Properties

Label 6930.2.a.br
Level $6930$
Weight $2$
Character orbit 6930.a
Self dual yes
Analytic conductor $55.336$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + q^{5} - q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + q^{5} - q^{7} - q^{8} - q^{10} + q^{11} + 2 q^{13} + q^{14} + q^{16} + ( -2 + 2 \beta ) q^{17} + \beta q^{19} + q^{20} - q^{22} -\beta q^{23} + q^{25} -2 q^{26} - q^{28} + ( -6 - \beta ) q^{29} + 2 \beta q^{31} - q^{32} + ( 2 - 2 \beta ) q^{34} - q^{35} + ( 6 + \beta ) q^{37} -\beta q^{38} - q^{40} + ( -2 - \beta ) q^{41} + ( -4 + 2 \beta ) q^{43} + q^{44} + \beta q^{46} + 8 q^{47} + q^{49} - q^{50} + 2 q^{52} + ( 2 - 3 \beta ) q^{53} + q^{55} + q^{56} + ( 6 + \beta ) q^{58} + 8 q^{59} + ( 2 + 4 \beta ) q^{61} -2 \beta q^{62} + q^{64} + 2 q^{65} + ( 4 - 4 \beta ) q^{67} + ( -2 + 2 \beta ) q^{68} + q^{70} -2 \beta q^{71} + ( 2 - 2 \beta ) q^{73} + ( -6 - \beta ) q^{74} + \beta q^{76} - q^{77} + ( -8 - \beta ) q^{79} + q^{80} + ( 2 + \beta ) q^{82} + ( 8 + 2 \beta ) q^{83} + ( -2 + 2 \beta ) q^{85} + ( 4 - 2 \beta ) q^{86} - q^{88} + ( 6 - 2 \beta ) q^{89} -2 q^{91} -\beta q^{92} -8 q^{94} + \beta q^{95} + ( 2 + \beta ) q^{97} - q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8} - 2 q^{10} + 2 q^{11} + 4 q^{13} + 2 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{20} - 2 q^{22} + 2 q^{25} - 4 q^{26} - 2 q^{28} - 12 q^{29} - 2 q^{32} + 4 q^{34} - 2 q^{35} + 12 q^{37} - 2 q^{40} - 4 q^{41} - 8 q^{43} + 2 q^{44} + 16 q^{47} + 2 q^{49} - 2 q^{50} + 4 q^{52} + 4 q^{53} + 2 q^{55} + 2 q^{56} + 12 q^{58} + 16 q^{59} + 4 q^{61} + 2 q^{64} + 4 q^{65} + 8 q^{67} - 4 q^{68} + 2 q^{70} + 4 q^{73} - 12 q^{74} - 2 q^{77} - 16 q^{79} + 2 q^{80} + 4 q^{82} + 16 q^{83} - 4 q^{85} + 8 q^{86} - 2 q^{88} + 12 q^{89} - 4 q^{91} - 16 q^{94} + 4 q^{97} - 2 q^{98} + O(q^{100}) \)

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
−1.41421
1.41421
−1.00000 0 1.00000 1.00000 0 −1.00000 −1.00000 0 −1.00000
1.2 −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.br 2
3.b odd 2 1 770.2.a.i 2
12.b even 2 1 6160.2.a.ba 2
15.d odd 2 1 3850.2.a.bi 2
15.e even 4 2 3850.2.c.u 4
21.c even 2 1 5390.2.a.bt 2
33.d even 2 1 8470.2.a.bo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.i 2 3.b odd 2 1
3850.2.a.bi 2 15.d odd 2 1
3850.2.c.u 4 15.e even 4 2
5390.2.a.bt 2 21.c even 2 1
6160.2.a.ba 2 12.b even 2 1
6930.2.a.br 2 1.a even 1 1 trivial
8470.2.a.bo 2 33.d even 2 1

Hecke kernels

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

\( T_{13} - 2 \)
\( T_{17}^{2} + 4 T_{17} - 28 \)
\( T_{19}^{2} - 8 \)
\( T_{23}^{2} - 8 \)
\( T_{29}^{2} + 12 T_{29} + 28 \)
\( T_{31}^{2} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( -28 + 4 T + T^{2} \)
$19$ \( -8 + T^{2} \)
$23$ \( -8 + T^{2} \)
$29$ \( 28 + 12 T + T^{2} \)
$31$ \( -32 + T^{2} \)
$37$ \( 28 - 12 T + T^{2} \)
$41$ \( -4 + 4 T + T^{2} \)
$43$ \( -16 + 8 T + T^{2} \)
$47$ \( ( -8 + T )^{2} \)
$53$ \( -68 - 4 T + T^{2} \)
$59$ \( ( -8 + T )^{2} \)
$61$ \( -124 - 4 T + T^{2} \)
$67$ \( -112 - 8 T + T^{2} \)
$71$ \( -32 + T^{2} \)
$73$ \( -28 - 4 T + T^{2} \)
$79$ \( 56 + 16 T + T^{2} \)
$83$ \( 32 - 16 T + T^{2} \)
$89$ \( 4 - 12 T + T^{2} \)
$97$ \( -4 - 4 T + T^{2} \)
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