Properties

Label 6930.2.a.by
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: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2310)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{33}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - q^{5} - q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} - q^{5} - q^{7} + q^{8} - q^{10} + q^{11} + ( 1 + \beta ) q^{13} - q^{14} + q^{16} + ( 1 + \beta ) q^{17} - q^{20} + q^{22} + ( 1 - \beta ) q^{23} + q^{25} + ( 1 + \beta ) q^{26} - q^{28} -2 q^{29} + q^{32} + ( 1 + \beta ) q^{34} + q^{35} + ( 3 - \beta ) q^{37} - q^{40} -6 q^{41} + 4 q^{43} + q^{44} + ( 1 - \beta ) q^{46} + 4 q^{47} + q^{49} + q^{50} + ( 1 + \beta ) q^{52} + 6 q^{53} - q^{55} - q^{56} -2 q^{58} -4 q^{59} + 6 q^{61} + q^{64} + ( -1 - \beta ) q^{65} + ( -7 - \beta ) q^{67} + ( 1 + \beta ) q^{68} + q^{70} + 6 q^{73} + ( 3 - \beta ) q^{74} - q^{77} + 12 q^{79} - q^{80} -6 q^{82} + 4 q^{83} + ( -1 - \beta ) q^{85} + 4 q^{86} + q^{88} + ( -1 - \beta ) q^{89} + ( -1 - \beta ) q^{91} + ( 1 - \beta ) q^{92} + 4 q^{94} + ( 4 + 2 \beta ) q^{97} + q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8} - 2 q^{10} + 2 q^{11} + 2 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{20} + 2 q^{22} + 2 q^{23} + 2 q^{25} + 2 q^{26} - 2 q^{28} - 4 q^{29} + 2 q^{32} + 2 q^{34} + 2 q^{35} + 6 q^{37} - 2 q^{40} - 12 q^{41} + 8 q^{43} + 2 q^{44} + 2 q^{46} + 8 q^{47} + 2 q^{49} + 2 q^{50} + 2 q^{52} + 12 q^{53} - 2 q^{55} - 2 q^{56} - 4 q^{58} - 8 q^{59} + 12 q^{61} + 2 q^{64} - 2 q^{65} - 14 q^{67} + 2 q^{68} + 2 q^{70} + 12 q^{73} + 6 q^{74} - 2 q^{77} + 24 q^{79} - 2 q^{80} - 12 q^{82} + 8 q^{83} - 2 q^{85} + 8 q^{86} + 2 q^{88} - 2 q^{89} - 2 q^{91} + 2 q^{92} + 8 q^{94} + 8 q^{97} + 2 q^{98} + O(q^{100}) \)

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
−2.37228
3.37228
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.by 2
3.b odd 2 1 2310.2.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2310.2.a.x 2 3.b odd 2 1
6930.2.a.by 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6930))\):

\( T_{13}^{2} - 2 T_{13} - 32 \)
\( T_{17}^{2} - 2 T_{17} - 32 \)
\( T_{19} \)
\( T_{23}^{2} - 2 T_{23} - 32 \)
\( T_{29} + 2 \)
\( T_{31} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( -32 - 2 T + T^{2} \)
$17$ \( -32 - 2 T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( -32 - 2 T + T^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( -24 - 6 T + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( ( -4 + T )^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( ( 4 + T )^{2} \)
$61$ \( ( -6 + T )^{2} \)
$67$ \( 16 + 14 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -6 + T )^{2} \)
$79$ \( ( -12 + T )^{2} \)
$83$ \( ( -4 + T )^{2} \)
$89$ \( -32 + 2 T + T^{2} \)
$97$ \( -116 - 8 T + T^{2} \)
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