Properties

Label 7623.2.a.cu
Level 7623
Weight 2
Character orbit 7623.a
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 8
CM no
Inner twists 2

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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: \(8\)
Coefficient field: 8.8.6988960000.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 693)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( \beta_{1} + \beta_{7} ) q^{5} - q^{7} + ( \beta_{1} + \beta_{5} + \beta_{6} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( \beta_{1} + \beta_{7} ) q^{5} - q^{7} + ( \beta_{1} + \beta_{5} + \beta_{6} ) q^{8} + ( 1 + \beta_{2} - \beta_{3} ) q^{10} + ( \beta_{3} + \beta_{4} ) q^{13} -\beta_{1} q^{14} + ( -\beta_{2} + \beta_{3} + \beta_{4} ) q^{16} + ( -2 \beta_{1} - 2 \beta_{5} + \beta_{7} ) q^{17} + ( -\beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{19} + ( 2 \beta_{1} + \beta_{6} - \beta_{7} ) q^{20} + ( -3 \beta_{1} - \beta_{6} - \beta_{7} ) q^{23} + ( -2 - \beta_{3} ) q^{25} + ( \beta_{1} + 3 \beta_{5} - 2 \beta_{7} ) q^{26} -\beta_{2} q^{28} + ( -3 \beta_{1} - 3 \beta_{6} ) q^{29} + ( 1 - \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{31} + ( -4 \beta_{1} - 3 \beta_{6} - 2 \beta_{7} ) q^{32} + ( -3 - 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} ) q^{34} + ( -\beta_{1} - \beta_{7} ) q^{35} + ( -5 - \beta_{2} + \beta_{4} ) q^{37} + ( -5 \beta_{1} - 6 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{38} + ( 2 + 3 \beta_{3} ) q^{40} + ( 4 \beta_{1} + 2 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{41} + ( 2 + 3 \beta_{3} + 3 \beta_{4} ) q^{43} + ( -4 - 3 \beta_{2} + \beta_{3} ) q^{46} + ( -5 \beta_{5} + \beta_{6} ) q^{47} + q^{49} + ( -2 \beta_{1} - \beta_{5} + \beta_{7} ) q^{50} + ( 1 + \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{52} + ( -2 \beta_{1} + \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{53} + ( -\beta_{1} - \beta_{5} - \beta_{6} ) q^{56} + ( -3 - 3 \beta_{2} ) q^{58} + ( -\beta_{1} + 3 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} ) q^{59} + ( 4 - 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{61} + ( -4 \beta_{1} - 4 \beta_{5} - \beta_{6} + \beta_{7} ) q^{62} + ( -3 - 2 \beta_{2} - 2 \beta_{4} ) q^{64} + ( 3 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{65} + ( -6 - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{67} + ( -7 \beta_{1} - 5 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} ) q^{68} + ( -1 - \beta_{2} + \beta_{3} ) q^{70} + ( 3 \beta_{1} + 4 \beta_{5} - \beta_{6} ) q^{71} + ( 3 + 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{73} + ( -7 \beta_{1} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{74} + ( -6 - 3 \beta_{2} - 7 \beta_{3} - 2 \beta_{4} ) q^{76} + ( -1 + 3 \beta_{2} - 7 \beta_{3} - \beta_{4} ) q^{79} + ( -2 \beta_{1} + 3 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{80} + ( 4 + 4 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{82} + ( -2 \beta_{1} + 2 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} ) q^{83} + ( -2 - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{85} + ( 5 \beta_{1} + 9 \beta_{5} - 6 \beta_{7} ) q^{86} + ( -\beta_{1} + \beta_{5} + 6 \beta_{6} + 3 \beta_{7} ) q^{89} + ( -\beta_{3} - \beta_{4} ) q^{91} + ( -7 \beta_{1} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{92} + ( 4 - 5 \beta_{3} - 5 \beta_{4} ) q^{94} + ( -2 \beta_{1} - 5 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{95} + ( -2 - 4 \beta_{2} + 5 \beta_{3} - \beta_{4} ) q^{97} + \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{4} - 8q^{7} + O(q^{10}) \) \( 8q + 2q^{4} - 8q^{7} + 14q^{10} - 2q^{16} - 6q^{19} - 12q^{25} - 2q^{28} - 6q^{31} - 24q^{34} - 38q^{37} + 4q^{40} + 16q^{43} - 42q^{46} + 8q^{49} + 2q^{52} - 30q^{58} + 28q^{61} - 36q^{64} - 36q^{67} - 14q^{70} + 14q^{73} - 34q^{76} + 22q^{79} + 36q^{82} - 18q^{85} + 32q^{94} - 48q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 9 x^{6} + 22 x^{4} - 11 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} - 8 \nu^{4} + 16 \nu^{2} - 5 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{6} + 10 \nu^{4} - 26 \nu^{2} + 9 \)\()/2\)
\(\beta_{5}\)\(=\)\( -\nu^{7} + 9 \nu^{5} - 21 \nu^{3} + 6 \nu \)
\(\beta_{6}\)\(=\)\( \nu^{7} - 9 \nu^{5} + 22 \nu^{3} - 11 \nu \)
\(\beta_{7}\)\(=\)\((\)\( -3 \nu^{7} + 26 \nu^{5} - 58 \nu^{3} + 17 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{6} + \beta_{5} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} + \beta_{3} + 5 \beta_{2} + 8\)
\(\nu^{5}\)\(=\)\(-2 \beta_{7} + 5 \beta_{6} + 8 \beta_{5} + 24 \beta_{1}\)
\(\nu^{6}\)\(=\)\(8 \beta_{4} + 10 \beta_{3} + 24 \beta_{2} + 37\)
\(\nu^{7}\)\(=\)\(-18 \beta_{7} + 24 \beta_{6} + 50 \beta_{5} + 117 \beta_{1}\)

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
−2.25947
−1.80692
−0.716111
−0.342036
0.342036
0.716111
1.80692
2.25947
−2.25947 0 3.10522 −1.54336 0 −1.00000 −2.49721 0 3.48718
1.2 −1.80692 0 1.26498 −2.14896 0 −1.00000 1.32813 0 3.88301
1.3 −0.716111 0 −1.48718 1.54336 0 −1.00000 2.49721 0 −1.10522
1.4 −0.342036 0 −1.88301 −2.14896 0 −1.00000 1.32813 0 0.735023
1.5 0.342036 0 −1.88301 2.14896 0 −1.00000 −1.32813 0 0.735023
1.6 0.716111 0 −1.48718 −1.54336 0 −1.00000 −2.49721 0 −1.10522
1.7 1.80692 0 1.26498 2.14896 0 −1.00000 −1.32813 0 3.88301
1.8 2.25947 0 3.10522 1.54336 0 −1.00000 2.49721 0 3.48718
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7623.2.a.cu 8
3.b odd 2 1 inner 7623.2.a.cu 8
11.b odd 2 1 7623.2.a.cv 8
11.c even 5 2 693.2.m.h 16
33.d even 2 1 7623.2.a.cv 8
33.h odd 10 2 693.2.m.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.m.h 16 11.c even 5 2
693.2.m.h 16 33.h odd 10 2
7623.2.a.cu 8 1.a even 1 1 trivial
7623.2.a.cu 8 3.b odd 2 1 inner
7623.2.a.cv 8 11.b odd 2 1
7623.2.a.cv 8 33.d even 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{8} - 9 T_{2}^{6} + 22 T_{2}^{4} - 11 T_{2}^{2} + 1 \)
\( T_{5}^{4} - 7 T_{5}^{2} + 11 \)
\( T_{13}^{4} - 11 T_{13}^{2} + 10 T_{13} + 4 \)