Properties

Label 678.2.a.e
Level 678
Weight 2
Character orbit 678.a
Self dual Yes
Analytic conductor 5.414
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 678 = 2 \cdot 3 \cdot 113 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 678.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(5.41385725704\)
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

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + q^{8} + q^{9} - q^{10} - 2q^{11} + q^{12} + 7q^{13} + q^{14} - q^{15} + q^{16} - 3q^{17} + q^{18} + 6q^{19} - q^{20} + q^{21} - 2q^{22} + 3q^{23} + q^{24} - 4q^{25} + 7q^{26} + q^{27} + q^{28} + 2q^{29} - q^{30} - 3q^{31} + q^{32} - 2q^{33} - 3q^{34} - q^{35} + q^{36} - 4q^{37} + 6q^{38} + 7q^{39} - q^{40} + q^{42} + 2q^{43} - 2q^{44} - q^{45} + 3q^{46} - 8q^{47} + q^{48} - 6q^{49} - 4q^{50} - 3q^{51} + 7q^{52} - 2q^{53} + q^{54} + 2q^{55} + q^{56} + 6q^{57} + 2q^{58} - 3q^{59} - q^{60} - q^{61} - 3q^{62} + q^{63} + q^{64} - 7q^{65} - 2q^{66} - 2q^{67} - 3q^{68} + 3q^{69} - q^{70} - 5q^{71} + q^{72} + 4q^{73} - 4q^{74} - 4q^{75} + 6q^{76} - 2q^{77} + 7q^{78} - 4q^{79} - q^{80} + q^{81} + q^{84} + 3q^{85} + 2q^{86} + 2q^{87} - 2q^{88} + 6q^{89} - q^{90} + 7q^{91} + 3q^{92} - 3q^{93} - 8q^{94} - 6q^{95} + q^{96} - 14q^{97} - 6q^{98} - 2q^{99} + O(q^{100}) \)

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 −1.00000 1.00000 1.00000 1.00000 1.00000 −1.00000
\(n\): 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
\(2\) \(-1\)
\(3\) \(-1\)
\(113\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(678))\):

\( T_{5} + 1 \)
\( T_{7} - 1 \)