Properties

Label 8034.2.a.bd
Level 8034
Weight 2
Character orbit 8034.a
Self dual Yes
Analytic conductor 64.152
Analytic rank 0
Dimension 16
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8034.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q\) \(+ q^{2}\) \(+ q^{3}\) \(+ q^{4}\) \( + \beta_{1} q^{5} \) \(+ q^{6}\) \( -\beta_{9} q^{7} \) \(+ q^{8}\) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{2}\) \(+ q^{3}\) \(+ q^{4}\) \( + \beta_{1} q^{5} \) \(+ q^{6}\) \( -\beta_{9} q^{7} \) \(+ q^{8}\) \(+ q^{9}\) \( + \beta_{1} q^{10} \) \( + ( 1 + \beta_{2} ) q^{11} \) \(+ q^{12}\) \(- q^{13}\) \( -\beta_{9} q^{14} \) \( + \beta_{1} q^{15} \) \(+ q^{16}\) \( + ( 1 - \beta_{15} ) q^{17} \) \(+ q^{18}\) \( + ( 1 - \beta_{11} ) q^{19} \) \( + \beta_{1} q^{20} \) \( -\beta_{9} q^{21} \) \( + ( 1 + \beta_{2} ) q^{22} \) \( + ( 1 + \beta_{4} ) q^{23} \) \(+ q^{24}\) \( + ( 1 + \beta_{11} - \beta_{12} ) q^{25} \) \(- q^{26}\) \(+ q^{27}\) \( -\beta_{9} q^{28} \) \( + ( 1 - \beta_{3} + \beta_{13} ) q^{29} \) \( + \beta_{1} q^{30} \) \( + ( \beta_{12} - \beta_{13} ) q^{31} \) \(+ q^{32}\) \( + ( 1 + \beta_{2} ) q^{33} \) \( + ( 1 - \beta_{15} ) q^{34} \) \( + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{13} + \beta_{15} ) q^{35} \) \(+ q^{36}\) \( + ( 2 + \beta_{5} + \beta_{12} + \beta_{13} ) q^{37} \) \( + ( 1 - \beta_{11} ) q^{38} \) \(- q^{39}\) \( + \beta_{1} q^{40} \) \( + ( 2 + \beta_{9} - \beta_{14} ) q^{41} \) \( -\beta_{9} q^{42} \) \( + ( 2 - \beta_{4} + \beta_{5} + \beta_{9} + \beta_{14} + \beta_{15} ) q^{43} \) \( + ( 1 + \beta_{2} ) q^{44} \) \( + \beta_{1} q^{45} \) \( + ( 1 + \beta_{4} ) q^{46} \) \( + ( 1 - \beta_{1} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{14} ) q^{47} \) \(+ q^{48}\) \( + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{5} + \beta_{6} - \beta_{9} - \beta_{14} ) q^{49} \) \( + ( 1 + \beta_{11} - \beta_{12} ) q^{50} \) \( + ( 1 - \beta_{15} ) q^{51} \) \(- q^{52}\) \( + ( -\beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} - \beta_{9} ) q^{53} \) \(+ q^{54}\) \( + ( 3 + \beta_{1} + \beta_{2} - \beta_{6} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{55} \) \( -\beta_{9} q^{56} \) \( + ( 1 - \beta_{11} ) q^{57} \) \( + ( 1 - \beta_{3} + \beta_{13} ) q^{58} \) \( + ( 3 - \beta_{1} - \beta_{3} - \beta_{5} - \beta_{7} - \beta_{9} - \beta_{10} + \beta_{15} ) q^{59} \) \( + \beta_{1} q^{60} \) \( + ( \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{12} - \beta_{15} ) q^{61} \) \( + ( \beta_{12} - \beta_{13} ) q^{62} \) \( -\beta_{9} q^{63} \) \(+ q^{64}\) \( -\beta_{1} q^{65} \) \( + ( 1 + \beta_{2} ) q^{66} \) \( + ( 2 - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{12} ) q^{67} \) \( + ( 1 - \beta_{15} ) q^{68} \) \( + ( 1 + \beta_{4} ) q^{69} \) \( + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{13} + \beta_{15} ) q^{70} \) \( + ( 1 - \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} ) q^{71} \) \(+ q^{72}\) \( + ( 2 - \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{10} + \beta_{11} ) q^{73} \) \( + ( 2 + \beta_{5} + \beta_{12} + \beta_{13} ) q^{74} \) \( + ( 1 + \beta_{11} - \beta_{12} ) q^{75} \) \( + ( 1 - \beta_{11} ) q^{76} \) \( + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{11} + \beta_{12} + \beta_{15} ) q^{77} \) \(- q^{78}\) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{7} - \beta_{8} - \beta_{10} + \beta_{12} ) q^{79} \) \( + \beta_{1} q^{80} \) \(+ q^{81}\) \( + ( 2 + \beta_{9} - \beta_{14} ) q^{82} \) \( + ( 3 - \beta_{2} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{83} \) \( -\beta_{9} q^{84} \) \( + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{11} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{85} \) \( + ( 2 - \beta_{4} + \beta_{5} + \beta_{9} + \beta_{14} + \beta_{15} ) q^{86} \) \( + ( 1 - \beta_{3} + \beta_{13} ) q^{87} \) \( + ( 1 + \beta_{2} ) q^{88} \) \( + ( 2 - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{12} ) q^{89} \) \( + \beta_{1} q^{90} \) \( + \beta_{9} q^{91} \) \( + ( 1 + \beta_{4} ) q^{92} \) \( + ( \beta_{12} - \beta_{13} ) q^{93} \) \( + ( 1 - \beta_{1} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{14} ) q^{94} \) \( + ( -1 + \beta_{2} - \beta_{3} + \beta_{8} - \beta_{11} + \beta_{12} + \beta_{15} ) q^{95} \) \(+ q^{96}\) \( + ( 2 + \beta_{1} - \beta_{2} + \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} ) q^{97} \) \( + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{5} + \beta_{6} - \beta_{9} - \beta_{14} ) q^{98} \) \( + ( 1 + \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(16q \) \(\mathstrut +\mathstrut 16q^{2} \) \(\mathstrut +\mathstrut 16q^{3} \) \(\mathstrut +\mathstrut 16q^{4} \) \(\mathstrut +\mathstrut 5q^{5} \) \(\mathstrut +\mathstrut 16q^{6} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 16q^{8} \) \(\mathstrut +\mathstrut 16q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(16q \) \(\mathstrut +\mathstrut 16q^{2} \) \(\mathstrut +\mathstrut 16q^{3} \) \(\mathstrut +\mathstrut 16q^{4} \) \(\mathstrut +\mathstrut 5q^{5} \) \(\mathstrut +\mathstrut 16q^{6} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 16q^{8} \) \(\mathstrut +\mathstrut 16q^{9} \) \(\mathstrut +\mathstrut 5q^{10} \) \(\mathstrut +\mathstrut 18q^{11} \) \(\mathstrut +\mathstrut 16q^{12} \) \(\mathstrut -\mathstrut 16q^{13} \) \(\mathstrut +\mathstrut 4q^{14} \) \(\mathstrut +\mathstrut 5q^{15} \) \(\mathstrut +\mathstrut 16q^{16} \) \(\mathstrut +\mathstrut 17q^{17} \) \(\mathstrut +\mathstrut 16q^{18} \) \(\mathstrut +\mathstrut 8q^{19} \) \(\mathstrut +\mathstrut 5q^{20} \) \(\mathstrut +\mathstrut 4q^{21} \) \(\mathstrut +\mathstrut 18q^{22} \) \(\mathstrut +\mathstrut 9q^{23} \) \(\mathstrut +\mathstrut 16q^{24} \) \(\mathstrut +\mathstrut 17q^{25} \) \(\mathstrut -\mathstrut 16q^{26} \) \(\mathstrut +\mathstrut 16q^{27} \) \(\mathstrut +\mathstrut 4q^{28} \) \(\mathstrut +\mathstrut 14q^{29} \) \(\mathstrut +\mathstrut 5q^{30} \) \(\mathstrut +\mathstrut 12q^{31} \) \(\mathstrut +\mathstrut 16q^{32} \) \(\mathstrut +\mathstrut 18q^{33} \) \(\mathstrut +\mathstrut 17q^{34} \) \(\mathstrut +\mathstrut 16q^{35} \) \(\mathstrut +\mathstrut 16q^{36} \) \(\mathstrut +\mathstrut 31q^{37} \) \(\mathstrut +\mathstrut 8q^{38} \) \(\mathstrut -\mathstrut 16q^{39} \) \(\mathstrut +\mathstrut 5q^{40} \) \(\mathstrut +\mathstrut 29q^{41} \) \(\mathstrut +\mathstrut 4q^{42} \) \(\mathstrut +\mathstrut 30q^{43} \) \(\mathstrut +\mathstrut 18q^{44} \) \(\mathstrut +\mathstrut 5q^{45} \) \(\mathstrut +\mathstrut 9q^{46} \) \(\mathstrut -\mathstrut q^{47} \) \(\mathstrut +\mathstrut 16q^{48} \) \(\mathstrut +\mathstrut 36q^{49} \) \(\mathstrut +\mathstrut 17q^{50} \) \(\mathstrut +\mathstrut 17q^{51} \) \(\mathstrut -\mathstrut 16q^{52} \) \(\mathstrut +\mathstrut 12q^{53} \) \(\mathstrut +\mathstrut 16q^{54} \) \(\mathstrut +\mathstrut 30q^{55} \) \(\mathstrut +\mathstrut 4q^{56} \) \(\mathstrut +\mathstrut 8q^{57} \) \(\mathstrut +\mathstrut 14q^{58} \) \(\mathstrut +\mathstrut 38q^{59} \) \(\mathstrut +\mathstrut 5q^{60} \) \(\mathstrut +\mathstrut 12q^{62} \) \(\mathstrut +\mathstrut 4q^{63} \) \(\mathstrut +\mathstrut 16q^{64} \) \(\mathstrut -\mathstrut 5q^{65} \) \(\mathstrut +\mathstrut 18q^{66} \) \(\mathstrut +\mathstrut 28q^{67} \) \(\mathstrut +\mathstrut 17q^{68} \) \(\mathstrut +\mathstrut 9q^{69} \) \(\mathstrut +\mathstrut 16q^{70} \) \(\mathstrut +\mathstrut 32q^{71} \) \(\mathstrut +\mathstrut 16q^{72} \) \(\mathstrut +\mathstrut 20q^{73} \) \(\mathstrut +\mathstrut 31q^{74} \) \(\mathstrut +\mathstrut 17q^{75} \) \(\mathstrut +\mathstrut 8q^{76} \) \(\mathstrut +\mathstrut 26q^{77} \) \(\mathstrut -\mathstrut 16q^{78} \) \(\mathstrut +\mathstrut 13q^{79} \) \(\mathstrut +\mathstrut 5q^{80} \) \(\mathstrut +\mathstrut 16q^{81} \) \(\mathstrut +\mathstrut 29q^{82} \) \(\mathstrut +\mathstrut 39q^{83} \) \(\mathstrut +\mathstrut 4q^{84} \) \(\mathstrut +\mathstrut 31q^{85} \) \(\mathstrut +\mathstrut 30q^{86} \) \(\mathstrut +\mathstrut 14q^{87} \) \(\mathstrut +\mathstrut 18q^{88} \) \(\mathstrut +\mathstrut 9q^{89} \) \(\mathstrut +\mathstrut 5q^{90} \) \(\mathstrut -\mathstrut 4q^{91} \) \(\mathstrut +\mathstrut 9q^{92} \) \(\mathstrut +\mathstrut 12q^{93} \) \(\mathstrut -\mathstrut q^{94} \) \(\mathstrut -\mathstrut 20q^{95} \) \(\mathstrut +\mathstrut 16q^{96} \) \(\mathstrut +\mathstrut 35q^{97} \) \(\mathstrut +\mathstrut 36q^{98} \) \(\mathstrut +\mathstrut 18q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16}\mathstrut -\mathstrut \) \(5\) \(x^{15}\mathstrut -\mathstrut \) \(36\) \(x^{14}\mathstrut +\mathstrut \) \(196\) \(x^{13}\mathstrut +\mathstrut \) \(498\) \(x^{12}\mathstrut -\mathstrut \) \(3101\) \(x^{11}\mathstrut -\mathstrut \) \(3150\) \(x^{10}\mathstrut +\mathstrut \) \(25368\) \(x^{9}\mathstrut +\mathstrut \) \(6763\) \(x^{8}\mathstrut -\mathstrut \) \(113788\) \(x^{7}\mathstrut +\mathstrut \) \(19731\) \(x^{6}\mathstrut +\mathstrut \) \(270913\) \(x^{5}\mathstrut -\mathstrut \) \(122680\) \(x^{4}\mathstrut -\mathstrut \) \(296326\) \(x^{3}\mathstrut +\mathstrut \) \(185524\) \(x^{2}\mathstrut +\mathstrut \) \(94528\) \(x\mathstrut -\mathstrut \) \(66432\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(90586583806696\) \(\nu^{15}\mathstrut +\mathstrut \) \(240231261271053\) \(\nu^{14}\mathstrut +\mathstrut \) \(3828384374947411\) \(\nu^{13}\mathstrut -\mathstrut \) \(8779176678679124\) \(\nu^{12}\mathstrut -\mathstrut \) \(65847084572606208\) \(\nu^{11}\mathstrut +\mathstrut \) \(126845466474286622\) \(\nu^{10}\mathstrut +\mathstrut \) \(584806300906520047\) \(\nu^{9}\mathstrut -\mathstrut \) \(933492931001692430\) \(\nu^{8}\mathstrut -\mathstrut \) \(2812402864763475044\) \(\nu^{7}\mathstrut +\mathstrut \) \(3770017974633780087\) \(\nu^{6}\mathstrut +\mathstrut \) \(7057641060272104164\) \(\nu^{5}\mathstrut -\mathstrut \) \(8218562721686531825\) \(\nu^{4}\mathstrut -\mathstrut \) \(8015723884984487975\) \(\nu^{3}\mathstrut +\mathstrut \) \(8434099748549824528\) \(\nu^{2}\mathstrut +\mathstrut \) \(2608177483732148142\) \(\nu\mathstrut -\mathstrut \) \(2641861718989240104\)\()/\)\(3920441740718852\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(548861320433565\) \(\nu^{15}\mathstrut +\mathstrut \) \(1432628181298625\) \(\nu^{14}\mathstrut +\mathstrut \) \(23250750972171732\) \(\nu^{13}\mathstrut -\mathstrut \) \(52222785574841876\) \(\nu^{12}\mathstrut -\mathstrut \) \(400715718523472458\) \(\nu^{11}\mathstrut +\mathstrut \) \(751279156738320473\) \(\nu^{10}\mathstrut +\mathstrut \) \(3564165190681044406\) \(\nu^{9}\mathstrut -\mathstrut \) \(5491103861096290984\) \(\nu^{8}\mathstrut -\mathstrut \) \(17153609961115061407\) \(\nu^{7}\mathstrut +\mathstrut \) \(21958491635614630188\) \(\nu^{6}\mathstrut +\mathstrut \) \(43039314676128770737\) \(\nu^{5}\mathstrut -\mathstrut \) \(47249773825946922341\) \(\nu^{4}\mathstrut -\mathstrut \) \(48815686138384895136\) \(\nu^{3}\mathstrut +\mathstrut \) \(47716865559993728766\) \(\nu^{2}\mathstrut +\mathstrut \) \(15822903042704832740\) \(\nu\mathstrut -\mathstrut \) \(14690943728166969488\)\()/\)\(7840883481437704\)
\(\beta_{4}\)\(=\)\((\)\(1804326041550225\) \(\nu^{15}\mathstrut -\mathstrut \) \(4966726230624225\) \(\nu^{14}\mathstrut -\mathstrut \) \(75869762801105224\) \(\nu^{13}\mathstrut +\mathstrut \) \(182455633738698884\) \(\nu^{12}\mathstrut +\mathstrut \) \(1298746915725286210\) \(\nu^{11}\mathstrut -\mathstrut \) \(2653626772894102149\) \(\nu^{10}\mathstrut -\mathstrut \) \(11487640535802356658\) \(\nu^{9}\mathstrut +\mathstrut \) \(19666961745324013104\) \(\nu^{8}\mathstrut +\mathstrut \) \(55060744838113792651\) \(\nu^{7}\mathstrut -\mathstrut \) \(79839053134054770160\) \(\nu^{6}\mathstrut -\mathstrut \) \(137710113475680299789\) \(\nu^{5}\mathstrut +\mathstrut \) \(174248716157357704813\) \(\nu^{4}\mathstrut +\mathstrut \) \(155576241533390408220\) \(\nu^{3}\mathstrut -\mathstrut \) \(178441043110187005830\) \(\nu^{2}\mathstrut -\mathstrut \) \(49887972149323407140\) \(\nu\mathstrut +\mathstrut \) \(55874993881846012448\)\()/\)\(15681766962875408\)
\(\beta_{5}\)\(=\)\((\)\(1835427375702341\) \(\nu^{15}\mathstrut -\mathstrut \) \(4835063132536829\) \(\nu^{14}\mathstrut -\mathstrut \) \(77588508853292304\) \(\nu^{13}\mathstrut +\mathstrut \) \(176229450470376708\) \(\nu^{12}\mathstrut +\mathstrut \) \(1334671116861721130\) \(\nu^{11}\mathstrut -\mathstrut \) \(2535841645574667481\) \(\nu^{10}\mathstrut -\mathstrut \) \(11852110311085567842\) \(\nu^{9}\mathstrut +\mathstrut \) \(18546698540088649616\) \(\nu^{8}\mathstrut +\mathstrut \) \(56965202389909391415\) \(\nu^{7}\mathstrut -\mathstrut \) \(74248219122217515688\) \(\nu^{6}\mathstrut -\mathstrut \) \(142761745839245510897\) \(\nu^{5}\mathstrut +\mathstrut \) \(160045111057739017785\) \(\nu^{4}\mathstrut +\mathstrut \) \(161726537131518925220\) \(\nu^{3}\mathstrut -\mathstrut \) \(162139203272019169390\) \(\nu^{2}\mathstrut -\mathstrut \) \(52289606909934558532\) \(\nu\mathstrut +\mathstrut \) \(50159255507717243952\)\()/\)\(15681766962875408\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(595029787973500\) \(\nu^{15}\mathstrut +\mathstrut \) \(1599200104156899\) \(\nu^{14}\mathstrut +\mathstrut \) \(25105417772097457\) \(\nu^{13}\mathstrut -\mathstrut \) \(58527974832066220\) \(\nu^{12}\mathstrut -\mathstrut \) \(431159357317343856\) \(\nu^{11}\mathstrut +\mathstrut \) \(846763531102328270\) \(\nu^{10}\mathstrut +\mathstrut \) \(3824729425828204465\) \(\nu^{9}\mathstrut -\mathstrut \) \(6233456803346647046\) \(\nu^{8}\mathstrut -\mathstrut \) \(18376977040077066764\) \(\nu^{7}\mathstrut +\mathstrut \) \(25116251602804037253\) \(\nu^{6}\mathstrut +\mathstrut \) \(46062057857375063812\) \(\nu^{5}\mathstrut -\mathstrut \) \(54416371328603985543\) \(\nu^{4}\mathstrut -\mathstrut \) \(52163888681140811825\) \(\nu^{3}\mathstrut +\mathstrut \) \(55327722525776352088\) \(\nu^{2}\mathstrut +\mathstrut \) \(16815865184121493238\) \(\nu\mathstrut -\mathstrut \) \(17193626618790198144\)\()/\)\(3920441740718852\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(2678489495580815\) \(\nu^{15}\mathstrut +\mathstrut \) \(7200273222605183\) \(\nu^{14}\mathstrut +\mathstrut \) \(113027570175626792\) \(\nu^{13}\mathstrut -\mathstrut \) \(263611332890321948\) \(\nu^{12}\mathstrut -\mathstrut \) \(1941458087986298718\) \(\nu^{11}\mathstrut +\mathstrut \) \(3815774167702393179\) \(\nu^{10}\mathstrut +\mathstrut \) \(17225700209520332286\) \(\nu^{9}\mathstrut -\mathstrut \) \(28110032331446166560\) \(\nu^{8}\mathstrut -\mathstrut \) \(82787094387047921925\) \(\nu^{7}\mathstrut +\mathstrut \) \(113376045704201381856\) \(\nu^{6}\mathstrut +\mathstrut \) \(207587078710285962883\) \(\nu^{5}\mathstrut -\mathstrut \) \(245983134707975915715\) \(\nu^{4}\mathstrut -\mathstrut \) \(235238717548900685604\) \(\nu^{3}\mathstrut +\mathstrut \) \(250609270874405645338\) \(\nu^{2}\mathstrut +\mathstrut \) \(75872710426262168508\) \(\nu\mathstrut -\mathstrut \) \(78040356907095709104\)\()/\)\(15681766962875408\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(1638703541012891\) \(\nu^{15}\mathstrut +\mathstrut \) \(4420432159038421\) \(\nu^{14}\mathstrut +\mathstrut \) \(69072367077303942\) \(\nu^{13}\mathstrut -\mathstrut \) \(161784188747179548\) \(\nu^{12}\mathstrut -\mathstrut \) \(1185014847148291254\) \(\nu^{11}\mathstrut +\mathstrut \) \(2341043276285811571\) \(\nu^{10}\mathstrut +\mathstrut \) \(10500262821630922156\) \(\nu^{9}\mathstrut -\mathstrut \) \(17239844280702946500\) \(\nu^{8}\mathstrut -\mathstrut \) \(50388167884484641945\) \(\nu^{7}\mathstrut +\mathstrut \) \(69503600178868582750\) \(\nu^{6}\mathstrut +\mathstrut \) \(126120647536918697015\) \(\nu^{5}\mathstrut -\mathstrut \) \(150709788164175297649\) \(\nu^{4}\mathstrut -\mathstrut \) \(142627782445967128474\) \(\nu^{3}\mathstrut +\mathstrut \) \(153429764577484876258\) \(\nu^{2}\mathstrut +\mathstrut \) \(45943002513497625888\) \(\nu\mathstrut -\mathstrut \) \(47772605674210336512\)\()/\)\(7840883481437704\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(3569097596475919\) \(\nu^{15}\mathstrut +\mathstrut \) \(9534940311240535\) \(\nu^{14}\mathstrut +\mathstrut \) \(150690531878100656\) \(\nu^{13}\mathstrut -\mathstrut \) \(348541387262623900\) \(\nu^{12}\mathstrut -\mathstrut \) \(2589606301822829662\) \(\nu^{11}\mathstrut +\mathstrut \) \(5034026218476398539\) \(\nu^{10}\mathstrut +\mathstrut \) \(22983893465451743718\) \(\nu^{9}\mathstrut -\mathstrut \) \(36974698692332377376\) \(\nu^{8}\mathstrut -\mathstrut \) \(110474820709431665877\) \(\nu^{7}\mathstrut +\mathstrut \) \(148581497019745053416\) \(\nu^{6}\mathstrut +\mathstrut \) \(277001407604204710851\) \(\nu^{5}\mathstrut -\mathstrut \) \(320980123683727999659\) \(\nu^{4}\mathstrut -\mathstrut \) \(313893272371210960748\) \(\nu^{3}\mathstrut +\mathstrut \) \(325396081581932857258\) \(\nu^{2}\mathstrut +\mathstrut \) \(101341860589668676780\) \(\nu\mathstrut -\mathstrut \) \(100871910618813386288\)\()/\)\(15681766962875408\)
\(\beta_{10}\)\(=\)\((\)\(4530672367783573\) \(\nu^{15}\mathstrut -\mathstrut \) \(12086915535739717\) \(\nu^{14}\mathstrut -\mathstrut \) \(191288673559836344\) \(\nu^{13}\mathstrut +\mathstrut \) \(441626866537395316\) \(\nu^{12}\mathstrut +\mathstrut \) \(3287022861871681850\) \(\nu^{11}\mathstrut -\mathstrut \) \(6374316318038130041\) \(\nu^{10}\mathstrut -\mathstrut \) \(29168492201670204122\) \(\nu^{9}\mathstrut +\mathstrut \) \(46778612431374204912\) \(\nu^{8}\mathstrut +\mathstrut \) \(140160717342289010247\) \(\nu^{7}\mathstrut -\mathstrut \) \(187791522521265099936\) \(\nu^{6}\mathstrut -\mathstrut \) \(351310754565662542289\) \(\nu^{5}\mathstrut +\mathstrut \) \(405296071724458119681\) \(\nu^{4}\mathstrut +\mathstrut \) \(398014867364435051884\) \(\nu^{3}\mathstrut -\mathstrut \) \(410559868342291139342\) \(\nu^{2}\mathstrut -\mathstrut \) \(128593948008476186420\) \(\nu\mathstrut +\mathstrut \) \(127255514637495805936\)\()/\)\(15681766962875408\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(653898814567821\) \(\nu^{15}\mathstrut +\mathstrut \) \(1739613880476431\) \(\nu^{14}\mathstrut +\mathstrut \) \(27620987107450456\) \(\nu^{13}\mathstrut -\mathstrut \) \(63529149729873054\) \(\nu^{12}\mathstrut -\mathstrut \) \(474890744134306278\) \(\nu^{11}\mathstrut +\mathstrut \) \(916354503792581425\) \(\nu^{10}\mathstrut +\mathstrut \) \(4216832882426115422\) \(\nu^{9}\mathstrut -\mathstrut \) \(6719299055051450480\) \(\nu^{8}\mathstrut -\mathstrut \) \(20278331057027016475\) \(\nu^{7}\mathstrut +\mathstrut \) \(26951008155860129116\) \(\nu^{6}\mathstrut +\mathstrut \) \(50875551644113476007\) \(\nu^{5}\mathstrut -\mathstrut \) \(58121177129369561431\) \(\nu^{4}\mathstrut -\mathstrut \) \(57709904926601002470\) \(\nu^{3}\mathstrut +\mathstrut \) \(58840600922971927372\) \(\nu^{2}\mathstrut +\mathstrut \) \(18677578787028375660\) \(\nu\mathstrut -\mathstrut \) \(18225241076803877394\)\()/\)\(1960220870359426\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(653898814567821\) \(\nu^{15}\mathstrut +\mathstrut \) \(1739613880476431\) \(\nu^{14}\mathstrut +\mathstrut \) \(27620987107450456\) \(\nu^{13}\mathstrut -\mathstrut \) \(63529149729873054\) \(\nu^{12}\mathstrut -\mathstrut \) \(474890744134306278\) \(\nu^{11}\mathstrut +\mathstrut \) \(916354503792581425\) \(\nu^{10}\mathstrut +\mathstrut \) \(4216832882426115422\) \(\nu^{9}\mathstrut -\mathstrut \) \(6719299055051450480\) \(\nu^{8}\mathstrut -\mathstrut \) \(20278331057027016475\) \(\nu^{7}\mathstrut +\mathstrut \) \(26951008155860129116\) \(\nu^{6}\mathstrut +\mathstrut \) \(50875551644113476007\) \(\nu^{5}\mathstrut -\mathstrut \) \(58121177129369561431\) \(\nu^{4}\mathstrut -\mathstrut \) \(57709904926601002470\) \(\nu^{3}\mathstrut +\mathstrut \) \(58838640702101567946\) \(\nu^{2}\mathstrut +\mathstrut \) \(18677578787028375660\) \(\nu\mathstrut -\mathstrut \) \(18213479751581720838\)\()/\)\(1960220870359426\)
\(\beta_{13}\)\(=\)\((\)\(3100242079114577\) \(\nu^{15}\mathstrut -\mathstrut \) \(8285924509613657\) \(\nu^{14}\mathstrut -\mathstrut \) \(130935794410224976\) \(\nu^{13}\mathstrut +\mathstrut \) \(303115201650576348\) \(\nu^{12}\mathstrut +\mathstrut \) \(2250734259315229658\) \(\nu^{11}\mathstrut -\mathstrut \) \(4382238950785207301\) \(\nu^{10}\mathstrut -\mathstrut \) \(19980925687065254106\) \(\nu^{9}\mathstrut +\mathstrut \) \(32228305300709613072\) \(\nu^{8}\mathstrut +\mathstrut \) \(96059841589747052611\) \(\nu^{7}\mathstrut -\mathstrut \) \(129712942536712800344\) \(\nu^{6}\mathstrut -\mathstrut \) \(240890934966007223645\) \(\nu^{5}\mathstrut +\mathstrut \) \(280741064934462981181\) \(\nu^{4}\mathstrut +\mathstrut \) \(272973521847255931684\) \(\nu^{3}\mathstrut -\mathstrut \) \(285212960447565670230\) \(\nu^{2}\mathstrut -\mathstrut \) \(88113331148205643228\) \(\nu\mathstrut +\mathstrut \) \(88599863268713580968\)\()/\)\(7840883481437704\)
\(\beta_{14}\)\(=\)\((\)\(12507240909772965\) \(\nu^{15}\mathstrut -\mathstrut \) \(33646326659730001\) \(\nu^{14}\mathstrut -\mathstrut \) \(527725179776986492\) \(\nu^{13}\mathstrut +\mathstrut \) \(1231914240821691860\) \(\nu^{12}\mathstrut +\mathstrut \) \(9063391346572318730\) \(\nu^{11}\mathstrut -\mathstrut \) \(17832062238083640769\) \(\nu^{10}\mathstrut -\mathstrut \) \(80401614076340790046\) \(\nu^{9}\mathstrut +\mathstrut \) \(131346321914925220744\) \(\nu^{8}\mathstrut +\mathstrut \) \(386326816150958387783\) \(\nu^{7}\mathstrut -\mathstrut \) \(529515818232642909092\) \(\nu^{6}\mathstrut -\mathstrut \) \(968407126713045024321\) \(\nu^{5}\mathstrut +\mathstrut \) \(1147728394339846755949\) \(\nu^{4}\mathstrut +\mathstrut \) \(1096868025231990569216\) \(\nu^{3}\mathstrut -\mathstrut \) \(1167474063125882771310\) \(\nu^{2}\mathstrut -\mathstrut \) \(353504367066187383884\) \(\nu\mathstrut +\mathstrut \) \(363101975930316935296\)\()/\)\(15681766962875408\)
\(\beta_{15}\)\(=\)\((\)\(3695268078821707\) \(\nu^{15}\mathstrut -\mathstrut \) \(9895392816158751\) \(\nu^{14}\mathstrut -\mathstrut \) \(156009228278353240\) \(\nu^{13}\mathstrut +\mathstrut \) \(362058147223910800\) \(\nu^{12}\mathstrut +\mathstrut \) \(2680768089249478598\) \(\nu^{11}\mathstrut -\mathstrut \) \(5235830759322990715\) \(\nu^{10}\mathstrut -\mathstrut \) \(23790467262191524218\) \(\nu^{9}\mathstrut +\mathstrut \) \(38519889889014706292\) \(\nu^{8}\mathstrut +\mathstrut \) \(114339342138814438977\) \(\nu^{7}\mathstrut -\mathstrut \) \(155097830554963060684\) \(\nu^{6}\mathstrut -\mathstrut \) \(286655332333952125587\) \(\nu^{5}\mathstrut +\mathstrut \) \(335808992886363284979\) \(\nu^{4}\mathstrut +\mathstrut \) \(324775010444080001192\) \(\nu^{3}\mathstrut -\mathstrut \) \(341269633737372277150\) \(\nu^{2}\mathstrut -\mathstrut \) \(104837318910714038052\) \(\nu\mathstrut +\mathstrut \) \(106042702582046298700\)\()/\)\(3920441740718852\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(6\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(9\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{13}\mathstrut -\mathstrut \) \(13\) \(\beta_{12}\mathstrut +\mathstrut \) \(14\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut -\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(53\)
\(\nu^{5}\)\(=\)\(-\)\(2\) \(\beta_{15}\mathstrut +\mathstrut \) \(2\) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(20\) \(\beta_{12}\mathstrut +\mathstrut \) \(23\) \(\beta_{11}\mathstrut +\mathstrut \) \(3\) \(\beta_{10}\mathstrut -\mathstrut \) \(3\) \(\beta_{9}\mathstrut -\mathstrut \) \(15\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(3\) \(\beta_{6}\mathstrut -\mathstrut \) \(15\) \(\beta_{5}\mathstrut -\mathstrut \) \(18\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(20\) \(\beta_{2}\mathstrut +\mathstrut \) \(95\) \(\beta_{1}\mathstrut +\mathstrut \) \(32\)
\(\nu^{6}\)\(=\)\(-\)\(6\) \(\beta_{15}\mathstrut +\mathstrut \) \(26\) \(\beta_{13}\mathstrut -\mathstrut \) \(166\) \(\beta_{12}\mathstrut +\mathstrut \) \(185\) \(\beta_{11}\mathstrut +\mathstrut \) \(22\) \(\beta_{10}\mathstrut -\mathstrut \) \(23\) \(\beta_{9}\mathstrut -\mathstrut \) \(5\) \(\beta_{8}\mathstrut +\mathstrut \) \(17\) \(\beta_{7}\mathstrut +\mathstrut \) \(20\) \(\beta_{6}\mathstrut -\mathstrut \) \(10\) \(\beta_{5}\mathstrut -\mathstrut \) \(18\) \(\beta_{4}\mathstrut +\mathstrut \) \(47\) \(\beta_{3}\mathstrut -\mathstrut \) \(67\) \(\beta_{2}\mathstrut +\mathstrut \) \(116\) \(\beta_{1}\mathstrut +\mathstrut \) \(557\)
\(\nu^{7}\)\(=\)\(-\)\(53\) \(\beta_{15}\mathstrut +\mathstrut \) \(37\) \(\beta_{14}\mathstrut +\mathstrut \) \(44\) \(\beta_{13}\mathstrut -\mathstrut \) \(338\) \(\beta_{12}\mathstrut +\mathstrut \) \(400\) \(\beta_{11}\mathstrut +\mathstrut \) \(72\) \(\beta_{10}\mathstrut -\mathstrut \) \(73\) \(\beta_{9}\mathstrut -\mathstrut \) \(203\) \(\beta_{8}\mathstrut -\mathstrut \) \(27\) \(\beta_{7}\mathstrut +\mathstrut \) \(81\) \(\beta_{6}\mathstrut -\mathstrut \) \(215\) \(\beta_{5}\mathstrut -\mathstrut \) \(248\) \(\beta_{4}\mathstrut +\mathstrut \) \(32\) \(\beta_{3}\mathstrut -\mathstrut \) \(325\) \(\beta_{2}\mathstrut +\mathstrut \) \(1105\) \(\beta_{1}\mathstrut +\mathstrut \) \(636\)
\(\nu^{8}\)\(=\)\(-\)\(179\) \(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(493\) \(\beta_{13}\mathstrut -\mathstrut \) \(2180\) \(\beta_{12}\mathstrut +\mathstrut \) \(2471\) \(\beta_{11}\mathstrut +\mathstrut \) \(385\) \(\beta_{10}\mathstrut -\mathstrut \) \(389\) \(\beta_{9}\mathstrut -\mathstrut \) \(159\) \(\beta_{8}\mathstrut +\mathstrut \) \(189\) \(\beta_{7}\mathstrut +\mathstrut \) \(332\) \(\beta_{6}\mathstrut -\mathstrut \) \(295\) \(\beta_{5}\mathstrut -\mathstrut \) \(277\) \(\beta_{4}\mathstrut +\mathstrut \) \(573\) \(\beta_{3}\mathstrut -\mathstrut \) \(1136\) \(\beta_{2}\mathstrut +\mathstrut \) \(2012\) \(\beta_{1}\mathstrut +\mathstrut \) \(6477\)
\(\nu^{9}\)\(=\)\(-\)\(1042\) \(\beta_{15}\mathstrut +\mathstrut \) \(504\) \(\beta_{14}\mathstrut +\mathstrut \) \(1108\) \(\beta_{13}\mathstrut -\mathstrut \) \(5323\) \(\beta_{12}\mathstrut +\mathstrut \) \(6331\) \(\beta_{11}\mathstrut +\mathstrut \) \(1308\) \(\beta_{10}\mathstrut -\mathstrut \) \(1351\) \(\beta_{9}\mathstrut -\mathstrut \) \(2730\) \(\beta_{8}\mathstrut -\mathstrut \) \(555\) \(\beta_{7}\mathstrut +\mathstrut \) \(1529\) \(\beta_{6}\mathstrut -\mathstrut \) \(3096\) \(\beta_{5}\mathstrut -\mathstrut \) \(3176\) \(\beta_{4}\mathstrut +\mathstrut \) \(371\) \(\beta_{3}\mathstrut -\mathstrut \) \(4909\) \(\beta_{2}\mathstrut +\mathstrut \) \(13660\) \(\beta_{1}\mathstrut +\mathstrut \) \(10766\)
\(\nu^{10}\)\(=\)\(-\)\(3702\) \(\beta_{15}\mathstrut +\mathstrut \) \(15\) \(\beta_{14}\mathstrut +\mathstrut \) \(8295\) \(\beta_{13}\mathstrut -\mathstrut \) \(29297\) \(\beta_{12}\mathstrut +\mathstrut \) \(33572\) \(\beta_{11}\mathstrut +\mathstrut \) \(6249\) \(\beta_{10}\mathstrut -\mathstrut \) \(6111\) \(\beta_{9}\mathstrut -\mathstrut \) \(3424\) \(\beta_{8}\mathstrut +\mathstrut \) \(1449\) \(\beta_{7}\mathstrut +\mathstrut \) \(5309\) \(\beta_{6}\mathstrut -\mathstrut \) \(6093\) \(\beta_{5}\mathstrut -\mathstrut \) \(4157\) \(\beta_{4}\mathstrut +\mathstrut \) \(6437\) \(\beta_{3}\mathstrut -\mathstrut \) \(17620\) \(\beta_{2}\mathstrut +\mathstrut \) \(31504\) \(\beta_{1}\mathstrut +\mathstrut \) \(80154\)
\(\nu^{11}\)\(=\)\(-\)\(18139\) \(\beta_{15}\mathstrut +\mathstrut \) \(6057\) \(\beta_{14}\mathstrut +\mathstrut \) \(22118\) \(\beta_{13}\mathstrut -\mathstrut \) \(80811\) \(\beta_{12}\mathstrut +\mathstrut \) \(96211\) \(\beta_{11}\mathstrut +\mathstrut \) \(21659\) \(\beta_{10}\mathstrut -\mathstrut \) \(22791\) \(\beta_{9}\mathstrut -\mathstrut \) \(36925\) \(\beta_{8}\mathstrut -\mathstrut \) \(10278\) \(\beta_{7}\mathstrut +\mathstrut \) \(25248\) \(\beta_{6}\mathstrut -\mathstrut \) \(44809\) \(\beta_{5}\mathstrut -\mathstrut \) \(39894\) \(\beta_{4}\mathstrut +\mathstrut \) \(3705\) \(\beta_{3}\mathstrut -\mathstrut \) \(72142\) \(\beta_{2}\mathstrut +\mathstrut \) \(175835\) \(\beta_{1}\mathstrut +\mathstrut \) \(169350\)
\(\nu^{12}\)\(=\)\(-\)\(65998\) \(\beta_{15}\mathstrut -\mathstrut \) \(242\) \(\beta_{14}\mathstrut +\mathstrut \) \(131574\) \(\beta_{13}\mathstrut -\mathstrut \) \(400190\) \(\beta_{12}\mathstrut +\mathstrut \) \(462969\) \(\beta_{11}\mathstrut +\mathstrut \) \(97786\) \(\beta_{10}\mathstrut -\mathstrut \) \(94813\) \(\beta_{9}\mathstrut -\mathstrut \) \(62592\) \(\beta_{8}\mathstrut +\mathstrut \) \(1478\) \(\beta_{7}\mathstrut +\mathstrut \) \(83562\) \(\beta_{6}\mathstrut -\mathstrut \) \(109098\) \(\beta_{5}\mathstrut -\mathstrut \) \(61819\) \(\beta_{4}\mathstrut +\mathstrut \) \(69544\) \(\beta_{3}\mathstrut -\mathstrut \) \(263646\) \(\beta_{2}\mathstrut +\mathstrut \) \(471316\) \(\beta_{1}\mathstrut +\mathstrut \) \(1032336\)
\(\nu^{13}\)\(=\)\(-\)\(295606\) \(\beta_{15}\mathstrut +\mathstrut \) \(66812\) \(\beta_{14}\mathstrut +\mathstrut \) \(391243\) \(\beta_{13}\mathstrut -\mathstrut \) \(1201340\) \(\beta_{12}\mathstrut +\mathstrut \) \(1433236\) \(\beta_{11}\mathstrut +\mathstrut \) \(344263\) \(\beta_{10}\mathstrut -\mathstrut \) \(369200\) \(\beta_{9}\mathstrut -\mathstrut \) \(502714\) \(\beta_{8}\mathstrut -\mathstrut \) \(180397\) \(\beta_{7}\mathstrut +\mathstrut \) \(391880\) \(\beta_{6}\mathstrut -\mathstrut \) \(650273\) \(\beta_{5}\mathstrut -\mathstrut \) \(501775\) \(\beta_{4}\mathstrut +\mathstrut \) \(32620\) \(\beta_{3}\mathstrut -\mathstrut \) \(1049040\) \(\beta_{2}\mathstrut +\mathstrut \) \(2327489\) \(\beta_{1}\mathstrut +\mathstrut \) \(2563372\)
\(\nu^{14}\)\(=\)\(-\)\(1088984\) \(\beta_{15}\mathstrut -\mathstrut \) \(17948\) \(\beta_{14}\mathstrut +\mathstrut \) \(2018086\) \(\beta_{13}\mathstrut -\mathstrut \) \(5529279\) \(\beta_{12}\mathstrut +\mathstrut \) \(6459587\) \(\beta_{11}\mathstrut +\mathstrut \) \(1498058\) \(\beta_{10}\mathstrut -\mathstrut \) \(1473675\) \(\beta_{9}\mathstrut -\mathstrut \) \(1048202\) \(\beta_{8}\mathstrut -\mathstrut \) \(242986\) \(\beta_{7}\mathstrut +\mathstrut \) \(1296392\) \(\beta_{6}\mathstrut -\mathstrut \) \(1813802\) \(\beta_{5}\mathstrut -\mathstrut \) \(912149\) \(\beta_{4}\mathstrut +\mathstrut \) \(732765\) \(\beta_{3}\mathstrut -\mathstrut \) \(3880474\) \(\beta_{2}\mathstrut +\mathstrut \) \(6894287\) \(\beta_{1}\mathstrut +\mathstrut \) \(13659793\)
\(\nu^{15}\)\(=\)\(-\)\(4625695\) \(\beta_{15}\mathstrut +\mathstrut \) \(668168\) \(\beta_{14}\mathstrut +\mathstrut \) \(6448737\) \(\beta_{13}\mathstrut -\mathstrut \) \(17630107\) \(\beta_{12}\mathstrut +\mathstrut \) \(21122914\) \(\beta_{11}\mathstrut +\mathstrut \) \(5351563\) \(\beta_{10}\mathstrut -\mathstrut \) \(5859785\) \(\beta_{9}\mathstrut -\mathstrut \) \(6884898\) \(\beta_{8}\mathstrut -\mathstrut \) \(3065032\) \(\beta_{7}\mathstrut +\mathstrut \) \(5892818\) \(\beta_{6}\mathstrut -\mathstrut \) \(9447603\) \(\beta_{5}\mathstrut -\mathstrut \) \(6377791\) \(\beta_{4}\mathstrut +\mathstrut \) \(238909\) \(\beta_{3}\mathstrut -\mathstrut \) \(15188046\) \(\beta_{2}\mathstrut +\mathstrut \) \(31424408\) \(\beta_{1}\mathstrut +\mathstrut \) \(37965894\)

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
−3.26657
−2.86082
−2.68392
−2.44876
−2.07316
−1.52621
−0.719032
0.771393
0.808021
1.64745
1.78856
2.32389
2.32411
3.35142
3.75055
3.81307
1.00000 1.00000 1.00000 −3.26657 1.00000 1.36909 1.00000 1.00000 −3.26657
1.2 1.00000 1.00000 1.00000 −2.86082 1.00000 2.85228 1.00000 1.00000 −2.86082
1.3 1.00000 1.00000 1.00000 −2.68392 1.00000 −4.10513 1.00000 1.00000 −2.68392
1.4 1.00000 1.00000 1.00000 −2.44876 1.00000 −1.82729 1.00000 1.00000 −2.44876
1.5 1.00000 1.00000 1.00000 −2.07316 1.00000 −0.689093 1.00000 1.00000 −2.07316
1.6 1.00000 1.00000 1.00000 −1.52621 1.00000 4.59536 1.00000 1.00000 −1.52621
1.7 1.00000 1.00000 1.00000 −0.719032 1.00000 −5.05986 1.00000 1.00000 −0.719032
1.8 1.00000 1.00000 1.00000 0.771393 1.00000 −0.402524 1.00000 1.00000 0.771393
1.9 1.00000 1.00000 1.00000 0.808021 1.00000 3.24300 1.00000 1.00000 0.808021
1.10 1.00000 1.00000 1.00000 1.64745 1.00000 3.14335 1.00000 1.00000 1.64745
1.11 1.00000 1.00000 1.00000 1.78856 1.00000 0.327596 1.00000 1.00000 1.78856
1.12 1.00000 1.00000 1.00000 2.32389 1.00000 −0.374750 1.00000 1.00000 2.32389
1.13 1.00000 1.00000 1.00000 2.32411 1.00000 −3.70744 1.00000 1.00000 2.32411
1.14 1.00000 1.00000 1.00000 3.35142 1.00000 2.91939 1.00000 1.00000 3.35142
1.15 1.00000 1.00000 1.00000 3.75055 1.00000 −2.75149 1.00000 1.00000 3.75055
1.16 1.00000 1.00000 1.00000 3.81307 1.00000 4.46750 1.00000 1.00000 3.81307
\(n\): 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
\(2\) \(-1\)
\(3\) \(-1\)
\(13\) \(1\)
\(103\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

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