Properties

Label 7623.2.a.cf
Level 7623
Weight 2
Character orbit 7623.a
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 4
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: \(4\)
Coefficient field: 4.4.7488.1
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 4 x^{2} + 2 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 4 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 3\)

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
0.698857
3.05896
−1.43091
−0.326909
−2.43091 0 3.90931 −3.43091 0 1.00000 −4.64136 0 8.34022
1.2 −1.32691 0 −0.239314 −2.32691 0 1.00000 2.97136 0 3.08759
1.3 −0.301143 0 −1.90931 −1.30114 0 1.00000 1.17726 0 0.391830
1.4 2.05896 0 2.23931 1.05896 0 1.00000 0.492737 0 2.18035
\(n\): 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.cf 4
3.b odd 2 1 2541.2.a.bo yes 4
11.b odd 2 1 7623.2.a.cm 4
33.d even 2 1 2541.2.a.bk 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.2.a.bk 4 33.d even 2 1
2541.2.a.bo yes 4 3.b odd 2 1
7623.2.a.cf 4 1.a even 1 1 trivial
7623.2.a.cm 4 11.b odd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{4} + 2 T_{2}^{3} - 4 T_{2}^{2} - 8 T_{2} - 2 \)
\( T_{5}^{4} + 6 T_{5}^{3} + 8 T_{5}^{2} - 6 T_{5} - 11 \)
\( T_{13}^{4} + 2 T_{13}^{3} - 40 T_{13}^{2} - 32 T_{13} + 358 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 4 T^{2} + 4 T^{3} + 6 T^{4} + 8 T^{5} + 16 T^{6} + 16 T^{7} + 16 T^{8} \)
$3$ \( \)
$5$ \( 1 + 6 T + 28 T^{2} + 84 T^{3} + 219 T^{4} + 420 T^{5} + 700 T^{6} + 750 T^{7} + 625 T^{8} \)
$7$ \( ( 1 - T )^{4} \)
$11$ \( \)
$13$ \( 1 + 2 T + 12 T^{2} + 46 T^{3} + 332 T^{4} + 598 T^{5} + 2028 T^{6} + 4394 T^{7} + 28561 T^{8} \)
$17$ \( 1 + 2 T + 56 T^{2} + 80 T^{3} + 1339 T^{4} + 1360 T^{5} + 16184 T^{6} + 9826 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 6 T + 72 T^{2} - 330 T^{3} + 2012 T^{4} - 6270 T^{5} + 25992 T^{6} - 41154 T^{7} + 130321 T^{8} \)
$23$ \( 1 + 10 T + 116 T^{2} + 694 T^{3} + 4276 T^{4} + 15962 T^{5} + 61364 T^{6} + 121670 T^{7} + 279841 T^{8} \)
$29$ \( 1 + 14 T + 164 T^{2} + 1238 T^{3} + 7756 T^{4} + 35902 T^{5} + 137924 T^{6} + 341446 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 108 T^{2} - 24 T^{3} + 4766 T^{4} - 744 T^{5} + 103788 T^{6} + 923521 T^{8} \)
$37$ \( 1 + 12 T + 168 T^{2} + 1284 T^{3} + 9650 T^{4} + 47508 T^{5} + 229992 T^{6} + 607836 T^{7} + 1874161 T^{8} \)
$41$ \( 1 + 16 T + 196 T^{2} + 1712 T^{3} + 11994 T^{4} + 70192 T^{5} + 329476 T^{6} + 1102736 T^{7} + 2825761 T^{8} \)
$43$ \( 1 - 20 T + 306 T^{2} - 2920 T^{3} + 22895 T^{4} - 125560 T^{5} + 565794 T^{6} - 1590140 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 2 T + 88 T^{2} + 568 T^{3} + 4023 T^{4} + 26696 T^{5} + 194392 T^{6} + 207646 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 16 T + 244 T^{2} - 2096 T^{3} + 18582 T^{4} - 111088 T^{5} + 685396 T^{6} - 2382032 T^{7} + 7890481 T^{8} \)
$59$ \( 1 + 2 T - 20 T^{2} + 136 T^{3} + 6195 T^{4} + 8024 T^{5} - 69620 T^{6} + 410758 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - 2 T + 180 T^{2} - 418 T^{3} + 14804 T^{4} - 25498 T^{5} + 669780 T^{6} - 453962 T^{7} + 13845841 T^{8} \)
$67$ \( 1 + 20 T + 282 T^{2} + 2776 T^{3} + 24707 T^{4} + 185992 T^{5} + 1265898 T^{6} + 6015260 T^{7} + 20151121 T^{8} \)
$71$ \( 1 - 10 T + 76 T^{2} + 790 T^{3} - 8172 T^{4} + 56090 T^{5} + 383116 T^{6} - 3579110 T^{7} + 25411681 T^{8} \)
$73$ \( 1 + 10 T + 52 T^{2} + 406 T^{3} + 7828 T^{4} + 29638 T^{5} + 277108 T^{6} + 3890170 T^{7} + 28398241 T^{8} \)
$79$ \( 1 - 16 T + 312 T^{2} - 3272 T^{3} + 37490 T^{4} - 258488 T^{5} + 1947192 T^{6} - 7888624 T^{7} + 38950081 T^{8} \)
$83$ \( 1 + 18 T + 400 T^{2} + 4404 T^{3} + 52635 T^{4} + 365532 T^{5} + 2755600 T^{6} + 10292166 T^{7} + 47458321 T^{8} \)
$89$ \( 1 - 14 T + 244 T^{2} - 2764 T^{3} + 32067 T^{4} - 245996 T^{5} + 1932724 T^{6} - 9869566 T^{7} + 62742241 T^{8} \)
$97$ \( 1 - 38 T + 844 T^{2} - 12458 T^{3} + 140980 T^{4} - 1208426 T^{5} + 7941196 T^{6} - 34681574 T^{7} + 88529281 T^{8} \)
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