Properties

Label 7623.2.a.bp
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.

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

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 0.381966 0 −1.00000 2.23607 0 −0.236068
1.2 1.61803 0 0.618034 2.61803 0 −1.00000 −2.23607 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.bp 2
3.b odd 2 1 2541.2.a.s 2
11.b odd 2 1 7623.2.a.ba 2
11.d odd 10 2 693.2.m.b 4
33.d even 2 1 2541.2.a.bb 2
33.f even 10 2 231.2.j.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.d 4 33.f even 10 2
693.2.m.b 4 11.d odd 10 2
2541.2.a.s 2 3.b odd 2 1
2541.2.a.bb 2 33.d even 2 1
7623.2.a.ba 2 11.b odd 2 1
7623.2.a.bp 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

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