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.

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

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

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

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 \(S_{60}^{\mathrm{new}}(\Gamma_0(1))\).