Properties

Label 6003.2.a.o
Level 6003
Weight 2
Character orbit 6003.a
Self dual Yes
Analytic conductor 47.934
Analytic rank 1
Dimension 13
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: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -1 + \beta_{8} ) q^{5} + \beta_{9} q^{7} + ( -\beta_{1} - \beta_{2} + \beta_{6} - \beta_{7} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -1 + \beta_{8} ) q^{5} + \beta_{9} q^{7} + ( -\beta_{1} - \beta_{2} + \beta_{6} - \beta_{7} ) q^{8} + ( \beta_{1} - \beta_{4} - \beta_{8} ) q^{10} + ( -1 - \beta_{11} ) q^{11} + ( 1 + \beta_{4} + \beta_{8} ) q^{13} + ( 1 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{9} + 2 \beta_{10} - \beta_{12} ) q^{14} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} ) q^{16} + ( -2 + \beta_{1} + \beta_{4} - \beta_{6} - \beta_{10} + \beta_{11} ) q^{17} + ( -\beta_{1} + 2 \beta_{6} - \beta_{7} - \beta_{10} - \beta_{11} ) q^{19} + ( -2 + \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{9} + \beta_{11} ) q^{20} + ( -2 + 3 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{7} + \beta_{11} ) q^{22} + q^{23} + ( 1 - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{8} - \beta_{9} + \beta_{12} ) q^{25} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{8} - \beta_{9} - \beta_{11} ) q^{26} + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} + 2 \beta_{12} ) q^{28} - q^{29} + ( -1 + 2 \beta_{1} + \beta_{3} - 2 \beta_{6} + 2 \beta_{7} + \beta_{11} ) q^{31} + ( -2 + \beta_{1} - \beta_{2} - \beta_{5} + \beta_{7} ) q^{32} + ( -2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{11} ) q^{34} + ( -\beta_{1} + \beta_{2} - \beta_{5} - \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{12} ) q^{35} + ( 1 - \beta_{3} + \beta_{5} - \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{12} ) q^{37} + ( 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} - \beta_{9} ) q^{38} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{40} + ( -1 + \beta_{2} - \beta_{4} + \beta_{8} - \beta_{9} - 2 \beta_{12} ) q^{41} + ( -\beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} ) q^{43} + ( -3 + \beta_{1} - \beta_{4} + 2 \beta_{5} - \beta_{8} - \beta_{9} - \beta_{11} ) q^{44} -\beta_{1} q^{46} + ( -2 + 3 \beta_{1} - \beta_{2} + \beta_{9} + \beta_{11} ) q^{47} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{49} + ( -2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{12} ) q^{50} + ( \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{52} + ( -4 + 3 \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{11} ) q^{53} + ( -\beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{11} ) q^{55} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{12} ) q^{56} + \beta_{1} q^{58} + ( \beta_{1} - \beta_{2} + \beta_{3} - 4 \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{12} ) q^{59} + ( 2 - \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{10} - \beta_{11} - 2 \beta_{12} ) q^{61} + ( -2 + \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{11} ) q^{62} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{7} + \beta_{9} - \beta_{11} + \beta_{12} ) q^{64} + ( 1 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{12} ) q^{65} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} ) q^{67} + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{5} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{12} ) q^{68} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{7} + \beta_{8} + 3 \beta_{9} - 4 \beta_{10} + \beta_{11} + 4 \beta_{12} ) q^{70} + ( -2 + 3 \beta_{1} - \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{12} ) q^{71} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + 2 \beta_{6} - 2 \beta_{8} + \beta_{9} + \beta_{11} ) q^{73} + ( -1 + 2 \beta_{4} + \beta_{8} - \beta_{9} - 2 \beta_{10} - 2 \beta_{12} ) q^{74} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{12} ) q^{76} + ( -2 + 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} ) q^{77} + ( 2 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} - \beta_{6} + 2 \beta_{8} + \beta_{10} + \beta_{12} ) q^{79} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{4} + 3 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{12} ) q^{80} + ( -2 + \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{7} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} + 3 \beta_{12} ) q^{82} + ( -1 - 2 \beta_{1} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{8} - \beta_{9} - \beta_{11} - 2 \beta_{12} ) q^{83} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 3 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} ) q^{85} + ( 2 - \beta_{2} + 2 \beta_{3} - 3 \beta_{5} - 2 \beta_{6} - \beta_{7} + 2 \beta_{8} + 3 \beta_{9} ) q^{86} + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} - 2 \beta_{10} - \beta_{12} ) q^{88} + ( -4 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - 3 \beta_{8} + 3 \beta_{10} - 2 \beta_{12} ) q^{89} + ( -2 + \beta_{2} - 2 \beta_{4} + \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} - 3 \beta_{12} ) q^{91} + ( 1 + \beta_{2} ) q^{92} + ( -6 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{6} - \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{12} ) q^{94} + ( -1 + 4 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 5 \beta_{6} + 2 \beta_{7} + \beta_{8} + 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} ) q^{95} + ( 4 + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{7} + \beta_{8} + \beta_{11} ) q^{97} + ( 3 \beta_{1} - 3 \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{8} + 4 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + \beta_{12} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 4q^{2} + 12q^{4} - 16q^{5} + q^{7} - 6q^{8} + O(q^{10}) \) \( 13q - 4q^{2} + 12q^{4} - 16q^{5} + q^{7} - 6q^{8} + 10q^{10} - 10q^{11} + 7q^{13} + 12q^{14} + 2q^{16} - 26q^{17} - 25q^{20} - 15q^{22} + 13q^{23} + 19q^{25} + 15q^{26} + 5q^{28} - 13q^{29} - 6q^{31} - 16q^{32} + 11q^{34} - q^{35} + 15q^{37} - 8q^{38} + 14q^{40} - 9q^{41} + q^{43} - 29q^{44} - 4q^{46} - 15q^{47} + 4q^{49} - 31q^{50} - 8q^{52} - 43q^{53} - 3q^{55} + 5q^{56} + 4q^{58} + 9q^{59} + 20q^{61} - 11q^{62} - 16q^{64} + 25q^{65} + q^{67} - 21q^{68} - 2q^{70} - 17q^{71} + 26q^{73} - 11q^{74} + 8q^{76} - 17q^{77} + 5q^{79} - 10q^{80} - 25q^{82} - 4q^{83} + 20q^{85} + 13q^{86} - 32q^{88} - 48q^{89} - 9q^{91} + 12q^{92} - 65q^{94} - 8q^{95} + 30q^{97} - 2q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13} - 4 x^{12} - 11 x^{11} + 58 x^{10} + 24 x^{9} - 298 x^{8} + 97 x^{7} + 641 x^{6} - 402 x^{5} - 547 x^{4} + 352 x^{3} + 219 x^{2} - 88 x - 40\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - \nu^{3} - 6 \nu^{2} + 4 \nu + 4 \)
\(\beta_{4}\)\(=\)\((\)\( 6 \nu^{12} - 22 \nu^{11} - 67 \nu^{10} + 313 \nu^{9} + 147 \nu^{8} - 1530 \nu^{7} + 604 \nu^{6} + 2882 \nu^{5} - 2433 \nu^{4} - 1566 \nu^{3} + 1894 \nu^{2} + 153 \nu - 363 \)\()/19\)
\(\beta_{5}\)\(=\)\((\)\( -11 \nu^{12} + 34 \nu^{11} + 145 \nu^{10} - 482 \nu^{9} - 621 \nu^{8} + 2387 \nu^{7} + 818 \nu^{6} - 4758 \nu^{5} + 233 \nu^{4} + 3232 \nu^{3} - 236 \nu^{2} - 613 \nu - 28 \)\()/19\)
\(\beta_{6}\)\(=\)\((\)\( -11 \nu^{12} + 34 \nu^{11} + 145 \nu^{10} - 482 \nu^{9} - 621 \nu^{8} + 2387 \nu^{7} + 818 \nu^{6} - 4777 \nu^{5} + 233 \nu^{4} + 3365 \nu^{3} - 198 \nu^{2} - 765 \nu - 104 \)\()/19\)
\(\beta_{7}\)\(=\)\((\)\( -11 \nu^{12} + 34 \nu^{11} + 145 \nu^{10} - 482 \nu^{9} - 621 \nu^{8} + 2387 \nu^{7} + 818 \nu^{6} - 4777 \nu^{5} + 233 \nu^{4} + 3384 \nu^{3} - 217 \nu^{2} - 860 \nu - 47 \)\()/19\)
\(\beta_{8}\)\(=\)\((\)\( -9 \nu^{12} + 33 \nu^{11} + 110 \nu^{10} - 479 \nu^{9} - 382 \nu^{8} + 2447 \nu^{7} + 44 \nu^{6} - 5121 \nu^{5} + 1379 \nu^{4} + 3869 \nu^{3} - 865 \nu^{2} - 942 \nu + 3 \)\()/19\)
\(\beta_{9}\)\(=\)\((\)\( 10 \nu^{12} - 24 \nu^{11} - 156 \nu^{10} + 357 \nu^{9} + 910 \nu^{8} - 1923 \nu^{7} - 2464 \nu^{6} + 4512 \nu^{5} + 3203 \nu^{4} - 4377 \nu^{3} - 2024 \nu^{2} + 1338 \nu + 516 \)\()/19\)
\(\beta_{10}\)\(=\)\((\)\( -31 \nu^{12} + 82 \nu^{11} + 457 \nu^{10} - 1196 \nu^{9} - 2422 \nu^{8} + 6176 \nu^{7} + 5613 \nu^{6} - 13307 \nu^{5} - 6078 \nu^{4} + 10979 \nu^{3} + 4344 \nu^{2} - 2985 \nu - 1326 \)\()/38\)
\(\beta_{11}\)\(=\)\((\)\( -21 \nu^{12} + 58 \nu^{11} + 301 \nu^{10} - 839 \nu^{9} - 1512 \nu^{8} + 4272 \nu^{7} + 3092 \nu^{6} - 8947 \nu^{5} - 2400 \nu^{4} + 6887 \nu^{3} + 1313 \nu^{2} - 1628 \nu - 449 \)\()/19\)
\(\beta_{12}\)\(=\)\((\)\( -31 \nu^{12} + 82 \nu^{11} + 457 \nu^{10} - 1196 \nu^{9} - 2422 \nu^{8} + 6195 \nu^{7} + 5575 \nu^{6} - 13478 \nu^{5} - 5774 \nu^{4} + 11397 \nu^{3} + 3736 \nu^{2} - 3175 \nu - 1136 \)\()/19\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{7} - \beta_{6} + \beta_{2} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{7} - \beta_{6} + \beta_{3} + 7 \beta_{2} + \beta_{1} + 14\)
\(\nu^{5}\)\(=\)\(7 \beta_{7} - 8 \beta_{6} + \beta_{5} + 9 \beta_{2} + 27 \beta_{1} + 2\)
\(\nu^{6}\)\(=\)\(\beta_{12} - \beta_{11} + \beta_{9} + 9 \beta_{7} - 10 \beta_{6} + \beta_{5} + 9 \beta_{3} + 44 \beta_{2} + 12 \beta_{1} + 74\)
\(\nu^{7}\)\(=\)\(3 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + 43 \beta_{7} - 54 \beta_{6} + 11 \beta_{5} + 2 \beta_{3} + 67 \beta_{2} + 151 \beta_{1} + 28\)
\(\nu^{8}\)\(=\)\(16 \beta_{12} - 13 \beta_{11} - 4 \beta_{10} + 15 \beta_{9} + 65 \beta_{7} - 80 \beta_{6} + 14 \beta_{5} + 64 \beta_{3} + 274 \beta_{2} + 106 \beta_{1} + 412\)
\(\nu^{9}\)\(=\)\(45 \beta_{12} - 28 \beta_{11} - 30 \beta_{10} + 33 \beta_{9} + 261 \beta_{7} - 352 \beta_{6} + 91 \beta_{5} + 2 \beta_{4} + 33 \beta_{3} + 473 \beta_{2} + 869 \beta_{1} + 264\)
\(\nu^{10}\)\(=\)\(169 \beta_{12} - 117 \beta_{11} - 66 \beta_{10} + 156 \beta_{9} + 2 \beta_{8} + 445 \beta_{7} - 599 \beta_{6} + 137 \beta_{5} + 5 \beta_{4} + 428 \beta_{3} + 1720 \beta_{2} + 837 \beta_{1} + 2367\)
\(\nu^{11}\)\(=\)\(462 \beta_{12} - 266 \beta_{11} - 312 \beta_{10} + 364 \beta_{9} + 7 \beta_{8} + 1603 \beta_{7} - 2290 \beta_{6} + 680 \beta_{5} + 41 \beta_{4} + 361 \beta_{3} + 3268 \beta_{2} + 5135 \beta_{1} + 2139\)
\(\nu^{12}\)\(=\)\(1506 \beta_{12} - 912 \beta_{11} - 728 \beta_{10} + 1397 \beta_{9} + 48 \beta_{8} + 3002 \beta_{7} - 4350 \beta_{6} + 1157 \beta_{5} + 105 \beta_{4} + 2823 \beta_{3} + 10918 \beta_{2} + 6258 \beta_{1} + 13929\)

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
2.60496
2.43510
2.24788
1.46878
1.30511
1.18857
0.775068
−0.397523
−0.552582
−0.806558
−1.78056
−2.22202
−2.26622
−2.60496 0 4.78583 −1.68070 0 2.80821 −7.25699 0 4.37815
1.2 −2.43510 0 3.92972 −3.49556 0 −4.87125 −4.69905 0 8.51205
1.3 −2.24788 0 3.05297 0.269352 0 −0.523800 −2.36696 0 −0.605470
1.4 −1.46878 0 0.157302 −4.31834 0 1.97692 2.70651 0 6.34267
1.5 −1.30511 0 −0.296685 −0.572849 0 1.21746 2.99743 0 0.747631
1.6 −1.18857 0 −0.587311 2.45282 0 −2.24002 3.07519 0 −2.91534
1.7 −0.775068 0 −1.39927 −3.48646 0 0.0624539 2.63467 0 2.70224
1.8 0.397523 0 −1.84198 3.16986 0 −1.67469 −1.52727 0 1.26009
1.9 0.552582 0 −1.69465 −3.05371 0 −3.72348 −2.04160 0 −1.68743
1.10 0.806558 0 −1.34946 −2.04515 0 4.05430 −2.70154 0 −1.64953
1.11 1.78056 0 1.17038 −0.267809 0 2.76268 −1.47718 0 −0.476849
1.12 2.22202 0 2.93739 −2.84468 0 −1.80107 2.08291 0 −6.32095
1.13 2.26622 0 3.13576 −0.126764 0 2.95229 2.57389 0 −0.287275
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

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