Properties

Label 354.6.a.a
Level 354
Weight 6
Character orbit 354.a
Self dual Yes
Analytic conductor 56.776
Analytic rank 0
Dimension 1
CM No
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 354.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(56.7758722138\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q + 4q^{2} - 9q^{3} + 16q^{4} + 10q^{5} - 36q^{6} + 144q^{7} + 64q^{8} + 81q^{9} + O(q^{10}) \) \( q + 4q^{2} - 9q^{3} + 16q^{4} + 10q^{5} - 36q^{6} + 144q^{7} + 64q^{8} + 81q^{9} + 40q^{10} + 668q^{11} - 144q^{12} - 270q^{13} + 576q^{14} - 90q^{15} + 256q^{16} - 758q^{17} + 324q^{18} + 868q^{19} + 160q^{20} - 1296q^{21} + 2672q^{22} + 784q^{23} - 576q^{24} - 3025q^{25} - 1080q^{26} - 729q^{27} + 2304q^{28} - 4574q^{29} - 360q^{30} + 8948q^{31} + 1024q^{32} - 6012q^{33} - 3032q^{34} + 1440q^{35} + 1296q^{36} - 670q^{37} + 3472q^{38} + 2430q^{39} + 640q^{40} - 7934q^{41} - 5184q^{42} + 4884q^{43} + 10688q^{44} + 810q^{45} + 3136q^{46} + 24280q^{47} - 2304q^{48} + 3929q^{49} - 12100q^{50} + 6822q^{51} - 4320q^{52} + 28962q^{53} - 2916q^{54} + 6680q^{55} + 9216q^{56} - 7812q^{57} - 18296q^{58} - 3481q^{59} - 1440q^{60} + 30490q^{61} + 35792q^{62} + 11664q^{63} + 4096q^{64} - 2700q^{65} - 24048q^{66} - 30764q^{67} - 12128q^{68} - 7056q^{69} + 5760q^{70} + 22452q^{71} + 5184q^{72} - 20966q^{73} - 2680q^{74} + 27225q^{75} + 13888q^{76} + 96192q^{77} + 9720q^{78} + 70520q^{79} + 2560q^{80} + 6561q^{81} - 31736q^{82} + 29756q^{83} - 20736q^{84} - 7580q^{85} + 19536q^{86} + 41166q^{87} + 42752q^{88} - 16470q^{89} + 3240q^{90} - 38880q^{91} + 12544q^{92} - 80532q^{93} + 97120q^{94} + 8680q^{95} - 9216q^{96} + 18506q^{97} + 15716q^{98} + 54108q^{99} + O(q^{100}) \)

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
4.00000 −9.00000 16.0000 10.0000 −36.0000 144.000 64.0000 81.0000 40.0000
\(n\): 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 \( T_{5} - 10 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(354))\).