Properties

Label 1.60.a.a
Level 1
Weight 60
Character orbit 1.a
Self dual Yes
Analytic conductor 22.046
Analytic rank 0
Dimension 5
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 60 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(22.045800551\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{39}\cdot 3^{13}\cdot 5^{3}\cdot 7^{2} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+(-89938373 - \beta_{1}) q^{2}\) \(+(16803326335969 - 70086 \beta_{1} + \beta_{2}) q^{3}\) \(+(347763875847469769 + 97803674 \beta_{1} - 529 \beta_{2} + \beta_{3}) q^{4}\) \(+(35992322464951803524 + 174673117149 \beta_{1} + 157945 \beta_{2} + 12 \beta_{3} + \beta_{4}) q^{5}\) \(+(\)\(62\!\cdots\!16\)\( - 7082751985044 \beta_{1} - 52897176 \beta_{2} + 188064 \beta_{3} - 120 \beta_{4}) q^{6}\) \(+(\)\(29\!\cdots\!62\)\( - 1403044288237600 \beta_{1} + 37036880006 \beta_{2} - 1472240 \beta_{3} + 7020 \beta_{4}) q^{7}\) \(+(-\)\(69\!\cdots\!00\)\( - 363735709336321072 \beta_{1} + 4052591058968 \beta_{2} - 360632280 \beta_{3} - 266560 \beta_{4}) q^{8}\) \(+(\)\(65\!\cdots\!09\)\( - 2757699652749137442 \beta_{1} + 50001513470982 \beta_{2} + 15750146952 \beta_{3} + 7379190 \beta_{4}) q^{9}\) \(+O(q^{10})\) \( q\) \(+(-89938373 - \beta_{1}) q^{2}\) \(+(16803326335969 - 70086 \beta_{1} + \beta_{2}) q^{3}\) \(+(347763875847469769 + 97803674 \beta_{1} - 529 \beta_{2} + \beta_{3}) q^{4}\) \(+(35992322464951803524 + 174673117149 \beta_{1} + 157945 \beta_{2} + 12 \beta_{3} + \beta_{4}) q^{5}\) \(+(\)\(62\!\cdots\!16\)\( - 7082751985044 \beta_{1} - 52897176 \beta_{2} + 188064 \beta_{3} - 120 \beta_{4}) q^{6}\) \(+(\)\(29\!\cdots\!62\)\( - 1403044288237600 \beta_{1} + 37036880006 \beta_{2} - 1472240 \beta_{3} + 7020 \beta_{4}) q^{7}\) \(+(-\)\(69\!\cdots\!00\)\( - 363735709336321072 \beta_{1} + 4052591058968 \beta_{2} - 360632280 \beta_{3} - 266560 \beta_{4}) q^{8}\) \(+(\)\(65\!\cdots\!09\)\( - 2757699652749137442 \beta_{1} + 50001513470982 \beta_{2} + 15750146952 \beta_{3} + 7379190 \beta_{4}) q^{9}\) \(+(-\)\(16\!\cdots\!06\)\( - 43242686857298718206 \beta_{1} - 1893316762650080 \beta_{2} - 323627897728 \beta_{3} - 158562144 \beta_{4}) q^{10}\) \(+(\)\(85\!\cdots\!87\)\( - \)\(37\!\cdots\!30\)\( \beta_{1} - 6563700232248445 \beta_{2} + 3717411735840 \beta_{3} + 2748847640 \beta_{4}) q^{11}\) \(+(-\)\(88\!\cdots\!28\)\( - \)\(13\!\cdots\!52\)\( \beta_{1} + 415207325503388164 \beta_{2} - 17027352061380 \beta_{3} - 39443189760 \beta_{4}) q^{12}\) \(+(-\)\(16\!\cdots\!96\)\( - \)\(58\!\cdots\!47\)\( \beta_{1} - 2297698997778049975 \beta_{2} - 198523391408180 \beta_{3} + 476711357265 \beta_{4}) q^{13}\) \(+(\)\(12\!\cdots\!04\)\( + \)\(91\!\cdots\!92\)\( \beta_{1} - 21464620088514798832 \beta_{2} + 4908054819167808 \beta_{3} - 4910778324400 \beta_{4}) q^{14}\) \(+(-\)\(81\!\cdots\!02\)\( + \)\(33\!\cdots\!48\)\( \beta_{1} + \)\(30\!\cdots\!90\)\( \beta_{2} - 53849714596780176 \beta_{3} + 43440479153652 \beta_{4}) q^{15}\) \(+(\)\(13\!\cdots\!24\)\( + \)\(26\!\cdots\!52\)\( \beta_{1} - \)\(84\!\cdots\!92\)\( \beta_{2} + 381700226166408768 \beta_{3} - 331129448133120 \beta_{4}) q^{16}\) \(+(-\)\(68\!\cdots\!18\)\( - \)\(21\!\cdots\!38\)\( \beta_{1} - \)\(60\!\cdots\!62\)\( \beta_{2} - 1778018042496512760 \beta_{3} + 2173189785854230 \beta_{4}) q^{17}\) \(+(\)\(19\!\cdots\!59\)\( - \)\(15\!\cdots\!81\)\( \beta_{1} + \)\(45\!\cdots\!36\)\( \beta_{2} + 4281238792208136960 \beta_{3} - 12199334429782080 \beta_{4}) q^{18}\) \(+(\)\(12\!\cdots\!69\)\( + \)\(87\!\cdots\!46\)\( \beta_{1} + \)\(32\!\cdots\!09\)\( \beta_{2} + 6757617129034581984 \beta_{3} + 57636170473930920 \beta_{4}) q^{19}\) \(+(\)\(33\!\cdots\!02\)\( + \)\(23\!\cdots\!52\)\( \beta_{1} - \)\(10\!\cdots\!90\)\( \beta_{2} - 95991735237875522274 \beta_{3} - 220943909849546752 \beta_{4}) q^{20}\) \(+(\)\(68\!\cdots\!20\)\( + \)\(34\!\cdots\!52\)\( \beta_{1} + \)\(32\!\cdots\!08\)\( \beta_{2} + \)\(20\!\cdots\!48\)\( \beta_{3} + 623388743422483500 \beta_{4}) q^{21}\) \(+(\)\(33\!\cdots\!84\)\( - \)\(30\!\cdots\!92\)\( \beta_{1} + \)\(73\!\cdots\!20\)\( \beta_{2} + \)\(17\!\cdots\!20\)\( \beta_{3} - 812126951793317160 \beta_{4}) q^{22}\) \(+(\)\(53\!\cdots\!14\)\( - \)\(83\!\cdots\!68\)\( \beta_{1} - \)\(54\!\cdots\!26\)\( \beta_{2} - \)\(17\!\cdots\!00\)\( \beta_{3} - 3523640114226532300 \beta_{4}) q^{23}\) \(+(\)\(86\!\cdots\!92\)\( + \)\(27\!\cdots\!88\)\( \beta_{1} - \)\(34\!\cdots\!48\)\( \beta_{2} + \)\(83\!\cdots\!92\)\( \beta_{3} + 30320518707908670720 \beta_{4}) q^{24}\) \(+(\)\(12\!\cdots\!35\)\( + \)\(38\!\cdots\!60\)\( \beta_{1} + \)\(92\!\cdots\!00\)\( \beta_{2} - \)\(22\!\cdots\!20\)\( \beta_{3} - \)\(12\!\cdots\!60\)\( \beta_{4}) q^{25}\) \(+(\)\(54\!\cdots\!94\)\( + \)\(26\!\cdots\!58\)\( \beta_{1} - \)\(20\!\cdots\!68\)\( \beta_{2} + \)\(27\!\cdots\!52\)\( \beta_{3} + \)\(27\!\cdots\!40\)\( \beta_{4}) q^{26}\) \(+(\)\(84\!\cdots\!50\)\( - \)\(10\!\cdots\!08\)\( \beta_{1} - \)\(26\!\cdots\!98\)\( \beta_{2} + \)\(53\!\cdots\!80\)\( \beta_{3} - 64684202649816639240 \beta_{4}) q^{27}\) \(+(-\)\(11\!\cdots\!44\)\( - \)\(35\!\cdots\!32\)\( \beta_{1} + \)\(90\!\cdots\!16\)\( \beta_{2} - \)\(27\!\cdots\!20\)\( \beta_{3} - \)\(24\!\cdots\!40\)\( \beta_{4}) q^{28}\) \(+(-\)\(35\!\cdots\!44\)\( + \)\(46\!\cdots\!89\)\( \beta_{1} + \)\(57\!\cdots\!81\)\( \beta_{2} + \)\(30\!\cdots\!96\)\( \beta_{3} + \)\(11\!\cdots\!65\)\( \beta_{4}) q^{29}\) \(+(-\)\(30\!\cdots\!12\)\( + \)\(42\!\cdots\!88\)\( \beta_{1} - \)\(29\!\cdots\!60\)\( \beta_{2} + \)\(57\!\cdots\!44\)\( \beta_{3} - \)\(24\!\cdots\!88\)\( \beta_{4}) q^{30}\) \(+(-\)\(71\!\cdots\!68\)\( - \)\(33\!\cdots\!40\)\( \beta_{1} + \)\(39\!\cdots\!40\)\( \beta_{2} + \)\(42\!\cdots\!20\)\( \beta_{3} + \)\(43\!\cdots\!20\)\( \beta_{4}) q^{31}\) \(+(-\)\(21\!\cdots\!88\)\( - \)\(16\!\cdots\!56\)\( \beta_{1} + \)\(13\!\cdots\!40\)\( \beta_{2} - \)\(17\!\cdots\!60\)\( \beta_{3} + \)\(17\!\cdots\!80\)\( \beta_{4}) q^{32}\) \(+(\)\(15\!\cdots\!48\)\( - \)\(41\!\cdots\!62\)\( \beta_{1} + \)\(18\!\cdots\!82\)\( \beta_{2} + \)\(63\!\cdots\!40\)\( \beta_{3} - \)\(65\!\cdots\!70\)\( \beta_{4}) q^{33}\) \(+(\)\(26\!\cdots\!78\)\( + \)\(16\!\cdots\!18\)\( \beta_{1} - \)\(11\!\cdots\!28\)\( \beta_{2} - \)\(49\!\cdots\!88\)\( \beta_{3} + \)\(10\!\cdots\!20\)\( \beta_{4}) q^{34}\) \(+(\)\(17\!\cdots\!44\)\( + \)\(10\!\cdots\!44\)\( \beta_{1} + \)\(18\!\cdots\!20\)\( \beta_{2} - \)\(33\!\cdots\!28\)\( \beta_{3} + \)\(10\!\cdots\!56\)\( \beta_{4}) q^{35}\) \(+(\)\(10\!\cdots\!49\)\( - \)\(23\!\cdots\!58\)\( \beta_{1} + \)\(42\!\cdots\!43\)\( \beta_{2} + \)\(12\!\cdots\!53\)\( \beta_{3} - \)\(94\!\cdots\!20\)\( \beta_{4}) q^{36}\) \(+(\)\(24\!\cdots\!72\)\( - \)\(68\!\cdots\!99\)\( \beta_{1} - \)\(11\!\cdots\!23\)\( \beta_{2} - \)\(15\!\cdots\!40\)\( \beta_{3} + \)\(22\!\cdots\!45\)\( \beta_{4}) q^{37}\) \(+(-\)\(91\!\cdots\!00\)\( - \)\(16\!\cdots\!08\)\( \beta_{1} - \)\(85\!\cdots\!88\)\( \beta_{2} - \)\(19\!\cdots\!60\)\( \beta_{3} - \)\(10\!\cdots\!20\)\( \beta_{4}) q^{38}\) \(+(-\)\(18\!\cdots\!58\)\( + \)\(44\!\cdots\!32\)\( \beta_{1} - \)\(10\!\cdots\!22\)\( \beta_{2} + \)\(93\!\cdots\!28\)\( \beta_{3} - \)\(88\!\cdots\!60\)\( \beta_{4}) q^{39}\) \(+(-\)\(12\!\cdots\!60\)\( + \)\(36\!\cdots\!40\)\( \beta_{1} + \)\(12\!\cdots\!00\)\( \beta_{2} - \)\(11\!\cdots\!80\)\( \beta_{3} + \)\(27\!\cdots\!60\)\( \beta_{4}) q^{40}\) \(+(-\)\(25\!\cdots\!78\)\( + \)\(22\!\cdots\!60\)\( \beta_{1} - \)\(20\!\cdots\!60\)\( \beta_{2} + \)\(42\!\cdots\!20\)\( \beta_{3} - \)\(28\!\cdots\!80\)\( \beta_{4}) q^{41}\) \(+(-\)\(37\!\cdots\!36\)\( - \)\(77\!\cdots\!88\)\( \beta_{1} - \)\(31\!\cdots\!36\)\( \beta_{2} - \)\(10\!\cdots\!40\)\( \beta_{3} - \)\(54\!\cdots\!80\)\( \beta_{4}) q^{42}\) \(+(\)\(27\!\cdots\!59\)\( + \)\(29\!\cdots\!10\)\( \beta_{1} - \)\(63\!\cdots\!93\)\( \beta_{2} + \)\(23\!\cdots\!00\)\( \beta_{3} + \)\(23\!\cdots\!00\)\( \beta_{4}) q^{43}\) \(+(\)\(20\!\cdots\!08\)\( - \)\(21\!\cdots\!32\)\( \beta_{1} + \)\(10\!\cdots\!72\)\( \beta_{2} + \)\(15\!\cdots\!12\)\( \beta_{3} - \)\(29\!\cdots\!80\)\( \beta_{4}) q^{44}\) \(+(\)\(20\!\cdots\!88\)\( + \)\(58\!\cdots\!13\)\( \beta_{1} + \)\(46\!\cdots\!65\)\( \beta_{2} - \)\(67\!\cdots\!56\)\( \beta_{3} - \)\(21\!\cdots\!63\)\( \beta_{4}) q^{45}\) \(+(\)\(71\!\cdots\!12\)\( + \)\(47\!\cdots\!40\)\( \beta_{1} - \)\(67\!\cdots\!40\)\( \beta_{2} + \)\(60\!\cdots\!40\)\( \beta_{3} + \)\(12\!\cdots\!20\)\( \beta_{4}) q^{46}\) \(+(\)\(11\!\cdots\!92\)\( + \)\(13\!\cdots\!28\)\( \beta_{1} + \)\(53\!\cdots\!84\)\( \beta_{2} + \)\(17\!\cdots\!00\)\( \beta_{3} - \)\(16\!\cdots\!00\)\( \beta_{4}) q^{47}\) \(+(-\)\(28\!\cdots\!36\)\( - \)\(59\!\cdots\!08\)\( \beta_{1} + \)\(47\!\cdots\!24\)\( \beta_{2} - \)\(43\!\cdots\!60\)\( \beta_{3} - \)\(48\!\cdots\!20\)\( \beta_{4}) q^{48}\) \(+(-\)\(34\!\cdots\!87\)\( + \)\(25\!\cdots\!00\)\( \beta_{1} + \)\(23\!\cdots\!00\)\( \beta_{2} - \)\(12\!\cdots\!20\)\( \beta_{3} + \)\(32\!\cdots\!20\)\( \beta_{4}) q^{49}\) \(+(-\)\(46\!\cdots\!15\)\( + \)\(17\!\cdots\!85\)\( \beta_{1} - \)\(66\!\cdots\!00\)\( \beta_{2} + \)\(13\!\cdots\!80\)\( \beta_{3} - \)\(18\!\cdots\!60\)\( \beta_{4}) q^{50}\) \(+(-\)\(93\!\cdots\!62\)\( + \)\(32\!\cdots\!24\)\( \beta_{1} - \)\(43\!\cdots\!54\)\( \beta_{2} - \)\(25\!\cdots\!04\)\( \beta_{3} - \)\(79\!\cdots\!20\)\( \beta_{4}) q^{51}\) \(+(-\)\(19\!\cdots\!98\)\( - \)\(39\!\cdots\!80\)\( \beta_{1} + \)\(20\!\cdots\!86\)\( \beta_{2} - \)\(49\!\cdots\!50\)\( \beta_{3} - \)\(15\!\cdots\!00\)\( \beta_{4}) q^{52}\) \(+(\)\(45\!\cdots\!44\)\( + \)\(11\!\cdots\!09\)\( \beta_{1} - \)\(39\!\cdots\!59\)\( \beta_{2} + \)\(64\!\cdots\!20\)\( \beta_{3} + \)\(41\!\cdots\!65\)\( \beta_{4}) q^{53}\) \(+(\)\(85\!\cdots\!44\)\( - \)\(11\!\cdots\!04\)\( \beta_{1} + \)\(17\!\cdots\!84\)\( \beta_{2} + \)\(60\!\cdots\!84\)\( \beta_{3} + \)\(14\!\cdots\!20\)\( \beta_{4}) q^{54}\) \(+(\)\(18\!\cdots\!58\)\( + \)\(59\!\cdots\!08\)\( \beta_{1} - \)\(12\!\cdots\!10\)\( \beta_{2} - \)\(23\!\cdots\!96\)\( \beta_{3} - \)\(21\!\cdots\!08\)\( \beta_{4}) q^{55}\) \(+(\)\(26\!\cdots\!56\)\( + \)\(23\!\cdots\!64\)\( \beta_{1} + \)\(42\!\cdots\!56\)\( \beta_{2} + \)\(27\!\cdots\!96\)\( \beta_{3} + \)\(29\!\cdots\!40\)\( \beta_{4}) q^{56}\) \(+(-\)\(29\!\cdots\!00\)\( + \)\(49\!\cdots\!74\)\( \beta_{1} + \)\(28\!\cdots\!34\)\( \beta_{2} - \)\(28\!\cdots\!00\)\( \beta_{3} + \)\(19\!\cdots\!50\)\( \beta_{4}) q^{57}\) \(+(-\)\(39\!\cdots\!50\)\( + \)\(22\!\cdots\!18\)\( \beta_{1} - \)\(89\!\cdots\!72\)\( \beta_{2} - \)\(39\!\cdots\!60\)\( \beta_{3} - \)\(93\!\cdots\!20\)\( \beta_{4}) q^{58}\) \(+(-\)\(43\!\cdots\!93\)\( - \)\(11\!\cdots\!22\)\( \beta_{1} - \)\(31\!\cdots\!13\)\( \beta_{2} + \)\(19\!\cdots\!72\)\( \beta_{3} + \)\(43\!\cdots\!00\)\( \beta_{4}) q^{59}\) \(+(-\)\(31\!\cdots\!96\)\( + \)\(42\!\cdots\!04\)\( \beta_{1} - \)\(74\!\cdots\!80\)\( \beta_{2} - \)\(42\!\cdots\!48\)\( \beta_{3} + \)\(11\!\cdots\!96\)\( \beta_{4}) q^{60}\) \(+(-\)\(11\!\cdots\!08\)\( - \)\(35\!\cdots\!75\)\( \beta_{1} + \)\(13\!\cdots\!25\)\( \beta_{2} + \)\(65\!\cdots\!00\)\( \beta_{3} + \)\(26\!\cdots\!25\)\( \beta_{4}) q^{61}\) \(+(\)\(36\!\cdots\!44\)\( + \)\(73\!\cdots\!28\)\( \beta_{1} - \)\(15\!\cdots\!40\)\( \beta_{2} + \)\(77\!\cdots\!60\)\( \beta_{3} - \)\(49\!\cdots\!80\)\( \beta_{4}) q^{62}\) \(+(\)\(36\!\cdots\!54\)\( - \)\(26\!\cdots\!84\)\( \beta_{1} + \)\(43\!\cdots\!78\)\( \beta_{2} - \)\(48\!\cdots\!60\)\( \beta_{3} - \)\(25\!\cdots\!20\)\( \beta_{4}) q^{63}\) \(+(\)\(88\!\cdots\!20\)\( + \)\(16\!\cdots\!08\)\( \beta_{1} - \)\(66\!\cdots\!68\)\( \beta_{2} - \)\(48\!\cdots\!48\)\( \beta_{3} + \)\(22\!\cdots\!40\)\( \beta_{4}) q^{64}\) \(+(\)\(21\!\cdots\!88\)\( - \)\(14\!\cdots\!12\)\( \beta_{1} - \)\(16\!\cdots\!60\)\( \beta_{2} - \)\(22\!\cdots\!56\)\( \beta_{3} - \)\(25\!\cdots\!88\)\( \beta_{4}) q^{65}\) \(+(\)\(37\!\cdots\!12\)\( - \)\(50\!\cdots\!08\)\( \beta_{1} + \)\(36\!\cdots\!68\)\( \beta_{2} + \)\(61\!\cdots\!68\)\( \beta_{3} - \)\(34\!\cdots\!60\)\( \beta_{4}) q^{66}\) \(+(\)\(50\!\cdots\!57\)\( + \)\(20\!\cdots\!22\)\( \beta_{1} - \)\(11\!\cdots\!27\)\( \beta_{2} + \)\(25\!\cdots\!40\)\( \beta_{3} + \)\(81\!\cdots\!80\)\( \beta_{4}) q^{67}\) \(+(-\)\(11\!\cdots\!34\)\( + \)\(34\!\cdots\!04\)\( \beta_{1} + \)\(50\!\cdots\!22\)\( \beta_{2} - \)\(20\!\cdots\!70\)\( \beta_{3} + \)\(15\!\cdots\!60\)\( \beta_{4}) q^{68}\) \(+(-\)\(23\!\cdots\!12\)\( + \)\(13\!\cdots\!16\)\( \beta_{1} + \)\(14\!\cdots\!64\)\( \beta_{2} + \)\(14\!\cdots\!04\)\( \beta_{3} - \)\(11\!\cdots\!20\)\( \beta_{4}) q^{69}\) \(+(-\)\(11\!\cdots\!36\)\( + \)\(37\!\cdots\!64\)\( \beta_{1} - \)\(15\!\cdots\!80\)\( \beta_{2} + \)\(16\!\cdots\!32\)\( \beta_{3} - \)\(12\!\cdots\!64\)\( \beta_{4}) q^{70}\) \(+(-\)\(12\!\cdots\!18\)\( + \)\(12\!\cdots\!00\)\( \beta_{1} + \)\(30\!\cdots\!50\)\( \beta_{2} - \)\(16\!\cdots\!00\)\( \beta_{3} + \)\(38\!\cdots\!00\)\( \beta_{4}) q^{71}\) \(+(\)\(10\!\cdots\!00\)\( - \)\(86\!\cdots\!24\)\( \beta_{1} + \)\(42\!\cdots\!36\)\( \beta_{2} - \)\(63\!\cdots\!80\)\( \beta_{3} + \)\(36\!\cdots\!40\)\( \beta_{4}) q^{72}\) \(+(\)\(24\!\cdots\!14\)\( - \)\(25\!\cdots\!98\)\( \beta_{1} - \)\(14\!\cdots\!66\)\( \beta_{2} - \)\(21\!\cdots\!60\)\( \beta_{3} - \)\(15\!\cdots\!70\)\( \beta_{4}) q^{73}\) \(+(\)\(60\!\cdots\!66\)\( + \)\(65\!\cdots\!50\)\( \beta_{1} - \)\(11\!\cdots\!00\)\( \beta_{2} + \)\(49\!\cdots\!20\)\( \beta_{3} + \)\(34\!\cdots\!80\)\( \beta_{4}) q^{74}\) \(+(\)\(14\!\cdots\!95\)\( + \)\(12\!\cdots\!70\)\( \beta_{1} + \)\(99\!\cdots\!75\)\( \beta_{2} - \)\(29\!\cdots\!40\)\( \beta_{3} + \)\(26\!\cdots\!80\)\( \beta_{4}) q^{75}\) \(+(\)\(88\!\cdots\!32\)\( + \)\(14\!\cdots\!08\)\( \beta_{1} - \)\(65\!\cdots\!68\)\( \beta_{2} + \)\(85\!\cdots\!12\)\( \beta_{3} - \)\(15\!\cdots\!20\)\( \beta_{4}) q^{76}\) \(+(\)\(34\!\cdots\!04\)\( - \)\(72\!\cdots\!00\)\( \beta_{1} - \)\(37\!\cdots\!28\)\( \beta_{2} + \)\(37\!\cdots\!40\)\( \beta_{3} - \)\(33\!\cdots\!20\)\( \beta_{4}) q^{77}\) \(+(-\)\(40\!\cdots\!12\)\( - \)\(54\!\cdots\!80\)\( \beta_{1} + \)\(57\!\cdots\!04\)\( \beta_{2} - \)\(60\!\cdots\!20\)\( \beta_{3} - \)\(58\!\cdots\!40\)\( \beta_{4}) q^{78}\) \(+(-\)\(39\!\cdots\!44\)\( - \)\(23\!\cdots\!56\)\( \beta_{1} - \)\(61\!\cdots\!24\)\( \beta_{2} + \)\(26\!\cdots\!36\)\( \beta_{3} + \)\(12\!\cdots\!20\)\( \beta_{4}) q^{79}\) \(+(-\)\(41\!\cdots\!36\)\( + \)\(69\!\cdots\!64\)\( \beta_{1} - \)\(43\!\cdots\!80\)\( \beta_{2} + \)\(14\!\cdots\!32\)\( \beta_{3} - \)\(27\!\cdots\!64\)\( \beta_{4}) q^{80}\) \(+(-\)\(54\!\cdots\!15\)\( - \)\(92\!\cdots\!54\)\( \beta_{1} + \)\(12\!\cdots\!34\)\( \beta_{2} - \)\(35\!\cdots\!96\)\( \beta_{3} - \)\(29\!\cdots\!50\)\( \beta_{4}) q^{81}\) \(+(-\)\(18\!\cdots\!26\)\( + \)\(20\!\cdots\!38\)\( \beta_{1} + \)\(89\!\cdots\!60\)\( \beta_{2} - \)\(42\!\cdots\!40\)\( \beta_{3} + \)\(26\!\cdots\!20\)\( \beta_{4}) q^{82}\) \(+(\)\(26\!\cdots\!49\)\( + \)\(15\!\cdots\!46\)\( \beta_{1} + \)\(10\!\cdots\!53\)\( \beta_{2} + \)\(16\!\cdots\!00\)\( \beta_{3} + \)\(20\!\cdots\!00\)\( \beta_{4}) q^{83}\) \(+(\)\(35\!\cdots\!72\)\( + \)\(24\!\cdots\!64\)\( \beta_{1} - \)\(16\!\cdots\!44\)\( \beta_{2} + \)\(73\!\cdots\!96\)\( \beta_{3} - \)\(21\!\cdots\!60\)\( \beta_{4}) q^{84}\) \(+(\)\(49\!\cdots\!44\)\( - \)\(61\!\cdots\!06\)\( \beta_{1} - \)\(74\!\cdots\!30\)\( \beta_{2} - \)\(23\!\cdots\!28\)\( \beta_{3} - \)\(25\!\cdots\!94\)\( \beta_{4}) q^{85}\) \(+(-\)\(29\!\cdots\!72\)\( - \)\(41\!\cdots\!96\)\( \beta_{1} - \)\(36\!\cdots\!84\)\( \beta_{2} - \)\(80\!\cdots\!64\)\( \beta_{3} - \)\(31\!\cdots\!40\)\( \beta_{4}) q^{86}\) \(+(\)\(54\!\cdots\!50\)\( + \)\(33\!\cdots\!96\)\( \beta_{1} + \)\(12\!\cdots\!46\)\( \beta_{2} - \)\(96\!\cdots\!40\)\( \beta_{3} + \)\(15\!\cdots\!20\)\( \beta_{4}) q^{87}\) \(+(-\)\(15\!\cdots\!00\)\( - \)\(10\!\cdots\!24\)\( \beta_{1} + \)\(67\!\cdots\!56\)\( \beta_{2} + \)\(25\!\cdots\!40\)\( \beta_{3} - \)\(94\!\cdots\!20\)\( \beta_{4}) q^{88}\) \(+(\)\(82\!\cdots\!98\)\( + \)\(24\!\cdots\!62\)\( \beta_{1} + \)\(11\!\cdots\!98\)\( \beta_{2} + \)\(84\!\cdots\!68\)\( \beta_{3} - \)\(92\!\cdots\!30\)\( \beta_{4}) q^{89}\) \(+(-\)\(55\!\cdots\!22\)\( + \)\(19\!\cdots\!78\)\( \beta_{1} - \)\(19\!\cdots\!60\)\( \beta_{2} - \)\(35\!\cdots\!36\)\( \beta_{3} + \)\(19\!\cdots\!72\)\( \beta_{4}) q^{90}\) \(+(\)\(40\!\cdots\!88\)\( + \)\(11\!\cdots\!44\)\( \beta_{1} - \)\(18\!\cdots\!24\)\( \beta_{2} + \)\(24\!\cdots\!16\)\( \beta_{3} - \)\(45\!\cdots\!60\)\( \beta_{4}) q^{91}\) \(+(-\)\(80\!\cdots\!68\)\( - \)\(65\!\cdots\!88\)\( \beta_{1} + \)\(33\!\cdots\!08\)\( \beta_{2} - \)\(57\!\cdots\!40\)\( \beta_{3} + \)\(64\!\cdots\!20\)\( \beta_{4}) q^{92}\) \(+(\)\(71\!\cdots\!68\)\( + \)\(40\!\cdots\!08\)\( \beta_{1} - \)\(56\!\cdots\!08\)\( \beta_{2} + \)\(71\!\cdots\!20\)\( \beta_{3} + \)\(21\!\cdots\!40\)\( \beta_{4}) q^{93}\) \(+(-\)\(22\!\cdots\!20\)\( - \)\(21\!\cdots\!44\)\( \beta_{1} + \)\(10\!\cdots\!24\)\( \beta_{2} + \)\(41\!\cdots\!24\)\( \beta_{3} - \)\(10\!\cdots\!80\)\( \beta_{4}) q^{94}\) \(+(\)\(17\!\cdots\!10\)\( + \)\(50\!\cdots\!60\)\( \beta_{1} + \)\(43\!\cdots\!50\)\( \beta_{2} - \)\(14\!\cdots\!20\)\( \beta_{3} + \)\(20\!\cdots\!40\)\( \beta_{4}) q^{95}\) \(+(\)\(70\!\cdots\!76\)\( + \)\(38\!\cdots\!16\)\( \beta_{1} - \)\(17\!\cdots\!36\)\( \beta_{2} + \)\(26\!\cdots\!44\)\( \beta_{3} - \)\(81\!\cdots\!60\)\( \beta_{4}) q^{96}\) \(+(\)\(23\!\cdots\!42\)\( - \)\(85\!\cdots\!62\)\( \beta_{1} + \)\(27\!\cdots\!74\)\( \beta_{2} - \)\(41\!\cdots\!20\)\( \beta_{3} + \)\(53\!\cdots\!10\)\( \beta_{4}) q^{97}\) \(+(-\)\(20\!\cdots\!89\)\( + \)\(43\!\cdots\!87\)\( \beta_{1} - \)\(10\!\cdots\!60\)\( \beta_{2} - \)\(10\!\cdots\!40\)\( \beta_{3} - \)\(28\!\cdots\!80\)\( \beta_{4}) q^{98}\) \(+(\)\(46\!\cdots\!43\)\( - \)\(57\!\cdots\!74\)\( \beta_{1} + \)\(19\!\cdots\!79\)\( \beta_{2} + \)\(46\!\cdots\!64\)\( \beta_{3} - \)\(10\!\cdots\!40\)\( \beta_{4}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(5q \) \(\mathstrut -\mathstrut 449691864q^{2} \) \(\mathstrut +\mathstrut 84016631749932q^{3} \) \(\mathstrut +\mathstrut 1738819379139544640q^{4} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!90\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!60\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!56\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!80\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!85\)\(q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(5q \) \(\mathstrut -\mathstrut 449691864q^{2} \) \(\mathstrut +\mathstrut 84016631749932q^{3} \) \(\mathstrut +\mathstrut 1738819379139544640q^{4} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!90\)\(q^{5} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!60\)\(q^{6} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!56\)\(q^{7} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!80\)\(q^{8} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!85\)\(q^{9} \) \(\mathstrut -\mathstrut \)\(81\!\cdots\!60\)\(q^{10} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!60\)\(q^{11} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!44\)\(q^{12} \) \(\mathstrut -\mathstrut \)\(84\!\cdots\!78\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(62\!\cdots\!80\)\(q^{14} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!20\)\(q^{15} \) \(\mathstrut +\mathstrut \)\(69\!\cdots\!80\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!54\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(96\!\cdots\!52\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!00\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!20\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!60\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!12\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!12\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!00\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(62\!\cdots\!75\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!60\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!80\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(56\!\cdots\!52\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!50\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!20\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!40\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!84\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(75\!\cdots\!44\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!80\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(87\!\cdots\!40\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!80\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!26\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!20\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(92\!\cdots\!80\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!00\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!90\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!88\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!92\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!80\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!30\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!60\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(58\!\cdots\!16\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!88\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!35\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!00\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!40\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(99\!\cdots\!24\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!82\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!00\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(90\!\cdots\!80\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!40\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!80\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!00\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!60\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!90\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!92\)\(q^{62} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!92\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!40\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!80\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!20\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!96\)\(q^{67} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!32\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!80\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(57\!\cdots\!60\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(63\!\cdots\!40\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!40\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!62\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!80\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!00\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!00\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!52\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!56\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!00\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!60\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!95\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(90\!\cdots\!68\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!52\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!80\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!40\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!40\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!40\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(76\!\cdots\!60\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!50\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!20\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!60\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!04\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!04\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!20\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(87\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!60\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!66\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!52\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!20\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5}\mathstrut -\mathstrut \) \(x^{4}\mathstrut -\mathstrut \) \(3976283494919360\) \(x^{3}\mathstrut +\mathstrut \) \(9065173660301515822976\) \(x^{2}\mathstrut +\mathstrut \) \(2677795447191606098169599438848\) \(x\mathstrut -\mathstrut \) \(23358185771581696169459363194340724736\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 24 \nu - 5 \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(7\) \(\nu^{4}\mathstrut -\mathstrut \) \(467245857\) \(\nu^{3}\mathstrut +\mathstrut \) \(26436728091610280\) \(\nu^{2}\mathstrut +\mathstrut \) \(1263143870442683559905088\) \(\nu\mathstrut -\mathstrut \) \(15314697613007994975674545652224\)\()/\)\(114888935397261312\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(3703\) \(\nu^{4}\mathstrut -\mathstrut \) \(247173058353\) \(\nu^{3}\mathstrut +\mathstrut \) \(80161055949284353832\) \(\nu^{2}\mathstrut +\mathstrut \) \(894505988695541738149247808\) \(\nu\mathstrut -\mathstrut \) \(113355332314436384988715808904765952\)\()/\)\(114888935397261312\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(507066253\) \(\nu^{4}\mathstrut -\mathstrut \) \(4055243770593579\) \(\nu^{3}\mathstrut +\mathstrut \) \(1802293574965541164562744\) \(\nu^{2}\mathstrut +\mathstrut \) \(12781327433864906041799038615488\) \(\nu\mathstrut -\mathstrut \) \(768029313405494928746068981312336287232\)\()/\)\(574444676986306560\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(5\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(529\) \(\beta_{2}\mathstrut -\mathstrut \) \(82073062\) \(\beta_{1}\mathstrut +\mathstrut \) \(916135717213006153\)\()/576\)
\(\nu^{3}\)\(=\)\((\)\(33320\) \(\beta_{4}\mathstrut +\mathstrut \) \(11352147\) \(\beta_{3}\mathstrut -\mathstrut \) \(488732358619\) \(\beta_{2}\mathstrut +\mathstrut \) \(189316879448348278\) \(\beta_{1}\mathstrut -\mathstrut \) \(9398767890526410432173157\)\()/1728\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(741363426440\) \(\beta_{4}\mathstrut +\mathstrut \) \(3524092410525041\) \(\beta_{3}\mathstrut -\mathstrut \) \(577380861966050953\) \(\beta_{2}\mathstrut -\mathstrut \) \(191447551512170411726190\) \(\beta_{1}\mathstrut +\mathstrut \) \(2408887185244043179434764083228937\)\()/576\)

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
5.54999e7
2.35709e7
9.74166e6
−3.26158e7
−5.61966e7
−1.42193e9 −1.64208e14 1.44544e18 2.46018e20 2.33493e23 −6.31094e24 −1.23563e27 1.28340e28 −3.49822e29
1.2 −6.55640e8 1.58783e14 −1.46597e17 7.23713e20 −1.04104e23 1.08717e25 4.74066e26 1.10817e28 −4.74495e29
1.3 −3.23738e8 −8.24972e12 −4.71654e17 −7.71676e20 2.67075e21 −4.97804e24 3.39315e26 −1.40623e28 2.49821e29
1.4 6.92842e8 −1.03284e14 −9.64308e16 3.87033e20 −7.15597e22 −7.47157e23 −4.66207e26 −3.46273e27 2.68153e29
1.5 1.25878e9 2.00976e14 1.00806e18 −4.05126e20 2.52984e23 2.66317e24 5.43293e26 2.62609e28 −5.09964e29
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{60}^{\mathrm{new}}(\Gamma_0(1))\).