Properties

Label 3.38.a.a
Level 3
Weight 38
Character orbit 3.a
Self dual Yes
Analytic conductor 26.014
Analytic rank 1
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 38 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(26.0142114374\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{5}\cdot 5 \)
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.

\(f(q)\) \(=\) \( q\) \( + ( -103636 - \beta_{1} ) q^{2} \) \( -387420489 q^{3} \) \( + ( 112825533616 + 140608 \beta_{1} + 2 \beta_{2} ) q^{4} \) \( + ( -3209572628930 - 13500494 \beta_{1} - 39 \beta_{2} ) q^{5} \) \( + ( 40150709798004 + 387420489 \beta_{1} ) q^{6} \) \( + ( -1540628114567248 - 442709162 \beta_{1} - 35413 \beta_{2} ) q^{7} \) \( + ( -31127969697970496 - 95903083520 \beta_{1} - 621816 \beta_{2} ) q^{8} \) \( + 150094635296999121 q^{9} \) \(+O(q^{10})\) \( q\) \(+(-103636 - \beta_{1}) q^{2}\) \(-387420489 q^{3}\) \(+(112825533616 + 140608 \beta_{1} + 2 \beta_{2}) q^{4}\) \(+(-3209572628930 - 13500494 \beta_{1} - 39 \beta_{2}) q^{5}\) \(+(40150709798004 + 387420489 \beta_{1}) q^{6}\) \(+(-1540628114567248 - 442709162 \beta_{1} - 35413 \beta_{2}) q^{7}\) \(+(-31127969697970496 - 95903083520 \beta_{1} - 621816 \beta_{2}) q^{8}\) \(+150094635296999121 q^{9}\) \(+(3566316712247265960 + 5957412808898 \beta_{1} + 33642688 \beta_{2}) q^{10}\) \(+(7557767785713209540 + 2973462571676 \beta_{1} + 34411854 \beta_{2}) q^{11}\) \(+(-43710923405196658224 - 54474420117312 \beta_{1} - 774840978 \beta_{2}) q^{12}\) \(+(-56090000067931697170 + 885146711869036 \beta_{1} - 657140938 \beta_{2}) q^{13}\) \(+(\)\(26\!\cdots\!76\)\( + 3598873140223312 \beta_{1} + 6916252224 \beta_{2}) q^{14}\) \(+(\)\(12\!\cdots\!70\)\( + 5230367987221566 \beta_{1} + 15109399071 \beta_{2}) q^{15}\) \(+(\)\(10\!\cdots\!52\)\( + 51201953589596160 \beta_{1} + 22823524896 \beta_{2}) q^{16}\) \(+(-\)\(90\!\cdots\!02\)\( + 30058128433440564 \beta_{1} - 514120264806 \beta_{2}) q^{17}\) \(+(-\)\(15\!\cdots\!56\)\( - 150094635296999121 \beta_{1}) q^{18}\) \(+(\)\(19\!\cdots\!68\)\( + 434381446040702124 \beta_{1} + 5732394190422 \beta_{2}) q^{19}\) \(+(-\)\(13\!\cdots\!60\)\( - 3870883224028101248 \beta_{1} - 12284056198788 \beta_{2}) q^{20}\) \(+(\)\(59\!\cdots\!72\)\( + 171514600026820218 \beta_{1} + 13719721776957 \beta_{2}) q^{21}\) \(+(-\)\(14\!\cdots\!84\)\( - 9651854777049673412 \beta_{1} - 11807263879552 \beta_{2}) q^{22}\) \(+(-\)\(57\!\cdots\!24\)\( + 8496445345062383212 \beta_{1} - 104980687284426 \beta_{2}) q^{23}\) \(+(\)\(12\!\cdots\!44\)\( + 37154819513926241280 \beta_{1} + 240904258788024 \beta_{2}) q^{24}\) \(+(-\)\(82\!\cdots\!25\)\( + \)\(13\!\cdots\!60\)\( \beta_{1} + 597373805525660 \beta_{2}) q^{25}\) \(+(-\)\(20\!\cdots\!48\)\( + 61254426912333621778 \beta_{1} - 1658382321996672 \beta_{2}) q^{26}\) \(-\)\(58\!\cdots\!69\)\( q^{27}\) \(+(-\)\(67\!\cdots\!96\)\( - \)\(73\!\cdots\!76\)\( \beta_{1} - 3508458374889888 \beta_{2}) q^{28}\) \(+(\)\(37\!\cdots\!26\)\( - \)\(82\!\cdots\!98\)\( \beta_{1} + 18257412042299043 \beta_{2}) q^{29}\) \(+(-\)\(13\!\cdots\!40\)\( - \)\(23\!\cdots\!22\)\( \beta_{1} - 13033866636234432 \beta_{2}) q^{30}\) \(+(-\)\(39\!\cdots\!12\)\( + \)\(13\!\cdots\!50\)\( \beta_{1} - 3843106330242889 \beta_{2}) q^{31}\) \(+(-\)\(90\!\cdots\!28\)\( - \)\(71\!\cdots\!32\)\( \beta_{1} - 20829013176835968 \beta_{2}) q^{32}\) \(+(-\)\(29\!\cdots\!60\)\( - \)\(11\!\cdots\!64\)\( \beta_{1} - 13331857304076606 \beta_{2}) q^{33}\) \(+(-\)\(62\!\cdots\!88\)\( + \)\(37\!\cdots\!94\)\( \beta_{1} + 27438424229580672 \beta_{2}) q^{34}\) \(+(\)\(15\!\cdots\!60\)\( + \)\(53\!\cdots\!88\)\( \beta_{1} + 139088055761777178 \beta_{2}) q^{35}\) \(+(\)\(16\!\cdots\!36\)\( + \)\(21\!\cdots\!68\)\( \beta_{1} + 300189270593998242 \beta_{2}) q^{36}\) \(+(\)\(20\!\cdots\!22\)\( - \)\(73\!\cdots\!20\)\( \beta_{1} - 401057286573669456 \beta_{2}) q^{37}\) \(+(-\)\(12\!\cdots\!92\)\( - \)\(54\!\cdots\!96\)\( \beta_{1} - 1844989622710270848 \beta_{2}) q^{38}\) \(+(\)\(21\!\cdots\!30\)\( - \)\(34\!\cdots\!04\)\( \beta_{1} + 254589863541878682 \beta_{2}) q^{39}\) \(+(\)\(57\!\cdots\!00\)\( + \)\(13\!\cdots\!60\)\( \beta_{1} + 5209925388004786160 \beta_{2}) q^{40}\) \(+(\)\(69\!\cdots\!78\)\( - \)\(44\!\cdots\!52\)\( \beta_{1} + 1156269906838666494 \beta_{2}) q^{41}\) \(+(-\)\(10\!\cdots\!64\)\( - \)\(13\!\cdots\!68\)\( \beta_{1} - 2679497818669417536 \beta_{2}) q^{42}\) \(+(\)\(82\!\cdots\!48\)\( + \)\(33\!\cdots\!16\)\( \beta_{1} - 18160112109281379310 \beta_{2}) q^{43}\) \(+(\)\(14\!\cdots\!24\)\( + \)\(21\!\cdots\!76\)\( \beta_{1} + 16584957391995795336 \beta_{2}) q^{44}\) \(+(-\)\(48\!\cdots\!30\)\( - \)\(20\!\cdots\!74\)\( \beta_{1} - 5853690776582965719 \beta_{2}) q^{45}\) \(+(-\)\(14\!\cdots\!52\)\( + \)\(11\!\cdots\!60\)\( \beta_{1} + 885320354412981376 \beta_{2}) q^{46}\) \(+(-\)\(39\!\cdots\!44\)\( - \)\(19\!\cdots\!56\)\( \beta_{1} + 49298541442184762322 \beta_{2}) q^{47}\) \(+(-\)\(41\!\cdots\!28\)\( - \)\(19\!\cdots\!40\)\( \beta_{1} - 8842301175911994144 \beta_{2}) q^{48}\) \(+(-\)\(74\!\cdots\!91\)\( + \)\(55\!\cdots\!72\)\( \beta_{1} + 39289783975049388704 \beta_{2}) q^{49}\) \(+(-\)\(32\!\cdots\!00\)\( - \)\(31\!\cdots\!95\)\( \beta_{1} - \)\(37\!\cdots\!20\)\( \beta_{2}) q^{50}\) \(+(\)\(35\!\cdots\!78\)\( - \)\(11\!\cdots\!96\)\( \beta_{1} + \)\(19\!\cdots\!34\)\( \beta_{2}) q^{51}\) \(+(\)\(14\!\cdots\!28\)\( + \)\(17\!\cdots\!40\)\( \beta_{1} + \)\(25\!\cdots\!80\)\( \beta_{2}) q^{52}\) \(+(-\)\(38\!\cdots\!94\)\( - \)\(17\!\cdots\!54\)\( \beta_{1} + \)\(11\!\cdots\!83\)\( \beta_{2}) q^{53}\) \(+(\)\(60\!\cdots\!84\)\( + \)\(58\!\cdots\!69\)\( \beta_{1}) q^{54}\) \(+(-\)\(43\!\cdots\!60\)\( - \)\(14\!\cdots\!88\)\( \beta_{1} - \)\(44\!\cdots\!28\)\( \beta_{2}) q^{55}\) \(+(\)\(21\!\cdots\!20\)\( + \)\(41\!\cdots\!04\)\( \beta_{1} + \)\(11\!\cdots\!24\)\( \beta_{2}) q^{56}\) \(+(-\)\(75\!\cdots\!52\)\( - \)\(16\!\cdots\!36\)\( \beta_{1} - \)\(22\!\cdots\!58\)\( \beta_{2}) q^{57}\) \(+(\)\(19\!\cdots\!96\)\( - \)\(10\!\cdots\!70\)\( \beta_{1} - \)\(14\!\cdots\!04\)\( \beta_{2}) q^{58}\) \(+(\)\(36\!\cdots\!12\)\( - \)\(73\!\cdots\!80\)\( \beta_{1} + \)\(19\!\cdots\!92\)\( \beta_{2}) q^{59}\) \(+(\)\(52\!\cdots\!40\)\( + \)\(14\!\cdots\!72\)\( \beta_{1} + \)\(47\!\cdots\!32\)\( \beta_{2}) q^{60}\) \(+(\)\(98\!\cdots\!34\)\( - \)\(26\!\cdots\!68\)\( \beta_{1} + \)\(71\!\cdots\!24\)\( \beta_{2}) q^{61}\) \(+(\)\(82\!\cdots\!64\)\( + \)\(41\!\cdots\!12\)\( \beta_{1} - \)\(20\!\cdots\!00\)\( \beta_{2}) q^{62}\) \(+(-\)\(23\!\cdots\!08\)\( - \)\(66\!\cdots\!02\)\( \beta_{1} - \)\(53\!\cdots\!73\)\( \beta_{2}) q^{63}\) \(+(-\)\(35\!\cdots\!08\)\( + \)\(32\!\cdots\!12\)\( \beta_{1} + \)\(18\!\cdots\!52\)\( \beta_{2}) q^{64}\) \(+(-\)\(25\!\cdots\!20\)\( - \)\(27\!\cdots\!16\)\( \beta_{1} - \)\(22\!\cdots\!46\)\( \beta_{2}) q^{65}\) \(+(\)\(57\!\cdots\!76\)\( + \)\(37\!\cdots\!68\)\( \beta_{1} + \)\(45\!\cdots\!28\)\( \beta_{2}) q^{66}\) \(+(\)\(12\!\cdots\!48\)\( - \)\(19\!\cdots\!72\)\( \beta_{1} + \)\(51\!\cdots\!28\)\( \beta_{2}) q^{67}\) \(+(-\)\(71\!\cdots\!72\)\( - \)\(84\!\cdots\!88\)\( \beta_{1} - \)\(91\!\cdots\!56\)\( \beta_{2}) q^{68}\) \(+(\)\(22\!\cdots\!36\)\( - \)\(32\!\cdots\!68\)\( \beta_{1} + \)\(40\!\cdots\!14\)\( \beta_{2}) q^{69}\) \(+(-\)\(14\!\cdots\!20\)\( - \)\(25\!\cdots\!96\)\( \beta_{1} - \)\(12\!\cdots\!76\)\( \beta_{2}) q^{70}\) \(+(-\)\(73\!\cdots\!48\)\( - \)\(97\!\cdots\!64\)\( \beta_{1} + \)\(10\!\cdots\!26\)\( \beta_{2}) q^{71}\) \(+(-\)\(46\!\cdots\!16\)\( - \)\(14\!\cdots\!20\)\( \beta_{1} - \)\(93\!\cdots\!36\)\( \beta_{2}) q^{72}\) \(+(\)\(10\!\cdots\!26\)\( + \)\(70\!\cdots\!28\)\( \beta_{1} - \)\(11\!\cdots\!80\)\( \beta_{2}) q^{73}\) \(+(\)\(15\!\cdots\!76\)\( + \)\(57\!\cdots\!18\)\( \beta_{1} + \)\(21\!\cdots\!40\)\( \beta_{2}) q^{74}\) \(+(\)\(32\!\cdots\!25\)\( - \)\(53\!\cdots\!40\)\( \beta_{1} - \)\(23\!\cdots\!40\)\( \beta_{2}) q^{75}\) \(+(\)\(11\!\cdots\!72\)\( + \)\(19\!\cdots\!76\)\( \beta_{1} + \)\(60\!\cdots\!08\)\( \beta_{2}) q^{76}\) \(+(-\)\(20\!\cdots\!56\)\( - \)\(17\!\cdots\!48\)\( \beta_{1} - \)\(26\!\cdots\!40\)\( \beta_{2}) q^{77}\) \(+(\)\(79\!\cdots\!72\)\( - \)\(23\!\cdots\!42\)\( \beta_{1} + \)\(64\!\cdots\!08\)\( \beta_{2}) q^{78}\) \(+(\)\(24\!\cdots\!80\)\( + \)\(70\!\cdots\!06\)\( \beta_{1} - \)\(62\!\cdots\!89\)\( \beta_{2}) q^{79}\) \(+(-\)\(20\!\cdots\!80\)\( - \)\(39\!\cdots\!64\)\( \beta_{1} - \)\(19\!\cdots\!84\)\( \beta_{2}) q^{80}\) \(+\)\(22\!\cdots\!41\)\( q^{81}\) \(+(\)\(34\!\cdots\!04\)\( - \)\(74\!\cdots\!34\)\( \beta_{1} + \)\(69\!\cdots\!04\)\( \beta_{2}) q^{82}\) \(+(-\)\(10\!\cdots\!68\)\( + \)\(56\!\cdots\!92\)\( \beta_{1} + \)\(34\!\cdots\!70\)\( \beta_{2}) q^{83}\) \(+(\)\(26\!\cdots\!44\)\( + \)\(28\!\cdots\!64\)\( \beta_{1} + \)\(13\!\cdots\!32\)\( \beta_{2}) q^{84}\) \(+(\)\(70\!\cdots\!60\)\( + \)\(42\!\cdots\!88\)\( \beta_{1} + \)\(42\!\cdots\!78\)\( \beta_{2}) q^{85}\) \(+(-\)\(88\!\cdots\!20\)\( + \)\(95\!\cdots\!00\)\( \beta_{1} - \)\(35\!\cdots\!32\)\( \beta_{2}) q^{86}\) \(+(-\)\(14\!\cdots\!14\)\( + \)\(31\!\cdots\!22\)\( \beta_{1} - \)\(70\!\cdots\!27\)\( \beta_{2}) q^{87}\) \(+(-\)\(45\!\cdots\!76\)\( - \)\(11\!\cdots\!32\)\( \beta_{1} - \)\(54\!\cdots\!08\)\( \beta_{2}) q^{88}\) \(+(-\)\(53\!\cdots\!42\)\( + \)\(25\!\cdots\!52\)\( \beta_{1} + \)\(13\!\cdots\!60\)\( \beta_{2}) q^{89}\) \(+(\)\(53\!\cdots\!60\)\( + \)\(89\!\cdots\!58\)\( \beta_{1} + \)\(50\!\cdots\!48\)\( \beta_{2}) q^{90}\) \(+(\)\(15\!\cdots\!68\)\( - \)\(30\!\cdots\!04\)\( \beta_{1} - \)\(11\!\cdots\!98\)\( \beta_{2}) q^{91}\) \(+(-\)\(18\!\cdots\!08\)\( - \)\(20\!\cdots\!32\)\( \beta_{1} - \)\(86\!\cdots\!48\)\( \beta_{2}) q^{92}\) \(+(\)\(15\!\cdots\!68\)\( - \)\(52\!\cdots\!50\)\( \beta_{1} + \)\(14\!\cdots\!21\)\( \beta_{2}) q^{93}\) \(+(\)\(50\!\cdots\!00\)\( + \)\(18\!\cdots\!76\)\( \beta_{1} + \)\(30\!\cdots\!12\)\( \beta_{2}) q^{94}\) \(+(-\)\(35\!\cdots\!00\)\( - \)\(94\!\cdots\!80\)\( \beta_{1} - \)\(31\!\cdots\!80\)\( \beta_{2}) q^{95}\) \(+(\)\(35\!\cdots\!92\)\( + \)\(27\!\cdots\!48\)\( \beta_{1} + \)\(80\!\cdots\!52\)\( \beta_{2}) q^{96}\) \(+(\)\(89\!\cdots\!38\)\( - \)\(47\!\cdots\!52\)\( \beta_{1} - \)\(40\!\cdots\!52\)\( \beta_{2}) q^{97}\) \(+(-\)\(55\!\cdots\!00\)\( + \)\(50\!\cdots\!07\)\( \beta_{1} - \)\(17\!\cdots\!44\)\( \beta_{2}) q^{98}\) \(+(\)\(11\!\cdots\!40\)\( + \)\(44\!\cdots\!96\)\( \beta_{1} + \)\(51\!\cdots\!34\)\( \beta_{2}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut -\mathstrut 310908q^{2} \) \(\mathstrut -\mathstrut 1162261467q^{3} \) \(\mathstrut +\mathstrut 338476600848q^{4} \) \(\mathstrut -\mathstrut 9628717886790q^{5} \) \(\mathstrut +\mathstrut 120452129394012q^{6} \) \(\mathstrut -\mathstrut 4621884343701744q^{7} \) \(\mathstrut -\mathstrut 93383909093911488q^{8} \) \(\mathstrut +\mathstrut 450283905890997363q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut -\mathstrut 310908q^{2} \) \(\mathstrut -\mathstrut 1162261467q^{3} \) \(\mathstrut +\mathstrut 338476600848q^{4} \) \(\mathstrut -\mathstrut 9628717886790q^{5} \) \(\mathstrut +\mathstrut 120452129394012q^{6} \) \(\mathstrut -\mathstrut 4621884343701744q^{7} \) \(\mathstrut -\mathstrut 93383909093911488q^{8} \) \(\mathstrut +\mathstrut 450283905890997363q^{9} \) \(\mathstrut +\mathstrut 10698950136741797880q^{10} \) \(\mathstrut +\mathstrut 22673303357139628620q^{11} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!72\)\(q^{12} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!10\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(79\!\cdots\!28\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!10\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!56\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!06\)\(q^{17} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!68\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(58\!\cdots\!04\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!80\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!16\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!52\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!72\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!32\)\(q^{24} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!75\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(61\!\cdots\!44\)\(q^{26} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!07\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!88\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!78\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!20\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!36\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!84\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!80\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!64\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!80\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!08\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(60\!\cdots\!66\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!76\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(65\!\cdots\!90\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!34\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!92\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!44\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!72\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!90\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!56\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!32\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!84\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!73\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(96\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!34\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!84\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!82\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!52\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!80\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(63\!\cdots\!60\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!56\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(58\!\cdots\!88\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!36\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!20\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!02\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!92\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(69\!\cdots\!24\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!24\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(75\!\cdots\!60\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!28\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!44\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!16\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(66\!\cdots\!08\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!60\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!44\)\(q^{71} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!48\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!78\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!28\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(96\!\cdots\!75\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!16\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(61\!\cdots\!68\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!16\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(74\!\cdots\!40\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(61\!\cdots\!40\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(67\!\cdots\!23\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!12\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(31\!\cdots\!04\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(78\!\cdots\!32\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!80\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!60\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!42\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!28\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!26\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!80\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!04\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!24\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!04\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!00\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!76\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!14\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!00\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!20\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(x^{2}\mathstrut -\mathstrut \) \(2495042360\) \(x\mathstrut +\mathstrut \) \(9241471873200\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 12 \nu - 4 \)
\(\beta_{2}\)\(=\)\( 72 \nu^{2} + 399936 \nu - 119762166616 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(4\)\()/12\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2}\mathstrut -\mathstrut \) \(33328\) \(\beta_{1}\mathstrut +\mathstrut \) \(119762033304\)\()/72\)

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
47984.3
3724.64
−51708.0
−679444. −3.87420e8 3.24205e11 −1.35264e13 2.63231e14 −4.10476e15 −1.26897e17 1.50095e17 9.19042e18
1.2 −148328. −3.87420e8 −1.15438e11 7.60742e11 5.74652e13 2.59260e15 3.75086e16 1.50095e17 −1.12839e17
1.3 516864. −3.87420e8 1.29709e11 3.13693e12 −2.00244e14 −3.10972e15 −3.99525e15 1.50095e17 1.62136e18
\(n\): 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 \(T_{2}^{3} \) \(\mathstrut +\mathstrut 310908 T_{2}^{2} \) \(\mathstrut -\mathstrut 327064838400 T_{2} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!24\)\( \) acting on \(S_{38}^{\mathrm{new}}(\Gamma_0(3))\).