Properties

Label 7623.2.a.bf
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: \(\Q(\sqrt{21}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} + ( 3 + \beta ) q^{4} -3 q^{5} - q^{7} + ( -5 - 2 \beta ) q^{8} +O(q^{10})\) \( q -\beta q^{2} + ( 3 + \beta ) q^{4} -3 q^{5} - q^{7} + ( -5 - 2 \beta ) q^{8} + 3 \beta q^{10} - q^{13} + \beta q^{14} + ( 4 + 5 \beta ) q^{16} + ( 4 - 2 \beta ) q^{17} + ( 3 - 2 \beta ) q^{19} + ( -9 - 3 \beta ) q^{20} + ( 2 - 2 \beta ) q^{23} + 4 q^{25} + \beta q^{26} + ( -3 - \beta ) q^{28} + ( -1 + 4 \beta ) q^{29} -2 \beta q^{31} + ( -15 - 5 \beta ) q^{32} + ( 10 - 2 \beta ) q^{34} + 3 q^{35} + q^{37} + ( 10 - \beta ) q^{38} + ( 15 + 6 \beta ) q^{40} + ( -4 + 4 \beta ) q^{41} + ( 2 + 2 \beta ) q^{43} + 10 q^{46} + ( -5 - 2 \beta ) q^{47} + q^{49} -4 \beta q^{50} + ( -3 - \beta ) q^{52} + ( 6 - 2 \beta ) q^{53} + ( 5 + 2 \beta ) q^{56} + ( -20 - 3 \beta ) q^{58} + ( -1 + 2 \beta ) q^{59} -10 q^{61} + ( 10 + 2 \beta ) q^{62} + ( 17 + 10 \beta ) q^{64} + 3 q^{65} + ( 5 - 2 \beta ) q^{67} + ( 2 - 4 \beta ) q^{68} -3 \beta q^{70} + ( -4 + 4 \beta ) q^{71} -7 q^{73} -\beta q^{74} + ( -1 - 5 \beta ) q^{76} + 4 \beta q^{79} + ( -12 - 15 \beta ) q^{80} -20 q^{82} + ( -8 + 2 \beta ) q^{83} + ( -12 + 6 \beta ) q^{85} + ( -10 - 4 \beta ) q^{86} + ( -2 + 4 \beta ) q^{89} + q^{91} + ( -4 - 6 \beta ) q^{92} + ( 10 + 7 \beta ) q^{94} + ( -9 + 6 \beta ) q^{95} + ( -6 - 2 \beta ) q^{97} -\beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + 7q^{4} - 6q^{5} - 2q^{7} - 12q^{8} + O(q^{10}) \) \( 2q - q^{2} + 7q^{4} - 6q^{5} - 2q^{7} - 12q^{8} + 3q^{10} - 2q^{13} + q^{14} + 13q^{16} + 6q^{17} + 4q^{19} - 21q^{20} + 2q^{23} + 8q^{25} + q^{26} - 7q^{28} + 2q^{29} - 2q^{31} - 35q^{32} + 18q^{34} + 6q^{35} + 2q^{37} + 19q^{38} + 36q^{40} - 4q^{41} + 6q^{43} + 20q^{46} - 12q^{47} + 2q^{49} - 4q^{50} - 7q^{52} + 10q^{53} + 12q^{56} - 43q^{58} - 20q^{61} + 22q^{62} + 44q^{64} + 6q^{65} + 8q^{67} - 3q^{70} - 4q^{71} - 14q^{73} - q^{74} - 7q^{76} + 4q^{79} - 39q^{80} - 40q^{82} - 14q^{83} - 18q^{85} - 24q^{86} + 2q^{91} - 14q^{92} + 27q^{94} - 12q^{95} - 14q^{97} - q^{98} + O(q^{100}) \)

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
2.79129
−1.79129
−2.79129 0 5.79129 −3.00000 0 −1.00000 −10.5826 0 8.37386
1.2 1.79129 0 1.20871 −3.00000 0 −1.00000 −1.41742 0 −5.37386
\(n\): 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.bf 2
3.b odd 2 1 2541.2.a.z 2
11.b odd 2 1 693.2.a.j 2
33.d even 2 1 231.2.a.b 2
77.b even 2 1 4851.2.a.ba 2
132.d odd 2 1 3696.2.a.bl 2
165.d even 2 1 5775.2.a.bn 2
231.h odd 2 1 1617.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.b 2 33.d even 2 1
693.2.a.j 2 11.b odd 2 1
1617.2.a.o 2 231.h odd 2 1
2541.2.a.z 2 3.b odd 2 1
3696.2.a.bl 2 132.d odd 2 1
4851.2.a.ba 2 77.b even 2 1
5775.2.a.bn 2 165.d even 2 1
7623.2.a.bf 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

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