Properties

Label 7623.2.a.ce
Level 7623
Weight 2
Character orbit 7623.a
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 3
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: \(3\)
Coefficient field: 3.3.568.1
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 3 + \beta_{2} ) q^{4} + \beta_{2} q^{5} - q^{7} + ( 3 - \beta_{1} + 2 \beta_{2} ) q^{8} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 3 + \beta_{2} ) q^{4} + \beta_{2} q^{5} - q^{7} + ( 3 - \beta_{1} + 2 \beta_{2} ) q^{8} + ( 2 + 2 \beta_{2} ) q^{10} + ( -2 - \beta_{1} + \beta_{2} ) q^{13} + ( -1 + \beta_{1} ) q^{14} + ( 5 - 2 \beta_{1} + 3 \beta_{2} ) q^{16} + ( 2 + \beta_{1} - \beta_{2} ) q^{17} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{19} + ( 6 - 2 \beta_{1} + 2 \beta_{2} ) q^{20} + ( -2 + \beta_{2} ) q^{23} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{25} + ( 4 + 3 \beta_{1} + 3 \beta_{2} ) q^{26} + ( -3 - \beta_{2} ) q^{28} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{29} + ( -4 - \beta_{1} ) q^{31} + ( 13 - \beta_{1} + 4 \beta_{2} ) q^{32} + ( -4 - 3 \beta_{1} - 3 \beta_{2} ) q^{34} -\beta_{2} q^{35} + ( -6 + 2 \beta_{1} + \beta_{2} ) q^{37} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{38} + ( 14 - 4 \beta_{1} + 2 \beta_{2} ) q^{40} + ( 6 - \beta_{1} + \beta_{2} ) q^{41} + ( -2 - 2 \beta_{2} ) q^{43} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{46} + ( -2 + \beta_{1} - \beta_{2} ) q^{47} + q^{49} + ( 7 + \beta_{1} ) q^{50} + ( 2 - 5 \beta_{1} + \beta_{2} ) q^{52} + ( 4 + 2 \beta_{2} ) q^{53} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{56} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{58} -\beta_{1} q^{59} + ( 6 - \beta_{1} + \beta_{2} ) q^{61} + ( 5 \beta_{1} + \beta_{2} ) q^{62} + ( 15 - 8 \beta_{1} + 3 \beta_{2} ) q^{64} + ( 8 - 2 \beta_{1} - 2 \beta_{2} ) q^{65} + ( -2 + 4 \beta_{1} + \beta_{2} ) q^{67} + ( -2 + 5 \beta_{1} - \beta_{2} ) q^{68} + ( -2 - 2 \beta_{2} ) q^{70} + ( -2 - 4 \beta_{1} - 5 \beta_{2} ) q^{71} + ( -6 + \beta_{1} - \beta_{2} ) q^{73} + ( -12 + 4 \beta_{1} ) q^{74} + ( 8 + 2 \beta_{1} + 2 \beta_{2} ) q^{76} + ( -10 - 2 \beta_{2} ) q^{79} + ( 22 - 6 \beta_{1} + 4 \beta_{2} ) q^{80} + ( 12 - 5 \beta_{1} + 3 \beta_{2} ) q^{82} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{83} + ( -8 + 2 \beta_{1} + 2 \beta_{2} ) q^{85} + ( -6 + 2 \beta_{1} - 4 \beta_{2} ) q^{86} + ( 8 - 2 \beta_{1} + \beta_{2} ) q^{89} + ( 2 + \beta_{1} - \beta_{2} ) q^{91} -2 \beta_{1} q^{92} + ( -8 + \beta_{1} - 3 \beta_{2} ) q^{94} + ( 8 - 4 \beta_{1} - 4 \beta_{2} ) q^{95} + ( -4 - 2 \beta_{1} - 3 \beta_{2} ) q^{97} + ( 1 - \beta_{1} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 2q^{2} + 8q^{4} - q^{5} - 3q^{7} + 6q^{8} + O(q^{10}) \) \( 3q + 2q^{2} + 8q^{4} - q^{5} - 3q^{7} + 6q^{8} + 4q^{10} - 8q^{13} - 2q^{14} + 10q^{16} + 8q^{17} + 14q^{20} - 7q^{23} + 2q^{25} + 12q^{26} - 8q^{28} - 13q^{31} + 34q^{32} - 12q^{34} + q^{35} - 17q^{37} - 16q^{38} + 36q^{40} + 16q^{41} - 4q^{43} - 4q^{47} + 3q^{49} + 22q^{50} + 10q^{53} - 6q^{56} - 16q^{58} - q^{59} + 16q^{61} + 4q^{62} + 34q^{64} + 24q^{65} - 3q^{67} - 4q^{70} - 5q^{71} - 16q^{73} - 32q^{74} + 24q^{76} - 28q^{79} + 56q^{80} + 28q^{82} + 8q^{83} - 24q^{85} - 12q^{86} + 21q^{89} + 8q^{91} - 2q^{92} - 20q^{94} + 24q^{95} - 11q^{97} + 2q^{98} + O(q^{100}) \)

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

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

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
3.12489
−0.363328
−1.76156
−2.12489 0 2.51514 −0.484862 0 −1.00000 −1.09461 0 1.03028
1.2 1.36333 0 −0.141336 −3.14134 0 −1.00000 −2.91934 0 −4.28267
1.3 2.76156 0 5.62620 2.62620 0 −1.00000 10.0140 0 7.25240
\(n\): 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.ce 3
3.b odd 2 1 847.2.a.i 3
11.b odd 2 1 7623.2.a.bz 3
21.c even 2 1 5929.2.a.t 3
33.d even 2 1 847.2.a.j yes 3
33.f even 10 4 847.2.f.t 12
33.h odd 10 4 847.2.f.u 12
231.h odd 2 1 5929.2.a.y 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.i 3 3.b odd 2 1
847.2.a.j yes 3 33.d even 2 1
847.2.f.t 12 33.f even 10 4
847.2.f.u 12 33.h odd 10 4
5929.2.a.t 3 21.c even 2 1
5929.2.a.y 3 231.h odd 2 1
7623.2.a.bz 3 11.b odd 2 1
7623.2.a.ce 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{3} - 2 T_{2}^{2} - 5 T_{2} + 8 \)
\( T_{5}^{3} + T_{5}^{2} - 8 T_{5} - 4 \)
\( T_{13}^{3} + 8 T_{13}^{2} + 2 T_{13} - 64 \)