Properties

Label 6930.2.a.cj
Level $6930$
Weight $2$
Character orbit 6930.a
Self dual yes
Analytic conductor $55.336$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - q^{5} + q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} - q^{5} + q^{7} + q^{8} - q^{10} - q^{11} + \beta_{1} q^{13} + q^{14} + q^{16} + ( -2 + \beta_{1} ) q^{17} + ( -2 - \beta_{1} - \beta_{2} ) q^{19} - q^{20} - q^{22} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{23} + q^{25} + \beta_{1} q^{26} + q^{28} + ( -2 - 2 \beta_{1} ) q^{29} + ( \beta_{1} + \beta_{2} ) q^{31} + q^{32} + ( -2 + \beta_{1} ) q^{34} - q^{35} + ( -2 - \beta_{1} ) q^{37} + ( -2 - \beta_{1} - \beta_{2} ) q^{38} - q^{40} + ( -2 + \beta_{1} + \beta_{2} ) q^{41} + ( -2 - \beta_{1} - \beta_{2} ) q^{43} - q^{44} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{46} -6 q^{47} + q^{49} + q^{50} + \beta_{1} q^{52} + ( -2 + \beta_{1} - \beta_{2} ) q^{53} + q^{55} + q^{56} + ( -2 - 2 \beta_{1} ) q^{58} + ( -4 - \beta_{1} - \beta_{2} ) q^{59} -4 q^{61} + ( \beta_{1} + \beta_{2} ) q^{62} + q^{64} -\beta_{1} q^{65} + ( 4 - \beta_{1} + 2 \beta_{2} ) q^{67} + ( -2 + \beta_{1} ) q^{68} - q^{70} + ( -2 + 4 \beta_{1} ) q^{71} + ( -2 - \beta_{1} - \beta_{2} ) q^{73} + ( -2 - \beta_{1} ) q^{74} + ( -2 - \beta_{1} - \beta_{2} ) q^{76} - q^{77} + 4 \beta_{1} q^{79} - q^{80} + ( -2 + \beta_{1} + \beta_{2} ) q^{82} + ( 3 \beta_{1} - \beta_{2} ) q^{83} + ( 2 - \beta_{1} ) q^{85} + ( -2 - \beta_{1} - \beta_{2} ) q^{86} - q^{88} + ( -4 - 4 \beta_{1} + \beta_{2} ) q^{89} + \beta_{1} q^{91} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{92} -6 q^{94} + ( 2 + \beta_{1} + \beta_{2} ) q^{95} + ( 2 + 3 \beta_{1} + \beta_{2} ) q^{97} + q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} + 3 q^{8} + O(q^{10}) \) \( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} + 3 q^{8} - 3 q^{10} - 3 q^{11} + 3 q^{14} + 3 q^{16} - 6 q^{17} - 6 q^{19} - 3 q^{20} - 3 q^{22} - 6 q^{23} + 3 q^{25} + 3 q^{28} - 6 q^{29} + 3 q^{32} - 6 q^{34} - 3 q^{35} - 6 q^{37} - 6 q^{38} - 3 q^{40} - 6 q^{41} - 6 q^{43} - 3 q^{44} - 6 q^{46} - 18 q^{47} + 3 q^{49} + 3 q^{50} - 6 q^{53} + 3 q^{55} + 3 q^{56} - 6 q^{58} - 12 q^{59} - 12 q^{61} + 3 q^{64} + 12 q^{67} - 6 q^{68} - 3 q^{70} - 6 q^{71} - 6 q^{73} - 6 q^{74} - 6 q^{76} - 3 q^{77} - 3 q^{80} - 6 q^{82} + 6 q^{85} - 6 q^{86} - 3 q^{88} - 12 q^{89} - 6 q^{92} - 18 q^{94} + 6 q^{95} + 6 q^{97} + 3 q^{98} + O(q^{100}) \)

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
−1.53209
−0.347296
1.87939
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.cj yes 3
3.b odd 2 1 6930.2.a.ci 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6930.2.a.ci 3 3.b odd 2 1
6930.2.a.cj yes 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6930))\):

\( T_{13}^{3} - 12 T_{13} - 8 \)
\( T_{17}^{3} + 6 T_{17}^{2} - 24 \)
\( T_{19}^{3} + 6 T_{19}^{2} - 36 T_{19} - 152 \)
\( T_{23}^{3} + 6 T_{23}^{2} - 72 T_{23} - 456 \)
\( T_{29}^{3} + 6 T_{29}^{2} - 36 T_{29} - 24 \)
\( T_{31}^{3} - 48 T_{31} + 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( T^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( ( 1 + T )^{3} \)
$13$ \( -8 - 12 T + T^{3} \)
$17$ \( -24 + 6 T^{2} + T^{3} \)
$19$ \( -152 - 36 T + 6 T^{2} + T^{3} \)
$23$ \( -456 - 72 T + 6 T^{2} + T^{3} \)
$29$ \( -24 - 36 T + 6 T^{2} + T^{3} \)
$31$ \( 64 - 48 T + T^{3} \)
$37$ \( -8 + 6 T^{2} + T^{3} \)
$41$ \( -24 - 36 T + 6 T^{2} + T^{3} \)
$43$ \( -152 - 36 T + 6 T^{2} + T^{3} \)
$47$ \( ( 6 + T )^{3} \)
$53$ \( -24 - 36 T + 6 T^{2} + T^{3} \)
$59$ \( -192 + 12 T^{2} + T^{3} \)
$61$ \( ( 4 + T )^{3} \)
$67$ \( 712 - 108 T - 12 T^{2} + T^{3} \)
$71$ \( -888 - 180 T + 6 T^{2} + T^{3} \)
$73$ \( -152 - 36 T + 6 T^{2} + T^{3} \)
$79$ \( -512 - 192 T + T^{3} \)
$83$ \( 576 - 144 T + T^{3} \)
$89$ \( -1704 - 180 T + 12 T^{2} + T^{3} \)
$97$ \( -296 - 132 T - 6 T^{2} + T^{3} \)
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