Properties

Label 7623.2.a.bu
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: \(\Z[a_1, \ldots, a_{13}]\)
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.

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

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
1.00000 0 −1.00000 −3.85410 0 1.00000 −3.00000 0 −3.85410
1.2 1.00000 0 −1.00000 2.85410 0 1.00000 −3.00000 0 2.85410
\(n\): 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.bu 2
3.b odd 2 1 2541.2.a.n 2
11.b odd 2 1 7623.2.a.w 2
11.c even 5 2 693.2.m.c 4
33.d even 2 1 2541.2.a.bd 2
33.h odd 10 2 231.2.j.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.c 4 33.h odd 10 2
693.2.m.c 4 11.c even 5 2
2541.2.a.n 2 3.b odd 2 1
2541.2.a.bd 2 33.d even 2 1
7623.2.a.w 2 11.b odd 2 1
7623.2.a.bu 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

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