Properties

Label 6930.2.a.bx
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: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
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 \(\beta = \sqrt{17}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - q^{5} - q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} - q^{5} - q^{7} + q^{8} - q^{10} - q^{11} + 2 q^{13} - q^{14} + q^{16} + ( -1 - \beta ) q^{17} + ( -3 - \beta ) q^{19} - q^{20} - q^{22} + ( 3 - \beta ) q^{23} + q^{25} + 2 q^{26} - q^{28} + 2 \beta q^{29} + ( -3 + \beta ) q^{31} + q^{32} + ( -1 - \beta ) q^{34} + q^{35} + ( 3 - \beta ) q^{37} + ( -3 - \beta ) q^{38} - q^{40} + ( 5 - \beta ) q^{41} + ( -1 + \beta ) q^{43} - q^{44} + ( 3 - \beta ) q^{46} + ( 1 + 3 \beta ) q^{47} + q^{49} + q^{50} + 2 q^{52} + 10 q^{53} + q^{55} - q^{56} + 2 \beta q^{58} + 4 q^{59} + ( -1 - \beta ) q^{61} + ( -3 + \beta ) q^{62} + q^{64} -2 q^{65} + ( -5 + \beta ) q^{67} + ( -1 - \beta ) q^{68} + q^{70} + ( 3 + \beta ) q^{71} + ( 8 + 2 \beta ) q^{73} + ( 3 - \beta ) q^{74} + ( -3 - \beta ) q^{76} + q^{77} + ( 1 + \beta ) q^{79} - q^{80} + ( 5 - \beta ) q^{82} + ( 7 - \beta ) q^{83} + ( 1 + \beta ) q^{85} + ( -1 + \beta ) q^{86} - q^{88} -2 \beta q^{89} -2 q^{91} + ( 3 - \beta ) q^{92} + ( 1 + 3 \beta ) q^{94} + ( 3 + \beta ) q^{95} + ( -7 - \beta ) q^{97} + q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8} - 2 q^{10} - 2 q^{11} + 4 q^{13} - 2 q^{14} + 2 q^{16} - 2 q^{17} - 6 q^{19} - 2 q^{20} - 2 q^{22} + 6 q^{23} + 2 q^{25} + 4 q^{26} - 2 q^{28} - 6 q^{31} + 2 q^{32} - 2 q^{34} + 2 q^{35} + 6 q^{37} - 6 q^{38} - 2 q^{40} + 10 q^{41} - 2 q^{43} - 2 q^{44} + 6 q^{46} + 2 q^{47} + 2 q^{49} + 2 q^{50} + 4 q^{52} + 20 q^{53} + 2 q^{55} - 2 q^{56} + 8 q^{59} - 2 q^{61} - 6 q^{62} + 2 q^{64} - 4 q^{65} - 10 q^{67} - 2 q^{68} + 2 q^{70} + 6 q^{71} + 16 q^{73} + 6 q^{74} - 6 q^{76} + 2 q^{77} + 2 q^{79} - 2 q^{80} + 10 q^{82} + 14 q^{83} + 2 q^{85} - 2 q^{86} - 2 q^{88} - 4 q^{91} + 6 q^{92} + 2 q^{94} + 6 q^{95} - 14 q^{97} + 2 q^{98} + O(q^{100}) \)

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
2.56155
−1.56155
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.bx yes 2
3.b odd 2 1 6930.2.a.bs 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6930.2.a.bs 2 3.b odd 2 1
6930.2.a.bx yes 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6930))\):

\( T_{13} - 2 \)
\( T_{17}^{2} + 2 T_{17} - 16 \)
\( T_{19}^{2} + 6 T_{19} - 8 \)
\( T_{23}^{2} - 6 T_{23} - 8 \)
\( T_{29}^{2} - 68 \)
\( T_{31}^{2} + 6 T_{31} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( -16 + 2 T + T^{2} \)
$19$ \( -8 + 6 T + T^{2} \)
$23$ \( -8 - 6 T + T^{2} \)
$29$ \( -68 + T^{2} \)
$31$ \( -8 + 6 T + T^{2} \)
$37$ \( -8 - 6 T + T^{2} \)
$41$ \( 8 - 10 T + T^{2} \)
$43$ \( -16 + 2 T + T^{2} \)
$47$ \( -152 - 2 T + T^{2} \)
$53$ \( ( -10 + T )^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( -16 + 2 T + T^{2} \)
$67$ \( 8 + 10 T + T^{2} \)
$71$ \( -8 - 6 T + T^{2} \)
$73$ \( -4 - 16 T + T^{2} \)
$79$ \( -16 - 2 T + T^{2} \)
$83$ \( 32 - 14 T + T^{2} \)
$89$ \( -68 + T^{2} \)
$97$ \( 32 + 14 T + T^{2} \)
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