Properties

Label 9360.2.a.be
Level $9360$
Weight $2$
Character orbit 9360.a
Self dual yes
Analytic conductor $74.740$
Analytic rank $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 585)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} - 2 q^{7} - 4 q^{11} - q^{13} + 4 q^{17} - 6 q^{19} + q^{25} + 4 q^{29} + 10 q^{31} - 2 q^{35} - 2 q^{37} + 6 q^{41} + 8 q^{43} - 8 q^{47} - 3 q^{49} + 4 q^{53} - 4 q^{55} + 12 q^{59} + 2 q^{61} - q^{65} + 10 q^{67} - 6 q^{73} + 8 q^{77} - 12 q^{79} - 4 q^{83} + 4 q^{85} - 14 q^{89} + 2 q^{91} - 6 q^{95} - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 −2.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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.be 1
3.b odd 2 1 9360.2.a.i 1
4.b odd 2 1 585.2.a.i yes 1
12.b even 2 1 585.2.a.d 1
20.d odd 2 1 2925.2.a.c 1
20.e even 4 2 2925.2.c.k 2
52.b odd 2 1 7605.2.a.c 1
60.h even 2 1 2925.2.a.m 1
60.l odd 4 2 2925.2.c.g 2
156.h even 2 1 7605.2.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.a.d 1 12.b even 2 1
585.2.a.i yes 1 4.b odd 2 1
2925.2.a.c 1 20.d odd 2 1
2925.2.a.m 1 60.h even 2 1
2925.2.c.g 2 60.l odd 4 2
2925.2.c.k 2 20.e even 4 2
7605.2.a.c 1 52.b odd 2 1
7605.2.a.q 1 156.h even 2 1
9360.2.a.i 1 3.b odd 2 1
9360.2.a.be 1 1.a even 1 1 trivial

Hecke kernels

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

\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display
\( T_{17} - 4 \) Copy content Toggle raw display
\( T_{19} + 6 \) Copy content Toggle raw display
\( T_{31} - 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T + 4 \) Copy content Toggle raw display
$13$ \( T + 1 \) Copy content Toggle raw display
$17$ \( T - 4 \) Copy content Toggle raw display
$19$ \( T + 6 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 4 \) Copy content Toggle raw display
$31$ \( T - 10 \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T - 8 \) Copy content Toggle raw display
$47$ \( T + 8 \) Copy content Toggle raw display
$53$ \( T - 4 \) Copy content Toggle raw display
$59$ \( T - 12 \) Copy content Toggle raw display
$61$ \( T - 2 \) Copy content Toggle raw display
$67$ \( T - 10 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 6 \) Copy content Toggle raw display
$79$ \( T + 12 \) Copy content Toggle raw display
$83$ \( T + 4 \) Copy content Toggle raw display
$89$ \( T + 14 \) Copy content Toggle raw display
$97$ \( T + 14 \) Copy content Toggle raw display
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