Properties

Label 354.3.b.a
Level 354
Weight 3
Character orbit 354.b
Analytic conductor 9.646
Analytic rank 0
Dimension 40
CM No

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Newspace parameters

Level: \( N \) = \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 354.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(9.64580135835\)
Analytic rank: \(0\)
Dimension: \(40\)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40q - 80q^{4} + 8q^{6} + 8q^{7} - 24q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 40q - 80q^{4} + 8q^{6} + 8q^{7} - 24q^{9} - 16q^{10} + 34q^{15} + 160q^{16} + 16q^{18} + 24q^{19} - 18q^{21} - 16q^{22} - 16q^{24} - 216q^{25} - 30q^{27} - 16q^{28} - 64q^{30} + 96q^{31} + 76q^{33} + 80q^{34} + 48q^{36} - 200q^{37} - 28q^{39} + 32q^{40} + 48q^{42} - 104q^{43} + 58q^{45} + 32q^{46} + 288q^{49} - 176q^{51} - 40q^{54} + 360q^{55} + 214q^{57} - 128q^{58} - 68q^{60} - 32q^{61} - 132q^{63} - 320q^{64} - 112q^{66} - 344q^{67} + 88q^{69} + 192q^{70} - 32q^{72} + 40q^{73} + 28q^{75} - 48q^{76} + 96q^{78} + 32q^{79} + 336q^{81} - 80q^{82} + 36q^{84} + 168q^{85} - 162q^{87} + 32q^{88} + 112q^{90} + 88q^{91} - 316q^{93} - 400q^{94} + 32q^{96} - 184q^{97} - 148q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
119.1 1.41421i −2.99974 + 0.0395926i −2.00000 8.98648i 0.0559924 + 4.24227i 1.46160 2.82843i 8.99686 0.237535i 12.7088
119.2 1.41421i −2.88248 + 0.831458i −2.00000 4.87722i 1.17586 + 4.07644i 5.25551 2.82843i 7.61735 4.79332i −6.89743
119.3 1.41421i −2.71554 1.27509i −2.00000 1.78565i −1.80326 + 3.84035i −11.5483 2.82843i 5.74827 + 6.92513i 2.52529
119.4 1.41421i −2.67506 1.35797i −2.00000 1.30422i −1.92046 + 3.78310i 6.99884 2.82843i 5.31185 + 7.26528i −1.84445
119.5 1.41421i −2.37497 + 1.83290i −2.00000 0.546180i 2.59211 + 3.35872i 0.308312 2.82843i 2.28097 8.70616i −0.772415
119.6 1.41421i −1.48935 + 2.60419i −2.00000 4.30678i 3.68289 + 2.10626i −7.55737 2.82843i −4.56365 7.75713i 6.09070
119.7 1.41421i −1.23071 2.73594i −2.00000 5.41438i −3.86920 + 1.74048i 0.567729 2.82843i −5.97072 + 6.73428i −7.65709
119.8 1.41421i −0.865873 2.87233i −2.00000 4.52476i −4.06208 + 1.22453i 8.90636 2.82843i −7.50053 + 4.97414i 6.39898
119.9 1.41421i −0.495685 + 2.95877i −2.00000 8.96910i 4.18433 + 0.701004i −9.25455 2.82843i −8.50859 2.93323i −12.6842
119.10 1.41421i −0.0610517 2.99938i −2.00000 5.31992i −4.24176 + 0.0863401i −5.79959 2.82843i −8.99255 + 0.366234i −7.52350
119.11 1.41421i −0.0106251 + 2.99998i −2.00000 7.50817i 4.24261 + 0.0150262i 12.9231 2.82843i −8.99977 0.0637503i −10.6182
119.12 1.41421i 0.306491 + 2.98430i −2.00000 9.44498i 4.22044 0.433443i 10.5753 2.82843i −8.81213 + 1.82932i 13.3572
119.13 1.41421i 1.10775 2.78799i −2.00000 5.86001i −3.94282 1.56659i −9.12163 2.82843i −6.54579 6.17678i 8.28731
119.14 1.41421i 1.45977 + 2.62089i −2.00000 1.67024i 3.70650 2.06443i −2.10379 2.82843i −4.73812 + 7.65181i 2.36208
119.15 1.41421i 1.89028 + 2.32956i −2.00000 1.71502i 3.29449 2.67326i −3.39354 2.82843i −1.85367 + 8.80704i −2.42540
119.16 1.41421i 1.91195 2.31181i −2.00000 3.94988i −3.26939 2.70390i 4.74743 2.82843i −1.68893 8.84011i 5.58597
119.17 1.41421i 2.49176 1.67067i −2.00000 7.29367i −2.36269 3.52388i 6.28110 2.82843i 3.41771 8.32582i −10.3148
119.18 1.41421i 2.68322 + 1.34177i −2.00000 2.14890i 1.89755 3.79464i 6.97234 2.82843i 5.39932 + 7.20051i −3.03900
119.19 1.41421i 2.95715 + 0.505227i −2.00000 6.09077i 0.714499 4.18204i −12.9107 2.82843i 8.48949 + 2.98806i −8.61364
119.20 1.41421i 2.99271 0.209029i −2.00000 5.00191i −0.295611 4.23233i 0.691870 2.82843i 8.91261 1.25112i 7.07377
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 119.40
Significant digits:
Format:

Inner twists

This newform does not have CM; other inner twists have not been computed.

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(354, [\chi])\).