Properties

Label 7920.2.a.cg
Level $7920$
Weight $2$
Character orbit 7920.a
Self dual yes
Analytic conductor $63.242$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + q^{5} + ( 2 + \beta ) q^{7} +O(q^{10})\) \( q + q^{5} + ( 2 + \beta ) q^{7} - q^{11} + 2 \beta q^{13} + ( 4 + \beta ) q^{17} + ( 4 - \beta ) q^{19} -4 q^{23} + q^{25} + ( 2 - \beta ) q^{29} + ( 2 + \beta ) q^{35} + ( 6 - 2 \beta ) q^{37} + ( -2 + \beta ) q^{41} + ( 6 - \beta ) q^{43} -4 q^{47} + ( 5 + 4 \beta ) q^{49} + ( 2 + 4 \beta ) q^{53} - q^{55} -4 q^{59} + ( -6 + 2 \beta ) q^{61} + 2 \beta q^{65} -2 \beta q^{67} + ( 8 + 2 \beta ) q^{71} -4 \beta q^{73} + ( -2 - \beta ) q^{77} + 3 \beta q^{79} -10 q^{83} + ( 4 + \beta ) q^{85} + ( 2 + 2 \beta ) q^{89} + ( 16 + 4 \beta ) q^{91} + ( 4 - \beta ) q^{95} + ( 6 - 2 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} + 4q^{7} + O(q^{10}) \) \( 2q + 2q^{5} + 4q^{7} - 2q^{11} + 8q^{17} + 8q^{19} - 8q^{23} + 2q^{25} + 4q^{29} + 4q^{35} + 12q^{37} - 4q^{41} + 12q^{43} - 8q^{47} + 10q^{49} + 4q^{53} - 2q^{55} - 8q^{59} - 12q^{61} + 16q^{71} - 4q^{77} - 20q^{83} + 8q^{85} + 4q^{89} + 32q^{91} + 8q^{95} + 12q^{97} + O(q^{100}) \)

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
−1.41421
1.41421
0 0 0 1.00000 0 −0.828427 0 0 0
1.2 0 0 0 1.00000 0 4.82843 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.cg 2
3.b odd 2 1 2640.2.a.bb 2
4.b odd 2 1 495.2.a.d 2
12.b even 2 1 165.2.a.a 2
20.d odd 2 1 2475.2.a.m 2
20.e even 4 2 2475.2.c.m 4
44.c even 2 1 5445.2.a.m 2
60.h even 2 1 825.2.a.g 2
60.l odd 4 2 825.2.c.e 4
84.h odd 2 1 8085.2.a.ba 2
132.d odd 2 1 1815.2.a.k 2
660.g odd 2 1 9075.2.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.a 2 12.b even 2 1
495.2.a.d 2 4.b odd 2 1
825.2.a.g 2 60.h even 2 1
825.2.c.e 4 60.l odd 4 2
1815.2.a.k 2 132.d odd 2 1
2475.2.a.m 2 20.d odd 2 1
2475.2.c.m 4 20.e even 4 2
2640.2.a.bb 2 3.b odd 2 1
5445.2.a.m 2 44.c even 2 1
7920.2.a.cg 2 1.a even 1 1 trivial
8085.2.a.ba 2 84.h odd 2 1
9075.2.a.v 2 660.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7920))\):

\( T_{7}^{2} - 4 T_{7} - 4 \)
\( T_{13}^{2} - 32 \)
\( T_{17}^{2} - 8 T_{17} + 8 \)
\( T_{19}^{2} - 8 T_{19} + 8 \)
\( T_{23} + 4 \)
\( T_{29}^{2} - 4 T_{29} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( -4 - 4 T + T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( -32 + T^{2} \)
$17$ \( 8 - 8 T + T^{2} \)
$19$ \( 8 - 8 T + T^{2} \)
$23$ \( ( 4 + T )^{2} \)
$29$ \( -4 - 4 T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( 4 - 12 T + T^{2} \)
$41$ \( -4 + 4 T + T^{2} \)
$43$ \( 28 - 12 T + T^{2} \)
$47$ \( ( 4 + T )^{2} \)
$53$ \( -124 - 4 T + T^{2} \)
$59$ \( ( 4 + T )^{2} \)
$61$ \( 4 + 12 T + T^{2} \)
$67$ \( -32 + T^{2} \)
$71$ \( 32 - 16 T + T^{2} \)
$73$ \( -128 + T^{2} \)
$79$ \( -72 + T^{2} \)
$83$ \( ( 10 + T )^{2} \)
$89$ \( -28 - 4 T + T^{2} \)
$97$ \( 4 - 12 T + T^{2} \)
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