Properties

Label 4020.2.q.j
Level 4020
Weight 2
Character orbit 4020.q
Analytic conductor 32.100
Analytic rank 0
Dimension 12
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4020.q (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(32.0998616126\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\)  \(=\)  \( q\) \(+ q^{3}\) \(- q^{5}\) \( + ( \beta_{1} - \beta_{6} - \beta_{9} ) q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{3}\) \(- q^{5}\) \( + ( \beta_{1} - \beta_{6} - \beta_{9} ) q^{7} \) \(+ q^{9}\) \( + ( -1 + \beta_{5} + \beta_{10} ) q^{11} \) \( + \beta_{5} q^{13} \) \(- q^{15}\) \( + ( \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{8} + \beta_{10} ) q^{17} \) \( + ( - \beta_{1} + \beta_{3} - 2 \beta_{5} + \beta_{7} + \beta_{9} + \beta_{11} ) q^{19} \) \( + ( \beta_{1} - \beta_{6} - \beta_{9} ) q^{21} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{5} - \beta_{8} ) q^{23} \) \(+ q^{25}\) \(+ q^{27}\) \( + ( 1 + \beta_{1} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} ) q^{29} \) \( + ( -1 + \beta_{5} + \beta_{10} + \beta_{11} ) q^{31} \) \( + ( -1 + \beta_{5} + \beta_{10} ) q^{33} \) \( + ( - \beta_{1} + \beta_{6} + \beta_{9} ) q^{35} \) \( + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{5} + 2 \beta_{8} + \beta_{9} ) q^{37} \) \( + \beta_{5} q^{39} \) \( + ( -1 + \beta_{1} + \beta_{5} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{41} \) \( + ( -1 - \beta_{3} + \beta_{6} - \beta_{7} ) q^{43} \) \(- q^{45}\) \( + ( -2 + 2 \beta_{5} + \beta_{8} + 2 \beta_{10} + 2 \beta_{11} ) q^{47} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} - 8 \beta_{5} - 2 \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{11} ) q^{49} \) \( + ( \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{8} + \beta_{10} ) q^{51} \) \( + ( -2 + \beta_{4} - \beta_{7} ) q^{53} \) \( + ( 1 - \beta_{5} - \beta_{10} ) q^{55} \) \( + ( - \beta_{1} + \beta_{3} - 2 \beta_{5} + \beta_{7} + \beta_{9} + \beta_{11} ) q^{57} \) \( + ( -2 \beta_{2} - 2 \beta_{3} - \beta_{6} ) q^{59} \) \( + ( - \beta_{2} + \beta_{4} + 4 \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{61} \) \( + ( \beta_{1} - \beta_{6} - \beta_{9} ) q^{63} \) \( - \beta_{5} q^{65} \) \( + ( - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} ) q^{67} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{5} - \beta_{8} ) q^{69} \) \( + ( 4 + \beta_{1} - 4 \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} ) q^{71} \) \( + ( 2 \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} + 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{73} \) \(+ q^{75}\) \( + ( 4 \beta_{1} - \beta_{2} - 4 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{77} \) \( + ( \beta_{1} - 2 \beta_{10} - \beta_{11} ) q^{79} \) \(+ q^{81}\) \( + ( - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{7} + \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{11} ) q^{83} \) \( + ( - \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{8} - \beta_{10} ) q^{85} \) \( + ( 1 + \beta_{1} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} ) q^{87} \) \( + ( 1 - \beta_{2} - 2 \beta_{4} ) q^{89} \) \( + ( \beta_{3} - \beta_{6} ) q^{91} \) \( + ( -1 + \beta_{5} + \beta_{10} + \beta_{11} ) q^{93} \) \( + ( \beta_{1} - \beta_{3} + 2 \beta_{5} - \beta_{7} - \beta_{9} - \beta_{11} ) q^{95} \) \( + ( - \beta_{2} - 3 \beta_{4} - \beta_{8} + 3 \beta_{10} ) q^{97} \) \( + ( -1 + \beta_{5} + \beta_{10} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)  \(=\)  \(12q \) \(\mathstrut +\mathstrut 12q^{3} \) \(\mathstrut -\mathstrut 12q^{5} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 12q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut +\mathstrut 12q^{3} \) \(\mathstrut -\mathstrut 12q^{5} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 12q^{9} \) \(\mathstrut -\mathstrut 5q^{11} \) \(\mathstrut +\mathstrut 6q^{13} \) \(\mathstrut -\mathstrut 12q^{15} \) \(\mathstrut +\mathstrut 9q^{17} \) \(\mathstrut -\mathstrut 11q^{19} \) \(\mathstrut +\mathstrut 2q^{21} \) \(\mathstrut +\mathstrut 19q^{23} \) \(\mathstrut +\mathstrut 12q^{25} \) \(\mathstrut +\mathstrut 12q^{27} \) \(\mathstrut +\mathstrut 8q^{29} \) \(\mathstrut -\mathstrut 4q^{31} \) \(\mathstrut -\mathstrut 5q^{33} \) \(\mathstrut -\mathstrut 2q^{35} \) \(\mathstrut +\mathstrut 5q^{37} \) \(\mathstrut +\mathstrut 6q^{39} \) \(\mathstrut -\mathstrut 9q^{41} \) \(\mathstrut -\mathstrut 14q^{43} \) \(\mathstrut -\mathstrut 12q^{45} \) \(\mathstrut -\mathstrut 6q^{47} \) \(\mathstrut -\mathstrut 46q^{49} \) \(\mathstrut +\mathstrut 9q^{51} \) \(\mathstrut -\mathstrut 20q^{53} \) \(\mathstrut +\mathstrut 5q^{55} \) \(\mathstrut -\mathstrut 11q^{57} \) \(\mathstrut +\mathstrut 6q^{59} \) \(\mathstrut +\mathstrut 27q^{61} \) \(\mathstrut +\mathstrut 2q^{63} \) \(\mathstrut -\mathstrut 6q^{65} \) \(\mathstrut +\mathstrut 3q^{67} \) \(\mathstrut +\mathstrut 19q^{69} \) \(\mathstrut +\mathstrut 25q^{71} \) \(\mathstrut -\mathstrut 8q^{73} \) \(\mathstrut +\mathstrut 12q^{75} \) \(\mathstrut +\mathstrut 10q^{77} \) \(\mathstrut -\mathstrut 2q^{79} \) \(\mathstrut +\mathstrut 12q^{81} \) \(\mathstrut +\mathstrut 20q^{83} \) \(\mathstrut -\mathstrut 9q^{85} \) \(\mathstrut +\mathstrut 8q^{87} \) \(\mathstrut +\mathstrut 12q^{89} \) \(\mathstrut +\mathstrut 4q^{91} \) \(\mathstrut -\mathstrut 4q^{93} \) \(\mathstrut +\mathstrut 11q^{95} \) \(\mathstrut -\mathstrut q^{97} \) \(\mathstrut -\mathstrut 5q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(x^{11}\mathstrut +\mathstrut \) \(30\) \(x^{10}\mathstrut -\mathstrut \) \(53\) \(x^{9}\mathstrut +\mathstrut \) \(798\) \(x^{8}\mathstrut -\mathstrut \) \(1096\) \(x^{7}\mathstrut +\mathstrut \) \(4060\) \(x^{6}\mathstrut -\mathstrut \) \(915\) \(x^{5}\mathstrut +\mathstrut \) \(10392\) \(x^{4}\mathstrut -\mathstrut \) \(7038\) \(x^{3}\mathstrut +\mathstrut \) \(4869\) \(x^{2}\mathstrut -\mathstrut \) \(675\) \(x\mathstrut +\mathstrut \) \(81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(3882223435\) \(\nu^{11}\mathstrut +\mathstrut \) \(10214206017\) \(\nu^{10}\mathstrut +\mathstrut \) \(117178408376\) \(\nu^{9}\mathstrut +\mathstrut \) \(172229907007\) \(\nu^{8}\mathstrut +\mathstrut \) \(2805124258771\) \(\nu^{7}\mathstrut +\mathstrut \) \(5330805726143\) \(\nu^{6}\mathstrut +\mathstrut \) \(13095529592739\) \(\nu^{5}\mathstrut +\mathstrut \) \(14682367453768\) \(\nu^{4}\mathstrut +\mathstrut \) \(104356504552668\) \(\nu^{3}\mathstrut +\mathstrut \) \(10811165099478\) \(\nu^{2}\mathstrut -\mathstrut \) \(1517742210897\) \(\nu\mathstrut -\mathstrut \) \(213693056523513\)\()/\)\(41206785731685\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(52241027\) \(\nu^{11}\mathstrut -\mathstrut \) \(8330484\) \(\nu^{10}\mathstrut -\mathstrut \) \(1527067746\) \(\nu^{9}\mathstrut +\mathstrut \) \(1015516512\) \(\nu^{8}\mathstrut -\mathstrut \) \(39224741894\) \(\nu^{7}\mathstrut +\mathstrut \) \(11206364358\) \(\nu^{6}\mathstrut -\mathstrut \) \(166492823853\) \(\nu^{5}\mathstrut -\mathstrut \) \(141488390232\) \(\nu^{4}\mathstrut -\mathstrut \) \(685298055296\) \(\nu^{3}\mathstrut -\mathstrut \) \(115195094163\) \(\nu^{2}\mathstrut +\mathstrut \) \(16205642379\) \(\nu\mathstrut +\mathstrut \) \(52425010656\)\()/\)\(392445578397\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(18208180335\) \(\nu^{11}\mathstrut +\mathstrut \) \(1146781944\) \(\nu^{10}\mathstrut -\mathstrut \) \(532350662473\) \(\nu^{9}\mathstrut +\mathstrut \) \(458507766964\) \(\nu^{8}\mathstrut -\mathstrut \) \(13717814681058\) \(\nu^{7}\mathstrut +\mathstrut \) \(6580973669456\) \(\nu^{6}\mathstrut -\mathstrut \) \(57724581657177\) \(\nu^{5}\mathstrut -\mathstrut \) \(47555282937464\) \(\nu^{4}\mathstrut -\mathstrut \) \(185567304670144\) \(\nu^{3}\mathstrut -\mathstrut \) \(39200281496859\) \(\nu^{2}\mathstrut +\mathstrut \) \(5516033458671\) \(\nu\mathstrut -\mathstrut \) \(55721026347906\)\()/\)\(13735595243895\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(5825001184\) \(\nu^{11}\mathstrut +\mathstrut \) \(5354831941\) \(\nu^{10}\mathstrut -\mathstrut \) \(174825009876\) \(\nu^{9}\mathstrut +\mathstrut \) \(294981453038\) \(\nu^{8}\mathstrut -\mathstrut \) \(4639211296224\) \(\nu^{7}\mathstrut +\mathstrut \) \(6031178620618\) \(\nu^{6}\mathstrut -\mathstrut \) \(23548647527818\) \(\nu^{5}\mathstrut +\mathstrut \) \(3831440668683\) \(\nu^{4}\mathstrut -\mathstrut \) \(61806807816216\) \(\nu^{3}\mathstrut +\mathstrut \) \(34828675835328\) \(\nu^{2}\mathstrut -\mathstrut \) \(29398686612363\) \(\nu\mathstrut +\mathstrut \) \(4077726580611\)\()/\)\(3532010205573\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(17210840888\) \(\nu^{11}\mathstrut +\mathstrut \) \(1079200536\) \(\nu^{10}\mathstrut -\mathstrut \) \(502794995191\) \(\nu^{9}\mathstrut +\mathstrut \) \(433270280680\) \(\nu^{8}\mathstrut -\mathstrut \) \(12966377092280\) \(\nu^{7}\mathstrut +\mathstrut \) \(6193265819117\) \(\nu^{6}\mathstrut -\mathstrut \) \(54563120045688\) \(\nu^{5}\mathstrut -\mathstrut \) \(44952548682560\) \(\nu^{4}\mathstrut -\mathstrut \) \(181135313239308\) \(\nu^{3}\mathstrut -\mathstrut \) \(37054233616944\) \(\nu^{2}\mathstrut +\mathstrut \) \(5214052586664\) \(\nu\mathstrut -\mathstrut \) \(37541561931450\)\()/\)\(8241357146337\)
\(\beta_{7}\)\(=\)\((\)\(99006849920\) \(\nu^{11}\mathstrut -\mathstrut \) \(12877775961\) \(\nu^{10}\mathstrut +\mathstrut \) \(2892833484967\) \(\nu^{9}\mathstrut -\mathstrut \) \(2664598755331\) \(\nu^{8}\mathstrut +\mathstrut \) \(74666534698007\) \(\nu^{7}\mathstrut -\mathstrut \) \(40028044437344\) \(\nu^{6}\mathstrut +\mathstrut \) \(313376622706938\) \(\nu^{5}\mathstrut +\mathstrut \) \(255696353386256\) \(\nu^{4}\mathstrut +\mathstrut \) \(958346242038351\) \(\nu^{3}\mathstrut +\mathstrut \) \(211593294448401\) \(\nu^{2}\mathstrut -\mathstrut \) \(29776415210874\) \(\nu\mathstrut +\mathstrut \) \(136395667228734\)\()/\)\(41206785731685\)
\(\beta_{8}\)\(=\)\((\)\(340088114552\) \(\nu^{11}\mathstrut -\mathstrut \) \(310530618669\) \(\nu^{10}\mathstrut +\mathstrut \) \(10196741983762\) \(\nu^{9}\mathstrut -\mathstrut \) \(17126290059559\) \(\nu^{8}\mathstrut +\mathstrut \) \(270520940621483\) \(\nu^{7}\mathstrut -\mathstrut \) \(349406047914461\) \(\nu^{6}\mathstrut +\mathstrut \) \(1366034458141293\) \(\nu^{5}\mathstrut -\mathstrut \) \(191204974749472\) \(\nu^{4}\mathstrut +\mathstrut \) \(3583136534671908\) \(\nu^{3}\mathstrut -\mathstrut \) \(2018885081453712\) \(\nu^{2}\mathstrut +\mathstrut \) \(1769719192585947\) \(\nu\mathstrut -\mathstrut \) \(31631837094135\)\()/\)\(41206785731685\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(208917470212\) \(\nu^{11}\mathstrut +\mathstrut \) \(225445934212\) \(\nu^{10}\mathstrut -\mathstrut \) \(6274826024079\) \(\nu^{9}\mathstrut +\mathstrut \) \(11552472669332\) \(\nu^{8}\mathstrut -\mathstrut \) \(167293976034885\) \(\nu^{7}\mathstrut +\mathstrut \) \(241497537648025\) \(\nu^{6}\mathstrut -\mathstrut \) \(858137457348736\) \(\nu^{5}\mathstrut +\mathstrut \) \(243087280642482\) \(\nu^{4}\mathstrut -\mathstrut \) \(2130621634155120\) \(\nu^{3}\mathstrut +\mathstrut \) \(1617132043932120\) \(\nu^{2}\mathstrut -\mathstrut \) \(983103667365429\) \(\nu\mathstrut +\mathstrut \) \(136216640185254\)\()/\)\(24724071439011\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(1061664495556\) \(\nu^{11}\mathstrut +\mathstrut \) \(934960473697\) \(\nu^{10}\mathstrut -\mathstrut \) \(31728339975486\) \(\nu^{9}\mathstrut +\mathstrut \) \(52500424408727\) \(\nu^{8}\mathstrut -\mathstrut \) \(840619144671909\) \(\nu^{7}\mathstrut +\mathstrut \) \(1063455151268233\) \(\nu^{6}\mathstrut -\mathstrut \) \(4175717123058529\) \(\nu^{5}\mathstrut +\mathstrut \) \(478464801171081\) \(\nu^{4}\mathstrut -\mathstrut \) \(10930594128742824\) \(\nu^{3}\mathstrut +\mathstrut \) \(6156279135530136\) \(\nu^{2}\mathstrut -\mathstrut \) \(4264311775658751\) \(\nu\mathstrut +\mathstrut \) \(96443745926355\)\()/\)\(123620357195055\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(361920964801\) \(\nu^{11}\mathstrut +\mathstrut \) \(304890086297\) \(\nu^{10}\mathstrut -\mathstrut \) \(10768625400391\) \(\nu^{9}\mathstrut +\mathstrut \) \(17453392390817\) \(\nu^{8}\mathstrut -\mathstrut \) \(284857156713719\) \(\nu^{7}\mathstrut +\mathstrut \) \(349950039715648\) \(\nu^{6}\mathstrut -\mathstrut \) \(1382628324291494\) \(\nu^{5}\mathstrut +\mathstrut \) \(78133074301936\) \(\nu^{4}\mathstrut -\mathstrut \) \(3608703960512079\) \(\nu^{3}\mathstrut +\mathstrut \) \(2031316101387126\) \(\nu^{2}\mathstrut -\mathstrut \) \(1184548981579686\) \(\nu\mathstrut +\mathstrut \) \(31816447355970\)\()/\)\(41206785731685\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{11}\mathstrut +\mathstrut \) \(2\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(10\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{2}\)
\(\nu^{3}\)\(=\)\(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(5\) \(\beta_{6}\mathstrut -\mathstrut \) \(25\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(11\)
\(\nu^{4}\)\(=\)\(28\) \(\beta_{11}\mathstrut -\mathstrut \) \(57\) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(17\) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(220\) \(\beta_{5}\mathstrut +\mathstrut \) \(18\) \(\beta_{1}\mathstrut -\mathstrut \) \(220\)
\(\nu^{5}\)\(=\)\(-\)\(103\) \(\beta_{11}\mathstrut +\mathstrut \) \(22\) \(\beta_{10}\mathstrut +\mathstrut \) \(144\) \(\beta_{9}\mathstrut +\mathstrut \) \(8\) \(\beta_{8}\mathstrut -\mathstrut \) \(103\) \(\beta_{7}\mathstrut -\mathstrut \) \(446\) \(\beta_{5}\mathstrut -\mathstrut \) \(22\) \(\beta_{4}\mathstrut +\mathstrut \) \(623\) \(\beta_{3}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut -\mathstrut \) \(623\) \(\beta_{1}\)
\(\nu^{6}\)\(=\)\(748\) \(\beta_{7}\mathstrut +\mathstrut \) \(3\) \(\beta_{6}\mathstrut +\mathstrut \) \(1463\) \(\beta_{4}\mathstrut -\mathstrut \) \(849\) \(\beta_{3}\mathstrut +\mathstrut \) \(376\) \(\beta_{2}\mathstrut +\mathstrut \) \(5536\)
\(\nu^{7}\)\(=\)\(3060\) \(\beta_{11}\mathstrut -\mathstrut \) \(1362\) \(\beta_{10}\mathstrut -\mathstrut \) \(3671\) \(\beta_{9}\mathstrut +\mathstrut \) \(98\) \(\beta_{8}\mathstrut -\mathstrut \) \(3671\) \(\beta_{6}\mathstrut +\mathstrut \) \(14744\) \(\beta_{5}\mathstrut +\mathstrut \) \(15847\) \(\beta_{1}\mathstrut -\mathstrut \) \(14744\)
\(\nu^{8}\)\(=\)\(-\)\(20269\) \(\beta_{11}\mathstrut +\mathstrut \) \(37161\) \(\beta_{10}\mathstrut +\mathstrut \) \(2113\) \(\beta_{9}\mathstrut -\mathstrut \) \(9116\) \(\beta_{8}\mathstrut -\mathstrut \) \(20269\) \(\beta_{7}\mathstrut -\mathstrut \) \(144691\) \(\beta_{5}\mathstrut -\mathstrut \) \(37161\) \(\beta_{4}\mathstrut +\mathstrut \) \(30930\) \(\beta_{3}\mathstrut -\mathstrut \) \(9116\) \(\beta_{2}\mathstrut -\mathstrut \) \(30930\) \(\beta_{1}\)
\(\nu^{9}\)\(=\)\(88360\) \(\beta_{7}\mathstrut +\mathstrut \) \(92478\) \(\beta_{6}\mathstrut +\mathstrut \) \(56197\) \(\beta_{4}\mathstrut -\mathstrut \) \(409535\) \(\beta_{3}\mathstrut +\mathstrut \) \(8548\) \(\beta_{2}\mathstrut +\mathstrut \) \(455798\)
\(\nu^{10}\)\(=\)\(554092\) \(\beta_{11}\mathstrut -\mathstrut \) \(952259\) \(\beta_{10}\mathstrut -\mathstrut \) \(108276\) \(\beta_{9}\mathstrut +\mathstrut \) \(228697\) \(\beta_{8}\mathstrut -\mathstrut \) \(108276\) \(\beta_{6}\mathstrut +\mathstrut \) \(3844291\) \(\beta_{5}\mathstrut +\mathstrut \) \(1018959\) \(\beta_{1}\mathstrut -\mathstrut \) \(3844291\)
\(\nu^{11}\)\(=\)\(-\)\(2525310\) \(\beta_{11}\mathstrut +\mathstrut \) \(1976724\) \(\beta_{10}\mathstrut +\mathstrut \) \(2350334\) \(\beta_{9}\mathstrut -\mathstrut \) \(356591\) \(\beta_{8}\mathstrut -\mathstrut \) \(2525310\) \(\beta_{7}\mathstrut -\mathstrut \) \(13648472\) \(\beta_{5}\mathstrut -\mathstrut \) \(1976724\) \(\beta_{4}\mathstrut +\mathstrut \) \(10725256\) \(\beta_{3}\mathstrut -\mathstrut \) \(356591\) \(\beta_{2}\mathstrut -\mathstrut \) \(10725256\) \(\beta_{1}\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times\).

\(n\) \(1141\) \(2011\) \(2681\) \(3217\)
\(\chi(n)\) \(-1 + \beta_{5}\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
841.1
−2.64313 + 4.57803i
1.25129 2.16730i
0.307496 0.532598i
−0.818190 + 1.41715i
0.0725333 0.125631i
2.33000 4.03568i
−2.64313 4.57803i
1.25129 + 2.16730i
0.307496 + 0.532598i
−0.818190 1.41715i
0.0725333 + 0.125631i
2.33000 + 4.03568i
0 1.00000 0 −1.00000 0 −2.33998 + 4.05297i 0 1.00000 0
841.2 0 1.00000 0 −1.00000 0 −1.97947 + 3.42854i 0 1.00000 0
841.3 0 1.00000 0 −1.00000 0 −0.550200 + 0.952975i 0 1.00000 0
841.4 0 1.00000 0 −1.00000 0 1.22030 2.11362i 0 1.00000 0
841.5 0 1.00000 0 −1.00000 0 2.26952 3.93092i 0 1.00000 0
841.6 0 1.00000 0 −1.00000 0 2.37984 4.12200i 0 1.00000 0
3781.1 0 1.00000 0 −1.00000 0 −2.33998 4.05297i 0 1.00000 0
3781.2 0 1.00000 0 −1.00000 0 −1.97947 3.42854i 0 1.00000 0
3781.3 0 1.00000 0 −1.00000 0 −0.550200 0.952975i 0 1.00000 0
3781.4 0 1.00000 0 −1.00000 0 1.22030 + 2.11362i 0 1.00000 0
3781.5 0 1.00000 0 −1.00000 0 2.26952 + 3.93092i 0 1.00000 0
3781.6 0 1.00000 0 −1.00000 0 2.37984 + 4.12200i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3781.6
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
67.c Even 1 no

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4020, \chi)\):

\(T_{7}^{12} - \cdots\)
\(T_{11}^{12} + \cdots\)
\(T_{17}^{12} - \cdots\)