Properties

Label 7488.2.a.bz
Level $7488$
Weight $2$
Character orbit 7488.a
Self dual yes
Analytic conductor $59.792$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7488 = 2^{6} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7488.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(59.7919810335\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{5} + 4 q^{7} - 4 q^{11} - q^{13} - 2 q^{17} + 8 q^{19} - q^{25} + 6 q^{29} - 4 q^{31} + 8 q^{35} + 2 q^{37} + 10 q^{41} - 4 q^{43} - 8 q^{47} + 9 q^{49} - 10 q^{53} - 8 q^{55} + 4 q^{59} + 2 q^{61} - 2 q^{65} + 16 q^{67} + 8 q^{71} + 2 q^{73} - 16 q^{77} + 8 q^{79} + 12 q^{83} - 4 q^{85} - 14 q^{89} - 4 q^{91} + 16 q^{95} + 10 q^{97}+O(q^{100}) \) 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 2.00000 0 4.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 7488.2.a.bz 1
3.b odd 2 1 2496.2.a.t 1
4.b odd 2 1 7488.2.a.bk 1
8.b even 2 1 234.2.a.c 1
8.d odd 2 1 1872.2.a.c 1
12.b even 2 1 2496.2.a.b 1
24.f even 2 1 624.2.a.h 1
24.h odd 2 1 78.2.a.a 1
40.f even 2 1 5850.2.a.d 1
40.i odd 4 2 5850.2.e.bb 2
72.j odd 6 2 2106.2.e.q 2
72.n even 6 2 2106.2.e.j 2
104.e even 2 1 3042.2.a.f 1
104.j odd 4 2 3042.2.b.g 2
120.i odd 2 1 1950.2.a.w 1
120.w even 4 2 1950.2.e.i 2
168.i even 2 1 3822.2.a.j 1
264.m even 2 1 9438.2.a.t 1
312.b odd 2 1 1014.2.a.d 1
312.h even 2 1 8112.2.a.v 1
312.y even 4 2 1014.2.b.b 2
312.bg odd 6 2 1014.2.e.c 2
312.bh odd 6 2 1014.2.e.f 2
312.bo even 12 4 1014.2.i.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.a.a 1 24.h odd 2 1
234.2.a.c 1 8.b even 2 1
624.2.a.h 1 24.f even 2 1
1014.2.a.d 1 312.b odd 2 1
1014.2.b.b 2 312.y even 4 2
1014.2.e.c 2 312.bg odd 6 2
1014.2.e.f 2 312.bh odd 6 2
1014.2.i.d 4 312.bo even 12 4
1872.2.a.c 1 8.d odd 2 1
1950.2.a.w 1 120.i odd 2 1
1950.2.e.i 2 120.w even 4 2
2106.2.e.j 2 72.n even 6 2
2106.2.e.q 2 72.j odd 6 2
2496.2.a.b 1 12.b even 2 1
2496.2.a.t 1 3.b odd 2 1
3042.2.a.f 1 104.e even 2 1
3042.2.b.g 2 104.j odd 4 2
3822.2.a.j 1 168.i even 2 1
5850.2.a.d 1 40.f even 2 1
5850.2.e.bb 2 40.i odd 4 2
7488.2.a.bk 1 4.b odd 2 1
7488.2.a.bz 1 1.a even 1 1 trivial
8112.2.a.v 1 312.h even 2 1
9438.2.a.t 1 264.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7488))\):

\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{7} - 4 \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display
\( T_{17} + 2 \) Copy content Toggle raw display
\( T_{19} - 8 \) Copy content Toggle raw display
\( T_{29} - 6 \) 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
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T - 4 \) Copy content Toggle raw display
$11$ \( T + 4 \) Copy content Toggle raw display
$13$ \( T + 1 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T - 8 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T - 10 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T + 8 \) Copy content Toggle raw display
$53$ \( T + 10 \) Copy content Toggle raw display
$59$ \( T - 4 \) Copy content Toggle raw display
$61$ \( T - 2 \) Copy content Toggle raw display
$67$ \( T - 16 \) Copy content Toggle raw display
$71$ \( T - 8 \) Copy content Toggle raw display
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T + 14 \) Copy content Toggle raw display
$97$ \( T - 10 \) Copy content Toggle raw display
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