Properties

Label 7623.2.a.cy
Level 7623
Weight 2
Character orbit 7623.a
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 10
CM no
Inner twists 1

Related objects

Downloads

Learn more about

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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{4} + \beta_{6} q^{5} + q^{7} + ( 1 + 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{4} + \beta_{6} q^{5} + q^{7} + ( 1 + 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{8} + ( -\beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{10} + ( -\beta_{3} + \beta_{5} - \beta_{9} ) q^{13} + \beta_{1} q^{14} + ( 4 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} - \beta_{7} + \beta_{8} ) q^{16} + ( -1 + \beta_{1} - \beta_{2} - \beta_{5} ) q^{17} + ( 1 + \beta_{3} + \beta_{4} - \beta_{7} ) q^{19} + ( 2 + 3 \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{20} + ( \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} ) q^{23} + ( 3 - \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{9} ) q^{25} + ( -3 - \beta_{1} - 4 \beta_{3} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{26} + ( 2 + \beta_{2} ) q^{28} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{8} - \beta_{9} ) q^{29} + ( 2 + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{31} + ( 7 + 3 \beta_{1} + 3 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{8} + 2 \beta_{9} ) q^{32} + ( 4 - 3 \beta_{1} + 3 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{9} ) q^{34} + \beta_{6} q^{35} + ( 3 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{37} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{5} + 2 \beta_{8} + \beta_{9} ) q^{38} + ( 1 + \beta_{1} - \beta_{2} + 3 \beta_{3} - 5 \beta_{5} + \beta_{8} + 2 \beta_{9} ) q^{40} + ( -2 + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{9} ) q^{41} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{8} - \beta_{9} ) q^{43} + ( 1 + \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{7} + \beta_{9} ) q^{46} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{9} ) q^{47} + q^{49} + ( -4 + 3 \beta_{1} - \beta_{2} - 6 \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{50} + ( -7 - 3 \beta_{1} + 2 \beta_{2} - 7 \beta_{3} + 5 \beta_{5} - 3 \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{52} + ( \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{53} + ( 1 + 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{56} + ( 2 + 2 \beta_{1} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{58} + ( -1 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{9} ) q^{59} + ( 2 \beta_{1} + \beta_{3} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{61} + ( 2 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{9} ) q^{62} + ( 6 + 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - 4 \beta_{5} + 2 \beta_{6} + \beta_{8} + \beta_{9} ) q^{64} + ( 3 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} + 2 \beta_{9} ) q^{65} + ( 5 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{67} + ( -8 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 3 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} ) q^{68} + ( -\beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{70} + ( -5 - \beta_{1} + \beta_{2} - 5 \beta_{3} - 2 \beta_{5} - \beta_{8} - \beta_{9} ) q^{71} + ( 1 + 3 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} ) q^{73} + ( 3 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{9} ) q^{74} + ( 5 + 2 \beta_{1} + 5 \beta_{3} + \beta_{4} + 2 \beta_{5} + 4 \beta_{6} - \beta_{7} + 2 \beta_{9} ) q^{76} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{79} + ( 7 + \beta_{1} - 2 \beta_{2} + 12 \beta_{3} - 2 \beta_{5} + 4 \beta_{6} + \beta_{7} + 3 \beta_{9} ) q^{80} + ( -3 - 3 \beta_{1} + \beta_{2} - 5 \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{8} - 2 \beta_{9} ) q^{82} + ( -2 - 4 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{83} + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{7} + \beta_{8} ) q^{85} + ( 4 + 2 \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{86} + ( 2 - \beta_{2} + 2 \beta_{3} + 3 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} ) q^{89} + ( -\beta_{3} + \beta_{5} - \beta_{9} ) q^{91} + ( 5 - \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} - 3 \beta_{7} - \beta_{9} ) q^{92} + ( 2 - 2 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{7} + \beta_{9} ) q^{94} + ( 6 - 2 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{95} + ( 2 - 2 \beta_{1} - \beta_{3} - \beta_{5} + \beta_{9} ) q^{97} + \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 18q^{4} - 5q^{5} + 10q^{7} + 3q^{8} + O(q^{10}) \) \( 10q + 18q^{4} - 5q^{5} + 10q^{7} + 3q^{8} - 6q^{10} + 6q^{13} + 38q^{16} - 8q^{17} - 7q^{20} + 31q^{25} - q^{26} + 18q^{28} + 14q^{29} + 26q^{31} + 41q^{32} + 21q^{34} - 5q^{35} + 24q^{37} - 8q^{38} - 5q^{40} - 19q^{41} - 6q^{43} - q^{46} - 15q^{47} + 10q^{49} + q^{50} - 25q^{52} + q^{53} + 3q^{56} + 11q^{58} - 23q^{59} - 11q^{62} + 53q^{64} + 29q^{65} + 38q^{67} - 87q^{68} - 6q^{70} - 26q^{71} - q^{73} + 39q^{74} - 2q^{76} + 5q^{79} - 6q^{80} + 5q^{82} - 6q^{83} - q^{85} + 41q^{86} + 9q^{89} + 6q^{91} + 48q^{92} + 42q^{95} + 24q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 19 x^{8} - x^{7} + 124 x^{6} + 6 x^{5} - 316 x^{4} + 17 x^{3} + 253 x^{2} - 70 x - 11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\( -25 \nu^{9} + 31 \nu^{8} + 558 \nu^{7} - 505 \nu^{6} - 4093 \nu^{5} + 2537 \nu^{4} + 11069 \nu^{3} - 4152 \nu^{2} - 8301 \nu + 385 \)\()/2024\)
\(\beta_{4}\)\(=\)\((\)\( 25 \nu^{9} - 31 \nu^{8} - 558 \nu^{7} + 505 \nu^{6} + 4093 \nu^{5} - 2537 \nu^{4} - 9045 \nu^{3} + 4152 \nu^{2} - 3843 \nu - 2409 \)\()/2024\)
\(\beta_{5}\)\(=\)\((\)\( -31 \nu^{9} - 83 \nu^{8} + 530 \nu^{7} + 993 \nu^{6} - 2687 \nu^{5} - 3169 \nu^{4} + 3727 \nu^{3} + 1976 \nu^{2} + 1365 \nu + 275 \)\()/2024\)
\(\beta_{6}\)\(=\)\((\)\( -97 \nu^{9} - 325 \nu^{8} + 2246 \nu^{7} + 5327 \nu^{6} - 16569 \nu^{5} - 27479 \nu^{4} + 43393 \nu^{3} + 43072 \nu^{2} - 29941 \nu - 5995 \)\()/2024\)
\(\beta_{7}\)\(=\)\((\)\( -229 \nu^{9} + 203 \nu^{8} + 3654 \nu^{7} - 2197 \nu^{6} - 19033 \nu^{5} + 4861 \nu^{4} + 34681 \nu^{3} + 5848 \nu^{2} - 13617 \nu - 7403 \)\()/2024\)
\(\beta_{8}\)\(=\)\((\)\( -285 \nu^{9} + 151 \nu^{8} + 4742 \nu^{7} - 1709 \nu^{6} - 25813 \nu^{5} + 6253 \nu^{4} + 49477 \nu^{3} - 12520 \nu^{2} - 24601 \nu + 9449 \)\()/2024\)
\(\beta_{9}\)\(=\)\((\)\( 70 \nu^{9} + 65 \nu^{8} - 1360 \nu^{7} - 1116 \nu^{6} + 8981 \nu^{5} + 5850 \nu^{4} - 22543 \nu^{3} - 8412 \nu^{2} + 16260 \nu - 1331 \)\()/506\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{4} + \beta_{3} + 6 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{8} - \beta_{7} - \beta_{5} - \beta_{3} + 8 \beta_{2} + 2 \beta_{1} + 24\)
\(\nu^{5}\)\(=\)\(2 \beta_{9} + \beta_{8} + 2 \beta_{6} + \beta_{5} + 9 \beta_{4} + 11 \beta_{3} + 39 \beta_{1} + 15\)
\(\nu^{6}\)\(=\)\(\beta_{9} + 11 \beta_{8} - 10 \beta_{7} + 2 \beta_{6} - 14 \beta_{5} + \beta_{4} - 12 \beta_{3} + 58 \beta_{2} + 26 \beta_{1} + 158\)
\(\nu^{7}\)\(=\)\(26 \beta_{9} + 13 \beta_{8} - \beta_{7} + 27 \beta_{6} + 10 \beta_{5} + 69 \beta_{4} + 104 \beta_{3} + 3 \beta_{2} + 266 \beta_{1} + 157\)
\(\nu^{8}\)\(=\)\(18 \beta_{9} + 97 \beta_{8} - 80 \beta_{7} + 30 \beta_{6} - 145 \beta_{5} + 16 \beta_{4} - 92 \beta_{3} + 412 \beta_{2} + 257 \beta_{1} + 1095\)
\(\nu^{9}\)\(=\)\(255 \beta_{9} + 126 \beta_{8} - 21 \beta_{7} + 272 \beta_{6} + 61 \beta_{5} + 509 \beta_{4} + 909 \beta_{3} + 52 \beta_{2} + 1873 \beta_{1} + 1444\)

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
−2.65195
−2.39396
−2.09767
−1.33330
−0.112481
0.473713
0.871604
1.80545
2.63994
2.79866
−2.65195 0 5.03286 −1.71311 0 1.00000 −8.04301 0 4.54308
1.2 −2.39396 0 3.73106 3.93829 0 1.00000 −4.14409 0 −9.42812
1.3 −2.09767 0 2.40021 −3.15947 0 1.00000 −0.839503 0 6.62751
1.4 −1.33330 0 −0.222305 0.873210 0 1.00000 2.96300 0 −1.16425
1.5 −0.112481 0 −1.98735 −1.06131 0 1.00000 0.448501 0 0.119378
1.6 0.473713 0 −1.77560 −3.75881 0 1.00000 −1.78855 0 −1.78060
1.7 0.871604 0 −1.24031 4.06436 0 1.00000 −2.82426 0 3.54252
1.8 1.80545 0 1.25966 −2.77715 0 1.00000 −1.33666 0 −5.01400
1.9 2.63994 0 4.96928 −3.08369 0 1.00000 7.83871 0 −8.14075
1.10 2.79866 0 5.83249 1.67767 0 1.00000 10.7258 0 4.69523
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.cy 10
3.b odd 2 1 2541.2.a.br 10
11.b odd 2 1 7623.2.a.cx 10
11.d odd 10 2 693.2.m.j 20
33.d even 2 1 2541.2.a.bq 10
33.f even 10 2 231.2.j.g 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.g 20 33.f even 10 2
693.2.m.j 20 11.d odd 10 2
2541.2.a.bq 10 33.d even 2 1
2541.2.a.br 10 3.b odd 2 1
7623.2.a.cx 10 11.b odd 2 1
7623.2.a.cy 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\(T_{2}^{10} - \cdots\)
\(T_{5}^{10} + \cdots\)
\(T_{13}^{10} - \cdots\)