Properties

Label 7650.2.a.ci
Level $7650$
Weight $2$
Character orbit 7650.a
Self dual yes
Analytic conductor $61.086$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7650 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7650.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.0855575463\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 34)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + 4q^{7} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} + 4q^{7} + q^{8} - 6q^{11} - 2q^{13} + 4q^{14} + q^{16} - q^{17} - 4q^{19} - 6q^{22} - 2q^{26} + 4q^{28} - 4q^{31} + q^{32} - q^{34} + 4q^{37} - 4q^{38} - 6q^{41} - 8q^{43} - 6q^{44} + 9q^{49} - 2q^{52} - 6q^{53} + 4q^{56} - 4q^{61} - 4q^{62} + q^{64} - 8q^{67} - q^{68} - 2q^{73} + 4q^{74} - 4q^{76} - 24q^{77} + 8q^{79} - 6q^{82} - 8q^{86} - 6q^{88} + 6q^{89} - 8q^{91} - 14q^{97} + 9q^{98} + O(q^{100}) \)

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
1.00000 0 1.00000 0 0 4.00000 1.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(17\) \(1\)

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 7650.2.a.ci 1
3.b odd 2 1 850.2.a.e 1
5.b even 2 1 306.2.a.a 1
12.b even 2 1 6800.2.a.b 1
15.d odd 2 1 34.2.a.a 1
15.e even 4 2 850.2.c.b 2
20.d odd 2 1 2448.2.a.k 1
40.e odd 2 1 9792.2.a.bj 1
40.f even 2 1 9792.2.a.y 1
60.h even 2 1 272.2.a.d 1
85.c even 2 1 5202.2.a.d 1
105.g even 2 1 1666.2.a.m 1
120.i odd 2 1 1088.2.a.l 1
120.m even 2 1 1088.2.a.d 1
165.d even 2 1 4114.2.a.a 1
195.e odd 2 1 5746.2.a.b 1
255.h odd 2 1 578.2.a.a 1
255.i odd 4 2 578.2.b.a 2
255.y odd 8 4 578.2.c.e 4
255.be even 16 8 578.2.d.e 8
1020.b even 2 1 4624.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
34.2.a.a 1 15.d odd 2 1
272.2.a.d 1 60.h even 2 1
306.2.a.a 1 5.b even 2 1
578.2.a.a 1 255.h odd 2 1
578.2.b.a 2 255.i odd 4 2
578.2.c.e 4 255.y odd 8 4
578.2.d.e 8 255.be even 16 8
850.2.a.e 1 3.b odd 2 1
850.2.c.b 2 15.e even 4 2
1088.2.a.d 1 120.m even 2 1
1088.2.a.l 1 120.i odd 2 1
1666.2.a.m 1 105.g even 2 1
2448.2.a.k 1 20.d odd 2 1
4114.2.a.a 1 165.d even 2 1
4624.2.a.a 1 1020.b even 2 1
5202.2.a.d 1 85.c even 2 1
5746.2.a.b 1 195.e odd 2 1
6800.2.a.b 1 12.b even 2 1
7650.2.a.ci 1 1.a even 1 1 trivial
9792.2.a.y 1 40.f even 2 1
9792.2.a.bj 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7650))\):

\( T_{7} - 4 \)
\( T_{11} + 6 \)
\( T_{13} + 2 \)
\( T_{19} + 4 \)
\( T_{23} \)
\( T_{29} \)

Hecke characteristic polynomials

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