Properties

Label 7623.2.a.cc
Level 7623
Weight 2
Character orbit 7623.a
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 3
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: \(3\)
Coefficient field: 3.3.316.1
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -\beta_{1} + \beta_{2} ) q^{5} - q^{7} + ( 1 + \beta_{2} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -\beta_{1} + \beta_{2} ) q^{5} - q^{7} + ( 1 + \beta_{2} ) q^{8} + ( -2 + \beta_{1} ) q^{10} + ( -2 - \beta_{1} ) q^{13} -\beta_{1} q^{14} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{16} + ( \beta_{1} - \beta_{2} ) q^{17} + ( -3 + \beta_{1} + \beta_{2} ) q^{19} + ( 3 - \beta_{2} ) q^{20} + ( 1 - \beta_{1} - 3 \beta_{2} ) q^{23} -2 \beta_{2} q^{25} + ( -3 - 2 \beta_{1} - \beta_{2} ) q^{26} + ( -1 - \beta_{2} ) q^{28} + ( 2 + \beta_{1} ) q^{29} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{31} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{32} + ( 2 - \beta_{1} ) q^{34} + ( \beta_{1} - \beta_{2} ) q^{35} + ( -1 - \beta_{2} ) q^{37} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{38} + ( 3 - \beta_{2} ) q^{40} + ( 5 - 2 \beta_{1} + \beta_{2} ) q^{41} + ( -4 + 4 \beta_{1} + \beta_{2} ) q^{43} + ( -6 - 2 \beta_{1} - 4 \beta_{2} ) q^{46} + ( 3 - 3 \beta_{1} + 2 \beta_{2} ) q^{47} + q^{49} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{50} + ( -3 - 2 \beta_{1} - 3 \beta_{2} ) q^{52} + ( 7 - \beta_{2} ) q^{53} + ( -1 - \beta_{2} ) q^{56} + ( 3 + 2 \beta_{1} + \beta_{2} ) q^{58} + ( -1 - 3 \beta_{1} ) q^{59} + ( -5 + 3 \beta_{1} - 3 \beta_{2} ) q^{61} + ( -4 + 2 \beta_{1} ) q^{62} + ( -5 - 2 \beta_{1} - \beta_{2} ) q^{64} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{65} + ( 2 \beta_{1} - 5 \beta_{2} ) q^{67} + ( -3 + \beta_{2} ) q^{68} + ( 2 - \beta_{1} ) q^{70} + ( -1 + 3 \beta_{1} - \beta_{2} ) q^{71} + ( 3 - 5 \beta_{1} - 3 \beta_{2} ) q^{73} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{74} + ( 2 + 4 \beta_{1} - 2 \beta_{2} ) q^{76} + ( -4 - 6 \beta_{1} + 2 \beta_{2} ) q^{79} + ( -7 + 2 \beta_{1} + \beta_{2} ) q^{80} + ( -5 + 6 \beta_{1} - \beta_{2} ) q^{82} + ( -5 + 3 \beta_{1} - 6 \beta_{2} ) q^{83} + ( -5 + 2 \beta_{2} ) q^{85} + ( 13 - 3 \beta_{1} + 5 \beta_{2} ) q^{86} + ( \beta_{1} + 3 \beta_{2} ) q^{89} + ( 2 + \beta_{1} ) q^{91} + ( -12 - 8 \beta_{1} ) q^{92} + ( -7 + 5 \beta_{1} - \beta_{2} ) q^{94} + ( 1 + 5 \beta_{1} - 5 \beta_{2} ) q^{95} + ( -6 - 7 \beta_{1} + 2 \beta_{2} ) q^{97} + \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{2} + 3q^{4} - q^{5} - 3q^{7} + 3q^{8} + O(q^{10}) \) \( 3q + q^{2} + 3q^{4} - q^{5} - 3q^{7} + 3q^{8} - 5q^{10} - 7q^{13} - q^{14} - q^{16} + q^{17} - 8q^{19} + 9q^{20} + 2q^{23} - 11q^{26} - 3q^{28} + 7q^{29} - 2q^{31} + 7q^{32} + 5q^{34} + q^{35} - 3q^{37} + 10q^{38} + 9q^{40} + 13q^{41} - 8q^{43} - 20q^{46} + 6q^{47} + 3q^{49} - 8q^{50} - 11q^{52} + 21q^{53} - 3q^{56} + 11q^{58} - 6q^{59} - 12q^{61} - 10q^{62} - 17q^{64} + 7q^{65} + 2q^{67} - 9q^{68} + 5q^{70} + 4q^{73} - 5q^{74} + 10q^{76} - 18q^{79} - 19q^{80} - 9q^{82} - 12q^{83} - 15q^{85} + 36q^{86} + q^{89} + 7q^{91} - 44q^{92} - 16q^{94} + 8q^{95} - 25q^{97} + q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)

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
−1.81361
0.470683
2.34292
−1.81361 0 1.28917 2.10278 0 −1.00000 1.28917 0 −3.81361
1.2 0.470683 0 −1.77846 −3.24914 0 −1.00000 −1.77846 0 −1.52932
1.3 2.34292 0 3.48929 0.146365 0 −1.00000 3.48929 0 0.342923
\(n\): 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.cc 3
3.b odd 2 1 2541.2.a.bh 3
11.b odd 2 1 7623.2.a.ca 3
33.d even 2 1 2541.2.a.bj yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.2.a.bh 3 3.b odd 2 1
2541.2.a.bj yes 3 33.d even 2 1
7623.2.a.ca 3 11.b odd 2 1
7623.2.a.cc 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{3} - T_{2}^{2} - 4 T_{2} + 2 \)
\( T_{5}^{3} + T_{5}^{2} - 7 T_{5} + 1 \)
\( T_{13}^{3} + 7 T_{13}^{2} + 12 T_{13} + 2 \)