Properties

Label 7623.2.a.bt
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: \(\Q(\sqrt{17}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2541)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( 2 + \beta ) q^{4} - q^{5} - q^{7} + ( 4 + \beta ) q^{8} +O(q^{10})\) \( q + \beta q^{2} + ( 2 + \beta ) q^{4} - q^{5} - q^{7} + ( 4 + \beta ) q^{8} -\beta q^{10} + ( -4 + \beta ) q^{13} -\beta q^{14} + 3 \beta q^{16} + ( 1 - 2 \beta ) q^{17} -6 q^{19} + ( -2 - \beta ) q^{20} + 4 q^{23} -4 q^{25} + ( 4 - 3 \beta ) q^{26} + ( -2 - \beta ) q^{28} + ( 2 - 3 \beta ) q^{29} + ( 2 - 4 \beta ) q^{31} + ( 4 + \beta ) q^{32} + ( -8 - \beta ) q^{34} + q^{35} + ( 2 + 3 \beta ) q^{37} -6 \beta q^{38} + ( -4 - \beta ) q^{40} + ( -6 + \beta ) q^{41} + ( 1 - 3 \beta ) q^{43} + 4 \beta q^{46} + ( -5 - \beta ) q^{47} + q^{49} -4 \beta q^{50} + ( -4 - \beta ) q^{52} + 5 \beta q^{53} + ( -4 - \beta ) q^{56} + ( -12 - \beta ) q^{58} + ( 11 - \beta ) q^{59} + ( 4 - 6 \beta ) q^{61} + ( -16 - 2 \beta ) q^{62} + ( 4 - \beta ) q^{64} + ( 4 - \beta ) q^{65} + ( 1 - 5 \beta ) q^{67} + ( -6 - 5 \beta ) q^{68} + \beta q^{70} + 2 \beta q^{71} + ( -4 + 2 \beta ) q^{73} + ( 12 + 5 \beta ) q^{74} + ( -12 - 6 \beta ) q^{76} + 2 \beta q^{79} -3 \beta q^{80} + ( 4 - 5 \beta ) q^{82} + ( -1 + 5 \beta ) q^{83} + ( -1 + 2 \beta ) q^{85} + ( -12 - 2 \beta ) q^{86} + ( 5 - 4 \beta ) q^{89} + ( 4 - \beta ) q^{91} + ( 8 + 4 \beta ) q^{92} + ( -4 - 6 \beta ) q^{94} + 6 q^{95} + ( 6 + 3 \beta ) q^{97} + \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + 5q^{4} - 2q^{5} - 2q^{7} + 9q^{8} + O(q^{10}) \) \( 2q + q^{2} + 5q^{4} - 2q^{5} - 2q^{7} + 9q^{8} - q^{10} - 7q^{13} - q^{14} + 3q^{16} - 12q^{19} - 5q^{20} + 8q^{23} - 8q^{25} + 5q^{26} - 5q^{28} + q^{29} + 9q^{32} - 17q^{34} + 2q^{35} + 7q^{37} - 6q^{38} - 9q^{40} - 11q^{41} - q^{43} + 4q^{46} - 11q^{47} + 2q^{49} - 4q^{50} - 9q^{52} + 5q^{53} - 9q^{56} - 25q^{58} + 21q^{59} + 2q^{61} - 34q^{62} + 7q^{64} + 7q^{65} - 3q^{67} - 17q^{68} + q^{70} + 2q^{71} - 6q^{73} + 29q^{74} - 30q^{76} + 2q^{79} - 3q^{80} + 3q^{82} + 3q^{83} - 26q^{86} + 6q^{89} + 7q^{91} + 20q^{92} - 14q^{94} + 12q^{95} + 15q^{97} + q^{98} + O(q^{100}) \)

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
−1.56155
2.56155
−1.56155 0 0.438447 −1.00000 0 −1.00000 2.43845 0 1.56155
1.2 2.56155 0 4.56155 −1.00000 0 −1.00000 6.56155 0 −2.56155
\(n\): 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.bt 2
3.b odd 2 1 2541.2.a.u 2
11.b odd 2 1 7623.2.a.be 2
33.d even 2 1 2541.2.a.bc yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.2.a.u 2 3.b odd 2 1
2541.2.a.bc yes 2 33.d even 2 1
7623.2.a.be 2 11.b odd 2 1
7623.2.a.bt 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{2} - T_{2} - 4 \)
\( T_{5} + 1 \)
\( T_{13}^{2} + 7 T_{13} + 8 \)