Properties

Label 8004.2.a.j
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: \(\Z[a_1, \ldots, a_{17}]\)
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.

\(f(q)\) \(=\) \( q\) \(+ q^{3}\) \( + \beta_{1} q^{5} \) \( -\beta_{7} q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{3}\) \( + \beta_{1} q^{5} \) \( -\beta_{7} q^{7} \) \(+ q^{9}\) \( -\beta_{6} q^{11} \) \( + \beta_{4} q^{13} \) \( + \beta_{1} q^{15} \) \( + \beta_{9} q^{17} \) \( + ( 1 + \beta_{8} ) q^{19} \) \( -\beta_{7} q^{21} \) \(+ q^{23}\) \( + ( 2 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{11} - \beta_{14} + \beta_{15} ) q^{25} \) \(+ q^{27}\) \(- q^{29}\) \( + ( 1 - \beta_{7} + \beta_{10} ) q^{31} \) \( -\beta_{6} q^{33} \) \( + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} + \beta_{14} ) q^{35} \) \( + ( \beta_{1} + \beta_{14} ) q^{37} \) \( + \beta_{4} q^{39} \) \( + ( -\beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{10} + \beta_{11} + \beta_{13} - \beta_{14} ) q^{41} \) \( + ( 1 + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} + \beta_{14} ) q^{43} \) \( + \beta_{1} q^{45} \) \( + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{11} - \beta_{15} ) q^{47} \) \( + ( 2 - \beta_{1} - \beta_{2} - \beta_{8} - \beta_{11} - \beta_{13} ) q^{49} \) \( + \beta_{9} q^{51} \) \( + ( 1 - \beta_{8} - \beta_{11} - \beta_{12} ) q^{53} \) \( + ( 2 + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{12} - \beta_{14} ) q^{55} \) \( + ( 1 + \beta_{8} ) q^{57} \) \( + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{7} + \beta_{12} - 2 \beta_{14} ) q^{59} \) \( + ( 2 + \beta_{2} - \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{13} ) q^{61} \) \( -\beta_{7} q^{63} \) \( + ( 2 \beta_{1} + \beta_{5} + 2 \beta_{7} - \beta_{12} + 2 \beta_{14} ) q^{65} \) \( + ( 3 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{15} ) q^{67} \) \(+ q^{69}\) \( + ( \beta_{1} - \beta_{3} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{12} - \beta_{15} ) q^{71} \) \( + ( 2 - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{10} + \beta_{13} - \beta_{14} ) q^{73} \) \( + ( 2 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{11} - \beta_{14} + \beta_{15} ) q^{75} \) \( + ( 3 + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{77} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{79} \) \(+ q^{81}\) \( + ( \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{14} ) q^{83} \) \( + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{85} \) \(- q^{87}\) \( + ( \beta_{7} - \beta_{11} + \beta_{12} ) q^{89} \) \( + ( 1 - \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - \beta_{10} - 2 \beta_{12} + \beta_{13} ) q^{91} \) \( + ( 1 - \beta_{7} + \beta_{10} ) q^{93} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{13} ) q^{95} \) \( + ( 2 + \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{97} \) \( -\beta_{6} q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(16q \) \(\mathstrut +\mathstrut 16q^{3} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 16q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(16q \) \(\mathstrut +\mathstrut 16q^{3} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 16q^{9} \) \(\mathstrut +\mathstrut 5q^{11} \) \(\mathstrut +\mathstrut 6q^{13} \) \(\mathstrut +\mathstrut 3q^{15} \) \(\mathstrut +\mathstrut 3q^{17} \) \(\mathstrut +\mathstrut 11q^{19} \) \(\mathstrut +\mathstrut 4q^{21} \) \(\mathstrut +\mathstrut 16q^{23} \) \(\mathstrut +\mathstrut 27q^{25} \) \(\mathstrut +\mathstrut 16q^{27} \) \(\mathstrut -\mathstrut 16q^{29} \) \(\mathstrut +\mathstrut 14q^{31} \) \(\mathstrut +\mathstrut 5q^{33} \) \(\mathstrut +\mathstrut 11q^{35} \) \(\mathstrut +\mathstrut 4q^{37} \) \(\mathstrut +\mathstrut 6q^{39} \) \(\mathstrut +\mathstrut 11q^{41} \) \(\mathstrut +\mathstrut 23q^{43} \) \(\mathstrut +\mathstrut 3q^{45} \) \(\mathstrut -\mathstrut 2q^{47} \) \(\mathstrut +\mathstrut 34q^{49} \) \(\mathstrut +\mathstrut 3q^{51} \) \(\mathstrut +\mathstrut 19q^{53} \) \(\mathstrut +\mathstrut 31q^{55} \) \(\mathstrut +\mathstrut 11q^{57} \) \(\mathstrut +\mathstrut 32q^{59} \) \(\mathstrut +\mathstrut 19q^{61} \) \(\mathstrut +\mathstrut 4q^{63} \) \(\mathstrut +\mathstrut 6q^{65} \) \(\mathstrut +\mathstrut 33q^{67} \) \(\mathstrut +\mathstrut 16q^{69} \) \(\mathstrut -\mathstrut 5q^{71} \) \(\mathstrut +\mathstrut 23q^{73} \) \(\mathstrut +\mathstrut 27q^{75} \) \(\mathstrut +\mathstrut 42q^{77} \) \(\mathstrut +\mathstrut 24q^{79} \) \(\mathstrut +\mathstrut 16q^{81} \) \(\mathstrut +\mathstrut 7q^{83} \) \(\mathstrut -\mathstrut 16q^{87} \) \(\mathstrut -\mathstrut 2q^{89} \) \(\mathstrut +\mathstrut 25q^{91} \) \(\mathstrut +\mathstrut 14q^{93} \) \(\mathstrut +\mathstrut 7q^{95} \) \(\mathstrut +\mathstrut 33q^{97} \) \(\mathstrut +\mathstrut 5q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16}\mathstrut -\mathstrut \) \(3\) \(x^{15}\mathstrut -\mathstrut \) \(49\) \(x^{14}\mathstrut +\mathstrut \) \(130\) \(x^{13}\mathstrut +\mathstrut \) \(932\) \(x^{12}\mathstrut -\mathstrut \) \(2028\) \(x^{11}\mathstrut -\mathstrut \) \(8965\) \(x^{10}\mathstrut +\mathstrut \) \(14400\) \(x^{9}\mathstrut +\mathstrut \) \(46229\) \(x^{8}\mathstrut -\mathstrut \) \(47547\) \(x^{7}\mathstrut -\mathstrut \) \(122604\) \(x^{6}\mathstrut +\mathstrut \) \(65278\) \(x^{5}\mathstrut +\mathstrut \) \(151028\) \(x^{4}\mathstrut -\mathstrut \) \(17988\) \(x^{3}\mathstrut -\mathstrut \) \(67608\) \(x^{2}\mathstrut -\mathstrut \) \(8424\) \(x\mathstrut +\mathstrut \) \(3888\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(128089526929863711\) \(\nu^{15}\mathstrut -\mathstrut \) \(1780989647760093687\) \(\nu^{14}\mathstrut -\mathstrut \) \(2343590043301536281\) \(\nu^{13}\mathstrut +\mathstrut \) \(82428009307722432792\) \(\nu^{12}\mathstrut -\mathstrut \) \(39868293740417462800\) \(\nu^{11}\mathstrut -\mathstrut \) \(1434850612960902643364\) \(\nu^{10}\mathstrut +\mathstrut \) \(1061787358987892121509\) \(\nu^{9}\mathstrut +\mathstrut \) \(12059212146105833185194\) \(\nu^{8}\mathstrut -\mathstrut \) \(6849714181952244815581\) \(\nu^{7}\mathstrut -\mathstrut \) \(49998222735113343315999\) \(\nu^{6}\mathstrut +\mathstrut \) \(12943701821822810175386\) \(\nu^{5}\mathstrut +\mathstrut \) \(87862981358437561842546\) \(\nu^{4}\mathstrut -\mathstrut \) \(985493942779777495944\) \(\nu^{3}\mathstrut -\mathstrut \) \(44010301598073058173068\) \(\nu^{2}\mathstrut -\mathstrut \) \(5734576008998611033216\) \(\nu\mathstrut +\mathstrut \) \(1436096932911927060552\)\()/\)\(60\!\cdots\!08\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(958087900246403592\) \(\nu^{15}\mathstrut +\mathstrut \) \(2000567955267874333\) \(\nu^{14}\mathstrut +\mathstrut \) \(49357290400200766149\) \(\nu^{13}\mathstrut -\mathstrut \) \(88974714684163101079\) \(\nu^{12}\mathstrut -\mathstrut \) \(963378769466844341132\) \(\nu^{11}\mathstrut +\mathstrut \) \(1415048553601951450082\) \(\nu^{10}\mathstrut +\mathstrut \) \(8893034704374805484880\) \(\nu^{9}\mathstrut -\mathstrut \) \(9960991537155374353069\) \(\nu^{8}\mathstrut -\mathstrut \) \(38488851452268663931116\) \(\nu^{7}\mathstrut +\mathstrut \) \(31123533131595448591055\) \(\nu^{6}\mathstrut +\mathstrut \) \(63124294156284455139465\) \(\nu^{5}\mathstrut -\mathstrut \) \(39843524562808100614170\) \(\nu^{4}\mathstrut -\mathstrut \) \(17036408551707978644276\) \(\nu^{3}\mathstrut +\mathstrut \) \(2778605399574854232452\) \(\nu^{2}\mathstrut +\mathstrut \) \(4140098642180225184060\) \(\nu\mathstrut +\mathstrut \) \(10187796734063718568848\)\()/\)\(18\!\cdots\!24\)
\(\beta_{4}\)\(=\)\((\)\(4486141389040236929\) \(\nu^{15}\mathstrut -\mathstrut \) \(16191524997021342828\) \(\nu^{14}\mathstrut -\mathstrut \) \(211861740633534736190\) \(\nu^{13}\mathstrut +\mathstrut \) \(721401882595358991197\) \(\nu^{12}\mathstrut +\mathstrut \) \(3820388029937201339968\) \(\nu^{11}\mathstrut -\mathstrut \) \(11804652191031635223666\) \(\nu^{10}\mathstrut -\mathstrut \) \(34218304176590055106445\) \(\nu^{9}\mathstrut +\mathstrut \) \(91025874133171328629977\) \(\nu^{8}\mathstrut +\mathstrut \) \(160820231928040018855981\) \(\nu^{7}\mathstrut -\mathstrut \) \(348051881480837120172498\) \(\nu^{6}\mathstrut -\mathstrut \) \(374464283660322591966429\) \(\nu^{5}\mathstrut +\mathstrut \) \(631086737220708958112600\) \(\nu^{4}\mathstrut +\mathstrut \) \(373450758905607661475524\) \(\nu^{3}\mathstrut -\mathstrut \) \(436002491424203250589680\) \(\nu^{2}\mathstrut -\mathstrut \) \(119911680097050427614324\) \(\nu\mathstrut +\mathstrut \) \(62507843402239991683728\)\()/\)\(54\!\cdots\!72\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(2170353582856352493\) \(\nu^{15}\mathstrut +\mathstrut \) \(6394679216719266718\) \(\nu^{14}\mathstrut +\mathstrut \) \(105192616124931196248\) \(\nu^{13}\mathstrut -\mathstrut \) \(273011890888993264339\) \(\nu^{12}\mathstrut -\mathstrut \) \(1963933514692243560836\) \(\nu^{11}\mathstrut +\mathstrut \) \(4159658356999503153722\) \(\nu^{10}\mathstrut +\mathstrut \) \(18290876230917575384385\) \(\nu^{9}\mathstrut -\mathstrut \) \(28530527789909789588287\) \(\nu^{8}\mathstrut -\mathstrut \) \(88888441699606672234641\) \(\nu^{7}\mathstrut +\mathstrut \) \(90582649286250348855860\) \(\nu^{6}\mathstrut +\mathstrut \) \(210029308309103838088587\) \(\nu^{5}\mathstrut -\mathstrut \) \(128266188808560877480848\) \(\nu^{4}\mathstrut -\mathstrut \) \(209073282062352857065304\) \(\nu^{3}\mathstrut +\mathstrut \) \(66867424884872498482256\) \(\nu^{2}\mathstrut +\mathstrut \) \(63961903110921104602836\) \(\nu\mathstrut -\mathstrut \) \(6638130556491638529144\)\()/\)\(18\!\cdots\!24\)
\(\beta_{6}\)\(=\)\((\)\(1181647930064972053\) \(\nu^{15}\mathstrut -\mathstrut \) \(4488671540266881279\) \(\nu^{14}\mathstrut -\mathstrut \) \(53334067423612189198\) \(\nu^{13}\mathstrut +\mathstrut \) \(193579150520849398669\) \(\nu^{12}\mathstrut +\mathstrut \) \(901332302367398689427\) \(\nu^{11}\mathstrut -\mathstrut \) \(3015546321065278419096\) \(\nu^{10}\mathstrut -\mathstrut \) \(7391050976096937823363\) \(\nu^{9}\mathstrut +\mathstrut \) \(21665705786600716578792\) \(\nu^{8}\mathstrut +\mathstrut \) \(30526909733759416311410\) \(\nu^{7}\mathstrut -\mathstrut \) \(74930972648825634056871\) \(\nu^{6}\mathstrut -\mathstrut \) \(56119572043326164111031\) \(\nu^{5}\mathstrut +\mathstrut \) \(117837296847755457978037\) \(\nu^{4}\mathstrut +\mathstrut \) \(33444989655224619931886\) \(\nu^{3}\mathstrut -\mathstrut \) \(61721460041205402182268\) \(\nu^{2}\mathstrut -\mathstrut \) \(7863407220333139897548\) \(\nu\mathstrut +\mathstrut \) \(3287618491616138537532\)\()/\)\(90\!\cdots\!12\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(8599410738229651769\) \(\nu^{15}\mathstrut +\mathstrut \) \(31111659866916402345\) \(\nu^{14}\mathstrut +\mathstrut \) \(397851089569802580887\) \(\nu^{13}\mathstrut -\mathstrut \) \(1357434095246951691440\) \(\nu^{12}\mathstrut -\mathstrut \) \(6950409487206069267520\) \(\nu^{11}\mathstrut +\mathstrut \) \(21480555688367312916876\) \(\nu^{10}\mathstrut +\mathstrut \) \(59301366750646584632885\) \(\nu^{9}\mathstrut -\mathstrut \) \(157147318825198786079022\) \(\nu^{8}\mathstrut -\mathstrut \) \(257042371188445739715205\) \(\nu^{7}\mathstrut +\mathstrut \) \(552656578883574074980929\) \(\nu^{6}\mathstrut +\mathstrut \) \(511728863085525731674458\) \(\nu^{5}\mathstrut -\mathstrut \) \(875606750046801999312014\) \(\nu^{4}\mathstrut -\mathstrut \) \(366980592422283677211496\) \(\nu^{3}\mathstrut +\mathstrut \) \(458996610472572057501996\) \(\nu^{2}\mathstrut +\mathstrut \) \(63732795840155393435880\) \(\nu\mathstrut -\mathstrut \) \(41744720482603951993704\)\()/\)\(54\!\cdots\!72\)
\(\beta_{8}\)\(=\)\((\)\(10166089516631326123\) \(\nu^{15}\mathstrut -\mathstrut \) \(39636043673251564446\) \(\nu^{14}\mathstrut -\mathstrut \) \(463149040304201162734\) \(\nu^{13}\mathstrut +\mathstrut \) \(1741577168182876800871\) \(\nu^{12}\mathstrut +\mathstrut \) \(7937358719018169248978\) \(\nu^{11}\mathstrut -\mathstrut \) \(27920257611472853652366\) \(\nu^{10}\mathstrut -\mathstrut \) \(66526615038749754213403\) \(\nu^{9}\mathstrut +\mathstrut \) \(209053689477697038444789\) \(\nu^{8}\mathstrut +\mathstrut \) \(286804021630695488156393\) \(\nu^{7}\mathstrut -\mathstrut \) \(763778398499676259472184\) \(\nu^{6}\mathstrut -\mathstrut \) \(590218313888419955236599\) \(\nu^{5}\mathstrut +\mathstrut \) \(1275883347371748259536286\) \(\nu^{4}\mathstrut +\mathstrut \) \(501014971463436711617372\) \(\nu^{3}\mathstrut -\mathstrut \) \(734875152379202721919584\) \(\nu^{2}\mathstrut -\mathstrut \) \(188592356674960101516348\) \(\nu\mathstrut +\mathstrut \) \(75381322683826038993408\)\()/\)\(54\!\cdots\!72\)
\(\beta_{9}\)\(=\)\((\)\(3833924860594975845\) \(\nu^{15}\mathstrut -\mathstrut \) \(15405971411407983416\) \(\nu^{14}\mathstrut -\mathstrut \) \(174748567588173763710\) \(\nu^{13}\mathstrut +\mathstrut \) \(683105279786040092525\) \(\nu^{12}\mathstrut +\mathstrut \) \(3000960197217652805056\) \(\nu^{11}\mathstrut -\mathstrut \) \(11102524477032892698670\) \(\nu^{10}\mathstrut -\mathstrut \) \(25318495554936950579361\) \(\nu^{9}\mathstrut +\mathstrut \) \(84691399626259266906305\) \(\nu^{8}\mathstrut +\mathstrut \) \(111080659953374901792933\) \(\nu^{7}\mathstrut -\mathstrut \) \(315727483540994364481618\) \(\nu^{6}\mathstrut -\mathstrut \) \(237361474884041971348545\) \(\nu^{5}\mathstrut +\mathstrut \) \(530394989727383036599272\) \(\nu^{4}\mathstrut +\mathstrut \) \(206406504455500178785888\) \(\nu^{3}\mathstrut -\mathstrut \) \(290455416050807336719528\) \(\nu^{2}\mathstrut -\mathstrut \) \(58482275477158954969404\) \(\nu\mathstrut +\mathstrut \) \(24368330568281801524128\)\()/\)\(18\!\cdots\!24\)
\(\beta_{10}\)\(=\)\((\)\(6055690973521494323\) \(\nu^{15}\mathstrut -\mathstrut \) \(21065591888740470630\) \(\nu^{14}\mathstrut -\mathstrut \) \(282138336758875451579\) \(\nu^{13}\mathstrut +\mathstrut \) \(904938036516128835734\) \(\nu^{12}\mathstrut +\mathstrut \) \(5019613746164753586637\) \(\nu^{11}\mathstrut -\mathstrut \) \(13994381788245243532422\) \(\nu^{10}\mathstrut -\mathstrut \) \(44614067074995671927885\) \(\nu^{9}\mathstrut +\mathstrut \) \(99213383165647751861865\) \(\nu^{8}\mathstrut +\mathstrut \) \(210015545205224104585822\) \(\nu^{7}\mathstrut -\mathstrut \) \(335507138819282232469092\) \(\nu^{6}\mathstrut -\mathstrut \) \(494990400061097230473426\) \(\nu^{5}\mathstrut +\mathstrut \) \(513309222385173256168109\) \(\nu^{4}\mathstrut +\mathstrut \) \(516577122221256491326894\) \(\nu^{3}\mathstrut -\mathstrut \) \(270657506289607141947036\) \(\nu^{2}\mathstrut -\mathstrut \) \(186121404779846364836280\) \(\nu\mathstrut +\mathstrut \) \(18853135064703346138200\)\()/\)\(27\!\cdots\!36\)
\(\beta_{11}\)\(=\)\((\)\(2103742732645796968\) \(\nu^{15}\mathstrut -\mathstrut \) \(7587807286478686338\) \(\nu^{14}\mathstrut -\mathstrut \) \(98318269336507859293\) \(\nu^{13}\mathstrut +\mathstrut \) \(333589527564189827935\) \(\nu^{12}\mathstrut +\mathstrut \) \(1746467731155074504273\) \(\nu^{11}\mathstrut -\mathstrut \) \(5342141610410401144176\) \(\nu^{10}\mathstrut -\mathstrut \) \(15313513871554454343490\) \(\nu^{9}\mathstrut +\mathstrut \) \(39771223605111128968242\) \(\nu^{8}\mathstrut +\mathstrut \) \(69581428527495728527523\) \(\nu^{7}\mathstrut -\mathstrut \) \(143065840395346199349174\) \(\nu^{6}\mathstrut -\mathstrut \) \(151733582643722727860283\) \(\nu^{5}\mathstrut +\mathstrut \) \(230632231887950394137473\) \(\nu^{4}\mathstrut +\mathstrut \) \(131479917376481033077250\) \(\nu^{3}\mathstrut -\mathstrut \) \(118630530405786786024672\) \(\nu^{2}\mathstrut -\mathstrut \) \(36722337309588432352164\) \(\nu\mathstrut +\mathstrut \) \(7696294671096247442904\)\()/\)\(90\!\cdots\!12\)
\(\beta_{12}\)\(=\)\((\)\(15643727938311237652\) \(\nu^{15}\mathstrut -\mathstrut \) \(56547423258025579917\) \(\nu^{14}\mathstrut -\mathstrut \) \(736811796288429578563\) \(\nu^{13}\mathstrut +\mathstrut \) \(2511604996502047287625\) \(\nu^{12}\mathstrut +\mathstrut \) \(13220075965007561551802\) \(\nu^{11}\mathstrut -\mathstrut \) \(40821134148960918199182\) \(\nu^{10}\mathstrut -\mathstrut \) \(117349102552007926729696\) \(\nu^{9}\mathstrut +\mathstrut \) \(310045122176464699783689\) \(\nu^{8}\mathstrut +\mathstrut \) \(543132566223574961590886\) \(\nu^{7}\mathstrut -\mathstrut \) \(1144928931914839786999539\) \(\nu^{6}\mathstrut -\mathstrut \) \(1234507192266499635469335\) \(\nu^{5}\mathstrut +\mathstrut \) \(1908106630392058166642008\) \(\nu^{4}\mathstrut +\mathstrut \) \(1205285884230039076834004\) \(\nu^{3}\mathstrut -\mathstrut \) \(1029539354898820273092228\) \(\nu^{2}\mathstrut -\mathstrut \) \(431698814922534389321508\) \(\nu\mathstrut +\mathstrut \) \(60226142682101943655944\)\()/\)\(54\!\cdots\!72\)
\(\beta_{13}\)\(=\)\((\)\(16902750278691257515\) \(\nu^{15}\mathstrut -\mathstrut \) \(66100600868765845944\) \(\nu^{14}\mathstrut -\mathstrut \) \(765501985844103172942\) \(\nu^{13}\mathstrut +\mathstrut \) \(2878724443289246873407\) \(\nu^{12}\mathstrut +\mathstrut \) \(13050836157168499238396\) \(\nu^{11}\mathstrut -\mathstrut \) \(45539064879024801708138\) \(\nu^{10}\mathstrut -\mathstrut \) \(109453059483503010262423\) \(\nu^{9}\mathstrut +\mathstrut \) \(334561429000218091255911\) \(\nu^{8}\mathstrut +\mathstrut \) \(478818628546944868872983\) \(\nu^{7}\mathstrut -\mathstrut \) \(1190123853407848150127478\) \(\nu^{6}\mathstrut -\mathstrut \) \(1026586069984405062827043\) \(\nu^{5}\mathstrut +\mathstrut \) \(1916731753201095741369916\) \(\nu^{4}\mathstrut +\mathstrut \) \(932234121437022270337808\) \(\nu^{3}\mathstrut -\mathstrut \) \(1043599212580558264999584\) \(\nu^{2}\mathstrut -\mathstrut \) \(292239039888885554789364\) \(\nu\mathstrut +\mathstrut \) \(83307899572874489830104\)\()/\)\(54\!\cdots\!72\)
\(\beta_{14}\)\(=\)\((\)\(18310254666719483314\) \(\nu^{15}\mathstrut -\mathstrut \) \(64997796606289655715\) \(\nu^{14}\mathstrut -\mathstrut \) \(854944160891350069327\) \(\nu^{13}\mathstrut +\mathstrut \) \(2842406557507315492831\) \(\nu^{12}\mathstrut +\mathstrut \) \(15152969014719198018848\) \(\nu^{11}\mathstrut -\mathstrut \) \(45130633610269192266030\) \(\nu^{10}\mathstrut -\mathstrut \) \(132199513307530952648122\) \(\nu^{9}\mathstrut +\mathstrut \) \(331680872425345273285917\) \(\nu^{8}\mathstrut +\mathstrut \) \(594566658421572741407942\) \(\nu^{7}\mathstrut -\mathstrut \) \(1173217721169586183526145\) \(\nu^{6}\mathstrut -\mathstrut \) \(1274906221868961120611337\) \(\nu^{5}\mathstrut +\mathstrut \) \(1870018247056491778135546\) \(\nu^{4}\mathstrut +\mathstrut \) \(1105580567571758766871040\) \(\nu^{3}\mathstrut -\mathstrut \) \(982373741782699986430740\) \(\nu^{2}\mathstrut -\mathstrut \) \(330829154134196740304148\) \(\nu\mathstrut +\mathstrut \) \(73093928798098161025416\)\()/\)\(54\!\cdots\!72\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(28101973564733554969\) \(\nu^{15}\mathstrut +\mathstrut \) \(108682923995574038268\) \(\nu^{14}\mathstrut +\mathstrut \) \(1292569355600345279962\) \(\nu^{13}\mathstrut -\mathstrut \) \(4807355654391981949669\) \(\nu^{12}\mathstrut -\mathstrut \) \(22457115817186195403960\) \(\nu^{11}\mathstrut +\mathstrut \) \(77765457791517503993622\) \(\nu^{10}\mathstrut +\mathstrut \) \(191723502462768280879765\) \(\nu^{9}\mathstrut -\mathstrut \) \(588180818937536644168137\) \(\nu^{8}\mathstrut -\mathstrut \) \(848072989308638681641817\) \(\nu^{7}\mathstrut +\mathstrut \) \(2164766731936135949379246\) \(\nu^{6}\mathstrut +\mathstrut \) \(1814874225627403113533529\) \(\nu^{5}\mathstrut -\mathstrut \) \(3591447951250514794518160\) \(\nu^{4}\mathstrut -\mathstrut \) \(1594204885835619991562096\) \(\nu^{3}\mathstrut +\mathstrut \) \(1969768145562148310520360\) \(\nu^{2}\mathstrut +\mathstrut \) \(517658891675791265035020\) \(\nu\mathstrut -\mathstrut \) \(193169117447690193359208\)\()/\)\(54\!\cdots\!72\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(7\)
\(\nu^{3}\)\(=\)\(\beta_{13}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(13\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{4}\)\(=\)\(17\) \(\beta_{15}\mathstrut -\mathstrut \) \(16\) \(\beta_{14}\mathstrut -\mathstrut \) \(5\) \(\beta_{13}\mathstrut +\mathstrut \) \(16\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(21\) \(\beta_{8}\mathstrut -\mathstrut \) \(22\) \(\beta_{7}\mathstrut -\mathstrut \) \(18\) \(\beta_{5}\mathstrut +\mathstrut \) \(16\) \(\beta_{4}\mathstrut -\mathstrut \) \(14\) \(\beta_{3}\mathstrut +\mathstrut \) \(18\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(92\)
\(\nu^{5}\)\(=\)\(6\) \(\beta_{14}\mathstrut +\mathstrut \) \(27\) \(\beta_{13}\mathstrut +\mathstrut \) \(3\) \(\beta_{12}\mathstrut +\mathstrut \) \(42\) \(\beta_{11}\mathstrut -\mathstrut \) \(27\) \(\beta_{10}\mathstrut -\mathstrut \) \(43\) \(\beta_{9}\mathstrut +\mathstrut \) \(18\) \(\beta_{8}\mathstrut +\mathstrut \) \(45\) \(\beta_{7}\mathstrut -\mathstrut \) \(4\) \(\beta_{6}\mathstrut +\mathstrut \) \(5\) \(\beta_{5}\mathstrut -\mathstrut \) \(6\) \(\beta_{4}\mathstrut +\mathstrut \) \(8\) \(\beta_{3}\mathstrut -\mathstrut \) \(6\) \(\beta_{2}\mathstrut +\mathstrut \) \(198\) \(\beta_{1}\mathstrut +\mathstrut \) \(48\)
\(\nu^{6}\)\(=\)\(292\) \(\beta_{15}\mathstrut -\mathstrut \) \(263\) \(\beta_{14}\mathstrut -\mathstrut \) \(140\) \(\beta_{13}\mathstrut +\mathstrut \) \(20\) \(\beta_{12}\mathstrut +\mathstrut \) \(250\) \(\beta_{11}\mathstrut +\mathstrut \) \(27\) \(\beta_{10}\mathstrut -\mathstrut \) \(22\) \(\beta_{9}\mathstrut +\mathstrut \) \(385\) \(\beta_{8}\mathstrut -\mathstrut \) \(435\) \(\beta_{7}\mathstrut +\mathstrut \) \(18\) \(\beta_{6}\mathstrut -\mathstrut \) \(310\) \(\beta_{5}\mathstrut +\mathstrut \) \(250\) \(\beta_{4}\mathstrut -\mathstrut \) \(196\) \(\beta_{3}\mathstrut +\mathstrut \) \(312\) \(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(1422\)
\(\nu^{7}\)\(=\)\(-\)\(18\) \(\beta_{15}\mathstrut +\mathstrut \) \(218\) \(\beta_{14}\mathstrut +\mathstrut \) \(566\) \(\beta_{13}\mathstrut +\mathstrut \) \(97\) \(\beta_{12}\mathstrut +\mathstrut \) \(742\) \(\beta_{11}\mathstrut -\mathstrut \) \(578\) \(\beta_{10}\mathstrut -\mathstrut \) \(802\) \(\beta_{9}\mathstrut +\mathstrut \) \(301\) \(\beta_{8}\mathstrut +\mathstrut \) \(923\) \(\beta_{7}\mathstrut -\mathstrut \) \(114\) \(\beta_{6}\mathstrut +\mathstrut \) \(160\) \(\beta_{5}\mathstrut -\mathstrut \) \(222\) \(\beta_{4}\mathstrut +\mathstrut \) \(266\) \(\beta_{3}\mathstrut -\mathstrut \) \(204\) \(\beta_{2}\mathstrut +\mathstrut \) \(3255\) \(\beta_{1}\mathstrut +\mathstrut \) \(626\)
\(\nu^{8}\)\(=\)\(5100\) \(\beta_{15}\mathstrut -\mathstrut \) \(4491\) \(\beta_{14}\mathstrut -\mathstrut \) \(3030\) \(\beta_{13}\mathstrut +\mathstrut \) \(664\) \(\beta_{12}\mathstrut +\mathstrut \) \(3993\) \(\beta_{11}\mathstrut +\mathstrut \) \(555\) \(\beta_{10}\mathstrut -\mathstrut \) \(404\) \(\beta_{9}\mathstrut +\mathstrut \) \(6878\) \(\beta_{8}\mathstrut -\mathstrut \) \(8259\) \(\beta_{7}\mathstrut +\mathstrut \) \(636\) \(\beta_{6}\mathstrut -\mathstrut \) \(5445\) \(\beta_{5}\mathstrut +\mathstrut \) \(4089\) \(\beta_{4}\mathstrut -\mathstrut \) \(2905\) \(\beta_{3}\mathstrut +\mathstrut \) \(5467\) \(\beta_{2}\mathstrut -\mathstrut \) \(413\) \(\beta_{1}\mathstrut +\mathstrut \) \(23542\)
\(\nu^{9}\)\(=\)\(-\)\(819\) \(\beta_{15}\mathstrut +\mathstrut \) \(5553\) \(\beta_{14}\mathstrut +\mathstrut \) \(10895\) \(\beta_{13}\mathstrut +\mathstrut \) \(2315\) \(\beta_{12}\mathstrut +\mathstrut \) \(12537\) \(\beta_{11}\mathstrut -\mathstrut \) \(11449\) \(\beta_{10}\mathstrut -\mathstrut \) \(14473\) \(\beta_{9}\mathstrut +\mathstrut \) \(4797\) \(\beta_{8}\mathstrut +\mathstrut \) \(18229\) \(\beta_{7}\mathstrut -\mathstrut \) \(2238\) \(\beta_{6}\mathstrut +\mathstrut \) \(3752\) \(\beta_{5}\mathstrut -\mathstrut \) \(5663\) \(\beta_{4}\mathstrut +\mathstrut \) \(6505\) \(\beta_{3}\mathstrut -\mathstrut \) \(5024\) \(\beta_{2}\mathstrut +\mathstrut \) \(55717\) \(\beta_{1}\mathstrut +\mathstrut \) \(6965\)
\(\nu^{10}\)\(=\)\(90066\) \(\beta_{15}\mathstrut -\mathstrut \) \(78119\) \(\beta_{14}\mathstrut -\mathstrut \) \(60472\) \(\beta_{13}\mathstrut +\mathstrut \) \(15735\) \(\beta_{12}\mathstrut +\mathstrut \) \(65104\) \(\beta_{11}\mathstrut +\mathstrut \) \(10695\) \(\beta_{10}\mathstrut -\mathstrut \) \(6786\) \(\beta_{9}\mathstrut +\mathstrut \) \(122233\) \(\beta_{8}\mathstrut -\mathstrut \) \(153768\) \(\beta_{7}\mathstrut +\mathstrut \) \(16003\) \(\beta_{6}\mathstrut -\mathstrut \) \(97105\) \(\beta_{5}\mathstrut +\mathstrut \) \(69364\) \(\beta_{4}\mathstrut -\mathstrut \) \(45408\) \(\beta_{3}\mathstrut +\mathstrut \) \(96976\) \(\beta_{2}\mathstrut -\mathstrut \) \(15564\) \(\beta_{1}\mathstrut +\mathstrut \) \(403524\)
\(\nu^{11}\)\(=\)\(-\)\(24915\) \(\beta_{15}\mathstrut +\mathstrut \) \(124569\) \(\beta_{14}\mathstrut +\mathstrut \) \(202728\) \(\beta_{13}\mathstrut +\mathstrut \) \(48849\) \(\beta_{12}\mathstrut +\mathstrut \) \(209058\) \(\beta_{11}\mathstrut -\mathstrut \) \(218564\) \(\beta_{10}\mathstrut -\mathstrut \) \(259251\) \(\beta_{9}\mathstrut +\mathstrut \) \(72776\) \(\beta_{8}\mathstrut +\mathstrut \) \(352935\) \(\beta_{7}\mathstrut -\mathstrut \) \(38377\) \(\beta_{6}\mathstrut +\mathstrut \) \(80018\) \(\beta_{5}\mathstrut -\mathstrut \) \(125013\) \(\beta_{4}\mathstrut +\mathstrut \) \(141337\) \(\beta_{3}\mathstrut -\mathstrut \) \(109660\) \(\beta_{2}\mathstrut +\mathstrut \) \(975756\) \(\beta_{1}\mathstrut +\mathstrut \) \(55139\)
\(\nu^{12}\)\(=\)\(1602574\) \(\beta_{15}\mathstrut -\mathstrut \) \(1369817\) \(\beta_{14}\mathstrut -\mathstrut \) \(1167210\) \(\beta_{13}\mathstrut +\mathstrut \) \(327646\) \(\beta_{12}\mathstrut +\mathstrut \) \(1078384\) \(\beta_{11}\mathstrut +\mathstrut \) \(203896\) \(\beta_{10}\mathstrut -\mathstrut \) \(105992\) \(\beta_{9}\mathstrut +\mathstrut \) \(2172396\) \(\beta_{8}\mathstrut -\mathstrut \) \(2833616\) \(\beta_{7}\mathstrut +\mathstrut \) \(353182\) \(\beta_{6}\mathstrut -\mathstrut \) \(1745053\) \(\beta_{5}\mathstrut +\mathstrut \) \(1206681\) \(\beta_{4}\mathstrut -\mathstrut \) \(739735\) \(\beta_{3}\mathstrut +\mathstrut \) \(1735350\) \(\beta_{2}\mathstrut -\mathstrut \) \(414872\) \(\beta_{1}\mathstrut +\mathstrut \) \(7050086\)
\(\nu^{13}\)\(=\)\(-\)\(638200\) \(\beta_{15}\mathstrut +\mathstrut \) \(2634590\) \(\beta_{14}\mathstrut +\mathstrut \) \(3719888\) \(\beta_{13}\mathstrut +\mathstrut \) \(964071\) \(\beta_{12}\mathstrut +\mathstrut \) \(3477248\) \(\beta_{11}\mathstrut -\mathstrut \) \(4089626\) \(\beta_{10}\mathstrut -\mathstrut \) \(4643295\) \(\beta_{9}\mathstrut +\mathstrut \) \(1043482\) \(\beta_{8}\mathstrut +\mathstrut \) \(6758749\) \(\beta_{7}\mathstrut -\mathstrut \) \(620343\) \(\beta_{6}\mathstrut +\mathstrut \) \(1646430\) \(\beta_{5}\mathstrut -\mathstrut \) \(2570854\) \(\beta_{4}\mathstrut +\mathstrut \) \(2894128\) \(\beta_{3}\mathstrut -\mathstrut \) \(2258798\) \(\beta_{2}\mathstrut +\mathstrut \) \(17322068\) \(\beta_{1}\mathstrut -\mathstrut \) \(143779\)
\(\nu^{14}\)\(=\)\(28665480\) \(\beta_{15}\mathstrut -\mathstrut \) \(24111775\) \(\beta_{14}\mathstrut -\mathstrut \) \(22171021\) \(\beta_{13}\mathstrut +\mathstrut \) \(6402010\) \(\beta_{12}\mathstrut +\mathstrut \) \(18076637\) \(\beta_{11}\mathstrut +\mathstrut \) \(3905179\) \(\beta_{10}\mathstrut -\mathstrut \) \(1539030\) \(\beta_{9}\mathstrut +\mathstrut \) \(38666159\) \(\beta_{8}\mathstrut -\mathstrut \) \(51935720\) \(\beta_{7}\mathstrut +\mathstrut \) \(7302225\) \(\beta_{6}\mathstrut -\mathstrut \) \(31465887\) \(\beta_{5}\mathstrut +\mathstrut \) \(21356178\) \(\beta_{4}\mathstrut -\mathstrut \) \(12425342\) \(\beta_{3}\mathstrut +\mathstrut \) \(31239529\) \(\beta_{2}\mathstrut -\mathstrut \) \(9671295\) \(\beta_{1}\mathstrut +\mathstrut \) \(124565501\)
\(\nu^{15}\)\(=\)\(-\)\(14893147\) \(\beta_{15}\mathstrut +\mathstrut \) \(53944933\) \(\beta_{14}\mathstrut +\mathstrut \) \(67887756\) \(\beta_{13}\mathstrut +\mathstrut \) \(18262187\) \(\beta_{12}\mathstrut +\mathstrut \) \(57915120\) \(\beta_{11}\mathstrut -\mathstrut \) \(75643357\) \(\beta_{10}\mathstrut -\mathstrut \) \(83307736\) \(\beta_{9}\mathstrut +\mathstrut \) \(13834904\) \(\beta_{8}\mathstrut +\mathstrut \) \(128598654\) \(\beta_{7}\mathstrut -\mathstrut \) \(9769310\) \(\beta_{6}\mathstrut +\mathstrut \) \(33307089\) \(\beta_{5}\mathstrut -\mathstrut \) \(50852857\) \(\beta_{4}\mathstrut +\mathstrut \) \(57298007\) \(\beta_{3}\mathstrut -\mathstrut \) \(45079355\) \(\beta_{2}\mathstrut +\mathstrut \) \(310116137\) \(\beta_{1}\mathstrut -\mathstrut \) \(21232931\)

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
−4.29726
−3.12066
−2.45101
−2.34509
−1.86734
−0.900032
−0.812118
−0.434018
0.187793
0.897331
1.63007
1.72426
2.96949
3.50323
4.08902
4.22634
0 1.00000 0 −4.29726 0 3.54512 0 1.00000 0
1.2 0 1.00000 0 −3.12066 0 −0.760812 0 1.00000 0
1.3 0 1.00000 0 −2.45101 0 3.23607 0 1.00000 0
1.4 0 1.00000 0 −2.34509 0 −1.25807 0 1.00000 0
1.5 0 1.00000 0 −1.86734 0 −4.84365 0 1.00000 0
1.6 0 1.00000 0 −0.900032 0 0.226627 0 1.00000 0
1.7 0 1.00000 0 −0.812118 0 −4.94593 0 1.00000 0
1.8 0 1.00000 0 −0.434018 0 −2.25969 0 1.00000 0
1.9 0 1.00000 0 0.187793 0 3.11113 0 1.00000 0
1.10 0 1.00000 0 0.897331 0 3.81467 0 1.00000 0
1.11 0 1.00000 0 1.63007 0 −0.577524 0 1.00000 0
1.12 0 1.00000 0 1.72426 0 4.78449 0 1.00000 0
1.13 0 1.00000 0 2.96949 0 −3.61534 0 1.00000 0
1.14 0 1.00000 0 3.50323 0 0.271751 0 1.00000 0
1.15 0 1.00000 0 4.08902 0 2.57555 0 1.00000 0
1.16 0 1.00000 0 4.22634 0 0.695618 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8004))\):

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