Properties

Label 7623.2.a.x
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: \(\Q(\sqrt{3}) \)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{2} + ( 2 - 2 \beta ) q^{4} + ( 2 - \beta ) q^{5} - q^{7} + ( -6 + 2 \beta ) q^{8} +O(q^{10})\) \( q + ( -1 + \beta ) q^{2} + ( 2 - 2 \beta ) q^{4} + ( 2 - \beta ) q^{5} - q^{7} + ( -6 + 2 \beta ) q^{8} + ( -5 + 3 \beta ) q^{10} + ( 1 + \beta ) q^{13} + ( 1 - \beta ) q^{14} + ( 8 - 4 \beta ) q^{16} + ( 2 + \beta ) q^{17} + ( 3 - 3 \beta ) q^{19} + ( 10 - 6 \beta ) q^{20} + ( -5 + \beta ) q^{23} + ( 2 - 4 \beta ) q^{25} + 2 q^{26} + ( -2 + 2 \beta ) q^{28} + ( -3 + \beta ) q^{29} + ( 4 + 2 \beta ) q^{31} + ( -8 + 8 \beta ) q^{32} + ( 1 + \beta ) q^{34} + ( -2 + \beta ) q^{35} + ( -6 - 2 \beta ) q^{37} + ( -12 + 6 \beta ) q^{38} + ( -18 + 10 \beta ) q^{40} -4 q^{41} + ( -3 + 2 \beta ) q^{43} + ( 8 - 6 \beta ) q^{46} + ( 4 + 3 \beta ) q^{47} + q^{49} + ( -14 + 6 \beta ) q^{50} -4 q^{52} + ( -6 + 2 \beta ) q^{53} + ( 6 - 2 \beta ) q^{56} + ( 6 - 4 \beta ) q^{58} + ( -2 + 7 \beta ) q^{59} + ( 3 + 3 \beta ) q^{61} + ( 2 + 2 \beta ) q^{62} + ( 16 - 8 \beta ) q^{64} + ( -1 + \beta ) q^{65} -7 q^{67} + ( -2 - 2 \beta ) q^{68} + ( 5 - 3 \beta ) q^{70} + ( 3 - 7 \beta ) q^{71} + ( -5 - \beta ) q^{73} -4 \beta q^{74} + ( 24 - 12 \beta ) q^{76} + ( -4 + 2 \beta ) q^{79} + ( 28 - 16 \beta ) q^{80} + ( 4 - 4 \beta ) q^{82} + ( -6 + \beta ) q^{83} + q^{85} + ( 9 - 5 \beta ) q^{86} + ( 6 + 5 \beta ) q^{89} + ( -1 - \beta ) q^{91} + ( -16 + 12 \beta ) q^{92} + ( 5 + \beta ) q^{94} + ( 15 - 9 \beta ) q^{95} + ( 5 - \beta ) q^{97} + ( -1 + \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 4q^{4} + 4q^{5} - 2q^{7} - 12q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 4q^{4} + 4q^{5} - 2q^{7} - 12q^{8} - 10q^{10} + 2q^{13} + 2q^{14} + 16q^{16} + 4q^{17} + 6q^{19} + 20q^{20} - 10q^{23} + 4q^{25} + 4q^{26} - 4q^{28} - 6q^{29} + 8q^{31} - 16q^{32} + 2q^{34} - 4q^{35} - 12q^{37} - 24q^{38} - 36q^{40} - 8q^{41} - 6q^{43} + 16q^{46} + 8q^{47} + 2q^{49} - 28q^{50} - 8q^{52} - 12q^{53} + 12q^{56} + 12q^{58} - 4q^{59} + 6q^{61} + 4q^{62} + 32q^{64} - 2q^{65} - 14q^{67} - 4q^{68} + 10q^{70} + 6q^{71} - 10q^{73} + 48q^{76} - 8q^{79} + 56q^{80} + 8q^{82} - 12q^{83} + 2q^{85} + 18q^{86} + 12q^{89} - 2q^{91} - 32q^{92} + 10q^{94} + 30q^{95} + 10q^{97} - 2q^{98} + O(q^{100}) \)

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
−1.73205
1.73205
−2.73205 0 5.46410 3.73205 0 −1.00000 −9.46410 0 −10.1962
1.2 0.732051 0 −1.46410 0.267949 0 −1.00000 −2.53590 0 0.196152
\(n\): 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.x 2
3.b odd 2 1 2541.2.a.be yes 2
11.b odd 2 1 7623.2.a.bv 2
33.d even 2 1 2541.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.2.a.o 2 33.d even 2 1
2541.2.a.be yes 2 3.b odd 2 1
7623.2.a.x 2 1.a even 1 1 trivial
7623.2.a.bv 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

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