Properties

Label 6003.2.a.k
Level 6003
Weight 2
Character orbit 6003.a
Self dual Yes
Analytic conductor 47.934
Analytic rank 1
Dimension 10
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6003.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(47.9341963334\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{2} + ( 2 + \beta_{6} ) q^{4} + ( -1 - \beta_{4} ) q^{5} + ( \beta_{3} + \beta_{4} - \beta_{7} + \beta_{9} ) q^{7} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{8} +O(q^{10})\) \( q -\beta_{3} q^{2} + ( 2 + \beta_{6} ) q^{4} + ( -1 - \beta_{4} ) q^{5} + ( \beta_{3} + \beta_{4} - \beta_{7} + \beta_{9} ) q^{7} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{8} + ( -1 + \beta_{3} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{10} + ( -\beta_{2} - \beta_{9} ) q^{11} + ( -2 - 2 \beta_{1} + 2 \beta_{3} + \beta_{4} + \beta_{9} ) q^{13} + ( -1 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} - 2 \beta_{9} ) q^{14} + ( 3 + \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{16} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{17} + ( \beta_{3} - \beta_{5} ) q^{19} + ( -3 - \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} ) q^{20} + ( 1 + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{22} + q^{23} + ( \beta_{2} + 2 \beta_{4} - \beta_{7} + \beta_{8} ) q^{25} + ( -5 + 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{8} + \beta_{9} ) q^{26} + ( -1 - \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{28} - q^{29} + ( 2 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{31} + ( -1 + 2 \beta_{1} - 4 \beta_{2} - \beta_{3} - \beta_{4} + 4 \beta_{5} + 2 \beta_{6} + \beta_{9} ) q^{32} + ( 1 - 3 \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - 3 \beta_{8} - \beta_{9} ) q^{34} + ( -4 + 2 \beta_{1} - \beta_{2} - 3 \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{35} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{8} ) q^{37} + ( -4 - \beta_{1} + \beta_{3} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{38} + ( \beta_{2} + 2 \beta_{3} + 2 \beta_{8} + \beta_{9} ) q^{40} + ( \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{41} + ( -\beta_{1} - \beta_{2} + 2 \beta_{5} - \beta_{8} - \beta_{9} ) q^{43} + ( -3 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{44} -\beta_{3} q^{46} + ( -2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{47} + ( 2 + 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{8} ) q^{49} + ( 4 - 3 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} ) q^{50} + ( -4 - 2 \beta_{1} - \beta_{2} + 7 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{52} + ( -4 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{9} ) q^{53} + ( -2 - \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{55} + ( -4 - 2 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} + \beta_{4} - 4 \beta_{5} - \beta_{6} - 3 \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{56} + \beta_{3} q^{58} + ( -5 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{9} ) q^{59} + ( -1 + 3 \beta_{1} + \beta_{2} - 3 \beta_{4} - 2 \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{61} + ( -1 - \beta_{1} + 3 \beta_{2} + \beta_{3} + 5 \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{8} ) q^{62} + ( 1 + 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + \beta_{6} + 2 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} ) q^{64} + ( 3 - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{65} + ( 5 - \beta_{3} + 2 \beta_{4} + \beta_{7} - \beta_{9} ) q^{67} + ( 2 + 5 \beta_{1} + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{68} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{5} + 5 \beta_{7} + 2 \beta_{8} ) q^{70} + ( -4 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{71} + ( -2 + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} ) q^{73} + ( -1 + 6 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 7 \beta_{5} + \beta_{6} + 4 \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{74} + ( -1 - 4 \beta_{1} + 4 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{76} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} - 3 \beta_{9} ) q^{77} + ( 1 - 2 \beta_{1} + 4 \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + \beta_{8} + \beta_{9} ) q^{79} + ( -1 - 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} - 2 \beta_{9} ) q^{80} + ( -1 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} ) q^{82} + ( -2 + 3 \beta_{1} + 3 \beta_{2} + \beta_{7} + 3 \beta_{8} + \beta_{9} ) q^{83} + ( -\beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} ) q^{85} + ( 1 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{7} - \beta_{9} ) q^{86} + ( 2 - 2 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} - 3 \beta_{9} ) q^{88} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{9} ) q^{89} + ( 4 - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{5} - \beta_{6} + 5 \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{91} + ( 2 + \beta_{6} ) q^{92} + ( 1 - 5 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - 3 \beta_{8} ) q^{94} + ( -\beta_{1} - \beta_{3} + \beta_{4} - \beta_{8} ) q^{95} + ( -4 + 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{97} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{8} - \beta_{9} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 3q^{2} + 17q^{4} - 6q^{5} + 3q^{7} + 6q^{8} + O(q^{10}) \) \( 10q - 3q^{2} + 17q^{4} - 6q^{5} + 3q^{7} + 6q^{8} - 4q^{10} - 9q^{11} - 16q^{13} - 16q^{14} + 27q^{16} + q^{19} - 21q^{20} + 17q^{22} + 10q^{23} - 4q^{25} - 28q^{26} - 14q^{28} - 10q^{29} + 17q^{31} - 21q^{32} - 3q^{34} - 29q^{35} + q^{37} - 32q^{38} + 13q^{40} - 5q^{43} - 33q^{44} - 3q^{46} - 15q^{47} + 31q^{49} + 22q^{50} - 21q^{52} - 35q^{53} - 20q^{55} - 18q^{56} + 3q^{58} - 49q^{59} + 8q^{61} - 15q^{62} + 12q^{64} + 3q^{65} + 35q^{67} + 18q^{68} - 16q^{70} - 30q^{71} - 15q^{73} - 23q^{74} + 10q^{76} - 23q^{77} + 24q^{79} - 23q^{80} - 5q^{82} - q^{83} + 10q^{86} + 18q^{88} - 15q^{89} + 26q^{91} + 17q^{92} + 3q^{94} - 7q^{95} - 35q^{97} - 3q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - x^{9} - 17 x^{8} + 23 x^{7} + 69 x^{6} - 88 x^{5} - 106 x^{4} + 101 x^{3} + 60 x^{2} - 23 x - 11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 31 \nu^{9} - 14 \nu^{8} - 481 \nu^{7} + 466 \nu^{6} + 1581 \nu^{5} - 1705 \nu^{4} - 1465 \nu^{3} + 1920 \nu^{2} + 172 \nu - 357 \)\()/52\)
\(\beta_{3}\)\(=\)\((\)\( 11 \nu^{9} - 23 \nu^{8} - 182 \nu^{7} + 448 \nu^{6} + 587 \nu^{5} - 1710 \nu^{4} - 493 \nu^{3} + 2027 \nu^{2} - 87 \nu - 450 \)\()/26\)
\(\beta_{4}\)\(=\)\((\)\( -55 \nu^{9} + 24 \nu^{8} + 871 \nu^{7} - 810 \nu^{6} - 3065 \nu^{5} + 3051 \nu^{4} + 3297 \nu^{3} - 3466 \nu^{2} - 566 \nu + 703 \)\()/52\)
\(\beta_{5}\)\(=\)\((\)\( 33 \nu^{9} - 30 \nu^{8} - 533 \nu^{7} + 720 \nu^{6} + 1865 \nu^{5} - 2647 \nu^{4} - 1999 \nu^{3} + 2870 \nu^{2} + 272 \nu - 505 \)\()/26\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{9} + 3 \nu^{8} + 48 \nu^{7} - 70 \nu^{6} - 161 \nu^{5} + 256 \nu^{4} + 155 \nu^{3} - 279 \nu^{2} - 3 \nu + 54 \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( 51 \nu^{9} - 148 \nu^{8} - 871 \nu^{7} + 2694 \nu^{6} + 2809 \nu^{5} - 10007 \nu^{4} - 2645 \nu^{3} + 11238 \nu^{2} + 2 \nu - 2227 \)\()/52\)
\(\beta_{8}\)\(=\)\((\)\( -41 \nu^{9} + 42 \nu^{8} + 663 \nu^{7} - 982 \nu^{6} - 2299 \nu^{5} + 3737 \nu^{4} + 2367 \nu^{3} - 4252 \nu^{2} - 256 \nu + 785 \)\()/26\)
\(\beta_{9}\)\(=\)\((\)\( 93 \nu^{9} - 120 \nu^{8} - 1521 \nu^{7} + 2594 \nu^{6} + 5263 \nu^{5} - 9717 \nu^{4} - 5435 \nu^{3} + 10830 \nu^{2} + 490 \nu - 2033 \)\()/52\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9} + \beta_{8} - \beta_{5} - \beta_{4} + 3\)
\(\nu^{3}\)\(=\)\(-3 \beta_{9} - 2 \beta_{8} - \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} + 7 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(14 \beta_{9} + 11 \beta_{8} - 14 \beta_{5} - 10 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 6 \beta_{1} + 23\)
\(\nu^{5}\)\(=\)\(-50 \beta_{9} - 34 \beta_{8} + 2 \beta_{7} - 12 \beta_{6} + 35 \beta_{5} + 17 \beta_{4} + 13 \beta_{3} - 27 \beta_{2} + 70 \beta_{1} - 29\)
\(\nu^{6}\)\(=\)\(186 \beta_{9} + 135 \beta_{8} - \beta_{7} + 8 \beta_{6} - 179 \beta_{5} - 114 \beta_{4} - 61 \beta_{3} + 43 \beta_{2} - 126 \beta_{1} + 241\)
\(\nu^{7}\)\(=\)\(-695 \beta_{9} - 480 \beta_{8} + 29 \beta_{7} - 136 \beta_{6} + 516 \beta_{5} + 268 \beta_{4} + 178 \beta_{3} - 324 \beta_{2} + 804 \beta_{1} - 510\)
\(\nu^{8}\)\(=\)\(2492 \beta_{9} + 1766 \beta_{8} - 31 \beta_{7} + 192 \beta_{6} - 2293 \beta_{5} - 1391 \beta_{4} - 804 \beta_{3} + 714 \beta_{2} - 2009 \beta_{1} + 2851\)
\(\nu^{9}\)\(=\)\(-9338 \beta_{9} - 6497 \beta_{8} + 349 \beta_{7} - 1579 \beta_{6} + 7263 \beta_{5} + 3936 \beta_{4} + 2480 \beta_{3} - 3957 \beta_{2} + 9887 \beta_{1} - 7726\)

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
1.19874
1.96386
−1.34161
1.92359
−1.52497
2.78422
−0.473620
−0.435319
−3.66000
0.565113
−2.74444 0 5.53194 −0.0839771 0 −2.30146 −9.69318 0 0.230470
1.2 −2.36221 0 3.58005 −3.25906 0 3.14721 −3.73241 0 7.69859
1.3 −2.08881 0 2.36311 −0.182168 0 1.65593 −0.758471 0 0.380514
1.4 −1.39479 0 −0.0545700 2.50233 0 4.61826 2.86569 0 −3.49022
1.5 −0.676519 0 −1.54232 1.80585 0 −4.44627 2.39645 0 −1.22169
1.6 −0.611391 0 −1.62620 −0.834863 0 −0.685491 2.21703 0 0.510427
1.7 −0.161383 0 −1.97396 −2.08087 0 3.66994 0.641331 0 0.335818
1.8 2.09108 0 2.37262 −4.27000 0 3.25035 0.779182 0 −8.92891
1.9 2.24431 0 3.03693 1.31370 0 −2.96709 2.32720 0 2.94835
1.10 2.70414 0 5.31240 −0.910949 0 −2.94138 8.95720 0 −2.46334
\(n\): 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.

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(23\) \(-1\)
\(29\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

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