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: \( 2^{12}\cdot 3^{3}\cdot 5\cdot 7 \)
Twist minimal: yes
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.

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

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

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

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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1.36.a.a 3
3.b odd 2 1 9.36.a.b 3
4.b odd 2 1 16.36.a.d 3
5.b even 2 1 25.36.a.a 3
5.c odd 4 2 25.36.b.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.36.a.a 3 1.a even 1 1 trivial
9.36.a.b 3 3.b odd 2 1
16.36.a.d 3 4.b odd 2 1
25.36.a.a 3 5.b even 2 1
25.36.b.a 6 5.c odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{36}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 139656 T + 43870875648 T^{2} - 11064712290631680 T^{3} + \)\(15\!\cdots\!64\)\( T^{4} - \)\(16\!\cdots\!44\)\( T^{5} + \)\(40\!\cdots\!32\)\( T^{6} \)
$3$ \( 1 + 104875308 T + 5500743324425217 T^{2} - \)\(10\!\cdots\!20\)\( T^{3} + \)\(27\!\cdots\!19\)\( T^{4} + \)\(26\!\cdots\!92\)\( T^{5} + \)\(12\!\cdots\!43\)\( T^{6} \)
$5$ \( 1 - 892652054010 T + \)\(44\!\cdots\!75\)\( T^{2} - \)\(16\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!75\)\( T^{4} - \)\(75\!\cdots\!50\)\( T^{5} + \)\(24\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 - 878422149346056 T + \)\(12\!\cdots\!93\)\( T^{2} - \)\(63\!\cdots\!00\)\( T^{3} + \)\(47\!\cdots\!99\)\( T^{4} - \)\(12\!\cdots\!44\)\( T^{5} + \)\(54\!\cdots\!07\)\( T^{6} \)
$11$ \( 1 + 1157945428549987044 T + \)\(66\!\cdots\!65\)\( T^{2} + \)\(50\!\cdots\!80\)\( T^{3} + \)\(18\!\cdots\!15\)\( T^{4} + \)\(91\!\cdots\!44\)\( T^{5} + \)\(22\!\cdots\!51\)\( T^{6} \)
$13$ \( 1 + 62139610550998650558 T + \)\(35\!\cdots\!27\)\( T^{2} + \)\(12\!\cdots\!60\)\( T^{3} + \)\(34\!\cdots\!39\)\( T^{4} + \)\(58\!\cdots\!42\)\( T^{5} + \)\(92\!\cdots\!93\)\( T^{6} \)
$17$ \( 1 + \)\(39\!\cdots\!94\)\( T + \)\(33\!\cdots\!83\)\( T^{2} + \)\(84\!\cdots\!60\)\( T^{3} + \)\(38\!\cdots\!19\)\( T^{4} + \)\(53\!\cdots\!06\)\( T^{5} + \)\(15\!\cdots\!57\)\( T^{6} \)
$19$ \( 1 + \)\(32\!\cdots\!40\)\( T + \)\(15\!\cdots\!97\)\( T^{2} + \)\(36\!\cdots\!20\)\( T^{3} + \)\(90\!\cdots\!03\)\( T^{4} + \)\(10\!\cdots\!40\)\( T^{5} + \)\(18\!\cdots\!99\)\( T^{6} \)
$23$ \( 1 + \)\(51\!\cdots\!08\)\( T + \)\(83\!\cdots\!37\)\( T^{2} + \)\(26\!\cdots\!40\)\( T^{3} + \)\(38\!\cdots\!59\)\( T^{4} + \)\(10\!\cdots\!92\)\( T^{5} + \)\(95\!\cdots\!43\)\( T^{6} \)
$29$ \( 1 + \)\(38\!\cdots\!10\)\( T + \)\(30\!\cdots\!47\)\( T^{2} + \)\(92\!\cdots\!80\)\( T^{3} + \)\(47\!\cdots\!03\)\( T^{4} + \)\(89\!\cdots\!10\)\( T^{5} + \)\(35\!\cdots\!49\)\( T^{6} \)
$31$ \( 1 - \)\(10\!\cdots\!56\)\( T + \)\(46\!\cdots\!65\)\( T^{2} - \)\(31\!\cdots\!20\)\( T^{3} + \)\(72\!\cdots\!15\)\( T^{4} - \)\(25\!\cdots\!56\)\( T^{5} + \)\(39\!\cdots\!51\)\( T^{6} \)
$37$ \( 1 - \)\(24\!\cdots\!06\)\( T + \)\(58\!\cdots\!63\)\( T^{2} - \)\(17\!\cdots\!20\)\( T^{3} + \)\(45\!\cdots\!59\)\( T^{4} - \)\(14\!\cdots\!94\)\( T^{5} + \)\(45\!\cdots\!57\)\( T^{6} \)
$41$ \( 1 - \)\(23\!\cdots\!06\)\( T + \)\(83\!\cdots\!15\)\( T^{2} - \)\(12\!\cdots\!20\)\( T^{3} + \)\(23\!\cdots\!15\)\( T^{4} - \)\(18\!\cdots\!06\)\( T^{5} + \)\(21\!\cdots\!01\)\( T^{6} \)
$43$ \( 1 + \)\(47\!\cdots\!08\)\( T + \)\(45\!\cdots\!57\)\( T^{2} + \)\(14\!\cdots\!00\)\( T^{3} + \)\(67\!\cdots\!99\)\( T^{4} + \)\(10\!\cdots\!92\)\( T^{5} + \)\(32\!\cdots\!43\)\( T^{6} \)
$47$ \( 1 - \)\(16\!\cdots\!56\)\( T + \)\(69\!\cdots\!53\)\( T^{2} - \)\(62\!\cdots\!60\)\( T^{3} + \)\(23\!\cdots\!79\)\( T^{4} - \)\(18\!\cdots\!44\)\( T^{5} + \)\(37\!\cdots\!07\)\( T^{6} \)
$53$ \( 1 + \)\(16\!\cdots\!58\)\( T + \)\(28\!\cdots\!67\)\( T^{2} + \)\(29\!\cdots\!80\)\( T^{3} + \)\(64\!\cdots\!19\)\( T^{4} + \)\(83\!\cdots\!42\)\( T^{5} + \)\(11\!\cdots\!93\)\( T^{6} \)
$59$ \( 1 - \)\(43\!\cdots\!80\)\( T + \)\(26\!\cdots\!97\)\( T^{2} - \)\(72\!\cdots\!40\)\( T^{3} + \)\(24\!\cdots\!03\)\( T^{4} - \)\(39\!\cdots\!80\)\( T^{5} + \)\(86\!\cdots\!99\)\( T^{6} \)
$61$ \( 1 - \)\(23\!\cdots\!06\)\( T + \)\(92\!\cdots\!15\)\( T^{2} - \)\(14\!\cdots\!20\)\( T^{3} + \)\(28\!\cdots\!15\)\( T^{4} - \)\(22\!\cdots\!06\)\( T^{5} + \)\(28\!\cdots\!01\)\( T^{6} \)
$67$ \( 1 + \)\(18\!\cdots\!44\)\( T + \)\(27\!\cdots\!33\)\( T^{2} + \)\(29\!\cdots\!60\)\( T^{3} + \)\(22\!\cdots\!19\)\( T^{4} + \)\(12\!\cdots\!56\)\( T^{5} + \)\(54\!\cdots\!07\)\( T^{6} \)
$71$ \( 1 - \)\(34\!\cdots\!56\)\( T + \)\(16\!\cdots\!65\)\( T^{2} - \)\(40\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!15\)\( T^{4} - \)\(13\!\cdots\!56\)\( T^{5} + \)\(24\!\cdots\!51\)\( T^{6} \)
$73$ \( 1 + \)\(28\!\cdots\!58\)\( T + \)\(22\!\cdots\!87\)\( T^{2} + \)\(11\!\cdots\!40\)\( T^{3} + \)\(37\!\cdots\!59\)\( T^{4} + \)\(77\!\cdots\!42\)\( T^{5} + \)\(44\!\cdots\!93\)\( T^{6} \)
$79$ \( 1 + \)\(42\!\cdots\!60\)\( T + \)\(17\!\cdots\!97\)\( T^{2} + \)\(13\!\cdots\!80\)\( T^{3} + \)\(44\!\cdots\!03\)\( T^{4} + \)\(29\!\cdots\!60\)\( T^{5} + \)\(17\!\cdots\!99\)\( T^{6} \)
$83$ \( 1 - \)\(14\!\cdots\!92\)\( T + \)\(11\!\cdots\!97\)\( T^{2} - \)\(54\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!79\)\( T^{4} - \)\(32\!\cdots\!08\)\( T^{5} + \)\(31\!\cdots\!43\)\( T^{6} \)
$89$ \( 1 - \)\(30\!\cdots\!70\)\( T + \)\(80\!\cdots\!47\)\( T^{2} - \)\(11\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!03\)\( T^{4} - \)\(87\!\cdots\!70\)\( T^{5} + \)\(48\!\cdots\!49\)\( T^{6} \)
$97$ \( 1 + \)\(10\!\cdots\!94\)\( T + \)\(12\!\cdots\!03\)\( T^{2} + \)\(72\!\cdots\!40\)\( T^{3} + \)\(42\!\cdots\!79\)\( T^{4} + \)\(12\!\cdots\!06\)\( T^{5} + \)\(40\!\cdots\!57\)\( T^{6} \)
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