Properties

Label 6930.2.a.cl
Level $6930$
Weight $2$
Character orbit 6930.a
Self dual yes
Analytic conductor $55.336$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.892.1
Defining polynomial: \(x^{3} - x^{2} - 8 x + 10\)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8} + q^{10} - q^{11} + 2 \beta_{1} q^{13} - q^{14} + q^{16} + ( 3 + \beta_{1} + \beta_{2} ) q^{17} + ( -1 - \beta_{2} ) q^{19} + q^{20} - q^{22} + ( -1 - \beta_{2} ) q^{23} + q^{25} + 2 \beta_{1} q^{26} - q^{28} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{29} + 6 q^{31} + q^{32} + ( 3 + \beta_{1} + \beta_{2} ) q^{34} - q^{35} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{37} + ( -1 - \beta_{2} ) q^{38} + q^{40} + ( 6 + \beta_{1} ) q^{41} + ( 3 - \beta_{1} + \beta_{2} ) q^{43} - q^{44} + ( -1 - \beta_{2} ) q^{46} + ( -1 + \beta_{1} - \beta_{2} ) q^{47} + q^{49} + q^{50} + 2 \beta_{1} q^{52} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{53} - q^{55} - q^{56} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{58} + ( -3 + 3 \beta_{1} + \beta_{2} ) q^{59} + ( 7 - \beta_{1} + \beta_{2} ) q^{61} + 6 q^{62} + q^{64} + 2 \beta_{1} q^{65} + 8 q^{67} + ( 3 + \beta_{1} + \beta_{2} ) q^{68} - q^{70} + ( -2 + 2 \beta_{2} ) q^{71} + ( -1 - 3 \beta_{1} + \beta_{2} ) q^{73} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{74} + ( -1 - \beta_{2} ) q^{76} + q^{77} + ( -3 - 2 \beta_{1} - 3 \beta_{2} ) q^{79} + q^{80} + ( 6 + \beta_{1} ) q^{82} + ( 8 - 2 \beta_{1} ) q^{83} + ( 3 + \beta_{1} + \beta_{2} ) q^{85} + ( 3 - \beta_{1} + \beta_{2} ) q^{86} - q^{88} + ( 2 + 2 \beta_{1} ) q^{89} -2 \beta_{1} q^{91} + ( -1 - \beta_{2} ) q^{92} + ( -1 + \beta_{1} - \beta_{2} ) q^{94} + ( -1 - \beta_{2} ) q^{95} + ( 3 + \beta_{2} ) q^{97} + q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} + 3 q^{8} + O(q^{10}) \) \( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} + 3 q^{8} + 3 q^{10} - 3 q^{11} - 3 q^{14} + 3 q^{16} + 8 q^{17} - 2 q^{19} + 3 q^{20} - 3 q^{22} - 2 q^{23} + 3 q^{25} - 3 q^{28} - 4 q^{29} + 18 q^{31} + 3 q^{32} + 8 q^{34} - 3 q^{35} + 4 q^{37} - 2 q^{38} + 3 q^{40} + 18 q^{41} + 8 q^{43} - 3 q^{44} - 2 q^{46} - 2 q^{47} + 3 q^{49} + 3 q^{50} - 4 q^{53} - 3 q^{55} - 3 q^{56} - 4 q^{58} - 10 q^{59} + 20 q^{61} + 18 q^{62} + 3 q^{64} + 24 q^{67} + 8 q^{68} - 3 q^{70} - 8 q^{71} - 4 q^{73} + 4 q^{74} - 2 q^{76} + 3 q^{77} - 6 q^{79} + 3 q^{80} + 18 q^{82} + 24 q^{83} + 8 q^{85} + 8 q^{86} - 3 q^{88} + 6 q^{89} - 2 q^{92} - 2 q^{94} - 2 q^{95} + 8 q^{97} + 3 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 8 x + 10\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + \nu - 6 \)
\(\beta_{2}\)\(=\)\( -\nu^{2} + \nu + 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{2} + \beta_{1} + 11\)\()/2\)

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
1.31955
−2.91729
2.59774
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
1.3 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.cl 3
3.b odd 2 1 770.2.a.l 3
12.b even 2 1 6160.2.a.bi 3
15.d odd 2 1 3850.2.a.bu 3
15.e even 4 2 3850.2.c.z 6
21.c even 2 1 5390.2.a.bz 3
33.d even 2 1 8470.2.a.cl 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.l 3 3.b odd 2 1
3850.2.a.bu 3 15.d odd 2 1
3850.2.c.z 6 15.e even 4 2
5390.2.a.bz 3 21.c even 2 1
6160.2.a.bi 3 12.b even 2 1
6930.2.a.cl 3 1.a even 1 1 trivial
8470.2.a.cl 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6930))\):

\( T_{13}^{3} - 40 T_{13} - 32 \)
\( T_{17}^{3} - 8 T_{17}^{2} - 12 T_{17} + 128 \)
\( T_{19}^{3} + 2 T_{19}^{2} - 30 T_{19} - 56 \)
\( T_{23}^{3} + 2 T_{23}^{2} - 30 T_{23} - 56 \)
\( T_{29}^{3} + 4 T_{29}^{2} - 82 T_{29} - 428 \)
\( T_{31} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( T^{3} \)
$5$ \( ( -1 + T )^{3} \)
$7$ \( ( 1 + T )^{3} \)
$11$ \( ( 1 + T )^{3} \)
$13$ \( -32 - 40 T + T^{3} \)
$17$ \( 128 - 12 T - 8 T^{2} + T^{3} \)
$19$ \( -56 - 30 T + 2 T^{2} + T^{3} \)
$23$ \( -56 - 30 T + 2 T^{2} + T^{3} \)
$29$ \( -428 - 82 T + 4 T^{2} + T^{3} \)
$31$ \( ( -6 + T )^{3} \)
$37$ \( 124 - 50 T - 4 T^{2} + T^{3} \)
$41$ \( -160 + 98 T - 18 T^{2} + T^{3} \)
$43$ \( 16 - 28 T - 8 T^{2} + T^{3} \)
$47$ \( 64 - 48 T + 2 T^{2} + T^{3} \)
$53$ \( -428 - 82 T + 4 T^{2} + T^{3} \)
$59$ \( -608 - 64 T + 10 T^{2} + T^{3} \)
$61$ \( -64 + 84 T - 20 T^{2} + T^{3} \)
$67$ \( ( -8 + T )^{3} \)
$71$ \( -32 - 104 T + 8 T^{2} + T^{3} \)
$73$ \( -784 - 140 T + 4 T^{2} + T^{3} \)
$79$ \( -2272 - 262 T + 6 T^{2} + T^{3} \)
$83$ \( -160 + 152 T - 24 T^{2} + T^{3} \)
$89$ \( 40 - 28 T - 6 T^{2} + T^{3} \)
$97$ \( 100 - 10 T - 8 T^{2} + T^{3} \)
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