Properties

Label 6930.2.a.ce
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: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + \nu - 3 \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 6\)\()/2\)

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
−1.81361
0.470683
2.34292
−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.ce 3
3.b odd 2 1 770.2.a.m 3
12.b even 2 1 6160.2.a.bf 3
15.d odd 2 1 3850.2.a.bt 3
15.e even 4 2 3850.2.c.ba 6
21.c even 2 1 5390.2.a.ca 3
33.d even 2 1 8470.2.a.ci 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.m 3 3.b odd 2 1
3850.2.a.bt 3 15.d odd 2 1
3850.2.c.ba 6 15.e even 4 2
5390.2.a.ca 3 21.c even 2 1
6160.2.a.bf 3 12.b even 2 1
6930.2.a.ce 3 1.a even 1 1 trivial
8470.2.a.ci 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6930))\):

\( T_{13}^{3} - 2 T_{13}^{2} - 16 T_{13} + 16 \)
\( T_{17}^{3} + 2 T_{17}^{2} - 28 T_{17} + 8 \)
\( T_{19}^{3} + 6 T_{19}^{2} - 46 T_{19} - 232 \)
\( T_{23}^{3} + 10 T_{23}^{2} + 26 T_{23} + 16 \)
\( T_{29}^{3} - 66 T_{29} + 92 \)
\( T_{31}^{3} + 12 T_{31}^{2} + 20 T_{31} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{3} \)
$3$ \( T^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( ( 1 + T )^{3} \)
$11$ \( ( 1 + T )^{3} \)
$13$ \( 16 - 16 T - 2 T^{2} + T^{3} \)
$17$ \( 8 - 28 T + 2 T^{2} + T^{3} \)
$19$ \( -232 - 46 T + 6 T^{2} + T^{3} \)
$23$ \( 16 + 26 T + 10 T^{2} + T^{3} \)
$29$ \( 92 - 66 T + T^{3} \)
$31$ \( -32 + 20 T + 12 T^{2} + T^{3} \)
$37$ \( 4 - 2 T - 4 T^{2} + T^{3} \)
$41$ \( 164 - 90 T + 4 T^{2} + T^{3} \)
$43$ \( 848 - 108 T - 8 T^{2} + T^{3} \)
$47$ \( 128 - 112 T + T^{3} \)
$53$ \( -292 - 18 T + 12 T^{2} + T^{3} \)
$59$ \( 128 - 64 T - 4 T^{2} + T^{3} \)
$61$ \( -344 - 100 T + 6 T^{2} + T^{3} \)
$67$ \( -64 - 112 T - 4 T^{2} + T^{3} \)
$71$ \( -64 + 200 T - 28 T^{2} + T^{3} \)
$73$ \( -8 + 36 T + 14 T^{2} + T^{3} \)
$79$ \( 256 + 50 T - 18 T^{2} + T^{3} \)
$83$ \( 32 - 8 T - 8 T^{2} + T^{3} \)
$89$ \( -1208 - 28 T + 18 T^{2} + T^{3} \)
$97$ \( 316 - 154 T + 4 T^{2} + T^{3} \)
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