Properties

Label 8004.2.a.i
Level 8004
Weight 2
Character orbit 8004.a
Self dual Yes
Analytic conductor 63.912
Analytic rank 0
Dimension 16
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8004.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(- q^{3}\) \( + \beta_{1} q^{5} \) \( -\beta_{12} q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(- q^{3}\) \( + \beta_{1} q^{5} \) \( -\beta_{12} q^{7} \) \(+ q^{9}\) \( -\beta_{4} q^{11} \) \( + \beta_{5} q^{13} \) \( -\beta_{1} q^{15} \) \( -\beta_{13} q^{17} \) \( + ( -\beta_{5} + \beta_{8} - \beta_{9} ) q^{19} \) \( + \beta_{12} q^{21} \) \(+ q^{23}\) \( + ( 1 + \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{10} - \beta_{13} + \beta_{14} ) q^{25} \) \(- q^{27}\) \(+ q^{29}\) \( + ( -\beta_{4} - \beta_{7} ) q^{31} \) \( + \beta_{4} q^{33} \) \( + ( 2 - \beta_{1} + \beta_{3} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{12} + \beta_{13} + \beta_{15} ) q^{35} \) \( + ( 1 - \beta_{14} ) q^{37} \) \( -\beta_{5} q^{39} \) \( + ( -1 - \beta_{3} + \beta_{8} ) q^{41} \) \( + ( -1 + \beta_{1} - \beta_{4} - \beta_{8} ) q^{43} \) \( + \beta_{1} q^{45} \) \( + ( -1 + \beta_{1} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{10} - \beta_{15} ) q^{47} \) \( + ( 1 + \beta_{1} + \beta_{4} + \beta_{7} + \beta_{9} - \beta_{13} - \beta_{15} ) q^{49} \) \( + \beta_{13} q^{51} \) \( + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{53} \) \( + ( -2 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{55} \) \( + ( \beta_{5} - \beta_{8} + \beta_{9} ) q^{57} \) \( + ( \beta_{1} + \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{15} ) q^{59} \) \( + ( 1 + \beta_{2} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{61} \) \( -\beta_{12} q^{63} \) \( + ( 2 + 3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{12} + \beta_{13} ) q^{65} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{7} + \beta_{9} + \beta_{11} + \beta_{13} + \beta_{15} ) q^{67} \) \(- q^{69}\) \( + ( \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{14} + \beta_{15} ) q^{71} \) \( + ( -2 + \beta_{1} + \beta_{2} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{13} ) q^{73} \) \( + ( -1 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{10} + \beta_{13} - \beta_{14} ) q^{75} \) \( + ( -\beta_{1} + \beta_{4} + \beta_{6} - \beta_{11} - \beta_{13} ) q^{77} \) \( + ( -1 + \beta_{1} + \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{13} + \beta_{15} ) q^{79} \) \(+ q^{81}\) \( + ( -\beta_{3} + \beta_{4} + 2 \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{83} \) \( + ( 3 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} - 2 \beta_{8} - \beta_{11} - 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{85} \) \(- q^{87}\) \( + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} + \beta_{13} - \beta_{14} ) q^{89} \) \( + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{4} - 2 \beta_{5} - 2 \beta_{7} - \beta_{9} - \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{91} \) \( + ( \beta_{4} + \beta_{7} ) q^{93} \) \( + ( 2 - \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{11} - 2 \beta_{12} - \beta_{15} ) q^{95} \) \( + ( 2 \beta_{1} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{97} \) \( -\beta_{4} q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(16q \) \(\mathstrut -\mathstrut 16q^{3} \) \(\mathstrut +\mathstrut 5q^{5} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 16q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(16q \) \(\mathstrut -\mathstrut 16q^{3} \) \(\mathstrut +\mathstrut 5q^{5} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 16q^{9} \) \(\mathstrut +\mathstrut 5q^{11} \) \(\mathstrut +\mathstrut 8q^{13} \) \(\mathstrut -\mathstrut 5q^{15} \) \(\mathstrut +\mathstrut 7q^{17} \) \(\mathstrut +\mathstrut q^{19} \) \(\mathstrut +\mathstrut 4q^{21} \) \(\mathstrut +\mathstrut 16q^{23} \) \(\mathstrut +\mathstrut 31q^{25} \) \(\mathstrut -\mathstrut 16q^{27} \) \(\mathstrut +\mathstrut 16q^{29} \) \(\mathstrut -\mathstrut 2q^{31} \) \(\mathstrut -\mathstrut 5q^{33} \) \(\mathstrut +\mathstrut 5q^{35} \) \(\mathstrut +\mathstrut 14q^{37} \) \(\mathstrut -\mathstrut 8q^{39} \) \(\mathstrut -\mathstrut q^{41} \) \(\mathstrut -\mathstrut 13q^{43} \) \(\mathstrut +\mathstrut 5q^{45} \) \(\mathstrut -\mathstrut 4q^{47} \) \(\mathstrut +\mathstrut 30q^{49} \) \(\mathstrut -\mathstrut 7q^{51} \) \(\mathstrut +\mathstrut 19q^{53} \) \(\mathstrut -\mathstrut 37q^{55} \) \(\mathstrut -\mathstrut q^{57} \) \(\mathstrut +\mathstrut 12q^{59} \) \(\mathstrut +\mathstrut 21q^{61} \) \(\mathstrut -\mathstrut 4q^{63} \) \(\mathstrut +\mathstrut 26q^{65} \) \(\mathstrut -\mathstrut 11q^{67} \) \(\mathstrut -\mathstrut 16q^{69} \) \(\mathstrut +\mathstrut 7q^{71} \) \(\mathstrut -\mathstrut 13q^{73} \) \(\mathstrut -\mathstrut 31q^{75} \) \(\mathstrut +\mathstrut 4q^{77} \) \(\mathstrut -\mathstrut 18q^{79} \) \(\mathstrut +\mathstrut 16q^{81} \) \(\mathstrut +\mathstrut 25q^{83} \) \(\mathstrut +\mathstrut 48q^{85} \) \(\mathstrut -\mathstrut 16q^{87} \) \(\mathstrut +\mathstrut 12q^{89} \) \(\mathstrut -\mathstrut 11q^{91} \) \(\mathstrut +\mathstrut 2q^{93} \) \(\mathstrut -\mathstrut q^{95} \) \(\mathstrut +\mathstrut 5q^{97} \) \(\mathstrut +\mathstrut 5q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16}\mathstrut -\mathstrut \) \(5\) \(x^{15}\mathstrut -\mathstrut \) \(43\) \(x^{14}\mathstrut +\mathstrut \) \(234\) \(x^{13}\mathstrut +\mathstrut \) \(634\) \(x^{12}\mathstrut -\mathstrut \) \(4048\) \(x^{11}\mathstrut -\mathstrut \) \(3483\) \(x^{10}\mathstrut +\mathstrut \) \(32512\) \(x^{9}\mathstrut -\mathstrut \) \(137\) \(x^{8}\mathstrut -\mathstrut \) \(121665\) \(x^{7}\mathstrut +\mathstrut \) \(60168\) \(x^{6}\mathstrut +\mathstrut \) \(172218\) \(x^{5}\mathstrut -\mathstrut \) \(138024\) \(x^{4}\mathstrut -\mathstrut \) \(17844\) \(x^{3}\mathstrut +\mathstrut \) \(14352\) \(x^{2}\mathstrut +\mathstrut \) \(72\) \(x\mathstrut -\mathstrut \) \(208\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(3040780320993915014\) \(\nu^{15}\mathstrut +\mathstrut \) \(11451848272286613279\) \(\nu^{14}\mathstrut -\mathstrut \) \(232887392825327605241\) \(\nu^{13}\mathstrut -\mathstrut \) \(552866610872686710043\) \(\nu^{12}\mathstrut +\mathstrut \) \(6701952172673579845542\) \(\nu^{11}\mathstrut +\mathstrut \) \(10014744839360784183052\) \(\nu^{10}\mathstrut -\mathstrut \) \(92816762444366568951910\) \(\nu^{9}\mathstrut -\mathstrut \) \(85086576675455429657055\) \(\nu^{8}\mathstrut +\mathstrut \) \(652807380563049726096164\) \(\nu^{7}\mathstrut +\mathstrut \) \(327739783995970876724887\) \(\nu^{6}\mathstrut -\mathstrut \) \(2206166321766519707780335\) \(\nu^{5}\mathstrut -\mathstrut \) \(345737863611574611652196\) \(\nu^{4}\mathstrut +\mathstrut \) \(2820087633149276715197960\) \(\nu^{3}\mathstrut -\mathstrut \) \(505057097539218663987944\) \(\nu^{2}\mathstrut -\mathstrut \) \(265212571902372090704692\) \(\nu\mathstrut +\mathstrut \) \(52042966114613282035144\)\()/\)\(82\!\cdots\!44\)
\(\beta_{3}\)\(=\)\((\)\(1133155627719690365\) \(\nu^{15}\mathstrut -\mathstrut \) \(6183981961343391225\) \(\nu^{14}\mathstrut -\mathstrut \) \(45962507843656667153\) \(\nu^{13}\mathstrut +\mathstrut \) \(285577742288708681086\) \(\nu^{12}\mathstrut +\mathstrut \) \(594553806911671718002\) \(\nu^{11}\mathstrut -\mathstrut \) \(4841993240933967346590\) \(\nu^{10}\mathstrut -\mathstrut \) \(1931243557888597965827\) \(\nu^{9}\mathstrut +\mathstrut \) \(37694840050022447013852\) \(\nu^{8}\mathstrut -\mathstrut \) \(15054873801417991803487\) \(\nu^{7}\mathstrut -\mathstrut \) \(133459508425382606930219\) \(\nu^{6}\mathstrut +\mathstrut \) \(118176839691438596424822\) \(\nu^{5}\mathstrut +\mathstrut \) \(162630129553180087835490\) \(\nu^{4}\mathstrut -\mathstrut \) \(217538448934933635766102\) \(\nu^{3}\mathstrut +\mathstrut \) \(28814439094940890866468\) \(\nu^{2}\mathstrut +\mathstrut \) \(20345563209993771194972\) \(\nu\mathstrut -\mathstrut \) \(181736188397342579200\)\()/\)\(13\!\cdots\!24\)
\(\beta_{4}\)\(=\)\((\)\(10327108820298630749\) \(\nu^{15}\mathstrut -\mathstrut \) \(46623029354138300703\) \(\nu^{14}\mathstrut -\mathstrut \) \(468648207638402451527\) \(\nu^{13}\mathstrut +\mathstrut \) \(2199331736699944090170\) \(\nu^{12}\mathstrut +\mathstrut \) \(7701026401458349054308\) \(\nu^{11}\mathstrut -\mathstrut \) \(38562172136739934951652\) \(\nu^{10}\mathstrut -\mathstrut \) \(55997241963193925731391\) \(\nu^{9}\mathstrut +\mathstrut \) \(317505626992755431744994\) \(\nu^{8}\mathstrut +\mathstrut \) \(160070546734796190670629\) \(\nu^{7}\mathstrut -\mathstrut \) \(1253648947739701211892215\) \(\nu^{6}\mathstrut +\mathstrut \) \(15498431467411784950720\) \(\nu^{5}\mathstrut +\mathstrut \) \(2076442970619250704115952\) \(\nu^{4}\mathstrut -\mathstrut \) \(572372392754759088010752\) \(\nu^{3}\mathstrut -\mathstrut \) \(871645844830868408937972\) \(\nu^{2}\mathstrut +\mathstrut \) \(75547605968724778430888\) \(\nu\mathstrut +\mathstrut \) \(39550714308552523443072\)\()/\)\(54\!\cdots\!96\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(15783173244444101129\) \(\nu^{15}\mathstrut +\mathstrut \) \(71891299794190027998\) \(\nu^{14}\mathstrut +\mathstrut \) \(710848605201316221158\) \(\nu^{13}\mathstrut -\mathstrut \) \(3385320715110873005891\) \(\nu^{12}\mathstrut -\mathstrut \) \(11475349920655983031686\) \(\nu^{11}\mathstrut +\mathstrut \) \(59064258354862485012260\) \(\nu^{10}\mathstrut +\mathstrut \) \(79549194381055089354751\) \(\nu^{9}\mathstrut -\mathstrut \) \(480110131920749519232459\) \(\nu^{8}\mathstrut -\mathstrut \) \(187404565471993343666351\) \(\nu^{7}\mathstrut +\mathstrut \) \(1833305564855172683452274\) \(\nu^{6}\mathstrut -\mathstrut \) \(268495880029625550510797\) \(\nu^{5}\mathstrut -\mathstrut \) \(2745160541549121158528824\) \(\nu^{4}\mathstrut +\mathstrut \) \(1224989810538650711651380\) \(\nu^{3}\mathstrut +\mathstrut \) \(637548574623595049191988\) \(\nu^{2}\mathstrut -\mathstrut \) \(28435539046608856852172\) \(\nu\mathstrut -\mathstrut \) \(48618750485435144983000\)\()/\)\(82\!\cdots\!44\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(43409510040342958591\) \(\nu^{15}\mathstrut +\mathstrut \) \(204436784173408593213\) \(\nu^{14}\mathstrut +\mathstrut \) \(1933384284718598061661\) \(\nu^{13}\mathstrut -\mathstrut \) \(9609952842332358293734\) \(\nu^{12}\mathstrut -\mathstrut \) \(30738946990350785496204\) \(\nu^{11}\mathstrut +\mathstrut \) \(167641296806600010566452\) \(\nu^{10}\mathstrut +\mathstrut \) \(209050863408559362327029\) \(\nu^{9}\mathstrut -\mathstrut \) \(1369734242025953674106622\) \(\nu^{8}\mathstrut -\mathstrut \) \(481998571182840860681887\) \(\nu^{7}\mathstrut +\mathstrut \) \(5334598757017222508511037\) \(\nu^{6}\mathstrut -\mathstrut \) \(669736051113647402565376\) \(\nu^{5}\mathstrut -\mathstrut \) \(8528134675336330855060592\) \(\nu^{4}\mathstrut +\mathstrut \) \(2968897987381385534458088\) \(\nu^{3}\mathstrut +\mathstrut \) \(2965875364568320786841260\) \(\nu^{2}\mathstrut +\mathstrut \) \(9531422201946669156536\) \(\nu\mathstrut -\mathstrut \) \(115448866564292248468160\)\()/\)\(16\!\cdots\!88\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(14876422501216426365\) \(\nu^{15}\mathstrut +\mathstrut \) \(63444477320010917489\) \(\nu^{14}\mathstrut +\mathstrut \) \(680444530030849926369\) \(\nu^{13}\mathstrut -\mathstrut \) \(2960246772843543741264\) \(\nu^{12}\mathstrut -\mathstrut \) \(11324818386317160387376\) \(\nu^{11}\mathstrut +\mathstrut \) \(50934833529316434627484\) \(\nu^{10}\mathstrut +\mathstrut \) \(84152959587324240424423\) \(\nu^{9}\mathstrut -\mathstrut \) \(405344514391113240217288\) \(\nu^{8}\mathstrut -\mathstrut \) \(252413267047417095561809\) \(\nu^{7}\mathstrut +\mathstrut \) \(1494626491148611763355897\) \(\nu^{6}\mathstrut +\mathstrut \) \(27305100885580799650810\) \(\nu^{5}\mathstrut -\mathstrut \) \(2074495686819045701186224\) \(\nu^{4}\mathstrut +\mathstrut \) \(799934920651196090914288\) \(\nu^{3}\mathstrut +\mathstrut \) \(246595002574247516330436\) \(\nu^{2}\mathstrut -\mathstrut \) \(32576648691634350903024\) \(\nu\mathstrut +\mathstrut \) \(430803475028090582512\)\()/\)\(54\!\cdots\!96\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(7603870522084452615\) \(\nu^{15}\mathstrut +\mathstrut \) \(32046172073721707913\) \(\nu^{14}\mathstrut +\mathstrut \) \(353456303235461540053\) \(\nu^{13}\mathstrut -\mathstrut \) \(1511395690869302031178\) \(\nu^{12}\mathstrut -\mathstrut \) \(6050691028509434703612\) \(\nu^{11}\mathstrut +\mathstrut \) \(26465002897883635403568\) \(\nu^{10}\mathstrut +\mathstrut \) \(47517289986971177956637\) \(\nu^{9}\mathstrut -\mathstrut \) \(217058424842370164675842\) \(\nu^{8}\mathstrut -\mathstrut \) \(165337223101262763914851\) \(\nu^{7}\mathstrut +\mathstrut \) \(849163607976476235455277\) \(\nu^{6}\mathstrut +\mathstrut \) \(154277444344167389409560\) \(\nu^{5}\mathstrut -\mathstrut \) \(1376527448283916327600720\) \(\nu^{4}\mathstrut +\mathstrut \) \(189737204442907705092132\) \(\nu^{3}\mathstrut +\mathstrut \) \(538400774762995113598740\) \(\nu^{2}\mathstrut +\mathstrut \) \(25492596062475603023440\) \(\nu\mathstrut -\mathstrut \) \(27182643118546951377232\)\()/\)\(27\!\cdots\!48\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(15813712956832157237\) \(\nu^{15}\mathstrut +\mathstrut \) \(70916958369897065433\) \(\nu^{14}\mathstrut +\mathstrut \) \(719012486847903488913\) \(\nu^{13}\mathstrut -\mathstrut \) \(3344745531914834911664\) \(\nu^{12}\mathstrut -\mathstrut \) \(11838796361473972966656\) \(\nu^{11}\mathstrut +\mathstrut \) \(58582378625148598351460\) \(\nu^{10}\mathstrut +\mathstrut \) \(86125788536977874427823\) \(\nu^{9}\mathstrut -\mathstrut \) \(480661070940613526614664\) \(\nu^{8}\mathstrut -\mathstrut \) \(244010727226114098109473\) \(\nu^{7}\mathstrut +\mathstrut \) \(1878957650573029373491257\) \(\nu^{6}\mathstrut -\mathstrut \) \(43320182299082538137646\) \(\nu^{5}\mathstrut -\mathstrut \) \(3016049778386644474401072\) \(\nu^{4}\mathstrut +\mathstrut \) \(910802804643754812688216\) \(\nu^{3}\mathstrut +\mathstrut \) \(1071305771301748204569172\) \(\nu^{2}\mathstrut -\mathstrut \) \(83620908096978852457952\) \(\nu\mathstrut -\mathstrut \) \(31358839502922458309168\)\()/\)\(54\!\cdots\!96\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(48750376286856419701\) \(\nu^{15}\mathstrut +\mathstrut \) \(229053636521692224375\) \(\nu^{14}\mathstrut +\mathstrut \) \(2151043186106097123631\) \(\nu^{13}\mathstrut -\mathstrut \) \(10697144276185392156634\) \(\nu^{12}\mathstrut -\mathstrut \) \(33512675596998272085780\) \(\nu^{11}\mathstrut +\mathstrut \) \(184493534161900674094564\) \(\nu^{10}\mathstrut +\mathstrut \) \(215733079036281931399607\) \(\nu^{9}\mathstrut -\mathstrut \) \(1476019645175755861347138\) \(\nu^{8}\mathstrut -\mathstrut \) \(369662096234080721052685\) \(\nu^{7}\mathstrut +\mathstrut \) \(5502983380583967143683615\) \(\nu^{6}\mathstrut -\mathstrut \) \(1501117875210636465270160\) \(\nu^{5}\mathstrut -\mathstrut \) \(7825684947137813046148304\) \(\nu^{4}\mathstrut +\mathstrut \) \(4671490927199216453556224\) \(\nu^{3}\mathstrut +\mathstrut \) \(1130638678095749386304068\) \(\nu^{2}\mathstrut -\mathstrut \) \(387205328131515237238792\) \(\nu\mathstrut -\mathstrut \) \(65047851739085236716416\)\()/\)\(16\!\cdots\!88\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(19819065957461023811\) \(\nu^{15}\mathstrut +\mathstrut \) \(105296243094269658907\) \(\nu^{14}\mathstrut +\mathstrut \) \(825924130302220654971\) \(\nu^{13}\mathstrut -\mathstrut \) \(4912947016947780053268\) \(\nu^{12}\mathstrut -\mathstrut \) \(11381973623510737504984\) \(\nu^{11}\mathstrut +\mathstrut \) \(84635925481199523199764\) \(\nu^{10}\mathstrut +\mathstrut \) \(49699417845194961077353\) \(\nu^{9}\mathstrut -\mathstrut \) \(675789922002448975704604\) \(\nu^{8}\mathstrut +\mathstrut \) \(145894747281510593499017\) \(\nu^{7}\mathstrut +\mathstrut \) \(2503760282664111920367379\) \(\nu^{6}\mathstrut -\mathstrut \) \(1670543280670879404138622\) \(\nu^{5}\mathstrut -\mathstrut \) \(3441103630357287161444048\) \(\nu^{4}\mathstrut +\mathstrut \) \(3297458934296158318389232\) \(\nu^{3}\mathstrut +\mathstrut \) \(124510648830211096930172\) \(\nu^{2}\mathstrut -\mathstrut \) \(256144554369799347906592\) \(\nu\mathstrut +\mathstrut \) \(83494722494782046960\)\()/\)\(54\!\cdots\!96\)
\(\beta_{12}\)\(=\)\((\)\(15344073927299171897\) \(\nu^{15}\mathstrut -\mathstrut \) \(76517980267269701535\) \(\nu^{14}\mathstrut -\mathstrut \) \(662035163879610543401\) \(\nu^{13}\mathstrut +\mathstrut \) \(3585945259827232385072\) \(\nu^{12}\mathstrut +\mathstrut \) \(9839491871465361076638\) \(\nu^{11}\mathstrut -\mathstrut \) \(62181408508523380600868\) \(\nu^{10}\mathstrut -\mathstrut \) \(55554094321534472370211\) \(\nu^{9}\mathstrut +\mathstrut \) \(501622126176053457192078\) \(\nu^{8}\mathstrut +\mathstrut \) \(17345193658302384622799\) \(\nu^{7}\mathstrut -\mathstrut \) \(1894731945704220803403863\) \(\nu^{6}\mathstrut +\mathstrut \) \(833413740406888323847562\) \(\nu^{5}\mathstrut +\mathstrut \) \(2755519788258439982119666\) \(\nu^{4}\mathstrut -\mathstrut \) \(1934597359389313500952252\) \(\nu^{3}\mathstrut -\mathstrut \) \(426428181856744945912580\) \(\nu^{2}\mathstrut +\mathstrut \) \(121576357609022372496944\) \(\nu\mathstrut +\mathstrut \) \(4443088667932115690440\)\()/\)\(41\!\cdots\!72\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(70933075913422809427\) \(\nu^{15}\mathstrut +\mathstrut \) \(319190212777976757561\) \(\nu^{14}\mathstrut +\mathstrut \) \(3190335724091810475001\) \(\nu^{13}\mathstrut -\mathstrut \) \(14936160860061056201230\) \(\nu^{12}\mathstrut -\mathstrut \) \(51494181033152023858860\) \(\nu^{11}\mathstrut +\mathstrut \) \(258321554986471115357812\) \(\nu^{10}\mathstrut +\mathstrut \) \(358664574332552237611265\) \(\nu^{9}\mathstrut -\mathstrut \) \(2075255844526952664807366\) \(\nu^{8}\mathstrut -\mathstrut \) \(869371686203195894627251\) \(\nu^{7}\mathstrut +\mathstrut \) \(7797419486270366874993913\) \(\nu^{6}\mathstrut -\mathstrut \) \(1086937561234508913856240\) \(\nu^{5}\mathstrut -\mathstrut \) \(11349378748818973102168016\) \(\nu^{4}\mathstrut +\mathstrut \) \(5541104129287872654006968\) \(\nu^{3}\mathstrut +\mathstrut \) \(2151780320120874407422252\) \(\nu^{2}\mathstrut -\mathstrut \) \(662086086998971531518376\) \(\nu\mathstrut -\mathstrut \) \(31359800394432217976480\)\()/\)\(16\!\cdots\!88\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(32225813913671340095\) \(\nu^{15}\mathstrut +\mathstrut \) \(159924296878592499289\) \(\nu^{14}\mathstrut +\mathstrut \) \(1389566703547393943857\) \(\nu^{13}\mathstrut -\mathstrut \) \(7479591954295351894242\) \(\nu^{12}\mathstrut -\mathstrut \) \(20617773136724243749644\) \(\nu^{11}\mathstrut +\mathstrut \) \(129273187698150131786804\) \(\nu^{10}\mathstrut +\mathstrut \) \(115592584583412047647733\) \(\nu^{9}\mathstrut -\mathstrut \) \(1037020682429358346607922\) \(\nu^{8}\mathstrut -\mathstrut \) \(23473862923459495877375\) \(\nu^{7}\mathstrut +\mathstrut \) \(3874435369925228060759257\) \(\nu^{6}\mathstrut -\mathstrut \) \(1830412119817231334118652\) \(\nu^{5}\mathstrut -\mathstrut \) \(5468684705975812969273176\) \(\nu^{4}\mathstrut +\mathstrut \) \(4274698916700503360392296\) \(\nu^{3}\mathstrut +\mathstrut \) \(543054530270901314400956\) \(\nu^{2}\mathstrut -\mathstrut \) \(386578636465775801362264\) \(\nu\mathstrut -\mathstrut \) \(12820290579032488312704\)\()/\)\(54\!\cdots\!96\)
\(\beta_{15}\)\(=\)\((\)\(45395358841873163252\) \(\nu^{15}\mathstrut -\mathstrut \) \(216603173312831897727\) \(\nu^{14}\mathstrut -\mathstrut \) \(1997551184811081394793\) \(\nu^{13}\mathstrut +\mathstrut \) \(10150483883027433779483\) \(\nu^{12}\mathstrut +\mathstrut \) \(30918959212470415787136\) \(\nu^{11}\mathstrut -\mathstrut \) \(175989519979563028243244\) \(\nu^{10}\mathstrut -\mathstrut \) \(195236339850544654368244\) \(\nu^{9}\mathstrut +\mathstrut \) \(1419614721106134505598337\) \(\nu^{8}\mathstrut +\mathstrut \) \(293196484616290703750216\) \(\nu^{7}\mathstrut -\mathstrut \) \(5367934888868512540382867\) \(\nu^{6}\mathstrut +\mathstrut \) \(1602889278627538618860671\) \(\nu^{5}\mathstrut +\mathstrut \) \(7880544085015915696563274\) \(\nu^{4}\mathstrut -\mathstrut \) \(4618652122090728228850804\) \(\nu^{3}\mathstrut -\mathstrut \) \(1480438220894389050010748\) \(\nu^{2}\mathstrut +\mathstrut \) \(319835708049430771492988\) \(\nu\mathstrut +\mathstrut \) \(39675257796285985779040\)\()/\)\(41\!\cdots\!72\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(6\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{4}\)\(=\)\(-\)\(3\) \(\beta_{15}\mathstrut +\mathstrut \) \(16\) \(\beta_{14}\mathstrut -\mathstrut \) \(20\) \(\beta_{13}\mathstrut -\mathstrut \) \(22\) \(\beta_{10}\mathstrut +\mathstrut \) \(3\) \(\beta_{9}\mathstrut -\mathstrut \) \(20\) \(\beta_{8}\mathstrut +\mathstrut \) \(3\) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(14\) \(\beta_{5}\mathstrut -\mathstrut \) \(18\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(18\) \(\beta_{2}\mathstrut +\mathstrut \) \(20\) \(\beta_{1}\mathstrut +\mathstrut \) \(72\)
\(\nu^{5}\)\(=\)\(-\)\(4\) \(\beta_{15}\mathstrut -\mathstrut \) \(2\) \(\beta_{14}\mathstrut -\mathstrut \) \(2\) \(\beta_{13}\mathstrut -\mathstrut \) \(13\) \(\beta_{12}\mathstrut +\mathstrut \) \(27\) \(\beta_{10}\mathstrut +\mathstrut \) \(20\) \(\beta_{9}\mathstrut +\mathstrut \) \(12\) \(\beta_{8}\mathstrut -\mathstrut \) \(43\) \(\beta_{7}\mathstrut -\mathstrut \) \(24\) \(\beta_{6}\mathstrut -\mathstrut \) \(9\) \(\beta_{5}\mathstrut +\mathstrut \) \(28\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(16\) \(\beta_{2}\mathstrut +\mathstrut \) \(143\) \(\beta_{1}\mathstrut +\mathstrut \) \(65\)
\(\nu^{6}\)\(=\)\(-\)\(96\) \(\beta_{15}\mathstrut +\mathstrut \) \(268\) \(\beta_{14}\mathstrut -\mathstrut \) \(381\) \(\beta_{13}\mathstrut +\mathstrut \) \(12\) \(\beta_{12}\mathstrut -\mathstrut \) \(22\) \(\beta_{11}\mathstrut -\mathstrut \) \(413\) \(\beta_{10}\mathstrut +\mathstrut \) \(68\) \(\beta_{9}\mathstrut -\mathstrut \) \(381\) \(\beta_{8}\mathstrut +\mathstrut \) \(70\) \(\beta_{7}\mathstrut -\mathstrut \) \(50\) \(\beta_{6}\mathstrut +\mathstrut \) \(214\) \(\beta_{5}\mathstrut -\mathstrut \) \(320\) \(\beta_{4}\mathstrut -\mathstrut \) \(34\) \(\beta_{3}\mathstrut -\mathstrut \) \(318\) \(\beta_{2}\mathstrut +\mathstrut \) \(345\) \(\beta_{1}\mathstrut +\mathstrut \) \(1014\)
\(\nu^{7}\)\(=\)\(-\)\(94\) \(\beta_{15}\mathstrut -\mathstrut \) \(82\) \(\beta_{14}\mathstrut -\mathstrut \) \(40\) \(\beta_{13}\mathstrut -\mathstrut \) \(131\) \(\beta_{12}\mathstrut +\mathstrut \) \(33\) \(\beta_{11}\mathstrut +\mathstrut \) \(593\) \(\beta_{10}\mathstrut +\mathstrut \) \(384\) \(\beta_{9}\mathstrut +\mathstrut \) \(385\) \(\beta_{8}\mathstrut -\mathstrut \) \(763\) \(\beta_{7}\mathstrut -\mathstrut \) \(484\) \(\beta_{6}\mathstrut -\mathstrut \) \(290\) \(\beta_{5}\mathstrut +\mathstrut \) \(591\) \(\beta_{4}\mathstrut -\mathstrut \) \(23\) \(\beta_{3}\mathstrut -\mathstrut \) \(200\) \(\beta_{2}\mathstrut +\mathstrut \) \(2058\) \(\beta_{1}\mathstrut +\mathstrut \) \(1084\)
\(\nu^{8}\)\(=\)\(-\)\(2185\) \(\beta_{15}\mathstrut +\mathstrut \) \(4636\) \(\beta_{14}\mathstrut -\mathstrut \) \(7081\) \(\beta_{13}\mathstrut +\mathstrut \) \(416\) \(\beta_{12}\mathstrut -\mathstrut \) \(736\) \(\beta_{11}\mathstrut -\mathstrut \) \(7451\) \(\beta_{10}\mathstrut +\mathstrut \) \(1312\) \(\beta_{9}\mathstrut -\mathstrut \) \(7090\) \(\beta_{8}\mathstrut +\mathstrut \) \(1421\) \(\beta_{7}\mathstrut -\mathstrut \) \(963\) \(\beta_{6}\mathstrut +\mathstrut \) \(3459\) \(\beta_{5}\mathstrut -\mathstrut \) \(5678\) \(\beta_{4}\mathstrut -\mathstrut \) \(761\) \(\beta_{3}\mathstrut -\mathstrut \) \(5594\) \(\beta_{2}\mathstrut +\mathstrut \) \(5735\) \(\beta_{1}\mathstrut +\mathstrut \) \(15516\)
\(\nu^{9}\)\(=\)\(-\)\(1625\) \(\beta_{15}\mathstrut -\mathstrut \) \(2295\) \(\beta_{14}\mathstrut -\mathstrut \) \(470\) \(\beta_{13}\mathstrut -\mathstrut \) \(1017\) \(\beta_{12}\mathstrut +\mathstrut \) \(1194\) \(\beta_{11}\mathstrut +\mathstrut \) \(12215\) \(\beta_{10}\mathstrut +\mathstrut \) \(7305\) \(\beta_{9}\mathstrut +\mathstrut \) \(9254\) \(\beta_{8}\mathstrut -\mathstrut \) \(12962\) \(\beta_{7}\mathstrut -\mathstrut \) \(9158\) \(\beta_{6}\mathstrut -\mathstrut \) \(6943\) \(\beta_{5}\mathstrut +\mathstrut \) \(11651\) \(\beta_{4}\mathstrut -\mathstrut \) \(485\) \(\beta_{3}\mathstrut -\mathstrut \) \(1982\) \(\beta_{2}\mathstrut +\mathstrut \) \(31367\) \(\beta_{1}\mathstrut +\mathstrut \) \(16430\)
\(\nu^{10}\)\(=\)\(-\)\(43998\) \(\beta_{15}\mathstrut +\mathstrut \) \(81591\) \(\beta_{14}\mathstrut -\mathstrut \) \(129666\) \(\beta_{13}\mathstrut +\mathstrut \) \(10392\) \(\beta_{12}\mathstrut -\mathstrut \) \(17679\) \(\beta_{11}\mathstrut -\mathstrut \) \(133030\) \(\beta_{10}\mathstrut +\mathstrut \) \(24157\) \(\beta_{9}\mathstrut -\mathstrut \) \(129934\) \(\beta_{8}\mathstrut +\mathstrut \) \(28119\) \(\beta_{7}\mathstrut -\mathstrut \) \(16928\) \(\beta_{6}\mathstrut +\mathstrut \) \(57781\) \(\beta_{5}\mathstrut -\mathstrut \) \(101022\) \(\beta_{4}\mathstrut -\mathstrut \) \(15125\) \(\beta_{3}\mathstrut -\mathstrut \) \(98180\) \(\beta_{2}\mathstrut +\mathstrut \) \(94302\) \(\beta_{1}\mathstrut +\mathstrut \) \(248780\)
\(\nu^{11}\)\(=\)\(-\)\(24501\) \(\beta_{15}\mathstrut -\mathstrut \) \(54903\) \(\beta_{14}\mathstrut -\mathstrut \) \(95\) \(\beta_{13}\mathstrut -\mathstrut \) \(1533\) \(\beta_{12}\mathstrut +\mathstrut \) \(29931\) \(\beta_{11}\mathstrut +\mathstrut \) \(244048\) \(\beta_{10}\mathstrut +\mathstrut \) \(136881\) \(\beta_{9}\mathstrut +\mathstrut \) \(200061\) \(\beta_{8}\mathstrut -\mathstrut \) \(219178\) \(\beta_{7}\mathstrut -\mathstrut \) \(167791\) \(\beta_{6}\mathstrut -\mathstrut \) \(148686\) \(\beta_{5}\mathstrut +\mathstrut \) \(224655\) \(\beta_{4}\mathstrut -\mathstrut \) \(10169\) \(\beta_{3}\mathstrut -\mathstrut \) \(8900\) \(\beta_{2}\mathstrut +\mathstrut \) \(494758\) \(\beta_{1}\mathstrut +\mathstrut \) \(235142\)
\(\nu^{12}\)\(=\)\(-\)\(837746\) \(\beta_{15}\mathstrut +\mathstrut \) \(1448480\) \(\beta_{14}\mathstrut -\mathstrut \) \(2352986\) \(\beta_{13}\mathstrut +\mathstrut \) \(229352\) \(\beta_{12}\mathstrut -\mathstrut \) \(373284\) \(\beta_{11}\mathstrut -\mathstrut \) \(2372878\) \(\beta_{10}\mathstrut +\mathstrut \) \(436378\) \(\beta_{9}\mathstrut -\mathstrut \) \(2360164\) \(\beta_{8}\mathstrut +\mathstrut \) \(551556\) \(\beta_{7}\mathstrut -\mathstrut \) \(284806\) \(\beta_{6}\mathstrut +\mathstrut \) \(986829\) \(\beta_{5}\mathstrut -\mathstrut \) \(1803570\) \(\beta_{4}\mathstrut -\mathstrut \) \(288483\) \(\beta_{3}\mathstrut -\mathstrut \) \(1722157\) \(\beta_{2}\mathstrut +\mathstrut \) \(1546720\) \(\beta_{1}\mathstrut +\mathstrut \) \(4102327\)
\(\nu^{13}\)\(=\)\(-\)\(331408\) \(\beta_{15}\mathstrut -\mathstrut \) \(1212971\) \(\beta_{14}\mathstrut +\mathstrut \) \(196253\) \(\beta_{13}\mathstrut +\mathstrut \) \(189349\) \(\beta_{12}\mathstrut +\mathstrut \) \(652250\) \(\beta_{11}\mathstrut +\mathstrut \) \(4785042\) \(\beta_{10}\mathstrut +\mathstrut \) \(2528884\) \(\beta_{9}\mathstrut +\mathstrut \) \(4105260\) \(\beta_{8}\mathstrut -\mathstrut \) \(3733385\) \(\beta_{7}\mathstrut -\mathstrut \) \(3017877\) \(\beta_{6}\mathstrut -\mathstrut \) \(3011680\) \(\beta_{5}\mathstrut +\mathstrut \) \(4293382\) \(\beta_{4}\mathstrut -\mathstrut \) \(208889\) \(\beta_{3}\mathstrut +\mathstrut \) \(274111\) \(\beta_{2}\mathstrut +\mathstrut \) \(7978295\) \(\beta_{1}\mathstrut +\mathstrut \) \(3180682\)
\(\nu^{14}\)\(=\)\(-\)\(15503403\) \(\beta_{15}\mathstrut +\mathstrut \) \(25829971\) \(\beta_{14}\mathstrut -\mathstrut \) \(42461733\) \(\beta_{13}\mathstrut +\mathstrut \) \(4762320\) \(\beta_{12}\mathstrut -\mathstrut \) \(7391012\) \(\beta_{11}\mathstrut -\mathstrut \) \(42417933\) \(\beta_{10}\mathstrut +\mathstrut \) \(7808976\) \(\beta_{9}\mathstrut -\mathstrut \) \(42674857\) \(\beta_{8}\mathstrut +\mathstrut \) \(10738719\) \(\beta_{7}\mathstrut -\mathstrut \) \(4664994\) \(\beta_{6}\mathstrut +\mathstrut \) \(17134871\) \(\beta_{5}\mathstrut -\mathstrut \) \(32299338\) \(\beta_{4}\mathstrut -\mathstrut \) \(5412584\) \(\beta_{3}\mathstrut -\mathstrut \) \(30230530\) \(\beta_{2}\mathstrut +\mathstrut \) \(25381774\) \(\beta_{1}\mathstrut +\mathstrut \) \(68876016\)
\(\nu^{15}\)\(=\)\(-\)\(3861844\) \(\beta_{15}\mathstrut -\mathstrut \) \(25604851\) \(\beta_{14}\mathstrut +\mathstrut \) \(7668573\) \(\beta_{13}\mathstrut +\mathstrut \) \(6343023\) \(\beta_{12}\mathstrut +\mathstrut \) \(13314661\) \(\beta_{11}\mathstrut +\mathstrut \) \(92598498\) \(\beta_{10}\mathstrut +\mathstrut \) \(46210791\) \(\beta_{9}\mathstrut +\mathstrut \) \(81763455\) \(\beta_{8}\mathstrut -\mathstrut \) \(64261165\) \(\beta_{7}\mathstrut -\mathstrut \) \(53681443\) \(\beta_{6}\mathstrut -\mathstrut \) \(59078108\) \(\beta_{5}\mathstrut +\mathstrut \) \(81652998\) \(\beta_{4}\mathstrut -\mathstrut \) \(4173985\) \(\beta_{3}\mathstrut +\mathstrut \) \(11987477\) \(\beta_{2}\mathstrut +\mathstrut \) \(130625667\) \(\beta_{1}\mathstrut +\mathstrut \) \(39683032\)

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
−4.28632
−3.33956
−2.66915
−2.58465
−1.83485
−0.309856
−0.123882
0.144883
0.282383
1.27509
1.38601
2.30095
3.01816
3.56414
4.02229
4.15437
0 −1.00000 0 −4.28632 0 1.70119 0 1.00000 0
1.2 0 −1.00000 0 −3.33956 0 −2.20985 0 1.00000 0
1.3 0 −1.00000 0 −2.66915 0 −2.40707 0 1.00000 0
1.4 0 −1.00000 0 −2.58465 0 −3.26725 0 1.00000 0
1.5 0 −1.00000 0 −1.83485 0 1.71276 0 1.00000 0
1.6 0 −1.00000 0 −0.309856 0 −1.15608 0 1.00000 0
1.7 0 −1.00000 0 −0.123882 0 3.12124 0 1.00000 0
1.8 0 −1.00000 0 0.144883 0 −2.05535 0 1.00000 0
1.9 0 −1.00000 0 0.282383 0 5.02462 0 1.00000 0
1.10 0 −1.00000 0 1.27509 0 1.05879 0 1.00000 0
1.11 0 −1.00000 0 1.38601 0 −3.43605 0 1.00000 0
1.12 0 −1.00000 0 2.30095 0 −4.14667 0 1.00000 0
1.13 0 −1.00000 0 3.01816 0 3.11628 0 1.00000 0
1.14 0 −1.00000 0 3.56414 0 −0.955621 0 1.00000 0
1.15 0 −1.00000 0 4.02229 0 4.04331 0 1.00000 0
1.16 0 −1.00000 0 4.15437 0 −4.14427 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(23\) \(-1\)
\(29\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8004))\):

\(T_{5}^{16} - \cdots\)
\(T_{7}^{16} + \cdots\)