Properties

Label 6039.2.a.h
Level 6039
Weight 2
Character orbit 6039.a
Self dual Yes
Analytic conductor 48.222
Analytic rank 1
Dimension 13
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6039.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2216577807\)
Analytic rank: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5 \)
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.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \( + ( -1 - \beta_{5} ) q^{5} \) \( + ( 1 - \beta_{6} ) q^{7} \) \( + ( -\beta_{1} - \beta_{3} ) q^{8} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \( + ( -1 - \beta_{5} ) q^{5} \) \( + ( 1 - \beta_{6} ) q^{7} \) \( + ( -\beta_{1} - \beta_{3} ) q^{8} \) \( + ( \beta_{1} + \beta_{5} - \beta_{8} ) q^{10} \) \(- q^{11}\) \( + ( 1 - \beta_{2} + \beta_{5} + \beta_{9} - \beta_{10} ) q^{13} \) \( + ( -\beta_{1} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{14} \) \( + ( \beta_{1} + \beta_{4} ) q^{16} \) \( + ( -2 - \beta_{2} + \beta_{5} + \beta_{6} + \beta_{11} + \beta_{12} ) q^{17} \) \( + ( 1 + \beta_{6} + \beta_{10} ) q^{19} \) \( + ( -1 - \beta_{2} - \beta_{6} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{20} \) \( + \beta_{1} q^{22} \) \( + ( -1 + \beta_{1} + \beta_{4} + \beta_{7} ) q^{23} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{11} ) q^{25} \) \( + ( \beta_{2} + \beta_{3} - \beta_{4} - \beta_{8} - \beta_{11} - \beta_{12} ) q^{26} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{8} - \beta_{9} + \beta_{12} ) q^{28} \) \( + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{7} - \beta_{12} ) q^{29} \) \( + ( \beta_{3} - \beta_{4} - \beta_{7} - \beta_{11} ) q^{31} \) \( + ( -2 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{11} ) q^{32} \) \( + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} + \beta_{8} - 2 \beta_{9} - \beta_{11} - \beta_{12} ) q^{34} \) \( + ( -1 + \beta_{1} + \beta_{3} - \beta_{7} + \beta_{9} - \beta_{11} ) q^{35} \) \( + ( \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{37} \) \( + ( 1 + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{38} \) \( + ( 2 \beta_{1} + \beta_{3} + 2 \beta_{6} + \beta_{8} + \beta_{10} + \beta_{11} - \beta_{12} ) q^{40} \) \( + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{6} - \beta_{8} - \beta_{10} - 2 \beta_{11} - \beta_{12} ) q^{41} \) \( + ( \beta_{2} - \beta_{5} + \beta_{6} + 2 \beta_{8} - 2 \beta_{9} ) q^{43} \) \( + ( -1 - \beta_{2} ) q^{44} \) \( + ( -1 - \beta_{3} - \beta_{5} + \beta_{7} - \beta_{11} ) q^{46} \) \( + ( -2 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} - \beta_{9} ) q^{47} \) \( + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{12} ) q^{49} \) \( + ( -2 + \beta_{1} + \beta_{3} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{50} \) \( + ( -\beta_{2} + \beta_{4} - \beta_{8} + 2 \beta_{9} + \beta_{12} ) q^{52} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} ) q^{53} \) \( + ( 1 + \beta_{5} ) q^{55} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{11} ) q^{56} \) \( + ( -2 + 3 \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{5} + 2 \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{58} \) \( + ( -2 - \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{59} \) \(+ q^{61}\) \( + ( -1 + 2 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{9} + 2 \beta_{11} ) q^{62} \) \( + ( -2 + 2 \beta_{1} - 4 \beta_{2} + \beta_{3} + 2 \beta_{5} + 3 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{11} ) q^{64} \) \( + ( -1 - \beta_{1} + 3 \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{7} - \beta_{9} + \beta_{10} ) q^{65} \) \( + ( 1 - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{9} + 2 \beta_{11} ) q^{67} \) \( + ( -2 + 2 \beta_{1} - 5 \beta_{2} + \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + 3 \beta_{11} + \beta_{12} ) q^{68} \) \( + ( -2 + 2 \beta_{1} - 2 \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{12} ) q^{70} \) \( + ( -2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} + \beta_{12} ) q^{71} \) \( + ( 2 \beta_{2} - \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{9} - \beta_{11} - \beta_{12} ) q^{73} \) \( + ( 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} - \beta_{7} + 3 \beta_{9} - \beta_{10} + 2 \beta_{11} + 2 \beta_{12} ) q^{74} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{7} - \beta_{8} - \beta_{10} - \beta_{12} ) q^{76} \) \( + ( -1 + \beta_{6} ) q^{77} \) \( + ( 1 - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{79} \) \( + ( -2 \beta_{1} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{11} + \beta_{12} ) q^{80} \) \( + ( -2 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{8} + 3 \beta_{9} + 2 \beta_{11} + \beta_{12} ) q^{82} \) \( + ( -4 + 2 \beta_{1} + \beta_{8} - \beta_{9} - \beta_{12} ) q^{83} \) \( + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{12} ) q^{85} \) \( + ( -2 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} + 3 \beta_{11} + 2 \beta_{12} ) q^{86} \) \( + ( \beta_{1} + \beta_{3} ) q^{88} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{5} - 2 \beta_{8} + 2 \beta_{9} - \beta_{11} + \beta_{12} ) q^{89} \) \( + ( -2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} - 3 \beta_{10} ) q^{91} \) \( + ( 2 + 2 \beta_{1} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{9} + \beta_{11} ) q^{92} \) \( + ( -2 - 3 \beta_{2} - \beta_{4} - \beta_{7} + \beta_{8} + 2 \beta_{11} + \beta_{12} ) q^{94} \) \( + ( -1 - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{9} + \beta_{11} - \beta_{12} ) q^{95} \) \( + ( -1 - \beta_{2} + \beta_{3} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} ) q^{97} \) \( + ( 1 - \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{8} - 3 \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(13q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 7q^{5} \) \(\mathstrut +\mathstrut 7q^{7} \) \(\mathstrut -\mathstrut 9q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(13q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 7q^{5} \) \(\mathstrut +\mathstrut 7q^{7} \) \(\mathstrut -\mathstrut 9q^{8} \) \(\mathstrut +\mathstrut 2q^{10} \) \(\mathstrut -\mathstrut 13q^{11} \) \(\mathstrut +\mathstrut 9q^{13} \) \(\mathstrut -\mathstrut 7q^{14} \) \(\mathstrut +\mathstrut 2q^{16} \) \(\mathstrut -\mathstrut 19q^{17} \) \(\mathstrut +\mathstrut 14q^{19} \) \(\mathstrut -\mathstrut 19q^{20} \) \(\mathstrut +\mathstrut 4q^{22} \) \(\mathstrut -\mathstrut 5q^{23} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut +\mathstrut 4q^{26} \) \(\mathstrut +\mathstrut 7q^{28} \) \(\mathstrut -\mathstrut 10q^{29} \) \(\mathstrut -\mathstrut q^{31} \) \(\mathstrut -\mathstrut 7q^{32} \) \(\mathstrut -\mathstrut 2q^{34} \) \(\mathstrut -\mathstrut 16q^{35} \) \(\mathstrut -\mathstrut 8q^{37} \) \(\mathstrut +\mathstrut 10q^{38} \) \(\mathstrut +\mathstrut 14q^{40} \) \(\mathstrut -\mathstrut 21q^{41} \) \(\mathstrut +\mathstrut 11q^{43} \) \(\mathstrut -\mathstrut 12q^{44} \) \(\mathstrut -\mathstrut 8q^{46} \) \(\mathstrut -\mathstrut 22q^{47} \) \(\mathstrut -\mathstrut 19q^{50} \) \(\mathstrut -\mathstrut q^{52} \) \(\mathstrut -\mathstrut 16q^{53} \) \(\mathstrut +\mathstrut 7q^{55} \) \(\mathstrut -\mathstrut 13q^{58} \) \(\mathstrut -\mathstrut 19q^{59} \) \(\mathstrut +\mathstrut 13q^{61} \) \(\mathstrut -\mathstrut 3q^{62} \) \(\mathstrut -\mathstrut 13q^{64} \) \(\mathstrut -\mathstrut 13q^{65} \) \(\mathstrut +\mathstrut 12q^{67} \) \(\mathstrut -\mathstrut 36q^{68} \) \(\mathstrut -\mathstrut 20q^{70} \) \(\mathstrut -\mathstrut 5q^{71} \) \(\mathstrut +\mathstrut 18q^{73} \) \(\mathstrut -\mathstrut 6q^{74} \) \(\mathstrut -\mathstrut 5q^{76} \) \(\mathstrut -\mathstrut 7q^{77} \) \(\mathstrut -\mathstrut q^{79} \) \(\mathstrut -\mathstrut 6q^{80} \) \(\mathstrut -\mathstrut 22q^{82} \) \(\mathstrut -\mathstrut 48q^{83} \) \(\mathstrut -\mathstrut 2q^{85} \) \(\mathstrut -\mathstrut 26q^{86} \) \(\mathstrut +\mathstrut 9q^{88} \) \(\mathstrut -\mathstrut 15q^{89} \) \(\mathstrut -\mathstrut 11q^{91} \) \(\mathstrut +\mathstrut 24q^{92} \) \(\mathstrut -\mathstrut 23q^{94} \) \(\mathstrut -\mathstrut 17q^{95} \) \(\mathstrut -\mathstrut 17q^{97} \) \(\mathstrut +\mathstrut 15q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13}\mathstrut -\mathstrut \) \(4\) \(x^{12}\mathstrut -\mathstrut \) \(11\) \(x^{11}\mathstrut +\mathstrut \) \(57\) \(x^{10}\mathstrut +\mathstrut \) \(28\) \(x^{9}\mathstrut -\mathstrut \) \(290\) \(x^{8}\mathstrut +\mathstrut \) \(51\) \(x^{7}\mathstrut +\mathstrut \) \(644\) \(x^{6}\mathstrut -\mathstrut \) \(259\) \(x^{5}\mathstrut -\mathstrut \) \(640\) \(x^{4}\mathstrut +\mathstrut \) \(274\) \(x^{3}\mathstrut +\mathstrut \) \(256\) \(x^{2}\mathstrut -\mathstrut \) \(74\) \(x\mathstrut -\mathstrut \) \(35\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 6 \nu^{2} - \nu + 4 \)
\(\beta_{5}\)\(=\)\((\)\( -8 \nu^{12} + 90 \nu^{11} - 26 \nu^{10} - 1165 \nu^{9} + 1132 \nu^{8} + 4883 \nu^{7} - 5564 \nu^{6} - 6816 \nu^{5} + 8767 \nu^{4} + 884 \nu^{3} - 4652 \nu^{2} + 1882 \nu + 410 \)\()/359\)
\(\beta_{6}\)\(=\)\((\)\( 7 \nu^{12} + 11 \nu^{11} - 67 \nu^{10} - 282 \nu^{9} - 93 \nu^{8} + 2324 \nu^{7} + 2894 \nu^{6} - 8037 \nu^{5} - 10588 \nu^{4} + 11612 \nu^{3} + 12866 \nu^{2} - 5506 \nu - 3500 \)\()/359\)
\(\beta_{7}\)\(=\)\((\)\( 29 \nu^{12} - 57 \nu^{11} - 534 \nu^{10} + 1037 \nu^{9} + 3615 \nu^{8} - 6886 \nu^{7} - 10884 \nu^{6} + 19682 \nu^{5} + 14037 \nu^{4} - 21334 \nu^{3} - 7010 \nu^{2} + 5294 \nu + 1655 \)\()/359\)
\(\beta_{8}\)\(=\)\((\)\( -66 \nu^{12} + 204 \nu^{11} + 683 \nu^{10} - 2521 \nu^{9} - 1431 \nu^{8} + 10039 \nu^{7} - 3900 \nu^{6} - 13511 \nu^{5} + 13003 \nu^{4} + 3344 \nu^{3} - 8582 \nu^{2} + 2064 \nu + 690 \)\()/359\)
\(\beta_{9}\)\(=\)\((\)\( -102 \nu^{12} + 250 \nu^{11} + 1284 \nu^{10} - 3276 \nu^{9} - 4953 \nu^{8} + 14601 \nu^{7} + 4449 \nu^{6} - 25874 \nu^{5} + 6682 \nu^{4} + 17733 \nu^{3} - 8694 \nu^{2} - 3109 \nu + 381 \)\()/359\)
\(\beta_{10}\)\(=\)\((\)\( -71 \nu^{12} + 350 \nu^{11} + 577 \nu^{10} - 4730 \nu^{9} + 533 \nu^{8} + 22021 \nu^{7} - 13660 \nu^{6} - 41824 \nu^{5} + 31900 \nu^{4} + 32078 \nu^{3} - 22798 \nu^{2} - 7440 \nu + 3190 \)\()/359\)
\(\beta_{11}\)\(=\)\((\)\( 96 \nu^{12} - 362 \nu^{11} - 1124 \nu^{10} + 5005 \nu^{9} + 4007 \nu^{8} - 24132 \nu^{7} - 4314 \nu^{6} + 48405 \nu^{5} + 2496 \nu^{4} - 39687 \nu^{3} - 4488 \nu^{2} + 10085 \nu + 1901 \)\()/359\)
\(\beta_{12}\)\(=\)\((\)\( -196 \nu^{12} + 410 \nu^{11} + 2953 \nu^{10} - 5746 \nu^{9} - 16423 \nu^{8} + 28627 \nu^{7} + 42464 \nu^{6} - 61446 \nu^{5} - 54279 \nu^{4} + 56481 \nu^{3} + 31780 \nu^{2} - 17434 \nu - 6469 \)\()/359\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(6\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(14\)
\(\nu^{5}\)\(=\)\(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(7\) \(\beta_{3}\mathstrut +\mathstrut \) \(27\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{6}\)\(=\)\(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(3\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(10\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(32\) \(\beta_{2}\mathstrut +\mathstrut \) \(12\) \(\beta_{1}\mathstrut +\mathstrut \) \(74\)
\(\nu^{7}\)\(=\)\(\beta_{12}\mathstrut +\mathstrut \) \(15\) \(\beta_{11}\mathstrut +\mathstrut \) \(2\) \(\beta_{10}\mathstrut +\mathstrut \) \(12\) \(\beta_{8}\mathstrut -\mathstrut \) \(11\) \(\beta_{7}\mathstrut +\mathstrut \) \(15\) \(\beta_{6}\mathstrut +\mathstrut \) \(12\) \(\beta_{5}\mathstrut +\mathstrut \) \(14\) \(\beta_{4}\mathstrut +\mathstrut \) \(42\) \(\beta_{3}\mathstrut +\mathstrut \) \(153\) \(\beta_{1}\mathstrut +\mathstrut \) \(25\)
\(\nu^{8}\)\(=\)\(\beta_{12}\mathstrut +\mathstrut \) \(31\) \(\beta_{11}\mathstrut +\mathstrut \) \(2\) \(\beta_{10}\mathstrut +\mathstrut \) \(14\) \(\beta_{9}\mathstrut +\mathstrut \) \(13\) \(\beta_{8}\mathstrut -\mathstrut \) \(14\) \(\beta_{7}\mathstrut +\mathstrut \) \(42\) \(\beta_{6}\mathstrut +\mathstrut \) \(30\) \(\beta_{5}\mathstrut +\mathstrut \) \(80\) \(\beta_{4}\mathstrut +\mathstrut \) \(14\) \(\beta_{3}\mathstrut +\mathstrut \) \(166\) \(\beta_{2}\mathstrut +\mathstrut \) \(109\) \(\beta_{1}\mathstrut +\mathstrut \) \(414\)
\(\nu^{9}\)\(=\)\(15\) \(\beta_{12}\mathstrut +\mathstrut \) \(153\) \(\beta_{11}\mathstrut +\mathstrut \) \(28\) \(\beta_{10}\mathstrut +\mathstrut \) \(4\) \(\beta_{9}\mathstrut +\mathstrut \) \(104\) \(\beta_{8}\mathstrut -\mathstrut \) \(91\) \(\beta_{7}\mathstrut +\mathstrut \) \(153\) \(\beta_{6}\mathstrut +\mathstrut \) \(115\) \(\beta_{5}\mathstrut +\mathstrut \) \(137\) \(\beta_{4}\mathstrut +\mathstrut \) \(246\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(900\) \(\beta_{1}\mathstrut +\mathstrut \) \(233\)
\(\nu^{10}\)\(=\)\(19\) \(\beta_{12}\mathstrut +\mathstrut \) \(328\) \(\beta_{11}\mathstrut +\mathstrut \) \(34\) \(\beta_{10}\mathstrut +\mathstrut \) \(134\) \(\beta_{9}\mathstrut +\mathstrut \) \(123\) \(\beta_{8}\mathstrut -\mathstrut \) \(137\) \(\beta_{7}\mathstrut +\mathstrut \) \(415\) \(\beta_{6}\mathstrut +\mathstrut \) \(312\) \(\beta_{5}\mathstrut +\mathstrut \) \(599\) \(\beta_{4}\mathstrut +\mathstrut \) \(135\) \(\beta_{3}\mathstrut +\mathstrut \) \(852\) \(\beta_{2}\mathstrut +\mathstrut \) \(890\) \(\beta_{1}\mathstrut +\mathstrut \) \(2406\)
\(\nu^{11}\)\(=\)\(153\) \(\beta_{12}\mathstrut +\mathstrut \) \(1334\) \(\beta_{11}\mathstrut +\mathstrut \) \(273\) \(\beta_{10}\mathstrut +\mathstrut \) \(74\) \(\beta_{9}\mathstrut +\mathstrut \) \(797\) \(\beta_{8}\mathstrut -\mathstrut \) \(686\) \(\beta_{7}\mathstrut +\mathstrut \) \(1337\) \(\beta_{6}\mathstrut +\mathstrut \) \(1001\) \(\beta_{5}\mathstrut +\mathstrut \) \(1165\) \(\beta_{4}\mathstrut +\mathstrut \) \(1451\) \(\beta_{3}\mathstrut -\mathstrut \) \(36\) \(\beta_{2}\mathstrut +\mathstrut \) \(5466\) \(\beta_{1}\mathstrut +\mathstrut \) \(1943\)
\(\nu^{12}\)\(=\)\(227\) \(\beta_{12}\mathstrut +\mathstrut \) \(2960\) \(\beta_{11}\mathstrut +\mathstrut \) \(387\) \(\beta_{10}\mathstrut +\mathstrut \) \(1100\) \(\beta_{9}\mathstrut +\mathstrut \) \(1038\) \(\beta_{8}\mathstrut -\mathstrut \) \(1168\) \(\beta_{7}\mathstrut +\mathstrut \) \(3572\) \(\beta_{6}\mathstrut +\mathstrut \) \(2782\) \(\beta_{5}\mathstrut +\mathstrut \) \(4363\) \(\beta_{4}\mathstrut +\mathstrut \) \(1129\) \(\beta_{3}\mathstrut +\mathstrut \) \(4344\) \(\beta_{2}\mathstrut +\mathstrut \) \(6882\) \(\beta_{1}\mathstrut +\mathstrut \) \(14427\)

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
2.65453
2.30577
2.25716
1.88247
1.31287
0.931518
0.669508
−0.312603
−0.638829
−1.17594
−1.38103
−2.17077
−2.33467
−2.65453 0 5.04652 −2.16368 0 −2.02198 −8.08709 0 5.74356
1.2 −2.30577 0 3.31659 2.09861 0 3.60849 −3.03576 0 −4.83892
1.3 −2.25716 0 3.09478 −1.69786 0 0.963803 −2.47110 0 3.83236
1.4 −1.88247 0 1.54370 −3.12160 0 2.89419 0.858967 0 5.87632
1.5 −1.31287 0 −0.276371 3.61438 0 −0.837447 2.98858 0 −4.74521
1.6 −0.931518 0 −1.13227 −1.13816 0 −2.87358 2.91777 0 1.06022
1.7 −0.669508 0 −1.55176 −2.40049 0 3.10183 2.37793 0 1.60715
1.8 0.312603 0 −1.90228 0.566394 0 3.64999 −1.21987 0 0.177056
1.9 0.638829 0 −1.59190 2.55169 0 −2.98273 −2.29461 0 1.63009
1.10 1.17594 0 −0.617176 −3.63888 0 −0.818481 −3.07763 0 −4.27908
1.11 1.38103 0 −0.0927606 0.547866 0 1.24123 −2.89016 0 0.756618
1.12 2.17077 0 2.71223 −2.18891 0 3.77754 1.54609 0 −4.75162
1.13 2.33467 0 3.45069 −0.0293558 0 −2.70286 3.38688 0 −0.0685361
\(n\): 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
\(3\) \(-1\)
\(11\) \(1\)
\(61\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{13} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\).