Properties

Label 7488.2.a.g
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 26)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{5} - q^{7} + 6 q^{11} - q^{13} + 3 q^{17} - 2 q^{19} + 4 q^{25} + 6 q^{29} - 4 q^{31} + 3 q^{35} + 7 q^{37} + q^{43} - 3 q^{47} - 6 q^{49} - 18 q^{55} - 6 q^{59} - 8 q^{61} + 3 q^{65} - 14 q^{67} + 3 q^{71} + 2 q^{73} - 6 q^{77} + 8 q^{79} + 12 q^{83} - 9 q^{85} + 6 q^{89} + q^{91} + 6 q^{95} - 10 q^{97}+O(q^{100}) \) 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 −3.00000 0 −1.00000 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.g 1
3.b odd 2 1 832.2.a.d 1
4.b odd 2 1 7488.2.a.h 1
8.b even 2 1 234.2.a.e 1
8.d odd 2 1 1872.2.a.q 1
12.b even 2 1 832.2.a.i 1
24.f even 2 1 208.2.a.a 1
24.h odd 2 1 26.2.a.a 1
40.f even 2 1 5850.2.a.p 1
40.i odd 4 2 5850.2.e.a 2
48.i odd 4 2 3328.2.b.m 2
48.k even 4 2 3328.2.b.j 2
72.j odd 6 2 2106.2.e.ba 2
72.n even 6 2 2106.2.e.b 2
104.e even 2 1 3042.2.a.a 1
104.j odd 4 2 3042.2.b.a 2
120.i odd 2 1 650.2.a.j 1
120.m even 2 1 5200.2.a.x 1
120.w even 4 2 650.2.b.d 2
168.i even 2 1 1274.2.a.d 1
168.s odd 6 2 1274.2.f.p 2
168.ba even 6 2 1274.2.f.r 2
264.m even 2 1 3146.2.a.n 1
312.b odd 2 1 338.2.a.f 1
312.h even 2 1 2704.2.a.f 1
312.w odd 4 2 2704.2.f.d 2
312.y even 4 2 338.2.b.c 2
312.bg odd 6 2 338.2.c.a 2
312.bh odd 6 2 338.2.c.d 2
312.bo even 12 4 338.2.e.a 4
408.b odd 2 1 7514.2.a.c 1
456.p even 2 1 9386.2.a.j 1
1560.y odd 2 1 8450.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.a 1 24.h odd 2 1
208.2.a.a 1 24.f even 2 1
234.2.a.e 1 8.b even 2 1
338.2.a.f 1 312.b odd 2 1
338.2.b.c 2 312.y even 4 2
338.2.c.a 2 312.bg odd 6 2
338.2.c.d 2 312.bh odd 6 2
338.2.e.a 4 312.bo even 12 4
650.2.a.j 1 120.i odd 2 1
650.2.b.d 2 120.w even 4 2
832.2.a.d 1 3.b odd 2 1
832.2.a.i 1 12.b even 2 1
1274.2.a.d 1 168.i even 2 1
1274.2.f.p 2 168.s odd 6 2
1274.2.f.r 2 168.ba even 6 2
1872.2.a.q 1 8.d odd 2 1
2106.2.e.b 2 72.n even 6 2
2106.2.e.ba 2 72.j odd 6 2
2704.2.a.f 1 312.h even 2 1
2704.2.f.d 2 312.w odd 4 2
3042.2.a.a 1 104.e even 2 1
3042.2.b.a 2 104.j odd 4 2
3146.2.a.n 1 264.m even 2 1
3328.2.b.j 2 48.k even 4 2
3328.2.b.m 2 48.i odd 4 2
5200.2.a.x 1 120.m even 2 1
5850.2.a.p 1 40.f even 2 1
5850.2.e.a 2 40.i odd 4 2
7488.2.a.g 1 1.a even 1 1 trivial
7488.2.a.h 1 4.b odd 2 1
7514.2.a.c 1 408.b odd 2 1
8450.2.a.c 1 1560.y odd 2 1
9386.2.a.j 1 456.p even 2 1

Hecke kernels

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

\( T_{5} + 3 \) Copy content Toggle raw display
\( T_{7} + 1 \) Copy content Toggle raw display
\( T_{11} - 6 \) Copy content Toggle raw display
\( T_{17} - 3 \) Copy content Toggle raw display
\( T_{19} + 2 \) 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
$5$ \( T + 3 \) Copy content Toggle raw display
$7$ \( T + 1 \) Copy content Toggle raw display
$11$ \( T - 6 \) Copy content Toggle raw display
$13$ \( T + 1 \) Copy content Toggle raw display
$17$ \( T - 3 \) Copy content Toggle raw display
$19$ \( T + 2 \) 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
$37$ \( T - 7 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T - 1 \) Copy content Toggle raw display
$47$ \( T + 3 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T + 6 \) Copy content Toggle raw display
$61$ \( T + 8 \) Copy content Toggle raw display
$67$ \( T + 14 \) Copy content Toggle raw display
$71$ \( T - 3 \) 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
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T + 10 \) Copy content Toggle raw display
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