Properties

Label 6930.2.a.bp
Level $6930$
Weight $2$
Character orbit 6930.a
Self dual yes
Analytic conductor $55.336$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{13}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{5} + q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} - q^{5} + q^{7} - q^{8} + q^{10} + q^{11} + ( 3 + \beta ) q^{13} - q^{14} + q^{16} + ( -1 - \beta ) q^{17} + 2 q^{19} - q^{20} - q^{22} + ( -1 - \beta ) q^{23} + q^{25} + ( -3 - \beta ) q^{26} + q^{28} -6 q^{29} -4 q^{31} - q^{32} + ( 1 + \beta ) q^{34} - q^{35} + ( -3 + \beta ) q^{37} -2 q^{38} + q^{40} + ( -4 + 2 \beta ) q^{41} -2 \beta q^{43} + q^{44} + ( 1 + \beta ) q^{46} + ( -4 + 2 \beta ) q^{47} + q^{49} - q^{50} + ( 3 + \beta ) q^{52} + ( -4 + 2 \beta ) q^{53} - q^{55} - q^{56} + 6 q^{58} + ( -6 - 2 \beta ) q^{61} + 4 q^{62} + q^{64} + ( -3 - \beta ) q^{65} + ( -1 - 3 \beta ) q^{67} + ( -1 - \beta ) q^{68} + q^{70} + ( 2 - 4 \beta ) q^{71} -10 q^{73} + ( 3 - \beta ) q^{74} + 2 q^{76} + q^{77} + ( 6 - 2 \beta ) q^{79} - q^{80} + ( 4 - 2 \beta ) q^{82} + ( 2 + 2 \beta ) q^{83} + ( 1 + \beta ) q^{85} + 2 \beta q^{86} - q^{88} + ( -5 + \beta ) q^{89} + ( 3 + \beta ) q^{91} + ( -1 - \beta ) q^{92} + ( 4 - 2 \beta ) q^{94} -2 q^{95} + ( 12 - 2 \beta ) q^{97} - q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 2 q^{10} + 2 q^{11} + 6 q^{13} - 2 q^{14} + 2 q^{16} - 2 q^{17} + 4 q^{19} - 2 q^{20} - 2 q^{22} - 2 q^{23} + 2 q^{25} - 6 q^{26} + 2 q^{28} - 12 q^{29} - 8 q^{31} - 2 q^{32} + 2 q^{34} - 2 q^{35} - 6 q^{37} - 4 q^{38} + 2 q^{40} - 8 q^{41} + 2 q^{44} + 2 q^{46} - 8 q^{47} + 2 q^{49} - 2 q^{50} + 6 q^{52} - 8 q^{53} - 2 q^{55} - 2 q^{56} + 12 q^{58} - 12 q^{61} + 8 q^{62} + 2 q^{64} - 6 q^{65} - 2 q^{67} - 2 q^{68} + 2 q^{70} + 4 q^{71} - 20 q^{73} + 6 q^{74} + 4 q^{76} + 2 q^{77} + 12 q^{79} - 2 q^{80} + 8 q^{82} + 4 q^{83} + 2 q^{85} - 2 q^{88} - 10 q^{89} + 6 q^{91} - 2 q^{92} + 8 q^{94} - 4 q^{95} + 24 q^{97} - 2 q^{98} + O(q^{100}) \)

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
−1.30278
2.30278
−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.bp 2
3.b odd 2 1 6930.2.a.cc yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6930.2.a.bp 2 1.a even 1 1 trivial
6930.2.a.cc yes 2 3.b odd 2 1

Hecke kernels

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

\( T_{13}^{2} - 6 T_{13} - 4 \)
\( T_{17}^{2} + 2 T_{17} - 12 \)
\( T_{19} - 2 \)
\( T_{23}^{2} + 2 T_{23} - 12 \)
\( T_{29} + 6 \)
\( T_{31} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( -4 - 6 T + T^{2} \)
$17$ \( -12 + 2 T + T^{2} \)
$19$ \( ( -2 + T )^{2} \)
$23$ \( -12 + 2 T + T^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( -4 + 6 T + T^{2} \)
$41$ \( -36 + 8 T + T^{2} \)
$43$ \( -52 + T^{2} \)
$47$ \( -36 + 8 T + T^{2} \)
$53$ \( -36 + 8 T + T^{2} \)
$59$ \( T^{2} \)
$61$ \( -16 + 12 T + T^{2} \)
$67$ \( -116 + 2 T + T^{2} \)
$71$ \( -204 - 4 T + T^{2} \)
$73$ \( ( 10 + T )^{2} \)
$79$ \( -16 - 12 T + T^{2} \)
$83$ \( -48 - 4 T + T^{2} \)
$89$ \( 12 + 10 T + T^{2} \)
$97$ \( 92 - 24 T + T^{2} \)
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