Properties

Label 1.36.a.a
Level 1
Weight 36
Character orbit 1.a
Self dual Yes
Analytic conductor 7.760
Analytic rank 0
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 36 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(7.75951306336\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{12}\cdot 3^{3}\cdot 5\cdot 7 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 46552 + \beta_{1} ) q^{2} \) \( + ( -34958436 + \beta_{1} - \beta_{2} ) q^{3} \) \( + ( 11613754048 + 194112 \beta_{1} + 72 \beta_{2} ) q^{4} \) \( + ( 297550684670 + 7444916 \beta_{1} - 2484 \beta_{2} ) q^{5} \) \( + ( -1595510188128 - 197319012 \beta_{1} + 54528 \beta_{2} ) q^{6} \) \( + ( 292807383115352 + 723239370 \beta_{1} - 852426 \beta_{2} ) q^{7} \) \( + ( 7445336641779200 + 17597768192 \beta_{1} + 10055232 \beta_{2} ) q^{8} \) \( + ( 50030659625414517 - 228209975496 \beta_{1} - 92389176 \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \(+(46552 + \beta_{1}) q^{2}\) \(+(-34958436 + \beta_{1} - \beta_{2}) q^{3}\) \(+(11613754048 + 194112 \beta_{1} + 72 \beta_{2}) q^{4}\) \(+(297550684670 + 7444916 \beta_{1} - 2484 \beta_{2}) q^{5}\) \(+(-1595510188128 - 197319012 \beta_{1} + 54528 \beta_{2}) q^{6}\) \(+(292807383115352 + 723239370 \beta_{1} - 852426 \beta_{2}) q^{7}\) \(+(7445336641779200 + 17597768192 \beta_{1} + 10055232 \beta_{2}) q^{8}\) \(+(50030659625414517 - 228209975496 \beta_{1} - 92389176 \beta_{2}) q^{9}\) \(+(339956937576432720 + 992452279806 \beta_{1} + 671302656 \beta_{2}) q^{10}\) \(+(-385981809516662348 + 583640593195 \beta_{1} - 3858182955 \beta_{2}) q^{11}\) \(+(-7516297175592162048 - 21885019697408 \beta_{1} + 17183392736 \beta_{2}) q^{12}\) \(+(-20713203516999550186 + 54628153929972 \beta_{1} - 55424393460 \beta_{2}) q^{13}\) \(+(45303114418357146176 + 261002424214616 \beta_{1} + 98492944896 \beta_{2}) q^{14}\) \(+(\)\(23\!\cdots\!40\)\( - 2035368980904018 \beta_{1} + 138398873682 \beta_{2}) q^{15}\) \(+(\)\(71\!\cdots\!16\)\( + 5006482791469056 \beta_{1} - 1754429566464 \beta_{2}) q^{16}\) \(+(-\)\(13\!\cdots\!98\)\( - 2825301075479112 \beta_{1} + 6543407393352 \beta_{2}) q^{17}\) \(+(-\)\(76\!\cdots\!36\)\( + 1342002862888821 \beta_{1} - 11399973267456 \beta_{2}) q^{18}\) \(+(-\)\(10\!\cdots\!80\)\( - 84000151153806507 \beta_{1} - 9872745919317 \beta_{2}) q^{19}\) \(+(\)\(49\!\cdots\!60\)\( + 339689973371278208 \beta_{1} + 120249696817008 \beta_{2}) q^{20}\) \(+(\)\(74\!\cdots\!92\)\( - 336783816338068624 \beta_{1} - 328342926322544 \beta_{2}) q^{21}\) \(+(\)\(75\!\cdots\!04\)\( - 926845923948830028 \beta_{1} + 252123333707520 \beta_{2}) q^{22}\) \(+(-\)\(17\!\cdots\!36\)\( + 2541836449593511358 \beta_{1} + 1160877406548546 \beta_{2}) q^{23}\) \(+(-\)\(12\!\cdots\!40\)\( - 1173379931106134016 \beta_{1} - 4385028066775296 \beta_{2}) q^{24}\) \(+(\)\(21\!\cdots\!75\)\( + 2612767956475898160 \beta_{1} + 5245695645374160 \beta_{2}) q^{25}\) \(+(\)\(14\!\cdots\!12\)\( - 21659187993208072426 \beta_{1} + 6951417853215744 \beta_{2}) q^{26}\) \(+(\)\(91\!\cdots\!00\)\( + 23978122270590027258 \beta_{1} - 34841458485708282 \beta_{2}) q^{27}\) \(+(\)\(34\!\cdots\!36\)\( + 74972221489028408832 \beta_{1} + 42717777074276544 \beta_{2}) q^{28}\) \(+(-\)\(12\!\cdots\!70\)\( - \)\(16\!\cdots\!08\)\( \beta_{1} + 30897215823693852 \beta_{2}) q^{29}\) \(+(-\)\(78\!\cdots\!60\)\( - 42493944051513394488 \beta_{1} - 154083215690316288 \beta_{2}) q^{30}\) \(+(\)\(34\!\cdots\!52\)\( + \)\(18\!\cdots\!60\)\( \beta_{1} + 125813443508636760 \beta_{2}) q^{31}\) \(+(-\)\(30\!\cdots\!48\)\( + \)\(56\!\cdots\!16\)\( \beta_{1} + 110510836707594240 \beta_{2}) q^{32}\) \(+(\)\(39\!\cdots\!28\)\( - \)\(99\!\cdots\!08\)\( \beta_{1} - 46744170434631912 \beta_{2}) q^{33}\) \(+(-\)\(18\!\cdots\!04\)\( - \)\(66\!\cdots\!46\)\( \beta_{1} - 559749470446872576 \beta_{2}) q^{34}\) \(+(\)\(53\!\cdots\!20\)\( + \)\(11\!\cdots\!96\)\( \beta_{1} - 114907039302000204 \beta_{2}) q^{35}\) \(+(-\)\(20\!\cdots\!84\)\( - \)\(14\!\cdots\!64\)\( \beta_{1} + 3891889065775683816 \beta_{2}) q^{36}\) \(+(\)\(82\!\cdots\!02\)\( + \)\(15\!\cdots\!64\)\( \beta_{1} - 5309319421076092452 \beta_{2}) q^{37}\) \(+(-\)\(41\!\cdots\!00\)\( - \)\(24\!\cdots\!12\)\( \beta_{1} - 5510380631291741952 \beta_{2}) q^{38}\) \(+(\)\(62\!\cdots\!04\)\( - \)\(23\!\cdots\!54\)\( \beta_{1} + 19171250864870765626 \beta_{2}) q^{39}\) \(+(\)\(54\!\cdots\!00\)\( + \)\(84\!\cdots\!20\)\( \beta_{1} - 5156423033038462080 \beta_{2}) q^{40}\) \(+(\)\(78\!\cdots\!02\)\( - \)\(39\!\cdots\!40\)\( \beta_{1} - 24089178969404126640 \beta_{2}) q^{41}\) \(+(-\)\(11\!\cdots\!96\)\( - \)\(29\!\cdots\!32\)\( \beta_{1} - 6368192380520484864 \beta_{2}) q^{42}\) \(+(-\)\(15\!\cdots\!36\)\( + \)\(17\!\cdots\!35\)\( \beta_{1} + 66376501078524728013 \beta_{2}) q^{43}\) \(+(-\)\(26\!\cdots\!04\)\( - \)\(10\!\cdots\!16\)\( \beta_{1} + 52103622124984646304 \beta_{2}) q^{44}\) \(+(-\)\(36\!\cdots\!10\)\( + \)\(61\!\cdots\!52\)\( \beta_{1} - \)\(29\!\cdots\!48\)\( \beta_{2}) q^{45}\) \(+(\)\(10\!\cdots\!52\)\( + \)\(39\!\cdots\!00\)\( \beta_{1} + \)\(11\!\cdots\!00\)\( \beta_{2}) q^{46}\) \(+(\)\(54\!\cdots\!52\)\( + \)\(86\!\cdots\!72\)\( \beta_{1} + \)\(51\!\cdots\!16\)\( \beta_{2}) q^{47}\) \(+(\)\(14\!\cdots\!64\)\( - \)\(13\!\cdots\!12\)\( \beta_{1} - \)\(43\!\cdots\!24\)\( \beta_{2}) q^{48}\) \(+(-\)\(19\!\cdots\!07\)\( + \)\(11\!\cdots\!80\)\( \beta_{1} - \)\(45\!\cdots\!20\)\( \beta_{2}) q^{49}\) \(+(\)\(12\!\cdots\!00\)\( + \)\(14\!\cdots\!35\)\( \beta_{1} - 97540309198230589440 \beta_{2}) q^{50}\) \(+(-\)\(60\!\cdots\!68\)\( + \)\(20\!\cdots\!62\)\( \beta_{1} + \)\(18\!\cdots\!22\)\( \beta_{2}) q^{51}\) \(+(-\)\(17\!\cdots\!48\)\( - \)\(25\!\cdots\!60\)\( \beta_{1} - 33640287635007496656 \beta_{2}) q^{52}\) \(+(-\)\(55\!\cdots\!86\)\( - \)\(47\!\cdots\!24\)\( \beta_{1} - \)\(46\!\cdots\!76\)\( \beta_{2}) q^{53}\) \(+(\)\(14\!\cdots\!20\)\( + \)\(70\!\cdots\!28\)\( \beta_{1} + \)\(36\!\cdots\!68\)\( \beta_{2}) q^{54}\) \(+(\)\(10\!\cdots\!40\)\( - \)\(93\!\cdots\!18\)\( \beta_{1} + \)\(16\!\cdots\!82\)\( \beta_{2}) q^{55}\) \(+(\)\(18\!\cdots\!80\)\( + \)\(12\!\cdots\!52\)\( \beta_{1} - \)\(31\!\cdots\!88\)\( \beta_{2}) q^{56}\) \(+(\)\(13\!\cdots\!00\)\( + \)\(14\!\cdots\!36\)\( \beta_{1} + \)\(10\!\cdots\!56\)\( \beta_{2}) q^{57}\) \(+(-\)\(79\!\cdots\!00\)\( - \)\(32\!\cdots\!78\)\( \beta_{1} - \)\(13\!\cdots\!88\)\( \beta_{2}) q^{58}\) \(+(\)\(14\!\cdots\!60\)\( + \)\(83\!\cdots\!79\)\( \beta_{1} + \)\(13\!\cdots\!49\)\( \beta_{2}) q^{59}\) \(+(-\)\(13\!\cdots\!80\)\( - \)\(39\!\cdots\!84\)\( \beta_{1} + \)\(57\!\cdots\!16\)\( \beta_{2}) q^{60}\) \(+(\)\(78\!\cdots\!02\)\( + \)\(49\!\cdots\!00\)\( \beta_{1} + \)\(10\!\cdots\!00\)\( \beta_{2}) q^{61}\) \(+(\)\(98\!\cdots\!04\)\( + \)\(82\!\cdots\!12\)\( \beta_{1} + \)\(66\!\cdots\!60\)\( \beta_{2}) q^{62}\) \(+(\)\(15\!\cdots\!64\)\( - \)\(44\!\cdots\!86\)\( \beta_{1} - \)\(10\!\cdots\!78\)\( \beta_{2}) q^{63}\) \(+(\)\(31\!\cdots\!28\)\( - \)\(73\!\cdots\!56\)\( \beta_{1} + \)\(95\!\cdots\!64\)\( \beta_{2}) q^{64}\) \(+(\)\(25\!\cdots\!40\)\( - \)\(21\!\cdots\!08\)\( \beta_{1} + \)\(97\!\cdots\!92\)\( \beta_{2}) q^{65}\) \(+(-\)\(25\!\cdots\!56\)\( + \)\(24\!\cdots\!16\)\( \beta_{1} - \)\(69\!\cdots\!04\)\( \beta_{2}) q^{66}\) \(+(-\)\(62\!\cdots\!48\)\( + \)\(32\!\cdots\!53\)\( \beta_{1} - \)\(10\!\cdots\!33\)\( \beta_{2}) q^{67}\) \(+(\)\(73\!\cdots\!36\)\( - \)\(27\!\cdots\!84\)\( \beta_{1} - \)\(24\!\cdots\!92\)\( \beta_{2}) q^{68}\) \(+(-\)\(10\!\cdots\!16\)\( - \)\(23\!\cdots\!32\)\( \beta_{1} + \)\(57\!\cdots\!08\)\( \beta_{2}) q^{69}\) \(+(\)\(74\!\cdots\!20\)\( + \)\(68\!\cdots\!36\)\( \beta_{1} + \)\(87\!\cdots\!36\)\( \beta_{2}) q^{70}\) \(+(\)\(11\!\cdots\!52\)\( - \)\(58\!\cdots\!50\)\( \beta_{1} - \)\(52\!\cdots\!50\)\( \beta_{2}) q^{71}\) \(+(\)\(10\!\cdots\!00\)\( - \)\(16\!\cdots\!76\)\( \beta_{1} + \)\(73\!\cdots\!04\)\( \beta_{2}) q^{72}\) \(+(-\)\(95\!\cdots\!86\)\( + \)\(17\!\cdots\!28\)\( \beta_{1} - \)\(75\!\cdots\!84\)\( \beta_{2}) q^{73}\) \(+(\)\(70\!\cdots\!36\)\( + \)\(21\!\cdots\!70\)\( \beta_{1} + \)\(13\!\cdots\!20\)\( \beta_{2}) q^{74}\) \(+(-\)\(52\!\cdots\!00\)\( + \)\(68\!\cdots\!95\)\( \beta_{1} + \)\(71\!\cdots\!45\)\( \beta_{2}) q^{75}\) \(+(-\)\(90\!\cdots\!40\)\( - \)\(58\!\cdots\!16\)\( \beta_{1} - \)\(11\!\cdots\!96\)\( \beta_{2}) q^{76}\) \(+(\)\(23\!\cdots\!04\)\( - \)\(13\!\cdots\!40\)\( \beta_{1} - \)\(97\!\cdots\!32\)\( \beta_{2}) q^{77}\) \(+(-\)\(73\!\cdots\!72\)\( + \)\(58\!\cdots\!00\)\( \beta_{1} - \)\(27\!\cdots\!44\)\( \beta_{2}) q^{78}\) \(+(-\)\(14\!\cdots\!20\)\( - \)\(26\!\cdots\!48\)\( \beta_{1} + \)\(62\!\cdots\!12\)\( \beta_{2}) q^{79}\) \(+(\)\(22\!\cdots\!20\)\( + \)\(54\!\cdots\!76\)\( \beta_{1} + \)\(22\!\cdots\!76\)\( \beta_{2}) q^{80}\) \(+(\)\(62\!\cdots\!21\)\( - \)\(12\!\cdots\!12\)\( \beta_{1} - \)\(65\!\cdots\!72\)\( \beta_{2}) q^{81}\) \(+(-\)\(13\!\cdots\!96\)\( - \)\(18\!\cdots\!38\)\( \beta_{1} - \)\(15\!\cdots\!40\)\( \beta_{2}) q^{82}\) \(+(\)\(49\!\cdots\!64\)\( - \)\(42\!\cdots\!51\)\( \beta_{1} - \)\(21\!\cdots\!93\)\( \beta_{2}) q^{83}\) \(+(-\)\(43\!\cdots\!84\)\( - \)\(50\!\cdots\!88\)\( \beta_{1} + \)\(95\!\cdots\!72\)\( \beta_{2}) q^{84}\) \(+(-\)\(29\!\cdots\!80\)\( + \)\(35\!\cdots\!76\)\( \beta_{1} + \)\(70\!\cdots\!76\)\( \beta_{2}) q^{85}\) \(+(\)\(49\!\cdots\!32\)\( - \)\(24\!\cdots\!68\)\( \beta_{1} - \)\(23\!\cdots\!08\)\( \beta_{2}) q^{86}\) \(+(-\)\(26\!\cdots\!00\)\( + \)\(40\!\cdots\!34\)\( \beta_{1} - \)\(13\!\cdots\!86\)\( \beta_{2}) q^{87}\) \(+(-\)\(62\!\cdots\!00\)\( - \)\(26\!\cdots\!16\)\( \beta_{1} - \)\(19\!\cdots\!36\)\( \beta_{2}) q^{88}\) \(+(\)\(10\!\cdots\!90\)\( - \)\(67\!\cdots\!44\)\( \beta_{1} + \)\(91\!\cdots\!36\)\( \beta_{2}) q^{89}\) \(+(\)\(95\!\cdots\!40\)\( - \)\(76\!\cdots\!18\)\( \beta_{1} + \)\(20\!\cdots\!32\)\( \beta_{2}) q^{90}\) \(+(\)\(33\!\cdots\!32\)\( - \)\(19\!\cdots\!48\)\( \beta_{1} + \)\(53\!\cdots\!12\)\( \beta_{2}) q^{91}\) \(+(\)\(27\!\cdots\!52\)\( + \)\(93\!\cdots\!08\)\( \beta_{1} - \)\(18\!\cdots\!28\)\( \beta_{2}) q^{92}\) \(+(-\)\(13\!\cdots\!72\)\( - \)\(81\!\cdots\!28\)\( \beta_{1} + \)\(35\!\cdots\!68\)\( \beta_{2}) q^{93}\) \(+(\)\(63\!\cdots\!56\)\( + \)\(15\!\cdots\!48\)\( \beta_{1} - \)\(21\!\cdots\!12\)\( \beta_{2}) q^{94}\) \(+(-\)\(28\!\cdots\!00\)\( - \)\(15\!\cdots\!70\)\( \beta_{1} - \)\(38\!\cdots\!70\)\( \beta_{2}) q^{95}\) \(+(-\)\(10\!\cdots\!28\)\( - \)\(86\!\cdots\!32\)\( \beta_{1} + \)\(74\!\cdots\!08\)\( \beta_{2}) q^{96}\) \(+(-\)\(35\!\cdots\!98\)\( + \)\(99\!\cdots\!92\)\( \beta_{1} + \)\(89\!\cdots\!36\)\( \beta_{2}) q^{97}\) \(+(-\)\(44\!\cdots\!64\)\( - \)\(25\!\cdots\!27\)\( \beta_{1} + \)\(32\!\cdots\!80\)\( \beta_{2}) q^{98}\) \(+(\)\(10\!\cdots\!84\)\( + \)\(15\!\cdots\!23\)\( \beta_{1} - \)\(30\!\cdots\!87\)\( \beta_{2}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut +\mathstrut 139656q^{2} \) \(\mathstrut -\mathstrut 104875308q^{3} \) \(\mathstrut +\mathstrut 34841262144q^{4} \) \(\mathstrut +\mathstrut 892652054010q^{5} \) \(\mathstrut -\mathstrut 4786530564384q^{6} \) \(\mathstrut +\mathstrut 878422149346056q^{7} \) \(\mathstrut +\mathstrut 22336009925337600q^{8} \) \(\mathstrut +\mathstrut 150091978876243551q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut +\mathstrut 139656q^{2} \) \(\mathstrut -\mathstrut 104875308q^{3} \) \(\mathstrut +\mathstrut 34841262144q^{4} \) \(\mathstrut +\mathstrut 892652054010q^{5} \) \(\mathstrut -\mathstrut 4786530564384q^{6} \) \(\mathstrut +\mathstrut 878422149346056q^{7} \) \(\mathstrut +\mathstrut 22336009925337600q^{8} \) \(\mathstrut +\mathstrut 150091978876243551q^{9} \) \(\mathstrut +\mathstrut 1019870812729298160q^{10} \) \(\mathstrut -\mathstrut 1157945428549987044q^{11} \) \(\mathstrut -\mathstrut 22548891526776486144q^{12} \) \(\mathstrut -\mathstrut 62139610550998650558q^{13} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!28\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(70\!\cdots\!20\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!48\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!94\)\(q^{17} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!08\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!40\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!80\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!76\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!12\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(51\!\cdots\!08\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!20\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!25\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!36\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!00\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!08\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!10\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!80\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!56\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(92\!\cdots\!44\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!84\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!12\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!60\)\(q^{35} \) \(\mathstrut -\mathstrut \)\(60\!\cdots\!52\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!06\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!00\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!12\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!06\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!88\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!08\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(80\!\cdots\!12\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!30\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!56\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!56\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!92\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(59\!\cdots\!21\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!04\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(51\!\cdots\!44\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!58\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!60\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!20\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(56\!\cdots\!40\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!00\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!00\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!80\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!40\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!06\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!12\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!92\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(93\!\cdots\!84\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(75\!\cdots\!20\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(75\!\cdots\!68\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!44\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!08\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!48\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!60\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!56\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!00\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!58\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!08\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!20\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(69\!\cdots\!12\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!16\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!60\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(68\!\cdots\!60\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!63\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!88\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!92\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!52\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!40\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!96\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(78\!\cdots\!00\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!00\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!70\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!20\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!96\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(83\!\cdots\!56\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!16\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!68\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(84\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!84\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!94\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!92\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!52\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(12422194\) \(x\mathstrut -\mathstrut \) \(2645665785\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 24 \nu^{2} + 44712 \nu - 198755104 \)\()/979\)
\(\beta_{2}\)\(=\)\((\)\( -39144 \nu^{2} + 84967848 \nu + 324169574624 \)\()/979\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(1631\) \(\beta_{1}\)\()/161280\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(621\) \(\beta_{2}\mathstrut +\mathstrut \) \(1180109\) \(\beta_{1}\mathstrut +\mathstrut \) \(445211432960\)\()/53760\)

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
−213.765
−3412.77
3626.53
−165109. −3.45913e8 −7.09870e9 −2.05014e12 5.71135e13 −1.25160e14 6.84517e15 6.96245e16 3.38496e17
1.2 −26808.0 3.95729e8 −3.36411e10 8.21401e11 −1.06087e13 6.06942e14 1.82297e15 1.06570e17 −2.20201e16
1.3 331573. −1.54691e8 7.55810e10 2.12139e12 −5.12913e13 3.96640e14 1.36679e16 −2.61023e16 7.03395e17
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{36}^{\mathrm{new}}(\Gamma_0(1))\).