Properties

Label 7623.2.a.bs
Level 7623
Weight 2
Character orbit 7623.a
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( 1 + \beta ) q^{4} + ( 1 - 2 \beta ) q^{5} + q^{7} + 3 q^{8} +O(q^{10})\) \( q + \beta q^{2} + ( 1 + \beta ) q^{4} + ( 1 - 2 \beta ) q^{5} + q^{7} + 3 q^{8} + ( -6 - \beta ) q^{10} + ( -2 - 2 \beta ) q^{13} + \beta q^{14} + ( -2 + \beta ) q^{16} + ( 5 - \beta ) q^{17} + 3 q^{19} + ( -5 - 3 \beta ) q^{20} + ( 5 - \beta ) q^{23} + 8 q^{25} + ( -6 - 4 \beta ) q^{26} + ( 1 + \beta ) q^{28} + ( 7 - \beta ) q^{29} + q^{31} + ( -3 - \beta ) q^{32} + ( -3 + 4 \beta ) q^{34} + ( 1 - 2 \beta ) q^{35} + ( 4 - 4 \beta ) q^{37} + 3 \beta q^{38} + ( 3 - 6 \beta ) q^{40} + 7 q^{41} + ( -4 + \beta ) q^{43} + ( -3 + 4 \beta ) q^{46} + ( -5 + 3 \beta ) q^{47} + q^{49} + 8 \beta q^{50} + ( -8 - 6 \beta ) q^{52} + ( 6 + 3 \beta ) q^{53} + 3 q^{56} + ( -3 + 6 \beta ) q^{58} + ( -9 + \beta ) q^{59} + ( 2 + \beta ) q^{61} + \beta q^{62} + ( 1 - 6 \beta ) q^{64} + ( 10 + 6 \beta ) q^{65} + ( -3 + 5 \beta ) q^{67} + ( 2 + 3 \beta ) q^{68} + ( -6 - \beta ) q^{70} + ( 2 + \beta ) q^{71} + 5 q^{73} -12 q^{74} + ( 3 + 3 \beta ) q^{76} + ( 6 + \beta ) q^{79} + ( -8 + 3 \beta ) q^{80} + 7 \beta q^{82} + 3 q^{83} + ( 11 - 9 \beta ) q^{85} + ( 3 - 3 \beta ) q^{86} + ( -6 + 9 \beta ) q^{89} + ( -2 - 2 \beta ) q^{91} + ( 2 + 3 \beta ) q^{92} + ( 9 - 2 \beta ) q^{94} + ( 3 - 6 \beta ) q^{95} + ( 1 - 2 \beta ) q^{97} + \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + 3q^{4} + 2q^{7} + 6q^{8} + O(q^{10}) \) \( 2q + q^{2} + 3q^{4} + 2q^{7} + 6q^{8} - 13q^{10} - 6q^{13} + q^{14} - 3q^{16} + 9q^{17} + 6q^{19} - 13q^{20} + 9q^{23} + 16q^{25} - 16q^{26} + 3q^{28} + 13q^{29} + 2q^{31} - 7q^{32} - 2q^{34} + 4q^{37} + 3q^{38} + 14q^{41} - 7q^{43} - 2q^{46} - 7q^{47} + 2q^{49} + 8q^{50} - 22q^{52} + 15q^{53} + 6q^{56} - 17q^{59} + 5q^{61} + q^{62} - 4q^{64} + 26q^{65} - q^{67} + 7q^{68} - 13q^{70} + 5q^{71} + 10q^{73} - 24q^{74} + 9q^{76} + 13q^{79} - 13q^{80} + 7q^{82} + 6q^{83} + 13q^{85} + 3q^{86} - 3q^{89} - 6q^{91} + 7q^{92} + 16q^{94} + q^{98} + O(q^{100}) \)

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
−1.30278
2.30278
−1.30278 0 −0.302776 3.60555 0 1.00000 3.00000 0 −4.69722
1.2 2.30278 0 3.30278 −3.60555 0 1.00000 3.00000 0 −8.30278
\(n\): 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.bs 2
3.b odd 2 1 847.2.a.e 2
11.b odd 2 1 7623.2.a.bc 2
21.c even 2 1 5929.2.a.k 2
33.d even 2 1 847.2.a.g yes 2
33.f even 10 4 847.2.f.o 8
33.h odd 10 4 847.2.f.r 8
231.h odd 2 1 5929.2.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.e 2 3.b odd 2 1
847.2.a.g yes 2 33.d even 2 1
847.2.f.o 8 33.f even 10 4
847.2.f.r 8 33.h odd 10 4
5929.2.a.k 2 21.c even 2 1
5929.2.a.p 2 231.h odd 2 1
7623.2.a.bc 2 11.b odd 2 1
7623.2.a.bs 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{2} - T_{2} - 3 \)
\( T_{5}^{2} - 13 \)
\( T_{13}^{2} + 6 T_{13} - 4 \)