Properties

Label 354.2.a.d
Level 354
Weight 2
Character orbit 354.a
Self dual Yes
Analytic conductor 2.827
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: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 354.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(2.82670423155\)
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

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - 4q^{5} - q^{6} - q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} - 4q^{5} - q^{6} - q^{7} + q^{8} + q^{9} - 4q^{10} - 3q^{11} - q^{12} - q^{13} - q^{14} + 4q^{15} + q^{16} - 7q^{17} + q^{18} - 4q^{19} - 4q^{20} + q^{21} - 3q^{22} + 2q^{23} - q^{24} + 11q^{25} - q^{26} - q^{27} - q^{28} - 2q^{29} + 4q^{30} + q^{32} + 3q^{33} - 7q^{34} + 4q^{35} + q^{36} + 7q^{37} - 4q^{38} + q^{39} - 4q^{40} + 3q^{41} + q^{42} + 5q^{43} - 3q^{44} - 4q^{45} + 2q^{46} + 12q^{47} - q^{48} - 6q^{49} + 11q^{50} + 7q^{51} - q^{52} - 8q^{53} - q^{54} + 12q^{55} - q^{56} + 4q^{57} - 2q^{58} + q^{59} + 4q^{60} - 14q^{61} - q^{63} + q^{64} + 4q^{65} + 3q^{66} - 4q^{67} - 7q^{68} - 2q^{69} + 4q^{70} - 15q^{71} + q^{72} - 4q^{73} + 7q^{74} - 11q^{75} - 4q^{76} + 3q^{77} + q^{78} + 5q^{79} - 4q^{80} + q^{81} + 3q^{82} + q^{83} + q^{84} + 28q^{85} + 5q^{86} + 2q^{87} - 3q^{88} + 4q^{89} - 4q^{90} + q^{91} + 2q^{92} + 12q^{94} + 16q^{95} - q^{96} - 4q^{97} - 6q^{98} - 3q^{99} + O(q^{100}) \)

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
1.00000 −1.00000 1.00000 −4.00000 −1.00000 −1.00000 1.00000 1.00000 −4.00000
\(n\): 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 intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(354))\):

\( T_{5} + 4 \)
\( T_{7} + 1 \)
\( T_{11} + 3 \)