Properties

Label 786.2.a.j
Level 786
Weight 2
Character orbit 786.a
Self dual Yes
Analytic conductor 6.276
Analytic rank 1
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: \(1\)
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

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - 3q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - 3q^{7} + q^{8} + q^{9} - q^{10} + q^{11} - q^{12} - 2q^{13} - 3q^{14} + q^{15} + q^{16} - 5q^{17} + q^{18} - 4q^{19} - q^{20} + 3q^{21} + q^{22} + 3q^{23} - q^{24} - 4q^{25} - 2q^{26} - q^{27} - 3q^{28} - 10q^{29} + q^{30} - 2q^{31} + q^{32} - q^{33} - 5q^{34} + 3q^{35} + q^{36} + 11q^{37} - 4q^{38} + 2q^{39} - q^{40} - 8q^{41} + 3q^{42} - 4q^{43} + q^{44} - q^{45} + 3q^{46} - q^{47} - q^{48} + 2q^{49} - 4q^{50} + 5q^{51} - 2q^{52} - 5q^{53} - q^{54} - q^{55} - 3q^{56} + 4q^{57} - 10q^{58} + 9q^{59} + q^{60} - 6q^{61} - 2q^{62} - 3q^{63} + q^{64} + 2q^{65} - q^{66} + 15q^{67} - 5q^{68} - 3q^{69} + 3q^{70} - q^{71} + q^{72} + 10q^{73} + 11q^{74} + 4q^{75} - 4q^{76} - 3q^{77} + 2q^{78} + 16q^{79} - q^{80} + q^{81} - 8q^{82} - 6q^{83} + 3q^{84} + 5q^{85} - 4q^{86} + 10q^{87} + q^{88} - 14q^{89} - q^{90} + 6q^{91} + 3q^{92} + 2q^{93} - q^{94} + 4q^{95} - q^{96} + 8q^{97} + 2q^{98} + q^{99} + O(q^{100}) \)

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(786))\):

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