Properties

Label 1.56.a.a
Level 1
Weight 56
Character orbit 1.a
Self dual Yes
Analytic conductor 19.158
Analytic rank 0
Dimension 4
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 56 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(19.1581467685\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{20}\cdot 3^{9}\cdot 5^{2}\cdot 7\cdot 11 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +(52155630 + \beta_{1}) q^{2} +(-1705367672820 + 3242 \beta_{1} + \beta_{2}) q^{3} +(9681939694685968 + 58418668 \beta_{1} + 746 \beta_{2} + \beta_{3}) q^{4} +(3599234490846791790 + 22664368232 \beta_{1} + 356196 \beta_{2} + 384 \beta_{3}) q^{5} +(50439136311928003752 + 1489816805772 \beta_{1} + 101562144 \beta_{2} + 20304 \beta_{3}) q^{6} +(-\)\(51\!\cdots\!00\)\( - 102563980249724 \beta_{1} + 12193230266 \beta_{2} - 2885120 \beta_{3}) q^{7} +(\)\(11\!\cdots\!60\)\( + 12391009158764960 \beta_{1} + 261776524848 \beta_{2} + 122207160 \beta_{3}) q^{8} +(\)\(11\!\cdots\!57\)\( + 209112115633326576 \beta_{1} - 13323630665448 \beta_{2} - 2974848768 \beta_{3}) q^{9} +O(q^{10})\) \( q +(52155630 + \beta_{1}) q^{2} +(-1705367672820 + 3242 \beta_{1} + \beta_{2}) q^{3} +(9681939694685968 + 58418668 \beta_{1} + 746 \beta_{2} + \beta_{3}) q^{4} +(3599234490846791790 + 22664368232 \beta_{1} + 356196 \beta_{2} + 384 \beta_{3}) q^{5} +(50439136311928003752 + 1489816805772 \beta_{1} + 101562144 \beta_{2} + 20304 \beta_{3}) q^{6} +(-\)\(51\!\cdots\!00\)\( - 102563980249724 \beta_{1} + 12193230266 \beta_{2} - 2885120 \beta_{3}) q^{7} +(\)\(11\!\cdots\!60\)\( + 12391009158764960 \beta_{1} + 261776524848 \beta_{2} + 122207160 \beta_{3}) q^{8} +(\)\(11\!\cdots\!57\)\( + 209112115633326576 \beta_{1} - 13323630665448 \beta_{2} - 2974848768 \beta_{3}) q^{9} +(\)\(11\!\cdots\!40\)\( + 18697416450078039662 \beta_{1} + 107608441284736 \beta_{2} + 48348916544 \beta_{3}) q^{10} +(\)\(48\!\cdots\!52\)\( + \)\(21\!\cdots\!50\)\( \beta_{1} + 1038417327392955 \beta_{2} - 549682252800 \beta_{3}) q^{11} +(\)\(12\!\cdots\!60\)\( + \)\(99\!\cdots\!96\)\( \beta_{1} - 21919607144427656 \beta_{2} + 4259397219660 \beta_{3}) q^{12} +(-\)\(11\!\cdots\!90\)\( - \)\(15\!\cdots\!88\)\( \beta_{1} + 93123620513895140 \beta_{2} - 18025859047040 \beta_{3}) q^{13} +(-\)\(70\!\cdots\!64\)\( - \)\(11\!\cdots\!76\)\( \beta_{1} + 716232693497310528 \beta_{2} - 41838005080032 \beta_{3}) q^{14} +(\)\(62\!\cdots\!80\)\( + \)\(39\!\cdots\!64\)\( \beta_{1} - 9345651062707231458 \beta_{2} + 1428278630266368 \beta_{3}) q^{15} +(\)\(24\!\cdots\!76\)\( + \)\(43\!\cdots\!24\)\( \beta_{1} + 25946238006477621888 \beta_{2} - 12931429873924032 \beta_{3}) q^{16} +(\)\(21\!\cdots\!90\)\( + \)\(26\!\cdots\!16\)\( \beta_{1} + \)\(17\!\cdots\!08\)\( \beta_{2} + 68245048336369920 \beta_{3}) q^{17} +(\)\(96\!\cdots\!30\)\( - \)\(13\!\cdots\!43\)\( \beta_{1} - \)\(15\!\cdots\!04\)\( \beta_{2} - 170112162838331520 \beta_{3}) q^{18} +(\)\(84\!\cdots\!00\)\( - \)\(13\!\cdots\!38\)\( \beta_{1} + \)\(31\!\cdots\!29\)\( \beta_{2} - 602771992327689216 \beta_{3}) q^{19} +(\)\(73\!\cdots\!20\)\( + \)\(25\!\cdots\!16\)\( \beta_{1} + \)\(18\!\cdots\!48\)\( \beta_{2} + 9167080902472652142 \beta_{3}) q^{20} +(\)\(23\!\cdots\!52\)\( + \)\(15\!\cdots\!64\)\( \beta_{1} - \)\(12\!\cdots\!92\)\( \beta_{2} - 54569280429890698752 \beta_{3}) q^{21} +(\)\(95\!\cdots\!60\)\( - \)\(10\!\cdots\!68\)\( \beta_{1} + \)\(18\!\cdots\!60\)\( \beta_{2} + \)\(20\!\cdots\!60\)\( \beta_{3}) q^{22} +(\)\(12\!\cdots\!40\)\( - \)\(86\!\cdots\!28\)\( \beta_{1} + \)\(88\!\cdots\!54\)\( \beta_{2} - \)\(45\!\cdots\!00\)\( \beta_{3}) q^{23} +(\)\(47\!\cdots\!00\)\( + \)\(16\!\cdots\!36\)\( \beta_{1} - \)\(44\!\cdots\!68\)\( \beta_{2} + \)\(10\!\cdots\!52\)\( \beta_{3}) q^{24} +(\)\(89\!\cdots\!75\)\( + \)\(10\!\cdots\!20\)\( \beta_{1} + \)\(65\!\cdots\!60\)\( \beta_{2} + \)\(36\!\cdots\!40\)\( \beta_{3}) q^{25} +(-\)\(73\!\cdots\!48\)\( - \)\(15\!\cdots\!94\)\( \beta_{1} - \)\(51\!\cdots\!88\)\( \beta_{2} - \)\(15\!\cdots\!08\)\( \beta_{3}) q^{26} +(-\)\(21\!\cdots\!60\)\( - \)\(52\!\cdots\!60\)\( \beta_{1} + \)\(81\!\cdots\!02\)\( \beta_{2} + \)\(28\!\cdots\!40\)\( \beta_{3}) q^{27} +(-\)\(35\!\cdots\!40\)\( - \)\(33\!\cdots\!52\)\( \beta_{1} - \)\(46\!\cdots\!24\)\( \beta_{2} - \)\(31\!\cdots\!60\)\( \beta_{3}) q^{28} +(-\)\(45\!\cdots\!50\)\( + \)\(40\!\cdots\!08\)\( \beta_{1} + \)\(10\!\cdots\!56\)\( \beta_{2} - \)\(13\!\cdots\!44\)\( \beta_{3}) q^{29} +(\)\(20\!\cdots\!80\)\( + \)\(86\!\cdots\!24\)\( \beta_{1} - \)\(42\!\cdots\!28\)\( \beta_{2} + \)\(31\!\cdots\!88\)\( \beta_{3}) q^{30} +(\)\(57\!\cdots\!52\)\( - \)\(24\!\cdots\!00\)\( \beta_{1} - \)\(84\!\cdots\!60\)\( \beta_{2} - \)\(15\!\cdots\!00\)\( \beta_{3}) q^{31} +(\)\(15\!\cdots\!80\)\( - \)\(55\!\cdots\!64\)\( \beta_{1} - \)\(54\!\cdots\!80\)\( \beta_{2} - \)\(27\!\cdots\!80\)\( \beta_{3}) q^{32} +(\)\(21\!\cdots\!60\)\( + \)\(11\!\cdots\!44\)\( \beta_{1} + \)\(17\!\cdots\!72\)\( \beta_{2} - \)\(17\!\cdots\!80\)\( \beta_{3}) q^{33} +(\)\(22\!\cdots\!36\)\( + \)\(51\!\cdots\!66\)\( \beta_{1} + \)\(29\!\cdots\!92\)\( \beta_{2} + \)\(90\!\cdots\!12\)\( \beta_{3}) q^{34} +(-\)\(12\!\cdots\!60\)\( - \)\(12\!\cdots\!08\)\( \beta_{1} - \)\(18\!\cdots\!24\)\( \beta_{2} - \)\(50\!\cdots\!96\)\( \beta_{3}) q^{35} +(-\)\(57\!\cdots\!24\)\( - \)\(99\!\cdots\!56\)\( \beta_{1} + \)\(19\!\cdots\!78\)\( \beta_{2} - \)\(64\!\cdots\!67\)\( \beta_{3}) q^{36} +(-\)\(80\!\cdots\!30\)\( - \)\(36\!\cdots\!64\)\( \beta_{1} + \)\(21\!\cdots\!92\)\( \beta_{2} + \)\(20\!\cdots\!80\)\( \beta_{3}) q^{37} +(-\)\(12\!\cdots\!60\)\( + \)\(72\!\cdots\!80\)\( \beta_{1} + \)\(12\!\cdots\!12\)\( \beta_{2} - \)\(10\!\cdots\!80\)\( \beta_{3}) q^{38} +(\)\(16\!\cdots\!64\)\( - \)\(18\!\cdots\!36\)\( \beta_{1} - \)\(25\!\cdots\!62\)\( \beta_{2} - \)\(69\!\cdots\!52\)\( \beta_{3}) q^{39} +(\)\(10\!\cdots\!00\)\( + \)\(46\!\cdots\!40\)\( \beta_{1} + \)\(12\!\cdots\!20\)\( \beta_{2} + \)\(15\!\cdots\!80\)\( \beta_{3}) q^{40} +(\)\(61\!\cdots\!02\)\( - \)\(10\!\cdots\!00\)\( \beta_{1} + \)\(12\!\cdots\!40\)\( \beta_{2} + \)\(20\!\cdots\!00\)\( \beta_{3}) q^{41} +(\)\(12\!\cdots\!40\)\( - \)\(57\!\cdots\!68\)\( \beta_{1} - \)\(20\!\cdots\!96\)\( \beta_{2} - \)\(47\!\cdots\!20\)\( \beta_{3}) q^{42} +(-\)\(21\!\cdots\!00\)\( - \)\(20\!\cdots\!98\)\( \beta_{1} - \)\(49\!\cdots\!13\)\( \beta_{2} + \)\(48\!\cdots\!00\)\( \beta_{3}) q^{43} +(-\)\(12\!\cdots\!64\)\( + \)\(96\!\cdots\!36\)\( \beta_{1} + \)\(27\!\cdots\!32\)\( \beta_{2} + \)\(23\!\cdots\!52\)\( \beta_{3}) q^{44} +(-\)\(23\!\cdots\!70\)\( - \)\(41\!\cdots\!96\)\( \beta_{1} + \)\(93\!\cdots\!12\)\( \beta_{2} - \)\(22\!\cdots\!52\)\( \beta_{3}) q^{45} +(-\)\(30\!\cdots\!48\)\( - \)\(17\!\cdots\!40\)\( \beta_{1} - \)\(42\!\cdots\!40\)\( \beta_{2} - \)\(95\!\cdots\!80\)\( \beta_{3}) q^{46} +(\)\(10\!\cdots\!60\)\( - \)\(25\!\cdots\!84\)\( \beta_{1} - \)\(34\!\cdots\!16\)\( \beta_{2} + \)\(26\!\cdots\!00\)\( \beta_{3}) q^{47} +(\)\(49\!\cdots\!40\)\( + \)\(25\!\cdots\!32\)\( \beta_{1} + \)\(48\!\cdots\!84\)\( \beta_{2} - \)\(59\!\cdots\!80\)\( \beta_{3}) q^{48} +(\)\(12\!\cdots\!93\)\( + \)\(96\!\cdots\!80\)\( \beta_{1} - \)\(46\!\cdots\!00\)\( \beta_{2} - \)\(77\!\cdots\!40\)\( \beta_{3}) q^{49} +(\)\(43\!\cdots\!50\)\( + \)\(16\!\cdots\!95\)\( \beta_{1} + \)\(19\!\cdots\!60\)\( \beta_{2} + \)\(13\!\cdots\!40\)\( \beta_{3}) q^{50} +(\)\(28\!\cdots\!52\)\( + \)\(97\!\cdots\!08\)\( \beta_{1} - \)\(15\!\cdots\!14\)\( \beta_{2} - \)\(11\!\cdots\!44\)\( \beta_{3}) q^{51} +(-\)\(65\!\cdots\!00\)\( - \)\(73\!\cdots\!64\)\( \beta_{1} - \)\(72\!\cdots\!44\)\( \beta_{2} - \)\(17\!\cdots\!50\)\( \beta_{3}) q^{52} +(-\)\(12\!\cdots\!70\)\( + \)\(17\!\cdots\!72\)\( \beta_{1} + \)\(10\!\cdots\!56\)\( \beta_{2} + \)\(17\!\cdots\!60\)\( \beta_{3}) q^{53} +(-\)\(33\!\cdots\!00\)\( - \)\(86\!\cdots\!48\)\( \beta_{1} + \)\(83\!\cdots\!84\)\( \beta_{2} - \)\(23\!\cdots\!36\)\( \beta_{3}) q^{54} +(-\)\(31\!\cdots\!20\)\( + \)\(37\!\cdots\!64\)\( \beta_{1} + \)\(22\!\cdots\!42\)\( \beta_{2} + \)\(87\!\cdots\!68\)\( \beta_{3}) q^{55} +(-\)\(74\!\cdots\!00\)\( - \)\(93\!\cdots\!52\)\( \beta_{1} - \)\(74\!\cdots\!64\)\( \beta_{2} - \)\(98\!\cdots\!64\)\( \beta_{3}) q^{56} +(\)\(40\!\cdots\!80\)\( + \)\(31\!\cdots\!40\)\( \beta_{1} + \)\(57\!\cdots\!44\)\( \beta_{2} - \)\(15\!\cdots\!00\)\( \beta_{3}) q^{57} +(\)\(15\!\cdots\!60\)\( - \)\(58\!\cdots\!10\)\( \beta_{1} + \)\(11\!\cdots\!48\)\( \beta_{2} + \)\(51\!\cdots\!20\)\( \beta_{3}) q^{58} +(\)\(20\!\cdots\!00\)\( - \)\(11\!\cdots\!94\)\( \beta_{1} - \)\(19\!\cdots\!93\)\( \beta_{2} - \)\(56\!\cdots\!08\)\( \beta_{3}) q^{59} +(\)\(25\!\cdots\!40\)\( + \)\(16\!\cdots\!32\)\( \beta_{1} + \)\(40\!\cdots\!96\)\( \beta_{2} + \)\(43\!\cdots\!84\)\( \beta_{3}) q^{60} +(\)\(17\!\cdots\!02\)\( + \)\(98\!\cdots\!00\)\( \beta_{1} - \)\(84\!\cdots\!00\)\( \beta_{2} + \)\(61\!\cdots\!00\)\( \beta_{3}) q^{61} +(-\)\(76\!\cdots\!40\)\( + \)\(47\!\cdots\!92\)\( \beta_{1} - \)\(29\!\cdots\!20\)\( \beta_{2} - \)\(26\!\cdots\!20\)\( \beta_{3}) q^{62} +(-\)\(17\!\cdots\!80\)\( - \)\(40\!\cdots\!28\)\( \beta_{1} + \)\(30\!\cdots\!38\)\( \beta_{2} + \)\(59\!\cdots\!20\)\( \beta_{3}) q^{63} +(-\)\(24\!\cdots\!32\)\( - \)\(29\!\cdots\!64\)\( \beta_{1} - \)\(19\!\cdots\!28\)\( \beta_{2} - \)\(20\!\cdots\!48\)\( \beta_{3}) q^{64} +(-\)\(24\!\cdots\!20\)\( - \)\(28\!\cdots\!16\)\( \beta_{1} - \)\(29\!\cdots\!48\)\( \beta_{2} - \)\(72\!\cdots\!92\)\( \beta_{3}) q^{65} +(\)\(16\!\cdots\!04\)\( + \)\(20\!\cdots\!44\)\( \beta_{1} + \)\(15\!\cdots\!48\)\( \beta_{2} + \)\(32\!\cdots\!08\)\( \beta_{3}) q^{66} +(\)\(10\!\cdots\!40\)\( + \)\(31\!\cdots\!66\)\( \beta_{1} + \)\(16\!\cdots\!93\)\( \beta_{2} + \)\(26\!\cdots\!20\)\( \beta_{3}) q^{67} +(\)\(15\!\cdots\!80\)\( + \)\(58\!\cdots\!08\)\( \beta_{1} + \)\(17\!\cdots\!12\)\( \beta_{2} + \)\(36\!\cdots\!90\)\( \beta_{3}) q^{68} +(\)\(12\!\cdots\!64\)\( - \)\(22\!\cdots\!08\)\( \beta_{1} + \)\(94\!\cdots\!84\)\( \beta_{2} - \)\(69\!\cdots\!56\)\( \beta_{3}) q^{69} +(-\)\(11\!\cdots\!60\)\( - \)\(19\!\cdots\!28\)\( \beta_{1} - \)\(20\!\cdots\!84\)\( \beta_{2} - \)\(47\!\cdots\!36\)\( \beta_{3}) q^{70} +(\)\(24\!\cdots\!52\)\( + \)\(59\!\cdots\!00\)\( \beta_{1} - \)\(28\!\cdots\!50\)\( \beta_{2} + \)\(18\!\cdots\!00\)\( \beta_{3}) q^{71} +(-\)\(10\!\cdots\!80\)\( - \)\(26\!\cdots\!60\)\( \beta_{1} + \)\(60\!\cdots\!76\)\( \beta_{2} - \)\(37\!\cdots\!40\)\( \beta_{3}) q^{72} +(-\)\(22\!\cdots\!10\)\( + \)\(64\!\cdots\!32\)\( \beta_{1} + \)\(36\!\cdots\!44\)\( \beta_{2} - \)\(11\!\cdots\!80\)\( \beta_{3}) q^{73} +(-\)\(19\!\cdots\!64\)\( - \)\(31\!\cdots\!30\)\( \beta_{1} + \)\(23\!\cdots\!00\)\( \beta_{2} - \)\(22\!\cdots\!60\)\( \beta_{3}) q^{74} +(\)\(13\!\cdots\!00\)\( + \)\(53\!\cdots\!90\)\( \beta_{1} - \)\(11\!\cdots\!05\)\( \beta_{2} + \)\(19\!\cdots\!80\)\( \beta_{3}) q^{75} +(-\)\(15\!\cdots\!00\)\( + \)\(42\!\cdots\!16\)\( \beta_{1} - \)\(62\!\cdots\!88\)\( \beta_{2} + \)\(90\!\cdots\!12\)\( \beta_{3}) q^{76} +(\)\(35\!\cdots\!00\)\( - \)\(26\!\cdots\!68\)\( \beta_{1} + \)\(24\!\cdots\!92\)\( \beta_{2} - \)\(87\!\cdots\!80\)\( \beta_{3}) q^{77} +(\)\(99\!\cdots\!00\)\( - \)\(16\!\cdots\!76\)\( \beta_{1} - \)\(36\!\cdots\!36\)\( \beta_{2} - \)\(96\!\cdots\!60\)\( \beta_{3}) q^{78} +(\)\(12\!\cdots\!00\)\( + \)\(12\!\cdots\!88\)\( \beta_{1} + \)\(75\!\cdots\!76\)\( \beta_{2} - \)\(42\!\cdots\!84\)\( \beta_{3}) q^{79} +(-\)\(88\!\cdots\!60\)\( + \)\(78\!\cdots\!52\)\( \beta_{1} + \)\(21\!\cdots\!56\)\( \beta_{2} + \)\(23\!\cdots\!24\)\( \beta_{3}) q^{80} +(\)\(15\!\cdots\!01\)\( - \)\(12\!\cdots\!68\)\( \beta_{1} - \)\(17\!\cdots\!96\)\( \beta_{2} + \)\(31\!\cdots\!24\)\( \beta_{3}) q^{81} +(-\)\(41\!\cdots\!40\)\( + \)\(10\!\cdots\!42\)\( \beta_{1} + \)\(48\!\cdots\!80\)\( \beta_{2} - \)\(80\!\cdots\!20\)\( \beta_{3}) q^{82} +(-\)\(17\!\cdots\!80\)\( - \)\(26\!\cdots\!38\)\( \beta_{1} + \)\(87\!\cdots\!93\)\( \beta_{2} - \)\(56\!\cdots\!00\)\( \beta_{3}) q^{83} +(-\)\(78\!\cdots\!64\)\( - \)\(11\!\cdots\!12\)\( \beta_{1} + \)\(17\!\cdots\!16\)\( \beta_{2} + \)\(13\!\cdots\!16\)\( \beta_{3}) q^{84} +(\)\(62\!\cdots\!40\)\( + \)\(23\!\cdots\!52\)\( \beta_{1} + \)\(56\!\cdots\!56\)\( \beta_{2} + \)\(14\!\cdots\!24\)\( \beta_{3}) q^{85} +(-\)\(97\!\cdots\!48\)\( - \)\(22\!\cdots\!72\)\( \beta_{1} - \)\(19\!\cdots\!64\)\( \beta_{2} - \)\(20\!\cdots\!04\)\( \beta_{3}) q^{86} +(\)\(19\!\cdots\!20\)\( + \)\(15\!\cdots\!00\)\( \beta_{1} - \)\(11\!\cdots\!74\)\( \beta_{2} - \)\(29\!\cdots\!20\)\( \beta_{3}) q^{87} +(\)\(65\!\cdots\!20\)\( + \)\(11\!\cdots\!20\)\( \beta_{1} + \)\(41\!\cdots\!96\)\( \beta_{2} + \)\(34\!\cdots\!20\)\( \beta_{3}) q^{88} +(\)\(39\!\cdots\!50\)\( + \)\(24\!\cdots\!44\)\( \beta_{1} + \)\(21\!\cdots\!08\)\( \beta_{2} + \)\(36\!\cdots\!08\)\( \beta_{3}) q^{89} +(-\)\(30\!\cdots\!20\)\( - \)\(29\!\cdots\!86\)\( \beta_{1} + \)\(28\!\cdots\!92\)\( \beta_{2} - \)\(37\!\cdots\!32\)\( \beta_{3}) q^{90} +(\)\(41\!\cdots\!52\)\( + \)\(18\!\cdots\!48\)\( \beta_{1} - \)\(15\!\cdots\!64\)\( \beta_{2} - \)\(52\!\cdots\!64\)\( \beta_{3}) q^{91} +(-\)\(68\!\cdots\!60\)\( - \)\(35\!\cdots\!04\)\( \beta_{1} - \)\(51\!\cdots\!32\)\( \beta_{2} + \)\(91\!\cdots\!80\)\( \beta_{3}) q^{92} +(-\)\(28\!\cdots\!40\)\( - \)\(43\!\cdots\!36\)\( \beta_{1} + \)\(58\!\cdots\!12\)\( \beta_{2} - \)\(32\!\cdots\!40\)\( \beta_{3}) q^{93} +(-\)\(10\!\cdots\!64\)\( + \)\(93\!\cdots\!32\)\( \beta_{1} - \)\(15\!\cdots\!56\)\( \beta_{2} - \)\(17\!\cdots\!76\)\( \beta_{3}) q^{94} +(\)\(22\!\cdots\!00\)\( + \)\(18\!\cdots\!60\)\( \beta_{1} - \)\(19\!\cdots\!70\)\( \beta_{2} + \)\(33\!\cdots\!20\)\( \beta_{3}) q^{95} +(-\)\(13\!\cdots\!48\)\( - \)\(15\!\cdots\!28\)\( \beta_{1} + \)\(20\!\cdots\!64\)\( \beta_{2} + \)\(38\!\cdots\!04\)\( \beta_{3}) q^{96} +(\)\(12\!\cdots\!10\)\( + \)\(24\!\cdots\!36\)\( \beta_{1} - \)\(17\!\cdots\!16\)\( \beta_{2} - \)\(32\!\cdots\!60\)\( \beta_{3}) q^{97} +(\)\(10\!\cdots\!90\)\( - \)\(16\!\cdots\!67\)\( \beta_{1} - \)\(15\!\cdots\!40\)\( \beta_{2} - \)\(37\!\cdots\!20\)\( \beta_{3}) q^{98} +(\)\(19\!\cdots\!64\)\( - \)\(34\!\cdots\!98\)\( \beta_{1} - \)\(12\!\cdots\!61\)\( \beta_{2} + \)\(36\!\cdots\!64\)\( \beta_{3}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 208622520q^{2} - 6821470691280q^{3} + 38727758778743872q^{4} + 14396937963387167160q^{5} + \)\(20\!\cdots\!08\)\(q^{6} - \)\(20\!\cdots\!00\)\(q^{7} + \)\(45\!\cdots\!40\)\(q^{8} + \)\(47\!\cdots\!28\)\(q^{9} + O(q^{10}) \) \( 4q + 208622520q^{2} - 6821470691280q^{3} + 38727758778743872q^{4} + 14396937963387167160q^{5} + \)\(20\!\cdots\!08\)\(q^{6} - \)\(20\!\cdots\!00\)\(q^{7} + \)\(45\!\cdots\!40\)\(q^{8} + \)\(47\!\cdots\!28\)\(q^{9} + \)\(46\!\cdots\!60\)\(q^{10} + \)\(19\!\cdots\!08\)\(q^{11} + \)\(51\!\cdots\!40\)\(q^{12} - \)\(44\!\cdots\!60\)\(q^{13} - \)\(28\!\cdots\!56\)\(q^{14} + \)\(24\!\cdots\!20\)\(q^{15} + \)\(97\!\cdots\!04\)\(q^{16} + \)\(85\!\cdots\!60\)\(q^{17} + \)\(38\!\cdots\!20\)\(q^{18} + \)\(33\!\cdots\!00\)\(q^{19} + \)\(29\!\cdots\!80\)\(q^{20} + \)\(92\!\cdots\!08\)\(q^{21} + \)\(38\!\cdots\!40\)\(q^{22} + \)\(49\!\cdots\!60\)\(q^{23} + \)\(19\!\cdots\!00\)\(q^{24} + \)\(35\!\cdots\!00\)\(q^{25} - \)\(29\!\cdots\!92\)\(q^{26} - \)\(85\!\cdots\!40\)\(q^{27} - \)\(14\!\cdots\!60\)\(q^{28} - \)\(18\!\cdots\!00\)\(q^{29} + \)\(81\!\cdots\!20\)\(q^{30} + \)\(22\!\cdots\!08\)\(q^{31} + \)\(63\!\cdots\!20\)\(q^{32} + \)\(84\!\cdots\!40\)\(q^{33} + \)\(89\!\cdots\!44\)\(q^{34} - \)\(48\!\cdots\!40\)\(q^{35} - \)\(23\!\cdots\!96\)\(q^{36} - \)\(32\!\cdots\!20\)\(q^{37} - \)\(49\!\cdots\!40\)\(q^{38} + \)\(66\!\cdots\!56\)\(q^{39} + \)\(42\!\cdots\!00\)\(q^{40} + \)\(24\!\cdots\!08\)\(q^{41} + \)\(50\!\cdots\!60\)\(q^{42} - \)\(86\!\cdots\!00\)\(q^{43} - \)\(48\!\cdots\!56\)\(q^{44} - \)\(95\!\cdots\!80\)\(q^{45} - \)\(12\!\cdots\!92\)\(q^{46} + \)\(42\!\cdots\!40\)\(q^{47} + \)\(19\!\cdots\!60\)\(q^{48} + \)\(51\!\cdots\!72\)\(q^{49} + \)\(17\!\cdots\!00\)\(q^{50} + \)\(11\!\cdots\!08\)\(q^{51} - \)\(26\!\cdots\!00\)\(q^{52} - \)\(48\!\cdots\!80\)\(q^{53} - \)\(13\!\cdots\!00\)\(q^{54} - \)\(12\!\cdots\!80\)\(q^{55} - \)\(29\!\cdots\!00\)\(q^{56} + \)\(16\!\cdots\!20\)\(q^{57} + \)\(60\!\cdots\!40\)\(q^{58} + \)\(81\!\cdots\!00\)\(q^{59} + \)\(10\!\cdots\!60\)\(q^{60} + \)\(70\!\cdots\!08\)\(q^{61} - \)\(30\!\cdots\!60\)\(q^{62} - \)\(71\!\cdots\!20\)\(q^{63} - \)\(98\!\cdots\!28\)\(q^{64} - \)\(96\!\cdots\!80\)\(q^{65} + \)\(64\!\cdots\!16\)\(q^{66} + \)\(41\!\cdots\!60\)\(q^{67} + \)\(62\!\cdots\!20\)\(q^{68} + \)\(51\!\cdots\!56\)\(q^{69} - \)\(47\!\cdots\!40\)\(q^{70} + \)\(99\!\cdots\!08\)\(q^{71} - \)\(43\!\cdots\!20\)\(q^{72} - \)\(89\!\cdots\!40\)\(q^{73} - \)\(79\!\cdots\!56\)\(q^{74} + \)\(52\!\cdots\!00\)\(q^{75} - \)\(60\!\cdots\!00\)\(q^{76} + \)\(14\!\cdots\!00\)\(q^{77} + \)\(39\!\cdots\!00\)\(q^{78} + \)\(48\!\cdots\!00\)\(q^{79} - \)\(35\!\cdots\!40\)\(q^{80} + \)\(62\!\cdots\!04\)\(q^{81} - \)\(16\!\cdots\!60\)\(q^{82} - \)\(71\!\cdots\!20\)\(q^{83} - \)\(31\!\cdots\!56\)\(q^{84} + \)\(24\!\cdots\!60\)\(q^{85} - \)\(39\!\cdots\!92\)\(q^{86} + \)\(79\!\cdots\!80\)\(q^{87} + \)\(26\!\cdots\!80\)\(q^{88} + \)\(15\!\cdots\!00\)\(q^{89} - \)\(12\!\cdots\!80\)\(q^{90} + \)\(16\!\cdots\!08\)\(q^{91} - \)\(27\!\cdots\!40\)\(q^{92} - \)\(11\!\cdots\!60\)\(q^{93} - \)\(40\!\cdots\!56\)\(q^{94} + \)\(90\!\cdots\!00\)\(q^{95} - \)\(54\!\cdots\!92\)\(q^{96} + \)\(51\!\cdots\!40\)\(q^{97} + \)\(43\!\cdots\!60\)\(q^{98} + \)\(79\!\cdots\!56\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 149272663100531 x^{2} + 190291401428579434725 x + 325546600176957146615614350\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 24 \nu - 6 \)
\(\beta_{2}\)\(=\)\((\)\( 3 \nu^{3} + 4282464 \nu^{2} - 426543633953529 \nu + 108528021030642975510 \)\()/37024736\)
\(\beta_{3}\)\(=\)\((\)\( -1119 \nu^{3} + 9065764896 \nu^{2} + 179490703370972877 \nu - 836337412779145122036366 \)\()/18512368\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 6\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 746 \beta_{2} - 45892580 \beta_{1} + 42990526972953072\)\()/576\)
\(\nu^{3}\)\(=\)\((\)\(-178436 \beta_{3} + 755480408 \beta_{2} + 434732522358409 \beta_{1} - 10275727616419482046458\)\()/72\)

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
−1.27360e7
−972934.
2.30346e6
1.14055e7
−2.53509e8 −1.66915e12 2.82382e16 1.07257e19 4.23145e20 −1.10352e23 1.97498e24 −1.71663e26 −2.71906e27
1.2 2.88052e7 1.23937e13 −3.51991e16 −1.26520e19 3.57004e20 2.79909e23 −2.05173e24 −2.08450e25 −3.64445e26
1.3 1.07439e8 −2.35279e13 −2.44858e16 −1.10426e19 −2.52780e21 −2.64783e23 −6.50160e24 3.79111e26 −1.18640e27
1.4 3.25888e8 5.98183e12 7.01743e16 2.73659e19 1.94941e21 −1.10303e23 1.11276e25 −1.38667e26 8.91821e27
\(n\): 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 \(S_{56}^{\mathrm{new}}(\Gamma_0(1))\).