Properties

Label 786.2.a.g
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

Related objects

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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

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

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 −2.00000 −1.00000 −2.00000 −1.00000 1.00000 2.00000
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(786))\):

\( T_{5} + 2 \)
\( T_{7} + 2 \)
\( T_{17} - 7 \)