Properties

Label 6930.2.a.w
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: no (minimal twist has level 2310)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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.w 1
3.b odd 2 1 2310.2.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2310.2.a.k 1 3.b odd 2 1
6930.2.a.w 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6930))\):

\( T_{13} - 2 \)
\( T_{17} - 2 \)
\( T_{19} + 8 \)
\( T_{23} - 8 \)
\( T_{29} - 6 \)
\( T_{31} \)

Hecke characteristic polynomials

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