Properties

Label 3.34.a.a
Level 3
Weight 34
Character orbit 3.a
Self dual Yes
Analytic conductor 20.695
Analytic rank 0
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 34 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(20.6948486643\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{6}\cdot 3^{6}\cdot 5\cdot 7\cdot 11 \)
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.

\(f(q)\) \(=\) \( q\) \( + ( 13734 - \beta_{1} ) q^{2} \) \( + 43046721 q^{3} \) \( + ( -369155924 - 63216 \beta_{1} + \beta_{2} ) q^{4} \) \( + ( 17087274630 + 280984 \beta_{1} + 108 \beta_{2} ) q^{5} \) \( + ( 591203666214 - 43046721 \beta_{1} ) q^{6} \) \( + ( 25371244671752 - 323610696 \beta_{1} - 6404 \beta_{2} ) q^{7} \) \( + ( 384716449496664 + 3092877952 \beta_{1} + 41202 \beta_{2} ) q^{8} \) \( + 1853020188851841 q^{9} \) \(+O(q^{10})\) \( q\) \(+(13734 - \beta_{1}) q^{2}\) \(+43046721 q^{3}\) \(+(-369155924 - 63216 \beta_{1} + \beta_{2}) q^{4}\) \(+(17087274630 + 280984 \beta_{1} + 108 \beta_{2}) q^{5}\) \(+(591203666214 - 43046721 \beta_{1}) q^{6}\) \(+(25371244671752 - 323610696 \beta_{1} - 6404 \beta_{2}) q^{7}\) \(+(384716449496664 + 3092877952 \beta_{1} + 41202 \beta_{2}) q^{8}\) \(+1853020188851841 q^{9}\) \(+(-2022248971884060 - 298904737158 \beta_{1} - 2658496 \beta_{2}) q^{10}\) \(+(-34507916295151860 - 1124553276112 \beta_{1} + 13291992 \beta_{2}) q^{11}\) \(+(-15890952065925204 - 2721241514736 \beta_{1} + 43046721 \beta_{2}) q^{12}\) \(+(638470867119282830 - 20317957756848 \beta_{1} - 460386584 \beta_{2}) q^{13}\) \(+(2947741324795972656 - 23848982404616 \beta_{1} + 464588352 \beta_{2}) q^{14}\) \(+(735551143647988230 + 12095439853464 \beta_{1} + 4649045868 \beta_{2}) q^{15}\) \(+(-16387763951764585808 + 198529037952768 \beta_{1} - 12589833372 \beta_{2}) q^{16}\) \(+(-5650418010665548782 + 3009063412790256 \beta_{1} - 5037146568 \beta_{2}) q^{17}\) \(+(25449379273691184294 - 1853020188851841 \beta_{1}) q^{18}\) \(+(\)\(87\!\cdots\!88\)\( - 3612192985132560 \beta_{1} + 273191138616 \beta_{2}) q^{19}\) \(+(\)\(22\!\cdots\!60\)\( - 7902406121558432 \beta_{1} - 570284067834 \beta_{2}) q^{20}\) \(+(\)\(10\!\cdots\!92\)\( - 13930379343327816 \beta_{1} - 275671201284 \beta_{2}) q^{21}\) \(+(\)\(85\!\cdots\!76\)\( - 57532809152138508 \beta_{1} + 831943364224 \beta_{2}) q^{22}\) \(+(\)\(26\!\cdots\!56\)\( - 20101240887203216 \beta_{1} + 2300566232952 \beta_{2}) q^{23}\) \(+(\)\(16\!\cdots\!44\)\( + 133138254286795392 \beta_{1} + 1773610998642 \beta_{2}) q^{24}\) \(+(\)\(14\!\cdots\!75\)\( + 1314683245233833760 \beta_{1} - 25998439792880 \beta_{2}) q^{25}\) \(+(\)\(17\!\cdots\!52\)\( - 383232636676627598 \beta_{1} + 30452908017024 \beta_{2}) q^{26}\) \(+\)\(79\!\cdots\!61\)\( q^{27}\) \(+(\)\(14\!\cdots\!44\)\( - 2620158482959136640 \beta_{1} + 68631475550856 \beta_{2}) q^{28}\) \(+(\)\(39\!\cdots\!86\)\( - 9122515363106364664 \beta_{1} - 173787588879516 \beta_{2}) q^{29}\) \(+(-\)\(87\!\cdots\!60\)\( - 12866868826018758918 \beta_{1} - 114439535591616 \beta_{2}) q^{30}\) \(+(-\)\(57\!\cdots\!92\)\( + 32335952639255595288 \beta_{1} + 532179934052908 \beta_{2}) q^{31}\) \(+(-\)\(51\!\cdots\!28\)\( + 34116717155837996544 \beta_{1} - 275298931161144 \beta_{2}) q^{32}\) \(+(-\)\(14\!\cdots\!60\)\( - 48408331126429228752 \beta_{1} + 572176671158232 \beta_{2}) q^{33}\) \(+(-\)\(24\!\cdots\!48\)\( + \)\(16\!\cdots\!10\)\( \beta_{1} - 2898175668242304 \beta_{2}) q^{34}\) \(+(-\)\(15\!\cdots\!60\)\( - \)\(16\!\cdots\!08\)\( \beta_{1} + 3119819378072904 \beta_{2}) q^{35}\) \(+(-\)\(68\!\cdots\!84\)\( - \)\(11\!\cdots\!56\)\( \beta_{1} + 1853020188851841 \beta_{2}) q^{36}\) \(+(\)\(76\!\cdots\!62\)\( + \)\(69\!\cdots\!92\)\( \beta_{1} - 1782384058562208 \beta_{2}) q^{37}\) \(+(\)\(41\!\cdots\!68\)\( - \)\(18\!\cdots\!40\)\( \beta_{1} - 2401836740360064 \beta_{2}) q^{38}\) \(+(\)\(27\!\cdots\!30\)\( - \)\(87\!\cdots\!08\)\( \beta_{1} - 19818132833591064 \beta_{2}) q^{39}\) \(+(\)\(11\!\cdots\!00\)\( + \)\(15\!\cdots\!20\)\( \beta_{1} + 43292946343949740 \beta_{2}) q^{40}\) \(+(\)\(14\!\cdots\!38\)\( + \)\(37\!\cdots\!80\)\( \beta_{1} - 7030519097942808 \beta_{2}) q^{41}\) \(+(\)\(12\!\cdots\!76\)\( - \)\(10\!\cdots\!36\)\( \beta_{1} + 19999005168393792 \beta_{2}) q^{42}\) \(+(\)\(31\!\cdots\!48\)\( + \)\(59\!\cdots\!56\)\( \beta_{1} - 91355347912557080 \beta_{2}) q^{43}\) \(+(\)\(87\!\cdots\!84\)\( - \)\(40\!\cdots\!76\)\( \beta_{1} - 74958933945275892 \beta_{2}) q^{44}\) \(+(\)\(31\!\cdots\!30\)\( + \)\(52\!\cdots\!44\)\( \beta_{1} + 200126180395998828 \beta_{2}) q^{45}\) \(+(\)\(52\!\cdots\!28\)\( - \)\(33\!\cdots\!72\)\( \beta_{1} - 30543424165002112 \beta_{2}) q^{46}\) \(+(-\)\(29\!\cdots\!84\)\( + \)\(95\!\cdots\!60\)\( \beta_{1} + 411342274012752936 \beta_{2}) q^{47}\) \(+(-\)\(70\!\cdots\!68\)\( + \)\(85\!\cdots\!28\)\( \beta_{1} - 541951044600973212 \beta_{2}) q^{48}\) \(+(-\)\(53\!\cdots\!91\)\( - \)\(48\!\cdots\!36\)\( \beta_{1} - 184357709370325568 \beta_{2}) q^{49}\) \(+(-\)\(86\!\cdots\!50\)\( - \)\(47\!\cdots\!95\)\( \beta_{1} - 742353591633373440 \beta_{2}) q^{50}\) \(+(-\)\(24\!\cdots\!22\)\( + \)\(12\!\cdots\!76\)\( \beta_{1} - 216832642948803528 \beta_{2}) q^{51}\) \(+(-\)\(44\!\cdots\!32\)\( - \)\(99\!\cdots\!20\)\( \beta_{1} + 3667532963184174990 \beta_{2}) q^{52}\) \(+(-\)\(39\!\cdots\!34\)\( + \)\(21\!\cdots\!20\)\( \beta_{1} + 996010849755971604 \beta_{2}) q^{53}\) \(+(\)\(10\!\cdots\!74\)\( - \)\(79\!\cdots\!61\)\( \beta_{1}) q^{54}\) \(+(\)\(28\!\cdots\!60\)\( - \)\(19\!\cdots\!72\)\( \beta_{1} - 11954579525329437664 \beta_{2}) q^{55}\) \(+(-\)\(40\!\cdots\!80\)\( - \)\(12\!\cdots\!64\)\( \beta_{1} - 2881478375702479728 \beta_{2}) q^{56}\) \(+(\)\(37\!\cdots\!48\)\( - \)\(15\!\cdots\!60\)\( \beta_{1} + 11759982723675278136 \beta_{2}) q^{57}\) \(+(\)\(78\!\cdots\!36\)\( - \)\(36\!\cdots\!02\)\( \beta_{1} + 12948275344700029888 \beta_{2}) q^{58}\) \(+(\)\(43\!\cdots\!52\)\( - \)\(40\!\cdots\!64\)\( \beta_{1} + 10719599070768289056 \beta_{2}) q^{59}\) \(+(\)\(95\!\cdots\!60\)\( - \)\(34\!\cdots\!72\)\( \beta_{1} - 24548859158795272314 \beta_{2}) q^{60}\) \(+(\)\(12\!\cdots\!54\)\( + \)\(37\!\cdots\!04\)\( \beta_{1} - 4256599947344600048 \beta_{2}) q^{61}\) \(+(-\)\(26\!\cdots\!56\)\( + \)\(71\!\cdots\!92\)\( \beta_{1} - 44051361707496312000 \beta_{2}) q^{62}\) \(+(\)\(47\!\cdots\!32\)\( - \)\(59\!\cdots\!36\)\( \beta_{1} - 11866741289407189764 \beta_{2}) q^{63}\) \(+(-\)\(20\!\cdots\!48\)\( + \)\(58\!\cdots\!68\)\( \beta_{1} + 80089558704402231696 \beta_{2}) q^{64}\) \(+(-\)\(11\!\cdots\!80\)\( - \)\(11\!\cdots\!24\)\( \beta_{1} + \)\(10\!\cdots\!12\)\( \beta_{2}) q^{65}\) \(+(\)\(36\!\cdots\!96\)\( - \)\(24\!\cdots\!68\)\( \beta_{1} + 35812433887551909504 \beta_{2}) q^{66}\) \(+(-\)\(52\!\cdots\!32\)\( - \)\(11\!\cdots\!68\)\( \beta_{1} - \)\(28\!\cdots\!16\)\( \beta_{2}) q^{67}\) \(+(-\)\(16\!\cdots\!72\)\( + \)\(14\!\cdots\!24\)\( \beta_{1} - 61268200541361628398 \beta_{2}) q^{68}\) \(+(\)\(11\!\cdots\!76\)\( - \)\(86\!\cdots\!36\)\( \beta_{1} + 99031832771905750392 \beta_{2}) q^{69}\) \(+(\)\(11\!\cdots\!20\)\( - \)\(11\!\cdots\!04\)\( \beta_{1} + 94795686836474695552 \beta_{2}) q^{70}\) \(+(-\)\(22\!\cdots\!28\)\( + \)\(63\!\cdots\!84\)\( \beta_{1} - 45679595098071025512 \beta_{2}) q^{71}\) \(+(\)\(71\!\cdots\!24\)\( + \)\(57\!\cdots\!32\)\( \beta_{1} + 76348137821073552882 \beta_{2}) q^{72}\) \(+(\)\(58\!\cdots\!06\)\( - \)\(18\!\cdots\!12\)\( \beta_{1} + \)\(86\!\cdots\!60\)\( \beta_{2}) q^{73}\) \(+(-\)\(54\!\cdots\!24\)\( + \)\(31\!\cdots\!98\)\( \beta_{1} - \)\(65\!\cdots\!80\)\( \beta_{2}) q^{74}\) \(+(\)\(60\!\cdots\!75\)\( + \)\(56\!\cdots\!60\)\( \beta_{1} - \)\(11\!\cdots\!80\)\( \beta_{2}) q^{75}\) \(+(\)\(75\!\cdots\!72\)\( - \)\(92\!\cdots\!00\)\( \beta_{1} - \)\(49\!\cdots\!36\)\( \beta_{2}) q^{76}\) \(+(\)\(17\!\cdots\!64\)\( - \)\(30\!\cdots\!68\)\( \beta_{1} + \)\(12\!\cdots\!80\)\( \beta_{2}) q^{77}\) \(+(\)\(74\!\cdots\!92\)\( - \)\(16\!\cdots\!58\)\( \beta_{1} + \)\(13\!\cdots\!04\)\( \beta_{2}) q^{78}\) \(+(\)\(15\!\cdots\!80\)\( - \)\(38\!\cdots\!16\)\( \beta_{1} - \)\(15\!\cdots\!32\)\( \beta_{2}) q^{79}\) \(+(-\)\(29\!\cdots\!20\)\( - \)\(87\!\cdots\!96\)\( \beta_{1} + \)\(24\!\cdots\!48\)\( \beta_{2}) q^{80}\) \(+\)\(34\!\cdots\!81\)\( q^{81}\) \(+(-\)\(27\!\cdots\!96\)\( + \)\(60\!\cdots\!38\)\( \beta_{1} - \)\(35\!\cdots\!68\)\( \beta_{2}) q^{82}\) \(+(-\)\(36\!\cdots\!88\)\( + \)\(13\!\cdots\!72\)\( \beta_{1} - \)\(53\!\cdots\!40\)\( \beta_{2}) q^{83}\) \(+(\)\(60\!\cdots\!24\)\( - \)\(11\!\cdots\!40\)\( \beta_{1} + \)\(29\!\cdots\!76\)\( \beta_{2}) q^{84}\) \(+(-\)\(52\!\cdots\!60\)\( + \)\(85\!\cdots\!52\)\( \beta_{1} + \)\(13\!\cdots\!24\)\( \beta_{2}) q^{85}\) \(+(-\)\(43\!\cdots\!20\)\( + \)\(23\!\cdots\!04\)\( \beta_{1} - \)\(39\!\cdots\!36\)\( \beta_{2}) q^{86}\) \(+(\)\(16\!\cdots\!06\)\( - \)\(39\!\cdots\!44\)\( \beta_{1} - \)\(74\!\cdots\!36\)\( \beta_{2}) q^{87}\) \(+(-\)\(29\!\cdots\!96\)\( - \)\(37\!\cdots\!96\)\( \beta_{1} - \)\(14\!\cdots\!44\)\( \beta_{2}) q^{88}\) \(+(\)\(93\!\cdots\!98\)\( - \)\(84\!\cdots\!68\)\( \beta_{1} + \)\(37\!\cdots\!80\)\( \beta_{2}) q^{89}\) \(+(-\)\(37\!\cdots\!60\)\( - \)\(55\!\cdots\!78\)\( \beta_{1} - \)\(49\!\cdots\!36\)\( \beta_{2}) q^{90}\) \(+(\)\(13\!\cdots\!28\)\( + \)\(95\!\cdots\!32\)\( \beta_{1} - \)\(72\!\cdots\!64\)\( \beta_{2}) q^{91}\) \(+(\)\(51\!\cdots\!72\)\( - \)\(19\!\cdots\!36\)\( \beta_{1} + \)\(14\!\cdots\!56\)\( \beta_{2}) q^{92}\) \(+(-\)\(24\!\cdots\!32\)\( + \)\(13\!\cdots\!48\)\( \beta_{1} + \)\(22\!\cdots\!68\)\( \beta_{2}) q^{93}\) \(+(-\)\(11\!\cdots\!00\)\( + \)\(22\!\cdots\!32\)\( \beta_{1} - \)\(18\!\cdots\!64\)\( \beta_{2}) q^{94}\) \(+(\)\(65\!\cdots\!00\)\( + \)\(22\!\cdots\!80\)\( \beta_{1} + \)\(63\!\cdots\!60\)\( \beta_{2}) q^{95}\) \(+(-\)\(22\!\cdots\!88\)\( + \)\(14\!\cdots\!24\)\( \beta_{1} - \)\(11\!\cdots\!24\)\( \beta_{2}) q^{96}\) \(+(\)\(27\!\cdots\!18\)\( - \)\(63\!\cdots\!48\)\( \beta_{1} - \)\(75\!\cdots\!76\)\( \beta_{2}) q^{97}\) \(+(-\)\(34\!\cdots\!50\)\( + \)\(56\!\cdots\!75\)\( \beta_{1} + \)\(88\!\cdots\!88\)\( \beta_{2}) q^{98}\) \(+(-\)\(63\!\cdots\!60\)\( - \)\(20\!\cdots\!92\)\( \beta_{1} + \)\(24\!\cdots\!72\)\( \beta_{2}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut +\mathstrut 41202q^{2} \) \(\mathstrut +\mathstrut 129140163q^{3} \) \(\mathstrut -\mathstrut 1107467772q^{4} \) \(\mathstrut +\mathstrut 51261823890q^{5} \) \(\mathstrut +\mathstrut 1773610998642q^{6} \) \(\mathstrut +\mathstrut 76113734015256q^{7} \) \(\mathstrut +\mathstrut 1154149348489992q^{8} \) \(\mathstrut +\mathstrut 5559060566555523q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut +\mathstrut 41202q^{2} \) \(\mathstrut +\mathstrut 129140163q^{3} \) \(\mathstrut -\mathstrut 1107467772q^{4} \) \(\mathstrut +\mathstrut 51261823890q^{5} \) \(\mathstrut +\mathstrut 1773610998642q^{6} \) \(\mathstrut +\mathstrut 76113734015256q^{7} \) \(\mathstrut +\mathstrut 1154149348489992q^{8} \) \(\mathstrut +\mathstrut 5559060566555523q^{9} \) \(\mathstrut -\mathstrut 6066746915652180q^{10} \) \(\mathstrut -\mathstrut 103523748885455580q^{11} \) \(\mathstrut -\mathstrut 47672856197775612q^{12} \) \(\mathstrut +\mathstrut 1915412601357848490q^{13} \) \(\mathstrut +\mathstrut 8843223974387917968q^{14} \) \(\mathstrut +\mathstrut 2206653430943964690q^{15} \) \(\mathstrut -\mathstrut 49163291855293757424q^{16} \) \(\mathstrut -\mathstrut 16951254031996646346q^{17} \) \(\mathstrut +\mathstrut 76348137821073552882q^{18} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!64\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(66\!\cdots\!80\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!76\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!28\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(79\!\cdots\!68\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!32\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!25\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!56\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!83\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!32\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!58\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!80\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!76\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!84\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!80\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(72\!\cdots\!44\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!80\)\(q^{35} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!52\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!86\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!04\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(82\!\cdots\!90\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!14\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!28\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(93\!\cdots\!44\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!52\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(94\!\cdots\!90\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!84\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!52\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!04\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!73\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!50\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(72\!\cdots\!66\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!96\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!02\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!22\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(85\!\cdots\!80\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!40\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!44\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!08\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!56\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!80\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!62\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(80\!\cdots\!68\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!96\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(61\!\cdots\!44\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!40\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!88\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!96\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(49\!\cdots\!16\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!28\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!60\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(66\!\cdots\!84\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!72\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!18\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!72\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!25\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!16\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(52\!\cdots\!92\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!76\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!40\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(89\!\cdots\!60\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!43\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(83\!\cdots\!88\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!64\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!72\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!80\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!60\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!18\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!88\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!94\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!80\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!84\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!16\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(74\!\cdots\!96\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!00\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(66\!\cdots\!64\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(81\!\cdots\!54\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!50\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!80\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(x^{2}\mathstrut -\mathstrut \) \(35900150\) \(x\mathstrut +\mathstrut \) \(10469144400\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{2} + 1429 \nu - 23933910 \)\()/195\)
\(\beta_{2}\)\(=\)\((\)\( -5046 \nu^{2} + 57654066 \nu + 120748888260 \)\()/65\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(15138\) \(\beta_{1}\mathstrut +\mathstrut \) \(332640\)\()/997920\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(1429\) \(\beta_{2}\mathstrut +\mathstrut \) \(172962198\) \(\beta_{1}\mathstrut +\mathstrut \) \(23883652124640\)\()/997920\)

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
5840.68
−6131.99
292.312
−81270.9 4.30467e7 −1.98497e9 5.17904e11 −3.49845e12 −3.34870e13 8.59432e14 1.85302e15 −4.20905e16
1.2 −11418.8 4.30467e7 −8.45955e9 −6.77881e11 −4.91541e11 5.88597e13 1.94684e14 1.85302e15 7.74057e15
1.3 133892. 4.30467e7 9.33705e9 2.11239e11 5.76360e12 5.07411e13 1.00033e14 1.85302e15 2.82832e16
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{3} \) \(\mathstrut -\mathstrut 41202 T_{2}^{2} \) \(\mathstrut -\mathstrut 11482365600 T_{2} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!84\)\( \) acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(3))\).