Properties

Label 354.8.a.a
Level 354
Weight 8
Character orbit 354.a
Self dual Yes
Analytic conductor 110.584
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 354.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(110.584299021\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 8q^{2} + 27q^{3} + 64q^{4} - 320q^{5} + 216q^{6} - 505q^{7} + 512q^{8} + 729q^{9} + O(q^{10}) \) \( q + 8q^{2} + 27q^{3} + 64q^{4} - 320q^{5} + 216q^{6} - 505q^{7} + 512q^{8} + 729q^{9} - 2560q^{10} + 2559q^{11} + 1728q^{12} + 7037q^{13} - 4040q^{14} - 8640q^{15} + 4096q^{16} - 551q^{17} + 5832q^{18} - 22144q^{19} - 20480q^{20} - 13635q^{21} + 20472q^{22} - 35966q^{23} + 13824q^{24} + 24275q^{25} + 56296q^{26} + 19683q^{27} - 32320q^{28} + 102278q^{29} - 69120q^{30} - 91872q^{31} + 32768q^{32} + 69093q^{33} - 4408q^{34} + 161600q^{35} + 46656q^{36} + 250189q^{37} - 177152q^{38} + 189999q^{39} - 163840q^{40} - 418053q^{41} - 109080q^{42} - 40225q^{43} + 163776q^{44} - 233280q^{45} - 287728q^{46} + 25656q^{47} + 110592q^{48} - 568518q^{49} + 194200q^{50} - 14877q^{51} + 450368q^{52} - 1681588q^{53} + 157464q^{54} - 818880q^{55} - 258560q^{56} - 597888q^{57} + 818224q^{58} + 205379q^{59} - 552960q^{60} - 1857650q^{61} - 734976q^{62} - 368145q^{63} + 262144q^{64} - 2251840q^{65} + 552744q^{66} + 3843284q^{67} - 35264q^{68} - 971082q^{69} + 1292800q^{70} - 3104607q^{71} + 373248q^{72} - 4328392q^{73} + 2001512q^{74} + 655425q^{75} - 1417216q^{76} - 1292295q^{77} + 1519992q^{78} - 4679059q^{79} - 1310720q^{80} + 531441q^{81} - 3344424q^{82} - 8910253q^{83} - 872640q^{84} + 176320q^{85} - 321800q^{86} + 2761506q^{87} + 1310208q^{88} + 785984q^{89} - 1866240q^{90} - 3553685q^{91} - 2301824q^{92} - 2480544q^{93} + 205248q^{94} + 7086080q^{95} + 884736q^{96} - 9841396q^{97} - 4548144q^{98} + 1865511q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
8.00000 27.0000 64.0000 −320.000 216.000 −505.000 512.000 729.000 −2560.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(59\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{5} + 320 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(354))\).