Properties

Label 7920.2.a.bz
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: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} -2 q^{7} +O(q^{10})\) \( q + q^{5} -2 q^{7} - q^{11} + ( 2 + \beta ) q^{13} + ( -2 - \beta ) q^{19} + 2 \beta q^{23} + q^{25} + \beta q^{29} + ( 4 + 2 \beta ) q^{31} -2 q^{35} + ( 2 - 2 \beta ) q^{37} -\beta q^{41} + ( -2 + 2 \beta ) q^{43} -2 \beta q^{47} -3 q^{49} + ( 6 - 2 \beta ) q^{53} - q^{55} -2 \beta q^{59} + 2 q^{61} + ( 2 + \beta ) q^{65} -8 q^{67} + 4 \beta q^{71} + ( 2 - 3 \beta ) q^{73} + 2 q^{77} + ( 10 - \beta ) q^{79} + ( 12 - \beta ) q^{83} + ( 6 - 2 \beta ) q^{89} + ( -4 - 2 \beta ) q^{91} + ( -2 - \beta ) q^{95} -10 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} - 4q^{7} + O(q^{10}) \) \( 2q + 2q^{5} - 4q^{7} - 2q^{11} + 4q^{13} - 4q^{19} + 2q^{25} + 8q^{31} - 4q^{35} + 4q^{37} - 4q^{43} - 6q^{49} + 12q^{53} - 2q^{55} + 4q^{61} + 4q^{65} - 16q^{67} + 4q^{73} + 4q^{77} + 20q^{79} + 24q^{83} + 12q^{89} - 8q^{91} - 4q^{95} - 20q^{97} + O(q^{100}) \)

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
−1.73205
1.73205
0 0 0 1.00000 0 −2.00000 0 0 0
1.2 0 0 0 1.00000 0 −2.00000 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.bz 2
3.b odd 2 1 2640.2.a.x 2
4.b odd 2 1 495.2.a.c 2
12.b even 2 1 165.2.a.b 2
20.d odd 2 1 2475.2.a.r 2
20.e even 4 2 2475.2.c.n 4
44.c even 2 1 5445.2.a.s 2
60.h even 2 1 825.2.a.e 2
60.l odd 4 2 825.2.c.c 4
84.h odd 2 1 8085.2.a.bd 2
132.d odd 2 1 1815.2.a.i 2
660.g odd 2 1 9075.2.a.bh 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.b 2 12.b even 2 1
495.2.a.c 2 4.b odd 2 1
825.2.a.e 2 60.h even 2 1
825.2.c.c 4 60.l odd 4 2
1815.2.a.i 2 132.d odd 2 1
2475.2.a.r 2 20.d odd 2 1
2475.2.c.n 4 20.e even 4 2
2640.2.a.x 2 3.b odd 2 1
5445.2.a.s 2 44.c even 2 1
7920.2.a.bz 2 1.a even 1 1 trivial
8085.2.a.bd 2 84.h odd 2 1
9075.2.a.bh 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7920))\):

\( T_{7} + 2 \)
\( T_{13}^{2} - 4 T_{13} - 8 \)
\( T_{17} \)
\( T_{19}^{2} + 4 T_{19} - 8 \)
\( T_{23}^{2} - 48 \)
\( T_{29}^{2} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( ( 2 + T )^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( -8 - 4 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( -8 + 4 T + T^{2} \)
$23$ \( -48 + T^{2} \)
$29$ \( -12 + T^{2} \)
$31$ \( -32 - 8 T + T^{2} \)
$37$ \( -44 - 4 T + T^{2} \)
$41$ \( -12 + T^{2} \)
$43$ \( -44 + 4 T + T^{2} \)
$47$ \( -48 + T^{2} \)
$53$ \( -12 - 12 T + T^{2} \)
$59$ \( -48 + T^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( ( 8 + T )^{2} \)
$71$ \( -192 + T^{2} \)
$73$ \( -104 - 4 T + T^{2} \)
$79$ \( 88 - 20 T + T^{2} \)
$83$ \( 132 - 24 T + T^{2} \)
$89$ \( -12 - 12 T + T^{2} \)
$97$ \( ( 10 + T )^{2} \)
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