Properties

Label 354.5.b.a
Level 354
Weight 5
Character orbit 354.b
Analytic conductor 36.593
Analytic rank 0
Dimension 76
CM No

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Newspace parameters

Level: \( N \) = \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 354.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(36.5929669317\)
Analytic rank: \(0\)
Dimension: \(76\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 76q - 608q^{4} - 64q^{6} - 184q^{7} + 168q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 76q - 608q^{4} - 64q^{6} - 184q^{7} + 168q^{9} + 256q^{10} - 200q^{13} - 26q^{15} + 4864q^{16} - 512q^{18} + 616q^{19} + 330q^{21} + 640q^{22} + 512q^{24} - 10540q^{25} - 354q^{27} + 1472q^{28} - 832q^{30} - 3920q^{31} - 188q^{33} + 2560q^{34} - 1344q^{36} - 1440q^{37} + 8204q^{39} - 2048q^{40} - 5760q^{42} - 1944q^{43} + 3886q^{45} + 4864q^{46} + 33636q^{49} - 7544q^{51} + 1600q^{52} + 3392q^{54} - 10536q^{55} - 12182q^{57} - 7168q^{58} + 208q^{60} + 6360q^{61} + 10860q^{63} - 38912q^{64} + 19712q^{66} + 30744q^{67} - 34208q^{69} - 23808q^{70} + 4096q^{72} + 4032q^{73} + 22324q^{75} - 4928q^{76} + 12864q^{78} - 29824q^{79} - 22584q^{81} + 13184q^{82} - 2640q^{84} + 9240q^{85} + 32850q^{87} - 5120q^{88} - 16448q^{90} - 31160q^{91} - 1780q^{93} + 5248q^{94} - 4096q^{96} + 77504q^{97} - 15412q^{99} + O(q^{100}) \)

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 2.82843i −8.93212 + 1.10332i −8.00000 5.83517i 3.12065 + 25.2638i −18.4051 22.6274i 78.5654 19.7099i 16.5043
119.2 2.82843i −8.91789 + 1.21294i −8.00000 12.8667i 3.43072 + 25.2236i −25.0597 22.6274i 78.0575 21.6338i 36.3926
119.3 2.82843i −8.83171 + 1.73229i −8.00000 40.6238i 4.89965 + 24.9799i 62.0657 22.6274i 74.9983 30.5982i −114.901
119.4 2.82843i −8.68666 + 2.35412i −8.00000 2.07514i 6.65846 + 24.5696i −53.2290 22.6274i 69.9162 40.8989i −5.86940
119.5 2.82843i −8.09468 3.93398i −8.00000 40.1568i −11.1270 + 22.8952i −61.8003 22.6274i 50.0476 + 63.6886i −113.581
119.6 2.82843i −8.08417 + 3.95552i −8.00000 31.0925i 11.1879 + 22.8655i 88.3750 22.6274i 49.7077 63.9542i 87.9429
119.7 2.82843i −7.91044 4.29243i −8.00000 19.2338i −12.1408 + 22.3741i 28.5043 22.6274i 44.1501 + 67.9100i −54.4013
119.8 2.82843i −7.33533 5.21468i −8.00000 49.1667i −14.7493 + 20.7475i −32.7313 22.6274i 26.6142 + 76.5028i 139.065
119.9 2.82843i −7.27743 5.29519i −8.00000 30.6811i −14.9771 + 20.5837i 36.2554 22.6274i 24.9218 + 77.0708i 86.7794
119.10 2.82843i −6.71583 + 5.99146i −8.00000 40.1609i 16.9464 + 18.9952i 33.0388 22.6274i 9.20482 80.4753i −113.592
119.11 2.82843i −6.39595 + 6.33181i −8.00000 30.3892i 17.9091 + 18.0905i −23.0970 22.6274i 0.816430 80.9959i 85.9536
119.12 2.82843i −4.28434 7.91482i −8.00000 3.69126i −22.3865 + 12.1179i 38.4283 22.6274i −44.2889 + 67.8196i 10.4405
119.13 2.82843i −4.23958 7.93889i −8.00000 15.5384i −22.4546 + 11.9913i −95.5047 22.6274i −45.0520 + 67.3151i 43.9493
119.14 2.82843i −4.23649 + 7.94054i −8.00000 25.1711i 22.4592 + 11.9826i −73.8012 22.6274i −45.1043 67.2801i −71.1945
119.15 2.82843i −3.88189 + 8.11979i −8.00000 43.7610i 22.9662 + 10.9796i −67.6729 22.6274i −50.8619 63.0402i 123.775
119.16 2.82843i −3.50299 + 8.29030i −8.00000 4.07079i 23.4485 + 9.90795i 46.9846 22.6274i −56.4581 58.0817i 11.5139
119.17 2.82843i −3.47377 8.30259i −8.00000 12.2691i −23.4833 + 9.82529i −13.7545 22.6274i −56.8659 + 57.6825i −34.7024
119.18 2.82843i −1.31589 8.90328i −8.00000 31.9146i −25.1823 + 3.72191i 95.4804 22.6274i −77.5369 + 23.4315i −90.2682
119.19 2.82843i 0.730836 + 8.97028i −8.00000 13.2882i 25.3718 2.06712i 42.9270 22.6274i −79.9318 + 13.1116i 37.5847
119.20 2.82843i 0.769993 8.96700i −8.00000 26.9261i −25.3625 2.17787i 20.9184 22.6274i −79.8142 13.8091i 76.1586
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 119.76
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 \(S_{5}^{\mathrm{new}}(354, [\chi])\).