Properties

Label 7623.2.a.cg
Level 7623
Weight 2
Character orbit 7623.a
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 4
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.7488.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2541)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( 1 - \beta_{2} ) q^{4} + ( 1 + \beta_{3} ) q^{5} - q^{7} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{3} q^{2} + ( 1 - \beta_{2} ) q^{4} + ( 1 + \beta_{3} ) q^{5} - q^{7} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{8} + ( 3 - \beta_{2} + \beta_{3} ) q^{10} + ( 3 - \beta_{1} ) q^{13} -\beta_{3} q^{14} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{16} + ( -1 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{17} + ( 5 - \beta_{2} + \beta_{3} ) q^{19} + ( 1 + \beta_{1} + 3 \beta_{3} ) q^{20} + ( 2 - 3 \beta_{1} + \beta_{2} ) q^{23} + ( -1 - \beta_{2} + 2 \beta_{3} ) q^{25} + ( 1 + \beta_{1} + 3 \beta_{3} ) q^{26} + ( -1 + \beta_{2} ) q^{28} + ( -\beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{29} + ( -\beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{31} + ( -4 - 2 \beta_{3} ) q^{32} + ( -3 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{34} + ( -1 - \beta_{3} ) q^{35} + ( 1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{37} + ( 3 + \beta_{1} + 7 \beta_{3} ) q^{38} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{40} + ( 1 + \beta_{1} + 3 \beta_{3} ) q^{41} + ( 4 - \beta_{1} - 3 \beta_{3} ) q^{43} + ( 3 + 2 \beta_{1} - \beta_{2} ) q^{46} + ( 2 - \beta_{1} - 4 \beta_{2} ) q^{47} + q^{49} + ( 6 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{50} + ( 2 + \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{52} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{53} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{56} + ( 7 + 4 \beta_{1} + \beta_{2} + 6 \beta_{3} ) q^{58} + ( 1 - 2 \beta_{1} - 3 \beta_{3} ) q^{59} + ( -5 + 4 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{61} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{62} + ( -6 + 4 \beta_{1} + 2 \beta_{2} ) q^{64} + ( 4 + 3 \beta_{3} ) q^{65} + ( -1 + 4 \beta_{1} - 2 \beta_{2} ) q^{67} + ( -9 + 3 \beta_{1} - \beta_{3} ) q^{68} + ( -3 + \beta_{2} - \beta_{3} ) q^{70} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{71} + ( 9 + \beta_{1} + 2 \beta_{3} ) q^{73} + ( 8 + \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{74} + ( 10 - \beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{76} + ( 9 - 6 \beta_{1} - 3 \beta_{2} ) q^{79} + ( -4 + 2 \beta_{2} - 2 \beta_{3} ) q^{80} + ( 8 - \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{82} + ( -2 + 4 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{83} + ( -4 - \beta_{1} + 2 \beta_{2} - 5 \beta_{3} ) q^{85} + ( -8 + \beta_{1} + 3 \beta_{2} + 4 \beta_{3} ) q^{86} + ( -5 - 4 \beta_{1} - 5 \beta_{3} ) q^{89} + ( -3 + \beta_{1} ) q^{91} + ( -6 + 5 \beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{92} + ( 1 + 5 \beta_{1} + 4 \beta_{2} + 10 \beta_{3} ) q^{94} + ( 8 + \beta_{1} - \beta_{2} + 8 \beta_{3} ) q^{95} + ( -1 - \beta_{1} + 2 \beta_{3} ) q^{97} + \beta_{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} + 4q^{4} + 2q^{5} - 4q^{7} + O(q^{10}) \) \( 4q - 2q^{2} + 4q^{4} + 2q^{5} - 4q^{7} + 10q^{10} + 10q^{13} + 2q^{14} - 6q^{17} + 18q^{19} + 2q^{23} - 8q^{25} - 4q^{28} - 6q^{29} - 12q^{32} - 2q^{34} - 2q^{35} - 4q^{37} + 8q^{40} + 20q^{43} + 16q^{46} + 6q^{47} + 4q^{49} + 24q^{50} + 8q^{52} + 24q^{58} + 6q^{59} - 10q^{61} - 16q^{64} + 10q^{65} + 4q^{67} - 28q^{68} - 10q^{70} + 6q^{71} + 34q^{73} + 36q^{74} + 36q^{76} + 24q^{79} - 12q^{80} + 28q^{82} - 6q^{83} - 8q^{85} - 38q^{86} - 18q^{89} - 10q^{91} - 24q^{92} - 6q^{94} + 18q^{95} - 10q^{97} - 2q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 4 x^{2} + 2 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 4 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 3\)

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
0.698857
3.05896
−1.43091
−0.326909
−2.43091 0 3.90931 −1.43091 0 −1.00000 −4.64136 0 3.47841
1.2 −1.32691 0 −0.239314 −0.326909 0 −1.00000 2.97136 0 0.433778
1.3 −0.301143 0 −1.90931 0.698857 0 −1.00000 1.17726 0 −0.210456
1.4 2.05896 0 2.23931 3.05896 0 −1.00000 0.492737 0 6.29827
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.cg 4
3.b odd 2 1 2541.2.a.bp yes 4
11.b odd 2 1 7623.2.a.cn 4
33.d even 2 1 2541.2.a.bl 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.2.a.bl 4 33.d even 2 1
2541.2.a.bp yes 4 3.b odd 2 1
7623.2.a.cg 4 1.a even 1 1 trivial
7623.2.a.cn 4 11.b odd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{4} + 2 T_{2}^{3} - 4 T_{2}^{2} - 8 T_{2} - 2 \)
\( T_{5}^{4} - 2 T_{5}^{3} - 4 T_{5}^{2} + 2 T_{5} + 1 \)
\( T_{13}^{4} - 10 T_{13}^{3} + 32 T_{13}^{2} - 32 T_{13} - 2 \)