 3 \zeta_{18}^{2}) q^{27} + (2 \zeta_{18}^{3} + 3 \zeta_{18}^{2} + 2 \zeta_{18}) q^{28} + (\zeta_{18}^{4} - \zeta_{18}^{3} + 5 \zeta_{18}^{2} - \zeta_{18} + 1) q^{29} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{2} - \zeta_{18}) q^{30} + (\zeta_{18}^{5} + \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 2 \zeta_{18} - 3) q^{31} + (3 \zeta_{18}^{3} - 3 \zeta_{18} - 3) q^{32} + (2 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} + 1) q^{33} + ( - 3 \zeta_{18}^{5} - 4 \zeta_{18}^{4} - 4 \zeta_{18}^{3} + 2 \zeta_{18} + 2) q^{34} + ( - 2 \zeta_{18}^{5} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - \zeta_{18}) q^{35} + (5 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 3 \zeta_{18} - 5) q^{36} + ( - 3 \zeta_{18}^{5} + \zeta_{18}^{4} + 2 \zeta_{18}^{2} + 2 \zeta_{18}) q^{37} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} - 4) q^{39} + (\zeta_{18}^{5} + 6 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 6 \zeta_{18} - 1) q^{40} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} - 3 \zeta_{18}^{2} - 4 \zeta_{18} - 4) q^{41} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}) q^{42} + ( - 3 \zeta_{18}^{5} - 5 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2) q^{43} + ( - 3 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + \zeta_{18}^{3} + 3 \zeta_{18}^{2} + \zeta_{18} + 1) q^{44} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + 5 \zeta_{18}^{3} + \zeta_{18}^{2} - 5) q^{45} + (2 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 6 \zeta_{18}^{3} + 2 \zeta_{18}^{2} + 2 \zeta_{18}) q^{46} + ( - 2 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 3 \zeta_{18} - 2) q^{47} + ( - 2 \zeta_{18}^{4} + 5 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 5 \zeta_{18} - 2) q^{48} + (\zeta_{18}^{5} - \zeta_{18}^{4} + 5 \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18}) q^{49} + ( - 2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 5 \zeta_{18}^{3} + 2 \zeta_{18}^{2} + 5) q^{50} + (\zeta_{18}^{5} + \zeta_{18}^{3} - \zeta_{18}^{2} - \zeta_{18} - 1) q^{51} + ( - 4 \zeta_{18}^{5} - 6 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{52} + ( - 2 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 2 \zeta_{18} - 1) q^{53} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{3} + 4 \zeta_{18}^{2} + \zeta_{18} + 6) q^{54} + ( - 3 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 3 \zeta_{18} + 3) q^{55} + ( - 2 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 5 \zeta_{18}^{2} + 5 \zeta_{18} + 1) q^{56} + (\zeta_{18}^{5} - 5 \zeta_{18}^{4} + 4 \zeta_{18}^{2} + 4 \zeta_{18} + 6) q^{58} + (2 \zeta_{18}^{5} - 7 \zeta_{18}^{4} + 7 \zeta_{18} - 2) q^{59} + (\zeta_{18}^{2} + 2 \zeta_{18} + 1) q^{60} + ( - 3 \zeta_{18}^{5} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{3} - 4) q^{61} + ( - \zeta_{18}^{5} + 7 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 3) q^{62} + (\zeta_{18}^{5} - \zeta_{18}^{4} - 2 \zeta_{18}^{3} - \zeta_{18}^{2} + 3 \zeta_{18} + 3) q^{63} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 4 \zeta_{18}^{3} + 3 \zeta_{18} - 4) q^{64} + ( - 5 \zeta_{18}^{5} + 4 \zeta_{18}^{3} - 5 \zeta_{18}) q^{65} + ( - 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 3 \zeta_{18} - 3) q^{66} + (4 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 6 \zeta_{18}^{2} + 2 \zeta_{18} + 4) q^{67} + ( - 3 \zeta_{18}^{5} - 4 \zeta_{18}^{4} - 5 \zeta_{18}^{3} - 4 \zeta_{18}^{2} - 3 \zeta_{18}) q^{68} + ( - 4 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 2 \zeta_{18} + 4) q^{69} + ( - 3 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + \zeta_{18}^{3} + 3 \zeta_{18}^{2} + \zeta_{18} + 1) q^{70} + (10 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2) q^{71} + (13 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 3 \zeta_{18} + 1) q^{72} + (4 \zeta_{18}^{5} - 4 \zeta_{18}^{3} - 4 \zeta_{18}^{2}) q^{73} + ( - 5 \zeta_{18}^{5} - \zeta_{18}^{4} - 5 \zeta_{18}^{3} + 5 \zeta_{18}^{2} + \zeta_{18} + 5) q^{74} + ( - 4 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} - 5) q^{75} + (\zeta_{18}^{5} - 2 \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} - 3) q^{77} + (\zeta_{18}^{5} + 4 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 4 \zeta_{18} - 1) q^{78} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} + 7 \zeta_{18}^{2} - 3 \zeta_{18} + 6) q^{79} + (4 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 2 \zeta_{18} - 1) q^{80} + (\zeta_{18}^{5} + 5 \zeta_{18}^{4} - \zeta_{18}^{2} + 1) q^{81} + (6 \zeta_{18}^{5} + 7 \zeta_{18}^{4} + 4 \zeta_{18}^{3} - 6 \zeta_{18}^{2} - 11 \zeta_{18} - 11) q^{82} + (6 \zeta_{18}^{5} + 6 \zeta_{18}^{4} + 3 \zeta_{18}^{2} - 9 \zeta_{18}) q^{83} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} - \zeta_{18}^{2} - \zeta_{18}) q^{84} + (\zeta_{18}^{4} - 2 \zeta_{18}^{3} - 3 \zeta_{18}^{2} - 2 \zeta_{18} + 1) q^{85} + ( - 8 \zeta_{18}^{3} - 7 \zeta_{18}^{2} - 8 \zeta_{18}) q^{86} + ( - 6 \zeta_{18}^{5} + 7 \zeta_{18}^{4} - 7 \zeta_{18}^{3} + 7 \zeta_{18}^{2} - 6 \zeta_{18}) q^{87} + ( - 6 \zeta_{18}^{5} - 6 \zeta_{18}^{4} + 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} + 3 \zeta_{18} - 3) q^{88} + (5 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 5 \zeta_{18}^{2} + \zeta_{18} + 1) q^{89} + ( - \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 4 \zeta_{18}^{3} + 5 \zeta_{18}^{2} - 5) q^{90} + (5 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + 3 \zeta_{18} - 2) q^{91} + ( - 2 \zeta_{18}^{5} + 2 \zeta_{18}^{3} + 8 \zeta_{18}^{2} + 8 \zeta_{18} + 6) q^{92} + (2 \zeta_{18}^{5} - 8 \zeta_{18}^{4} + 7 \zeta_{18}^{3} - 7 \zeta_{18}^{2} + 8 \zeta_{18} - 2) q^{93} + ( - 4 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{2} + 2 \zeta_{18} - 3) q^{94} + 3 q^{96} + (2 \zeta_{18}^{5} + 5 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 5 \zeta_{18} - 2) q^{97} + ( - 5 \zeta_{18}^{5} + 5 \zeta_{18}^{3} + 4 \zeta_{18}^{2} + 6 \zeta_{18} - 1) q^{98} + (6 \zeta_{18}^{5} - 5 \zeta_{18}^{4} - 5 \zeta_{18}^{3} + \zeta_{18} + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 3 q^{3} - 6 q^{5} + 3 q^{6} - 6 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{2} + 3 q^{3} - 6 q^{5} + 3 q^{6} - 6 q^{8} + 3 q^{9} - 9 q^{10} - 3 q^{12} + 3 q^{13} + 3 q^{14} - 3 q^{15} - 18 q^{16} + 3 q^{17} + 6 q^{18} - 6 q^{20} + 6 q^{23} + 15 q^{24} + 15 q^{26} - 6 q^{27} + 6 q^{28} + 3 q^{29} - 9 q^{31} - 9 q^{32} + 9 q^{33} + 6 q^{35} - 24 q^{36} - 24 q^{39} - 21 q^{41} - 3 q^{42} - 3 q^{43} + 9 q^{44} - 15 q^{45} + 18 q^{46} - 3 q^{47} + 3 q^{48} + 15 q^{49} + 15 q^{50} - 3 q^{51} - 15 q^{52} + 3 q^{53} + 30 q^{54} + 18 q^{55} + 6 q^{56} + 36 q^{58} - 12 q^{59} + 6 q^{60} - 12 q^{61} - 12 q^{62} + 12 q^{63} - 12 q^{64} + 12 q^{65} - 9 q^{66} + 30 q^{67} - 15 q^{68} + 12 q^{69} + 9 q^{70} + 6 q^{71} + 12 q^{72} - 12 q^{73} + 15 q^{74} - 30 q^{75} - 18 q^{77} - 15 q^{78} + 39 q^{79} + 3 q^{80} + 6 q^{81} - 54 q^{82} - 3 q^{84} - 24 q^{86} - 21 q^{87} - 9 q^{88} + 12 q^{89} - 18 q^{90} - 15 q^{91} + 42 q^{92} + 9 q^{93} - 18 q^{94} + 18 q^{96} - 18 q^{97} + 9 q^{98} + 9 q^{99}+O(q^{100}) \) 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
0.233956 + 1.32683i 2.20574 1.85083i 0.173648 0.0632028i −0.826352 0.300767i 2.97178 + 2.49362i −0.173648 + 0.300767i 1.47178 + 2.54920i 0.918748 5.21048i 0.205737 1.16679i
54.1 1.93969 + 1.62760i −0.613341 + 0.223238i 0.766044 + 4.34445i −0.233956 + 1.32683i −1.55303 0.565258i −0.766044 1.32683i −3.05303 + 5.28801i −1.97178 + 1.65452i −2.61334 + 2.19285i
62.1 0.826352 0.300767i −0.0923963 + 0.524005i −0.939693 + 0.788496i −1.93969 1.62760i 0.0812519 + 0.460802i 0.939693 + 1.62760i −1.41875 + 2.45734i 2.55303 + 0.929228i −2.09240 0.761570i
99.1 0.826352 + 0.300767i −0.0923963 0.524005i −0.939693 0.788496i −1.93969 + 1.62760i 0.0812519 0.460802i 0.939693 1.62760i −1.41875 2.45734i 2.55303 0.929228i −2.09240 + 0.761570i
234.1 1.93969 1.62760i −0.613341 0.223238i 0.766044 4.34445i −0.233956 1.32683i −1.55303 + 0.565258i −0.766044 + 1.32683i −3.05303 5.28801i −1.97178 1.65452i −2.61334 2.19285i
245.1 0.233956 1.32683i 2.20574 + 1.85083i 0.173648 + 0.0632028i −0.826352 + 0.300767i 2.97178 2.49362i −0.173648 0.300767i 1.47178 2.54920i 0.918748 + 5.21048i 0.205737 + 1.16679i
\(n\): 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.h 6
19.b odd 2 1 19.2.e.a 6
19.c even 3 1 361.2.e.a 6
19.c even 3 1 361.2.e.b 6
19.d odd 6 1 361.2.e.f 6
19.d odd 6 1 361.2.e.g 6
19.e even 9 1 361.2.a.h 3
19.e even 9 2 361.2.c.h 6
19.e even 9 1 361.2.e.a 6
19.e even 9 1 361.2.e.b 6
19.e even 9 1 inner 361.2.e.h 6
19.f odd 18 1 19.2.e.a 6
19.f odd 18 1 361.2.a.g 3
19.f odd 18 2 361.2.c.i 6
19.f odd 18 1 361.2.e.f 6
19.f odd 18 1 361.2.e.g 6
57.d even 2 1 171.2.u.c 6
57.j even 18 1 171.2.u.c 6
57.j even 18 1 3249.2.a.z 3
57.l odd 18 1 3249.2.a.s 3
76.d even 2 1 304.2.u.b 6
76.k even 18 1 304.2.u.b 6
76.k even 18 1 5776.2.a.br 3
76.l odd 18 1 5776.2.a.bi 3
95.d odd 2 1 475.2.l.a 6
95.g even 4 2 475.2.u.a 12
95.o odd 18 1 475.2.l.a 6
95.o odd 18 1 9025.2.a.bd 3
95.p even 18 1 9025.2.a.x 3
95.r even 36 2 475.2.u.a 12
133.c even 2 1 931.2.w.a 6
133.o even 6 1 931.2.v.a 6
133.o even 6 1 931.2.x.b 6
133.r odd 6 1 931.2.v.b 6
133.r odd 6 1 931.2.x.a 6
133.ba even 18 1 931.2.w.a 6
133.bb even 18 1 931.2.x.b 6
133.bd odd 18 1 931.2.x.a 6
133.be odd 18 1 931.2.v.b 6
133.bf even 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.b odd 2 1
19.2.e.a 6 19.f odd 18 1
171.2.u.c 6 57.d even 2 1
171.2.u.c 6 57.j even 18 1
304.2.u.b 6 76.d even 2 1
304.2.u.b 6 76.k even 18 1
361.2.a.g 3 19.f odd 18 1
361.2.a.h 3 19.e even 9 1
361.2.c.h 6 19.e even 9 2
361.2.c.i 6 19.f odd 18 2
361.2.e.a 6 19.c even 3 1
361.2.e.a 6 19.e even 9 1
361.2.e.b 6 19.c even 3 1
361.2.e.b 6 19.e even 9 1
361.2.e.f 6 19.d odd 6 1
361.2.e.f 6 19.f odd 18 1
361.2.e.g 6 19.d odd 6 1
361.2.e.g 6 19.f odd 18 1
361.2.e.h 6 1.a even 1 1 trivial
361.2.e.h 6 19.e even 9 1 inner
475.2.l.a 6 95.d odd 2 1
475.2.l.a 6 95.o odd 18 1
475.2.u.a 12 95.g even 4 2
475.2.u.a 12 95.r even 36 2
931.2.v.a 6 133.o even 6 1
931.2.v.a 6 133.bf even 18 1
931.2.v.b 6 133.r odd 6 1
931.2.v.b 6 133.be odd 18 1
931.2.w.a 6 133.c even 2 1
931.2.w.a 6 133.ba even 18 1
931.2.x.a 6 133.r odd 6 1
931.2.x.a 6 133.bd odd 18 1
931.2.x.b 6 133.o even 6 1
931.2.x.b 6 133.bb even 18 1
3249.2.a.s 3 57.l odd 18 1
3249.2.a.z 3 57.j even 18 1
5776.2.a.bi 3 76.l odd 18 1
5776.2.a.br 3 76.k even 18 1
9025.2.a.x 3 95.p even 18 1
9025.2.a.bd 3 95.o odd 18 1

Hecke kernels

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

\( T_{2}^{6} - 6T_{2}^{5} + 18T_{2}^{4} - 30T_{2}^{3} + 36T_{2}^{2} - 27T_{2} + 9 \) Copy content Toggle raw display
\( T_{3}^{6} - 3T_{3}^{5} + 3T_{3}^{4} + 8T_{3}^{3} + 6T_{3}^{2} + 3T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 6 T^{5} + 18 T^{4} - 30 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{6} - 3 T^{5} + 3 T^{4} + 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} + 6 T^{5} + 18 T^{4} + 30 T^{3} + \cdots + 9 \) 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
$13$ \( T^{6} - 3 T^{5} + 24 T^{4} + \cdots + 1369 \) Copy content Toggle raw display
$17$ \( T^{6} - 3 T^{5} + 30 T^{3} + 36 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 6 T^{5} + 36 T^{4} - 192 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$29$ \( T^{6} - 3 T^{5} + 36 T^{4} + \cdots + 12321 \) 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
$41$ \( T^{6} + 21 T^{5} + 162 T^{4} + \cdots + 12321 \) Copy content Toggle raw display
$43$ \( T^{6} + 3 T^{5} + 60 T^{4} + \cdots + 26569 \) Copy content Toggle raw display
$47$ \( T^{6} + 3 T^{5} + 54 T^{4} + 24 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$53$ \( T^{6} - 3 T^{5} + 84 T^{3} + \cdots + 2601 \) Copy content Toggle raw display
$59$ \( T^{6} + 12 T^{5} + 18 T^{4} + \cdots + 71289 \) Copy content Toggle raw display
$61$ \( T^{6} + 12 T^{5} + 24 T^{4} + \cdots + 32761 \) Copy content Toggle raw display
$67$ \( T^{6} - 30 T^{5} + 348 T^{4} + \cdots + 179776 \) Copy content Toggle raw display
$71$ \( T^{6} - 6 T^{5} - 36 T^{4} + \cdots + 788544 \) Copy content Toggle raw display
$73$ \( T^{6} + 12 T^{5} + 96 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$79$ \( T^{6} - 39 T^{5} + 708 T^{4} + \cdots + 654481 \) Copy content Toggle raw display
$83$ \( T^{6} + 189 T^{4} + 918 T^{3} + \cdots + 210681 \) Copy content Toggle raw display
$89$ \( T^{6} - 12 T^{5} + 54 T^{4} + \cdots + 3249 \) Copy content Toggle raw display
$97$ \( T^{6} + 18 T^{5} + 234 T^{4} + \cdots + 16129 \) Copy content Toggle raw display
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