Properties

Label 7623.2.a.z
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 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.

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

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
1.61803
−0.618034
−1.61803 0 0.618034 −0.618034 0 1.00000 2.23607 0 1.00000
1.2 0.618034 0 −1.61803 1.61803 0 1.00000 −2.23607 0 1.00000
\(n\): 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.z 2
3.b odd 2 1 2541.2.a.x 2
11.b odd 2 1 7623.2.a.bo 2
11.d odd 10 2 693.2.m.d 4
33.d even 2 1 2541.2.a.p 2
33.f even 10 2 231.2.j.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.b 4 33.f even 10 2
693.2.m.d 4 11.d odd 10 2
2541.2.a.p 2 33.d even 2 1
2541.2.a.x 2 3.b odd 2 1
7623.2.a.z 2 1.a even 1 1 trivial
7623.2.a.bo 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

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