Properties

Label 7920.2.a.bq
Level $7920$
Weight $2$
Character orbit 7920.a
Self dual yes
Analytic conductor $63.242$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.2415184009\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 110)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{5} -\beta q^{7} +O(q^{10})\) \( q - q^{5} -\beta q^{7} - q^{11} + 2 q^{13} + ( 2 - \beta ) q^{17} + ( -4 + \beta ) q^{19} + ( -4 + 2 \beta ) q^{23} + q^{25} + ( 2 - \beta ) q^{29} -\beta q^{31} + \beta q^{35} + ( 6 + \beta ) q^{37} + ( -2 + 4 \beta ) q^{41} + 4 q^{43} + ( -4 + 2 \beta ) q^{47} + ( 1 + \beta ) q^{49} + ( -6 + 3 \beta ) q^{53} + q^{55} + ( 4 - 2 \beta ) q^{59} + ( -2 - \beta ) q^{61} -2 q^{65} -8 q^{67} + 3 \beta q^{71} + ( -2 - 4 \beta ) q^{73} + \beta q^{77} + ( 8 - 2 \beta ) q^{79} + ( 4 - 2 \beta ) q^{83} + ( -2 + \beta ) q^{85} + ( -2 + \beta ) q^{89} -2 \beta q^{91} + ( 4 - \beta ) q^{95} + ( -6 - 2 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} - q^{7} + O(q^{10}) \) \( 2q - 2q^{5} - q^{7} - 2q^{11} + 4q^{13} + 3q^{17} - 7q^{19} - 6q^{23} + 2q^{25} + 3q^{29} - q^{31} + q^{35} + 13q^{37} + 8q^{43} - 6q^{47} + 3q^{49} - 9q^{53} + 2q^{55} + 6q^{59} - 5q^{61} - 4q^{65} - 16q^{67} + 3q^{71} - 8q^{73} + q^{77} + 14q^{79} + 6q^{83} - 3q^{85} - 3q^{89} - 2q^{91} + 7q^{95} - 14q^{97} + O(q^{100}) \)

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
3.37228
−2.37228
0 0 0 −1.00000 0 −3.37228 0 0 0
1.2 0 0 0 −1.00000 0 2.37228 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(11\) \(1\)

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 7920.2.a.bq 2
3.b odd 2 1 880.2.a.n 2
4.b odd 2 1 990.2.a.m 2
12.b even 2 1 110.2.a.d 2
15.d odd 2 1 4400.2.a.bl 2
15.e even 4 2 4400.2.b.p 4
20.d odd 2 1 4950.2.a.bw 2
20.e even 4 2 4950.2.c.bc 4
24.f even 2 1 3520.2.a.bq 2
24.h odd 2 1 3520.2.a.bj 2
33.d even 2 1 9680.2.a.bt 2
60.h even 2 1 550.2.a.n 2
60.l odd 4 2 550.2.b.f 4
84.h odd 2 1 5390.2.a.bp 2
132.d odd 2 1 1210.2.a.r 2
660.g odd 2 1 6050.2.a.cb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.a.d 2 12.b even 2 1
550.2.a.n 2 60.h even 2 1
550.2.b.f 4 60.l odd 4 2
880.2.a.n 2 3.b odd 2 1
990.2.a.m 2 4.b odd 2 1
1210.2.a.r 2 132.d odd 2 1
3520.2.a.bj 2 24.h odd 2 1
3520.2.a.bq 2 24.f even 2 1
4400.2.a.bl 2 15.d odd 2 1
4400.2.b.p 4 15.e even 4 2
4950.2.a.bw 2 20.d odd 2 1
4950.2.c.bc 4 20.e even 4 2
5390.2.a.bp 2 84.h odd 2 1
6050.2.a.cb 2 660.g odd 2 1
7920.2.a.bq 2 1.a even 1 1 trivial
9680.2.a.bt 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7920))\):

\( T_{7}^{2} + T_{7} - 8 \)
\( T_{13} - 2 \)
\( T_{17}^{2} - 3 T_{17} - 6 \)
\( T_{19}^{2} + 7 T_{19} + 4 \)
\( T_{23}^{2} + 6 T_{23} - 24 \)
\( T_{29}^{2} - 3 T_{29} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -8 + T + T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( -6 - 3 T + T^{2} \)
$19$ \( 4 + 7 T + T^{2} \)
$23$ \( -24 + 6 T + T^{2} \)
$29$ \( -6 - 3 T + T^{2} \)
$31$ \( -8 + T + T^{2} \)
$37$ \( 34 - 13 T + T^{2} \)
$41$ \( -132 + T^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( -24 + 6 T + T^{2} \)
$53$ \( -54 + 9 T + T^{2} \)
$59$ \( -24 - 6 T + T^{2} \)
$61$ \( -2 + 5 T + T^{2} \)
$67$ \( ( 8 + T )^{2} \)
$71$ \( -72 - 3 T + T^{2} \)
$73$ \( -116 + 8 T + T^{2} \)
$79$ \( 16 - 14 T + T^{2} \)
$83$ \( -24 - 6 T + T^{2} \)
$89$ \( -6 + 3 T + T^{2} \)
$97$ \( 16 + 14 T + T^{2} \)
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