Properties

Label 7623.2.a.ct
Level 7623
Weight 2
Character orbit 7623.a
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 8
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{5} + \beta_{6} ) q^{4} + ( -2 - \beta_{2} - \beta_{3} ) q^{5} + q^{7} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{5} + \beta_{6} ) q^{4} + ( -2 - \beta_{2} - \beta_{3} ) q^{5} + q^{7} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{8} + ( 1 + 2 \beta_{1} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{10} + ( -1 + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{13} -\beta_{1} q^{14} + ( \beta_{1} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{16} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{17} + ( -2 - \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} ) q^{19} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{20} + ( \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{23} + ( 2 + \beta_{2} + 3 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{25} + ( 2 \beta_{1} + \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{26} + ( 1 + \beta_{5} + \beta_{6} ) q^{28} + ( \beta_{1} - \beta_{2} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{29} + ( -\beta_{1} - 3 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{31} + ( -3 - \beta_{2} - 2 \beta_{3} + \beta_{6} - \beta_{7} ) q^{32} + ( -2 - \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{34} + ( -2 - \beta_{2} - \beta_{3} ) q^{35} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{7} ) q^{37} + ( 2 + \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{38} + ( 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{5} + 3 \beta_{6} ) q^{40} + ( 1 + \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{41} + ( -2 - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{43} + ( -1 + 3 \beta_{1} - 2 \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{46} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{6} - 2 \beta_{7} ) q^{47} + q^{49} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 5 \beta_{5} - 4 \beta_{7} ) q^{50} + ( -4 - \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{52} + ( -4 - \beta_{1} - 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{7} ) q^{53} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{56} + ( -2 + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{58} + ( 1 - 3 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{59} + ( 3 + 3 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{61} + ( 3 + \beta_{2} - \beta_{3} - \beta_{4} + 6 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{62} + ( 1 + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{64} + ( -3 + 2 \beta_{1} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + 3 \beta_{7} ) q^{65} + ( -3 + 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{67} + ( -1 + 5 \beta_{1} - \beta_{2} + 4 \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{68} + ( 1 + 2 \beta_{1} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{70} + ( 1 + \beta_{1} + 3 \beta_{2} - 3 \beta_{4} - 3 \beta_{6} - \beta_{7} ) q^{71} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{4} + 5 \beta_{5} - 2 \beta_{6} ) q^{73} + ( -4 - 2 \beta_{2} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{74} + ( 3 - 2 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{76} + ( 2 \beta_{1} - \beta_{3} - 4 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{79} + ( -3 - 4 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 10 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{80} + ( -3 \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{82} + ( 4 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{83} + ( -1 - 3 \beta_{1} - 3 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{85} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + 5 \beta_{5} - \beta_{6} + \beta_{7} ) q^{86} + ( -4 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{89} + ( -1 + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{91} + ( -7 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 5 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{92} + ( -5 + 2 \beta_{2} + 4 \beta_{3} + \beta_{4} - 6 \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{94} + ( 2 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 5 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{95} + ( 5 - 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{7} ) q^{97} -\beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - q^{2} + 7q^{4} - 10q^{5} + 8q^{7} + O(q^{10}) \) \( 8q - q^{2} + 7q^{4} - 10q^{5} + 8q^{7} + 6q^{10} - 6q^{13} - q^{14} + q^{16} + 5q^{17} - 13q^{19} - 23q^{20} - 16q^{23} + 16q^{25} + 6q^{26} + 7q^{28} - 9q^{29} + 9q^{31} - 16q^{32} - 12q^{34} - 10q^{35} + 7q^{37} + 10q^{38} + 5q^{40} + 10q^{41} - 4q^{43} + 4q^{46} - 16q^{47} + 8q^{49} - 6q^{50} - 41q^{52} - 37q^{53} - 15q^{58} - q^{59} + 19q^{61} + 18q^{62} - 4q^{64} + 4q^{65} - 19q^{67} - 9q^{68} + 6q^{70} - 13q^{71} - 25q^{73} - 33q^{74} + 26q^{76} - 4q^{80} - 13q^{82} + 25q^{83} + 3q^{85} - 4q^{86} - 37q^{89} - 6q^{91} - 35q^{92} - 42q^{94} - 21q^{95} + 15q^{97} - q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - 11 x^{6} + 10 x^{5} + 35 x^{4} - 30 x^{3} - 30 x^{2} + 30 x - 5\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + 2 \nu^{6} - 5 \nu^{5} - 30 \nu^{4} - 30 \nu^{3} + 130 \nu^{2} + 135 \nu - 140 \)\()/25\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{7} - 4 \nu^{6} + 35 \nu^{5} + 35 \nu^{4} - 165 \nu^{3} - 60 \nu^{2} + 180 \nu - 45 \)\()/25\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{7} + 6 \nu^{6} - 40 \nu^{5} - 65 \nu^{4} + 160 \nu^{3} + 165 \nu^{2} - 170 \nu - 20 \)\()/25\)
\(\beta_{5}\)\(=\)\((\)\( -9 \nu^{7} + 7 \nu^{6} + 95 \nu^{5} - 55 \nu^{4} - 280 \nu^{3} + 105 \nu^{2} + 210 \nu - 90 \)\()/25\)
\(\beta_{6}\)\(=\)\((\)\( 9 \nu^{7} - 7 \nu^{6} - 95 \nu^{5} + 55 \nu^{4} + 280 \nu^{3} - 80 \nu^{2} - 210 \nu + 15 \)\()/25\)
\(\beta_{7}\)\(=\)\((\)\( 12 \nu^{7} - \nu^{6} - 135 \nu^{5} + 15 \nu^{4} + 440 \nu^{3} - 90 \nu^{2} - 405 \nu + 170 \)\()/25\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + \beta_{5} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{7} + 6 \beta_{6} + 7 \beta_{5} - \beta_{4} + \beta_{1} + 14\)
\(\nu^{5}\)\(=\)\(\beta_{7} + 7 \beta_{6} + 8 \beta_{5} + 8 \beta_{4} + 10 \beta_{3} - 7 \beta_{2} + 28 \beta_{1} + 3\)
\(\nu^{6}\)\(=\)\(11 \beta_{7} + 35 \beta_{6} + 47 \beta_{5} - 7 \beta_{4} + 2 \beta_{3} + \beta_{2} + 10 \beta_{1} + 77\)
\(\nu^{7}\)\(=\)\(13 \beta_{7} + 45 \beta_{6} + 56 \beta_{5} + 54 \beta_{4} + 76 \beta_{3} - 42 \beta_{2} + 165 \beta_{1} + 31\)

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
2.55194
1.98451
1.11447
0.669744
0.226211
−1.40927
−1.70716
−2.43045
−2.55194 0 4.51241 −3.42898 0 1.00000 −6.41153 0 8.75055
1.2 −1.98451 0 1.93830 0.0269243 0 1.00000 0.122446 0 −0.0534317
1.3 −1.11447 0 −0.757964 −3.45608 0 1.00000 3.07366 0 3.85168
1.4 −0.669744 0 −1.55144 −2.14378 0 1.00000 2.37856 0 1.43578
1.5 −0.226211 0 −1.94883 2.49552 0 1.00000 0.893270 0 −0.564516
1.6 1.40927 0 −0.0139645 1.83139 0 1.00000 −2.83822 0 2.58091
1.7 1.70716 0 0.914391 −4.06637 0 1.00000 −1.85331 0 −6.94194
1.8 2.43045 0 3.90710 −1.25863 0 1.00000 4.63512 0 −3.05904
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.ct 8
3.b odd 2 1 847.2.a.p 8
11.b odd 2 1 7623.2.a.cw 8
11.c even 5 2 693.2.m.i 16
21.c even 2 1 5929.2.a.bt 8
33.d even 2 1 847.2.a.o 8
33.f even 10 2 847.2.f.v 16
33.f even 10 2 847.2.f.x 16
33.h odd 10 2 77.2.f.b 16
33.h odd 10 2 847.2.f.w 16
231.h odd 2 1 5929.2.a.bs 8
231.u even 10 2 539.2.f.e 16
231.z odd 30 4 539.2.q.g 32
231.bc even 30 4 539.2.q.f 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.b 16 33.h odd 10 2
539.2.f.e 16 231.u even 10 2
539.2.q.f 32 231.bc even 30 4
539.2.q.g 32 231.z odd 30 4
693.2.m.i 16 11.c even 5 2
847.2.a.o 8 33.d even 2 1
847.2.a.p 8 3.b odd 2 1
847.2.f.v 16 33.f even 10 2
847.2.f.w 16 33.h odd 10 2
847.2.f.x 16 33.f even 10 2
5929.2.a.bs 8 231.h odd 2 1
5929.2.a.bt 8 21.c even 2 1
7623.2.a.ct 8 1.a even 1 1 trivial
7623.2.a.cw 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\(T_{2}^{8} + \cdots\)
\(T_{5}^{8} + \cdots\)
\(T_{13}^{8} + \cdots\)