Properties

Label 7650.2.a.s
Level $7650$
Weight $2$
Character orbit 7650.a
Self dual yes
Analytic conductor $61.086$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 170)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{8} + O(q^{10}) \) \( q - q^{2} + q^{4} - q^{8} + 6q^{11} - 3q^{13} + q^{16} + q^{17} - 7q^{19} - 6q^{22} - 8q^{23} + 3q^{26} + 5q^{29} + 5q^{31} - q^{32} - q^{34} - 8q^{37} + 7q^{38} + 4q^{43} + 6q^{44} + 8q^{46} + 3q^{47} - 7q^{49} - 3q^{52} + 9q^{53} - 5q^{58} - 5q^{59} - 3q^{61} - 5q^{62} + q^{64} + 2q^{67} + q^{68} + 15q^{71} + 11q^{73} + 8q^{74} - 7q^{76} + 8q^{79} + 4q^{83} - 4q^{86} - 6q^{88} + q^{89} - 8q^{92} - 3q^{94} + 9q^{97} + 7q^{98} + O(q^{100}) \)

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 0 −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.s 1
3.b odd 2 1 850.2.a.h 1
5.b even 2 1 7650.2.a.cb 1
5.c odd 4 2 1530.2.d.b 2
12.b even 2 1 6800.2.a.r 1
15.d odd 2 1 850.2.a.d 1
15.e even 4 2 170.2.c.a 2
60.h even 2 1 6800.2.a.g 1
60.l odd 4 2 1360.2.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.c.a 2 15.e even 4 2
850.2.a.d 1 15.d odd 2 1
850.2.a.h 1 3.b odd 2 1
1360.2.e.b 2 60.l odd 4 2
1530.2.d.b 2 5.c odd 4 2
6800.2.a.g 1 60.h even 2 1
6800.2.a.r 1 12.b even 2 1
7650.2.a.s 1 1.a even 1 1 trivial
7650.2.a.cb 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7650))\):

\( T_{7} \)
\( T_{11} - 6 \)
\( T_{13} + 3 \)
\( T_{19} + 7 \)
\( T_{23} + 8 \)
\( T_{29} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( -6 + T \)
$13$ \( 3 + T \)
$17$ \( -1 + T \)
$19$ \( 7 + T \)
$23$ \( 8 + T \)
$29$ \( -5 + T \)
$31$ \( -5 + T \)
$37$ \( 8 + T \)
$41$ \( T \)
$43$ \( -4 + T \)
$47$ \( -3 + T \)
$53$ \( -9 + T \)
$59$ \( 5 + T \)
$61$ \( 3 + T \)
$67$ \( -2 + T \)
$71$ \( -15 + T \)
$73$ \( -11 + T \)
$79$ \( -8 + T \)
$83$ \( -4 + T \)
$89$ \( -1 + T \)
$97$ \( -9 + T \)
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