Properties

Label 7623.2.a.bd
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 693)
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.

\(f(q)\) \(=\) \( q -\beta q^{2} + ( 1 + \beta ) q^{4} + q^{5} + q^{7} -3 q^{8} +O(q^{10})\) \( q -\beta q^{2} + ( 1 + \beta ) q^{4} + q^{5} + q^{7} -3 q^{8} -\beta q^{10} + ( 1 - 2 \beta ) q^{13} -\beta q^{14} + ( -2 + \beta ) q^{16} -4 q^{17} -3 q^{19} + ( 1 + \beta ) q^{20} + 2 q^{23} -4 q^{25} + ( 6 + \beta ) q^{26} + ( 1 + \beta ) q^{28} + ( 1 + 2 \beta ) q^{29} -2 q^{31} + ( 3 + \beta ) q^{32} + 4 \beta q^{34} + q^{35} + ( 1 - 4 \beta ) q^{37} + 3 \beta q^{38} -3 q^{40} + ( -2 + 4 \beta ) q^{41} + ( -4 + 4 \beta ) q^{43} -2 \beta q^{46} + ( 7 - 2 \beta ) q^{47} + q^{49} + 4 \beta q^{50} + ( -5 - 3 \beta ) q^{52} -3 q^{56} + ( -6 - 3 \beta ) q^{58} + ( 3 + 2 \beta ) q^{59} + ( 2 + 4 \beta ) q^{61} + 2 \beta q^{62} + ( 1 - 6 \beta ) q^{64} + ( 1 - 2 \beta ) q^{65} + ( -3 + 2 \beta ) q^{67} + ( -4 - 4 \beta ) q^{68} -\beta q^{70} + ( 2 + 4 \beta ) q^{71} + ( -1 + 6 \beta ) q^{73} + ( 12 + 3 \beta ) q^{74} + ( -3 - 3 \beta ) q^{76} + ( -6 + 4 \beta ) q^{79} + ( -2 + \beta ) q^{80} + ( -12 - 2 \beta ) q^{82} -12 q^{83} -4 q^{85} -12 q^{86} + 6 q^{89} + ( 1 - 2 \beta ) q^{91} + ( 2 + 2 \beta ) q^{92} + ( 6 - 5 \beta ) q^{94} -3 q^{95} + ( -8 + 4 \beta ) q^{97} -\beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + 3q^{4} + 2q^{5} + 2q^{7} - 6q^{8} + O(q^{10}) \) \( 2q - q^{2} + 3q^{4} + 2q^{5} + 2q^{7} - 6q^{8} - q^{10} - q^{14} - 3q^{16} - 8q^{17} - 6q^{19} + 3q^{20} + 4q^{23} - 8q^{25} + 13q^{26} + 3q^{28} + 4q^{29} - 4q^{31} + 7q^{32} + 4q^{34} + 2q^{35} - 2q^{37} + 3q^{38} - 6q^{40} - 4q^{43} - 2q^{46} + 12q^{47} + 2q^{49} + 4q^{50} - 13q^{52} - 6q^{56} - 15q^{58} + 8q^{59} + 8q^{61} + 2q^{62} - 4q^{64} - 4q^{67} - 12q^{68} - q^{70} + 8q^{71} + 4q^{73} + 27q^{74} - 9q^{76} - 8q^{79} - 3q^{80} - 26q^{82} - 24q^{83} - 8q^{85} - 24q^{86} + 12q^{89} + 6q^{92} + 7q^{94} - 6q^{95} - 12q^{97} - q^{98} + O(q^{100}) \)

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 1.00000 0 1.00000 −3.00000 0 −2.30278
1.2 1.30278 0 −0.302776 1.00000 0 1.00000 −3.00000 0 1.30278
\(n\): 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.bd 2
3.b odd 2 1 7623.2.a.br 2
11.b odd 2 1 693.2.a.i yes 2
33.d even 2 1 693.2.a.g 2
77.b even 2 1 4851.2.a.z 2
231.h odd 2 1 4851.2.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.a.g 2 33.d even 2 1
693.2.a.i yes 2 11.b odd 2 1
4851.2.a.x 2 231.h odd 2 1
4851.2.a.z 2 77.b even 2 1
7623.2.a.bd 2 1.a even 1 1 trivial
7623.2.a.br 2 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{2} + T_{2} - 3 \)
\( T_{5} - 1 \)
\( T_{13}^{2} - 13 \)