Properties

Label 6930.2.a.bo
Level $6930$
Weight $2$
Character orbit 6930.a
Self dual yes
Analytic conductor $55.336$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 770)
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.

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

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.bo 2
3.b odd 2 1 770.2.a.k 2
12.b even 2 1 6160.2.a.r 2
15.d odd 2 1 3850.2.a.bc 2
15.e even 4 2 3850.2.c.y 4
21.c even 2 1 5390.2.a.bq 2
33.d even 2 1 8470.2.a.bu 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.k 2 3.b odd 2 1
3850.2.a.bc 2 15.d odd 2 1
3850.2.c.y 4 15.e even 4 2
5390.2.a.bq 2 21.c even 2 1
6160.2.a.r 2 12.b even 2 1
6930.2.a.bo 2 1.a even 1 1 trivial
8470.2.a.bu 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6930))\):

\( T_{13}^{2} - 2 T_{13} - 32 \)
\( T_{17}^{2} - 2 T_{17} - 32 \)
\( T_{19}^{2} + 2 T_{19} - 32 \)
\( T_{23}^{2} - 2 T_{23} - 32 \)
\( T_{29}^{2} + 6 T_{29} - 24 \)
\( T_{31}^{2} + 2 T_{31} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( -32 - 2 T + T^{2} \)
$17$ \( -32 - 2 T + T^{2} \)
$19$ \( -32 + 2 T + T^{2} \)
$23$ \( -32 - 2 T + T^{2} \)
$29$ \( -24 + 6 T + T^{2} \)
$31$ \( -32 + 2 T + T^{2} \)
$37$ \( -8 + 10 T + T^{2} \)
$41$ \( ( -4 + T )^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( -32 - 2 T + T^{2} \)
$53$ \( 16 - 14 T + T^{2} \)
$59$ \( -24 - 6 T + T^{2} \)
$61$ \( 16 - 14 T + T^{2} \)
$67$ \( ( 4 + T )^{2} \)
$71$ \( ( -4 + T )^{2} \)
$73$ \( -8 - 10 T + T^{2} \)
$79$ \( -32 - 2 T + T^{2} \)
$83$ \( ( 8 + T )^{2} \)
$89$ \( -116 + 8 T + T^{2} \)
$97$ \( 88 + 22 T + T^{2} \)
show more
show less