Properties

Label 8016.2.a.bg
Level 8016
Weight 2
Character orbit 8016.a
Self dual Yes
Analytic conductor 64.008
Analytic rank 0
Dimension 13
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8016.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.0080822603\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(- q^{3}\) \( + \beta_{1} q^{5} \) \( + \beta_{9} q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(- q^{3}\) \( + \beta_{1} q^{5} \) \( + \beta_{9} q^{7} \) \(+ q^{9}\) \( + ( -1 - \beta_{5} ) q^{11} \) \( + ( 1 + \beta_{3} ) q^{13} \) \( -\beta_{1} q^{15} \) \( + ( 1 - \beta_{2} ) q^{17} \) \( + ( -1 + \beta_{9} - \beta_{10} ) q^{19} \) \( -\beta_{9} q^{21} \) \( + ( -2 - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{23} \) \( + ( 3 - \beta_{5} + \beta_{6} ) q^{25} \) \(- q^{27}\) \( -\beta_{7} q^{29} \) \( + ( 1 - \beta_{4} - \beta_{10} ) q^{31} \) \( + ( 1 + \beta_{5} ) q^{33} \) \( + ( -\beta_{1} + \beta_{3} + \beta_{4} - \beta_{7} - \beta_{8} + \beta_{10} ) q^{35} \) \( + ( 1 - \beta_{5} + \beta_{7} - \beta_{10} - \beta_{11} ) q^{37} \) \( + ( -1 - \beta_{3} ) q^{39} \) \( + ( \beta_{1} - \beta_{4} + \beta_{10} - \beta_{12} ) q^{41} \) \( + ( \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} ) q^{43} \) \( + \beta_{1} q^{45} \) \( + ( 1 + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{11} + \beta_{12} ) q^{47} \) \( + ( \beta_{1} - \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{12} ) q^{49} \) \( + ( -1 + \beta_{2} ) q^{51} \) \( + ( -1 + \beta_{1} + \beta_{9} + \beta_{10} ) q^{53} \) \( + ( 2 \beta_{4} + \beta_{7} + 2 \beta_{9} ) q^{55} \) \( + ( 1 - \beta_{9} + \beta_{10} ) q^{57} \) \( + ( -2 + \beta_{1} - \beta_{4} + \beta_{8} - \beta_{9} ) q^{59} \) \( + ( 3 + \beta_{2} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} ) q^{61} \) \( + \beta_{9} q^{63} \) \( + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{65} \) \( + ( -2 + 2 \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} ) q^{67} \) \( + ( 2 + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} ) q^{69} \) \( + ( -1 + \beta_{1} + \beta_{5} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{71} \) \( + ( 4 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{7} + \beta_{9} + \beta_{12} ) q^{73} \) \( + ( -3 + \beta_{5} - \beta_{6} ) q^{75} \) \( + ( 2 + \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} ) q^{77} \) \( + ( -1 + \beta_{1} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{10} + \beta_{11} ) q^{79} \) \(+ q^{81}\) \( + ( -4 + 2 \beta_{1} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{9} - \beta_{11} ) q^{83} \) \( + ( 3 + 2 \beta_{4} + \beta_{5} - 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{11} ) q^{85} \) \( + \beta_{7} q^{87} \) \( + ( 3 + \beta_{2} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} ) q^{89} \) \( + ( -2 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{10} - \beta_{11} ) q^{91} \) \( + ( -1 + \beta_{4} + \beta_{10} ) q^{93} \) \( + ( 2 - 2 \beta_{1} + \beta_{4} - \beta_{7} - 2 \beta_{10} ) q^{95} \) \( + ( 3 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} - 2 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{97} \) \( + ( -1 - \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(13q \) \(\mathstrut -\mathstrut 13q^{3} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut 13q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(13q \) \(\mathstrut -\mathstrut 13q^{3} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut 13q^{9} \) \(\mathstrut -\mathstrut 11q^{11} \) \(\mathstrut +\mathstrut 12q^{13} \) \(\mathstrut -\mathstrut 2q^{15} \) \(\mathstrut +\mathstrut 15q^{17} \) \(\mathstrut -\mathstrut 14q^{19} \) \(\mathstrut +\mathstrut q^{21} \) \(\mathstrut -\mathstrut 9q^{23} \) \(\mathstrut +\mathstrut 37q^{25} \) \(\mathstrut -\mathstrut 13q^{27} \) \(\mathstrut -\mathstrut 3q^{29} \) \(\mathstrut +\mathstrut 17q^{31} \) \(\mathstrut +\mathstrut 11q^{33} \) \(\mathstrut -\mathstrut 15q^{35} \) \(\mathstrut +\mathstrut 16q^{37} \) \(\mathstrut -\mathstrut 12q^{39} \) \(\mathstrut +\mathstrut 12q^{41} \) \(\mathstrut -\mathstrut 20q^{43} \) \(\mathstrut +\mathstrut 2q^{45} \) \(\mathstrut +\mathstrut 6q^{47} \) \(\mathstrut +\mathstrut 26q^{49} \) \(\mathstrut -\mathstrut 15q^{51} \) \(\mathstrut -\mathstrut 12q^{53} \) \(\mathstrut -\mathstrut 7q^{55} \) \(\mathstrut +\mathstrut 14q^{57} \) \(\mathstrut -\mathstrut 14q^{59} \) \(\mathstrut +\mathstrut 24q^{61} \) \(\mathstrut -\mathstrut q^{63} \) \(\mathstrut +\mathstrut 8q^{65} \) \(\mathstrut -\mathstrut 3q^{67} \) \(\mathstrut +\mathstrut 9q^{69} \) \(\mathstrut -\mathstrut 17q^{71} \) \(\mathstrut +\mathstrut 34q^{73} \) \(\mathstrut -\mathstrut 37q^{75} \) \(\mathstrut +\mathstrut 30q^{77} \) \(\mathstrut -\mathstrut 10q^{79} \) \(\mathstrut +\mathstrut 13q^{81} \) \(\mathstrut -\mathstrut 44q^{83} \) \(\mathstrut +\mathstrut 25q^{85} \) \(\mathstrut +\mathstrut 3q^{87} \) \(\mathstrut +\mathstrut 25q^{89} \) \(\mathstrut -\mathstrut 29q^{91} \) \(\mathstrut -\mathstrut 17q^{93} \) \(\mathstrut +\mathstrut 15q^{95} \) \(\mathstrut +\mathstrut 38q^{97} \) \(\mathstrut -\mathstrut 11q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13}\mathstrut -\mathstrut \) \(2\) \(x^{12}\mathstrut -\mathstrut \) \(49\) \(x^{11}\mathstrut +\mathstrut \) \(99\) \(x^{10}\mathstrut +\mathstrut \) \(901\) \(x^{9}\mathstrut -\mathstrut \) \(1879\) \(x^{8}\mathstrut -\mathstrut \) \(7582\) \(x^{7}\mathstrut +\mathstrut \) \(16968\) \(x^{6}\mathstrut +\mathstrut \) \(26911\) \(x^{5}\mathstrut -\mathstrut \) \(72240\) \(x^{4}\mathstrut -\mathstrut \) \(14532\) \(x^{3}\mathstrut +\mathstrut \) \(112850\) \(x^{2}\mathstrut -\mathstrut \) \(72184\) \(x\mathstrut +\mathstrut \) \(12144\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(6207348777\) \(\nu^{12}\mathstrut -\mathstrut \) \(12640530894\) \(\nu^{11}\mathstrut +\mathstrut \) \(576861363065\) \(\nu^{10}\mathstrut +\mathstrut \) \(295838273673\) \(\nu^{9}\mathstrut -\mathstrut \) \(16627326024925\) \(\nu^{8}\mathstrut +\mathstrut \) \(5016057310831\) \(\nu^{7}\mathstrut +\mathstrut \) \(201417980697138\) \(\nu^{6}\mathstrut -\mathstrut \) \(196205928405368\) \(\nu^{5}\mathstrut -\mathstrut \) \(1037037612348531\) \(\nu^{4}\mathstrut +\mathstrut \) \(1724063123712248\) \(\nu^{3}\mathstrut +\mathstrut \) \(1650853947740980\) \(\nu^{2}\mathstrut -\mathstrut \) \(4373968597429142\) \(\nu\mathstrut +\mathstrut \) \(1211669325949820\)\()/\)\(111451400959900\)
\(\beta_{3}\)\(=\)\((\)\(126516435801\) \(\nu^{12}\mathstrut -\mathstrut \) \(351643713690\) \(\nu^{11}\mathstrut -\mathstrut \) \(4532098501229\) \(\nu^{10}\mathstrut +\mathstrut \) \(14015645726951\) \(\nu^{9}\mathstrut +\mathstrut \) \(48247617829913\) \(\nu^{8}\mathstrut -\mathstrut \) \(193085396549923\) \(\nu^{7}\mathstrut -\mathstrut \) \(90263735212158\) \(\nu^{6}\mathstrut +\mathstrut \) \(1012871433621172\) \(\nu^{5}\mathstrut -\mathstrut \) \(714691137288825\) \(\nu^{4}\mathstrut -\mathstrut \) \(993417294727940\) \(\nu^{3}\mathstrut +\mathstrut \) \(598916455112988\) \(\nu^{2}\mathstrut -\mathstrut \) \(3707736754746094\) \(\nu\mathstrut +\mathstrut \) \(3605634089719788\)\()/\)\(557257004799500\)
\(\beta_{4}\)\(=\)\((\)\(8155241826\) \(\nu^{12}\mathstrut -\mathstrut \) \(24439858116\) \(\nu^{11}\mathstrut -\mathstrut \) \(363301449706\) \(\nu^{10}\mathstrut +\mathstrut \) \(1102737562371\) \(\nu^{9}\mathstrut +\mathstrut \) \(5918945839452\) \(\nu^{8}\mathstrut -\mathstrut \) \(18714943276923\) \(\nu^{7}\mathstrut -\mathstrut \) \(42939671149010\) \(\nu^{6}\mathstrut +\mathstrut \) \(149114772131121\) \(\nu^{5}\mathstrut +\mathstrut \) \(129168526434186\) \(\nu^{4}\mathstrut -\mathstrut \) \(552823809736903\) \(\nu^{3}\mathstrut -\mathstrut \) \(61985566741568\) \(\nu^{2}\mathstrut +\mathstrut \) \(719430531278666\) \(\nu\mathstrut -\mathstrut \) \(261781210799648\)\()/\)\(27862850239975\)
\(\beta_{5}\)\(=\)\((\)\(42253210649\) \(\nu^{12}\mathstrut -\mathstrut \) \(42904580362\) \(\nu^{11}\mathstrut -\mathstrut \) \(2192557864275\) \(\nu^{10}\mathstrut +\mathstrut \) \(1857640717639\) \(\nu^{9}\mathstrut +\mathstrut \) \(43420560752365\) \(\nu^{8}\mathstrut -\mathstrut \) \(30214877144177\) \(\nu^{7}\mathstrut -\mathstrut \) \(406570644479846\) \(\nu^{6}\mathstrut +\mathstrut \) \(236986143576626\) \(\nu^{5}\mathstrut +\mathstrut \) \(1772258179746447\) \(\nu^{4}\mathstrut -\mathstrut \) \(967840084997036\) \(\nu^{3}\mathstrut -\mathstrut \) \(2780474603245950\) \(\nu^{2}\mathstrut +\mathstrut \) \(1829362384230014\) \(\nu\mathstrut -\mathstrut \) \(9603799562310\)\()/\)\(55725700479950\)
\(\beta_{6}\)\(=\)\((\)\(42253210649\) \(\nu^{12}\mathstrut -\mathstrut \) \(42904580362\) \(\nu^{11}\mathstrut -\mathstrut \) \(2192557864275\) \(\nu^{10}\mathstrut +\mathstrut \) \(1857640717639\) \(\nu^{9}\mathstrut +\mathstrut \) \(43420560752365\) \(\nu^{8}\mathstrut -\mathstrut \) \(30214877144177\) \(\nu^{7}\mathstrut -\mathstrut \) \(406570644479846\) \(\nu^{6}\mathstrut +\mathstrut \) \(236986143576626\) \(\nu^{5}\mathstrut +\mathstrut \) \(1772258179746447\) \(\nu^{4}\mathstrut -\mathstrut \) \(967840084997036\) \(\nu^{3}\mathstrut -\mathstrut \) \(2724748902766000\) \(\nu^{2}\mathstrut +\mathstrut \) \(1829362384230014\) \(\nu\mathstrut -\mathstrut \) \(455409403401910\)\()/\)\(55725700479950\)
\(\beta_{7}\)\(=\)\((\)\(138385219703\) \(\nu^{12}\mathstrut -\mathstrut \) \(105435570025\) \(\nu^{11}\mathstrut -\mathstrut \) \(6791649579897\) \(\nu^{10}\mathstrut +\mathstrut \) \(5771355044103\) \(\nu^{9}\mathstrut +\mathstrut \) \(127066606088584\) \(\nu^{8}\mathstrut -\mathstrut \) \(121899916781369\) \(\nu^{7}\mathstrut -\mathstrut \) \(1131795821929059\) \(\nu^{6}\mathstrut +\mathstrut \) \(1217996115089036\) \(\nu^{5}\mathstrut +\mathstrut \) \(4728856566083805\) \(\nu^{4}\mathstrut -\mathstrut \) \(5762733210647085\) \(\nu^{3}\mathstrut -\mathstrut \) \(6707972583182866\) \(\nu^{2}\mathstrut +\mathstrut \) \(10417867358098218\) \(\nu\mathstrut -\mathstrut \) \(2901404480931816\)\()/\)\(139314251199875\)
\(\beta_{8}\)\(=\)\((\)\(111023509273\) \(\nu^{12}\mathstrut -\mathstrut \) \(148746957474\) \(\nu^{11}\mathstrut -\mathstrut \) \(5533191476745\) \(\nu^{10}\mathstrut +\mathstrut \) \(7117312024823\) \(\nu^{9}\mathstrut +\mathstrut \) \(104000386651945\) \(\nu^{8}\mathstrut -\mathstrut \) \(129559753690019\) \(\nu^{7}\mathstrut -\mathstrut \) \(912294162144422\) \(\nu^{6}\mathstrut +\mathstrut \) \(1109495790415652\) \(\nu^{5}\mathstrut +\mathstrut \) \(3686070397711199\) \(\nu^{4}\mathstrut -\mathstrut \) \(4408101894281592\) \(\nu^{3}\mathstrut -\mathstrut \) \(5320985228785100\) \(\nu^{2}\mathstrut +\mathstrut \) \(6347787481933558\) \(\nu\mathstrut -\mathstrut \) \(629228580752360\)\()/\)\(111451400959900\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(323942240303\) \(\nu^{12}\mathstrut +\mathstrut \) \(655210507370\) \(\nu^{11}\mathstrut +\mathstrut \) \(16238231918487\) \(\nu^{10}\mathstrut -\mathstrut \) \(30174775561853\) \(\nu^{9}\mathstrut -\mathstrut \) \(309203328646339\) \(\nu^{8}\mathstrut +\mathstrut \) \(524566352898419\) \(\nu^{7}\mathstrut +\mathstrut \) \(2761108370208174\) \(\nu^{6}\mathstrut -\mathstrut \) \(4297098906328266\) \(\nu^{5}\mathstrut -\mathstrut \) \(11231871461142475\) \(\nu^{4}\mathstrut +\mathstrut \) \(16707098673252820\) \(\nu^{3}\mathstrut +\mathstrut \) \(14675109344372636\) \(\nu^{2}\mathstrut -\mathstrut \) \(25352491816897518\) \(\nu\mathstrut +\mathstrut \) \(6802024064231936\)\()/\)\(278628502399750\)
\(\beta_{10}\)\(=\)\((\)\(208422301596\) \(\nu^{12}\mathstrut +\mathstrut \) \(101272023955\) \(\nu^{11}\mathstrut -\mathstrut \) \(9810853937919\) \(\nu^{10}\mathstrut -\mathstrut \) \(3015289235129\) \(\nu^{9}\mathstrut +\mathstrut \) \(173349897433318\) \(\nu^{8}\mathstrut +\mathstrut \) \(10485608743617\) \(\nu^{7}\mathstrut -\mathstrut \) \(1435122721494978\) \(\nu^{6}\mathstrut +\mathstrut \) \(348943217873482\) \(\nu^{5}\mathstrut +\mathstrut \) \(5522030332256130\) \(\nu^{4}\mathstrut -\mathstrut \) \(3276596851987855\) \(\nu^{3}\mathstrut -\mathstrut \) \(7596810398586632\) \(\nu^{2}\mathstrut +\mathstrut \) \(7530986189108326\) \(\nu\mathstrut -\mathstrut \) \(1313465392184157\)\()/\)\(139314251199875\)
\(\beta_{11}\)\(=\)\((\)\(889692540773\) \(\nu^{12}\mathstrut -\mathstrut \) \(1710596413190\) \(\nu^{11}\mathstrut -\mathstrut \) \(43690158347557\) \(\nu^{10}\mathstrut +\mathstrut \) \(79678278406623\) \(\nu^{9}\mathstrut +\mathstrut \) \(807810238699029\) \(\nu^{8}\mathstrut -\mathstrut \) \(1397243777337779\) \(\nu^{7}\mathstrut -\mathstrut \) \(6930554296860074\) \(\nu^{6}\mathstrut +\mathstrut \) \(11473935236033836\) \(\nu^{5}\mathstrut +\mathstrut \) \(26745397139049795\) \(\nu^{4}\mathstrut -\mathstrut \) \(44471075457654780\) \(\nu^{3}\mathstrut -\mathstrut \) \(31856351450553896\) \(\nu^{2}\mathstrut +\mathstrut \) \(67210663867287138\) \(\nu\mathstrut -\mathstrut \) \(20950363454455596\)\()/\)\(557257004799500\)
\(\beta_{12}\)\(=\)\((\)\(224267911141\) \(\nu^{12}\mathstrut +\mathstrut \) \(63390661414\) \(\nu^{11}\mathstrut -\mathstrut \) \(11028950583521\) \(\nu^{10}\mathstrut -\mathstrut \) \(1496522255749\) \(\nu^{9}\mathstrut +\mathstrut \) \(205661421513157\) \(\nu^{8}\mathstrut -\mathstrut \) \(13254732421323\) \(\nu^{7}\mathstrut -\mathstrut \) \(1812887391077730\) \(\nu^{6}\mathstrut +\mathstrut \) \(558179494985656\) \(\nu^{5}\mathstrut +\mathstrut \) \(7475418034130291\) \(\nu^{4}\mathstrut -\mathstrut \) \(4406202874244528\) \(\nu^{3}\mathstrut -\mathstrut \) \(10980877487070868\) \(\nu^{2}\mathstrut +\mathstrut \) \(10312125439960266\) \(\nu\mathstrut -\mathstrut \) \(2219186123392628\)\()/\)\(111451400959900\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(8\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(5\) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(5\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(19\) \(\beta_{6}\mathstrut -\mathstrut \) \(17\) \(\beta_{5}\mathstrut -\mathstrut \) \(4\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(4\) \(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(91\)
\(\nu^{5}\)\(=\)\(-\)\(30\) \(\beta_{12}\mathstrut -\mathstrut \) \(18\) \(\beta_{11}\mathstrut +\mathstrut \) \(29\) \(\beta_{10}\mathstrut +\mathstrut \) \(7\) \(\beta_{9}\mathstrut +\mathstrut \) \(20\) \(\beta_{8}\mathstrut +\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(46\) \(\beta_{6}\mathstrut -\mathstrut \) \(16\) \(\beta_{5}\mathstrut +\mathstrut \) \(43\) \(\beta_{4}\mathstrut -\mathstrut \) \(21\) \(\beta_{3}\mathstrut -\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(143\) \(\beta_{1}\mathstrut -\mathstrut \) \(7\)
\(\nu^{6}\)\(=\)\(-\)\(137\) \(\beta_{12}\mathstrut -\mathstrut \) \(28\) \(\beta_{11}\mathstrut +\mathstrut \) \(144\) \(\beta_{10}\mathstrut -\mathstrut \) \(42\) \(\beta_{9}\mathstrut +\mathstrut \) \(41\) \(\beta_{8}\mathstrut -\mathstrut \) \(12\) \(\beta_{7}\mathstrut +\mathstrut \) \(335\) \(\beta_{6}\mathstrut -\mathstrut \) \(280\) \(\beta_{5}\mathstrut -\mathstrut \) \(100\) \(\beta_{4}\mathstrut +\mathstrut \) \(40\) \(\beta_{3}\mathstrut -\mathstrut \) \(102\) \(\beta_{2}\mathstrut +\mathstrut \) \(49\) \(\beta_{1}\mathstrut +\mathstrut \) \(1184\)
\(\nu^{7}\)\(=\)\(-\)\(665\) \(\beta_{12}\mathstrut -\mathstrut \) \(281\) \(\beta_{11}\mathstrut +\mathstrut \) \(632\) \(\beta_{10}\mathstrut +\mathstrut \) \(213\) \(\beta_{9}\mathstrut +\mathstrut \) \(350\) \(\beta_{8}\mathstrut +\mathstrut \) \(97\) \(\beta_{7}\mathstrut +\mathstrut \) \(905\) \(\beta_{6}\mathstrut -\mathstrut \) \(247\) \(\beta_{5}\mathstrut +\mathstrut \) \(753\) \(\beta_{4}\mathstrut -\mathstrut \) \(381\) \(\beta_{3}\mathstrut -\mathstrut \) \(199\) \(\beta_{2}\mathstrut +\mathstrut \) \(2061\) \(\beta_{1}\mathstrut +\mathstrut \) \(6\)
\(\nu^{8}\)\(=\)\(-\)\(2931\) \(\beta_{12}\mathstrut -\mathstrut \) \(676\) \(\beta_{11}\mathstrut +\mathstrut \) \(3142\) \(\beta_{10}\mathstrut -\mathstrut \) \(1097\) \(\beta_{9}\mathstrut +\mathstrut \) \(716\) \(\beta_{8}\mathstrut -\mathstrut \) \(320\) \(\beta_{7}\mathstrut +\mathstrut \) \(5857\) \(\beta_{6}\mathstrut -\mathstrut \) \(4654\) \(\beta_{5}\mathstrut -\mathstrut \) \(1956\) \(\beta_{4}\mathstrut +\mathstrut \) \(668\) \(\beta_{3}\mathstrut -\mathstrut \) \(2055\) \(\beta_{2}\mathstrut +\mathstrut \) \(1017\) \(\beta_{1}\mathstrut +\mathstrut \) \(16711\)
\(\nu^{9}\)\(=\)\(-\)\(13266\) \(\beta_{12}\mathstrut -\mathstrut \) \(4390\) \(\beta_{11}\mathstrut +\mathstrut \) \(12482\) \(\beta_{10}\mathstrut +\mathstrut \) \(4644\) \(\beta_{9}\mathstrut +\mathstrut \) \(6123\) \(\beta_{8}\mathstrut +\mathstrut \) \(2314\) \(\beta_{7}\mathstrut +\mathstrut \) \(16957\) \(\beta_{6}\mathstrut -\mathstrut \) \(4186\) \(\beta_{5}\mathstrut +\mathstrut \) \(12395\) \(\beta_{4}\mathstrut -\mathstrut \) \(6687\) \(\beta_{3}\mathstrut -\mathstrut \) \(4266\) \(\beta_{2}\mathstrut +\mathstrut \) \(31753\) \(\beta_{1}\mathstrut +\mathstrut \) \(1836\)
\(\nu^{10}\)\(=\)\(-\)\(57319\) \(\beta_{12}\mathstrut -\mathstrut \) \(14661\) \(\beta_{11}\mathstrut +\mathstrut \) \(61925\) \(\beta_{10}\mathstrut -\mathstrut \) \(23592\) \(\beta_{9}\mathstrut +\mathstrut \) \(12118\) \(\beta_{8}\mathstrut -\mathstrut \) \(6102\) \(\beta_{7}\mathstrut +\mathstrut \) \(102280\) \(\beta_{6}\mathstrut -\mathstrut \) \(78127\) \(\beta_{5}\mathstrut -\mathstrut \) \(34966\) \(\beta_{4}\mathstrut +\mathstrut \) \(10790\) \(\beta_{3}\mathstrut -\mathstrut \) \(38346\) \(\beta_{2}\mathstrut +\mathstrut \) \(20530\) \(\beta_{1}\mathstrut +\mathstrut \) \(249558\)
\(\nu^{11}\)\(=\)\(-\)\(251329\) \(\beta_{12}\mathstrut -\mathstrut \) \(70391\) \(\beta_{11}\mathstrut +\mathstrut \) \(235453\) \(\beta_{10}\mathstrut +\mathstrut \) \(89347\) \(\beta_{9}\mathstrut +\mathstrut \) \(107880\) \(\beta_{8}\mathstrut +\mathstrut \) \(48175\) \(\beta_{7}\mathstrut +\mathstrut \) \(310864\) \(\beta_{6}\mathstrut -\mathstrut \) \(76539\) \(\beta_{5}\mathstrut +\mathstrut \) \(199760\) \(\beta_{4}\mathstrut -\mathstrut \) \(116067\) \(\beta_{3}\mathstrut -\mathstrut \) \(83231\) \(\beta_{2}\mathstrut +\mathstrut \) \(510061\) \(\beta_{1}\mathstrut +\mathstrut \) \(60252\)
\(\nu^{12}\)\(=\)\(-\)\(1073038\) \(\beta_{12}\mathstrut -\mathstrut \) \(294937\) \(\beta_{11}\mathstrut +\mathstrut \) \(1162932\) \(\beta_{10}\mathstrut -\mathstrut \) \(459487\) \(\beta_{9}\mathstrut +\mathstrut \) \(205019\) \(\beta_{8}\mathstrut -\mathstrut \) \(103543\) \(\beta_{7}\mathstrut +\mathstrut \) \(1786032\) \(\beta_{6}\mathstrut -\mathstrut \) \(1320651\) \(\beta_{5}\mathstrut -\mathstrut \) \(597828\) \(\beta_{4}\mathstrut +\mathstrut \) \(173012\) \(\beta_{3}\mathstrut -\mathstrut \) \(691732\) \(\beta_{2}\mathstrut +\mathstrut \) \(410169\) \(\beta_{1}\mathstrut +\mathstrut \) \(3880740\)

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
−4.02595
−3.60065
−3.05600
−2.71099
−2.04519
0.284022
0.649069
1.25667
1.68837
2.79706
3.18209
3.38131
4.20017
0 −1.00000 0 −4.02595 0 −0.910115 0 1.00000 0
1.2 0 −1.00000 0 −3.60065 0 4.85776 0 1.00000 0
1.3 0 −1.00000 0 −3.05600 0 −2.59989 0 1.00000 0
1.4 0 −1.00000 0 −2.71099 0 0.775439 0 1.00000 0
1.5 0 −1.00000 0 −2.04519 0 −1.81275 0 1.00000 0
1.6 0 −1.00000 0 0.284022 0 3.90646 0 1.00000 0
1.7 0 −1.00000 0 0.649069 0 −4.19646 0 1.00000 0
1.8 0 −1.00000 0 1.25667 0 4.55521 0 1.00000 0
1.9 0 −1.00000 0 1.68837 0 −0.722089 0 1.00000 0
1.10 0 −1.00000 0 2.79706 0 −4.56518 0 1.00000 0
1.11 0 −1.00000 0 3.18209 0 1.39408 0 1.00000 0
1.12 0 −1.00000 0 3.38131 0 −2.18014 0 1.00000 0
1.13 0 −1.00000 0 4.20017 0 0.497670 0 1.00000 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
\(2\) \(1\)
\(3\) \(1\)
\(167\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\(T_{5}^{13} - \cdots\)
\(T_{7}^{13} + \cdots\)
\(T_{11}^{13} + \cdots\)
\(T_{13}^{13} - \cdots\)