Properties

Label 9360.2.a.o
Level $9360$
Weight $2$
Character orbit 9360.a
Self dual yes
Analytic conductor $74.740$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9360.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(74.7399762919\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{5} + O(q^{10}) \) \( q - q^{5} + 4q^{11} + q^{13} - 2q^{17} + 4q^{19} + 8q^{23} + q^{25} + 2q^{29} + 8q^{31} + 6q^{37} + 6q^{41} + 4q^{43} - 8q^{47} - 7q^{49} - 6q^{53} - 4q^{55} - 12q^{59} - 2q^{61} - q^{65} + 4q^{67} - 6q^{73} - 16q^{79} - 4q^{83} + 2q^{85} - 10q^{89} - 4q^{95} + 18q^{97} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(13\) \(-1\)

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 9360.2.a.o 1
3.b odd 2 1 3120.2.a.k 1
4.b odd 2 1 585.2.a.g 1
12.b even 2 1 195.2.a.a 1
20.d odd 2 1 2925.2.a.d 1
20.e even 4 2 2925.2.c.f 2
52.b odd 2 1 7605.2.a.h 1
60.h even 2 1 975.2.a.i 1
60.l odd 4 2 975.2.c.e 2
84.h odd 2 1 9555.2.a.b 1
156.h even 2 1 2535.2.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.a.a 1 12.b even 2 1
585.2.a.g 1 4.b odd 2 1
975.2.a.i 1 60.h even 2 1
975.2.c.e 2 60.l odd 4 2
2535.2.a.k 1 156.h even 2 1
2925.2.a.d 1 20.d odd 2 1
2925.2.c.f 2 20.e even 4 2
3120.2.a.k 1 3.b odd 2 1
7605.2.a.h 1 52.b odd 2 1
9360.2.a.o 1 1.a even 1 1 trivial
9555.2.a.b 1 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9360))\):

\( T_{7} \)
\( T_{11} - 4 \)
\( T_{17} + 2 \)
\( T_{19} - 4 \)
\( T_{31} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 1 + T \)
$7$ \( T \)
$11$ \( -4 + T \)
$13$ \( -1 + T \)
$17$ \( 2 + T \)
$19$ \( -4 + T \)
$23$ \( -8 + T \)
$29$ \( -2 + T \)
$31$ \( -8 + T \)
$37$ \( -6 + T \)
$41$ \( -6 + T \)
$43$ \( -4 + T \)
$47$ \( 8 + T \)
$53$ \( 6 + T \)
$59$ \( 12 + T \)
$61$ \( 2 + T \)
$67$ \( -4 + T \)
$71$ \( T \)
$73$ \( 6 + T \)
$79$ \( 16 + T \)
$83$ \( 4 + T \)
$89$ \( 10 + T \)
$97$ \( -18 + T \)
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