Properties

Label 7623.2.a.cn
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} + ( -3 + \beta_{1} ) q^{13} -\beta_{3} q^{14} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{16} + ( 1 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{17} + ( -5 + \beta_{2} - \beta_{3} ) q^{19} + ( 1 + \beta_{1} + 3 \beta_{3} ) q^{20} + ( 2 - 3 \beta_{1} + \beta_{2} ) q^{23} + ( -1 - \beta_{2} + 2 \beta_{3} ) q^{25} + ( 1 + \beta_{1} + 3 \beta_{3} ) q^{26} + ( 1 - \beta_{2} ) q^{28} + ( \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{29} + ( -\beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{31} + ( 4 + 2 \beta_{3} ) q^{32} + ( -3 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{34} + ( 1 + \beta_{3} ) q^{35} + ( 1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{37} + ( 3 + \beta_{1} + 7 \beta_{3} ) q^{38} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{40} + ( -1 - \beta_{1} - 3 \beta_{3} ) q^{41} + ( -4 + \beta_{1} + 3 \beta_{3} ) q^{43} + ( -3 - 2 \beta_{1} + \beta_{2} ) q^{46} + ( 2 - \beta_{1} - 4 \beta_{2} ) q^{47} + q^{49} + ( -6 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{50} + ( -2 - \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{52} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{53} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{56} + ( 7 + 4 \beta_{1} + \beta_{2} + 6 \beta_{3} ) q^{58} + ( 1 - 2 \beta_{1} - 3 \beta_{3} ) q^{59} + ( 5 - 4 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{61} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} ) q^{62} + ( -6 + 4 \beta_{1} + 2 \beta_{2} ) q^{64} + ( -4 - 3 \beta_{3} ) q^{65} + ( -1 + 4 \beta_{1} - 2 \beta_{2} ) q^{67} + ( 9 - 3 \beta_{1} + \beta_{3} ) q^{68} + ( -3 + \beta_{2} - \beta_{3} ) q^{70} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{71} + ( -9 - \beta_{1} - 2 \beta_{3} ) q^{73} + ( -8 - \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{74} + ( -10 + \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{76} + ( -9 + 6 \beta_{1} + 3 \beta_{2} ) q^{79} + ( -4 + 2 \beta_{2} - 2 \beta_{3} ) q^{80} + ( 8 - \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{82} + ( 2 - 4 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{83} + ( 4 + \beta_{1} - 2 \beta_{2} + 5 \beta_{3} ) q^{85} + ( -8 + \beta_{1} + 3 \beta_{2} + 4 \beta_{3} ) q^{86} + ( -5 - 4 \beta_{1} - 5 \beta_{3} ) q^{89} + ( -3 + \beta_{1} ) q^{91} + ( -6 + 5 \beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{92} + ( -1 - 5 \beta_{1} - 4 \beta_{2} - 10 \beta_{3} ) q^{94} + ( -8 - \beta_{1} + \beta_{2} - 8 \beta_{3} ) q^{95} + ( -1 - \beta_{1} + 2 \beta_{3} ) q^{97} -\beta_{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} + 4q^{4} + 2q^{5} + 4q^{7} + O(q^{10}) \) \( 4q + 2q^{2} + 4q^{4} + 2q^{5} + 4q^{7} - 10q^{10} - 10q^{13} + 2q^{14} + 6q^{17} - 18q^{19} + 2q^{23} - 8q^{25} + 4q^{28} + 6q^{29} + 12q^{32} - 2q^{34} + 2q^{35} - 4q^{37} - 8q^{40} - 20q^{43} - 16q^{46} + 6q^{47} + 4q^{49} - 24q^{50} - 8q^{52} + 24q^{58} + 6q^{59} + 10q^{61} - 16q^{64} - 10q^{65} + 4q^{67} + 28q^{68} - 10q^{70} + 6q^{71} - 34q^{73} - 36q^{74} - 36q^{76} - 24q^{79} - 12q^{80} + 28q^{82} + 6q^{83} + 8q^{85} - 38q^{86} - 18q^{89} - 10q^{91} - 24q^{92} + 6q^{94} - 18q^{95} - 10q^{97} + 2q^{98} + O(q^{100}) \)

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
−0.326909
−1.43091
3.05896
0.698857
−2.05896 0 2.23931 3.05896 0 1.00000 −0.492737 0 −6.29827
1.2 0.301143 0 −1.90931 0.698857 0 1.00000 −1.17726 0 0.210456
1.3 1.32691 0 −0.239314 −0.326909 0 1.00000 −2.97136 0 −0.433778
1.4 2.43091 0 3.90931 −1.43091 0 1.00000 4.64136 0 −3.47841
\(n\): 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.cn 4
3.b odd 2 1 2541.2.a.bl 4
11.b odd 2 1 7623.2.a.cg 4
33.d even 2 1 2541.2.a.bp yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.2.a.bl 4 3.b odd 2 1
2541.2.a.bp yes 4 33.d even 2 1
7623.2.a.cg 4 11.b odd 2 1
7623.2.a.cn 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{4} - 2 T_{2}^{3} - 4 T_{2}^{2} + 8 T_{2} - 2 \)
\( T_{5}^{4} - 2 T_{5}^{3} - 4 T_{5}^{2} + 2 T_{5} + 1 \)
\( T_{13}^{4} + 10 T_{13}^{3} + 32 T_{13}^{2} + 32 T_{13} - 2 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 4 T^{2} - 4 T^{3} + 6 T^{4} - 8 T^{5} + 16 T^{6} - 16 T^{7} + 16 T^{8} \)
$3$ \( \)
$5$ \( 1 - 2 T + 16 T^{2} - 28 T^{3} + 111 T^{4} - 140 T^{5} + 400 T^{6} - 250 T^{7} + 625 T^{8} \)
$7$ \( ( 1 - T )^{4} \)
$11$ \( \)
$13$ \( 1 + 10 T + 84 T^{2} + 422 T^{3} + 1844 T^{4} + 5486 T^{5} + 14196 T^{6} + 21970 T^{7} + 28561 T^{8} \)
$17$ \( 1 - 6 T + 28 T^{2} - 24 T^{3} + 111 T^{4} - 408 T^{5} + 8092 T^{6} - 29478 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 18 T + 184 T^{2} + 1278 T^{3} + 6468 T^{4} + 24282 T^{5} + 66424 T^{6} + 123462 T^{7} + 130321 T^{8} \)
$23$ \( 1 - 2 T + 28 T^{2} - 190 T^{3} + 516 T^{4} - 4370 T^{5} + 14812 T^{6} - 24334 T^{7} + 279841 T^{8} \)
$29$ \( 1 - 6 T + 20 T^{2} - 126 T^{3} + 1404 T^{4} - 3654 T^{5} + 16820 T^{6} - 146334 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 28 T^{2} - 216 T^{3} + 606 T^{4} - 6696 T^{5} + 26908 T^{6} + 923521 T^{8} \)
$37$ \( 1 + 4 T + 72 T^{2} + 284 T^{3} + 4082 T^{4} + 10508 T^{5} + 98568 T^{6} + 202612 T^{7} + 1874161 T^{8} \)
$41$ \( 1 + 124 T^{2} + 48 T^{3} + 6810 T^{4} + 1968 T^{5} + 208444 T^{6} + 2825761 T^{8} \)
$43$ \( 1 + 20 T + 282 T^{2} + 2632 T^{3} + 19943 T^{4} + 113176 T^{5} + 521418 T^{6} + 1590140 T^{7} + 3418801 T^{8} \)
$47$ \( 1 - 6 T + 44 T^{2} - 504 T^{3} + 4371 T^{4} - 23688 T^{5} + 97196 T^{6} - 622938 T^{7} + 4879681 T^{8} \)
$53$ \( 1 + 148 T^{2} + 192 T^{3} + 9942 T^{4} + 10176 T^{5} + 415732 T^{6} + 7890481 T^{8} \)
$59$ \( 1 - 6 T + 208 T^{2} - 936 T^{3} + 17751 T^{4} - 55224 T^{5} + 724048 T^{6} - 1232274 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - 10 T + 108 T^{2} - 314 T^{3} + 3596 T^{4} - 19154 T^{5} + 401868 T^{6} - 2269810 T^{7} + 13845841 T^{8} \)
$67$ \( 1 - 4 T + 130 T^{2} + 56 T^{3} + 8299 T^{4} + 3752 T^{5} + 583570 T^{6} - 1203052 T^{7} + 20151121 T^{8} \)
$71$ \( 1 - 6 T + 220 T^{2} - 1014 T^{3} + 20916 T^{4} - 71994 T^{5} + 1109020 T^{6} - 2147466 T^{7} + 25411681 T^{8} \)
$73$ \( 1 + 34 T + 708 T^{2} + 9614 T^{3} + 96764 T^{4} + 701822 T^{5} + 3772932 T^{6} + 13226578 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 24 T + 280 T^{2} + 1584 T^{3} + 8754 T^{4} + 125136 T^{5} + 1747480 T^{6} + 11832936 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 6 T + 260 T^{2} - 1188 T^{3} + 30039 T^{4} - 98604 T^{5} + 1791140 T^{6} - 3430722 T^{7} + 47458321 T^{8} \)
$89$ \( 1 + 18 T + 352 T^{2} + 4068 T^{3} + 48255 T^{4} + 362052 T^{5} + 2788192 T^{6} + 12689442 T^{7} + 62742241 T^{8} \)
$97$ \( 1 + 10 T + 388 T^{2} + 2758 T^{3} + 56428 T^{4} + 267526 T^{5} + 3650692 T^{6} + 9126730 T^{7} + 88529281 T^{8} \)
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