Properties

Label 6040.2.a.n
Level 6040
Weight 2
Character orbit 6040.a
Self dual Yes
Analytic conductor 48.230
Analytic rank 1
Dimension 13
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6040.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2296428209\)
Analytic rank: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{3} \) \(- q^{5}\) \( -\beta_{7} q^{7} \) \( + \beta_{2} q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{3} \) \(- q^{5}\) \( -\beta_{7} q^{7} \) \( + \beta_{2} q^{9} \) \( + ( -1 + \beta_{1} - \beta_{9} ) q^{11} \) \( -\beta_{11} q^{13} \) \( + \beta_{1} q^{15} \) \( + ( -1 + \beta_{3} ) q^{17} \) \( + ( 2 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} ) q^{19} \) \( + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{9} + \beta_{11} ) q^{21} \) \( + ( -1 - \beta_{1} + \beta_{3} + \beta_{9} + \beta_{10} - \beta_{12} ) q^{23} \) \(+ q^{25}\) \( + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{27} \) \( + ( \beta_{1} - \beta_{4} - \beta_{5} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} ) q^{29} \) \( + ( -\beta_{1} - \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{31} \) \( + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{4} ) q^{33} \) \( + \beta_{7} q^{35} \) \( + ( 1 - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} - \beta_{12} ) q^{37} \) \( + ( 2 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{39} \) \( + ( -\beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{41} \) \( + ( 2 - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{9} - \beta_{10} ) q^{43} \) \( -\beta_{2} q^{45} \) \( + ( -2 + 2 \beta_{1} - \beta_{4} + \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} + \beta_{12} ) q^{47} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - 2 \beta_{8} + \beta_{11} ) q^{49} \) \( + ( 2 \beta_{1} - \beta_{4} - \beta_{8} + \beta_{10} ) q^{51} \) \( + ( -4 - 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + 2 \beta_{9} + 3 \beta_{10} - \beta_{11} - 2 \beta_{12} ) q^{53} \) \( + ( 1 - \beta_{1} + \beta_{9} ) q^{55} \) \( + ( -2 - 2 \beta_{1} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{8} - \beta_{11} - \beta_{12} ) q^{57} \) \( + ( -1 + \beta_{1} + \beta_{2} - \beta_{6} - \beta_{10} + \beta_{11} + \beta_{12} ) q^{59} \) \( + ( 1 + 3 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{9} + \beta_{11} ) q^{61} \) \( + ( -1 + 4 \beta_{1} - \beta_{4} + \beta_{5} - 2 \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{63} \) \( + \beta_{11} q^{65} \) \( + ( \beta_{1} - \beta_{4} + \beta_{6} + \beta_{9} - \beta_{11} ) q^{67} \) \( + ( 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{12} ) q^{69} \) \( + ( \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{7} - \beta_{8} + \beta_{11} + 2 \beta_{12} ) q^{71} \) \( + ( -1 - 2 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{12} ) q^{73} \) \( -\beta_{1} q^{75} \) \( + ( -1 + \beta_{1} + \beta_{5} + \beta_{7} - \beta_{8} + \beta_{12} ) q^{77} \) \( + ( 3 + 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{8} - \beta_{9} - \beta_{12} ) q^{79} \) \( + ( \beta_{1} - 2 \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{12} ) q^{81} \) \( + ( -3 - \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + 4 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{12} ) q^{83} \) \( + ( 1 - \beta_{3} ) q^{85} \) \( + ( -2 + 3 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} - 4 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} ) q^{87} \) \( + ( -5 + 3 \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} - \beta_{11} ) q^{89} \) \( + ( -1 + 2 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{12} ) q^{91} \) \( + ( 4 \beta_{1} - \beta_{4} + 2 \beta_{5} - 2 \beta_{7} - \beta_{8} - 4 \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} ) q^{93} \) \( + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{95} \) \( + ( 3 - 2 \beta_{2} - 3 \beta_{3} + \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} + 3 \beta_{12} ) q^{97} \) \( + ( -1 + 4 \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(13q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut -\mathstrut 13q^{5} \) \(\mathstrut +\mathstrut 5q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(13q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut -\mathstrut 13q^{5} \) \(\mathstrut +\mathstrut 5q^{9} \) \(\mathstrut -\mathstrut 14q^{11} \) \(\mathstrut +\mathstrut 5q^{13} \) \(\mathstrut +\mathstrut 4q^{15} \) \(\mathstrut -\mathstrut 8q^{17} \) \(\mathstrut +\mathstrut 16q^{19} \) \(\mathstrut -\mathstrut 5q^{21} \) \(\mathstrut -\mathstrut 4q^{23} \) \(\mathstrut +\mathstrut 13q^{25} \) \(\mathstrut +\mathstrut 2q^{27} \) \(\mathstrut -\mathstrut 6q^{29} \) \(\mathstrut +\mathstrut 11q^{31} \) \(\mathstrut -\mathstrut 19q^{33} \) \(\mathstrut +\mathstrut 6q^{37} \) \(\mathstrut +\mathstrut 7q^{39} \) \(\mathstrut -\mathstrut 18q^{41} \) \(\mathstrut +\mathstrut 7q^{43} \) \(\mathstrut -\mathstrut 5q^{45} \) \(\mathstrut -\mathstrut 22q^{47} \) \(\mathstrut -\mathstrut q^{49} \) \(\mathstrut +\mathstrut 12q^{51} \) \(\mathstrut -\mathstrut 17q^{53} \) \(\mathstrut +\mathstrut 14q^{55} \) \(\mathstrut -\mathstrut 16q^{57} \) \(\mathstrut -\mathstrut 6q^{59} \) \(\mathstrut +\mathstrut 10q^{61} \) \(\mathstrut -\mathstrut 5q^{65} \) \(\mathstrut +\mathstrut 12q^{67} \) \(\mathstrut +\mathstrut 13q^{69} \) \(\mathstrut -\mathstrut 16q^{71} \) \(\mathstrut -\mathstrut 24q^{73} \) \(\mathstrut -\mathstrut 4q^{75} \) \(\mathstrut -\mathstrut 11q^{77} \) \(\mathstrut +\mathstrut 36q^{79} \) \(\mathstrut -\mathstrut 19q^{81} \) \(\mathstrut +\mathstrut q^{83} \) \(\mathstrut +\mathstrut 8q^{85} \) \(\mathstrut -\mathstrut 8q^{87} \) \(\mathstrut -\mathstrut 53q^{89} \) \(\mathstrut +\mathstrut 23q^{91} \) \(\mathstrut -\mathstrut 9q^{93} \) \(\mathstrut -\mathstrut 16q^{95} \) \(\mathstrut -\mathstrut 21q^{97} \) \(\mathstrut -\mathstrut 9q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13}\mathstrut -\mathstrut \) \(4\) \(x^{12}\mathstrut -\mathstrut \) \(14\) \(x^{11}\mathstrut +\mathstrut \) \(70\) \(x^{10}\mathstrut +\mathstrut \) \(41\) \(x^{9}\mathstrut -\mathstrut \) \(403\) \(x^{8}\mathstrut +\mathstrut \) \(109\) \(x^{7}\mathstrut +\mathstrut \) \(870\) \(x^{6}\mathstrut -\mathstrut \) \(444\) \(x^{5}\mathstrut -\mathstrut \) \(708\) \(x^{4}\mathstrut +\mathstrut \) \(373\) \(x^{3}\mathstrut +\mathstrut \) \(108\) \(x^{2}\mathstrut -\mathstrut \) \(82\) \(x\mathstrut +\mathstrut \) \(11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( -49 \nu^{12} + 146 \nu^{11} + 769 \nu^{10} - 2522 \nu^{9} - 3344 \nu^{8} + 14165 \nu^{7} + 1109 \nu^{6} - 28959 \nu^{5} + 13459 \nu^{4} + 21771 \nu^{3} - 19240 \nu^{2} - 5889 \nu + 4040 \)\()/212\)
\(\beta_{4}\)\(=\)\((\)\( 178 \nu^{12} - 711 \nu^{11} - 2588 \nu^{10} + 12639 \nu^{9} + 9026 \nu^{8} - 74902 \nu^{7} + 8885 \nu^{6} + 171713 \nu^{5} - 53107 \nu^{4} - 154817 \nu^{3} + 38301 \nu^{2} + 31798 \nu - 8423 \)\()/212\)
\(\beta_{5}\)\(=\)\((\)\( 486 \nu^{12} - 1765 \nu^{11} - 7452 \nu^{10} + 31293 \nu^{9} + 31378 \nu^{8} - 184594 \nu^{7} - 14213 \nu^{6} + 419143 \nu^{5} - 64733 \nu^{4} - 370815 \nu^{3} + 49407 \nu^{2} + 72146 \nu - 15237 \)\()/212\)
\(\beta_{6}\)\(=\)\((\)\( 741 \nu^{12} - 2766 \nu^{11} - 11097 \nu^{10} + 48834 \nu^{9} + 43236 \nu^{8} - 285965 \nu^{7} + 4559 \nu^{6} + 640815 \nu^{5} - 154375 \nu^{4} - 559995 \nu^{3} + 120244 \nu^{2} + 112017 \nu - 28888 \)\()/212\)
\(\beta_{7}\)\(=\)\((\)\( -379 \nu^{12} + 1404 \nu^{11} + 5723 \nu^{10} - 24848 \nu^{9} - 22886 \nu^{8} + 146093 \nu^{7} + 1545 \nu^{6} - 329565 \nu^{5} + 72987 \nu^{4} + 289139 \nu^{3} - 58934 \nu^{2} - 57175 \nu + 14604 \)\()/106\)
\(\beta_{8}\)\(=\)\((\)\( 779 \nu^{12} - 2916 \nu^{11} - 11715 \nu^{10} + 51640 \nu^{9} + 46236 \nu^{8} - 304011 \nu^{7} + 1023 \nu^{6} + 687971 \nu^{5} - 157027 \nu^{4} - 607915 \nu^{3} + 124130 \nu^{2} + 122083 \nu - 31106 \)\()/212\)
\(\beta_{9}\)\(=\)\((\)\( -785 \nu^{12} + 2962 \nu^{11} + 11701 \nu^{10} - 52362 \nu^{9} - 44824 \nu^{8} + 307329 \nu^{7} - 10663 \nu^{6} - 691835 \nu^{5} + 176827 \nu^{4} + 608887 \nu^{3} - 137988 \nu^{2} - 123081 \nu + 32784 \)\()/212\)
\(\beta_{10}\)\(=\)\((\)\( 19 \nu^{12} - 70 \nu^{11} - 287 \nu^{10} + 1238 \nu^{9} + 1148 \nu^{8} - 7271 \nu^{7} - 71 \nu^{6} + 16377 \nu^{5} - 3721 \nu^{4} - 14373 \nu^{3} + 3076 \nu^{2} + 2899 \nu - 756 \)\()/4\)
\(\beta_{11}\)\(=\)\((\)\( 1092 \nu^{12} - 4079 \nu^{11} - 16426 \nu^{10} + 72203 \nu^{9} + 64938 \nu^{8} - 424736 \nu^{7} + 127 \nu^{6} + 959715 \nu^{5} - 214621 \nu^{4} - 845711 \nu^{3} + 166041 \nu^{2} + 167008 \nu - 42039 \)\()/212\)
\(\beta_{12}\)\(=\)\((\)\( -1991 \nu^{12} + 7385 \nu^{11} + 29981 \nu^{10} - 130545 \nu^{9} - 118974 \nu^{8} + 766063 \nu^{7} + 3104 \nu^{6} - 1722952 \nu^{5} + 385094 \nu^{4} + 1510060 \nu^{3} - 302861 \nu^{2} - 299723 \nu + 75647 \)\()/212\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(2\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(18\)
\(\nu^{5}\)\(=\)\(10\) \(\beta_{12}\mathstrut +\mathstrut \) \(8\) \(\beta_{11}\mathstrut -\mathstrut \) \(10\) \(\beta_{10}\mathstrut -\mathstrut \) \(20\) \(\beta_{9}\mathstrut -\mathstrut \) \(9\) \(\beta_{8}\mathstrut -\mathstrut \) \(10\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(11\) \(\beta_{5}\mathstrut -\mathstrut \) \(9\) \(\beta_{4}\mathstrut -\mathstrut \) \(10\) \(\beta_{3}\mathstrut -\mathstrut \) \(10\) \(\beta_{2}\mathstrut +\mathstrut \) \(61\) \(\beta_{1}\mathstrut +\mathstrut \) \(8\)
\(\nu^{6}\)\(=\)\(12\) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(15\) \(\beta_{9}\mathstrut -\mathstrut \) \(13\) \(\beta_{8}\mathstrut -\mathstrut \) \(13\) \(\beta_{7}\mathstrut +\mathstrut \) \(9\) \(\beta_{6}\mathstrut +\mathstrut \) \(11\) \(\beta_{5}\mathstrut -\mathstrut \) \(3\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(50\) \(\beta_{2}\mathstrut +\mathstrut \) \(13\) \(\beta_{1}\mathstrut +\mathstrut \) \(120\)
\(\nu^{7}\)\(=\)\(82\) \(\beta_{12}\mathstrut +\mathstrut \) \(57\) \(\beta_{11}\mathstrut -\mathstrut \) \(86\) \(\beta_{10}\mathstrut -\mathstrut \) \(166\) \(\beta_{9}\mathstrut -\mathstrut \) \(71\) \(\beta_{8}\mathstrut -\mathstrut \) \(86\) \(\beta_{7}\mathstrut +\mathstrut \) \(12\) \(\beta_{6}\mathstrut +\mathstrut \) \(96\) \(\beta_{5}\mathstrut -\mathstrut \) \(69\) \(\beta_{4}\mathstrut -\mathstrut \) \(85\) \(\beta_{3}\mathstrut -\mathstrut \) \(82\) \(\beta_{2}\mathstrut +\mathstrut \) \(458\) \(\beta_{1}\mathstrut +\mathstrut \) \(58\)
\(\nu^{8}\)\(=\)\(108\) \(\beta_{12}\mathstrut -\mathstrut \) \(3\) \(\beta_{11}\mathstrut +\mathstrut \) \(20\) \(\beta_{10}\mathstrut -\mathstrut \) \(154\) \(\beta_{9}\mathstrut -\mathstrut \) \(128\) \(\beta_{8}\mathstrut -\mathstrut \) \(125\) \(\beta_{7}\mathstrut +\mathstrut \) \(63\) \(\beta_{6}\mathstrut +\mathstrut \) \(93\) \(\beta_{5}\mathstrut -\mathstrut \) \(44\) \(\beta_{4}\mathstrut +\mathstrut \) \(37\) \(\beta_{3}\mathstrut +\mathstrut \) \(370\) \(\beta_{2}\mathstrut +\mathstrut \) \(127\) \(\beta_{1}\mathstrut +\mathstrut \) \(829\)
\(\nu^{9}\)\(=\)\(634\) \(\beta_{12}\mathstrut +\mathstrut \) \(398\) \(\beta_{11}\mathstrut -\mathstrut \) \(703\) \(\beta_{10}\mathstrut -\mathstrut \) \(1310\) \(\beta_{9}\mathstrut -\mathstrut \) \(543\) \(\beta_{8}\mathstrut -\mathstrut \) \(699\) \(\beta_{7}\mathstrut +\mathstrut \) \(104\) \(\beta_{6}\mathstrut +\mathstrut \) \(782\) \(\beta_{5}\mathstrut -\mathstrut \) \(508\) \(\beta_{4}\mathstrut -\mathstrut \) \(688\) \(\beta_{3}\mathstrut -\mathstrut \) \(637\) \(\beta_{2}\mathstrut +\mathstrut \) \(3430\) \(\beta_{1}\mathstrut +\mathstrut \) \(432\)
\(\nu^{10}\)\(=\)\(883\) \(\beta_{12}\mathstrut -\mathstrut \) \(52\) \(\beta_{11}\mathstrut +\mathstrut \) \(258\) \(\beta_{10}\mathstrut -\mathstrut \) \(1386\) \(\beta_{9}\mathstrut -\mathstrut \) \(1148\) \(\beta_{8}\mathstrut -\mathstrut \) \(1080\) \(\beta_{7}\mathstrut +\mathstrut \) \(398\) \(\beta_{6}\mathstrut +\mathstrut \) \(723\) \(\beta_{5}\mathstrut -\mathstrut \) \(458\) \(\beta_{4}\mathstrut +\mathstrut \) \(451\) \(\beta_{3}\mathstrut +\mathstrut \) \(2794\) \(\beta_{2}\mathstrut +\mathstrut \) \(1118\) \(\beta_{1}\mathstrut +\mathstrut \) \(5848\)
\(\nu^{11}\)\(=\)\(4798\) \(\beta_{12}\mathstrut +\mathstrut \) \(2783\) \(\beta_{11}\mathstrut -\mathstrut \) \(5615\) \(\beta_{10}\mathstrut -\mathstrut \) \(10164\) \(\beta_{9}\mathstrut -\mathstrut \) \(4133\) \(\beta_{8}\mathstrut -\mathstrut \) \(5532\) \(\beta_{7}\mathstrut +\mathstrut \) \(797\) \(\beta_{6}\mathstrut +\mathstrut \) \(6204\) \(\beta_{5}\mathstrut -\mathstrut \) \(3695\) \(\beta_{4}\mathstrut -\mathstrut \) \(5463\) \(\beta_{3}\mathstrut -\mathstrut \) \(4862\) \(\beta_{2}\mathstrut +\mathstrut \) \(25735\) \(\beta_{1}\mathstrut +\mathstrut \) \(3311\)
\(\nu^{12}\)\(=\)\(6937\) \(\beta_{12}\mathstrut -\mathstrut \) \(610\) \(\beta_{11}\mathstrut +\mathstrut \) \(2764\) \(\beta_{10}\mathstrut -\mathstrut \) \(11772\) \(\beta_{9}\mathstrut -\mathstrut \) \(9867\) \(\beta_{8}\mathstrut -\mathstrut \) \(8889\) \(\beta_{7}\mathstrut +\mathstrut \) \(2325\) \(\beta_{6}\mathstrut +\mathstrut \) \(5455\) \(\beta_{5}\mathstrut -\mathstrut \) \(4188\) \(\beta_{4}\mathstrut +\mathstrut \) \(4621\) \(\beta_{3}\mathstrut +\mathstrut \) \(21320\) \(\beta_{2}\mathstrut +\mathstrut \) \(9370\) \(\beta_{1}\mathstrut +\mathstrut \) \(41881\)

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
2.79963
2.64979
2.12021
1.97615
1.74594
0.429184
0.291870
0.278392
−0.565329
−1.14309
−1.28532
−2.51561
−2.78181
0 −2.79963 0 −1.00000 0 2.90399 0 4.83794 0
1.2 0 −2.64979 0 −1.00000 0 1.08783 0 4.02136 0
1.3 0 −2.12021 0 −1.00000 0 −4.61476 0 1.49528 0
1.4 0 −1.97615 0 −1.00000 0 −2.05862 0 0.905164 0
1.5 0 −1.74594 0 −1.00000 0 3.52041 0 0.0483066 0
1.6 0 −0.429184 0 −1.00000 0 −1.03124 0 −2.81580 0
1.7 0 −0.291870 0 −1.00000 0 0.544006 0 −2.91481 0
1.8 0 −0.278392 0 −1.00000 0 −2.48850 0 −2.92250 0
1.9 0 0.565329 0 −1.00000 0 3.63919 0 −2.68040 0
1.10 0 1.14309 0 −1.00000 0 3.09474 0 −1.69334 0
1.11 0 1.28532 0 −1.00000 0 −3.04407 0 −1.34796 0
1.12 0 2.51561 0 −1.00000 0 0.120976 0 3.32830 0
1.13 0 2.78181 0 −1.00000 0 −1.67394 0 4.73847 0
\(n\): 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
\(2\) \(-1\)
\(5\) \(1\)
\(151\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6040))\):

\(T_{3}^{13} + \cdots\)
\(T_{7}^{13} - \cdots\)
\(T_{11}^{13} + \cdots\)