Properties

Label 7623.2.a.bq
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 2541)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−0.618034
1.61803
−0.618034 0 −1.61803 −0.236068 0 1.00000 2.23607 0 0.145898
1.2 1.61803 0 0.618034 4.23607 0 1.00000 −2.23607 0 6.85410
\(n\): 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.bq 2
3.b odd 2 1 2541.2.a.r 2
11.b odd 2 1 7623.2.a.bb 2
33.d even 2 1 2541.2.a.ba yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.2.a.r 2 3.b odd 2 1
2541.2.a.ba yes 2 33.d even 2 1
7623.2.a.bb 2 11.b odd 2 1
7623.2.a.bq 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{2} - T_{2} - 1 \)
\( T_{5}^{2} - 4 T_{5} - 1 \)
\( T_{13}^{2} + 8 T_{13} + 11 \)