Properties

Label 7623.2.a.cq
Level 7623
Weight 2
Character orbit 7623.a
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 6
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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.3829849.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -\beta_{1} + \beta_{3} ) q^{5} - q^{7} + ( \beta_{1} - \beta_{3} + \beta_{5} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -\beta_{1} + \beta_{3} ) q^{5} - q^{7} + ( \beta_{1} - \beta_{3} + \beta_{5} ) q^{8} + ( -2 - \beta_{2} ) q^{10} + ( \beta_{2} + \beta_{4} ) q^{13} -\beta_{1} q^{14} + ( 1 - \beta_{4} ) q^{16} + 2 \beta_{3} q^{17} + ( -\beta_{2} + \beta_{4} ) q^{19} + ( -2 \beta_{1} - \beta_{3} - \beta_{5} ) q^{20} + ( -2 \beta_{3} + 2 \beta_{5} ) q^{23} + ( -1 - \beta_{2} - \beta_{4} ) q^{25} + ( \beta_{1} - \beta_{5} ) q^{26} + ( -1 - \beta_{2} ) q^{28} + ( \beta_{1} - \beta_{3} - 2 \beta_{5} ) q^{29} -2 \beta_{4} q^{31} + \beta_{3} q^{32} + 2 q^{34} + ( \beta_{1} - \beta_{3} ) q^{35} + ( 2 + \beta_{2} + \beta_{4} ) q^{37} + ( -3 \beta_{1} + 2 \beta_{3} - 3 \beta_{5} ) q^{38} + ( -4 - \beta_{2} + \beta_{4} ) q^{40} + ( 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} ) q^{41} + ( -4 - 2 \beta_{2} ) q^{43} + ( 2 \beta_{2} - 2 \beta_{4} ) q^{46} + ( -\beta_{1} - \beta_{3} - 2 \beta_{5} ) q^{47} + q^{49} + ( -2 \beta_{1} + \beta_{5} ) q^{50} + ( 2 - 2 \beta_{2} - \beta_{4} ) q^{52} + ( -4 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} ) q^{53} + ( -\beta_{1} + \beta_{3} - \beta_{5} ) q^{56} + ( -\beta_{2} + 2 \beta_{4} ) q^{58} + ( 3 \beta_{1} + 3 \beta_{3} - 2 \beta_{5} ) q^{59} + ( -4 - 2 \beta_{2} - 2 \beta_{4} ) q^{61} + ( 2 \beta_{1} - 2 \beta_{3} + 4 \beta_{5} ) q^{62} + ( -1 + 2 \beta_{4} ) q^{64} + ( \beta_{1} - 5 \beta_{3} ) q^{65} + ( 4 + \beta_{2} - \beta_{4} ) q^{67} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{68} + ( 2 + \beta_{2} ) q^{70} + ( -6 \beta_{1} - 2 \beta_{3} ) q^{71} + ( -8 - \beta_{2} - \beta_{4} ) q^{73} + ( 3 \beta_{1} - \beta_{5} ) q^{74} + ( -10 - 4 \beta_{2} + \beta_{4} ) q^{76} + 4 \beta_{2} q^{79} + ( -3 \beta_{1} + 4 \beta_{3} - \beta_{5} ) q^{80} + ( 2 + 2 \beta_{4} ) q^{82} + ( -4 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} ) q^{83} + ( 4 - 4 \beta_{2} - 2 \beta_{4} ) q^{85} + ( -8 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} ) q^{86} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{89} + ( -\beta_{2} - \beta_{4} ) q^{91} + ( 6 \beta_{1} + 2 \beta_{5} ) q^{92} + ( -6 - 3 \beta_{2} + 2 \beta_{4} ) q^{94} + ( 3 \beta_{1} - \beta_{3} + 2 \beta_{5} ) q^{95} + ( -2 + 2 \beta_{4} ) q^{97} + \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 4q^{4} - 6q^{7} + O(q^{10}) \) \( 6q + 4q^{4} - 6q^{7} - 10q^{10} - 4q^{13} + 8q^{16} - 2q^{25} - 4q^{28} + 4q^{31} + 12q^{34} + 8q^{37} - 24q^{40} - 20q^{43} + 6q^{49} + 18q^{52} - 2q^{58} - 16q^{61} - 10q^{64} + 24q^{67} + 10q^{70} - 44q^{73} - 54q^{76} - 8q^{79} + 8q^{82} + 36q^{85} + 4q^{91} - 34q^{94} - 16q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 6 x^{4} + 15 x^{3} - 5 x^{2} - 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{4} - 6 \nu^{2} + 3 \nu + 1 \)
\(\beta_{2}\)\(=\)\( -\nu^{5} + \nu^{4} + 7 \nu^{3} - 8 \nu^{2} - 2 \nu \)
\(\beta_{3}\)\(=\)\( \nu^{5} - \nu^{4} - 7 \nu^{3} + 8 \nu^{2} + 4 \nu - 1 \)
\(\beta_{4}\)\(=\)\( -2 \nu^{5} + 3 \nu^{4} + 14 \nu^{3} - 24 \nu^{2} - 5 \nu + 8 \)
\(\beta_{5}\)\(=\)\( 3 \nu^{5} - 4 \nu^{4} - 19 \nu^{3} + 32 \nu^{2} - 3 \nu - 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{4} - 2 \beta_{3} + \beta_{1} + 5\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{5} + \beta_{4} + 5 \beta_{3} + 6 \beta_{2} + 2\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-6 \beta_{4} - 15 \beta_{3} - 3 \beta_{2} + 8 \beta_{1} + 25\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(7 \beta_{5} + 9 \beta_{4} + 34 \beta_{3} + 35 \beta_{2} - 3\)\()/2\)

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
1.81705
−0.430314
−2.63651
0.725011
0.300853
2.22391
−2.45784 0 4.04096 2.05098 0 −1.00000 −5.01636 0 −5.04096
1.2 −1.36768 0 −0.129461 0.636509 0 −1.00000 2.91241 0 −0.870539
1.3 −0.297484 0 −1.91150 −3.06404 0 −1.00000 1.16361 0 0.911503
1.4 0.297484 0 −1.91150 3.06404 0 −1.00000 −1.16361 0 0.911503
1.5 1.36768 0 −0.129461 −0.636509 0 −1.00000 −2.91241 0 −0.870539
1.6 2.45784 0 4.04096 −2.05098 0 −1.00000 5.01636 0 −5.04096
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.cq 6
3.b odd 2 1 inner 7623.2.a.cq 6
11.b odd 2 1 7623.2.a.cr yes 6
33.d even 2 1 7623.2.a.cr yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7623.2.a.cq 6 1.a even 1 1 trivial
7623.2.a.cq 6 3.b odd 2 1 inner
7623.2.a.cr yes 6 11.b odd 2 1
7623.2.a.cr yes 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{6} - 8 T_{2}^{4} + 12 T_{2}^{2} - 1 \)
\( T_{5}^{6} - 14 T_{5}^{4} + 45 T_{5}^{2} - 16 \)
\( T_{13}^{3} + 2 T_{13}^{2} - 19 T_{13} - 4 \)