Properties

Label 4020.2.q.k
Level 4020
Weight 2
Character orbit 4020.q
Analytic conductor 32.100
Analytic rank 0
Dimension 14
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: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(- q^{3}\) \(+ q^{5}\) \( -\beta_{6} q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(- q^{3}\) \(+ q^{5}\) \( -\beta_{6} q^{7} \) \(+ q^{9}\) \( + ( 1 - \beta_{5} - \beta_{10} - \beta_{12} ) q^{11} \) \( + ( -\beta_{1} - \beta_{3} + \beta_{5} ) q^{13} \) \(- q^{15}\) \( + ( -\beta_{2} - 2 \beta_{5} - \beta_{11} + \beta_{13} ) q^{17} \) \( + ( -\beta_{2} + 2 \beta_{5} - \beta_{11} ) q^{19} \) \( + \beta_{6} q^{21} \) \( + ( -\beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} - \beta_{13} ) q^{23} \) \(+ q^{25}\) \(- q^{27}\) \( + ( -1 - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{29} \) \( + ( 1 + \beta_{1} - \beta_{5} - \beta_{6} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{31} \) \( + ( -1 + \beta_{5} + \beta_{10} + \beta_{12} ) q^{33} \) \( -\beta_{6} q^{35} \) \( + ( -2 \beta_{2} + \beta_{9} - 2 \beta_{11} - \beta_{12} ) q^{37} \) \( + ( \beta_{1} + \beta_{3} - \beta_{5} ) q^{39} \) \( + ( -2 - \beta_{4} + 2 \beta_{5} + \beta_{12} + \beta_{13} ) q^{41} \) \( + ( 1 - \beta_{2} - 2 \beta_{8} + 2 \beta_{9} ) q^{43} \) \(+ q^{45}\) \( + ( 2 - 2 \beta_{1} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{47} \) \( + ( -\beta_{2} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{49} \) \( + ( \beta_{2} + 2 \beta_{5} + \beta_{11} - \beta_{13} ) q^{51} \) \( + ( \beta_{2} + \beta_{3} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{53} \) \( + ( 1 - \beta_{5} - \beta_{10} - \beta_{12} ) q^{55} \) \( + ( \beta_{2} - 2 \beta_{5} + \beta_{11} ) q^{57} \) \( + ( -1 - \beta_{3} + 2 \beta_{8} + \beta_{9} ) q^{59} \) \( + ( \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{12} + \beta_{13} ) q^{61} \) \( -\beta_{6} q^{63} \) \( + ( -\beta_{1} - \beta_{3} + \beta_{5} ) q^{65} \) \( + ( 3 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{12} ) q^{67} \) \( + ( \beta_{2} - 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{69} \) \( + ( 4 - \beta_{1} + 2 \beta_{4} - 4 \beta_{5} + 3 \beta_{6} + \beta_{10} + 2 \beta_{11} + \beta_{12} - 2 \beta_{13} ) q^{71} \) \( + ( -\beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{73} \) \(- q^{75}\) \( + ( \beta_{1} + \beta_{3} + 2 \beta_{5} - 3 \beta_{6} - 3 \beta_{8} + 2 \beta_{9} - 2 \beta_{12} + \beta_{13} ) q^{77} \) \( + ( 1 + \beta_{1} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{11} + \beta_{12} + 2 \beta_{13} ) q^{79} \) \(+ q^{81}\) \( + ( -2 \beta_{1} - 2 \beta_{3} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{12} - \beta_{13} ) q^{83} \) \( + ( -\beta_{2} - 2 \beta_{5} - \beta_{11} + \beta_{13} ) q^{85} \) \( + ( 1 + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{87} \) \( + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} + \beta_{9} ) q^{89} \) \( + ( -4 - \beta_{3} + 2 \beta_{4} + \beta_{8} ) q^{91} \) \( + ( -1 - \beta_{1} + \beta_{5} + \beta_{6} + \beta_{10} + \beta_{11} - \beta_{12} ) q^{93} \) \( + ( -\beta_{2} + 2 \beta_{5} - \beta_{11} ) q^{95} \) \( + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{10} + \beta_{11} - \beta_{13} ) q^{97} \) \( + ( 1 - \beta_{5} - \beta_{10} - \beta_{12} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(14q \) \(\mathstrut -\mathstrut 14q^{3} \) \(\mathstrut +\mathstrut 14q^{5} \) \(\mathstrut +\mathstrut 3q^{7} \) \(\mathstrut +\mathstrut 14q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(14q \) \(\mathstrut -\mathstrut 14q^{3} \) \(\mathstrut +\mathstrut 14q^{5} \) \(\mathstrut +\mathstrut 3q^{7} \) \(\mathstrut +\mathstrut 14q^{9} \) \(\mathstrut +\mathstrut 6q^{11} \) \(\mathstrut +\mathstrut 9q^{13} \) \(\mathstrut -\mathstrut 14q^{15} \) \(\mathstrut -\mathstrut 12q^{17} \) \(\mathstrut +\mathstrut 14q^{19} \) \(\mathstrut -\mathstrut 3q^{21} \) \(\mathstrut +\mathstrut 6q^{23} \) \(\mathstrut +\mathstrut 14q^{25} \) \(\mathstrut -\mathstrut 14q^{27} \) \(\mathstrut -\mathstrut q^{29} \) \(\mathstrut +\mathstrut 7q^{31} \) \(\mathstrut -\mathstrut 6q^{33} \) \(\mathstrut +\mathstrut 3q^{35} \) \(\mathstrut -\mathstrut 2q^{37} \) \(\mathstrut -\mathstrut 9q^{39} \) \(\mathstrut -\mathstrut 18q^{41} \) \(\mathstrut -\mathstrut 6q^{43} \) \(\mathstrut +\mathstrut 14q^{45} \) \(\mathstrut +\mathstrut 7q^{47} \) \(\mathstrut +\mathstrut 12q^{51} \) \(\mathstrut -\mathstrut 12q^{53} \) \(\mathstrut +\mathstrut 6q^{55} \) \(\mathstrut -\mathstrut 14q^{57} \) \(\mathstrut -\mathstrut 2q^{59} \) \(\mathstrut +\mathstrut 3q^{63} \) \(\mathstrut +\mathstrut 9q^{65} \) \(\mathstrut +\mathstrut 25q^{67} \) \(\mathstrut -\mathstrut 6q^{69} \) \(\mathstrut +\mathstrut 22q^{71} \) \(\mathstrut -\mathstrut 15q^{73} \) \(\mathstrut -\mathstrut 14q^{75} \) \(\mathstrut +\mathstrut q^{77} \) \(\mathstrut +\mathstrut 9q^{79} \) \(\mathstrut +\mathstrut 14q^{81} \) \(\mathstrut -\mathstrut q^{83} \) \(\mathstrut -\mathstrut 12q^{85} \) \(\mathstrut +\mathstrut q^{87} \) \(\mathstrut -\mathstrut 12q^{89} \) \(\mathstrut -\mathstrut 38q^{91} \) \(\mathstrut -\mathstrut 7q^{93} \) \(\mathstrut +\mathstrut 14q^{95} \) \(\mathstrut +\mathstrut 16q^{97} \) \(\mathstrut +\mathstrut 6q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14}\mathstrut -\mathstrut \) \(x^{13}\mathstrut +\mathstrut \) \(11\) \(x^{12}\mathstrut -\mathstrut \) \(8\) \(x^{11}\mathstrut +\mathstrut \) \(88\) \(x^{10}\mathstrut -\mathstrut \) \(57\) \(x^{9}\mathstrut +\mathstrut \) \(270\) \(x^{8}\mathstrut +\mathstrut \) \(17\) \(x^{7}\mathstrut +\mathstrut \) \(458\) \(x^{6}\mathstrut -\mathstrut \) \(101\) \(x^{5}\mathstrut +\mathstrut \) \(189\) \(x^{4}\mathstrut -\mathstrut \) \(30\) \(x^{3}\mathstrut +\mathstrut \) \(54\) \(x^{2}\mathstrut -\mathstrut \) \(12\) \(x\mathstrut +\mathstrut \) \(4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(2680061185\) \(\nu^{13}\mathstrut +\mathstrut \) \(4118856992\) \(\nu^{12}\mathstrut -\mathstrut \) \(29747186705\) \(\nu^{11}\mathstrut +\mathstrut \) \(35610122429\) \(\nu^{10}\mathstrut -\mathstrut \) \(234940121607\) \(\nu^{9}\mathstrut +\mathstrut \) \(266952713308\) \(\nu^{8}\mathstrut -\mathstrut \) \(708369775075\) \(\nu^{7}\mathstrut +\mathstrut \) \(250820920885\) \(\nu^{6}\mathstrut -\mathstrut \) \(934856019808\) \(\nu^{5}\mathstrut +\mathstrut \) \(984764699106\) \(\nu^{4}\mathstrut -\mathstrut \) \(308324347672\) \(\nu^{3}\mathstrut +\mathstrut \) \(320958040428\) \(\nu^{2}\mathstrut -\mathstrut \) \(76957145272\) \(\nu\mathstrut +\mathstrut \) \(970249939441\)\()/\)\(325648644013\)
\(\beta_{3}\)\(=\)\((\)\(3347996299\) \(\nu^{13}\mathstrut -\mathstrut \) \(8708118669\) \(\nu^{12}\mathstrut +\mathstrut \) \(45065673273\) \(\nu^{11}\mathstrut -\mathstrut \) \(86278343802\) \(\nu^{10}\mathstrut +\mathstrut \) \(365843919170\) \(\nu^{9}\mathstrut -\mathstrut \) \(660716032257\) \(\nu^{8}\mathstrut +\mathstrut \) \(1437864427346\) \(\nu^{7}\mathstrut -\mathstrut \) \(1359823613067\) \(\nu^{6}\mathstrut +\mathstrut \) \(2035024146712\) \(\nu^{5}\mathstrut -\mathstrut \) \(2207859665815\) \(\nu^{4}\mathstrut +\mathstrut \) \(2602300698723\) \(\nu^{3}\mathstrut -\mathstrut \) \(717088584314\) \(\nu^{2}\mathstrut +\mathstrut \) \(171410592976\) \(\nu\mathstrut -\mathstrut \) \(194090246132\)\()/\)\(325648644013\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(10892153560\) \(\nu^{13}\mathstrut +\mathstrut \) \(9758640678\) \(\nu^{12}\mathstrut -\mathstrut \) \(105683331260\) \(\nu^{11}\mathstrut +\mathstrut \) \(64289364146\) \(\nu^{10}\mathstrut -\mathstrut \) \(815634907035\) \(\nu^{9}\mathstrut +\mathstrut \) \(455144021782\) \(\nu^{8}\mathstrut -\mathstrut \) \(1814707096630\) \(\nu^{7}\mathstrut -\mathstrut \) \(908790740124\) \(\nu^{6}\mathstrut -\mathstrut \) \(2130418252432\) \(\nu^{5}\mathstrut +\mathstrut \) \(2120608715454\) \(\nu^{4}\mathstrut +\mathstrut \) \(2831612352657\) \(\nu^{3}\mathstrut +\mathstrut \) \(695977413972\) \(\nu^{2}\mathstrut -\mathstrut \) \(167893036528\) \(\nu\mathstrut +\mathstrut \) \(861786952374\)\()/\)\(325648644013\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(48522561533\) \(\nu^{13}\mathstrut +\mathstrut \) \(45174565234\) \(\nu^{12}\mathstrut -\mathstrut \) \(525040058194\) \(\nu^{11}\mathstrut +\mathstrut \) \(343114818991\) \(\nu^{10}\mathstrut -\mathstrut \) \(4183707071102\) \(\nu^{9}\mathstrut +\mathstrut \) \(2399942088211\) \(\nu^{8}\mathstrut -\mathstrut \) \(12440375581653\) \(\nu^{7}\mathstrut -\mathstrut \) \(2262747973407\) \(\nu^{6}\mathstrut -\mathstrut \) \(20863509569047\) \(\nu^{5}\mathstrut +\mathstrut \) \(2865754568121\) \(\nu^{4}\mathstrut -\mathstrut \) \(6962904463922\) \(\nu^{3}\mathstrut -\mathstrut \) \(1146623852733\) \(\nu^{2}\mathstrut -\mathstrut \) \(1903129738468\) \(\nu\mathstrut +\mathstrut \) \(410860145420\)\()/\)\(651297288026\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(809674271\) \(\nu^{13}\mathstrut -\mathstrut \) \(252390382\) \(\nu^{12}\mathstrut -\mathstrut \) \(8122293536\) \(\nu^{11}\mathstrut -\mathstrut \) \(4713678201\) \(\nu^{10}\mathstrut -\mathstrut \) \(65999965440\) \(\nu^{9}\mathstrut -\mathstrut \) \(42770485013\) \(\nu^{8}\mathstrut -\mathstrut \) \(184014082317\) \(\nu^{7}\mathstrut -\mathstrut \) \(266207177033\) \(\nu^{6}\mathstrut -\mathstrut \) \(473551890151\) \(\nu^{5}\mathstrut -\mathstrut \) \(353375346337\) \(\nu^{4}\mathstrut -\mathstrut \) \(160280487010\) \(\nu^{3}\mathstrut -\mathstrut \) \(54504988027\) \(\nu^{2}\mathstrut -\mathstrut \) \(43501677200\) \(\nu\mathstrut -\mathstrut \) \(7944658082\)\()/\)\(8458406338\)
\(\beta_{7}\)\(=\)\((\)\(37194535803\) \(\nu^{13}\mathstrut -\mathstrut \) \(47919288537\) \(\nu^{12}\mathstrut +\mathstrut \) \(395084796401\) \(\nu^{11}\mathstrut -\mathstrut \) \(400339609502\) \(\nu^{10}\mathstrut +\mathstrut \) \(3098569859104\) \(\nu^{9}\mathstrut -\mathstrut \) \(2969882527405\) \(\nu^{8}\mathstrut +\mathstrut \) \(8589017027582\) \(\nu^{7}\mathstrut -\mathstrut \) \(1664041174406\) \(\nu^{6}\mathstrut +\mathstrut \) \(11026504067640\) \(\nu^{5}\mathstrut -\mathstrut \) \(11470995916307\) \(\nu^{4}\mathstrut -\mathstrut \) \(2133066776665\) \(\nu^{3}\mathstrut -\mathstrut \) \(3744294132514\) \(\nu^{2}\mathstrut +\mathstrut \) \(898967472816\) \(\nu\mathstrut -\mathstrut \) \(841108812233\)\()/\)\(325648644013\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(45711237420\) \(\nu^{13}\mathstrut +\mathstrut \) \(74034574851\) \(\nu^{12}\mathstrut -\mathstrut \) \(518899664240\) \(\nu^{11}\mathstrut +\mathstrut \) \(668330017670\) \(\nu^{10}\mathstrut -\mathstrut \) \(4117338117621\) \(\nu^{9}\mathstrut +\mathstrut \) \(5030477527030\) \(\nu^{8}\mathstrut -\mathstrut \) \(12888553943330\) \(\nu^{7}\mathstrut +\mathstrut \) \(6408314158958\) \(\nu^{6}\mathstrut -\mathstrut \) \(17209864321680\) \(\nu^{5}\mathstrut +\mathstrut \) \(18222273890390\) \(\nu^{4}\mathstrut -\mathstrut \) \(5135681407065\) \(\nu^{3}\mathstrut +\mathstrut \) \(5935414947940\) \(\nu^{2}\mathstrut -\mathstrut \) \(1422383351760\) \(\nu\mathstrut +\mathstrut \) \(974875365383\)\()/\)\(325648644013\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(58435287502\) \(\nu^{13}\mathstrut +\mathstrut \) \(104277787850\) \(\nu^{12}\mathstrut -\mathstrut \) \(683843870764\) \(\nu^{11}\mathstrut +\mathstrut \) \(962775171505\) \(\nu^{10}\mathstrut -\mathstrut \) \(5449809996738\) \(\nu^{9}\mathstrut +\mathstrut \) \(7279578337109\) \(\nu^{8}\mathstrut -\mathstrut \) \(17910517000443\) \(\nu^{7}\mathstrut +\mathstrut \) \(10738605919044\) \(\nu^{6}\mathstrut -\mathstrut \) \(24249792781624\) \(\nu^{5}\mathstrut +\mathstrut \) \(25830617586941\) \(\nu^{4}\mathstrut -\mathstrut \) \(12925259206893\) \(\nu^{3}\mathstrut +\mathstrut \) \(8407638741310\) \(\nu^{2}\mathstrut -\mathstrut \) \(2013572275960\) \(\nu\mathstrut +\mathstrut \) \(2201747399207\)\()/\)\(325648644013\)
\(\beta_{10}\)\(=\)\((\)\(126618741863\) \(\nu^{13}\mathstrut -\mathstrut \) \(49216475610\) \(\nu^{12}\mathstrut +\mathstrut \) \(1335886940970\) \(\nu^{11}\mathstrut -\mathstrut \) \(183514685193\) \(\nu^{10}\mathstrut +\mathstrut \) \(10733169032550\) \(\nu^{9}\mathstrut -\mathstrut \) \(585966489615\) \(\nu^{8}\mathstrut +\mathstrut \) \(31403882537963\) \(\nu^{7}\mathstrut +\mathstrut \) \(21731699467035\) \(\nu^{6}\mathstrut +\mathstrut \) \(63503082991395\) \(\nu^{5}\mathstrut +\mathstrut \) \(22467398449373\) \(\nu^{4}\mathstrut +\mathstrut \) \(21347976231450\) \(\nu^{3}\mathstrut +\mathstrut \) \(5457745711665\) \(\nu^{2}\mathstrut +\mathstrut \) \(2242764960296\) \(\nu\mathstrut +\mathstrut \) \(903683328390\)\()/\)\(651297288026\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(20029651747\) \(\nu^{13}\mathstrut +\mathstrut \) \(18183711674\) \(\nu^{12}\mathstrut -\mathstrut \) \(216517971596\) \(\nu^{11}\mathstrut +\mathstrut \) \(136874887445\) \(\nu^{10}\mathstrut -\mathstrut \) \(1725891567156\) \(\nu^{9}\mathstrut +\mathstrut \) \(952274405431\) \(\nu^{8}\mathstrut -\mathstrut \) \(5129198170687\) \(\nu^{7}\mathstrut -\mathstrut \) \(1041412251713\) \(\nu^{6}\mathstrut -\mathstrut \) \(8674402381075\) \(\nu^{5}\mathstrut +\mathstrut \) \(946819186593\) \(\nu^{4}\mathstrut -\mathstrut \) \(2896009242346\) \(\nu^{3}\mathstrut -\mathstrut \) \(490070050147\) \(\nu^{2}\mathstrut -\mathstrut \) \(793639274980\) \(\nu\mathstrut -\mathstrut \) \(101131348946\)\()/\)\(93042469718\)
\(\beta_{12}\)\(=\)\((\)\(156907511177\) \(\nu^{13}\mathstrut -\mathstrut \) \(63210384410\) \(\nu^{12}\mathstrut +\mathstrut \) \(1644341611148\) \(\nu^{11}\mathstrut -\mathstrut \) \(250245646301\) \(\nu^{10}\mathstrut +\mathstrut \) \(13207909689396\) \(\nu^{9}\mathstrut -\mathstrut \) \(942157914055\) \(\nu^{8}\mathstrut +\mathstrut \) \(38225859873625\) \(\nu^{7}\mathstrut +\mathstrut \) \(26110314425969\) \(\nu^{6}\mathstrut +\mathstrut \) \(77748959678947\) \(\nu^{5}\mathstrut +\mathstrut \) \(23844336362505\) \(\nu^{4}\mathstrut +\mathstrut \) \(26131870142026\) \(\nu^{3}\mathstrut +\mathstrut \) \(6616282893811\) \(\nu^{2}\mathstrut +\mathstrut \) \(10832044605312\) \(\nu\mathstrut +\mathstrut \) \(1100661461810\)\()/\)\(651297288026\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(87738052524\) \(\nu^{13}\mathstrut +\mathstrut \) \(71009258532\) \(\nu^{12}\mathstrut -\mathstrut \) \(933363507764\) \(\nu^{11}\mathstrut +\mathstrut \) \(502153407798\) \(\nu^{10}\mathstrut -\mathstrut \) \(7424278972227\) \(\nu^{9}\mathstrut +\mathstrut \) \(3401605949923\) \(\nu^{8}\mathstrut -\mathstrut \) \(21440487873381\) \(\nu^{7}\mathstrut -\mathstrut \) \(6897421130211\) \(\nu^{6}\mathstrut -\mathstrut \) \(36599366732449\) \(\nu^{5}\mathstrut +\mathstrut \) \(1482620818260\) \(\nu^{4}\mathstrut -\mathstrut \) \(8695369936605\) \(\nu^{3}\mathstrut -\mathstrut \) \(1496669907681\) \(\nu^{2}\mathstrut -\mathstrut \) \(2171714647326\) \(\nu\mathstrut +\mathstrut \) \(438504860088\)\()/\)\(325648644013\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\(\beta_{11}\mathstrut -\mathstrut \) \(3\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(8\) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(18\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(18\)
\(\nu^{5}\)\(=\)\(8\) \(\beta_{12}\mathstrut +\mathstrut \) \(10\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(8\) \(\beta_{9}\mathstrut +\mathstrut \) \(9\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(9\) \(\beta_{6}\mathstrut -\mathstrut \) \(12\) \(\beta_{5}\mathstrut -\mathstrut \) \(20\) \(\beta_{3}\mathstrut +\mathstrut \) \(10\) \(\beta_{2}\mathstrut -\mathstrut \) \(20\) \(\beta_{1}\)
\(\nu^{6}\)\(=\)\(10\) \(\beta_{9}\mathstrut -\mathstrut \) \(10\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(10\) \(\beta_{4}\mathstrut +\mathstrut \) \(13\) \(\beta_{3}\mathstrut -\mathstrut \) \(58\) \(\beta_{2}\mathstrut +\mathstrut \) \(119\)
\(\nu^{7}\)\(=\)\(-\)\(\beta_{13}\mathstrut -\mathstrut \) \(58\) \(\beta_{12}\mathstrut -\mathstrut \) \(84\) \(\beta_{11}\mathstrut -\mathstrut \) \(11\) \(\beta_{10}\mathstrut -\mathstrut \) \(68\) \(\beta_{6}\mathstrut +\mathstrut \) \(117\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(136\) \(\beta_{1}\mathstrut -\mathstrut \) \(117\)
\(\nu^{8}\)\(=\)\(-\)\(80\) \(\beta_{13}\mathstrut +\mathstrut \) \(84\) \(\beta_{12}\mathstrut +\mathstrut \) \(415\) \(\beta_{11}\mathstrut -\mathstrut \) \(12\) \(\beta_{10}\mathstrut -\mathstrut \) \(84\) \(\beta_{9}\mathstrut +\mathstrut \) \(83\) \(\beta_{8}\mathstrut -\mathstrut \) \(12\) \(\beta_{7}\mathstrut +\mathstrut \) \(83\) \(\beta_{6}\mathstrut -\mathstrut \) \(812\) \(\beta_{5}\mathstrut -\mathstrut \) \(127\) \(\beta_{3}\mathstrut +\mathstrut \) \(415\) \(\beta_{2}\mathstrut -\mathstrut \) \(127\) \(\beta_{1}\)
\(\nu^{9}\)\(=\)\(415\) \(\beta_{9}\mathstrut -\mathstrut \) \(495\) \(\beta_{8}\mathstrut -\mathstrut \) \(92\) \(\beta_{7}\mathstrut -\mathstrut \) \(9\) \(\beta_{4}\mathstrut +\mathstrut \) \(939\) \(\beta_{3}\mathstrut -\mathstrut \) \(673\) \(\beta_{2}\mathstrut +\mathstrut \) \(1027\)
\(\nu^{10}\)\(=\)\(596\) \(\beta_{13}\mathstrut -\mathstrut \) \(673\) \(\beta_{12}\mathstrut -\mathstrut \) \(2975\) \(\beta_{11}\mathstrut +\mathstrut \) \(101\) \(\beta_{10}\mathstrut -\mathstrut \) \(664\) \(\beta_{6}\mathstrut +\mathstrut \) \(5649\) \(\beta_{5}\mathstrut -\mathstrut \) \(596\) \(\beta_{4}\mathstrut +\mathstrut \) \(1114\) \(\beta_{1}\mathstrut -\mathstrut \) \(5649\)
\(\nu^{11}\)\(=\)\(33\) \(\beta_{13}\mathstrut +\mathstrut \) \(2975\) \(\beta_{12}\mathstrut +\mathstrut \) \(5280\) \(\beta_{11}\mathstrut +\mathstrut \) \(697\) \(\beta_{10}\mathstrut -\mathstrut \) \(2975\) \(\beta_{9}\mathstrut +\mathstrut \) \(3571\) \(\beta_{8}\mathstrut +\mathstrut \) \(697\) \(\beta_{7}\mathstrut +\mathstrut \) \(3571\) \(\beta_{6}\mathstrut -\mathstrut \) \(8509\) \(\beta_{5}\mathstrut -\mathstrut \) \(6568\) \(\beta_{3}\mathstrut +\mathstrut \) \(5280\) \(\beta_{2}\mathstrut -\mathstrut \) \(6568\) \(\beta_{1}\)
\(\nu^{12}\)\(=\)\(5280\) \(\beta_{9}\mathstrut -\mathstrut \) \(5247\) \(\beta_{8}\mathstrut +\mathstrut \) \(730\) \(\beta_{7}\mathstrut +\mathstrut \) \(4301\) \(\beta_{4}\mathstrut +\mathstrut \) \(9250\) \(\beta_{3}\mathstrut -\mathstrut \) \(21424\) \(\beta_{2}\mathstrut +\mathstrut \) \(39857\)
\(\nu^{13}\)\(=\)\(216\) \(\beta_{13}\mathstrut -\mathstrut \) \(21424\) \(\beta_{12}\mathstrut -\mathstrut \) \(40903\) \(\beta_{11}\mathstrut -\mathstrut \) \(5031\) \(\beta_{10}\mathstrut -\mathstrut \) \(25725\) \(\beta_{6}\mathstrut +\mathstrut \) \(68139\) \(\beta_{5}\mathstrut -\mathstrut \) \(216\) \(\beta_{4}\mathstrut +\mathstrut \) \(46436\) \(\beta_{1}\mathstrut -\mathstrut \) \(68139\)

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
0.295122 0.511167i
−0.288886 + 0.500366i
1.10425 1.91261i
0.150493 0.260662i
1.24942 2.16407i
−1.36264 + 2.36015i
−0.647766 + 1.12196i
0.295122 + 0.511167i
−0.288886 0.500366i
1.10425 + 1.91261i
0.150493 + 0.260662i
1.24942 + 2.16407i
−1.36264 2.36015i
−0.647766 1.12196i
0 −1.00000 0 1.00000 0 −1.59463 + 2.76199i 0 1.00000 0
841.2 0 −1.00000 0 1.00000 0 −1.13810 + 1.97125i 0 1.00000 0
841.3 0 −1.00000 0 1.00000 0 −0.952306 + 1.64944i 0 1.00000 0
841.4 0 −1.00000 0 1.00000 0 0.827889 1.43395i 0 1.00000 0
841.5 0 −1.00000 0 1.00000 0 1.07806 1.86725i 0 1.00000 0
841.6 0 −1.00000 0 1.00000 0 1.26463 2.19041i 0 1.00000 0
841.7 0 −1.00000 0 1.00000 0 2.01447 3.48916i 0 1.00000 0
3781.1 0 −1.00000 0 1.00000 0 −1.59463 2.76199i 0 1.00000 0
3781.2 0 −1.00000 0 1.00000 0 −1.13810 1.97125i 0 1.00000 0
3781.3 0 −1.00000 0 1.00000 0 −0.952306 1.64944i 0 1.00000 0
3781.4 0 −1.00000 0 1.00000 0 0.827889 + 1.43395i 0 1.00000 0
3781.5 0 −1.00000 0 1.00000 0 1.07806 + 1.86725i 0 1.00000 0
3781.6 0 −1.00000 0 1.00000 0 1.26463 + 2.19041i 0 1.00000 0
3781.7 0 −1.00000 0 1.00000 0 2.01447 + 3.48916i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3781.7
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
67.c Even 1 yes

Hecke kernels

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

\(T_{7}^{14} - \cdots\)
\(T_{11}^{14} - \cdots\)
\(T_{17}^{14} + \cdots\)