Properties

Label 786.2.a.m
Level 786
Weight 2
Character orbit 786.a
Self dual Yes
Analytic conductor 6.276
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 786 = 2 \cdot 3 \cdot 131 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 786.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(6.27624159887\)
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

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + 3q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + 3q^{7} + q^{8} + q^{9} + q^{10} - 3q^{11} + q^{12} + 4q^{13} + 3q^{14} + q^{15} + q^{16} - 7q^{17} + q^{18} + q^{20} + 3q^{21} - 3q^{22} - q^{23} + q^{24} - 4q^{25} + 4q^{26} + q^{27} + 3q^{28} + q^{30} + 2q^{31} + q^{32} - 3q^{33} - 7q^{34} + 3q^{35} + q^{36} + 3q^{37} + 4q^{39} + q^{40} + 2q^{41} + 3q^{42} - 6q^{43} - 3q^{44} + q^{45} - q^{46} + 3q^{47} + q^{48} + 2q^{49} - 4q^{50} - 7q^{51} + 4q^{52} + 9q^{53} + q^{54} - 3q^{55} + 3q^{56} - 15q^{59} + q^{60} + 2q^{61} + 2q^{62} + 3q^{63} + q^{64} + 4q^{65} - 3q^{66} - 7q^{67} - 7q^{68} - q^{69} + 3q^{70} + 7q^{71} + q^{72} + 4q^{73} + 3q^{74} - 4q^{75} - 9q^{77} + 4q^{78} + 10q^{79} + q^{80} + q^{81} + 2q^{82} - 6q^{83} + 3q^{84} - 7q^{85} - 6q^{86} - 3q^{88} + q^{90} + 12q^{91} - q^{92} + 2q^{93} + 3q^{94} + q^{96} - 2q^{97} + 2q^{98} - 3q^{99} + O(q^{100}) \)

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 1.00000 1.00000 3.00000 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
\(2\) \(-1\)
\(3\) \(-1\)
\(131\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(786))\):

\( T_{5} - 1 \)
\( T_{7} - 3 \)
\( T_{17} + 7 \)