Properties

Label 7623.2.a.f
Level $7623$
Weight $2$
Character orbit 7623.a
Self dual yes
Analytic conductor $60.870$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - q^{4} + 2 q^{5} - q^{7} + 3 q^{8} + O(q^{10}) \) \( q - q^{2} - q^{4} + 2 q^{5} - q^{7} + 3 q^{8} - 2 q^{10} - 6 q^{13} + q^{14} - q^{16} + 2 q^{17} - 4 q^{19} - 2 q^{20} - q^{25} + 6 q^{26} + q^{28} - 2 q^{29} + 8 q^{31} - 5 q^{32} - 2 q^{34} - 2 q^{35} + 6 q^{37} + 4 q^{38} + 6 q^{40} + 10 q^{41} + 4 q^{43} + 8 q^{47} + q^{49} + q^{50} + 6 q^{52} - 6 q^{53} - 3 q^{56} + 2 q^{58} - 4 q^{59} + 10 q^{61} - 8 q^{62} + 7 q^{64} - 12 q^{65} - 12 q^{67} - 2 q^{68} + 2 q^{70} - 2 q^{73} - 6 q^{74} + 4 q^{76} - 16 q^{79} - 2 q^{80} - 10 q^{82} + 4 q^{83} + 4 q^{85} - 4 q^{86} - 18 q^{89} + 6 q^{91} - 8 q^{94} - 8 q^{95} + 2 q^{97} - q^{98} + O(q^{100}) \)

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 2.00000 0 −1.00000 3.00000 0 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(11\) \(-1\)

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 7623.2.a.f 1
3.b odd 2 1 2541.2.a.h 1
11.b odd 2 1 693.2.a.d 1
33.d even 2 1 231.2.a.a 1
77.b even 2 1 4851.2.a.p 1
132.d odd 2 1 3696.2.a.t 1
165.d even 2 1 5775.2.a.t 1
231.h odd 2 1 1617.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.a 1 33.d even 2 1
693.2.a.d 1 11.b odd 2 1
1617.2.a.e 1 231.h odd 2 1
2541.2.a.h 1 3.b odd 2 1
3696.2.a.t 1 132.d odd 2 1
4851.2.a.p 1 77.b even 2 1
5775.2.a.t 1 165.d even 2 1
7623.2.a.f 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2} + 1 \)
\( T_{5} - 2 \)
\( T_{13} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( T \)
$5$ \( -2 + T \)
$7$ \( 1 + T \)
$11$ \( T \)
$13$ \( 6 + T \)
$17$ \( -2 + T \)
$19$ \( 4 + T \)
$23$ \( T \)
$29$ \( 2 + T \)
$31$ \( -8 + T \)
$37$ \( -6 + T \)
$41$ \( -10 + T \)
$43$ \( -4 + T \)
$47$ \( -8 + T \)
$53$ \( 6 + T \)
$59$ \( 4 + T \)
$61$ \( -10 + T \)
$67$ \( 12 + T \)
$71$ \( T \)
$73$ \( 2 + T \)
$79$ \( 16 + T \)
$83$ \( -4 + T \)
$89$ \( 18 + T \)
$97$ \( -2 + T \)
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