Properties

Label 786.2.a.a
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: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 786.2.a.a 1
3.b odd 2 1 2358.2.a.w 1
4.b odd 2 1 6288.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
786.2.a.a 1 1.a even 1 1 trivial
2358.2.a.w 1 3.b odd 2 1
6288.2.a.i 1 4.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(131\) \(1\)

Hecke kernels

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 1 + T \)
$5$ \( 1 + 2 T + 5 T^{2} \)
$7$ \( 1 - 2 T + 7 T^{2} \)
$11$ \( 1 + 3 T + 11 T^{2} \)
$13$ \( 1 - 3 T + 13 T^{2} \)
$17$ \( 1 + 5 T + 17 T^{2} \)
$19$ \( 1 - T + 19 T^{2} \)
$23$ \( 1 - 4 T + 23 T^{2} \)
$29$ \( 1 - 9 T + 29 T^{2} \)
$31$ \( 1 + 5 T + 31 T^{2} \)
$37$ \( 1 + 8 T + 37 T^{2} \)
$41$ \( 1 + 12 T + 41 T^{2} \)
$43$ \( 1 + 6 T + 43 T^{2} \)
$47$ \( 1 + 8 T + 47 T^{2} \)
$53$ \( 1 - 12 T + 53 T^{2} \)
$59$ \( 1 + 5 T + 59 T^{2} \)
$61$ \( 1 + 3 T + 61 T^{2} \)
$67$ \( 1 + 67 T^{2} \)
$71$ \( 1 + 8 T + 71 T^{2} \)
$73$ \( 1 + 2 T + 73 T^{2} \)
$79$ \( 1 + 8 T + 79 T^{2} \)
$83$ \( 1 + 14 T + 83 T^{2} \)
$89$ \( 1 + 14 T + 89 T^{2} \)
$97$ \( 1 - 12 T + 97 T^{2} \)
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