Properties

Label 9360.2.a.b
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 1560)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{5} - 4q^{7} + O(q^{10}) \) \( q - q^{5} - 4q^{7} + 4q^{11} + q^{13} - 6q^{17} - 4q^{23} + q^{25} + 6q^{29} + 8q^{31} + 4q^{35} - 2q^{37} - 10q^{41} + 4q^{43} + 8q^{47} + 9q^{49} + 2q^{53} - 4q^{55} + 4q^{59} + 14q^{61} - q^{65} + 12q^{67} - 8q^{71} - 10q^{73} - 16q^{77} - 4q^{83} + 6q^{85} - 10q^{89} - 4q^{91} - 2q^{97} + O(q^{100}) \)

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 −4.00000 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.b 1
3.b odd 2 1 3120.2.a.s 1
4.b odd 2 1 4680.2.a.k 1
12.b even 2 1 1560.2.a.f 1
60.h even 2 1 7800.2.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.a.f 1 12.b even 2 1
3120.2.a.s 1 3.b odd 2 1
4680.2.a.k 1 4.b odd 2 1
7800.2.a.o 1 60.h even 2 1
9360.2.a.b 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9360))\):

\( T_{7} + 4 \)
\( T_{11} - 4 \)
\( T_{17} + 6 \)
\( T_{19} \)
\( T_{31} - 8 \)

Hecke characteristic polynomials

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