Properties

Label 7623.2.a.cj
Level 7623
Weight 2
Character orbit 7623.a
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 4
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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{7})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 5 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 11 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{2} - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 4\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{2} + 11 \beta_{1}\)\()/2\)

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
−2.57794
1.16372
2.57794
−1.16372
−1.41421 0 0 −2.64575 0 −1.00000 2.82843 0 3.74166
1.2 −1.41421 0 0 2.64575 0 −1.00000 2.82843 0 −3.74166
1.3 1.41421 0 0 −2.64575 0 −1.00000 −2.82843 0 −3.74166
1.4 1.41421 0 0 2.64575 0 −1.00000 −2.82843 0 3.74166
\(n\): e.g. 2-40 or 990-1000
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.cj 4
3.b odd 2 1 inner 7623.2.a.cj 4
11.b odd 2 1 7623.2.a.ck yes 4
33.d even 2 1 7623.2.a.ck yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7623.2.a.cj 4 1.a even 1 1 trivial
7623.2.a.cj 4 3.b odd 2 1 inner
7623.2.a.ck yes 4 11.b odd 2 1
7623.2.a.ck yes 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{2} - 2 \)
\( T_{5}^{2} - 7 \)
\( T_{13}^{2} - 4 T_{13} - 10 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T^{2} + 4 T^{4} )^{2} \)
$3$ \( \)
$5$ \( ( 1 + 3 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 + T )^{4} \)
$11$ \( \)
$13$ \( ( 1 - 4 T + 16 T^{2} - 52 T^{3} + 169 T^{4} )^{2} \)
$17$ \( 1 + 38 T^{2} + 715 T^{4} + 10982 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 + 4 T + 28 T^{2} + 76 T^{3} + 361 T^{4} )^{2} \)
$23$ \( 1 + 32 T^{2} + 1090 T^{4} + 16928 T^{6} + 279841 T^{8} \)
$29$ \( 1 + 56 T^{2} + 2242 T^{4} + 47096 T^{6} + 707281 T^{8} \)
$31$ \( ( 1 + 4 T + 10 T^{2} + 124 T^{3} + 961 T^{4} )^{2} \)
$37$ \( ( 1 + 8 T + 34 T^{2} + 296 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 50 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - T + 43 T^{2} )^{4} \)
$47$ \( 1 + 158 T^{2} + 10435 T^{4} + 349022 T^{6} + 4879681 T^{8} \)
$53$ \( 1 - 28 T^{2} + 2230 T^{4} - 78652 T^{6} + 7890481 T^{8} \)
$59$ \( 1 + 158 T^{2} + 12307 T^{4} + 549998 T^{6} + 12117361 T^{8} \)
$61$ \( ( 1 - 4 T + 112 T^{2} - 244 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 2 T + 79 T^{2} - 134 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 + 140 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 24 T + 276 T^{2} - 1752 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 12 T + 138 T^{2} - 948 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( 1 + 174 T^{2} + 19331 T^{4} + 1198686 T^{6} + 47458321 T^{8} \)
$89$ \( 1 + 198 T^{2} + 23627 T^{4} + 1568358 T^{6} + 62742241 T^{8} \)
$97$ \( ( 1 - 8 T + 84 T^{2} - 776 T^{3} + 9409 T^{4} )^{2} \)
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