Properties

Label 7623.2.a.by
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: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
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.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{2} + 3 \beta q^{4} -\beta q^{5} - q^{7} + ( 1 + 4 \beta ) q^{8} +O(q^{10})\) \( q + ( 1 + \beta ) q^{2} + 3 \beta q^{4} -\beta q^{5} - q^{7} + ( 1 + 4 \beta ) q^{8} + ( -1 - 2 \beta ) q^{10} + ( 1 - 4 \beta ) q^{13} + ( -1 - \beta ) q^{14} + ( 5 + 3 \beta ) q^{16} -3 q^{17} -4 q^{19} + ( -3 - 3 \beta ) q^{20} + ( 4 + 2 \beta ) q^{23} + ( -4 + \beta ) q^{25} + ( -3 - 7 \beta ) q^{26} -3 \beta q^{28} + 3 q^{29} + ( -3 + 2 \beta ) q^{31} + ( 6 + 3 \beta ) q^{32} + ( -3 - 3 \beta ) q^{34} + \beta q^{35} + ( -2 - 4 \beta ) q^{37} + ( -4 - 4 \beta ) q^{38} + ( -4 - 5 \beta ) q^{40} + ( 5 - 5 \beta ) q^{41} + 9 q^{43} + ( 6 + 8 \beta ) q^{46} + ( -2 - 5 \beta ) q^{47} + q^{49} + ( -3 - 2 \beta ) q^{50} + ( -12 - 9 \beta ) q^{52} + ( 4 - 5 \beta ) q^{53} + ( -1 - 4 \beta ) q^{56} + ( 3 + 3 \beta ) q^{58} + ( -11 - \beta ) q^{59} + ( -3 + 6 \beta ) q^{61} + ( -1 + \beta ) q^{62} + ( -1 + 6 \beta ) q^{64} + ( 4 + 3 \beta ) q^{65} + ( 6 - 6 \beta ) q^{67} -9 \beta q^{68} + ( 1 + 2 \beta ) q^{70} + ( -9 - 2 \beta ) q^{71} + ( -1 + 3 \beta ) q^{73} + ( -6 - 10 \beta ) q^{74} -12 \beta q^{76} + ( -2 - 3 \beta ) q^{79} + ( -3 - 8 \beta ) q^{80} -5 \beta q^{82} + ( -10 + 8 \beta ) q^{83} + 3 \beta q^{85} + ( 9 + 9 \beta ) q^{86} + ( -14 + 7 \beta ) q^{89} + ( -1 + 4 \beta ) q^{91} + ( 6 + 18 \beta ) q^{92} + ( -7 - 12 \beta ) q^{94} + 4 \beta q^{95} + ( 3 - 12 \beta ) q^{97} + ( 1 + \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{2} + 3q^{4} - q^{5} - 2q^{7} + 6q^{8} + O(q^{10}) \) \( 2q + 3q^{2} + 3q^{4} - q^{5} - 2q^{7} + 6q^{8} - 4q^{10} - 2q^{13} - 3q^{14} + 13q^{16} - 6q^{17} - 8q^{19} - 9q^{20} + 10q^{23} - 7q^{25} - 13q^{26} - 3q^{28} + 6q^{29} - 4q^{31} + 15q^{32} - 9q^{34} + q^{35} - 8q^{37} - 12q^{38} - 13q^{40} + 5q^{41} + 18q^{43} + 20q^{46} - 9q^{47} + 2q^{49} - 8q^{50} - 33q^{52} + 3q^{53} - 6q^{56} + 9q^{58} - 23q^{59} - q^{62} + 4q^{64} + 11q^{65} + 6q^{67} - 9q^{68} + 4q^{70} - 20q^{71} + q^{73} - 22q^{74} - 12q^{76} - 7q^{79} - 14q^{80} - 5q^{82} - 12q^{83} + 3q^{85} + 27q^{86} - 21q^{89} + 2q^{91} + 30q^{92} - 26q^{94} + 4q^{95} - 6q^{97} + 3q^{98} + O(q^{100}) \)

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 0.618034 0 −1.00000 −1.47214 0 0.236068
1.2 2.61803 0 4.85410 −1.61803 0 −1.00000 7.47214 0 −4.23607
\(n\): 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.by 2
3.b odd 2 1 2541.2.a.m 2
11.b odd 2 1 7623.2.a.u 2
11.d odd 10 2 693.2.m.e 4
33.d even 2 1 2541.2.a.bf 2
33.f even 10 2 231.2.j.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.a 4 33.f even 10 2
693.2.m.e 4 11.d odd 10 2
2541.2.a.m 2 3.b odd 2 1
2541.2.a.bf 2 33.d even 2 1
7623.2.a.u 2 11.b odd 2 1
7623.2.a.by 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{2} - 3 T_{2} + 1 \)
\( T_{5}^{2} + T_{5} - 1 \)
\( T_{13}^{2} + 2 T_{13} - 19 \)