Properties

Label 7623.2.a.bw
Level 7623
Weight 2
Character orbit 7623.a
Self dual yes
Analytic conductor 60.870
Analytic rank 0
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{2} + 3 \beta q^{4} - q^{5} - q^{7} + ( 1 + 4 \beta ) q^{8} +O(q^{10})\) \( q + ( 1 + \beta ) q^{2} + 3 \beta q^{4} - q^{5} - q^{7} + ( 1 + 4 \beta ) q^{8} + ( -1 - \beta ) q^{10} + ( 1 + 2 \beta ) q^{13} + ( -1 - \beta ) q^{14} + ( 5 + 3 \beta ) q^{16} + ( 4 - 4 \beta ) q^{17} + ( -1 + 4 \beta ) q^{19} -3 \beta q^{20} + ( -6 + 4 \beta ) q^{23} -4 q^{25} + ( 3 + 5 \beta ) q^{26} -3 \beta q^{28} + ( 3 + 2 \beta ) q^{29} + ( 2 + 4 \beta ) q^{31} + ( 6 + 3 \beta ) q^{32} -4 \beta q^{34} + q^{35} + ( -3 + 4 \beta ) q^{37} + ( 3 + 7 \beta ) q^{38} + ( -1 - 4 \beta ) q^{40} + ( 2 - 4 \beta ) q^{41} + 4 q^{43} + ( -2 + 2 \beta ) q^{46} + ( -1 - 2 \beta ) q^{47} + q^{49} + ( -4 - 4 \beta ) q^{50} + ( 6 + 9 \beta ) q^{52} + ( -8 - 4 \beta ) q^{53} + ( -1 - 4 \beta ) q^{56} + ( 5 + 7 \beta ) q^{58} + ( -9 + 10 \beta ) q^{59} + ( -6 + 4 \beta ) q^{61} + ( 6 + 10 \beta ) q^{62} + ( -1 + 6 \beta ) q^{64} + ( -1 - 2 \beta ) q^{65} + ( 3 - 10 \beta ) q^{67} -12 q^{68} + ( 1 + \beta ) q^{70} + ( 2 - 4 \beta ) q^{71} + ( 11 + 2 \beta ) q^{73} + ( 1 + 5 \beta ) q^{74} + ( 12 + 9 \beta ) q^{76} + ( -2 + 4 \beta ) q^{79} + ( -5 - 3 \beta ) q^{80} + ( -2 - 6 \beta ) q^{82} + ( -4 + 12 \beta ) q^{83} + ( -4 + 4 \beta ) q^{85} + ( 4 + 4 \beta ) q^{86} + 10 q^{89} + ( -1 - 2 \beta ) q^{91} + ( 12 - 6 \beta ) q^{92} + ( -3 - 5 \beta ) q^{94} + ( 1 - 4 \beta ) q^{95} -8 q^{97} + ( 1 + \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{2} + 3q^{4} - 2q^{5} - 2q^{7} + 6q^{8} + O(q^{10}) \) \( 2q + 3q^{2} + 3q^{4} - 2q^{5} - 2q^{7} + 6q^{8} - 3q^{10} + 4q^{13} - 3q^{14} + 13q^{16} + 4q^{17} + 2q^{19} - 3q^{20} - 8q^{23} - 8q^{25} + 11q^{26} - 3q^{28} + 8q^{29} + 8q^{31} + 15q^{32} - 4q^{34} + 2q^{35} - 2q^{37} + 13q^{38} - 6q^{40} + 8q^{43} - 2q^{46} - 4q^{47} + 2q^{49} - 12q^{50} + 21q^{52} - 20q^{53} - 6q^{56} + 17q^{58} - 8q^{59} - 8q^{61} + 22q^{62} + 4q^{64} - 4q^{65} - 4q^{67} - 24q^{68} + 3q^{70} + 24q^{73} + 7q^{74} + 33q^{76} - 13q^{80} - 10q^{82} + 4q^{83} - 4q^{85} + 12q^{86} + 20q^{89} - 4q^{91} + 18q^{92} - 11q^{94} - 2q^{95} - 16q^{97} + 3q^{98} + O(q^{100}) \)

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
−0.618034
1.61803
0.381966 0 −1.85410 −1.00000 0 −1.00000 −1.47214 0 −0.381966
1.2 2.61803 0 4.85410 −1.00000 0 −1.00000 7.47214 0 −2.61803
\(n\): 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.bw 2
3.b odd 2 1 7623.2.a.v 2
11.b odd 2 1 693.2.a.e 2
33.d even 2 1 693.2.a.k yes 2
77.b even 2 1 4851.2.a.u 2
231.h odd 2 1 4851.2.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.a.e 2 11.b odd 2 1
693.2.a.k yes 2 33.d even 2 1
4851.2.a.u 2 77.b even 2 1
4851.2.a.bf 2 231.h odd 2 1
7623.2.a.v 2 3.b odd 2 1
7623.2.a.bw 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{2} - 3 T_{2} + 1 \)
\( T_{5} + 1 \)
\( T_{13}^{2} - 4 T_{13} - 1 \)