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

Related objects

Downloads

Learn more about

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.

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

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

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

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 \(S_{56}^{\mathrm{new}}(\Gamma_0(1))\).