Properties

Label 1.42.a.a
Level 1
Weight 42
Character orbit 1.a
Self dual Yes
Analytic conductor 10.647
Analytic rank 1
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 42 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(10.6471670456\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{11}\cdot 3^{3}\cdot 5\cdot 7 \)
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.

\(f(q)\) \(=\) \( q\) \( + ( -114896 + \beta_{1} ) q^{2} \) \( + ( -3606984348 + 291 \beta_{1} - \beta_{2} ) q^{3} \) \( + ( 2090568301312 - 280368 \beta_{1} + 576 \beta_{2} ) q^{4} \) \( + ( -70767450093850 - 72947980 \beta_{1} - 22140 \beta_{2} ) q^{5} \) \( + ( 1656731371660992 - 7362840348 \beta_{1} + 231936 \beta_{2} ) q^{6} \) \( + ( 19292805419413064 - 51223345002 \beta_{1} + 4657806 \beta_{2} ) q^{7} \) \( + ( -1185277237061079040 + 2073575400704 \beta_{1} - 198540288 \beta_{2} ) q^{8} \) \( + ( 4425668036977292973 - 3932182720296 \beta_{1} + 3525606072 \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \(+(-114896 + \beta_{1}) q^{2}\) \(+(-3606984348 + 291 \beta_{1} - \beta_{2}) q^{3}\) \(+(2090568301312 - 280368 \beta_{1} + 576 \beta_{2}) q^{4}\) \(+(-70767450093850 - 72947980 \beta_{1} - 22140 \beta_{2}) q^{5}\) \(+(1656731371660992 - 7362840348 \beta_{1} + 231936 \beta_{2}) q^{6}\) \(+(19292805419413064 - 51223345002 \beta_{1} + 4657806 \beta_{2}) q^{7}\) \(+(-1185277237061079040 + 2073575400704 \beta_{1} - 198540288 \beta_{2}) q^{8}\) \(+(4425668036977292973 - 3932182720296 \beta_{1} + 3525606072 \beta_{2}) q^{9}\) \(+(-\)\(30\!\cdots\!00\)\( - 140785160714010 \beta_{1} - 40593991680 \beta_{2}) q^{10}\) \(+(-\)\(10\!\cdots\!88\)\( + 759168084431785 \beta_{1} + 335559667005 \beta_{2}) q^{11}\) \(+(-\)\(23\!\cdots\!16\)\( + 3095109475662144 \beta_{1} - 2056890908416 \beta_{2}) q^{12}\) \(+(-\)\(32\!\cdots\!98\)\( - 35612693181898092 \beta_{1} + 9313221642660 \beta_{2}) q^{13}\) \(+(-\)\(22\!\cdots\!36\)\( + 45038599061460296 \beta_{1} - 29804236803072 \beta_{2}) q^{14}\) \(+(\)\(77\!\cdots\!00\)\( + 497473797642290370 \beta_{1} + 59673845194410 \beta_{2}) q^{15}\) \(+(\)\(44\!\cdots\!16\)\( - 1647988703746338816 \beta_{1} - 59487853068288 \beta_{2}) q^{16}\) \(+(\)\(11\!\cdots\!34\)\( - 2997874477951097448 \beta_{1} + 215575732336248 \beta_{2}) q^{17}\) \(+(-\)\(17\!\cdots\!68\)\( + 18148236670635463341 \beta_{1} - 2491704229441536 \beta_{2}) q^{18}\) \(+(-\)\(77\!\cdots\!60\)\( + 27857044325303505423 \beta_{1} + 12913807288600539 \beta_{2}) q^{19}\) \(+(-\)\(41\!\cdots\!00\)\( - \)\(27\!\cdots\!60\)\( \beta_{1} - 29794872148490880 \beta_{2}) q^{20}\) \(+(-\)\(26\!\cdots\!08\)\( + \)\(38\!\cdots\!64\)\( \beta_{1} - 24750302958480848 \beta_{2}) q^{21}\) \(+(\)\(33\!\cdots\!48\)\( + 97174430374741228332 \beta_{1} + 415697618850946560 \beta_{2}) q^{22}\) \(+(\)\(93\!\cdots\!52\)\( + \)\(26\!\cdots\!90\)\( \beta_{1} - 1342498814797114134 \beta_{2}) q^{23}\) \(+(\)\(12\!\cdots\!20\)\( - \)\(15\!\cdots\!56\)\( \beta_{1} + 1405049623411003392 \beta_{2}) q^{24}\) \(+(-\)\(42\!\cdots\!25\)\( + \)\(22\!\cdots\!00\)\( \beta_{1} + 4544640270218142000 \beta_{2}) q^{25}\) \(+(-\)\(14\!\cdots\!68\)\( + \)\(75\!\cdots\!06\)\( \beta_{1} - 21111937688829192192 \beta_{2}) q^{26}\) \(+(\)\(13\!\cdots\!60\)\( + \)\(22\!\cdots\!14\)\( \beta_{1} + 31994090766592603542 \beta_{2}) q^{27}\) \(+(\)\(17\!\cdots\!88\)\( - \)\(22\!\cdots\!00\)\( \beta_{1} + 17616617856725082624 \beta_{2}) q^{28}\) \(+(-\)\(42\!\cdots\!90\)\( + \)\(82\!\cdots\!32\)\( \beta_{1} - \)\(16\!\cdots\!24\)\( \beta_{2}) q^{29}\) \(+(\)\(20\!\cdots\!00\)\( + \)\(91\!\cdots\!40\)\( \beta_{1} + \)\(28\!\cdots\!20\)\( \beta_{2}) q^{30}\) \(+(-\)\(18\!\cdots\!68\)\( - \)\(93\!\cdots\!20\)\( \beta_{1} - 81913420829731658760 \beta_{2}) q^{31}\) \(+(-\)\(49\!\cdots\!16\)\( - \)\(10\!\cdots\!64\)\( \beta_{1} - \)\(50\!\cdots\!80\)\( \beta_{2}) q^{32}\) \(+(-\)\(46\!\cdots\!76\)\( - \)\(56\!\cdots\!48\)\( \beta_{1} + \)\(11\!\cdots\!68\)\( \beta_{2}) q^{33}\) \(+(-\)\(14\!\cdots\!36\)\( + \)\(13\!\cdots\!94\)\( \beta_{1} - \)\(17\!\cdots\!08\)\( \beta_{2}) q^{34}\) \(+(\)\(11\!\cdots\!00\)\( + \)\(49\!\cdots\!40\)\( \beta_{1} + \)\(18\!\cdots\!20\)\( \beta_{2}) q^{35}\) \(+(\)\(69\!\cdots\!76\)\( - \)\(20\!\cdots\!56\)\( \beta_{1} + \)\(28\!\cdots\!92\)\( \beta_{2}) q^{36}\) \(+(\)\(16\!\cdots\!74\)\( - \)\(19\!\cdots\!80\)\( \beta_{1} - \)\(15\!\cdots\!88\)\( \beta_{2}) q^{37}\) \(+(\)\(12\!\cdots\!40\)\( - \)\(34\!\cdots\!44\)\( \beta_{1} + \)\(15\!\cdots\!68\)\( \beta_{2}) q^{38}\) \(+(-\)\(18\!\cdots\!04\)\( + \)\(25\!\cdots\!46\)\( \beta_{1} + \)\(29\!\cdots\!78\)\( \beta_{2}) q^{39}\) \(+(-\)\(44\!\cdots\!00\)\( - \)\(16\!\cdots\!00\)\( \beta_{1} - \)\(64\!\cdots\!00\)\( \beta_{2}) q^{40}\) \(+(-\)\(10\!\cdots\!58\)\( - \)\(73\!\cdots\!20\)\( \beta_{1} - \)\(57\!\cdots\!60\)\( \beta_{2}) q^{41}\) \(+(\)\(16\!\cdots\!08\)\( - \)\(41\!\cdots\!20\)\( \beta_{1} + \)\(22\!\cdots\!24\)\( \beta_{2}) q^{42}\) \(+(\)\(49\!\cdots\!52\)\( - \)\(35\!\cdots\!11\)\( \beta_{1} + \)\(85\!\cdots\!33\)\( \beta_{2}) q^{43}\) \(+(\)\(22\!\cdots\!44\)\( + \)\(32\!\cdots\!04\)\( \beta_{1} - \)\(70\!\cdots\!28\)\( \beta_{2}) q^{44}\) \(+(-\)\(12\!\cdots\!50\)\( - \)\(71\!\cdots\!40\)\( \beta_{1} + \)\(98\!\cdots\!80\)\( \beta_{2}) q^{45}\) \(+(\)\(11\!\cdots\!72\)\( - \)\(44\!\cdots\!60\)\( \beta_{1} + \)\(16\!\cdots\!20\)\( \beta_{2}) q^{46}\) \(+(-\)\(21\!\cdots\!56\)\( - \)\(46\!\cdots\!16\)\( \beta_{1} - \)\(22\!\cdots\!16\)\( \beta_{2}) q^{47}\) \(+(-\)\(16\!\cdots\!48\)\( + \)\(13\!\cdots\!80\)\( \beta_{1} - \)\(46\!\cdots\!64\)\( \beta_{2}) q^{48}\) \(+(-\)\(32\!\cdots\!43\)\( - \)\(38\!\cdots\!60\)\( \beta_{1} + \)\(15\!\cdots\!20\)\( \beta_{2}) q^{49}\) \(+(\)\(95\!\cdots\!00\)\( + \)\(89\!\cdots\!75\)\( \beta_{1} + \)\(12\!\cdots\!00\)\( \beta_{2}) q^{50}\) \(+(-\)\(52\!\cdots\!08\)\( + \)\(25\!\cdots\!98\)\( \beta_{1} - \)\(12\!\cdots\!86\)\( \beta_{2}) q^{51}\) \(+(\)\(12\!\cdots\!84\)\( - \)\(14\!\cdots\!32\)\( \beta_{1} - \)\(14\!\cdots\!24\)\( \beta_{2}) q^{52}\) \(+(\)\(26\!\cdots\!02\)\( + \)\(10\!\cdots\!56\)\( \beta_{1} + \)\(22\!\cdots\!44\)\( \beta_{2}) q^{53}\) \(+(\)\(93\!\cdots\!40\)\( + \)\(12\!\cdots\!68\)\( \beta_{1} + \)\(10\!\cdots\!24\)\( \beta_{2}) q^{54}\) \(+(-\)\(36\!\cdots\!00\)\( - \)\(13\!\cdots\!10\)\( \beta_{1} - \)\(75\!\cdots\!30\)\( \beta_{2}) q^{55}\) \(+(-\)\(50\!\cdots\!60\)\( + \)\(17\!\cdots\!48\)\( \beta_{1} - \)\(66\!\cdots\!36\)\( \beta_{2}) q^{56}\) \(+(-\)\(40\!\cdots\!80\)\( - \)\(21\!\cdots\!32\)\( \beta_{1} + \)\(82\!\cdots\!04\)\( \beta_{2}) q^{57}\) \(+(\)\(35\!\cdots\!60\)\( - \)\(62\!\cdots\!46\)\( \beta_{1} + \)\(58\!\cdots\!12\)\( \beta_{2}) q^{58}\) \(+(\)\(64\!\cdots\!20\)\( + \)\(37\!\cdots\!09\)\( \beta_{1} - \)\(28\!\cdots\!63\)\( \beta_{2}) q^{59}\) \(+(\)\(19\!\cdots\!00\)\( + \)\(18\!\cdots\!40\)\( \beta_{1} + \)\(37\!\cdots\!20\)\( \beta_{2}) q^{60}\) \(+(\)\(29\!\cdots\!62\)\( - \)\(12\!\cdots\!00\)\( \beta_{1} + \)\(48\!\cdots\!00\)\( \beta_{2}) q^{61}\) \(+(-\)\(40\!\cdots\!72\)\( - \)\(16\!\cdots\!08\)\( \beta_{1} - \)\(53\!\cdots\!20\)\( \beta_{2}) q^{62}\) \(+(\)\(13\!\cdots\!12\)\( - \)\(10\!\cdots\!18\)\( \beta_{1} + \)\(13\!\cdots\!82\)\( \beta_{2}) q^{63}\) \(+(-\)\(95\!\cdots\!88\)\( - \)\(31\!\cdots\!16\)\( \beta_{1} + \)\(10\!\cdots\!12\)\( \beta_{2}) q^{64}\) \(+(\)\(77\!\cdots\!00\)\( + \)\(51\!\cdots\!20\)\( \beta_{1} + \)\(23\!\cdots\!60\)\( \beta_{2}) q^{65}\) \(+(-\)\(23\!\cdots\!96\)\( + \)\(57\!\cdots\!44\)\( \beta_{1} - \)\(33\!\cdots\!08\)\( \beta_{2}) q^{66}\) \(+(\)\(37\!\cdots\!84\)\( + \)\(13\!\cdots\!47\)\( \beta_{1} - \)\(30\!\cdots\!17\)\( \beta_{2}) q^{67}\) \(+(\)\(31\!\cdots\!28\)\( - \)\(16\!\cdots\!92\)\( \beta_{1} + \)\(72\!\cdots\!08\)\( \beta_{2}) q^{68}\) \(+(\)\(36\!\cdots\!16\)\( - \)\(20\!\cdots\!72\)\( \beta_{1} - \)\(77\!\cdots\!96\)\( \beta_{2}) q^{69}\) \(+(\)\(19\!\cdots\!00\)\( + \)\(17\!\cdots\!80\)\( \beta_{1} + \)\(27\!\cdots\!40\)\( \beta_{2}) q^{70}\) \(+(-\)\(46\!\cdots\!28\)\( - \)\(11\!\cdots\!50\)\( \beta_{1} - \)\(13\!\cdots\!50\)\( \beta_{2}) q^{71}\) \(+(-\)\(59\!\cdots\!20\)\( + \)\(44\!\cdots\!92\)\( \beta_{1} - \)\(67\!\cdots\!24\)\( \beta_{2}) q^{72}\) \(+(\)\(15\!\cdots\!02\)\( + \)\(41\!\cdots\!60\)\( \beta_{1} + \)\(33\!\cdots\!76\)\( \beta_{2}) q^{73}\) \(+(-\)\(85\!\cdots\!36\)\( - \)\(51\!\cdots\!90\)\( \beta_{1} - \)\(10\!\cdots\!20\)\( \beta_{2}) q^{74}\) \(+(-\)\(82\!\cdots\!00\)\( - \)\(16\!\cdots\!75\)\( \beta_{1} + \)\(73\!\cdots\!25\)\( \beta_{2}) q^{75}\) \(+(\)\(87\!\cdots\!80\)\( + \)\(12\!\cdots\!76\)\( \beta_{1} - \)\(49\!\cdots\!32\)\( \beta_{2}) q^{76}\) \(+(-\)\(14\!\cdots\!32\)\( + \)\(18\!\cdots\!96\)\( \beta_{1} - \)\(27\!\cdots\!68\)\( \beta_{2}) q^{77}\) \(+(\)\(11\!\cdots\!44\)\( - \)\(11\!\cdots\!72\)\( \beta_{1} + \)\(14\!\cdots\!36\)\( \beta_{2}) q^{78}\) \(+(-\)\(17\!\cdots\!40\)\( + \)\(41\!\cdots\!32\)\( \beta_{1} - \)\(44\!\cdots\!24\)\( \beta_{2}) q^{79}\) \(+(\)\(23\!\cdots\!00\)\( - \)\(59\!\cdots\!80\)\( \beta_{1} - \)\(26\!\cdots\!40\)\( \beta_{2}) q^{80}\) \(+(-\)\(10\!\cdots\!99\)\( + \)\(91\!\cdots\!92\)\( \beta_{1} - \)\(13\!\cdots\!44\)\( \beta_{2}) q^{81}\) \(+(-\)\(19\!\cdots\!32\)\( - \)\(12\!\cdots\!98\)\( \beta_{1} - \)\(38\!\cdots\!20\)\( \beta_{2}) q^{82}\) \(+(-\)\(20\!\cdots\!48\)\( + \)\(12\!\cdots\!87\)\( \beta_{1} + \)\(44\!\cdots\!67\)\( \beta_{2}) q^{83}\) \(+(-\)\(13\!\cdots\!96\)\( + \)\(17\!\cdots\!72\)\( \beta_{1} - \)\(20\!\cdots\!04\)\( \beta_{2}) q^{84}\) \(+(-\)\(35\!\cdots\!00\)\( - \)\(47\!\cdots\!60\)\( \beta_{1} - \)\(13\!\cdots\!80\)\( \beta_{2}) q^{85}\) \(+(-\)\(15\!\cdots\!48\)\( + \)\(87\!\cdots\!28\)\( \beta_{1} - \)\(21\!\cdots\!96\)\( \beta_{2}) q^{86}\) \(+(\)\(46\!\cdots\!80\)\( - \)\(73\!\cdots\!38\)\( \beta_{1} - \)\(51\!\cdots\!14\)\( \beta_{2}) q^{87}\) \(+(\)\(61\!\cdots\!20\)\( - \)\(10\!\cdots\!52\)\( \beta_{1} + \)\(98\!\cdots\!44\)\( \beta_{2}) q^{88}\) \(+(-\)\(47\!\cdots\!70\)\( - \)\(15\!\cdots\!04\)\( \beta_{1} - \)\(87\!\cdots\!72\)\( \beta_{2}) q^{89}\) \(+(-\)\(29\!\cdots\!00\)\( - \)\(75\!\cdots\!30\)\( \beta_{1} - \)\(41\!\cdots\!40\)\( \beta_{2}) q^{90}\) \(+(\)\(83\!\cdots\!32\)\( - \)\(14\!\cdots\!12\)\( \beta_{1} + \)\(88\!\cdots\!84\)\( \beta_{2}) q^{91}\) \(+(-\)\(22\!\cdots\!16\)\( + \)\(12\!\cdots\!72\)\( \beta_{1} + \)\(26\!\cdots\!08\)\( \beta_{2}) q^{92}\) \(+(\)\(11\!\cdots\!64\)\( + \)\(67\!\cdots\!92\)\( \beta_{1} - \)\(10\!\cdots\!92\)\( \beta_{2}) q^{93}\) \(+(-\)\(17\!\cdots\!36\)\( - \)\(21\!\cdots\!72\)\( \beta_{1} - \)\(26\!\cdots\!96\)\( \beta_{2}) q^{94}\) \(+(-\)\(11\!\cdots\!00\)\( - \)\(20\!\cdots\!50\)\( \beta_{1} + \)\(61\!\cdots\!50\)\( \beta_{2}) q^{95}\) \(+(\)\(31\!\cdots\!32\)\( - \)\(10\!\cdots\!68\)\( \beta_{1} + \)\(48\!\cdots\!76\)\( \beta_{2}) q^{96}\) \(+(\)\(38\!\cdots\!94\)\( + \)\(14\!\cdots\!84\)\( \beta_{1} + \)\(16\!\cdots\!84\)\( \beta_{2}) q^{97}\) \(+(-\)\(12\!\cdots\!72\)\( - \)\(25\!\cdots\!63\)\( \beta_{1} - \)\(23\!\cdots\!60\)\( \beta_{2}) q^{98}\) \(+(\)\(15\!\cdots\!76\)\( + \)\(12\!\cdots\!53\)\( \beta_{1} - \)\(81\!\cdots\!71\)\( \beta_{2}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut -\mathstrut 344688q^{2} \) \(\mathstrut -\mathstrut 10820953044q^{3} \) \(\mathstrut +\mathstrut 6271704903936q^{4} \) \(\mathstrut -\mathstrut 212302350281550q^{5} \) \(\mathstrut +\mathstrut 4970194114982976q^{6} \) \(\mathstrut +\mathstrut 57878416258239192q^{7} \) \(\mathstrut -\mathstrut 3555831711183237120q^{8} \) \(\mathstrut +\mathstrut 13277004110931878919q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut -\mathstrut 344688q^{2} \) \(\mathstrut -\mathstrut 10820953044q^{3} \) \(\mathstrut +\mathstrut 6271704903936q^{4} \) \(\mathstrut -\mathstrut 212302350281550q^{5} \) \(\mathstrut +\mathstrut 4970194114982976q^{6} \) \(\mathstrut +\mathstrut 57878416258239192q^{7} \) \(\mathstrut -\mathstrut 3555831711183237120q^{8} \) \(\mathstrut +\mathstrut 13277004110931878919q^{9} \) \(\mathstrut -\mathstrut \)\(91\!\cdots\!00\)\(q^{10} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!64\)\(q^{11} \) \(\mathstrut -\mathstrut \)\(71\!\cdots\!48\)\(q^{12} \) \(\mathstrut -\mathstrut \)\(98\!\cdots\!94\)\(q^{13} \) \(\mathstrut -\mathstrut \)\(66\!\cdots\!08\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!00\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!48\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!02\)\(q^{17} \) \(\mathstrut -\mathstrut \)\(51\!\cdots\!04\)\(q^{18} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!80\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!00\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(78\!\cdots\!24\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!44\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!56\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!60\)\(q^{24} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!75\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!04\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!80\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(52\!\cdots\!64\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!70\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!00\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!04\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!48\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!28\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!08\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!00\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!28\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!22\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!20\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!12\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!00\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!74\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!24\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!56\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(68\!\cdots\!32\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!50\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!16\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(63\!\cdots\!68\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(48\!\cdots\!44\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(97\!\cdots\!29\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!24\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!52\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(79\!\cdots\!06\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!20\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!00\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!80\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!40\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!80\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!60\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!00\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(87\!\cdots\!86\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!16\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!36\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!64\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!00\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(70\!\cdots\!88\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!52\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(95\!\cdots\!84\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!48\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!00\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!84\)\(q^{71} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!60\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!06\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!08\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!00\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!40\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!96\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!32\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!20\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!00\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!97\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(57\!\cdots\!96\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(61\!\cdots\!44\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!88\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!00\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!44\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!40\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!60\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!10\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!00\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!96\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(67\!\cdots\!48\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!92\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(53\!\cdots\!08\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(95\!\cdots\!96\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!82\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!16\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!28\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(x^{2}\mathstrut -\mathstrut \) \(2784108376\) \(x\mathstrut +\mathstrut \) \(1945534874860\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 48 \nu - 16 \)
\(\beta_{2}\)\(=\)\( 4 \nu^{2} + 4212 \nu - 7424290408 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(16\)\()/48\)
\(\nu^{2}\)\(=\)\((\)\(4\) \(\beta_{2}\mathstrut -\mathstrut \) \(351\) \(\beta_{1}\mathstrut +\mathstrut \) \(29697156016\)\()/16\)

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
−53110.1
698.922
52412.2
−2.66420e6 −7.98359e9 4.89893e12 3.47255e13 2.12699e16 1.66807e17 −7.19310e18 2.72647e19 −9.25156e19
1.2 −81363.7 3.82217e9 −2.19240e12 9.10518e13 −3.10986e14 −1.69829e16 3.57303e17 −2.18640e19 −7.40831e18
1.3 2.40087e6 −6.65953e9 3.56518e12 −3.38080e14 −1.59887e16 −9.19453e16 3.27996e18 7.87633e18 −8.11687e20
\(n\): 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 \(S_{42}^{\mathrm{new}}(\Gamma_0(1))\).