Properties

Label 786.2.a.k
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

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

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 −4.00000 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
\(2\) \(-1\)
\(3\) \(1\)
\(131\) \(1\)

Hecke kernels

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

\( T_{5} - 4 \)
\( T_{7} + 4 \)
\( T_{17} + 2 \)