Properties

Label 6042.2.a.bh
Level 6042
Weight 2
Character orbit 6042.a
Self dual Yes
Analytic conductor 48.246
Analytic rank 0
Dimension 13
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6042.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2456129013\)
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.

\(f(q)\) \(=\) \( q\) \(+ q^{2}\) \(+ q^{3}\) \(+ q^{4}\) \( + \beta_{1} q^{5} \) \(+ q^{6}\) \( + ( 1 - \beta_{2} ) q^{7} \) \(+ q^{8}\) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{2}\) \(+ q^{3}\) \(+ q^{4}\) \( + \beta_{1} q^{5} \) \(+ q^{6}\) \( + ( 1 - \beta_{2} ) q^{7} \) \(+ q^{8}\) \(+ q^{9}\) \( + \beta_{1} q^{10} \) \( -\beta_{5} q^{11} \) \(+ q^{12}\) \( + ( 1 + \beta_{10} ) q^{13} \) \( + ( 1 - \beta_{2} ) q^{14} \) \( + \beta_{1} q^{15} \) \(+ q^{16}\) \( + \beta_{12} q^{17} \) \(+ q^{18}\) \(+ q^{19}\) \( + \beta_{1} q^{20} \) \( + ( 1 - \beta_{2} ) q^{21} \) \( -\beta_{5} q^{22} \) \( + ( 1 + \beta_{6} ) q^{23} \) \(+ q^{24}\) \( + ( 2 - \beta_{1} - \beta_{6} + \beta_{8} + \beta_{11} - \beta_{12} ) q^{25} \) \( + ( 1 + \beta_{10} ) q^{26} \) \(+ q^{27}\) \( + ( 1 - \beta_{2} ) q^{28} \) \( + ( 1 + \beta_{3} + \beta_{5} ) q^{29} \) \( + \beta_{1} q^{30} \) \( -\beta_{8} q^{31} \) \(+ q^{32}\) \( -\beta_{5} q^{33} \) \( + \beta_{12} q^{34} \) \( + ( 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{11} ) q^{35} \) \(+ q^{36}\) \( + ( 2 - \beta_{1} + \beta_{2} - \beta_{6} + \beta_{7} - \beta_{10} ) q^{37} \) \(+ q^{38}\) \( + ( 1 + \beta_{10} ) q^{39} \) \( + \beta_{1} q^{40} \) \( + ( -\beta_{1} + \beta_{4} ) q^{41} \) \( + ( 1 - \beta_{2} ) q^{42} \) \( + ( 2 + \beta_{1} - \beta_{3} - \beta_{7} - \beta_{11} ) q^{43} \) \( -\beta_{5} q^{44} \) \( + \beta_{1} q^{45} \) \( + ( 1 + \beta_{6} ) q^{46} \) \( + ( 1 + \beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} - \beta_{12} ) q^{47} \) \(+ q^{48}\) \( + ( 4 - 2 \beta_{3} - \beta_{6} + \beta_{8} - 2 \beta_{10} ) q^{49} \) \( + ( 2 - \beta_{1} - \beta_{6} + \beta_{8} + \beta_{11} - \beta_{12} ) q^{50} \) \( + \beta_{12} q^{51} \) \( + ( 1 + \beta_{10} ) q^{52} \) \(- q^{53}\) \(+ q^{54}\) \( + ( 2 + \beta_{2} - \beta_{6} + \beta_{9} - \beta_{10} ) q^{55} \) \( + ( 1 - \beta_{2} ) q^{56} \) \(+ q^{57}\) \( + ( 1 + \beta_{3} + \beta_{5} ) q^{58} \) \( + ( 1 - \beta_{1} + \beta_{2} + \beta_{7} - \beta_{10} ) q^{59} \) \( + \beta_{1} q^{60} \) \( + ( 2 - \beta_{3} + \beta_{4} + \beta_{8} - \beta_{9} - \beta_{12} ) q^{61} \) \( -\beta_{8} q^{62} \) \( + ( 1 - \beta_{2} ) q^{63} \) \(+ q^{64}\) \( + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{12} ) q^{65} \) \( -\beta_{5} q^{66} \) \( + ( -1 - \beta_{2} - \beta_{9} + \beta_{10} ) q^{67} \) \( + \beta_{12} q^{68} \) \( + ( 1 + \beta_{6} ) q^{69} \) \( + ( 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{11} ) q^{70} \) \( + ( 2 + \beta_{2} + \beta_{3} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{71} \) \(+ q^{72}\) \( + ( 1 - \beta_{1} + 2 \beta_{3} + \beta_{5} + \beta_{7} - \beta_{8} + \beta_{11} ) q^{73} \) \( + ( 2 - \beta_{1} + \beta_{2} - \beta_{6} + \beta_{7} - \beta_{10} ) q^{74} \) \( + ( 2 - \beta_{1} - \beta_{6} + \beta_{8} + \beta_{11} - \beta_{12} ) q^{75} \) \(+ q^{76}\) \( + ( 1 + \beta_{2} - \beta_{4} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{77} \) \( + ( 1 + \beta_{10} ) q^{78} \) \( + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{8} + \beta_{9} - \beta_{10} ) q^{79} \) \( + \beta_{1} q^{80} \) \(+ q^{81}\) \( + ( -\beta_{1} + \beta_{4} ) q^{82} \) \( + ( 1 + 2 \beta_{2} + \beta_{5} - \beta_{8} - \beta_{10} - \beta_{11} - \beta_{12} ) q^{83} \) \( + ( 1 - \beta_{2} ) q^{84} \) \( + ( -\beta_{1} - \beta_{2} - \beta_{7} + \beta_{8} + 2 \beta_{10} - \beta_{12} ) q^{85} \) \( + ( 2 + \beta_{1} - \beta_{3} - \beta_{7} - \beta_{11} ) q^{86} \) \( + ( 1 + \beta_{3} + \beta_{5} ) q^{87} \) \( -\beta_{5} q^{88} \) \( + ( \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} ) q^{89} \) \( + \beta_{1} q^{90} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{91} \) \( + ( 1 + \beta_{6} ) q^{92} \) \( -\beta_{8} q^{93} \) \( + ( 1 + \beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} - \beta_{12} ) q^{94} \) \( + \beta_{1} q^{95} \) \(+ q^{96}\) \( + ( 4 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{7} + \beta_{11} ) q^{97} \) \( + ( 4 - 2 \beta_{3} - \beta_{6} + \beta_{8} - 2 \beta_{10} ) q^{98} \) \( -\beta_{5} q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(13q \) \(\mathstrut +\mathstrut 13q^{2} \) \(\mathstrut +\mathstrut 13q^{3} \) \(\mathstrut +\mathstrut 13q^{4} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 13q^{6} \) \(\mathstrut +\mathstrut 12q^{7} \) \(\mathstrut +\mathstrut 13q^{8} \) \(\mathstrut +\mathstrut 13q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(13q \) \(\mathstrut +\mathstrut 13q^{2} \) \(\mathstrut +\mathstrut 13q^{3} \) \(\mathstrut +\mathstrut 13q^{4} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 13q^{6} \) \(\mathstrut +\mathstrut 12q^{7} \) \(\mathstrut +\mathstrut 13q^{8} \) \(\mathstrut +\mathstrut 13q^{9} \) \(\mathstrut +\mathstrut 3q^{10} \) \(\mathstrut +\mathstrut 4q^{11} \) \(\mathstrut +\mathstrut 13q^{12} \) \(\mathstrut +\mathstrut 18q^{13} \) \(\mathstrut +\mathstrut 12q^{14} \) \(\mathstrut +\mathstrut 3q^{15} \) \(\mathstrut +\mathstrut 13q^{16} \) \(\mathstrut +\mathstrut 3q^{17} \) \(\mathstrut +\mathstrut 13q^{18} \) \(\mathstrut +\mathstrut 13q^{19} \) \(\mathstrut +\mathstrut 3q^{20} \) \(\mathstrut +\mathstrut 12q^{21} \) \(\mathstrut +\mathstrut 4q^{22} \) \(\mathstrut +\mathstrut 8q^{23} \) \(\mathstrut +\mathstrut 13q^{24} \) \(\mathstrut +\mathstrut 24q^{25} \) \(\mathstrut +\mathstrut 18q^{26} \) \(\mathstrut +\mathstrut 13q^{27} \) \(\mathstrut +\mathstrut 12q^{28} \) \(\mathstrut +\mathstrut 11q^{29} \) \(\mathstrut +\mathstrut 3q^{30} \) \(\mathstrut +\mathstrut 2q^{31} \) \(\mathstrut +\mathstrut 13q^{32} \) \(\mathstrut +\mathstrut 4q^{33} \) \(\mathstrut +\mathstrut 3q^{34} \) \(\mathstrut -\mathstrut 6q^{35} \) \(\mathstrut +\mathstrut 13q^{36} \) \(\mathstrut +\mathstrut 26q^{37} \) \(\mathstrut +\mathstrut 13q^{38} \) \(\mathstrut +\mathstrut 18q^{39} \) \(\mathstrut +\mathstrut 3q^{40} \) \(\mathstrut +\mathstrut 3q^{41} \) \(\mathstrut +\mathstrut 12q^{42} \) \(\mathstrut +\mathstrut 24q^{43} \) \(\mathstrut +\mathstrut 4q^{44} \) \(\mathstrut +\mathstrut 3q^{45} \) \(\mathstrut +\mathstrut 8q^{46} \) \(\mathstrut +\mathstrut 4q^{47} \) \(\mathstrut +\mathstrut 13q^{48} \) \(\mathstrut +\mathstrut 41q^{49} \) \(\mathstrut +\mathstrut 24q^{50} \) \(\mathstrut +\mathstrut 3q^{51} \) \(\mathstrut +\mathstrut 18q^{52} \) \(\mathstrut -\mathstrut 13q^{53} \) \(\mathstrut +\mathstrut 13q^{54} \) \(\mathstrut +\mathstrut 23q^{55} \) \(\mathstrut +\mathstrut 12q^{56} \) \(\mathstrut +\mathstrut 13q^{57} \) \(\mathstrut +\mathstrut 11q^{58} \) \(\mathstrut +\mathstrut 8q^{59} \) \(\mathstrut +\mathstrut 3q^{60} \) \(\mathstrut +\mathstrut 29q^{61} \) \(\mathstrut +\mathstrut 2q^{62} \) \(\mathstrut +\mathstrut 12q^{63} \) \(\mathstrut +\mathstrut 13q^{64} \) \(\mathstrut +\mathstrut 13q^{65} \) \(\mathstrut +\mathstrut 4q^{66} \) \(\mathstrut -\mathstrut 5q^{67} \) \(\mathstrut +\mathstrut 3q^{68} \) \(\mathstrut +\mathstrut 8q^{69} \) \(\mathstrut -\mathstrut 6q^{70} \) \(\mathstrut +\mathstrut 29q^{71} \) \(\mathstrut +\mathstrut 13q^{72} \) \(\mathstrut +\mathstrut 15q^{73} \) \(\mathstrut +\mathstrut 26q^{74} \) \(\mathstrut +\mathstrut 24q^{75} \) \(\mathstrut +\mathstrut 13q^{76} \) \(\mathstrut +\mathstrut 3q^{77} \) \(\mathstrut +\mathstrut 18q^{78} \) \(\mathstrut +\mathstrut 15q^{79} \) \(\mathstrut +\mathstrut 3q^{80} \) \(\mathstrut +\mathstrut 13q^{81} \) \(\mathstrut +\mathstrut 3q^{82} \) \(\mathstrut +\mathstrut 4q^{83} \) \(\mathstrut +\mathstrut 12q^{84} \) \(\mathstrut -\mathstrut q^{85} \) \(\mathstrut +\mathstrut 24q^{86} \) \(\mathstrut +\mathstrut 11q^{87} \) \(\mathstrut +\mathstrut 4q^{88} \) \(\mathstrut +\mathstrut 3q^{89} \) \(\mathstrut +\mathstrut 3q^{90} \) \(\mathstrut -\mathstrut 29q^{91} \) \(\mathstrut +\mathstrut 8q^{92} \) \(\mathstrut +\mathstrut 2q^{93} \) \(\mathstrut +\mathstrut 4q^{94} \) \(\mathstrut +\mathstrut 3q^{95} \) \(\mathstrut +\mathstrut 13q^{96} \) \(\mathstrut +\mathstrut 50q^{97} \) \(\mathstrut +\mathstrut 41q^{98} \) \(\mathstrut +\mathstrut 4q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13}\mathstrut -\mathstrut \) \(3\) \(x^{12}\mathstrut -\mathstrut \) \(40\) \(x^{11}\mathstrut +\mathstrut \) \(123\) \(x^{10}\mathstrut +\mathstrut \) \(537\) \(x^{9}\mathstrut -\mathstrut \) \(1707\) \(x^{8}\mathstrut -\mathstrut \) \(2914\) \(x^{7}\mathstrut +\mathstrut \) \(9639\) \(x^{6}\mathstrut +\mathstrut \) \(6938\) \(x^{5}\mathstrut -\mathstrut \) \(22200\) \(x^{4}\mathstrut -\mathstrut \) \(9466\) \(x^{3}\mathstrut +\mathstrut \) \(16812\) \(x^{2}\mathstrut +\mathstrut \) \(9304\) \(x\mathstrut +\mathstrut \) \(1200\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(1104465971\) \(\nu^{12}\mathstrut +\mathstrut \) \(6679376015\) \(\nu^{11}\mathstrut -\mathstrut \) \(55180929014\) \(\nu^{10}\mathstrut -\mathstrut \) \(276650415923\) \(\nu^{9}\mathstrut +\mathstrut \) \(1022243116203\) \(\nu^{8}\mathstrut +\mathstrut \) \(3980299416517\) \(\nu^{7}\mathstrut -\mathstrut \) \(8515787410070\) \(\nu^{6}\mathstrut -\mathstrut \) \(24476327772653\) \(\nu^{5}\mathstrut +\mathstrut \) \(30037232426544\) \(\nu^{4}\mathstrut +\mathstrut \) \(66746578762844\) \(\nu^{3}\mathstrut -\mathstrut \) \(30721425263802\) \(\nu^{2}\mathstrut -\mathstrut \) \(69505164946224\) \(\nu\mathstrut -\mathstrut \) \(15131253915640\)\()/\)\(887834031664\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(2632284209\) \(\nu^{12}\mathstrut +\mathstrut \) \(18882316439\) \(\nu^{11}\mathstrut +\mathstrut \) \(89540587722\) \(\nu^{10}\mathstrut -\mathstrut \) \(780424603231\) \(\nu^{9}\mathstrut -\mathstrut \) \(786204074125\) \(\nu^{8}\mathstrut +\mathstrut \) \(11062731147833\) \(\nu^{7}\mathstrut -\mathstrut \) \(443211211426\) \(\nu^{6}\mathstrut -\mathstrut \) \(65484111774973\) \(\nu^{5}\mathstrut +\mathstrut \) \(20272416631968\) \(\nu^{4}\mathstrut +\mathstrut \) \(165465205024724\) \(\nu^{3}\mathstrut -\mathstrut \) \(30222105954378\) \(\nu^{2}\mathstrut -\mathstrut \) \(150515869605488\) \(\nu\mathstrut -\mathstrut \) \(33713612784760\)\()/\)\(887834031664\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(2727750661\) \(\nu^{12}\mathstrut +\mathstrut \) \(14646261136\) \(\nu^{11}\mathstrut +\mathstrut \) \(102820445846\) \(\nu^{10}\mathstrut -\mathstrut \) \(605287792477\) \(\nu^{9}\mathstrut -\mathstrut \) \(1209915311616\) \(\nu^{8}\mathstrut +\mathstrut \) \(8558511877653\) \(\nu^{7}\mathstrut +\mathstrut \) \(4585058152073\) \(\nu^{6}\mathstrut -\mathstrut \) \(50218247672710\) \(\nu^{5}\mathstrut -\mathstrut \) \(2416172669298\) \(\nu^{4}\mathstrut +\mathstrut \) \(123875804125670\) \(\nu^{3}\mathstrut -\mathstrut \) \(60768123000\) \(\nu^{2}\mathstrut -\mathstrut \) \(106411914676176\) \(\nu\mathstrut -\mathstrut \) \(25057438289328\)\()/\)\(443917015832\)
\(\beta_{5}\)\(=\)\((\)\(3825967735\) \(\nu^{12}\mathstrut -\mathstrut \) \(5504342876\) \(\nu^{11}\mathstrut -\mathstrut \) \(156363041234\) \(\nu^{10}\mathstrut +\mathstrut \) \(219870360311\) \(\nu^{9}\mathstrut +\mathstrut \) \(2181905208796\) \(\nu^{8}\mathstrut -\mathstrut \) \(2859488827735\) \(\nu^{7}\mathstrut -\mathstrut \) \(12585440057303\) \(\nu^{6}\mathstrut +\mathstrut \) \(13755460366038\) \(\nu^{5}\mathstrut +\mathstrut \) \(30424461859070\) \(\nu^{4}\mathstrut -\mathstrut \) \(20469284564146\) \(\nu^{3}\mathstrut -\mathstrut \) \(25651497176616\) \(\nu^{2}\mathstrut -\mathstrut \) \(1427999257888\) \(\nu\mathstrut -\mathstrut \) \(955171318320\)\()/\)\(443917015832\)
\(\beta_{6}\)\(=\)\((\)\(4180312254\) \(\nu^{12}\mathstrut +\mathstrut \) \(7962412967\) \(\nu^{11}\mathstrut -\mathstrut \) \(186403032696\) \(\nu^{10}\mathstrut -\mathstrut \) \(333593208456\) \(\nu^{9}\mathstrut +\mathstrut \) \(2999393383979\) \(\nu^{8}\mathstrut +\mathstrut \) \(4949970794130\) \(\nu^{7}\mathstrut -\mathstrut \) \(21545221318909\) \(\nu^{6}\mathstrut -\mathstrut \) \(32224399272003\) \(\nu^{5}\mathstrut +\mathstrut \) \(68240550554438\) \(\nu^{4}\mathstrut +\mathstrut \) \(94219150878618\) \(\nu^{3}\mathstrut -\mathstrut \) \(69773601238538\) \(\nu^{2}\mathstrut -\mathstrut \) \(102911563743592\) \(\nu\mathstrut -\mathstrut \) \(20324471216384\)\()/\)\(443917015832\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(9076089647\) \(\nu^{12}\mathstrut +\mathstrut \) \(14744154019\) \(\nu^{11}\mathstrut +\mathstrut \) \(376726997738\) \(\nu^{10}\mathstrut -\mathstrut \) \(592303202345\) \(\nu^{9}\mathstrut -\mathstrut \) \(5407907514181\) \(\nu^{8}\mathstrut +\mathstrut \) \(7818704648743\) \(\nu^{7}\mathstrut +\mathstrut \) \(33013088468736\) \(\nu^{6}\mathstrut -\mathstrut \) \(39050414261665\) \(\nu^{5}\mathstrut -\mathstrut \) \(89720193617028\) \(\nu^{4}\mathstrut +\mathstrut \) \(64874222641024\) \(\nu^{3}\mathstrut +\mathstrut \) \(99166364858438\) \(\nu^{2}\mathstrut -\mathstrut \) \(8967190274096\) \(\nu\mathstrut -\mathstrut \) \(15375912065432\)\()/\)\(887834031664\)
\(\beta_{8}\)\(=\)\((\)\(10002084743\) \(\nu^{12}\mathstrut +\mathstrut \) \(15569453759\) \(\nu^{11}\mathstrut -\mathstrut \) \(450595476454\) \(\nu^{10}\mathstrut -\mathstrut \) \(664510529295\) \(\nu^{9}\mathstrut +\mathstrut \) \(7352334071027\) \(\nu^{8}\mathstrut +\mathstrut \) \(10188017922425\) \(\nu^{7}\mathstrut -\mathstrut \) \(53794158630402\) \(\nu^{6}\mathstrut -\mathstrut \) \(69934638097797\) \(\nu^{5}\mathstrut +\mathstrut \) \(174198902752088\) \(\nu^{4}\mathstrut +\mathstrut \) \(219914719422692\) \(\nu^{3}\mathstrut -\mathstrut \) \(182388318974730\) \(\nu^{2}\mathstrut -\mathstrut \) \(261526193845840\) \(\nu\mathstrut -\mathstrut \) \(57489181360808\)\()/\)\(887834031664\)
\(\beta_{9}\)\(=\)\((\)\(2761861453\) \(\nu^{12}\mathstrut -\mathstrut \) \(2173491833\) \(\nu^{11}\mathstrut -\mathstrut \) \(115007316091\) \(\nu^{10}\mathstrut +\mathstrut \) \(84482827509\) \(\nu^{9}\mathstrut +\mathstrut \) \(1657562870917\) \(\nu^{8}\mathstrut -\mathstrut \) \(1017129760174\) \(\nu^{7}\mathstrut -\mathstrut \) \(10090594369310\) \(\nu^{6}\mathstrut +\mathstrut \) \(3874256295364\) \(\nu^{5}\mathstrut +\mathstrut \) \(26156616453175\) \(\nu^{4}\mathstrut -\mathstrut \) \(909737621208\) \(\nu^{3}\mathstrut -\mathstrut \) \(23030561263496\) \(\nu^{2}\mathstrut -\mathstrut \) \(9936848006134\) \(\nu\mathstrut -\mathstrut \) \(1802645531584\)\()/\)\(221958507916\)
\(\beta_{10}\)\(=\)\((\)\(15738407933\) \(\nu^{12}\mathstrut -\mathstrut \) \(24588080919\) \(\nu^{11}\mathstrut -\mathstrut \) \(643851470174\) \(\nu^{10}\mathstrut +\mathstrut \) \(983188381227\) \(\nu^{9}\mathstrut +\mathstrut \) \(8996584023733\) \(\nu^{8}\mathstrut -\mathstrut \) \(12861301367465\) \(\nu^{7}\mathstrut -\mathstrut \) \(52033807512746\) \(\nu^{6}\mathstrut +\mathstrut \) \(63229291380089\) \(\nu^{5}\mathstrut +\mathstrut \) \(127116995436076\) \(\nu^{4}\mathstrut -\mathstrut \) \(104200446673436\) \(\nu^{3}\mathstrut -\mathstrut \) \(114796805478126\) \(\nu^{2}\mathstrut +\mathstrut \) \(23466620266904\) \(\nu\mathstrut +\mathstrut \) \(10900451890120\)\()/\)\(887834031664\)
\(\beta_{11}\)\(=\)\((\)\(4706616569\) \(\nu^{12}\mathstrut -\mathstrut \) \(8016452396\) \(\nu^{11}\mathstrut -\mathstrut \) \(190884176433\) \(\nu^{10}\mathstrut +\mathstrut \) \(324090176126\) \(\nu^{9}\mathstrut +\mathstrut \) \(2627378123161\) \(\nu^{8}\mathstrut -\mathstrut \) \(4339680041921\) \(\nu^{7}\mathstrut -\mathstrut \) \(14800691726179\) \(\nu^{6}\mathstrut +\mathstrut \) \(22580897844015\) \(\nu^{5}\mathstrut +\mathstrut \) \(34875401841134\) \(\nu^{4}\mathstrut -\mathstrut \) \(43741985100670\) \(\nu^{3}\mathstrut -\mathstrut \) \(31658973603658\) \(\nu^{2}\mathstrut +\mathstrut \) \(22658914077896\) \(\nu\mathstrut +\mathstrut \) \(7501450230024\)\()/\)\(221958507916\)
\(\beta_{12}\)\(=\)\((\)\(20467926511\) \(\nu^{12}\mathstrut -\mathstrut \) \(32421181759\) \(\nu^{11}\mathstrut -\mathstrut \) \(841326116794\) \(\nu^{10}\mathstrut +\mathstrut \) \(1299036592121\) \(\nu^{9}\mathstrut +\mathstrut \) \(11863059795713\) \(\nu^{8}\mathstrut -\mathstrut \) \(17070643833519\) \(\nu^{7}\mathstrut -\mathstrut \) \(69906482897300\) \(\nu^{6}\mathstrut +\mathstrut \) \(84837751822269\) \(\nu^{5}\mathstrut +\mathstrut \) \(177219409007748\) \(\nu^{4}\mathstrut -\mathstrut \) \(143491522737224\) \(\nu^{3}\mathstrut -\mathstrut \) \(170364844943950\) \(\nu^{2}\mathstrut +\mathstrut \) \(34044755921264\) \(\nu\mathstrut +\mathstrut \) \(19380400213704\)\()/\)\(887834031664\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(7\)
\(\nu^{3}\)\(=\)\(2\) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(2\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{2}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\mathstrut -\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(-\)\(18\) \(\beta_{12}\mathstrut +\mathstrut \) \(17\) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(3\) \(\beta_{9}\mathstrut +\mathstrut \) \(15\) \(\beta_{8}\mathstrut -\mathstrut \) \(3\) \(\beta_{7}\mathstrut -\mathstrut \) \(17\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(20\) \(\beta_{1}\mathstrut +\mathstrut \) \(89\)
\(\nu^{5}\)\(=\)\(44\) \(\beta_{12}\mathstrut +\mathstrut \) \(23\) \(\beta_{11}\mathstrut -\mathstrut \) \(39\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(24\) \(\beta_{8}\mathstrut +\mathstrut \) \(27\) \(\beta_{7}\mathstrut -\mathstrut \) \(20\) \(\beta_{6}\mathstrut -\mathstrut \) \(4\) \(\beta_{5}\mathstrut -\mathstrut \) \(5\) \(\beta_{4}\mathstrut +\mathstrut \) \(55\) \(\beta_{3}\mathstrut +\mathstrut \) \(83\) \(\beta_{2}\mathstrut +\mathstrut \) \(128\) \(\beta_{1}\mathstrut -\mathstrut \) \(57\)
\(\nu^{6}\)\(=\)\(-\)\(313\) \(\beta_{12}\mathstrut +\mathstrut \) \(279\) \(\beta_{11}\mathstrut -\mathstrut \) \(39\) \(\beta_{10}\mathstrut +\mathstrut \) \(91\) \(\beta_{9}\mathstrut +\mathstrut \) \(237\) \(\beta_{8}\mathstrut -\mathstrut \) \(78\) \(\beta_{7}\mathstrut -\mathstrut \) \(283\) \(\beta_{6}\mathstrut +\mathstrut \) \(21\) \(\beta_{5}\mathstrut -\mathstrut \) \(5\) \(\beta_{4}\mathstrut +\mathstrut \) \(52\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(351\) \(\beta_{1}\mathstrut +\mathstrut \) \(1380\)
\(\nu^{7}\)\(=\)\(830\) \(\beta_{12}\mathstrut +\mathstrut \) \(422\) \(\beta_{11}\mathstrut -\mathstrut \) \(650\) \(\beta_{10}\mathstrut -\mathstrut \) \(42\) \(\beta_{9}\mathstrut -\mathstrut \) \(512\) \(\beta_{8}\mathstrut +\mathstrut \) \(568\) \(\beta_{7}\mathstrut -\mathstrut \) \(323\) \(\beta_{6}\mathstrut -\mathstrut \) \(89\) \(\beta_{5}\mathstrut -\mathstrut \) \(178\) \(\beta_{4}\mathstrut +\mathstrut \) \(1220\) \(\beta_{3}\mathstrut +\mathstrut \) \(1502\) \(\beta_{2}\mathstrut +\mathstrut \) \(1891\) \(\beta_{1}\mathstrut -\mathstrut \) \(1159\)
\(\nu^{8}\)\(=\)\(-\)\(5483\) \(\beta_{12}\mathstrut +\mathstrut \) \(4669\) \(\beta_{11}\mathstrut -\mathstrut \) \(1013\) \(\beta_{10}\mathstrut +\mathstrut \) \(2055\) \(\beta_{9}\mathstrut +\mathstrut \) \(3938\) \(\beta_{8}\mathstrut -\mathstrut \) \(1660\) \(\beta_{7}\mathstrut -\mathstrut \) \(4780\) \(\beta_{6}\mathstrut +\mathstrut \) \(729\) \(\beta_{5}\mathstrut -\mathstrut \) \(148\) \(\beta_{4}\mathstrut +\mathstrut \) \(1051\) \(\beta_{3}\mathstrut -\mathstrut \) \(493\) \(\beta_{2}\mathstrut -\mathstrut \) \(6057\) \(\beta_{1}\mathstrut +\mathstrut \) \(23136\)
\(\nu^{9}\)\(=\)\(15171\) \(\beta_{12}\mathstrut +\mathstrut \) \(7406\) \(\beta_{11}\mathstrut -\mathstrut \) \(10670\) \(\beta_{10}\mathstrut -\mathstrut \) \(1190\) \(\beta_{9}\mathstrut -\mathstrut \) \(10321\) \(\beta_{8}\mathstrut +\mathstrut \) \(11105\) \(\beta_{7}\mathstrut -\mathstrut \) \(4992\) \(\beta_{6}\mathstrut -\mathstrut \) \(1516\) \(\beta_{5}\mathstrut -\mathstrut \) \(4464\) \(\beta_{4}\mathstrut +\mathstrut \) \(24829\) \(\beta_{3}\mathstrut +\mathstrut \) \(26700\) \(\beta_{2}\mathstrut +\mathstrut \) \(30208\) \(\beta_{1}\mathstrut -\mathstrut \) \(22032\)
\(\nu^{10}\)\(=\)\(-\)\(97176\) \(\beta_{12}\mathstrut +\mathstrut \) \(79790\) \(\beta_{11}\mathstrut -\mathstrut \) \(22368\) \(\beta_{10}\mathstrut +\mathstrut \) \(41914\) \(\beta_{9}\mathstrut +\mathstrut \) \(67591\) \(\beta_{8}\mathstrut -\mathstrut \) \(33120\) \(\beta_{7}\mathstrut -\mathstrut \) \(82132\) \(\beta_{6}\mathstrut +\mathstrut \) \(18057\) \(\beta_{5}\mathstrut -\mathstrut \) \(3145\) \(\beta_{4}\mathstrut +\mathstrut \) \(19465\) \(\beta_{3}\mathstrut -\mathstrut \) \(16520\) \(\beta_{2}\mathstrut -\mathstrut \) \(105479\) \(\beta_{1}\mathstrut +\mathstrut \) \(401275\)
\(\nu^{11}\)\(=\)\(276242\) \(\beta_{12}\mathstrut +\mathstrut \) \(128935\) \(\beta_{11}\mathstrut -\mathstrut \) \(177279\) \(\beta_{10}\mathstrut -\mathstrut \) \(28567\) \(\beta_{9}\mathstrut -\mathstrut \) \(201325\) \(\beta_{8}\mathstrut +\mathstrut \) \(210979\) \(\beta_{7}\mathstrut -\mathstrut \) \(76793\) \(\beta_{6}\mathstrut -\mathstrut \) \(24672\) \(\beta_{5}\mathstrut -\mathstrut \) \(97122\) \(\beta_{4}\mathstrut +\mathstrut \) \(483242\) \(\beta_{3}\mathstrut +\mathstrut \) \(476005\) \(\beta_{2}\mathstrut +\mathstrut \) \(505032\) \(\beta_{1}\mathstrut -\mathstrut \) \(414057\)
\(\nu^{12}\)\(=\)\(-\)\(1739338\) \(\beta_{12}\mathstrut +\mathstrut \) \(1385481\) \(\beta_{11}\mathstrut -\mathstrut \) \(454943\) \(\beta_{10}\mathstrut +\mathstrut \) \(816015\) \(\beta_{9}\mathstrut +\mathstrut \) \(1185360\) \(\beta_{8}\mathstrut -\mathstrut \) \(641477\) \(\beta_{7}\mathstrut -\mathstrut \) \(1431558\) \(\beta_{6}\mathstrut +\mathstrut \) \(390914\) \(\beta_{5}\mathstrut -\mathstrut \) \(58827\) \(\beta_{4}\mathstrut +\mathstrut \) \(344423\) \(\beta_{3}\mathstrut -\mathstrut \) \(409247\) \(\beta_{2}\mathstrut -\mathstrut \) \(1861262\) \(\beta_{1}\mathstrut +\mathstrut \) \(7082839\)

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
−4.29179
−3.54278
−2.34301
−1.64331
−0.942914
−0.312107
−0.229612
1.76686
2.08556
2.23588
2.48443
3.50660
4.22619
1.00000 1.00000 1.00000 −4.29179 1.00000 4.11570 1.00000 1.00000 −4.29179
1.2 1.00000 1.00000 1.00000 −3.54278 1.00000 −0.903867 1.00000 1.00000 −3.54278
1.3 1.00000 1.00000 1.00000 −2.34301 1.00000 4.92290 1.00000 1.00000 −2.34301
1.4 1.00000 1.00000 1.00000 −1.64331 1.00000 −4.90896 1.00000 1.00000 −1.64331
1.5 1.00000 1.00000 1.00000 −0.942914 1.00000 −0.429327 1.00000 1.00000 −0.942914
1.6 1.00000 1.00000 1.00000 −0.312107 1.00000 −1.12710 1.00000 1.00000 −0.312107
1.7 1.00000 1.00000 1.00000 −0.229612 1.00000 2.69176 1.00000 1.00000 −0.229612
1.8 1.00000 1.00000 1.00000 1.76686 1.00000 1.80214 1.00000 1.00000 1.76686
1.9 1.00000 1.00000 1.00000 2.08556 1.00000 1.83609 1.00000 1.00000 2.08556
1.10 1.00000 1.00000 1.00000 2.23588 1.00000 4.85906 1.00000 1.00000 2.23588
1.11 1.00000 1.00000 1.00000 2.48443 1.00000 3.12498 1.00000 1.00000 2.48443
1.12 1.00000 1.00000 1.00000 3.50660 1.00000 −4.14603 1.00000 1.00000 3.50660
1.13 1.00000 1.00000 1.00000 4.22619 1.00000 0.162672 1.00000 1.00000 4.22619
\(n\): 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
\(2\) \(-1\)
\(3\) \(-1\)
\(19\) \(-1\)
\(53\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

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