Properties

Label 7623.2.a.bg
Level 7623
Weight 2
Character orbit 7623.a
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{13}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 q^{4} -\beta q^{5} - q^{7} +O(q^{10})\) \( q -2 q^{4} -\beta q^{5} - q^{7} + 2 q^{13} + 4 q^{16} -\beta q^{17} + 6 q^{19} + 2 \beta q^{20} -2 \beta q^{23} + 8 q^{25} + 2 q^{28} -2 \beta q^{29} -2 q^{31} + \beta q^{35} -2 q^{37} -2 \beta q^{41} -5 q^{43} -\beta q^{47} + q^{49} -4 q^{52} + 3 \beta q^{59} -14 q^{61} -8 q^{64} -2 \beta q^{65} -15 q^{67} + 2 \beta q^{68} + 4 \beta q^{71} + 4 q^{73} -12 q^{76} -4 \beta q^{80} -3 \beta q^{83} + 13 q^{85} -3 \beta q^{89} -2 q^{91} + 4 \beta q^{92} -6 \beta q^{95} -8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{4} - 2q^{7} + O(q^{10}) \) \( 2q - 4q^{4} - 2q^{7} + 4q^{13} + 8q^{16} + 12q^{19} + 16q^{25} + 4q^{28} - 4q^{31} - 4q^{37} - 10q^{43} + 2q^{49} - 8q^{52} - 28q^{61} - 16q^{64} - 30q^{67} + 8q^{73} - 24q^{76} + 26q^{85} - 4q^{91} - 16q^{97} + O(q^{100}) \)

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
2.30278
−1.30278
0 0 −2.00000 −3.60555 0 −1.00000 0 0 0
1.2 0 0 −2.00000 3.60555 0 −1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7623.2.a.bg 2
3.b odd 2 1 inner 7623.2.a.bg 2
11.b odd 2 1 7623.2.a.bh yes 2
33.d even 2 1 7623.2.a.bh yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7623.2.a.bg 2 1.a even 1 1 trivial
7623.2.a.bg 2 3.b odd 2 1 inner
7623.2.a.bh yes 2 11.b odd 2 1
7623.2.a.bh yes 2 33.d even 2 1

Atkin-Lehner signs

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

Hecke kernels

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

\( T_{2} \)
\( T_{5}^{2} - 13 \)
\( T_{13} - 2 \)