Properties

Label 7623.2.a.bi
Level 7623
Weight 2
Character orbit 7623.a
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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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: \(\Q(\sqrt{3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + q^{4} + 2 \beta q^{5} - q^{7} -\beta q^{8} +O(q^{10})\) \( q + \beta q^{2} + q^{4} + 2 \beta q^{5} - q^{7} -\beta q^{8} + 6 q^{10} -2 q^{13} -\beta q^{14} -5 q^{16} + 2 \beta q^{17} + 4 q^{19} + 2 \beta q^{20} + 2 \beta q^{23} + 7 q^{25} -2 \beta q^{26} - q^{28} -4 q^{31} -3 \beta q^{32} + 6 q^{34} -2 \beta q^{35} + 2 q^{37} + 4 \beta q^{38} -6 q^{40} + 6 \beta q^{41} + 4 q^{43} + 6 q^{46} -4 \beta q^{47} + q^{49} + 7 \beta q^{50} -2 q^{52} + 4 \beta q^{53} + \beta q^{56} + 4 \beta q^{59} + 10 q^{61} -4 \beta q^{62} + q^{64} -4 \beta q^{65} -4 q^{67} + 2 \beta q^{68} -6 q^{70} + 6 \beta q^{71} -14 q^{73} + 2 \beta q^{74} + 4 q^{76} -8 q^{79} -10 \beta q^{80} + 18 q^{82} + 12 q^{85} + 4 \beta q^{86} + 2 \beta q^{89} + 2 q^{91} + 2 \beta q^{92} -12 q^{94} + 8 \beta q^{95} + 14 q^{97} + \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} - 2q^{7} + O(q^{10}) \) \( 2q + 2q^{4} - 2q^{7} + 12q^{10} - 4q^{13} - 10q^{16} + 8q^{19} + 14q^{25} - 2q^{28} - 8q^{31} + 12q^{34} + 4q^{37} - 12q^{40} + 8q^{43} + 12q^{46} + 2q^{49} - 4q^{52} + 20q^{61} + 2q^{64} - 8q^{67} - 12q^{70} - 28q^{73} + 8q^{76} - 16q^{79} + 36q^{82} + 24q^{85} + 4q^{91} - 24q^{94} + 28q^{97} + O(q^{100}) \)

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
−1.73205
1.73205
−1.73205 0 1.00000 −3.46410 0 −1.00000 1.73205 0 6.00000
1.2 1.73205 0 1.00000 3.46410 0 −1.00000 −1.73205 0 6.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7623.2.a.bi 2
3.b odd 2 1 inner 7623.2.a.bi 2
11.b odd 2 1 63.2.a.b 2
33.d even 2 1 63.2.a.b 2
44.c even 2 1 1008.2.a.n 2
55.d odd 2 1 1575.2.a.q 2
55.e even 4 2 1575.2.d.i 4
77.b even 2 1 441.2.a.g 2
77.h odd 6 2 441.2.e.j 4
77.i even 6 2 441.2.e.i 4
88.b odd 2 1 4032.2.a.bt 2
88.g even 2 1 4032.2.a.bq 2
99.g even 6 2 567.2.f.j 4
99.h odd 6 2 567.2.f.j 4
132.d odd 2 1 1008.2.a.n 2
165.d even 2 1 1575.2.a.q 2
165.l odd 4 2 1575.2.d.i 4
231.h odd 2 1 441.2.a.g 2
231.k odd 6 2 441.2.e.i 4
231.l even 6 2 441.2.e.j 4
264.m even 2 1 4032.2.a.bt 2
264.p odd 2 1 4032.2.a.bq 2
308.g odd 2 1 7056.2.a.cm 2
924.n even 2 1 7056.2.a.cm 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.a.b 2 11.b odd 2 1
63.2.a.b 2 33.d even 2 1
441.2.a.g 2 77.b even 2 1
441.2.a.g 2 231.h odd 2 1
441.2.e.i 4 77.i even 6 2
441.2.e.i 4 231.k odd 6 2
441.2.e.j 4 77.h odd 6 2
441.2.e.j 4 231.l even 6 2
567.2.f.j 4 99.g even 6 2
567.2.f.j 4 99.h odd 6 2
1008.2.a.n 2 44.c even 2 1
1008.2.a.n 2 132.d odd 2 1
1575.2.a.q 2 55.d odd 2 1
1575.2.a.q 2 165.d even 2 1
1575.2.d.i 4 55.e even 4 2
1575.2.d.i 4 165.l odd 4 2
4032.2.a.bq 2 88.g even 2 1
4032.2.a.bq 2 264.p odd 2 1
4032.2.a.bt 2 88.b odd 2 1
4032.2.a.bt 2 264.m even 2 1
7056.2.a.cm 2 308.g odd 2 1
7056.2.a.cm 2 924.n even 2 1
7623.2.a.bi 2 1.a even 1 1 trivial
7623.2.a.bi 2 3.b odd 2 1 inner

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

\( T_{2}^{2} - 3 \)
\( T_{5}^{2} - 12 \)
\( T_{13} + 2 \)