Properties

Label 8022.2.a.ba
Level 8022
Weight 2
Character orbit 8022.a
Self dual Yes
Analytic conductor 64.056
Analytic rank 0
Dimension 16
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8022 = 2 \cdot 3 \cdot 7 \cdot 191 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8022.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.0559925015\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16}\mathstrut -\mathstrut \) \(5\) \(x^{15}\mathstrut -\mathstrut \) \(39\) \(x^{14}\mathstrut +\mathstrut \) \(165\) \(x^{13}\mathstrut +\mathstrut \) \(711\) \(x^{12}\mathstrut -\mathstrut \) \(1949\) \(x^{11}\mathstrut -\mathstrut \) \(7497\) \(x^{10}\mathstrut +\mathstrut \) \(8238\) \(x^{9}\mathstrut +\mathstrut \) \(42708\) \(x^{8}\mathstrut +\mathstrut \) \(7872\) \(x^{7}\mathstrut -\mathstrut \) \(97852\) \(x^{6}\mathstrut -\mathstrut \) \(109082\) \(x^{5}\mathstrut -\mathstrut \) \(6830\) \(x^{4}\mathstrut +\mathstrut \) \(49478\) \(x^{3}\mathstrut +\mathstrut \) \(28368\) \(x^{2}\mathstrut +\mathstrut \) \(4947\) \(x\mathstrut +\mathstrut \) \(30\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 6 \)
\(\beta_{3}\)\(=\)\((\)\(360816121747607\) \(\nu^{15}\mathstrut -\mathstrut \) \(1858971373994450\) \(\nu^{14}\mathstrut -\mathstrut \) \(13805565080432691\) \(\nu^{13}\mathstrut +\mathstrut \) \(61635111797442546\) \(\nu^{12}\mathstrut +\mathstrut \) \(248530144229996007\) \(\nu^{11}\mathstrut -\mathstrut \) \(742890387441907954\) \(\nu^{10}\mathstrut -\mathstrut \) \(2623802025941701713\) \(\nu^{9}\mathstrut +\mathstrut \) \(3415010954098860007\) \(\nu^{8}\mathstrut +\mathstrut \) \(15223820634780724357\) \(\nu^{7}\mathstrut +\mathstrut \) \(211795117501297639\) \(\nu^{6}\mathstrut -\mathstrut \) \(37080512123104918339\) \(\nu^{5}\mathstrut -\mathstrut \) \(33232480427324300463\) \(\nu^{4}\mathstrut +\mathstrut \) \(6613899958671435893\) \(\nu^{3}\mathstrut +\mathstrut \) \(18337341498917888609\) \(\nu^{2}\mathstrut +\mathstrut \) \(5645750959318185599\) \(\nu\mathstrut -\mathstrut \) \(786237527426014\)\()/\)\(15467244183346168\)
\(\beta_{4}\)\(=\)\((\)\(2404738159108160\) \(\nu^{15}\mathstrut -\mathstrut \) \(16987479144925981\) \(\nu^{14}\mathstrut -\mathstrut \) \(63359138466328507\) \(\nu^{13}\mathstrut +\mathstrut \) \(556926640317639770\) \(\nu^{12}\mathstrut +\mathstrut \) \(704031743998173156\) \(\nu^{11}\mathstrut -\mathstrut \) \(7108828222870656881\) \(\nu^{10}\mathstrut -\mathstrut \) \(5425902390484073439\) \(\nu^{9}\mathstrut +\mathstrut \) \(43204386200753250296\) \(\nu^{8}\mathstrut +\mathstrut \) \(32963521934699762419\) \(\nu^{7}\mathstrut -\mathstrut \) \(118193251096346846144\) \(\nu^{6}\mathstrut -\mathstrut \) \(105767928309356796181\) \(\nu^{5}\mathstrut +\mathstrut \) \(99995425241484980304\) \(\nu^{4}\mathstrut +\mathstrut \) \(87147191499674307957\) \(\nu^{3}\mathstrut -\mathstrut \) \(26080361166734809590\) \(\nu^{2}\mathstrut -\mathstrut \) \(20396488439231843633\) \(\nu\mathstrut -\mathstrut \) \(197929973671692594\)\()/\)\(15467244183346168\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(2558980742353787\) \(\nu^{15}\mathstrut +\mathstrut \) \(12232491470418130\) \(\nu^{14}\mathstrut +\mathstrut \) \(104371850077057335\) \(\nu^{13}\mathstrut -\mathstrut \) \(411065037244848482\) \(\nu^{12}\mathstrut -\mathstrut \) \(1969401565764964083\) \(\nu^{11}\mathstrut +\mathstrut \) \(4945956493117627826\) \(\nu^{10}\mathstrut +\mathstrut \) \(21147008604277527989\) \(\nu^{9}\mathstrut -\mathstrut \) \(21392200004168577315\) \(\nu^{8}\mathstrut -\mathstrut \) \(122249419155915594369\) \(\nu^{7}\mathstrut -\mathstrut \) \(18783933861075731563\) \(\nu^{6}\mathstrut +\mathstrut \) \(294552047494053709207\) \(\nu^{5}\mathstrut +\mathstrut \) \(285006020020213608459\) \(\nu^{4}\mathstrut -\mathstrut \) \(49774140720887313721\) \(\nu^{3}\mathstrut -\mathstrut \) \(152600456535277116125\) \(\nu^{2}\mathstrut -\mathstrut \) \(46218272612677154851\) \(\nu\mathstrut -\mathstrut \) \(194353346975362418\)\()/\)\(15467244183346168\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(2991332914508731\) \(\nu^{15}\mathstrut +\mathstrut \) \(15464967044133522\) \(\nu^{14}\mathstrut +\mathstrut \) \(114786322332772087\) \(\nu^{13}\mathstrut -\mathstrut \) \(518000325996077466\) \(\nu^{12}\mathstrut -\mathstrut \) \(2060790796879351723\) \(\nu^{11}\mathstrut +\mathstrut \) \(6339871045544305826\) \(\nu^{10}\mathstrut +\mathstrut \) \(21645920032065511477\) \(\nu^{9}\mathstrut -\mathstrut \) \(30297133756549208803\) \(\nu^{8}\mathstrut -\mathstrut \) \(125337516182122262673\) \(\nu^{7}\mathstrut +\mathstrut \) \(8829062928396849189\) \(\nu^{6}\mathstrut +\mathstrut \) \(307628718039426194215\) \(\nu^{5}\mathstrut +\mathstrut \) \(250781864272867259211\) \(\nu^{4}\mathstrut -\mathstrut \) \(67780218347351236265\) \(\nu^{3}\mathstrut -\mathstrut \) \(140394519662950836693\) \(\nu^{2}\mathstrut -\mathstrut \) \(39915087438466782467\) \(\nu\mathstrut -\mathstrut \) \(216429364161712730\)\()/\)\(15467244183346168\)
\(\beta_{7}\)\(=\)\((\)\(5079188575302172\) \(\nu^{15}\mathstrut -\mathstrut \) \(24836877130054383\) \(\nu^{14}\mathstrut -\mathstrut \) \(203784340124885733\) \(\nu^{13}\mathstrut +\mathstrut \) \(834413914156353318\) \(\nu^{12}\mathstrut +\mathstrut \) \(3794756418582891544\) \(\nu^{11}\mathstrut -\mathstrut \) \(10100920875896730387\) \(\nu^{10}\mathstrut -\mathstrut \) \(40505641517835511729\) \(\nu^{9}\mathstrut +\mathstrut \) \(45177506828371373076\) \(\nu^{8}\mathstrut +\mathstrut \) \(234219062222588163093\) \(\nu^{7}\mathstrut +\mathstrut \) \(21584289840736892620\) \(\nu^{6}\mathstrut -\mathstrut \) \(567182384518172851387\) \(\nu^{5}\mathstrut -\mathstrut \) \(524032562856090369444\) \(\nu^{4}\mathstrut +\mathstrut \) \(104336936618704405859\) \(\nu^{3}\mathstrut +\mathstrut \) \(283467616674894662498\) \(\nu^{2}\mathstrut +\mathstrut \) \(84674555479338101721\) \(\nu\mathstrut +\mathstrut \) \(486441051260577946\)\()/\)\(15467244183346168\)
\(\beta_{8}\)\(=\)\((\)\(8004691284267924\) \(\nu^{15}\mathstrut -\mathstrut \) \(44624892068500975\) \(\nu^{14}\mathstrut -\mathstrut \) \(286332989876117789\) \(\nu^{13}\mathstrut +\mathstrut \) \(1484026363776827382\) \(\nu^{12}\mathstrut +\mathstrut \) \(4832906612301154232\) \(\nu^{11}\mathstrut -\mathstrut \) \(18336730513490002251\) \(\nu^{10}\mathstrut -\mathstrut \) \(49404537451860076681\) \(\nu^{9}\mathstrut +\mathstrut \) \(93826817662890676700\) \(\nu^{8}\mathstrut +\mathstrut \) \(287342956291792079285\) \(\nu^{7}\mathstrut -\mathstrut \) \(99320255363589946260\) \(\nu^{6}\mathstrut -\mathstrut \) \(722538951525587731931\) \(\nu^{5}\mathstrut -\mathstrut \) \(463819096298434583292\) \(\nu^{4}\mathstrut +\mathstrut \) \(201474426481247595027\) \(\nu^{3}\mathstrut +\mathstrut \) \(279778476919650493954\) \(\nu^{2}\mathstrut +\mathstrut \) \(71142288813654706193\) \(\nu\mathstrut +\mathstrut \) \(415075293243338258\)\()/\)\(15467244183346168\)
\(\beta_{9}\)\(=\)\((\)\(579567055437717\) \(\nu^{15}\mathstrut -\mathstrut \) \(3203689799549224\) \(\nu^{14}\mathstrut -\mathstrut \) \(20910475164102167\) \(\nu^{13}\mathstrut +\mathstrut \) \(106648118442492682\) \(\nu^{12}\mathstrut +\mathstrut \) \(355756747599959133\) \(\nu^{11}\mathstrut -\mathstrut \) \(1316874030372372472\) \(\nu^{10}\mathstrut -\mathstrut \) \(3649917894744421345\) \(\nu^{9}\mathstrut +\mathstrut \) \(6695797723592915445\) \(\nu^{8}\mathstrut +\mathstrut \) \(21217851306646318517\) \(\nu^{7}\mathstrut -\mathstrut \) \(6610735426192869659\) \(\nu^{6}\mathstrut -\mathstrut \) \(53209921440128660667\) \(\nu^{5}\mathstrut -\mathstrut \) \(35187849036739123293\) \(\nu^{4}\mathstrut +\mathstrut \) \(14552094429002839169\) \(\nu^{3}\mathstrut +\mathstrut \) \(20993552981857519847\) \(\nu^{2}\mathstrut +\mathstrut \) \(5392259823645795683\) \(\nu\mathstrut +\mathstrut \) \(34037890251575594\)\()/\)\(1066706495403184\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(10941133433520240\) \(\nu^{15}\mathstrut +\mathstrut \) \(59823595444910515\) \(\nu^{14}\mathstrut +\mathstrut \) \(399018860499802485\) \(\nu^{13}\mathstrut -\mathstrut \) \(1994016571440352278\) \(\nu^{12}\mathstrut -\mathstrut \) \(6854037643024684044\) \(\nu^{11}\mathstrut +\mathstrut \) \(24595365120234200111\) \(\nu^{10}\mathstrut +\mathstrut \) \(70607849740783514617\) \(\nu^{9}\mathstrut -\mathstrut \) \(123938037126623810104\) \(\nu^{8}\mathstrut -\mathstrut \) \(410051923081018477293\) \(\nu^{7}\mathstrut +\mathstrut \) \(109923251269844873624\) \(\nu^{6}\mathstrut +\mathstrut \) \(1024041605799865759835\) \(\nu^{5}\mathstrut +\mathstrut \) \(705522686501094250800\) \(\nu^{4}\mathstrut -\mathstrut \) \(269739526255688620755\) \(\nu^{3}\mathstrut -\mathstrut \) \(415583115800183400870\) \(\nu^{2}\mathstrut -\mathstrut \) \(109271632660308650817\) \(\nu\mathstrut -\mathstrut \) \(651614662119315114\)\()/\)\(15467244183346168\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(9527443508841001\) \(\nu^{15}\mathstrut +\mathstrut \) \(52287797891647549\) \(\nu^{14}\mathstrut +\mathstrut \) \(346145108227398466\) \(\nu^{13}\mathstrut -\mathstrut \) \(1741603069018825640\) \(\nu^{12}\mathstrut -\mathstrut \) \(5926979203878492461\) \(\nu^{11}\mathstrut +\mathstrut \) \(21482466332667875025\) \(\nu^{10}\mathstrut +\mathstrut \) \(60986239736629291532\) \(\nu^{9}\mathstrut -\mathstrut \) \(108514030981227852405\) \(\nu^{8}\mathstrut -\mathstrut \) \(354354444624857857452\) \(\nu^{7}\mathstrut +\mathstrut \) \(99436858022080508743\) \(\nu^{6}\mathstrut +\mathstrut \) \(886226559210986271304\) \(\nu^{5}\mathstrut +\mathstrut \) \(603629949625618748029\) \(\nu^{4}\mathstrut -\mathstrut \) \(236263351028544581554\) \(\nu^{3}\mathstrut -\mathstrut \) \(356893752487047328733\) \(\nu^{2}\mathstrut -\mathstrut \) \(93021626809886463686\) \(\nu\mathstrut -\mathstrut \) \(506557334692391336\)\()/\)\(7733622091673084\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(21591937970527439\) \(\nu^{15}\mathstrut +\mathstrut \) \(118594897929666243\) \(\nu^{14}\mathstrut +\mathstrut \) \(783854693111930074\) \(\nu^{13}\mathstrut -\mathstrut \) \(3949983512265484620\) \(\nu^{12}\mathstrut -\mathstrut \) \(13411476319069546827\) \(\nu^{11}\mathstrut +\mathstrut \) \(48727854369508488559\) \(\nu^{10}\mathstrut +\mathstrut \) \(137941047793451810060\) \(\nu^{9}\mathstrut -\mathstrut \) \(246301040459188928927\) \(\nu^{8}\mathstrut -\mathstrut \) \(801439546379944839316\) \(\nu^{7}\mathstrut +\mathstrut \) \(227484916135141617793\) \(\nu^{6}\mathstrut +\mathstrut \) \(2004453200882562867940\) \(\nu^{5}\mathstrut +\mathstrut \) \(1362204620259165093191\) \(\nu^{4}\mathstrut -\mathstrut \) \(533895543805275133670\) \(\nu^{3}\mathstrut -\mathstrut \) \(805685162035829231443\) \(\nu^{2}\mathstrut -\mathstrut \) \(210684441264238947514\) \(\nu\mathstrut -\mathstrut \) \(1403691847395937672\)\()/\)\(15467244183346168\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(44947398674292021\) \(\nu^{15}\mathstrut +\mathstrut \) \(246623488343178522\) \(\nu^{14}\mathstrut +\mathstrut \) \(1633374529348339397\) \(\nu^{13}\mathstrut -\mathstrut \) \(8215010915928139134\) \(\nu^{12}\mathstrut -\mathstrut \) \(27972810516124342485\) \(\nu^{11}\mathstrut +\mathstrut \) \(101332815183607452210\) \(\nu^{10}\mathstrut +\mathstrut \) \(287839209913755729415\) \(\nu^{9}\mathstrut -\mathstrut \) \(511802272066713628341\) \(\nu^{8}\mathstrut -\mathstrut \) \(1672326835389688788195\) \(\nu^{7}\mathstrut +\mathstrut \) \(468205730834366520795\) \(\nu^{6}\mathstrut +\mathstrut \) \(4181602957527339146637\) \(\nu^{5}\mathstrut +\mathstrut \) \(2850421849361708684477\) \(\nu^{4}\mathstrut -\mathstrut \) \(1112198856101897371635\) \(\nu^{3}\mathstrut -\mathstrut \) \(1684429569743577204395\) \(\nu^{2}\mathstrut -\mathstrut \) \(440785885246598371193\) \(\nu\mathstrut -\mathstrut \) \(2822759804203384646\)\()/\)\(30934488366692336\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(49898010571860411\) \(\nu^{15}\mathstrut +\mathstrut \) \(280491031767307208\) \(\nu^{14}\mathstrut +\mathstrut \) \(1770524347055971001\) \(\nu^{13}\mathstrut -\mathstrut \) \(9325177630628903142\) \(\nu^{12}\mathstrut -\mathstrut \) \(29647305713795325235\) \(\nu^{11}\mathstrut +\mathstrut \) \(115410508178166476984\) \(\nu^{10}\mathstrut +\mathstrut \) \(301879302328150439743\) \(\nu^{9}\mathstrut -\mathstrut \) \(595378998924216414363\) \(\nu^{8}\mathstrut -\mathstrut \) \(1756493120817045170267\) \(\nu^{7}\mathstrut +\mathstrut \) \(680308513212160985605\) \(\nu^{6}\mathstrut +\mathstrut \) \(4432177236973505793733\) \(\nu^{5}\mathstrut +\mathstrut \) \(2727470476886055307219\) \(\nu^{4}\mathstrut -\mathstrut \) \(1277094424599821151071\) \(\nu^{3}\mathstrut -\mathstrut \) \(1668618823913183530761\) \(\nu^{2}\mathstrut -\mathstrut \) \(414614960134121586781\) \(\nu\mathstrut -\mathstrut \) \(2589367118518574470\)\()/\)\(30934488366692336\)
\(\beta_{15}\)\(=\)\((\)\(43747026888993092\) \(\nu^{15}\mathstrut -\mathstrut \) \(241477203218813639\) \(\nu^{14}\mathstrut -\mathstrut \) \(1580609691770587901\) \(\nu^{13}\mathstrut +\mathstrut \) \(8040114225572168558\) \(\nu^{12}\mathstrut +\mathstrut \) \(26923740586190315000\) \(\nu^{11}\mathstrut -\mathstrut \) \(99262659708869913203\) \(\nu^{10}\mathstrut -\mathstrut \) \(276345446226916614673\) \(\nu^{9}\mathstrut +\mathstrut \) \(504062365321008933244\) \(\nu^{8}\mathstrut +\mathstrut \) \(1606046117832215286373\) \(\nu^{7}\mathstrut -\mathstrut \) \(490508165139974594836\) \(\nu^{6}\mathstrut -\mathstrut \) \(4024450124744608891107\) \(\nu^{5}\mathstrut -\mathstrut \) \(2679956369515080705364\) \(\nu^{4}\mathstrut +\mathstrut \) \(1091672282942228029875\) \(\nu^{3}\mathstrut +\mathstrut \) \(1596239087488088143754\) \(\nu^{2}\mathstrut +\mathstrut \) \(412433814356981047569\) \(\nu\mathstrut +\mathstrut \) \(2420062394674856346\)\()/\)\(15467244183346168\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(6\)
\(\nu^{3}\)\(=\)\(2\) \(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(2\) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(12\) \(\beta_{1}\mathstrut +\mathstrut \) \(10\)
\(\nu^{4}\)\(=\)\(8\) \(\beta_{15}\mathstrut +\mathstrut \) \(2\) \(\beta_{14}\mathstrut +\mathstrut \) \(5\) \(\beta_{12}\mathstrut +\mathstrut \) \(11\) \(\beta_{11}\mathstrut +\mathstrut \) \(7\) \(\beta_{10}\mathstrut +\mathstrut \) \(6\) \(\beta_{9}\mathstrut +\mathstrut \) \(3\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(4\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(7\) \(\beta_{3}\mathstrut +\mathstrut \) \(14\) \(\beta_{2}\mathstrut +\mathstrut \) \(37\) \(\beta_{1}\mathstrut +\mathstrut \) \(84\)
\(\nu^{5}\)\(=\)\(62\) \(\beta_{15}\mathstrut +\mathstrut \) \(15\) \(\beta_{14}\mathstrut +\mathstrut \) \(5\) \(\beta_{13}\mathstrut +\mathstrut \) \(42\) \(\beta_{12}\mathstrut +\mathstrut \) \(72\) \(\beta_{11}\mathstrut +\mathstrut \) \(54\) \(\beta_{10}\mathstrut +\mathstrut \) \(46\) \(\beta_{9}\mathstrut +\mathstrut \) \(21\) \(\beta_{8}\mathstrut +\mathstrut \) \(4\) \(\beta_{7}\mathstrut -\mathstrut \) \(31\) \(\beta_{6}\mathstrut -\mathstrut \) \(4\) \(\beta_{5}\mathstrut -\mathstrut \) \(8\) \(\beta_{4}\mathstrut -\mathstrut \) \(41\) \(\beta_{3}\mathstrut +\mathstrut \) \(32\) \(\beta_{2}\mathstrut +\mathstrut \) \(190\) \(\beta_{1}\mathstrut +\mathstrut \) \(260\)
\(\nu^{6}\)\(=\)\(291\) \(\beta_{15}\mathstrut +\mathstrut \) \(34\) \(\beta_{14}\mathstrut +\mathstrut \) \(36\) \(\beta_{13}\mathstrut +\mathstrut \) \(216\) \(\beta_{12}\mathstrut +\mathstrut \) \(377\) \(\beta_{11}\mathstrut +\mathstrut \) \(231\) \(\beta_{10}\mathstrut +\mathstrut \) \(260\) \(\beta_{9}\mathstrut +\mathstrut \) \(77\) \(\beta_{8}\mathstrut +\mathstrut \) \(27\) \(\beta_{7}\mathstrut -\mathstrut \) \(147\) \(\beta_{6}\mathstrut -\mathstrut \) \(48\) \(\beta_{5}\mathstrut -\mathstrut \) \(48\) \(\beta_{4}\mathstrut -\mathstrut \) \(232\) \(\beta_{3}\mathstrut +\mathstrut \) \(232\) \(\beta_{2}\mathstrut +\mathstrut \) \(726\) \(\beta_{1}\mathstrut +\mathstrut \) \(1522\)
\(\nu^{7}\)\(=\)\(1678\) \(\beta_{15}\mathstrut +\mathstrut \) \(153\) \(\beta_{14}\mathstrut +\mathstrut \) \(283\) \(\beta_{13}\mathstrut +\mathstrut \) \(1294\) \(\beta_{12}\mathstrut +\mathstrut \) \(2039\) \(\beta_{11}\mathstrut +\mathstrut \) \(1319\) \(\beta_{10}\mathstrut +\mathstrut \) \(1524\) \(\beta_{9}\mathstrut +\mathstrut \) \(382\) \(\beta_{8}\mathstrut +\mathstrut \) \(115\) \(\beta_{7}\mathstrut -\mathstrut \) \(843\) \(\beta_{6}\mathstrut -\mathstrut \) \(157\) \(\beta_{5}\mathstrut -\mathstrut \) \(391\) \(\beta_{4}\mathstrut -\mathstrut \) \(1215\) \(\beta_{3}\mathstrut +\mathstrut \) \(795\) \(\beta_{2}\mathstrut +\mathstrut \) \(3545\) \(\beta_{1}\mathstrut +\mathstrut \) \(6080\)
\(\nu^{8}\)\(=\)\(8237\) \(\beta_{15}\mathstrut +\mathstrut \) \(199\) \(\beta_{14}\mathstrut +\mathstrut \) \(1696\) \(\beta_{13}\mathstrut +\mathstrut \) \(6635\) \(\beta_{12}\mathstrut +\mathstrut \) \(10412\) \(\beta_{11}\mathstrut +\mathstrut \) \(6139\) \(\beta_{10}\mathstrut +\mathstrut \) \(8193\) \(\beta_{9}\mathstrut +\mathstrut \) \(1481\) \(\beta_{8}\mathstrut +\mathstrut \) \(615\) \(\beta_{7}\mathstrut -\mathstrut \) \(4187\) \(\beta_{6}\mathstrut -\mathstrut \) \(1075\) \(\beta_{5}\mathstrut -\mathstrut \) \(2183\) \(\beta_{4}\mathstrut -\mathstrut \) \(6349\) \(\beta_{3}\mathstrut +\mathstrut \) \(4505\) \(\beta_{2}\mathstrut +\mathstrut \) \(15264\) \(\beta_{1}\mathstrut +\mathstrut \) \(31547\)
\(\nu^{9}\)\(=\)\(43381\) \(\beta_{15}\mathstrut -\mathstrut \) \(33\) \(\beta_{14}\mathstrut +\mathstrut \) \(10150\) \(\beta_{13}\mathstrut +\mathstrut \) \(35715\) \(\beta_{12}\mathstrut +\mathstrut \) \(53505\) \(\beta_{11}\mathstrut +\mathstrut \) \(31962\) \(\beta_{10}\mathstrut +\mathstrut \) \(44023\) \(\beta_{9}\mathstrut +\mathstrut \) \(6470\) \(\beta_{8}\mathstrut +\mathstrut \) \(2749\) \(\beta_{7}\mathstrut -\mathstrut \) \(22034\) \(\beta_{6}\mathstrut -\mathstrut \) \(4512\) \(\beta_{5}\mathstrut -\mathstrut \) \(13155\) \(\beta_{4}\mathstrut -\mathstrut \) \(32287\) \(\beta_{3}\mathstrut +\mathstrut \) \(18793\) \(\beta_{2}\mathstrut +\mathstrut \) \(73570\) \(\beta_{1}\mathstrut +\mathstrut \) \(141163\)
\(\nu^{10}\)\(=\)\(215505\) \(\beta_{15}\mathstrut -\mathstrut \) \(8312\) \(\beta_{14}\mathstrut +\mathstrut \) \(55969\) \(\beta_{13}\mathstrut +\mathstrut \) \(181403\) \(\beta_{12}\mathstrut +\mathstrut \) \(269716\) \(\beta_{11}\mathstrut +\mathstrut \) \(154291\) \(\beta_{10}\mathstrut +\mathstrut \) \(229173\) \(\beta_{9}\mathstrut +\mathstrut \) \(25364\) \(\beta_{8}\mathstrut +\mathstrut \) \(13862\) \(\beta_{7}\mathstrut -\mathstrut \) \(110529\) \(\beta_{6}\mathstrut -\mathstrut \) \(25347\) \(\beta_{5}\mathstrut -\mathstrut \) \(70120\) \(\beta_{4}\mathstrut -\mathstrut \) \(163785\) \(\beta_{3}\mathstrut +\mathstrut \) \(96856\) \(\beta_{2}\mathstrut +\mathstrut \) \(337655\) \(\beta_{1}\mathstrut +\mathstrut \) \(703146\)
\(\nu^{11}\)\(=\)\(1097010\) \(\beta_{15}\mathstrut -\mathstrut \) \(63609\) \(\beta_{14}\mathstrut +\mathstrut \) \(304939\) \(\beta_{13}\mathstrut +\mathstrut \) \(935508\) \(\beta_{12}\mathstrut +\mathstrut \) \(1360126\) \(\beta_{11}\mathstrut +\mathstrut \) \(778151\) \(\beta_{10}\mathstrut +\mathstrut \) \(1186154\) \(\beta_{9}\mathstrut +\mathstrut \) \(104128\) \(\beta_{8}\mathstrut +\mathstrut \) \(64341\) \(\beta_{7}\mathstrut -\mathstrut \) \(564141\) \(\beta_{6}\mathstrut -\mathstrut \) \(118642\) \(\beta_{5}\mathstrut -\mathstrut \) \(381387\) \(\beta_{4}\mathstrut -\mathstrut \) \(822828\) \(\beta_{3}\mathstrut +\mathstrut \) \(442747\) \(\beta_{2}\mathstrut +\mathstrut \) \(1630170\) \(\beta_{1}\mathstrut +\mathstrut \) \(3312519\)
\(\nu^{12}\)\(=\)\(5460398\) \(\beta_{15}\mathstrut -\mathstrut \) \(438297\) \(\beta_{14}\mathstrut +\mathstrut \) \(1609003\) \(\beta_{13}\mathstrut +\mathstrut \) \(4712176\) \(\beta_{12}\mathstrut +\mathstrut \) \(6807850\) \(\beta_{11}\mathstrut +\mathstrut \) \(3815343\) \(\beta_{10}\mathstrut +\mathstrut \) \(6055538\) \(\beta_{9}\mathstrut +\mathstrut \) \(405390\) \(\beta_{8}\mathstrut +\mathstrut \) \(318663\) \(\beta_{7}\mathstrut -\mathstrut \) \(2829558\) \(\beta_{6}\mathstrut -\mathstrut \) \(619493\) \(\beta_{5}\mathstrut -\mathstrut \) \(1976443\) \(\beta_{4}\mathstrut -\mathstrut \) \(4124608\) \(\beta_{3}\mathstrut +\mathstrut \) \(2206904\) \(\beta_{2}\mathstrut +\mathstrut \) \(7738192\) \(\beta_{1}\mathstrut +\mathstrut \) \(16308444\)
\(\nu^{13}\)\(=\)\(27405035\) \(\beta_{15}\mathstrut -\mathstrut \) \(2578991\) \(\beta_{14}\mathstrut +\mathstrut \) \(8403668\) \(\beta_{13}\mathstrut +\mathstrut \) \(23835478\) \(\beta_{12}\mathstrut +\mathstrut \) \(34046890\) \(\beta_{11}\mathstrut +\mathstrut \) \(19028939\) \(\beta_{10}\mathstrut +\mathstrut \) \(30761221\) \(\beta_{9}\mathstrut +\mathstrut \) \(1598303\) \(\beta_{8}\mathstrut +\mathstrut \) \(1518339\) \(\beta_{7}\mathstrut -\mathstrut \) \(14257582\) \(\beta_{6}\mathstrut -\mathstrut \) \(3024082\) \(\beta_{5}\mathstrut -\mathstrut \) \(10283226\) \(\beta_{4}\mathstrut -\mathstrut \) \(20599071\) \(\beta_{3}\mathstrut +\mathstrut \) \(10517337\) \(\beta_{2}\mathstrut +\mathstrut \) \(37579539\) \(\beta_{1}\mathstrut +\mathstrut \) \(78702569\)
\(\nu^{14}\)\(=\)\(136313625\) \(\beta_{15}\mathstrut -\mathstrut \) \(14681193\) \(\beta_{14}\mathstrut +\mathstrut \) \(43249623\) \(\beta_{13}\mathstrut +\mathstrut \) \(119348311\) \(\beta_{12}\mathstrut +\mathstrut \) \(169721546\) \(\beta_{11}\mathstrut +\mathstrut \) \(93912573\) \(\beta_{10}\mathstrut +\mathstrut \) \(155213426\) \(\beta_{9}\mathstrut +\mathstrut \) \(6102307\) \(\beta_{8}\mathstrut +\mathstrut \) \(7481057\) \(\beta_{7}\mathstrut -\mathstrut \) \(71339170\) \(\beta_{6}\mathstrut -\mathstrut \) \(15356828\) \(\beta_{5}\mathstrut -\mathstrut \) \(52384820\) \(\beta_{4}\mathstrut -\mathstrut \) \(102718038\) \(\beta_{3}\mathstrut +\mathstrut \) \(51915818\) \(\beta_{2}\mathstrut +\mathstrut \) \(181645646\) \(\beta_{1}\mathstrut +\mathstrut \) \(386866234\)
\(\nu^{15}\)\(=\)\(679813166\) \(\beta_{15}\mathstrut -\mathstrut \) \(79637715\) \(\beta_{14}\mathstrut +\mathstrut \) \(221020013\) \(\beta_{13}\mathstrut +\mathstrut \) \(598017898\) \(\beta_{12}\mathstrut +\mathstrut \) \(845330606\) \(\beta_{11}\mathstrut +\mathstrut \) \(466564740\) \(\beta_{10}\mathstrut +\mathstrut \) \(780468878\) \(\beta_{9}\mathstrut +\mathstrut \) \(23128932\) \(\beta_{8}\mathstrut +\mathstrut \) \(36231015\) \(\beta_{7}\mathstrut -\mathstrut \) \(357194170\) \(\beta_{6}\mathstrut -\mathstrut \) \(76050960\) \(\beta_{5}\mathstrut -\mathstrut \) \(266700278\) \(\beta_{4}\mathstrut -\mathstrut \) \(511398393\) \(\beta_{3}\mathstrut +\mathstrut \) \(252210831\) \(\beta_{2}\mathstrut +\mathstrut \) \(887344384\) \(\beta_{1}\mathstrut +\mathstrut \) \(1889208920\)

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
4.95576
3.97835
3.85724
3.15148
2.58623
0.754154
−0.00628856
−0.510835
−0.518182
−0.811854
−0.831480
−1.54551
−1.98989
−2.05325
−2.95354
−3.06239
1.00000 1.00000 1.00000 −3.95576 1.00000 1.00000 1.00000 1.00000 −3.95576
1.2 1.00000 1.00000 1.00000 −2.97835 1.00000 1.00000 1.00000 1.00000 −2.97835
1.3 1.00000 1.00000 1.00000 −2.85724 1.00000 1.00000 1.00000 1.00000 −2.85724
1.4 1.00000 1.00000 1.00000 −2.15148 1.00000 1.00000 1.00000 1.00000 −2.15148
1.5 1.00000 1.00000 1.00000 −1.58623 1.00000 1.00000 1.00000 1.00000 −1.58623
1.6 1.00000 1.00000 1.00000 0.245846 1.00000 1.00000 1.00000 1.00000 0.245846
1.7 1.00000 1.00000 1.00000 1.00629 1.00000 1.00000 1.00000 1.00000 1.00629
1.8 1.00000 1.00000 1.00000 1.51084 1.00000 1.00000 1.00000 1.00000 1.51084
1.9 1.00000 1.00000 1.00000 1.51818 1.00000 1.00000 1.00000 1.00000 1.51818
1.10 1.00000 1.00000 1.00000 1.81185 1.00000 1.00000 1.00000 1.00000 1.81185
1.11 1.00000 1.00000 1.00000 1.83148 1.00000 1.00000 1.00000 1.00000 1.83148
1.12 1.00000 1.00000 1.00000 2.54551 1.00000 1.00000 1.00000 1.00000 2.54551
1.13 1.00000 1.00000 1.00000 2.98989 1.00000 1.00000 1.00000 1.00000 2.98989
1.14 1.00000 1.00000 1.00000 3.05325 1.00000 1.00000 1.00000 1.00000 3.05325
1.15 1.00000 1.00000 1.00000 3.95354 1.00000 1.00000 1.00000 1.00000 3.95354
1.16 1.00000 1.00000 1.00000 4.06239 1.00000 1.00000 1.00000 1.00000 4.06239
\(n\): 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
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(191\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8022))\):

\(T_{5}^{16} - \cdots\)
\(T_{11}^{16} - \cdots\)
\(T_{13}^{16} - \cdots\)