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.

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

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

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

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