Properties

Label 7360.2.a.ba
Level $7360$
Weight $2$
Character orbit 7360.a
Self dual yes
Analytic conductor $58.770$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7360 = 2^{6} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7360.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(58.7698958877\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3 q^{3} - q^{5} - 2 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} - q^{5} - 2 q^{7} + 6 q^{9} - q^{13} - 3 q^{15} - 6 q^{21} + q^{23} + q^{25} + 9 q^{27} + 3 q^{29} + 3 q^{31} + 2 q^{35} + 8 q^{37} - 3 q^{39} + 3 q^{41} + 2 q^{43} - 6 q^{45} - 11 q^{47} - 3 q^{49} + 14 q^{53} + 8 q^{59} + 4 q^{61} - 12 q^{63} + q^{65} + 4 q^{67} + 3 q^{69} + 7 q^{71} - 9 q^{73} + 3 q^{75} + 9 q^{81} - 4 q^{83} + 9 q^{87} - 2 q^{89} + 2 q^{91} + 9 q^{93} + 18 q^{97}+O(q^{100}) \) 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 3.00000 0 −1.00000 0 −2.00000 0 6.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(23\) \(-1\)

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 7360.2.a.ba 1
4.b odd 2 1 7360.2.a.a 1
8.b even 2 1 920.2.a.a 1
8.d odd 2 1 1840.2.a.i 1
24.h odd 2 1 8280.2.a.d 1
40.e odd 2 1 9200.2.a.c 1
40.f even 2 1 4600.2.a.p 1
40.i odd 4 2 4600.2.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.a 1 8.b even 2 1
1840.2.a.i 1 8.d odd 2 1
4600.2.a.p 1 40.f even 2 1
4600.2.e.b 2 40.i odd 4 2
7360.2.a.a 1 4.b odd 2 1
7360.2.a.ba 1 1.a even 1 1 trivial
8280.2.a.d 1 24.h odd 2 1
9200.2.a.c 1 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7360))\):

\( T_{3} - 3 \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} + 1 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 1 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 1 \) Copy content Toggle raw display
$29$ \( T - 3 \) Copy content Toggle raw display
$31$ \( T - 3 \) Copy content Toggle raw display
$37$ \( T - 8 \) Copy content Toggle raw display
$41$ \( T - 3 \) Copy content Toggle raw display
$43$ \( T - 2 \) Copy content Toggle raw display
$47$ \( T + 11 \) Copy content Toggle raw display
$53$ \( T - 14 \) Copy content Toggle raw display
$59$ \( T - 8 \) Copy content Toggle raw display
$61$ \( T - 4 \) Copy content Toggle raw display
$67$ \( T - 4 \) Copy content Toggle raw display
$71$ \( T - 7 \) Copy content Toggle raw display
$73$ \( T + 9 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T + 4 \) Copy content Toggle raw display
$89$ \( T + 2 \) Copy content Toggle raw display
$97$ \( T - 18 \) Copy content Toggle raw display
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