Properties

Label 7623.2.a.cs
Level 7623
Weight 2
Character orbit 7623.a
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 6
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: \(6\)
Coefficient field: 6.6.7674048.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 847)
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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 1 - \beta_{1} + \beta_{2} ) q^{4} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{5} - q^{7} + ( 2 + 2 \beta_{2} - \beta_{3} ) q^{8} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 1 - \beta_{1} + \beta_{2} ) q^{4} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{5} - q^{7} + ( 2 + 2 \beta_{2} - \beta_{3} ) q^{8} + ( 1 - \beta_{2} - \beta_{5} ) q^{10} + ( -1 - \beta_{2} + \beta_{4} - \beta_{5} ) q^{13} + ( -1 + \beta_{1} ) q^{14} + ( 2 - 2 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{16} + ( 4 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{17} + ( -1 - \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{19} + ( -1 + \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} ) q^{20} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{23} + ( -4 \beta_{2} - 2 \beta_{5} ) q^{25} + ( -2 + \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{26} + ( -1 + \beta_{1} - \beta_{2} ) q^{28} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{29} + ( -1 + \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{31} + ( 2 - 3 \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{32} + ( 5 - 3 \beta_{1} + \beta_{2} + \beta_{4} ) q^{34} + ( -1 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{35} + ( 4 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{37} + ( 4 + 2 \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{38} + ( -2 - \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{40} + ( 4 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{41} + ( 2 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{43} + ( -1 + \beta_{2} + 3 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{46} + ( 3 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{47} + q^{49} + ( -2 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{50} + ( -3 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{52} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{53} + ( -2 - 2 \beta_{2} + \beta_{3} ) q^{56} + ( 1 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{58} + ( 3 - 4 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{5} ) q^{59} + ( 4 - 6 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{61} + ( 4 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{62} + ( 3 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{64} + ( 3 - 3 \beta_{2} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{65} + ( -1 + 4 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{67} + ( 3 - 4 \beta_{1} + 5 \beta_{2} - \beta_{5} ) q^{68} + ( -1 + \beta_{2} + \beta_{5} ) q^{70} + ( -4 - \beta_{1} + \beta_{2} - 2 \beta_{5} ) q^{71} + ( -1 - \beta_{1} + 4 \beta_{2} + \beta_{3} + 2 \beta_{4} + 3 \beta_{5} ) q^{73} + ( 9 - 4 \beta_{1} + 3 \beta_{2} - 3 \beta_{4} + \beta_{5} ) q^{74} + ( 7 - 3 \beta_{1} - \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{76} + ( 6 - 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{79} + ( 2 - \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{80} + ( -3 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{82} + ( 3 + 3 \beta_{1} + 3 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{83} + ( 6 - 5 \beta_{1} - 5 \beta_{2} - \beta_{3} + 5 \beta_{4} + \beta_{5} ) q^{85} + ( 5 - 2 \beta_{1} + 3 \beta_{2} - 5 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{86} + ( \beta_{1} + 4 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{89} + ( 1 + \beta_{2} - \beta_{4} + \beta_{5} ) q^{91} + ( -\beta_{1} - 3 \beta_{2} + 4 \beta_{3} + 4 \beta_{5} ) q^{92} + ( 7 - \beta_{1} + 5 \beta_{3} + \beta_{5} ) q^{94} + ( -5 + \beta_{2} + 3 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{95} + ( -1 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{97} + ( 1 - \beta_{1} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 4q^{2} + 4q^{4} + 4q^{5} - 6q^{7} + 12q^{8} + O(q^{10}) \) \( 6q + 4q^{2} + 4q^{4} + 4q^{5} - 6q^{7} + 12q^{8} + 8q^{10} - 4q^{13} - 4q^{14} + 8q^{16} + 22q^{17} - 6q^{19} - 2q^{20} - 2q^{23} + 4q^{25} - 6q^{26} - 4q^{28} + 12q^{29} - 2q^{31} + 8q^{32} + 24q^{34} - 4q^{35} + 14q^{37} + 22q^{38} - 18q^{40} + 26q^{41} + 4q^{43} - 12q^{46} + 16q^{47} + 6q^{49} - 4q^{50} - 12q^{52} - 4q^{53} - 12q^{56} - 2q^{58} + 4q^{59} + 8q^{61} + 20q^{62} + 26q^{64} + 24q^{65} + 6q^{67} + 12q^{68} - 8q^{70} - 22q^{71} - 14q^{73} + 44q^{74} + 30q^{76} + 28q^{79} + 4q^{80} - 4q^{82} + 22q^{83} + 24q^{85} + 30q^{86} + 4q^{91} - 10q^{92} + 38q^{94} - 24q^{95} - 4q^{97} + 4q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 5 x^{4} + 8 x^{3} + 7 x^{2} - 6 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 3 \nu + 1 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - \nu^{3} - 4 \nu^{2} + 2 \nu + 2 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - 2 \nu^{4} - 4 \nu^{3} + 6 \nu^{2} + 4 \nu - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{4} + \beta_{3} + 5 \beta_{2} + 6 \beta_{1} + 7\)
\(\nu^{5}\)\(=\)\(\beta_{5} + 2 \beta_{4} + 6 \beta_{3} + 8 \beta_{2} + 18 \beta_{1} + 8\)

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
2.38595
1.82356
0.879640
−0.276564
−1.10939
−1.70320
−1.38595 0 −0.0791355 0.133004 0 −1.00000 2.88158 0 −0.184338
1.2 −0.823556 0 −1.32176 −2.98565 0 −1.00000 2.73565 0 2.45885
1.3 0.120360 0 −1.98551 2.80853 0 −1.00000 −0.479696 0 0.338034
1.4 1.27656 0 −0.370384 4.09144 0 −1.00000 −3.02595 0 5.22298
1.5 2.10939 0 2.44952 −0.492391 0 −1.00000 0.948212 0 −1.03864
1.6 2.70320 0 5.30727 0.445072 0 −1.00000 8.94020 0 1.20312
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.cs 6
3.b odd 2 1 847.2.a.m 6
11.b odd 2 1 7623.2.a.cp 6
21.c even 2 1 5929.2.a.bj 6
33.d even 2 1 847.2.a.n yes 6
33.f even 10 4 847.2.f.y 24
33.h odd 10 4 847.2.f.z 24
231.h odd 2 1 5929.2.a.bm 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.m 6 3.b odd 2 1
847.2.a.n yes 6 33.d even 2 1
847.2.f.y 24 33.f even 10 4
847.2.f.z 24 33.h odd 10 4
5929.2.a.bj 6 21.c even 2 1
5929.2.a.bm 6 231.h odd 2 1
7623.2.a.cp 6 11.b odd 2 1
7623.2.a.cs 6 1.a even 1 1 trivial

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{6} - 4 T_{2}^{5} + 12 T_{2}^{3} - 4 T_{2}^{2} - 8 T_{2} + 1 \)
\( T_{5}^{6} - 4 T_{5}^{5} - 9 T_{5}^{4} + 36 T_{5}^{3} - T_{5}^{2} - 8 T_{5} + 1 \)
\( T_{13}^{6} + 4 T_{13}^{5} - 19 T_{13}^{4} - 72 T_{13}^{3} - 36 T_{13}^{2} + 32 T_{13} + 16 \)