Properties

Label 7623.2.a.bc
Level 7623
Weight 2
Character orbit 7623.a
Self dual yes
Analytic conductor 60.870
Analytic rank 1
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: \(1\)
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
2.30278
−1.30278
−2.30278 0 3.30278 −3.60555 0 −1.00000 −3.00000 0 8.30278
1.2 1.30278 0 −0.302776 3.60555 0 −1.00000 −3.00000 0 4.69722
\(n\): 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.bc 2
3.b odd 2 1 847.2.a.g yes 2
11.b odd 2 1 7623.2.a.bs 2
21.c even 2 1 5929.2.a.p 2
33.d even 2 1 847.2.a.e 2
33.f even 10 4 847.2.f.r 8
33.h odd 10 4 847.2.f.o 8
231.h odd 2 1 5929.2.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.e 2 33.d even 2 1
847.2.a.g yes 2 3.b odd 2 1
847.2.f.o 8 33.h odd 10 4
847.2.f.r 8 33.f even 10 4
5929.2.a.k 2 231.h odd 2 1
5929.2.a.p 2 21.c even 2 1
7623.2.a.bc 2 1.a even 1 1 trivial
7623.2.a.bs 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

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