Properties

Label 6040.2.a.m
Level 6040
Weight 2
Character orbit 6040.a
Self dual Yes
Analytic conductor 48.230
Analytic rank 0
Dimension 12
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: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{3} \) \(- q^{5}\) \( + \beta_{9} q^{7} \) \( + ( 1 - \beta_{3} + \beta_{10} + \beta_{11} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{3} \) \(- q^{5}\) \( + \beta_{9} q^{7} \) \( + ( 1 - \beta_{3} + \beta_{10} + \beta_{11} ) q^{9} \) \( + ( 1 + \beta_{5} ) q^{11} \) \( + ( 1 + \beta_{10} ) q^{13} \) \( -\beta_{1} q^{15} \) \( + ( -\beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{10} ) q^{17} \) \( + ( 1 - \beta_{1} - \beta_{7} + \beta_{11} ) q^{19} \) \( + ( -\beta_{3} - \beta_{7} + \beta_{9} ) q^{21} \) \( + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{7} + \beta_{10} + \beta_{11} ) q^{23} \) \(+ q^{25}\) \( + ( 2 \beta_{1} - 2 \beta_{2} - \beta_{5} + \beta_{7} + \beta_{9} - \beta_{11} ) q^{27} \) \( + ( 1 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{11} ) q^{29} \) \( + ( \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{9} ) q^{31} \) \( + ( 2 \beta_{1} - \beta_{4} + \beta_{9} - \beta_{11} ) q^{33} \) \( -\beta_{9} q^{35} \) \( + ( -\beta_{2} + \beta_{4} - \beta_{6} - \beta_{10} + \beta_{11} ) q^{37} \) \( + ( 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{6} + \beta_{9} - \beta_{11} ) q^{39} \) \( + ( -\beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} + 2 \beta_{11} ) q^{41} \) \( + ( 2 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{43} \) \( + ( -1 + \beta_{3} - \beta_{10} - \beta_{11} ) q^{45} \) \( + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{47} \) \( + ( -1 + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{49} \) \( + ( -2 + \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{51} \) \( + ( 4 - \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{8} - \beta_{9} + 2 \beta_{11} ) q^{53} \) \( + ( -1 - \beta_{5} ) q^{55} \) \( + ( -3 + \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{57} \) \( + ( -2 \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{59} \) \( + ( -3 + \beta_{1} + \beta_{3} + \beta_{6} - \beta_{10} + \beta_{11} ) q^{61} \) \( + ( 1 + \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + 2 \beta_{10} ) q^{63} \) \( + ( -1 - \beta_{10} ) q^{65} \) \( + ( \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{67} \) \( + ( -1 + 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{69} \) \( + ( 1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{5} + 2 \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{71} \) \( + ( -1 + 2 \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{73} \) \( + \beta_{1} q^{75} \) \( + ( 2 + \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{9} ) q^{77} \) \( + ( 3 + \beta_{2} - \beta_{6} - 2 \beta_{9} ) q^{79} \) \( + ( 3 \beta_{1} - 4 \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + 3 \beta_{8} - 3 \beta_{10} - 2 \beta_{11} ) q^{81} \) \( + ( 2 \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{83} \) \( + ( \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{10} ) q^{85} \) \( + ( 2 + \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{87} \) \( + ( 1 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} - 3 \beta_{11} ) q^{89} \) \( + ( 1 + \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} + 3 \beta_{10} ) q^{91} \) \( + ( 2 + 4 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{6} + 3 \beta_{7} - \beta_{11} ) q^{93} \) \( + ( -1 + \beta_{1} + \beta_{7} - \beta_{11} ) q^{95} \) \( + ( 3 + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{97} \) \( + ( 4 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{7} + 2 \beta_{8} + \beta_{10} + 2 \beta_{11} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 12q^{5} \) \(\mathstrut +\mathstrut 5q^{7} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 12q^{5} \) \(\mathstrut +\mathstrut 5q^{7} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut +\mathstrut 10q^{11} \) \(\mathstrut +\mathstrut 11q^{13} \) \(\mathstrut -\mathstrut 3q^{15} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 5q^{19} \) \(\mathstrut -\mathstrut q^{21} \) \(\mathstrut +\mathstrut 18q^{23} \) \(\mathstrut +\mathstrut 12q^{25} \) \(\mathstrut +\mathstrut 9q^{27} \) \(\mathstrut +\mathstrut 16q^{29} \) \(\mathstrut -\mathstrut q^{31} \) \(\mathstrut +\mathstrut 8q^{33} \) \(\mathstrut -\mathstrut 5q^{35} \) \(\mathstrut +\mathstrut 2q^{37} \) \(\mathstrut +\mathstrut 6q^{39} \) \(\mathstrut +\mathstrut 4q^{41} \) \(\mathstrut +\mathstrut 7q^{43} \) \(\mathstrut -\mathstrut 9q^{45} \) \(\mathstrut +\mathstrut 3q^{49} \) \(\mathstrut -\mathstrut 4q^{51} \) \(\mathstrut +\mathstrut 39q^{53} \) \(\mathstrut -\mathstrut 10q^{55} \) \(\mathstrut -\mathstrut 15q^{57} \) \(\mathstrut -\mathstrut 4q^{59} \) \(\mathstrut -\mathstrut 32q^{61} \) \(\mathstrut +\mathstrut 3q^{63} \) \(\mathstrut -\mathstrut 11q^{65} \) \(\mathstrut +\mathstrut 4q^{67} \) \(\mathstrut +\mathstrut 12q^{69} \) \(\mathstrut +\mathstrut 24q^{71} \) \(\mathstrut -\mathstrut 10q^{73} \) \(\mathstrut +\mathstrut 3q^{75} \) \(\mathstrut +\mathstrut 38q^{77} \) \(\mathstrut +\mathstrut 32q^{79} \) \(\mathstrut -\mathstrut 8q^{81} \) \(\mathstrut +\mathstrut 9q^{83} \) \(\mathstrut +\mathstrut 4q^{85} \) \(\mathstrut +\mathstrut 3q^{87} \) \(\mathstrut +\mathstrut 15q^{89} \) \(\mathstrut +\mathstrut 18q^{91} \) \(\mathstrut +\mathstrut 36q^{93} \) \(\mathstrut -\mathstrut 5q^{95} \) \(\mathstrut +\mathstrut 15q^{97} \) \(\mathstrut +\mathstrut 28q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(3\) \(x^{11}\mathstrut -\mathstrut \) \(18\) \(x^{10}\mathstrut +\mathstrut \) \(54\) \(x^{9}\mathstrut +\mathstrut \) \(110\) \(x^{8}\mathstrut -\mathstrut \) \(335\) \(x^{7}\mathstrut -\mathstrut \) \(258\) \(x^{6}\mathstrut +\mathstrut \) \(825\) \(x^{5}\mathstrut +\mathstrut \) \(168\) \(x^{4}\mathstrut -\mathstrut \) \(669\) \(x^{3}\mathstrut +\mathstrut \) \(39\) \(x^{2}\mathstrut +\mathstrut \) \(106\) \(x\mathstrut -\mathstrut \) \(16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 2008 \nu^{11} + 1424 \nu^{10} + 2835 \nu^{9} - 76496 \nu^{8} - 561574 \nu^{7} + 560150 \nu^{6} + 4976515 \nu^{5} + 254206 \nu^{4} - 15096580 \nu^{3} - 6948313 \nu^{2} + 13947213 \nu + 4169518 \)\()/1691597\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(10958\) \(\nu^{11}\mathstrut +\mathstrut \) \(39405\) \(\nu^{10}\mathstrut +\mathstrut \) \(302967\) \(\nu^{9}\mathstrut -\mathstrut \) \(957392\) \(\nu^{8}\mathstrut -\mathstrut \) \(2950336\) \(\nu^{7}\mathstrut +\mathstrut \) \(7985722\) \(\nu^{6}\mathstrut +\mathstrut \) \(11939425\) \(\nu^{5}\mathstrut -\mathstrut \) \(26852183\) \(\nu^{4}\mathstrut -\mathstrut \) \(17581346\) \(\nu^{3}\mathstrut +\mathstrut \) \(30742326\) \(\nu^{2}\mathstrut +\mathstrut \) \(3916719\) \(\nu\mathstrut -\mathstrut \) \(4607858\)\()/1691597\)
\(\beta_{4}\)\(=\)\((\)\(87088\) \(\nu^{11}\mathstrut -\mathstrut \) \(120205\) \(\nu^{10}\mathstrut -\mathstrut \) \(1528205\) \(\nu^{9}\mathstrut +\mathstrut \) \(1743641\) \(\nu^{8}\mathstrut +\mathstrut \) \(8148517\) \(\nu^{7}\mathstrut -\mathstrut \) \(8028312\) \(\nu^{6}\mathstrut -\mathstrut \) \(9822311\) \(\nu^{5}\mathstrut +\mathstrut \) \(13188403\) \(\nu^{4}\mathstrut -\mathstrut \) \(20498710\) \(\nu^{3}\mathstrut -\mathstrut \) \(9965926\) \(\nu^{2}\mathstrut +\mathstrut \) \(19457022\) \(\nu\mathstrut +\mathstrut \) \(1976415\)\()/1691597\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(179015\) \(\nu^{11}\mathstrut +\mathstrut \) \(513295\) \(\nu^{10}\mathstrut +\mathstrut \) \(3434568\) \(\nu^{9}\mathstrut -\mathstrut \) \(9422340\) \(\nu^{8}\mathstrut -\mathstrut \) \(23531438\) \(\nu^{7}\mathstrut +\mathstrut \) \(60233277\) \(\nu^{6}\mathstrut +\mathstrut \) \(69392678\) \(\nu^{5}\mathstrut -\mathstrut \) \(156627403\) \(\nu^{4}\mathstrut -\mathstrut \) \(84391842\) \(\nu^{3}\mathstrut +\mathstrut \) \(142631113\) \(\nu^{2}\mathstrut +\mathstrut \) \(35453427\) \(\nu\mathstrut -\mathstrut \) \(25948112\)\()/3383194\)
\(\beta_{6}\)\(=\)\((\)\(141470\) \(\nu^{11}\mathstrut -\mathstrut \) \(108597\) \(\nu^{10}\mathstrut -\mathstrut \) \(2902088\) \(\nu^{9}\mathstrut +\mathstrut \) \(1424177\) \(\nu^{8}\mathstrut +\mathstrut \) \(20904860\) \(\nu^{7}\mathstrut -\mathstrut \) \(5571890\) \(\nu^{6}\mathstrut -\mathstrut \) \(62739819\) \(\nu^{5}\mathstrut +\mathstrut \) \(6954679\) \(\nu^{4}\mathstrut +\mathstrut \) \(70967431\) \(\nu^{3}\mathstrut -\mathstrut \) \(6820788\) \(\nu^{2}\mathstrut -\mathstrut \) \(23778569\) \(\nu\mathstrut +\mathstrut \) \(2912349\)\()/1691597\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(321409\) \(\nu^{11}\mathstrut +\mathstrut \) \(547103\) \(\nu^{10}\mathstrut +\mathstrut \) \(6257848\) \(\nu^{9}\mathstrut -\mathstrut \) \(9375818\) \(\nu^{8}\mathstrut -\mathstrut \) \(42338020\) \(\nu^{7}\mathstrut +\mathstrut \) \(55323693\) \(\nu^{6}\mathstrut +\mathstrut \) \(117883388\) \(\nu^{5}\mathstrut -\mathstrut \) \(127670063\) \(\nu^{4}\mathstrut -\mathstrut \) \(123267626\) \(\nu^{3}\mathstrut +\mathstrut \) \(92334701\) \(\nu^{2}\mathstrut +\mathstrut \) \(35618399\) \(\nu\mathstrut -\mathstrut \) \(10358630\)\()/3383194\)
\(\beta_{8}\)\(=\)\((\)\(334279\) \(\nu^{11}\mathstrut -\mathstrut \) \(834511\) \(\nu^{10}\mathstrut -\mathstrut \) \(6126792\) \(\nu^{9}\mathstrut +\mathstrut \) \(14627522\) \(\nu^{8}\mathstrut +\mathstrut \) \(38485960\) \(\nu^{7}\mathstrut -\mathstrut \) \(88399359\) \(\nu^{6}\mathstrut -\mathstrut \) \(96185542\) \(\nu^{5}\mathstrut +\mathstrut \) \(213363645\) \(\nu^{4}\mathstrut +\mathstrut \) \(81414310\) \(\nu^{3}\mathstrut -\mathstrut \) \(176404141\) \(\nu^{2}\mathstrut -\mathstrut \) \(12047985\) \(\nu\mathstrut +\mathstrut \) \(28264038\)\()/3383194\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(358363\) \(\nu^{11}\mathstrut +\mathstrut \) \(1060051\) \(\nu^{10}\mathstrut +\mathstrut \) \(6884672\) \(\nu^{9}\mathstrut -\mathstrut \) \(19330712\) \(\nu^{8}\mathstrut -\mathstrut \) \(47460040\) \(\nu^{7}\mathstrut +\mathstrut \) \(120830257\) \(\nu^{6}\mathstrut +\mathstrut \) \(141992774\) \(\nu^{5}\mathstrut -\mathstrut \) \(295418939\) \(\nu^{4}\mathstrut -\mathstrut \) \(174249494\) \(\nu^{3}\mathstrut +\mathstrut \) \(223925671\) \(\nu^{2}\mathstrut +\mathstrut \) \(56130161\) \(\nu\mathstrut -\mathstrut \) \(25308154\)\()/3383194\)
\(\beta_{10}\)\(=\)\((\)\(486873\) \(\nu^{11}\mathstrut -\mathstrut \) \(1009353\) \(\nu^{10}\mathstrut -\mathstrut \) \(9090678\) \(\nu^{9}\mathstrut +\mathstrut \) \(17063422\) \(\nu^{8}\mathstrut +\mathstrut \) \(58119654\) \(\nu^{7}\mathstrut -\mathstrut \) \(97708629\) \(\nu^{6}\mathstrut -\mathstrut \) \(146698574\) \(\nu^{5}\mathstrut +\mathstrut \) \(213774057\) \(\nu^{4}\mathstrut +\mathstrut \) \(120959460\) \(\nu^{3}\mathstrut -\mathstrut \) \(136554665\) \(\nu^{2}\mathstrut -\mathstrut \) \(19738395\) \(\nu\mathstrut +\mathstrut \) \(3648252\)\()/3383194\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(508789\) \(\nu^{11}\mathstrut +\mathstrut \) \(1088163\) \(\nu^{10}\mathstrut +\mathstrut \) \(9696612\) \(\nu^{9}\mathstrut -\mathstrut \) \(18978206\) \(\nu^{8}\mathstrut -\mathstrut \) \(64020326\) \(\nu^{7}\mathstrut +\mathstrut \) \(113680073\) \(\nu^{6}\mathstrut +\mathstrut \) \(170577424\) \(\nu^{5}\mathstrut -\mathstrut \) \(267478423\) \(\nu^{4}\mathstrut -\mathstrut \) \(156122152\) \(\nu^{3}\mathstrut +\mathstrut \) \(201422511\) \(\nu^{2}\mathstrut +\mathstrut \) \(27571833\) \(\nu\mathstrut -\mathstrut \) \(26396744\)\()/3383194\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(7\) \(\beta_{11}\mathstrut +\mathstrut \) \(6\) \(\beta_{10}\mathstrut +\mathstrut \) \(3\) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(9\) \(\beta_{3}\mathstrut -\mathstrut \) \(4\) \(\beta_{2}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\mathstrut +\mathstrut \) \(27\)
\(\nu^{5}\)\(=\)\(-\)\(13\) \(\beta_{11}\mathstrut +\mathstrut \) \(12\) \(\beta_{9}\mathstrut +\mathstrut \) \(4\) \(\beta_{8}\mathstrut +\mathstrut \) \(15\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(6\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(4\) \(\beta_{3}\mathstrut -\mathstrut \) \(25\) \(\beta_{2}\mathstrut +\mathstrut \) \(67\) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)
\(\nu^{6}\)\(=\)\(43\) \(\beta_{11}\mathstrut +\mathstrut \) \(36\) \(\beta_{10}\mathstrut +\mathstrut \) \(4\) \(\beta_{9}\mathstrut +\mathstrut \) \(42\) \(\beta_{8}\mathstrut +\mathstrut \) \(32\) \(\beta_{7}\mathstrut +\mathstrut \) \(13\) \(\beta_{6}\mathstrut +\mathstrut \) \(29\) \(\beta_{5}\mathstrut +\mathstrut \) \(12\) \(\beta_{4}\mathstrut -\mathstrut \) \(78\) \(\beta_{3}\mathstrut -\mathstrut \) \(64\) \(\beta_{2}\mathstrut +\mathstrut \) \(55\) \(\beta_{1}\mathstrut +\mathstrut \) \(205\)
\(\nu^{7}\)\(=\)\(-\)\(129\) \(\beta_{11}\mathstrut +\mathstrut \) \(3\) \(\beta_{10}\mathstrut +\mathstrut \) \(123\) \(\beta_{9}\mathstrut +\mathstrut \) \(74\) \(\beta_{8}\mathstrut +\mathstrut \) \(170\) \(\beta_{7}\mathstrut +\mathstrut \) \(18\) \(\beta_{6}\mathstrut -\mathstrut \) \(23\) \(\beta_{5}\mathstrut -\mathstrut \) \(16\) \(\beta_{4}\mathstrut -\mathstrut \) \(71\) \(\beta_{3}\mathstrut -\mathstrut \) \(272\) \(\beta_{2}\mathstrut +\mathstrut \) \(587\) \(\beta_{1}\mathstrut +\mathstrut \) \(101\)
\(\nu^{8}\)\(=\)\(243\) \(\beta_{11}\mathstrut +\mathstrut \) \(231\) \(\beta_{10}\mathstrut +\mathstrut \) \(93\) \(\beta_{9}\mathstrut +\mathstrut \) \(470\) \(\beta_{8}\mathstrut +\mathstrut \) \(390\) \(\beta_{7}\mathstrut +\mathstrut \) \(134\) \(\beta_{6}\mathstrut +\mathstrut \) \(315\) \(\beta_{5}\mathstrut +\mathstrut \) \(109\) \(\beta_{4}\mathstrut -\mathstrut \) \(700\) \(\beta_{3}\mathstrut -\mathstrut \) \(776\) \(\beta_{2}\mathstrut +\mathstrut \) \(740\) \(\beta_{1}\mathstrut +\mathstrut \) \(1674\)
\(\nu^{9}\)\(=\)\(-\)\(1186\) \(\beta_{11}\mathstrut +\mathstrut \) \(69\) \(\beta_{10}\mathstrut +\mathstrut \) \(1214\) \(\beta_{9}\mathstrut +\mathstrut \) \(990\) \(\beta_{8}\mathstrut +\mathstrut \) \(1773\) \(\beta_{7}\mathstrut +\mathstrut \) \(231\) \(\beta_{6}\mathstrut +\mathstrut \) \(38\) \(\beta_{5}\mathstrut -\mathstrut \) \(177\) \(\beta_{4}\mathstrut -\mathstrut \) \(943\) \(\beta_{3}\mathstrut -\mathstrut \) \(2848\) \(\beta_{2}\mathstrut +\mathstrut \) \(5336\) \(\beta_{1}\mathstrut +\mathstrut \) \(1463\)
\(\nu^{10}\)\(=\)\(1160\) \(\beta_{11}\mathstrut +\mathstrut \) \(1613\) \(\beta_{10}\mathstrut +\mathstrut \) \(1436\) \(\beta_{9}\mathstrut +\mathstrut \) \(4923\) \(\beta_{8}\mathstrut +\mathstrut \) \(4350\) \(\beta_{7}\mathstrut +\mathstrut \) \(1309\) \(\beta_{6}\mathstrut +\mathstrut \) \(3136\) \(\beta_{5}\mathstrut +\mathstrut \) \(891\) \(\beta_{4}\mathstrut -\mathstrut \) \(6489\) \(\beta_{3}\mathstrut -\mathstrut \) \(8573\) \(\beta_{2}\mathstrut +\mathstrut \) \(8865\) \(\beta_{1}\mathstrut +\mathstrut \) \(14457\)
\(\nu^{11}\)\(=\)\(-\)\(10689\) \(\beta_{11}\mathstrut +\mathstrut \) \(1056\) \(\beta_{10}\mathstrut +\mathstrut \) \(11872\) \(\beta_{9}\mathstrut +\mathstrut \) \(11702\) \(\beta_{8}\mathstrut +\mathstrut \) \(17976\) \(\beta_{7}\mathstrut +\mathstrut \) \(2653\) \(\beta_{6}\mathstrut +\mathstrut \) \(2299\) \(\beta_{5}\mathstrut -\mathstrut \) \(1700\) \(\beta_{4}\mathstrut -\mathstrut \) \(11239\) \(\beta_{3}\mathstrut -\mathstrut \) \(29407\) \(\beta_{2}\mathstrut +\mathstrut \) \(49964\) \(\beta_{1}\mathstrut +\mathstrut \) \(18469\)

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
−2.73116
−2.38247
−1.89604
−1.08777
−0.467276
0.168382
0.416740
0.983156
2.09968
2.24403
2.45327
3.19946
0 −2.73116 0 −1.00000 0 −0.736166 0 4.45925 0
1.2 0 −2.38247 0 −1.00000 0 3.11118 0 2.67614 0
1.3 0 −1.89604 0 −1.00000 0 −1.82001 0 0.594974 0
1.4 0 −1.08777 0 −1.00000 0 2.80815 0 −1.81675 0
1.5 0 −0.467276 0 −1.00000 0 −0.199223 0 −2.78165 0
1.6 0 0.168382 0 −1.00000 0 −3.12001 0 −2.97165 0
1.7 0 0.416740 0 −1.00000 0 5.24669 0 −2.82633 0
1.8 0 0.983156 0 −1.00000 0 −2.43489 0 −2.03340 0
1.9 0 2.09968 0 −1.00000 0 1.50711 0 1.40866 0
1.10 0 2.24403 0 −1.00000 0 3.06714 0 2.03565 0
1.11 0 2.45327 0 −1.00000 0 −3.17952 0 3.01854 0
1.12 0 3.19946 0 −1.00000 0 0.749556 0 7.23656 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6040))\):

\(T_{3}^{12} - \cdots\)
\(T_{7}^{12} - \cdots\)
\(T_{11}^{12} - \cdots\)