Properties

Label 6930.2.a.bh
Level $6930$
Weight $2$
Character orbit 6930.a
Self dual yes
Analytic conductor $55.336$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6930.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.3363286007\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{5} + q^{7} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} + q^{5} + q^{7} + q^{8} + q^{10} - q^{11} - 2 q^{13} + q^{14} + q^{16} + 2 q^{17} - 4 q^{19} + q^{20} - q^{22} + 2 q^{23} + q^{25} - 2 q^{26} + q^{28} - 6 q^{29} + 2 q^{31} + q^{32} + 2 q^{34} + q^{35} + 10 q^{37} - 4 q^{38} + q^{40} + 8 q^{41} + 4 q^{43} - q^{44} + 2 q^{46} - 4 q^{47} + q^{49} + q^{50} - 2 q^{52} + 2 q^{53} - q^{55} + q^{56} - 6 q^{58} + 12 q^{59} + 10 q^{61} + 2 q^{62} + q^{64} - 2 q^{65} - 4 q^{67} + 2 q^{68} + q^{70} + 8 q^{71} + 14 q^{73} + 10 q^{74} - 4 q^{76} - q^{77} - 2 q^{79} + q^{80} + 8 q^{82} + 2 q^{83} + 2 q^{85} + 4 q^{86} - q^{88} + 10 q^{89} - 2 q^{91} + 2 q^{92} - 4 q^{94} - 4 q^{95} - 8 q^{97} + q^{98} + O(q^{100}) \)

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 0 1.00000 1.00000 0 1.00000 1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)
\(7\) \(-1\)
\(11\) \(1\)

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 6930.2.a.bh yes 1
3.b odd 2 1 6930.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6930.2.a.e 1 3.b odd 2 1
6930.2.a.bh yes 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6930))\):

\( T_{13} + 2 \)
\( T_{17} - 2 \)
\( T_{19} + 4 \)
\( T_{23} - 2 \)
\( T_{29} + 6 \)
\( T_{31} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( -1 + T \)
$7$ \( -1 + T \)
$11$ \( 1 + T \)
$13$ \( 2 + T \)
$17$ \( -2 + T \)
$19$ \( 4 + T \)
$23$ \( -2 + T \)
$29$ \( 6 + T \)
$31$ \( -2 + T \)
$37$ \( -10 + T \)
$41$ \( -8 + T \)
$43$ \( -4 + T \)
$47$ \( 4 + T \)
$53$ \( -2 + T \)
$59$ \( -12 + T \)
$61$ \( -10 + T \)
$67$ \( 4 + T \)
$71$ \( -8 + T \)
$73$ \( -14 + T \)
$79$ \( 2 + T \)
$83$ \( -2 + T \)
$89$ \( -10 + T \)
$97$ \( 8 + T \)
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