Properties

Label 8036.2.a.t
Level 8036
Weight 2
Character orbit 8036.a
Self dual Yes
Analytic conductor 64.168
Analytic rank 0
Dimension 20
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8036.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1677830643\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{3} \) \( -\beta_{11} q^{5} \) \( + ( 1 + \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{3} \) \( -\beta_{11} q^{5} \) \( + ( 1 + \beta_{2} ) q^{9} \) \( + ( -1 + \beta_{1} - \beta_{7} ) q^{11} \) \( + ( 1 + \beta_{14} ) q^{13} \) \( + ( \beta_{1} + \beta_{6} - \beta_{11} - \beta_{15} ) q^{15} \) \( + \beta_{18} q^{17} \) \( + ( 1 - \beta_{5} - \beta_{11} - \beta_{15} + \beta_{16} ) q^{19} \) \( + ( -\beta_{2} + \beta_{4} - \beta_{6} + \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{16} ) q^{23} \) \( + ( \beta_{1} - \beta_{2} - \beta_{6} - \beta_{10} - 2 \beta_{11} - \beta_{12} + \beta_{18} - \beta_{19} ) q^{25} \) \( + ( 1 + \beta_{1} + \beta_{3} - \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{15} + \beta_{16} - \beta_{17} ) q^{27} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{10} + \beta_{11} + 2 \beta_{12} + \beta_{13} + \beta_{15} ) q^{29} \) \( + ( -\beta_{3} - \beta_{4} + \beta_{8} + \beta_{11} + \beta_{12} - \beta_{18} ) q^{31} \) \( + ( 2 + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{11} - \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} - \beta_{19} ) q^{33} \) \( + ( -\beta_{2} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{37} \) \( + ( 2 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{16} - \beta_{17} + \beta_{19} ) q^{39} \) \(- q^{41}\) \( + ( \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - \beta_{11} + \beta_{13} + \beta_{14} - 2 \beta_{15} + \beta_{16} + \beta_{19} ) q^{43} \) \( + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{16} - \beta_{19} ) q^{45} \) \( + ( \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} - \beta_{13} + \beta_{19} ) q^{47} \) \( + ( 1 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} + 2 \beta_{11} + \beta_{12} - \beta_{14} + 2 \beta_{15} - \beta_{16} - \beta_{18} - \beta_{19} ) q^{51} \) \( + ( -2 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{11} - 2 \beta_{12} - \beta_{13} - \beta_{16} + \beta_{17} + \beta_{18} ) q^{53} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{11} + \beta_{12} - \beta_{16} ) q^{55} \) \( + ( 1 + \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} - 2 \beta_{11} - 2 \beta_{12} + \beta_{15} + \beta_{18} ) q^{57} \) \( + ( \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} + 2 \beta_{11} + \beta_{12} + \beta_{13} + \beta_{15} - \beta_{16} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{59} \) \( + ( 3 - \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{13} - \beta_{15} + \beta_{17} + \beta_{19} ) q^{61} \) \( + ( -1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} + \beta_{13} + \beta_{15} - \beta_{16} + \beta_{18} + \beta_{19} ) q^{65} \) \( + ( 2 \beta_{1} + \beta_{2} - \beta_{6} - \beta_{7} + \beta_{11} + \beta_{12} - \beta_{18} ) q^{67} \) \( + ( 3 - 2 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + \beta_{12} - 2 \beta_{14} + \beta_{15} - \beta_{17} ) q^{69} \) \( + ( \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} - \beta_{16} - \beta_{18} + \beta_{19} ) q^{71} \) \( + ( 3 - \beta_{2} - \beta_{4} - \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} - 2 \beta_{17} ) q^{73} \) \( + ( 4 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} + \beta_{11} + \beta_{13} + \beta_{17} ) q^{75} \) \( + ( -3 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{13} - \beta_{14} ) q^{79} \) \( + ( 3 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} - 2 \beta_{10} - \beta_{11} - \beta_{13} - \beta_{16} + \beta_{17} - \beta_{18} - 2 \beta_{19} ) q^{81} \) \( + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{7} + \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} - \beta_{19} ) q^{83} \) \( + ( -2 + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} + \beta_{12} + \beta_{14} + \beta_{15} - \beta_{16} - \beta_{18} + \beta_{19} ) q^{85} \) \( + ( 2 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{6} + 3 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{14} - \beta_{15} + \beta_{16} + \beta_{19} ) q^{87} \) \( + ( 4 - \beta_{3} + \beta_{5} + 2 \beta_{11} + \beta_{12} + \beta_{14} + \beta_{15} + \beta_{17} - \beta_{19} ) q^{89} \) \( + ( \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} + 2 \beta_{11} + \beta_{12} + \beta_{15} - \beta_{16} - \beta_{18} - \beta_{19} ) q^{93} \) \( + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 4 \beta_{6} + \beta_{7} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} - \beta_{18} ) q^{95} \) \( + ( 1 - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{12} + \beta_{17} + \beta_{18} ) q^{97} \) \( + ( -2 + 4 \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{9} - 3 \beta_{10} - 4 \beta_{11} - 3 \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} + \beta_{18} - \beta_{19} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(20q \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 8q^{5} \) \(\mathstrut +\mathstrut 16q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(20q \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 8q^{5} \) \(\mathstrut +\mathstrut 16q^{9} \) \(\mathstrut -\mathstrut 8q^{11} \) \(\mathstrut +\mathstrut 12q^{13} \) \(\mathstrut +\mathstrut 8q^{15} \) \(\mathstrut +\mathstrut 8q^{17} \) \(\mathstrut +\mathstrut 24q^{19} \) \(\mathstrut +\mathstrut 8q^{23} \) \(\mathstrut +\mathstrut 20q^{25} \) \(\mathstrut +\mathstrut 16q^{27} \) \(\mathstrut -\mathstrut 12q^{29} \) \(\mathstrut +\mathstrut 44q^{33} \) \(\mathstrut +\mathstrut 12q^{37} \) \(\mathstrut +\mathstrut 12q^{39} \) \(\mathstrut -\mathstrut 20q^{41} \) \(\mathstrut +\mathstrut 4q^{43} \) \(\mathstrut +\mathstrut 40q^{45} \) \(\mathstrut +\mathstrut 4q^{47} \) \(\mathstrut +\mathstrut 4q^{51} \) \(\mathstrut -\mathstrut 12q^{53} \) \(\mathstrut -\mathstrut 16q^{55} \) \(\mathstrut +\mathstrut 28q^{57} \) \(\mathstrut +\mathstrut 16q^{59} \) \(\mathstrut +\mathstrut 68q^{61} \) \(\mathstrut -\mathstrut 8q^{65} \) \(\mathstrut +\mathstrut 4q^{67} \) \(\mathstrut +\mathstrut 32q^{69} \) \(\mathstrut +\mathstrut 8q^{71} \) \(\mathstrut +\mathstrut 48q^{73} \) \(\mathstrut +\mathstrut 60q^{75} \) \(\mathstrut -\mathstrut 20q^{79} \) \(\mathstrut +\mathstrut 32q^{81} \) \(\mathstrut -\mathstrut 8q^{83} \) \(\mathstrut -\mathstrut 28q^{85} \) \(\mathstrut +\mathstrut 60q^{89} \) \(\mathstrut -\mathstrut 16q^{93} \) \(\mathstrut +\mathstrut 20q^{95} \) \(\mathstrut +\mathstrut 40q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20}\mathstrut -\mathstrut \) \(4\) \(x^{19}\mathstrut -\mathstrut \) \(30\) \(x^{18}\mathstrut +\mathstrut \) \(128\) \(x^{17}\mathstrut +\mathstrut \) \(348\) \(x^{16}\mathstrut -\mathstrut \) \(1644\) \(x^{15}\mathstrut -\mathstrut \) \(1934\) \(x^{14}\mathstrut +\mathstrut \) \(10948\) \(x^{13}\mathstrut +\mathstrut \) \(4748\) \(x^{12}\mathstrut -\mathstrut \) \(40524\) \(x^{11}\mathstrut -\mathstrut \) \(220\) \(x^{10}\mathstrut +\mathstrut \) \(82500\) \(x^{9}\mathstrut -\mathstrut \) \(21992\) \(x^{8}\mathstrut -\mathstrut \) \(84720\) \(x^{7}\mathstrut +\mathstrut \) \(37544\) \(x^{6}\mathstrut +\mathstrut \) \(34656\) \(x^{5}\mathstrut -\mathstrut \) \(18823\) \(x^{4}\mathstrut -\mathstrut \) \(2276\) \(x^{3}\mathstrut +\mathstrut \) \(1130\) \(x^{2}\mathstrut +\mathstrut \) \(204\) \(x\mathstrut +\mathstrut \) \(9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\(693771077180\) \(\nu^{19}\mathstrut -\mathstrut \) \(2778645197913\) \(\nu^{18}\mathstrut -\mathstrut \) \(20725264188092\) \(\nu^{17}\mathstrut +\mathstrut \) \(88680236252730\) \(\nu^{16}\mathstrut +\mathstrut \) \(238604415918489\) \(\nu^{15}\mathstrut -\mathstrut \) \(1134655223791407\) \(\nu^{14}\mathstrut -\mathstrut \) \(1304931363722533\) \(\nu^{13}\mathstrut +\mathstrut \) \(7513711673122948\) \(\nu^{12}\mathstrut +\mathstrut \) \(3042707531442513\) \(\nu^{11}\mathstrut -\mathstrut \) \(27567693836690814\) \(\nu^{10}\mathstrut +\mathstrut \) \(811667273331286\) \(\nu^{9}\mathstrut +\mathstrut \) \(55269502351641049\) \(\nu^{8}\mathstrut -\mathstrut \) \(17304021386138605\) \(\nu^{7}\mathstrut -\mathstrut \) \(54976710977041287\) \(\nu^{6}\mathstrut +\mathstrut \) \(28198262762586312\) \(\nu^{5}\mathstrut +\mathstrut \) \(20402313321661661\) \(\nu^{4}\mathstrut -\mathstrut \) \(13852123920661924\) \(\nu^{3}\mathstrut -\mathstrut \) \(196305191328821\) \(\nu^{2}\mathstrut +\mathstrut \) \(738083784597603\) \(\nu\mathstrut +\mathstrut \) \(59952362841513\)\()/\)\(8907432980628\)
\(\beta_{4}\)\(=\)\((\)\(1567892632533415234\) \(\nu^{19}\mathstrut -\mathstrut \) \(6277851071446228846\) \(\nu^{18}\mathstrut -\mathstrut \) \(47074158679250201267\) \(\nu^{17}\mathstrut +\mathstrut \) \(200987274480806224623\) \(\nu^{16}\mathstrut +\mathstrut \) \(547064973518762056275\) \(\nu^{15}\mathstrut -\mathstrut \) \(2583040292296088360850\) \(\nu^{14}\mathstrut -\mathstrut \) \(3055149489179038791221\) \(\nu^{13}\mathstrut +\mathstrut \) \(17214742740867450156004\) \(\nu^{12}\mathstrut +\mathstrut \) \(7637148660948016713900\) \(\nu^{11}\mathstrut -\mathstrut \) \(63775711863953965507911\) \(\nu^{10}\mathstrut -\mathstrut \) \(1268832263838809426215\) \(\nu^{9}\mathstrut +\mathstrut \) \(129922976197309212858705\) \(\nu^{8}\mathstrut -\mathstrut \) \(31996738533222421425408\) \(\nu^{7}\mathstrut -\mathstrut \) \(133301820620661470919197\) \(\nu^{6}\mathstrut +\mathstrut \) \(55417302249307152484678\) \(\nu^{5}\mathstrut +\mathstrut \) \(53967376288662842383274\) \(\nu^{4}\mathstrut -\mathstrut \) \(27515570123960339467871\) \(\nu^{3}\mathstrut -\mathstrut \) \(2950763457356350118969\) \(\nu^{2}\mathstrut +\mathstrut \) \(1555738005610136792040\) \(\nu\mathstrut +\mathstrut \) \(190983464209108996263\)\()/\)\(7045699320779922348\)
\(\beta_{5}\)\(=\)\((\)\(2338757660528812226\) \(\nu^{19}\mathstrut -\mathstrut \) \(9262887357902091940\) \(\nu^{18}\mathstrut -\mathstrut \) \(70424247764763687101\) \(\nu^{17}\mathstrut +\mathstrut \) \(296231174702212346977\) \(\nu^{16}\mathstrut +\mathstrut \) \(822244089607423348755\) \(\nu^{15}\mathstrut -\mathstrut \) \(3801108458647015368204\) \(\nu^{14}\mathstrut -\mathstrut \) \(4630220852871028587469\) \(\nu^{13}\mathstrut +\mathstrut \) \(25275099625931947407732\) \(\nu^{12}\mathstrut +\mathstrut \) \(11815139014102216431238\) \(\nu^{11}\mathstrut -\mathstrut \) \(93326413146857308210869\) \(\nu^{10}\mathstrut -\mathstrut \) \(3140475534235154283977\) \(\nu^{9}\mathstrut +\mathstrut \) \(189167220552118841239433\) \(\nu^{8}\mathstrut -\mathstrut \) \(46054021734106426917342\) \(\nu^{7}\mathstrut -\mathstrut \) \(192459021916121267807269\) \(\nu^{6}\mathstrut +\mathstrut \) \(82085159614420114983234\) \(\nu^{5}\mathstrut +\mathstrut \) \(76448600437103562826964\) \(\nu^{4}\mathstrut -\mathstrut \) \(41370391953693733063937\) \(\nu^{3}\mathstrut -\mathstrut \) \(3540129341608367956689\) \(\nu^{2}\mathstrut +\mathstrut \) \(2351603977580673921478\) \(\nu\mathstrut +\mathstrut \) \(237506550933062053047\)\()/\)\(7045699320779922348\)
\(\beta_{6}\)\(=\)\((\)\(6661373649057\) \(\nu^{19}\mathstrut -\mathstrut \) \(27339265673408\) \(\nu^{18}\mathstrut -\mathstrut \) \(197062564273797\) \(\nu^{17}\mathstrut +\mathstrut \) \(873381091267388\) \(\nu^{16}\mathstrut +\mathstrut \) \(2229477793619106\) \(\nu^{15}\mathstrut -\mathstrut \) \(11189902694968197\) \(\nu^{14}\mathstrut -\mathstrut \) \(11748441413484831\) \(\nu^{13}\mathstrut +\mathstrut \) \(74233650073598569\) \(\nu^{12}\mathstrut +\mathstrut \) \(24114490412599688\) \(\nu^{11}\mathstrut -\mathstrut \) \(272988213285828381\) \(\nu^{10}\mathstrut +\mathstrut \) \(26102191633898274\) \(\nu^{9}\mathstrut +\mathstrut \) \(548751658773871214\) \(\nu^{8}\mathstrut -\mathstrut \) \(201766431641702593\) \(\nu^{7}\mathstrut -\mathstrut \) \(547047554161970435\) \(\nu^{6}\mathstrut +\mathstrut \) \(305071323257237295\) \(\nu^{5}\mathstrut +\mathstrut \) \(202658302419133080\) \(\nu^{4}\mathstrut -\mathstrut \) \(145789349517861572\) \(\nu^{3}\mathstrut -\mathstrut \) \(1309162504591808\) \(\nu^{2}\mathstrut +\mathstrut \) \(7723657414763231\) \(\nu\mathstrut +\mathstrut \) \(620836439810025\)\()/\)\(8907432980628\)
\(\beta_{7}\)\(=\)\((\)\(1049275988912470441\) \(\nu^{19}\mathstrut -\mathstrut \) \(4334271434248616952\) \(\nu^{18}\mathstrut -\mathstrut \) \(30930718070891675108\) \(\nu^{17}\mathstrut +\mathstrut \) \(138415002843547076609\) \(\nu^{16}\mathstrut +\mathstrut \) \(347664092105664073677\) \(\nu^{15}\mathstrut -\mathstrut \) \(1772518274441041104831\) \(\nu^{14}\mathstrut -\mathstrut \) \(1805440454952041490734\) \(\nu^{13}\mathstrut +\mathstrut \) \(11749994362571690048087\) \(\nu^{12}\mathstrut +\mathstrut \) \(3498116413657850939414\) \(\nu^{11}\mathstrut -\mathstrut \) \(43154858634515689790568\) \(\nu^{10}\mathstrut +\mathstrut \) \(5221070179901194950473\) \(\nu^{9}\mathstrut +\mathstrut \) \(86537534252271920654531\) \(\nu^{8}\mathstrut -\mathstrut \) \(34029275141695079410549\) \(\nu^{7}\mathstrut -\mathstrut \) \(85783168225253530008394\) \(\nu^{6}\mathstrut +\mathstrut \) \(50327088938842591365967\) \(\nu^{5}\mathstrut +\mathstrut \) \(31172753632958704540280\) \(\nu^{4}\mathstrut -\mathstrut \) \(23837476286048645159857\) \(\nu^{3}\mathstrut +\mathstrut \) \(137683984888334896715\) \(\nu^{2}\mathstrut +\mathstrut \) \(1234511067892056717935\) \(\nu\mathstrut +\mathstrut \) \(97226989580590920354\)\()/\)\(1006528474397131764\)
\(\beta_{8}\)\(=\)\((\)\(7576293982927979121\) \(\nu^{19}\mathstrut -\mathstrut \) \(31073635796039525650\) \(\nu^{18}\mathstrut -\mathstrut \) \(224074168773683551559\) \(\nu^{17}\mathstrut +\mathstrut \) \(992270980545485712530\) \(\nu^{16}\mathstrut +\mathstrut \) \(2533932755487489006282\) \(\nu^{15}\mathstrut -\mathstrut \) \(12705398753421574017087\) \(\nu^{14}\mathstrut -\mathstrut \) \(13339100027251658888871\) \(\nu^{13}\mathstrut +\mathstrut \) \(84209031937717862545217\) \(\nu^{12}\mathstrut +\mathstrut \) \(27270563865313329311810\) \(\nu^{11}\mathstrut -\mathstrut \) \(309205145409647896750815\) \(\nu^{10}\mathstrut +\mathstrut \) \(30268230823871797593612\) \(\nu^{9}\mathstrut +\mathstrut \) \(619876660982428449866914\) \(\nu^{8}\mathstrut -\mathstrut \) \(230588328077451908360595\) \(\nu^{7}\mathstrut -\mathstrut \) \(614376030427052055814071\) \(\nu^{6}\mathstrut +\mathstrut \) \(347741351999178390256559\) \(\nu^{5}\mathstrut +\mathstrut \) \(223334005665434926549898\) \(\nu^{4}\mathstrut -\mathstrut \) \(165493117300867307298002\) \(\nu^{3}\mathstrut +\mathstrut \) \(1023954424120529904920\) \(\nu^{2}\mathstrut +\mathstrut \) \(8395092563731168662665\) \(\nu\mathstrut +\mathstrut \) \(606826360165697573025\)\()/\)\(7045699320779922348\)
\(\beta_{9}\)\(=\)\((\)\(3972783117852940809\) \(\nu^{19}\mathstrut -\mathstrut \) \(16342846128512544173\) \(\nu^{18}\mathstrut -\mathstrut \) \(117320671722447788928\) \(\nu^{17}\mathstrut +\mathstrut \) \(521906431334303239643\) \(\nu^{16}\mathstrut +\mathstrut \) \(1322869278130680137994\) \(\nu^{15}\mathstrut -\mathstrut \) \(6683420464723393238238\) \(\nu^{14}\mathstrut -\mathstrut \) \(6916239982613718566301\) \(\nu^{13}\mathstrut +\mathstrut \) \(44305336615419018612667\) \(\nu^{12}\mathstrut +\mathstrut \) \(13746983099618899161683\) \(\nu^{11}\mathstrut -\mathstrut \) \(162744505113663406493844\) \(\nu^{10}\mathstrut +\mathstrut \) \(18087044949039785162649\) \(\nu^{9}\mathstrut +\mathstrut \) \(326510789581180900501616\) \(\nu^{8}\mathstrut -\mathstrut \) \(125963171001350030109214\) \(\nu^{7}\mathstrut -\mathstrut \) \(324232804881156119466287\) \(\nu^{6}\mathstrut +\mathstrut \) \(188504634894224254795827\) \(\nu^{5}\mathstrut +\mathstrut \) \(118753928893976335890255\) \(\nu^{4}\mathstrut -\mathstrut \) \(90208099965535405712597\) \(\nu^{3}\mathstrut -\mathstrut \) \(113306502518376301934\) \(\nu^{2}\mathstrut +\mathstrut \) \(4967650318979614979306\) \(\nu\mathstrut +\mathstrut \) \(392584122713171677173\)\()/\)\(3522849660389961174\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(2047786292360142275\) \(\nu^{19}\mathstrut +\mathstrut \) \(8453306354806390649\) \(\nu^{18}\mathstrut +\mathstrut \) \(60415489542289066625\) \(\nu^{17}\mathstrut -\mathstrut \) \(270070665887929370564\) \(\nu^{16}\mathstrut -\mathstrut \) \(680096131542676025598\) \(\nu^{15}\mathstrut +\mathstrut \) \(3460621390478264793615\) \(\nu^{14}\mathstrut +\mathstrut \) \(3543492889243349642386\) \(\nu^{13}\mathstrut -\mathstrut \) \(22962070603119437409617\) \(\nu^{12}\mathstrut -\mathstrut \) \(6956162240324603946659\) \(\nu^{11}\mathstrut +\mathstrut \) \(84464400288516335583243\) \(\nu^{10}\mathstrut -\mathstrut \) \(9732855884360263281280\) \(\nu^{9}\mathstrut -\mathstrut \) \(169854154542396568640853\) \(\nu^{8}\mathstrut +\mathstrut \) \(65568896606482854054425\) \(\nu^{7}\mathstrut +\mathstrut \) \(169432892648625526742675\) \(\nu^{6}\mathstrut -\mathstrut \) \(97578804334811055019104\) \(\nu^{5}\mathstrut -\mathstrut \) \(62888363036257849808159\) \(\nu^{4}\mathstrut +\mathstrut \) \(46575172685757514399803\) \(\nu^{3}\mathstrut +\mathstrut \) \(517517724798107266852\) \(\nu^{2}\mathstrut -\mathstrut \) \(2586400883220466564331\) \(\nu\mathstrut -\mathstrut \) \(221061471807410959458\)\()/\)\(1761424830194980587\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(8356272662031765421\) \(\nu^{19}\mathstrut +\mathstrut \) \(34202040308351693999\) \(\nu^{18}\mathstrut +\mathstrut \) \(247380582079000993072\) \(\nu^{17}\mathstrut -\mathstrut \) \(1092181398511623458951\) \(\nu^{16}\mathstrut -\mathstrut \) \(2802339477585508903152\) \(\nu^{15}\mathstrut +\mathstrut \) \(13984839560922314660460\) \(\nu^{14}\mathstrut +\mathstrut \) \(14807867321195626880525\) \(\nu^{13}\mathstrut -\mathstrut \) \(92690601820563143085051\) \(\nu^{12}\mathstrut -\mathstrut \) \(30703096120479886653617\) \(\nu^{11}\mathstrut +\mathstrut \) \(340365600693759113900646\) \(\nu^{10}\mathstrut -\mathstrut \) \(31116542935035778457819\) \(\nu^{9}\mathstrut -\mathstrut \) \(682452603951447075395134\) \(\nu^{8}\mathstrut +\mathstrut \) \(249783826314754461858204\) \(\nu^{7}\mathstrut +\mathstrut \) \(676734395670792749969903\) \(\nu^{6}\mathstrut -\mathstrut \) \(378819093668096210188203\) \(\nu^{5}\mathstrut -\mathstrut \) \(246474731929496125697395\) \(\nu^{4}\mathstrut +\mathstrut \) \(180338782805028994078333\) \(\nu^{3}\mathstrut -\mathstrut \) \(880481015365818753984\) \(\nu^{2}\mathstrut -\mathstrut \) \(8943772883317561833176\) \(\nu\mathstrut -\mathstrut \) \(667661155757941536921\)\()/\)\(7045699320779922348\)
\(\beta_{12}\)\(=\)\((\)\(10817735530643672823\) \(\nu^{19}\mathstrut -\mathstrut \) \(44004533606033770380\) \(\nu^{18}\mathstrut -\mathstrut \) \(321427966399360503593\) \(\nu^{17}\mathstrut +\mathstrut \) \(1406026625803290483046\) \(\nu^{16}\mathstrut +\mathstrut \) \(3665451754025270244468\) \(\nu^{15}\mathstrut -\mathstrut \) \(18018706532048828189043\) \(\nu^{14}\mathstrut -\mathstrut \) \(19652386941621891075063\) \(\nu^{13}\mathstrut +\mathstrut \) \(119580968135206563719079\) \(\nu^{12}\mathstrut +\mathstrut \) \(42952487320825958297536\) \(\nu^{11}\mathstrut -\mathstrut \) \(440051048998962173817801\) \(\nu^{10}\mathstrut +\mathstrut \) \(28483713793780474325016\) \(\nu^{9}\mathstrut +\mathstrut \) \(885892877952153906591132\) \(\nu^{8}\mathstrut -\mathstrut \) \(299668100923119368276203\) \(\nu^{7}\mathstrut -\mathstrut \) \(886587875671824490647875\) \(\nu^{6}\mathstrut +\mathstrut \) \(467029295366688741017921\) \(\nu^{5}\mathstrut +\mathstrut \) \(333187098654750974129600\) \(\nu^{4}\mathstrut -\mathstrut \) \(225340677579528815198014\) \(\nu^{3}\mathstrut -\mathstrut \) \(5004868760721322981926\) \(\nu^{2}\mathstrut +\mathstrut \) \(11973135027759598136281\) \(\nu\mathstrut +\mathstrut \) \(1039311719576074586769\)\()/\)\(7045699320779922348\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(5765110608764984219\) \(\nu^{19}\mathstrut +\mathstrut \) \(23774153729099490769\) \(\nu^{18}\mathstrut +\mathstrut \) \(170121063113279701288\) \(\nu^{17}\mathstrut -\mathstrut \) \(759339338545804496015\) \(\nu^{16}\mathstrut -\mathstrut \) \(1915869049907838609096\) \(\nu^{15}\mathstrut +\mathstrut \) \(9725925495596380955028\) \(\nu^{14}\mathstrut +\mathstrut \) \(9993361151771601407425\) \(\nu^{13}\mathstrut -\mathstrut \) \(64491433648002404390229\) \(\nu^{12}\mathstrut -\mathstrut \) \(19717931987831196765635\) \(\nu^{11}\mathstrut +\mathstrut \) \(236964510849042737550060\) \(\nu^{10}\mathstrut -\mathstrut \) \(26798116659087406948195\) \(\nu^{9}\mathstrut -\mathstrut \) \(475532953075794779675528\) \(\nu^{8}\mathstrut +\mathstrut \) \(182985431498476594699216\) \(\nu^{7}\mathstrut +\mathstrut \) \(472103252678673209814171\) \(\nu^{6}\mathstrut -\mathstrut \) \(272203242371124194650313\) \(\nu^{5}\mathstrut -\mathstrut \) \(172340908041091230121837\) \(\nu^{4}\mathstrut +\mathstrut \) \(129044554135267262643513\) \(\nu^{3}\mathstrut -\mathstrut \) \(395578244725691148738\) \(\nu^{2}\mathstrut -\mathstrut \) \(6622263148757199924236\) \(\nu\mathstrut -\mathstrut \) \(508723097353615002549\)\()/\)\(3522849660389961174\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(6253881503724915265\) \(\nu^{19}\mathstrut +\mathstrut \) \(25710747398443785873\) \(\nu^{18}\mathstrut +\mathstrut \) \(184817918043869208801\) \(\nu^{17}\mathstrut -\mathstrut \) \(821215965080328317242\) \(\nu^{16}\mathstrut -\mathstrut \) \(2087033475527660694129\) \(\nu^{15}\mathstrut +\mathstrut \) \(10518852033194202600960\) \(\nu^{14}\mathstrut +\mathstrut \) \(10952270309572306681580\) \(\nu^{13}\mathstrut -\mathstrut \) \(69753480654678076097837\) \(\nu^{12}\mathstrut -\mathstrut \) \(22128310064864091219739\) \(\nu^{11}\mathstrut +\mathstrut \) \(256332262566733625164623\) \(\nu^{10}\mathstrut -\mathstrut \) \(26358642828281747248718\) \(\nu^{9}\mathstrut -\mathstrut \) \(514564778294223298658171\) \(\nu^{8}\mathstrut +\mathstrut \) \(193026483803831537370618\) \(\nu^{7}\mathstrut +\mathstrut \) \(511321403595229157488586\) \(\nu^{6}\mathstrut -\mathstrut \) \(289687900253439035789069\) \(\nu^{5}\mathstrut -\mathstrut \) \(187322137052218898966457\) \(\nu^{4}\mathstrut +\mathstrut \) \(137687461555365045233094\) \(\nu^{3}\mathstrut -\mathstrut \) \(20762070898140250163\) \(\nu^{2}\mathstrut -\mathstrut \) \(7045871191970967584026\) \(\nu\mathstrut -\mathstrut \) \(562332436760355025320\)\()/\)\(3522849660389961174\)
\(\beta_{15}\)\(=\)\((\)\(14402308926297643223\) \(\nu^{19}\mathstrut -\mathstrut \) \(59134751184578330885\) \(\nu^{18}\mathstrut -\mathstrut \) \(425833794628053440962\) \(\nu^{17}\mathstrut +\mathstrut \) \(1888661390075851423815\) \(\nu^{16}\mathstrut +\mathstrut \) \(4812963651580171485534\) \(\nu^{15}\mathstrut -\mathstrut \) \(24189115890691711217376\) \(\nu^{14}\mathstrut -\mathstrut \) \(25306907459928782093989\) \(\nu^{13}\mathstrut +\mathstrut \) \(160381237404775884342221\) \(\nu^{12}\mathstrut +\mathstrut \) \(51513448344016378444467\) \(\nu^{11}\mathstrut -\mathstrut \) \(589251743429612557666656\) \(\nu^{10}\mathstrut +\mathstrut \) \(58703032263898179944719\) \(\nu^{9}\mathstrut +\mathstrut \) \(1182522905208002117547780\) \(\nu^{8}\mathstrut -\mathstrut \) \(440588282101994854095084\) \(\nu^{7}\mathstrut -\mathstrut \) \(1174535280429699515543167\) \(\nu^{6}\mathstrut +\mathstrut \) \(663248018168538332130329\) \(\nu^{5}\mathstrut +\mathstrut \) \(429824287705913693338525\) \(\nu^{4}\mathstrut -\mathstrut \) \(315556203763661953628365\) \(\nu^{3}\mathstrut +\mathstrut \) \(343760481474573044810\) \(\nu^{2}\mathstrut +\mathstrut \) \(16097180552095863216408\) \(\nu\mathstrut +\mathstrut \) \(1233943646077998910485\)\()/\)\(7045699320779922348\)
\(\beta_{16}\)\(=\)\((\)\(14645155040509963820\) \(\nu^{19}\mathstrut -\mathstrut \) \(59642851508940305086\) \(\nu^{18}\mathstrut -\mathstrut \) \(434846673787457040185\) \(\nu^{17}\mathstrut +\mathstrut \) \(1905411720410527343023\) \(\nu^{16}\mathstrut +\mathstrut \) \(4952712998950562047623\) \(\nu^{15}\mathstrut -\mathstrut \) \(24412967211259506148740\) \(\nu^{14}\mathstrut -\mathstrut \) \(26485294281914839965835\) \(\nu^{13}\mathstrut +\mathstrut \) \(161958144182482346030238\) \(\nu^{12}\mathstrut +\mathstrut \) \(57379737864677684063536\) \(\nu^{11}\mathstrut -\mathstrut \) \(595622318194356679148397\) \(\nu^{10}\mathstrut +\mathstrut \) \(41227112868720385687825\) \(\nu^{9}\mathstrut +\mathstrut \) \(1197582146857521448758737\) \(\nu^{8}\mathstrut -\mathstrut \) \(410474329960614160479102\) \(\nu^{7}\mathstrut -\mathstrut \) \(1194877112155448350622119\) \(\nu^{6}\mathstrut +\mathstrut \) \(635842430113801342281372\) \(\nu^{5}\mathstrut +\mathstrut \) \(444117927600826721299226\) \(\nu^{4}\mathstrut -\mathstrut \) \(304997202980737096627727\) \(\nu^{3}\mathstrut -\mathstrut \) \(3554413158860711081553\) \(\nu^{2}\mathstrut +\mathstrut \) \(15402396422972692780726\) \(\nu\mathstrut +\mathstrut \) \(1234585521558462095301\)\()/\)\(7045699320779922348\)
\(\beta_{17}\)\(=\)\((\)\(7438783877591216548\) \(\nu^{19}\mathstrut -\mathstrut \) \(30459378048999778099\) \(\nu^{18}\mathstrut -\mathstrut \) \(220200766266302213723\) \(\nu^{17}\mathstrut +\mathstrut \) \(972757216930066495875\) \(\nu^{16}\mathstrut +\mathstrut \) \(2494047510119311879458\) \(\nu^{15}\mathstrut -\mathstrut \) \(12457405771199322026019\) \(\nu^{14}\mathstrut -\mathstrut \) \(13173947522025534613790\) \(\nu^{13}\mathstrut +\mathstrut \) \(82584351503069563769488\) \(\nu^{12}\mathstrut +\mathstrut \) \(27275698837548390607893\) \(\nu^{11}\mathstrut -\mathstrut \) \(303357063011471710727439\) \(\nu^{10}\mathstrut +\mathstrut \) \(27923448863376964351637\) \(\nu^{9}\mathstrut +\mathstrut \) \(608602934352236377481532\) \(\nu^{8}\mathstrut -\mathstrut \) \(222874255585591719674739\) \(\nu^{7}\mathstrut -\mathstrut \) \(604220200791226407993500\) \(\nu^{6}\mathstrut +\mathstrut \) \(337911546657425909550568\) \(\nu^{5}\mathstrut +\mathstrut \) \(220873751730400091677289\) \(\nu^{4}\mathstrut -\mathstrut \) \(161021264521041724703129\) \(\nu^{3}\mathstrut +\mathstrut \) \(321155306165435922256\) \(\nu^{2}\mathstrut +\mathstrut \) \(8099183834371161701361\) \(\nu\mathstrut +\mathstrut \) \(617682916703533320726\)\()/\)\(3522849660389961174\)
\(\beta_{18}\)\(=\)\((\)\(-\)\(8138886928753770041\) \(\nu^{19}\mathstrut +\mathstrut \) \(33499956946277000400\) \(\nu^{18}\mathstrut +\mathstrut \) \(240323297279872800415\) \(\nu^{17}\mathstrut -\mathstrut \) \(1069812371476842628582\) \(\nu^{16}\mathstrut -\mathstrut \) \(2709600815139216588042\) \(\nu^{15}\mathstrut +\mathstrut \) \(13699510172653880436333\) \(\nu^{14}\mathstrut +\mathstrut \) \(14168966215045114054639\) \(\nu^{13}\mathstrut -\mathstrut \) \(90809978718344201679229\) \(\nu^{12}\mathstrut -\mathstrut \) \(28224573672325185426058\) \(\nu^{11}\mathstrut +\mathstrut \) \(333503362373389497835863\) \(\nu^{10}\mathstrut -\mathstrut \) \(36486799032282948761686\) \(\nu^{9}\mathstrut -\mathstrut \) \(668728059025171509164860\) \(\nu^{8}\mathstrut +\mathstrut \) \(255826579517142651304643\) \(\nu^{7}\mathstrut +\mathstrut \) \(662877997391653078852985\) \(\nu^{6}\mathstrut -\mathstrut \) \(381979586086710828489095\) \(\nu^{5}\mathstrut -\mathstrut \) \(240832709198133636810616\) \(\nu^{4}\mathstrut +\mathstrut \) \(181302476507307694588730\) \(\nu^{3}\mathstrut -\mathstrut \) \(1221891996360336135274\) \(\nu^{2}\mathstrut -\mathstrut \) \(9235939572202007595577\) \(\nu\mathstrut -\mathstrut \) \(699634305926123616771\)\()/\)\(3522849660389961174\)
\(\beta_{19}\)\(=\)\((\)\(9044475989189897959\) \(\nu^{19}\mathstrut -\mathstrut \) \(37172639749616416747\) \(\nu^{18}\mathstrut -\mathstrut \) \(267337054794349577899\) \(\nu^{17}\mathstrut +\mathstrut \) \(1187394854241713040880\) \(\nu^{16}\mathstrut +\mathstrut \) \(3019802809987906445103\) \(\nu^{15}\mathstrut -\mathstrut \) \(15210701217109761007314\) \(\nu^{14}\mathstrut -\mathstrut \) \(15856792379304385164044\) \(\nu^{13}\mathstrut +\mathstrut \) \(100882261019315843361703\) \(\nu^{12}\mathstrut +\mathstrut \) \(32101655974673253786169\) \(\nu^{11}\mathstrut -\mathstrut \) \(370826360968046011930875\) \(\nu^{10}\mathstrut +\mathstrut \) \(37844023940778257308124\) \(\nu^{9}\mathstrut +\mathstrut \) \(744805250324323650729783\) \(\nu^{8}\mathstrut -\mathstrut \) \(278897838495269416179886\) \(\nu^{7}\mathstrut -\mathstrut \) \(741081009576028826055736\) \(\nu^{6}\mathstrut +\mathstrut \) \(419276455516957466764863\) \(\nu^{5}\mathstrut +\mathstrut \) \(272777471804394727207141\) \(\nu^{4}\mathstrut -\mathstrut \) \(199891469248486819547262\) \(\nu^{3}\mathstrut -\mathstrut \) \(748060606226458740485\) \(\nu^{2}\mathstrut +\mathstrut \) \(10524879908602517748106\) \(\nu\mathstrut +\mathstrut \) \(829762774988511791610\)\()/\)\(3522849660389961174\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{17}\mathstrut +\mathstrut \) \(\beta_{16}\mathstrut -\mathstrut \) \(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(2\) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(2\) \(\beta_{19}\mathstrut -\mathstrut \) \(\beta_{18}\mathstrut +\mathstrut \) \(\beta_{17}\mathstrut -\mathstrut \) \(\beta_{16}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(2\) \(\beta_{10}\mathstrut -\mathstrut \) \(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\mathstrut +\mathstrut \) \(27\)
\(\nu^{5}\)\(=\)\(\beta_{19}\mathstrut -\mathstrut \) \(10\) \(\beta_{17}\mathstrut +\mathstrut \) \(10\) \(\beta_{16}\mathstrut -\mathstrut \) \(14\) \(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(12\) \(\beta_{12}\mathstrut -\mathstrut \) \(23\) \(\beta_{11}\mathstrut -\mathstrut \) \(12\) \(\beta_{10}\mathstrut -\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(3\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(11\) \(\beta_{3}\mathstrut +\mathstrut \) \(61\) \(\beta_{1}\mathstrut +\mathstrut \) \(14\)
\(\nu^{6}\)\(=\)\(-\)\(31\) \(\beta_{19}\mathstrut -\mathstrut \) \(16\) \(\beta_{18}\mathstrut +\mathstrut \) \(13\) \(\beta_{17}\mathstrut -\mathstrut \) \(18\) \(\beta_{16}\mathstrut +\mathstrut \) \(2\) \(\beta_{15}\mathstrut -\mathstrut \) \(4\) \(\beta_{14}\mathstrut -\mathstrut \) \(15\) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(18\) \(\beta_{11}\mathstrut -\mathstrut \) \(29\) \(\beta_{10}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(37\) \(\beta_{7}\mathstrut -\mathstrut \) \(8\) \(\beta_{6}\mathstrut +\mathstrut \) \(16\) \(\beta_{5}\mathstrut +\mathstrut \) \(13\) \(\beta_{4}\mathstrut -\mathstrut \) \(16\) \(\beta_{3}\mathstrut +\mathstrut \) \(85\) \(\beta_{2}\mathstrut +\mathstrut \) \(48\) \(\beta_{1}\mathstrut +\mathstrut \) \(211\)
\(\nu^{7}\)\(=\)\(9\) \(\beta_{19}\mathstrut +\mathstrut \) \(\beta_{18}\mathstrut -\mathstrut \) \(87\) \(\beta_{17}\mathstrut +\mathstrut \) \(91\) \(\beta_{16}\mathstrut -\mathstrut \) \(165\) \(\beta_{15}\mathstrut -\mathstrut \) \(14\) \(\beta_{14}\mathstrut +\mathstrut \) \(14\) \(\beta_{13}\mathstrut -\mathstrut \) \(130\) \(\beta_{12}\mathstrut -\mathstrut \) \(244\) \(\beta_{11}\mathstrut -\mathstrut \) \(134\) \(\beta_{10}\mathstrut -\mathstrut \) \(35\) \(\beta_{9}\mathstrut +\mathstrut \) \(37\) \(\beta_{8}\mathstrut -\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(67\) \(\beta_{6}\mathstrut +\mathstrut \) \(38\) \(\beta_{5}\mathstrut -\mathstrut \) \(17\) \(\beta_{4}\mathstrut +\mathstrut \) \(95\) \(\beta_{3}\mathstrut -\mathstrut \) \(8\) \(\beta_{2}\mathstrut +\mathstrut \) \(581\) \(\beta_{1}\mathstrut +\mathstrut \) \(151\)
\(\nu^{8}\)\(=\)\(-\)\(378\) \(\beta_{19}\mathstrut -\mathstrut \) \(197\) \(\beta_{18}\mathstrut +\mathstrut \) \(141\) \(\beta_{17}\mathstrut -\mathstrut \) \(233\) \(\beta_{16}\mathstrut +\mathstrut \) \(29\) \(\beta_{15}\mathstrut -\mathstrut \) \(78\) \(\beta_{14}\mathstrut -\mathstrut \) \(166\) \(\beta_{13}\mathstrut +\mathstrut \) \(15\) \(\beta_{12}\mathstrut -\mathstrut \) \(238\) \(\beta_{11}\mathstrut -\mathstrut \) \(343\) \(\beta_{10}\mathstrut -\mathstrut \) \(7\) \(\beta_{9}\mathstrut -\mathstrut \) \(28\) \(\beta_{8}\mathstrut -\mathstrut \) \(475\) \(\beta_{7}\mathstrut -\mathstrut \) \(25\) \(\beta_{6}\mathstrut +\mathstrut \) \(206\) \(\beta_{5}\mathstrut +\mathstrut \) \(131\) \(\beta_{4}\mathstrut -\mathstrut \) \(221\) \(\beta_{3}\mathstrut +\mathstrut \) \(824\) \(\beta_{2}\mathstrut +\mathstrut \) \(596\) \(\beta_{1}\mathstrut +\mathstrut \) \(1806\)
\(\nu^{9}\)\(=\)\(38\) \(\beta_{19}\mathstrut +\mathstrut \) \(7\) \(\beta_{18}\mathstrut -\mathstrut \) \(748\) \(\beta_{17}\mathstrut +\mathstrut \) \(827\) \(\beta_{16}\mathstrut -\mathstrut \) \(1839\) \(\beta_{15}\mathstrut -\mathstrut \) \(152\) \(\beta_{14}\mathstrut +\mathstrut \) \(178\) \(\beta_{13}\mathstrut -\mathstrut \) \(1361\) \(\beta_{12}\mathstrut -\mathstrut \) \(2555\) \(\beta_{11}\mathstrut -\mathstrut \) \(1473\) \(\beta_{10}\mathstrut -\mathstrut \) \(474\) \(\beta_{9}\mathstrut +\mathstrut \) \(518\) \(\beta_{8}\mathstrut -\mathstrut \) \(63\) \(\beta_{7}\mathstrut +\mathstrut \) \(1034\) \(\beta_{6}\mathstrut +\mathstrut \) \(519\) \(\beta_{5}\mathstrut -\mathstrut \) \(235\) \(\beta_{4}\mathstrut +\mathstrut \) \(726\) \(\beta_{3}\mathstrut -\mathstrut \) \(167\) \(\beta_{2}\mathstrut +\mathstrut \) \(5766\) \(\beta_{1}\mathstrut +\mathstrut \) \(1521\)
\(\nu^{10}\)\(=\)\(-\)\(4243\) \(\beta_{19}\mathstrut -\mathstrut \) \(2238\) \(\beta_{18}\mathstrut +\mathstrut \) \(1492\) \(\beta_{17}\mathstrut -\mathstrut \) \(2696\) \(\beta_{16}\mathstrut +\mathstrut \) \(233\) \(\beta_{15}\mathstrut -\mathstrut \) \(1046\) \(\beta_{14}\mathstrut -\mathstrut \) \(1629\) \(\beta_{13}\mathstrut +\mathstrut \) \(169\) \(\beta_{12}\mathstrut -\mathstrut \) \(2819\) \(\beta_{11}\mathstrut -\mathstrut \) \(3837\) \(\beta_{10}\mathstrut -\mathstrut \) \(236\) \(\beta_{9}\mathstrut -\mathstrut \) \(217\) \(\beta_{8}\mathstrut -\mathstrut \) \(5363\) \(\beta_{7}\mathstrut +\mathstrut \) \(473\) \(\beta_{6}\mathstrut +\mathstrut \) \(2452\) \(\beta_{5}\mathstrut +\mathstrut \) \(1173\) \(\beta_{4}\mathstrut -\mathstrut \) \(2886\) \(\beta_{3}\mathstrut +\mathstrut \) \(8071\) \(\beta_{2}\mathstrut +\mathstrut \) \(6869\) \(\beta_{1}\mathstrut +\mathstrut \) \(16429\)
\(\nu^{11}\)\(=\)\(-\)\(277\) \(\beta_{19}\mathstrut -\mathstrut \) \(133\) \(\beta_{18}\mathstrut -\mathstrut \) \(6518\) \(\beta_{17}\mathstrut +\mathstrut \) \(7570\) \(\beta_{16}\mathstrut -\mathstrut \) \(19975\) \(\beta_{15}\mathstrut -\mathstrut \) \(1542\) \(\beta_{14}\mathstrut +\mathstrut \) \(2282\) \(\beta_{13}\mathstrut -\mathstrut \) \(13950\) \(\beta_{12}\mathstrut -\mathstrut \) \(26591\) \(\beta_{11}\mathstrut -\mathstrut \) \(16019\) \(\beta_{10}\mathstrut -\mathstrut \) \(5916\) \(\beta_{9}\mathstrut +\mathstrut \) \(6526\) \(\beta_{8}\mathstrut -\mathstrut \) \(896\) \(\beta_{7}\mathstrut +\mathstrut \) \(13810\) \(\beta_{6}\mathstrut +\mathstrut \) \(6318\) \(\beta_{5}\mathstrut -\mathstrut \) \(3104\) \(\beta_{4}\mathstrut +\mathstrut \) \(4784\) \(\beta_{3}\mathstrut -\mathstrut \) \(2376\) \(\beta_{2}\mathstrut +\mathstrut \) \(58474\) \(\beta_{1}\mathstrut +\mathstrut \) \(15151\)
\(\nu^{12}\)\(=\)\(-\)\(45923\) \(\beta_{19}\mathstrut -\mathstrut \) \(24655\) \(\beta_{18}\mathstrut +\mathstrut \) \(15782\) \(\beta_{17}\mathstrut -\mathstrut \) \(29677\) \(\beta_{16}\mathstrut +\mathstrut \) \(569\) \(\beta_{15}\mathstrut -\mathstrut \) \(12100\) \(\beta_{14}\mathstrut -\mathstrut \) \(14928\) \(\beta_{13}\mathstrut +\mathstrut \) \(1815\) \(\beta_{12}\mathstrut -\mathstrut \) \(31788\) \(\beta_{11}\mathstrut -\mathstrut \) \(42101\) \(\beta_{10}\mathstrut -\mathstrut \) \(4876\) \(\beta_{9}\mathstrut -\mathstrut \) \(417\) \(\beta_{8}\mathstrut -\mathstrut \) \(57145\) \(\beta_{7}\mathstrut +\mathstrut \) \(12998\) \(\beta_{6}\mathstrut +\mathstrut \) \(28063\) \(\beta_{5}\mathstrut +\mathstrut \) \(9426\) \(\beta_{4}\mathstrut -\mathstrut \) \(36181\) \(\beta_{3}\mathstrut +\mathstrut \) \(79438\) \(\beta_{2}\mathstrut +\mathstrut \) \(77074\) \(\beta_{1}\mathstrut +\mathstrut \) \(155705\)
\(\nu^{13}\)\(=\)\(-\)\(9828\) \(\beta_{19}\mathstrut -\mathstrut \) \(5006\) \(\beta_{18}\mathstrut -\mathstrut \) \(57756\) \(\beta_{17}\mathstrut +\mathstrut \) \(69586\) \(\beta_{16}\mathstrut -\mathstrut \) \(214052\) \(\beta_{15}\mathstrut -\mathstrut \) \(15246\) \(\beta_{14}\mathstrut +\mathstrut \) \(29053\) \(\beta_{13}\mathstrut -\mathstrut \) \(140895\) \(\beta_{12}\mathstrut -\mathstrut \) \(275443\) \(\beta_{11}\mathstrut -\mathstrut \) \(172867\) \(\beta_{10}\mathstrut -\mathstrut \) \(71231\) \(\beta_{9}\mathstrut +\mathstrut \) \(78089\) \(\beta_{8}\mathstrut -\mathstrut \) \(10859\) \(\beta_{7}\mathstrut +\mathstrut \) \(171522\) \(\beta_{6}\mathstrut +\mathstrut \) \(73218\) \(\beta_{5}\mathstrut -\mathstrut \) \(40130\) \(\beta_{4}\mathstrut +\mathstrut \) \(22103\) \(\beta_{3}\mathstrut -\mathstrut \) \(28888\) \(\beta_{2}\mathstrut +\mathstrut \) \(600588\) \(\beta_{1}\mathstrut +\mathstrut \) \(152541\)
\(\nu^{14}\)\(=\)\(-\)\(488333\) \(\beta_{19}\mathstrut -\mathstrut \) \(268379\) \(\beta_{18}\mathstrut +\mathstrut \) \(166948\) \(\beta_{17}\mathstrut -\mathstrut \) \(318472\) \(\beta_{16}\mathstrut -\mathstrut \) \(20717\) \(\beta_{15}\mathstrut -\mathstrut \) \(130109\) \(\beta_{14}\mathstrut -\mathstrut \) \(129557\) \(\beta_{13}\mathstrut +\mathstrut \) \(20182\) \(\beta_{12}\mathstrut -\mathstrut \) \(349438\) \(\beta_{11}\mathstrut -\mathstrut \) \(458410\) \(\beta_{10}\mathstrut -\mathstrut \) \(80985\) \(\beta_{9}\mathstrut +\mathstrut \) \(21669\) \(\beta_{8}\mathstrut -\mathstrut \) \(591235\) \(\beta_{7}\mathstrut +\mathstrut \) \(219551\) \(\beta_{6}\mathstrut +\mathstrut \) \(314111\) \(\beta_{5}\mathstrut +\mathstrut \) \(64153\) \(\beta_{4}\mathstrut -\mathstrut \) \(439351\) \(\beta_{3}\mathstrut +\mathstrut \) \(784451\) \(\beta_{2}\mathstrut +\mathstrut \) \(855988\) \(\beta_{1}\mathstrut +\mathstrut \) \(1517375\)
\(\nu^{15}\)\(=\)\(-\)\(171630\) \(\beta_{19}\mathstrut -\mathstrut \) \(101254\) \(\beta_{18}\mathstrut -\mathstrut \) \(518951\) \(\beta_{17}\mathstrut +\mathstrut \) \(639870\) \(\beta_{16}\mathstrut -\mathstrut \) \(2276953\) \(\beta_{15}\mathstrut -\mathstrut \) \(148336\) \(\beta_{14}\mathstrut +\mathstrut \) \(362583\) \(\beta_{13}\mathstrut -\mathstrut \) \(1408178\) \(\beta_{12}\mathstrut -\mathstrut \) \(2843005\) \(\beta_{11}\mathstrut -\mathstrut \) \(1856361\) \(\beta_{10}\mathstrut -\mathstrut \) \(840656\) \(\beta_{9}\mathstrut +\mathstrut \) \(908234\) \(\beta_{8}\mathstrut -\mathstrut \) \(121133\) \(\beta_{7}\mathstrut +\mathstrut \) \(2043303\) \(\beta_{6}\mathstrut +\mathstrut \) \(828805\) \(\beta_{5}\mathstrut -\mathstrut \) \(508524\) \(\beta_{4}\mathstrut -\mathstrut \) \(43294\) \(\beta_{3}\mathstrut -\mathstrut \) \(323319\) \(\beta_{2}\mathstrut +\mathstrut \) \(6219868\) \(\beta_{1}\mathstrut +\mathstrut \) \(1564378\)
\(\nu^{16}\)\(=\)\(-\)\(5147120\) \(\beta_{19}\mathstrut -\mathstrut \) \(2909362\) \(\beta_{18}\mathstrut +\mathstrut \) \(1762319\) \(\beta_{17}\mathstrut -\mathstrut \) \(3370083\) \(\beta_{16}\mathstrut -\mathstrut \) \(536726\) \(\beta_{15}\mathstrut -\mathstrut \) \(1343624\) \(\beta_{14}\mathstrut -\mathstrut \) \(1059434\) \(\beta_{13}\mathstrut +\mathstrut \) \(236052\) \(\beta_{12}\mathstrut -\mathstrut \) \(3785785\) \(\beta_{11}\mathstrut -\mathstrut \) \(4974420\) \(\beta_{10}\mathstrut -\mathstrut \) \(1191284\) \(\beta_{9}\mathstrut +\mathstrut \) \(537912\) \(\beta_{8}\mathstrut -\mathstrut \) \(6019735\) \(\beta_{7}\mathstrut +\mathstrut \) \(3143784\) \(\beta_{6}\mathstrut +\mathstrut \) \(3469453\) \(\beta_{5}\mathstrut +\mathstrut \) \(281084\) \(\beta_{4}\mathstrut -\mathstrut \) \(5205679\) \(\beta_{3}\mathstrut +\mathstrut \) \(7769680\) \(\beta_{2}\mathstrut +\mathstrut \) \(9469298\) \(\beta_{1}\mathstrut +\mathstrut \) \(15077038\)
\(\nu^{17}\)\(=\)\(-\)\(2477970\) \(\beta_{19}\mathstrut -\mathstrut \) \(1649754\) \(\beta_{18}\mathstrut -\mathstrut \) \(4708169\) \(\beta_{17}\mathstrut +\mathstrut \) \(5865640\) \(\beta_{16}\mathstrut -\mathstrut \) \(24125681\) \(\beta_{15}\mathstrut -\mathstrut \) \(1422288\) \(\beta_{14}\mathstrut +\mathstrut \) \(4424234\) \(\beta_{13}\mathstrut -\mathstrut \) \(13966392\) \(\beta_{12}\mathstrut -\mathstrut \) \(29273289\) \(\beta_{11}\mathstrut -\mathstrut \) \(19880479\) \(\beta_{10}\mathstrut -\mathstrut \) \(9789062\) \(\beta_{9}\mathstrut +\mathstrut \) \(10384354\) \(\beta_{8}\mathstrut -\mathstrut \) \(1288423\) \(\beta_{7}\mathstrut +\mathstrut \) \(23714425\) \(\beta_{6}\mathstrut +\mathstrut \) \(9266739\) \(\beta_{5}\mathstrut -\mathstrut \) \(6315612\) \(\beta_{4}\mathstrut -\mathstrut \) \(3152036\) \(\beta_{3}\mathstrut -\mathstrut \) \(3439771\) \(\beta_{2}\mathstrut +\mathstrut \) \(64793238\) \(\beta_{1}\mathstrut +\mathstrut \) \(16359136\)
\(\nu^{18}\)\(=\)\(-\)\(54018690\) \(\beta_{19}\mathstrut -\mathstrut \) \(31510453\) \(\beta_{18}\mathstrut +\mathstrut \) \(18543655\) \(\beta_{17}\mathstrut -\mathstrut \) \(35377845\) \(\beta_{16}\mathstrut -\mathstrut \) \(9178237\) \(\beta_{15}\mathstrut -\mathstrut \) \(13545029\) \(\beta_{14}\mathstrut -\mathstrut \) \(7953251\) \(\beta_{13}\mathstrut +\mathstrut \) \(2855543\) \(\beta_{12}\mathstrut -\mathstrut \) \(40659906\) \(\beta_{11}\mathstrut -\mathstrut \) \(53893287\) \(\beta_{10}\mathstrut -\mathstrut \) \(16244890\) \(\beta_{9}\mathstrut +\mathstrut \) \(9093699\) \(\beta_{8}\mathstrut -\mathstrut \) \(60737222\) \(\beta_{7}\mathstrut +\mathstrut \) \(41391847\) \(\beta_{6}\mathstrut +\mathstrut \) \(38011629\) \(\beta_{5}\mathstrut -\mathstrut \) \(1322899\) \(\beta_{4}\mathstrut -\mathstrut \) \(60534715\) \(\beta_{3}\mathstrut +\mathstrut \) \(77183689\) \(\beta_{2}\mathstrut +\mathstrut \) \(104595921\) \(\beta_{1}\mathstrut +\mathstrut \) \(151922809\)
\(\nu^{19}\)\(=\)\(-\)\(32748466\) \(\beta_{19}\mathstrut -\mathstrut \) \(23979501\) \(\beta_{18}\mathstrut -\mathstrut \) \(42949033\) \(\beta_{17}\mathstrut +\mathstrut \) \(53432566\) \(\beta_{16}\mathstrut -\mathstrut \) \(255133669\) \(\beta_{15}\mathstrut -\mathstrut \) \(13432468\) \(\beta_{14}\mathstrut +\mathstrut \) \(52891047\) \(\beta_{13}\mathstrut -\mathstrut \) \(137716944\) \(\beta_{12}\mathstrut -\mathstrut \) \(301000564\) \(\beta_{11}\mathstrut -\mathstrut \) \(212638933\) \(\beta_{10}\mathstrut -\mathstrut \) \(112832331\) \(\beta_{9}\mathstrut +\mathstrut \) \(117414503\) \(\beta_{8}\mathstrut -\mathstrut \) \(13323065\) \(\beta_{7}\mathstrut +\mathstrut \) \(270495307\) \(\beta_{6}\mathstrut +\mathstrut \) \(102870573\) \(\beta_{5}\mathstrut -\mathstrut \) \(76968461\) \(\beta_{4}\mathstrut -\mathstrut \) \(59643231\) \(\beta_{3}\mathstrut -\mathstrut \) \(35332232\) \(\beta_{2}\mathstrut +\mathstrut \) \(677975640\) \(\beta_{1}\mathstrut +\mathstrut \) \(174055949\)

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
−3.15720
−2.45978
−2.32542
−2.15199
−1.77641
−1.37724
−0.915292
−0.146508
−0.103746
−0.0954148
0.476040
0.538932
1.06664
1.15563
1.53485
2.15119
2.36764
2.83122
3.09326
3.29362
0 −3.15720 0 2.22019 0 0 0 6.96793 0
1.2 0 −2.45978 0 −2.10149 0 0 0 3.05053 0
1.3 0 −2.32542 0 1.16439 0 0 0 2.40759 0
1.4 0 −2.15199 0 −0.0591445 0 0 0 1.63107 0
1.5 0 −1.77641 0 −0.716761 0 0 0 0.155650 0
1.6 0 −1.37724 0 3.83294 0 0 0 −1.10321 0
1.7 0 −0.915292 0 −1.31004 0 0 0 −2.16224 0
1.8 0 −0.146508 0 3.56882 0 0 0 −2.97854 0
1.9 0 −0.103746 0 −3.71451 0 0 0 −2.98924 0
1.10 0 −0.0954148 0 −0.581124 0 0 0 −2.99090 0
1.11 0 0.476040 0 −2.17736 0 0 0 −2.77339 0
1.12 0 0.538932 0 0.810745 0 0 0 −2.70955 0
1.13 0 1.06664 0 −0.715473 0 0 0 −1.86229 0
1.14 0 1.15563 0 3.94572 0 0 0 −1.66452 0
1.15 0 1.53485 0 1.74990 0 0 0 −0.644239 0
1.16 0 2.15119 0 −3.75581 0 0 0 1.62763 0
1.17 0 2.36764 0 2.42438 0 0 0 2.60574 0
1.18 0 2.83122 0 −2.22134 0 0 0 5.01578 0
1.19 0 3.09326 0 2.53432 0 0 0 6.56828 0
1.20 0 3.29362 0 3.10165 0 0 0 7.84790 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(41\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8036))\):

\(T_{3}^{20} - \cdots\)
\(T_{5}^{20} - \cdots\)
\(T_{11}^{20} + \cdots\)