Properties

Label 1.78.a.a
Level 1
Weight 78
Character orbit 1.a
Self dual Yes
Analytic conductor 37.548
Analytic rank 1
Dimension 6
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 78 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(37.5479417817\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{64}\cdot 3^{20}\cdot 5^{8}\cdot 7^{3}\cdot 11^{2}\cdot 13^{2}\cdot 19 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +(44120315520 - \beta_{1}) q^{2} +(240268562631348180 + 328682 \beta_{1} - \beta_{2}) q^{3} +(\)\(69\!\cdots\!12\)\( - 376906182 \beta_{1} - 13307 \beta_{2} + \beta_{3}) q^{4} +(-\)\(43\!\cdots\!50\)\( + 712041485938330 \beta_{1} + 16345778 \beta_{2} - 850 \beta_{3} - \beta_{4}) q^{5} +(-\)\(61\!\cdots\!48\)\( - 533724841483345995 \beta_{1} - 43249157770 \beta_{2} + 41034 \beta_{3} + 116 \beta_{4} - \beta_{5}) q^{6} +(\)\(45\!\cdots\!00\)\( + \)\(23\!\cdots\!08\)\( \beta_{1} + 25717152674278 \beta_{2} + 87615016 \beta_{3} + 81756 \beta_{4} + 648 \beta_{5}) q^{7} +(-\)\(35\!\cdots\!40\)\( - \)\(82\!\cdots\!68\)\( \beta_{1} - 3477188673986912 \beta_{2} + 24525571168 \beta_{3} + 74235328 \beta_{4} - 66096 \beta_{5}) q^{8} +(-\)\(81\!\cdots\!07\)\( + \)\(27\!\cdots\!80\)\( \beta_{1} + 331144208395396860 \beta_{2} - 14388605026812 \beta_{3} + 1712740242 \beta_{4} + 3311328 \beta_{5}) q^{9} +O(q^{10})\) \( q +(44120315520 - \beta_{1}) q^{2} +(240268562631348180 + 328682 \beta_{1} - \beta_{2}) q^{3} +(\)\(69\!\cdots\!12\)\( - 376906182 \beta_{1} - 13307 \beta_{2} + \beta_{3}) q^{4} +(-\)\(43\!\cdots\!50\)\( + 712041485938330 \beta_{1} + 16345778 \beta_{2} - 850 \beta_{3} - \beta_{4}) q^{5} +(-\)\(61\!\cdots\!48\)\( - 533724841483345995 \beta_{1} - 43249157770 \beta_{2} + 41034 \beta_{3} + 116 \beta_{4} - \beta_{5}) q^{6} +(\)\(45\!\cdots\!00\)\( + \)\(23\!\cdots\!08\)\( \beta_{1} + 25717152674278 \beta_{2} + 87615016 \beta_{3} + 81756 \beta_{4} + 648 \beta_{5}) q^{7} +(-\)\(35\!\cdots\!40\)\( - \)\(82\!\cdots\!68\)\( \beta_{1} - 3477188673986912 \beta_{2} + 24525571168 \beta_{3} + 74235328 \beta_{4} - 66096 \beta_{5}) q^{8} +(-\)\(81\!\cdots\!07\)\( + \)\(27\!\cdots\!80\)\( \beta_{1} + 331144208395396860 \beta_{2} - 14388605026812 \beta_{3} + 1712740242 \beta_{4} + 3311328 \beta_{5}) q^{9} +(-\)\(15\!\cdots\!00\)\( + \)\(15\!\cdots\!10\)\( \beta_{1} + 30995076281577436616 \beta_{2} - 2234992092877000 \beta_{3} - 279959488272 \beta_{4} - 99297900 \beta_{5}) q^{10} +(-\)\(30\!\cdots\!28\)\( - \)\(52\!\cdots\!54\)\( \beta_{1} + \)\(13\!\cdots\!01\)\( \beta_{2} - 41184440803446896 \beta_{3} + 8205331093528 \beta_{4} + 1750529232 \beta_{5}) q^{11} +(\)\(77\!\cdots\!40\)\( + \)\(16\!\cdots\!12\)\( \beta_{1} + \)\(48\!\cdots\!48\)\( \beta_{2} + 1348666694002532052 \beta_{3} - 58990685419008 \beta_{4} - 8013423744 \beta_{5}) q^{12} +(\)\(39\!\cdots\!10\)\( + \)\(50\!\cdots\!94\)\( \beta_{1} + \)\(26\!\cdots\!46\)\( \beta_{2} + 15576878904383732158 \beta_{3} - 2327267424252537 \beta_{4} - 567373968576 \beta_{5}) q^{13} +(-\)\(49\!\cdots\!76\)\( - \)\(62\!\cdots\!26\)\( \beta_{1} + \)\(61\!\cdots\!64\)\( \beta_{2} - \)\(52\!\cdots\!52\)\( \beta_{3} + 80734291965230440 \beta_{4} + 22044177153630 \beta_{5}) q^{14} +(-\)\(34\!\cdots\!00\)\( + \)\(76\!\cdots\!80\)\( \beta_{1} + \)\(21\!\cdots\!98\)\( \beta_{2} + \)\(59\!\cdots\!00\)\( \beta_{3} - 1296965416112245916 \beta_{4} - 496842620136200 \beta_{5}) q^{15} +(\)\(74\!\cdots\!96\)\( + \)\(22\!\cdots\!08\)\( \beta_{1} - \)\(66\!\cdots\!12\)\( \beta_{2} + \)\(85\!\cdots\!60\)\( \beta_{3} + 12155424611832446976 \beta_{4} + 8385394073186304 \beta_{5}) q^{16} +(-\)\(59\!\cdots\!90\)\( + \)\(13\!\cdots\!80\)\( \beta_{1} - \)\(53\!\cdots\!84\)\( \beta_{2} - \)\(88\!\cdots\!36\)\( \beta_{3} - 48926160579634451326 \beta_{4} - 114323688049864608 \beta_{5}) q^{17} +(-\)\(96\!\cdots\!20\)\( + \)\(32\!\cdots\!83\)\( \beta_{1} + \)\(32\!\cdots\!52\)\( \beta_{2} - \)\(17\!\cdots\!36\)\( \beta_{3} - \)\(40\!\cdots\!76\)\( \beta_{4} + 1303429551045135192 \beta_{5}) q^{18} +(\)\(93\!\cdots\!60\)\( + \)\(18\!\cdots\!54\)\( \beta_{1} + \)\(32\!\cdots\!39\)\( \beta_{2} + \)\(63\!\cdots\!52\)\( \beta_{3} + \)\(91\!\cdots\!96\)\( \beta_{4} - 12661980422829440976 \beta_{5}) q^{19} +(-\)\(33\!\cdots\!00\)\( + \)\(41\!\cdots\!60\)\( \beta_{1} + \)\(35\!\cdots\!86\)\( \beta_{2} - \)\(43\!\cdots\!50\)\( \beta_{3} - \)\(84\!\cdots\!12\)\( \beta_{4} + \)\(10\!\cdots\!00\)\( \beta_{5}) q^{20} +(-\)\(90\!\cdots\!28\)\( + \)\(13\!\cdots\!76\)\( \beta_{1} - \)\(24\!\cdots\!04\)\( \beta_{2} - \)\(74\!\cdots\!88\)\( \beta_{3} + \)\(46\!\cdots\!80\)\( \beta_{4} - \)\(76\!\cdots\!20\)\( \beta_{5}) q^{21} +(\)\(10\!\cdots\!40\)\( + \)\(10\!\cdots\!51\)\( \beta_{1} + \)\(11\!\cdots\!74\)\( \beta_{2} + \)\(15\!\cdots\!42\)\( \beta_{3} - \)\(12\!\cdots\!88\)\( \beta_{4} + \)\(48\!\cdots\!01\)\( \beta_{5}) q^{22} +(-\)\(79\!\cdots\!60\)\( - \)\(13\!\cdots\!88\)\( \beta_{1} + \)\(66\!\cdots\!66\)\( \beta_{2} - \)\(70\!\cdots\!60\)\( \beta_{3} - \)\(43\!\cdots\!60\)\( \beta_{4} - \)\(25\!\cdots\!80\)\( \beta_{5}) q^{23} +(\)\(91\!\cdots\!80\)\( - \)\(20\!\cdots\!52\)\( \beta_{1} + \)\(56\!\cdots\!68\)\( \beta_{2} - \)\(97\!\cdots\!20\)\( \beta_{3} + \)\(61\!\cdots\!56\)\( \beta_{4} + \)\(11\!\cdots\!44\)\( \beta_{5}) q^{24} +(\)\(20\!\cdots\!75\)\( - \)\(12\!\cdots\!00\)\( \beta_{1} - \)\(20\!\cdots\!00\)\( \beta_{2} + \)\(19\!\cdots\!00\)\( \beta_{3} - \)\(28\!\cdots\!00\)\( \beta_{4} - \)\(45\!\cdots\!00\)\( \beta_{5}) q^{25} +(-\)\(91\!\cdots\!28\)\( - \)\(22\!\cdots\!30\)\( \beta_{1} + \)\(77\!\cdots\!80\)\( \beta_{2} - \)\(43\!\cdots\!08\)\( \beta_{3} + \)\(27\!\cdots\!48\)\( \beta_{4} + \)\(14\!\cdots\!52\)\( \beta_{5}) q^{26} +(-\)\(30\!\cdots\!40\)\( + \)\(38\!\cdots\!12\)\( \beta_{1} + \)\(31\!\cdots\!38\)\( \beta_{2} - \)\(22\!\cdots\!12\)\( \beta_{3} + \)\(50\!\cdots\!48\)\( \beta_{4} - \)\(31\!\cdots\!36\)\( \beta_{5}) q^{27} +(\)\(46\!\cdots\!60\)\( + \)\(11\!\cdots\!56\)\( \beta_{1} + \)\(22\!\cdots\!68\)\( \beta_{2} + \)\(12\!\cdots\!12\)\( \beta_{3} - \)\(38\!\cdots\!48\)\( \beta_{4} + \)\(37\!\cdots\!36\)\( \beta_{5}) q^{28} +(\)\(27\!\cdots\!90\)\( + \)\(15\!\cdots\!22\)\( \beta_{1} - \)\(47\!\cdots\!38\)\( \beta_{2} + \)\(11\!\cdots\!42\)\( \beta_{3} + \)\(14\!\cdots\!87\)\( \beta_{4} + \)\(36\!\cdots\!88\)\( \beta_{5}) q^{29} +(-\)\(16\!\cdots\!00\)\( - \)\(34\!\cdots\!90\)\( \beta_{1} - \)\(25\!\cdots\!44\)\( \beta_{2} - \)\(24\!\cdots\!00\)\( \beta_{3} - \)\(30\!\cdots\!52\)\( \beta_{4} - \)\(12\!\cdots\!50\)\( \beta_{5}) q^{30} +(\)\(29\!\cdots\!12\)\( - \)\(44\!\cdots\!52\)\( \beta_{1} + \)\(47\!\cdots\!68\)\( \beta_{2} + \)\(58\!\cdots\!52\)\( \beta_{3} + \)\(25\!\cdots\!04\)\( \beta_{4} - \)\(87\!\cdots\!84\)\( \beta_{5}) q^{31} +(\)\(36\!\cdots\!20\)\( - \)\(10\!\cdots\!60\)\( \beta_{1} + \)\(53\!\cdots\!68\)\( \beta_{2} + \)\(11\!\cdots\!24\)\( \beta_{3} - \)\(79\!\cdots\!16\)\( \beta_{4} + \)\(55\!\cdots\!72\)\( \beta_{5}) q^{32} +(-\)\(72\!\cdots\!40\)\( - \)\(34\!\cdots\!12\)\( \beta_{1} - \)\(50\!\cdots\!00\)\( \beta_{2} - \)\(58\!\cdots\!64\)\( \beta_{3} + \)\(13\!\cdots\!86\)\( \beta_{4} - \)\(24\!\cdots\!92\)\( \beta_{5}) q^{33} +(-\)\(32\!\cdots\!96\)\( + \)\(18\!\cdots\!58\)\( \beta_{1} - \)\(76\!\cdots\!32\)\( \beta_{2} - \)\(13\!\cdots\!20\)\( \beta_{3} - \)\(79\!\cdots\!04\)\( \beta_{4} - \)\(12\!\cdots\!76\)\( \beta_{5}) q^{34} +(-\)\(25\!\cdots\!00\)\( + \)\(20\!\cdots\!60\)\( \beta_{1} + \)\(33\!\cdots\!76\)\( \beta_{2} + \)\(13\!\cdots\!00\)\( \beta_{3} + \)\(24\!\cdots\!08\)\( \beta_{4} + \)\(78\!\cdots\!00\)\( \beta_{5}) q^{35} +(-\)\(58\!\cdots\!84\)\( + \)\(33\!\cdots\!78\)\( \beta_{1} + \)\(13\!\cdots\!53\)\( \beta_{2} - \)\(25\!\cdots\!35\)\( \beta_{3} - \)\(27\!\cdots\!04\)\( \beta_{4} - \)\(25\!\cdots\!56\)\( \beta_{5}) q^{36} +(-\)\(12\!\cdots\!70\)\( - \)\(21\!\cdots\!42\)\( \beta_{1} - \)\(30\!\cdots\!66\)\( \beta_{2} - \)\(14\!\cdots\!54\)\( \beta_{3} - \)\(95\!\cdots\!09\)\( \beta_{4} + \)\(27\!\cdots\!88\)\( \beta_{5}) q^{37} +(-\)\(39\!\cdots\!60\)\( - \)\(11\!\cdots\!71\)\( \beta_{1} - \)\(11\!\cdots\!70\)\( \beta_{2} + \)\(35\!\cdots\!26\)\( \beta_{3} + \)\(50\!\cdots\!56\)\( \beta_{4} + \)\(15\!\cdots\!03\)\( \beta_{5}) q^{38} +(-\)\(11\!\cdots\!84\)\( + \)\(68\!\cdots\!08\)\( \beta_{1} + \)\(63\!\cdots\!78\)\( \beta_{2} + \)\(65\!\cdots\!84\)\( \beta_{3} - \)\(92\!\cdots\!88\)\( \beta_{4} - \)\(98\!\cdots\!72\)\( \beta_{5}) q^{39} +(-\)\(67\!\cdots\!00\)\( + \)\(64\!\cdots\!00\)\( \beta_{1} + \)\(13\!\cdots\!40\)\( \beta_{2} - \)\(27\!\cdots\!00\)\( \beta_{3} + \)\(15\!\cdots\!20\)\( \beta_{4} + \)\(26\!\cdots\!00\)\( \beta_{5}) q^{40} +(-\)\(10\!\cdots\!18\)\( + \)\(14\!\cdots\!08\)\( \beta_{1} - \)\(99\!\cdots\!92\)\( \beta_{2} + \)\(36\!\cdots\!92\)\( \beta_{3} + \)\(13\!\cdots\!24\)\( \beta_{4} - \)\(15\!\cdots\!64\)\( \beta_{5}) q^{41} +(-\)\(29\!\cdots\!40\)\( - \)\(37\!\cdots\!08\)\( \beta_{1} - \)\(72\!\cdots\!24\)\( \beta_{2} - \)\(70\!\cdots\!84\)\( \beta_{3} + \)\(59\!\cdots\!56\)\( \beta_{4} - \)\(15\!\cdots\!52\)\( \beta_{5}) q^{42} +(-\)\(43\!\cdots\!00\)\( - \)\(30\!\cdots\!78\)\( \beta_{1} + \)\(12\!\cdots\!33\)\( \beta_{2} + \)\(15\!\cdots\!00\)\( \beta_{3} - \)\(30\!\cdots\!60\)\( \beta_{4} + \)\(69\!\cdots\!00\)\( \beta_{5}) q^{43} +(-\)\(17\!\cdots\!36\)\( - \)\(30\!\cdots\!12\)\( \beta_{1} + \)\(35\!\cdots\!68\)\( \beta_{2} - \)\(86\!\cdots\!20\)\( \beta_{3} + \)\(31\!\cdots\!16\)\( \beta_{4} - \)\(12\!\cdots\!36\)\( \beta_{5}) q^{44} +(-\)\(71\!\cdots\!50\)\( - \)\(39\!\cdots\!10\)\( \beta_{1} + \)\(38\!\cdots\!54\)\( \beta_{2} + \)\(11\!\cdots\!50\)\( \beta_{3} + \)\(29\!\cdots\!07\)\( \beta_{4} - \)\(68\!\cdots\!00\)\( \beta_{5}) q^{45} +(\)\(25\!\cdots\!92\)\( + \)\(21\!\cdots\!78\)\( \beta_{1} - \)\(19\!\cdots\!72\)\( \beta_{2} + \)\(18\!\cdots\!92\)\( \beta_{3} - \)\(92\!\cdots\!36\)\( \beta_{4} + \)\(10\!\cdots\!46\)\( \beta_{5}) q^{46} +(\)\(17\!\cdots\!40\)\( + \)\(14\!\cdots\!44\)\( \beta_{1} - \)\(20\!\cdots\!76\)\( \beta_{2} - \)\(54\!\cdots\!00\)\( \beta_{3} + \)\(72\!\cdots\!20\)\( \beta_{4} - \)\(26\!\cdots\!00\)\( \beta_{5}) q^{47} +(\)\(32\!\cdots\!40\)\( + \)\(14\!\cdots\!36\)\( \beta_{1} + \)\(21\!\cdots\!28\)\( \beta_{2} + \)\(10\!\cdots\!36\)\( \beta_{3} + \)\(25\!\cdots\!76\)\( \beta_{4} + \)\(19\!\cdots\!08\)\( \beta_{5}) q^{48} +(\)\(78\!\cdots\!57\)\( - \)\(10\!\cdots\!48\)\( \beta_{1} + \)\(41\!\cdots\!52\)\( \beta_{2} - \)\(16\!\cdots\!12\)\( \beta_{3} - \)\(76\!\cdots\!84\)\( \beta_{4} + \)\(90\!\cdots\!24\)\( \beta_{5}) q^{49} +(\)\(27\!\cdots\!00\)\( - \)\(51\!\cdots\!75\)\( \beta_{1} - \)\(67\!\cdots\!00\)\( \beta_{2} + \)\(11\!\cdots\!00\)\( \beta_{3} + \)\(60\!\cdots\!00\)\( \beta_{4} - \)\(34\!\cdots\!00\)\( \beta_{5}) q^{50} +(\)\(23\!\cdots\!12\)\( - \)\(13\!\cdots\!96\)\( \beta_{1} + \)\(35\!\cdots\!14\)\( \beta_{2} - \)\(19\!\cdots\!88\)\( \beta_{3} + \)\(10\!\cdots\!36\)\( \beta_{4} + \)\(45\!\cdots\!84\)\( \beta_{5}) q^{51} +(\)\(41\!\cdots\!00\)\( + \)\(82\!\cdots\!44\)\( \beta_{1} - \)\(28\!\cdots\!14\)\( \beta_{2} - \)\(73\!\cdots\!50\)\( \beta_{3} + \)\(21\!\cdots\!20\)\( \beta_{4} + \)\(40\!\cdots\!00\)\( \beta_{5}) q^{52} +(-\)\(22\!\cdots\!70\)\( + \)\(19\!\cdots\!14\)\( \beta_{1} + \)\(65\!\cdots\!30\)\( \beta_{2} + \)\(31\!\cdots\!78\)\( \beta_{3} - \)\(66\!\cdots\!37\)\( \beta_{4} - \)\(29\!\cdots\!16\)\( \beta_{5}) q^{53} +(-\)\(96\!\cdots\!40\)\( + \)\(72\!\cdots\!34\)\( \beta_{1} - \)\(11\!\cdots\!36\)\( \beta_{2} + \)\(40\!\cdots\!32\)\( \beta_{3} - \)\(11\!\cdots\!64\)\( \beta_{4} + \)\(51\!\cdots\!94\)\( \beta_{5}) q^{54} +(-\)\(53\!\cdots\!00\)\( - \)\(10\!\cdots\!40\)\( \beta_{1} + \)\(13\!\cdots\!66\)\( \beta_{2} + \)\(87\!\cdots\!00\)\( \beta_{3} + \)\(80\!\cdots\!28\)\( \beta_{4} - \)\(84\!\cdots\!00\)\( \beta_{5}) q^{55} +(-\)\(18\!\cdots\!40\)\( - \)\(20\!\cdots\!72\)\( \beta_{1} - \)\(70\!\cdots\!32\)\( \beta_{2} - \)\(95\!\cdots\!52\)\( \beta_{3} - \)\(11\!\cdots\!12\)\( \beta_{4} - \)\(15\!\cdots\!48\)\( \beta_{5}) q^{56} +(-\)\(13\!\cdots\!80\)\( - \)\(17\!\cdots\!20\)\( \beta_{1} + \)\(51\!\cdots\!84\)\( \beta_{2} + \)\(13\!\cdots\!00\)\( \beta_{3} - \)\(10\!\cdots\!90\)\( \beta_{4} + \)\(36\!\cdots\!00\)\( \beta_{5}) q^{57} +(-\)\(33\!\cdots\!40\)\( - \)\(40\!\cdots\!26\)\( \beta_{1} - \)\(28\!\cdots\!16\)\( \beta_{2} + \)\(83\!\cdots\!36\)\( \beta_{3} + \)\(45\!\cdots\!96\)\( \beta_{4} - \)\(29\!\cdots\!92\)\( \beta_{5}) q^{58} +(-\)\(52\!\cdots\!20\)\( + \)\(13\!\cdots\!66\)\( \beta_{1} + \)\(37\!\cdots\!61\)\( \beta_{2} - \)\(10\!\cdots\!08\)\( \beta_{3} - \)\(21\!\cdots\!20\)\( \beta_{4} - \)\(35\!\cdots\!20\)\( \beta_{5}) q^{59} +(\)\(52\!\cdots\!00\)\( + \)\(45\!\cdots\!60\)\( \beta_{1} - \)\(21\!\cdots\!24\)\( \beta_{2} - \)\(55\!\cdots\!00\)\( \beta_{3} - \)\(51\!\cdots\!92\)\( \beta_{4} + \)\(15\!\cdots\!00\)\( \beta_{5}) q^{60} +(\)\(20\!\cdots\!22\)\( + \)\(93\!\cdots\!90\)\( \beta_{1} - \)\(94\!\cdots\!50\)\( \beta_{2} - \)\(39\!\cdots\!90\)\( \beta_{3} - \)\(73\!\cdots\!25\)\( \beta_{4} - \)\(26\!\cdots\!20\)\( \beta_{5}) q^{61} +(\)\(97\!\cdots\!40\)\( - \)\(91\!\cdots\!88\)\( \beta_{1} - \)\(65\!\cdots\!28\)\( \beta_{2} + \)\(47\!\cdots\!96\)\( \beta_{3} + \)\(41\!\cdots\!36\)\( \beta_{4} + \)\(20\!\cdots\!88\)\( \beta_{5}) q^{62} +(\)\(96\!\cdots\!20\)\( - \)\(22\!\cdots\!84\)\( \beta_{1} + \)\(16\!\cdots\!34\)\( \beta_{2} - \)\(44\!\cdots\!64\)\( \beta_{3} - \)\(10\!\cdots\!44\)\( \beta_{4} + \)\(30\!\cdots\!08\)\( \beta_{5}) q^{63} +(\)\(12\!\cdots\!32\)\( - \)\(18\!\cdots\!00\)\( \beta_{1} + \)\(92\!\cdots\!00\)\( \beta_{2} - \)\(37\!\cdots\!92\)\( \beta_{3} - \)\(12\!\cdots\!28\)\( \beta_{4} - \)\(14\!\cdots\!92\)\( \beta_{5}) q^{64} +(\)\(14\!\cdots\!00\)\( + \)\(21\!\cdots\!80\)\( \beta_{1} - \)\(10\!\cdots\!72\)\( \beta_{2} - \)\(60\!\cdots\!00\)\( \beta_{3} + \)\(13\!\cdots\!24\)\( \beta_{4} + \)\(21\!\cdots\!00\)\( \beta_{5}) q^{65} +(\)\(43\!\cdots\!44\)\( + \)\(15\!\cdots\!92\)\( \beta_{1} - \)\(38\!\cdots\!28\)\( \beta_{2} + \)\(24\!\cdots\!16\)\( \beta_{3} - \)\(86\!\cdots\!12\)\( \beta_{4} + \)\(20\!\cdots\!72\)\( \beta_{5}) q^{66} +(-\)\(33\!\cdots\!40\)\( + \)\(64\!\cdots\!90\)\( \beta_{1} - \)\(56\!\cdots\!49\)\( \beta_{2} + \)\(50\!\cdots\!04\)\( \beta_{3} + \)\(95\!\cdots\!24\)\( \beta_{4} - \)\(51\!\cdots\!88\)\( \beta_{5}) q^{67} +(-\)\(32\!\cdots\!20\)\( + \)\(23\!\cdots\!16\)\( \beta_{1} + \)\(10\!\cdots\!42\)\( \beta_{2} - \)\(15\!\cdots\!78\)\( \beta_{3} - \)\(19\!\cdots\!88\)\( \beta_{4} + \)\(42\!\cdots\!16\)\( \beta_{5}) q^{68} +(-\)\(33\!\cdots\!24\)\( - \)\(57\!\cdots\!60\)\( \beta_{1} + \)\(46\!\cdots\!60\)\( \beta_{2} + \)\(99\!\cdots\!56\)\( \beta_{3} - \)\(27\!\cdots\!56\)\( \beta_{4} + \)\(59\!\cdots\!76\)\( \beta_{5}) q^{69} +(-\)\(45\!\cdots\!00\)\( - \)\(18\!\cdots\!80\)\( \beta_{1} + \)\(32\!\cdots\!72\)\( \beta_{2} + \)\(12\!\cdots\!00\)\( \beta_{3} + \)\(15\!\cdots\!76\)\( \beta_{4} + \)\(27\!\cdots\!00\)\( \beta_{5}) q^{70} +(-\)\(14\!\cdots\!08\)\( - \)\(90\!\cdots\!80\)\( \beta_{1} - \)\(31\!\cdots\!50\)\( \beta_{2} + \)\(37\!\cdots\!80\)\( \beta_{3} - \)\(14\!\cdots\!00\)\( \beta_{4} - \)\(52\!\cdots\!60\)\( \beta_{5}) q^{71} +(-\)\(18\!\cdots\!20\)\( + \)\(54\!\cdots\!08\)\( \beta_{1} - \)\(53\!\cdots\!20\)\( \beta_{2} - \)\(36\!\cdots\!48\)\( \beta_{3} - \)\(22\!\cdots\!68\)\( \beta_{4} + \)\(34\!\cdots\!56\)\( \beta_{5}) q^{72} +(\)\(52\!\cdots\!90\)\( - \)\(10\!\cdots\!04\)\( \beta_{1} + \)\(10\!\cdots\!28\)\( \beta_{2} - \)\(11\!\cdots\!24\)\( \beta_{3} + \)\(58\!\cdots\!06\)\( \beta_{4} + \)\(21\!\cdots\!28\)\( \beta_{5}) q^{73} +(\)\(40\!\cdots\!64\)\( + \)\(14\!\cdots\!78\)\( \beta_{1} + \)\(37\!\cdots\!88\)\( \beta_{2} + \)\(59\!\cdots\!32\)\( \beta_{3} - \)\(25\!\cdots\!96\)\( \beta_{4} - \)\(38\!\cdots\!64\)\( \beta_{5}) q^{74} +(\)\(90\!\cdots\!00\)\( - \)\(13\!\cdots\!50\)\( \beta_{1} + \)\(33\!\cdots\!25\)\( \beta_{2} + \)\(81\!\cdots\!00\)\( \beta_{3} - \)\(33\!\cdots\!00\)\( \beta_{4} - \)\(31\!\cdots\!00\)\( \beta_{5}) q^{75} +(\)\(20\!\cdots\!20\)\( + \)\(78\!\cdots\!36\)\( \beta_{1} - \)\(64\!\cdots\!84\)\( \beta_{2} + \)\(84\!\cdots\!96\)\( \beta_{3} + \)\(24\!\cdots\!36\)\( \beta_{4} + \)\(14\!\cdots\!44\)\( \beta_{5}) q^{76} +(\)\(58\!\cdots\!00\)\( - \)\(83\!\cdots\!36\)\( \beta_{1} + \)\(21\!\cdots\!80\)\( \beta_{2} - \)\(16\!\cdots\!96\)\( \beta_{3} - \)\(69\!\cdots\!36\)\( \beta_{4} + \)\(24\!\cdots\!12\)\( \beta_{5}) q^{77} +(-\)\(53\!\cdots\!00\)\( + \)\(13\!\cdots\!22\)\( \beta_{1} - \)\(88\!\cdots\!20\)\( \beta_{2} - \)\(66\!\cdots\!08\)\( \beta_{3} + \)\(17\!\cdots\!52\)\( \beta_{4} - \)\(41\!\cdots\!74\)\( \beta_{5}) q^{78} +(-\)\(74\!\cdots\!60\)\( - \)\(93\!\cdots\!40\)\( \beta_{1} + \)\(13\!\cdots\!40\)\( \beta_{2} + \)\(29\!\cdots\!44\)\( \beta_{3} + \)\(17\!\cdots\!56\)\( \beta_{4} + \)\(18\!\cdots\!24\)\( \beta_{5}) q^{79} +(-\)\(11\!\cdots\!00\)\( + \)\(50\!\cdots\!80\)\( \beta_{1} + \)\(40\!\cdots\!88\)\( \beta_{2} - \)\(22\!\cdots\!00\)\( \beta_{3} - \)\(61\!\cdots\!96\)\( \beta_{4} + \)\(11\!\cdots\!00\)\( \beta_{5}) q^{80} +(-\)\(10\!\cdots\!79\)\( - \)\(82\!\cdots\!72\)\( \beta_{1} - \)\(35\!\cdots\!92\)\( \beta_{2} + \)\(82\!\cdots\!56\)\( \beta_{3} - \)\(93\!\cdots\!70\)\( \beta_{4} - \)\(11\!\cdots\!40\)\( \beta_{5}) q^{81} +(-\)\(35\!\cdots\!60\)\( + \)\(35\!\cdots\!22\)\( \beta_{1} - \)\(12\!\cdots\!28\)\( \beta_{2} - \)\(10\!\cdots\!84\)\( \beta_{3} + \)\(29\!\cdots\!36\)\( \beta_{4} - \)\(27\!\cdots\!52\)\( \beta_{5}) q^{82} +(-\)\(24\!\cdots\!80\)\( - \)\(13\!\cdots\!38\)\( \beta_{1} - \)\(15\!\cdots\!73\)\( \beta_{2} - \)\(20\!\cdots\!00\)\( \beta_{3} + \)\(11\!\cdots\!00\)\( \beta_{4} + \)\(42\!\cdots\!00\)\( \beta_{5}) q^{83} +(\)\(85\!\cdots\!64\)\( + \)\(87\!\cdots\!52\)\( \beta_{1} + \)\(18\!\cdots\!52\)\( \beta_{2} + \)\(25\!\cdots\!32\)\( \beta_{3} - \)\(69\!\cdots\!88\)\( \beta_{4} + \)\(59\!\cdots\!68\)\( \beta_{5}) q^{84} +(\)\(81\!\cdots\!00\)\( - \)\(39\!\cdots\!40\)\( \beta_{1} - \)\(69\!\cdots\!04\)\( \beta_{2} + \)\(53\!\cdots\!00\)\( \beta_{3} - \)\(22\!\cdots\!82\)\( \beta_{4} - \)\(13\!\cdots\!00\)\( \beta_{5}) q^{85} +(\)\(47\!\cdots\!32\)\( + \)\(20\!\cdots\!11\)\( \beta_{1} + \)\(60\!\cdots\!46\)\( \beta_{2} - \)\(55\!\cdots\!70\)\( \beta_{3} + \)\(10\!\cdots\!32\)\( \beta_{4} - \)\(13\!\cdots\!47\)\( \beta_{5}) q^{86} +(\)\(22\!\cdots\!80\)\( + \)\(35\!\cdots\!04\)\( \beta_{1} - \)\(52\!\cdots\!62\)\( \beta_{2} - \)\(76\!\cdots\!64\)\( \beta_{3} + \)\(24\!\cdots\!36\)\( \beta_{4} + \)\(45\!\cdots\!08\)\( \beta_{5}) q^{87} +(\)\(44\!\cdots\!20\)\( + \)\(18\!\cdots\!44\)\( \beta_{1} - \)\(13\!\cdots\!04\)\( \beta_{2} + \)\(13\!\cdots\!56\)\( \beta_{3} - \)\(59\!\cdots\!04\)\( \beta_{4} + \)\(11\!\cdots\!68\)\( \beta_{5}) q^{88} +(\)\(40\!\cdots\!70\)\( - \)\(51\!\cdots\!84\)\( \beta_{1} - \)\(47\!\cdots\!24\)\( \beta_{2} - \)\(43\!\cdots\!84\)\( \beta_{3} - \)\(15\!\cdots\!34\)\( \beta_{4} - \)\(11\!\cdots\!96\)\( \beta_{5}) q^{89} +(\)\(82\!\cdots\!00\)\( - \)\(80\!\cdots\!70\)\( \beta_{1} - \)\(18\!\cdots\!12\)\( \beta_{2} + \)\(89\!\cdots\!00\)\( \beta_{3} + \)\(13\!\cdots\!04\)\( \beta_{4} + \)\(28\!\cdots\!00\)\( \beta_{5}) q^{90} +(\)\(10\!\cdots\!92\)\( - \)\(12\!\cdots\!12\)\( \beta_{1} + \)\(74\!\cdots\!28\)\( \beta_{2} + \)\(53\!\cdots\!48\)\( \beta_{3} - \)\(13\!\cdots\!92\)\( \beta_{4} + \)\(23\!\cdots\!32\)\( \beta_{5}) q^{91} +(-\)\(34\!\cdots\!40\)\( - \)\(50\!\cdots\!52\)\( \beta_{1} + \)\(22\!\cdots\!80\)\( \beta_{2} - \)\(30\!\cdots\!64\)\( \beta_{3} + \)\(11\!\cdots\!56\)\( \beta_{4} - \)\(68\!\cdots\!92\)\( \beta_{5}) q^{92} +(-\)\(24\!\cdots\!40\)\( - \)\(26\!\cdots\!24\)\( \beta_{1} + \)\(33\!\cdots\!04\)\( \beta_{2} + \)\(15\!\cdots\!68\)\( \beta_{3} - \)\(54\!\cdots\!92\)\( \beta_{4} - \)\(56\!\cdots\!96\)\( \beta_{5}) q^{93} +(-\)\(29\!\cdots\!56\)\( + \)\(58\!\cdots\!04\)\( \beta_{1} - \)\(23\!\cdots\!16\)\( \beta_{2} + \)\(13\!\cdots\!12\)\( \beta_{3} - \)\(46\!\cdots\!04\)\( \beta_{4} + \)\(24\!\cdots\!84\)\( \beta_{5}) q^{94} +(-\)\(54\!\cdots\!00\)\( + \)\(23\!\cdots\!00\)\( \beta_{1} - \)\(27\!\cdots\!70\)\( \beta_{2} - \)\(24\!\cdots\!00\)\( \beta_{3} + \)\(84\!\cdots\!40\)\( \beta_{4} + \)\(14\!\cdots\!00\)\( \beta_{5}) q^{95} +(-\)\(31\!\cdots\!68\)\( - \)\(44\!\cdots\!16\)\( \beta_{1} + \)\(64\!\cdots\!84\)\( \beta_{2} + \)\(26\!\cdots\!40\)\( \beta_{3} - \)\(12\!\cdots\!32\)\( \beta_{4} - \)\(14\!\cdots\!48\)\( \beta_{5}) q^{96} +(-\)\(20\!\cdots\!10\)\( + \)\(18\!\cdots\!76\)\( \beta_{1} + \)\(18\!\cdots\!20\)\( \beta_{2} - \)\(14\!\cdots\!72\)\( \beta_{3} + \)\(39\!\cdots\!38\)\( \beta_{4} - \)\(21\!\cdots\!16\)\( \beta_{5}) q^{97} +(\)\(57\!\cdots\!40\)\( - \)\(40\!\cdots\!01\)\( \beta_{1} + \)\(11\!\cdots\!28\)\( \beta_{2} - \)\(17\!\cdots\!76\)\( \beta_{3} - \)\(24\!\cdots\!96\)\( \beta_{4} + \)\(36\!\cdots\!72\)\( \beta_{5}) q^{98} +(\)\(37\!\cdots\!96\)\( + \)\(19\!\cdots\!18\)\( \beta_{1} - \)\(73\!\cdots\!67\)\( \beta_{2} + \)\(40\!\cdots\!28\)\( \beta_{3} - \)\(14\!\cdots\!12\)\( \beta_{4} + \)\(20\!\cdots\!52\)\( \beta_{5}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 264721893120q^{2} + 1441611375788089080q^{3} + \)\(41\!\cdots\!72\)\(q^{4} - \)\(26\!\cdots\!00\)\(q^{5} - \)\(36\!\cdots\!88\)\(q^{6} + \)\(27\!\cdots\!00\)\(q^{7} - \)\(21\!\cdots\!40\)\(q^{8} - \)\(48\!\cdots\!42\)\(q^{9} + O(q^{10}) \) \( 6q + 264721893120q^{2} + 1441611375788089080q^{3} + \)\(41\!\cdots\!72\)\(q^{4} - \)\(26\!\cdots\!00\)\(q^{5} - \)\(36\!\cdots\!88\)\(q^{6} + \)\(27\!\cdots\!00\)\(q^{7} - \)\(21\!\cdots\!40\)\(q^{8} - \)\(48\!\cdots\!42\)\(q^{9} - \)\(94\!\cdots\!00\)\(q^{10} - \)\(18\!\cdots\!68\)\(q^{11} + \)\(46\!\cdots\!40\)\(q^{12} + \)\(23\!\cdots\!60\)\(q^{13} - \)\(29\!\cdots\!56\)\(q^{14} - \)\(20\!\cdots\!00\)\(q^{15} + \)\(44\!\cdots\!76\)\(q^{16} - \)\(35\!\cdots\!40\)\(q^{17} - \)\(57\!\cdots\!20\)\(q^{18} + \)\(56\!\cdots\!60\)\(q^{19} - \)\(20\!\cdots\!00\)\(q^{20} - \)\(54\!\cdots\!68\)\(q^{21} + \)\(61\!\cdots\!40\)\(q^{22} - \)\(47\!\cdots\!60\)\(q^{23} + \)\(54\!\cdots\!80\)\(q^{24} + \)\(12\!\cdots\!50\)\(q^{25} - \)\(55\!\cdots\!68\)\(q^{26} - \)\(18\!\cdots\!40\)\(q^{27} + \)\(27\!\cdots\!60\)\(q^{28} + \)\(16\!\cdots\!40\)\(q^{29} - \)\(10\!\cdots\!00\)\(q^{30} + \)\(17\!\cdots\!72\)\(q^{31} + \)\(21\!\cdots\!20\)\(q^{32} - \)\(43\!\cdots\!40\)\(q^{33} - \)\(19\!\cdots\!76\)\(q^{34} - \)\(15\!\cdots\!00\)\(q^{35} - \)\(34\!\cdots\!04\)\(q^{36} - \)\(74\!\cdots\!20\)\(q^{37} - \)\(23\!\cdots\!60\)\(q^{38} - \)\(70\!\cdots\!04\)\(q^{39} - \)\(40\!\cdots\!00\)\(q^{40} - \)\(62\!\cdots\!08\)\(q^{41} - \)\(17\!\cdots\!40\)\(q^{42} - \)\(26\!\cdots\!00\)\(q^{43} - \)\(10\!\cdots\!16\)\(q^{44} - \)\(42\!\cdots\!00\)\(q^{45} + \)\(15\!\cdots\!52\)\(q^{46} + \)\(10\!\cdots\!40\)\(q^{47} + \)\(19\!\cdots\!40\)\(q^{48} + \)\(47\!\cdots\!42\)\(q^{49} + \)\(16\!\cdots\!00\)\(q^{50} + \)\(14\!\cdots\!72\)\(q^{51} + \)\(24\!\cdots\!00\)\(q^{52} - \)\(13\!\cdots\!20\)\(q^{53} - \)\(58\!\cdots\!40\)\(q^{54} - \)\(31\!\cdots\!00\)\(q^{55} - \)\(10\!\cdots\!40\)\(q^{56} - \)\(80\!\cdots\!80\)\(q^{57} - \)\(19\!\cdots\!40\)\(q^{58} - \)\(31\!\cdots\!20\)\(q^{59} + \)\(31\!\cdots\!00\)\(q^{60} + \)\(12\!\cdots\!32\)\(q^{61} + \)\(58\!\cdots\!40\)\(q^{62} + \)\(57\!\cdots\!20\)\(q^{63} + \)\(74\!\cdots\!92\)\(q^{64} + \)\(88\!\cdots\!00\)\(q^{65} + \)\(26\!\cdots\!64\)\(q^{66} - \)\(20\!\cdots\!40\)\(q^{67} - \)\(19\!\cdots\!20\)\(q^{68} - \)\(20\!\cdots\!44\)\(q^{69} - \)\(27\!\cdots\!00\)\(q^{70} - \)\(84\!\cdots\!48\)\(q^{71} - \)\(11\!\cdots\!20\)\(q^{72} + \)\(31\!\cdots\!40\)\(q^{73} + \)\(24\!\cdots\!84\)\(q^{74} + \)\(54\!\cdots\!00\)\(q^{75} + \)\(12\!\cdots\!20\)\(q^{76} + \)\(35\!\cdots\!00\)\(q^{77} - \)\(32\!\cdots\!00\)\(q^{78} - \)\(44\!\cdots\!60\)\(q^{79} - \)\(71\!\cdots\!00\)\(q^{80} - \)\(62\!\cdots\!74\)\(q^{81} - \)\(21\!\cdots\!60\)\(q^{82} - \)\(14\!\cdots\!80\)\(q^{83} + \)\(51\!\cdots\!84\)\(q^{84} + \)\(49\!\cdots\!00\)\(q^{85} + \)\(28\!\cdots\!92\)\(q^{86} + \)\(13\!\cdots\!80\)\(q^{87} + \)\(26\!\cdots\!20\)\(q^{88} + \)\(24\!\cdots\!20\)\(q^{89} + \)\(49\!\cdots\!00\)\(q^{90} + \)\(63\!\cdots\!52\)\(q^{91} - \)\(20\!\cdots\!40\)\(q^{92} - \)\(14\!\cdots\!40\)\(q^{93} - \)\(17\!\cdots\!36\)\(q^{94} - \)\(32\!\cdots\!00\)\(q^{95} - \)\(18\!\cdots\!08\)\(q^{96} - \)\(12\!\cdots\!60\)\(q^{97} + \)\(34\!\cdots\!40\)\(q^{98} + \)\(22\!\cdots\!76\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 17761277524319496093 x^{4} - 5418653098480012458842128200 x^{3} + 63613941732383091007079230349679850008 x^{2} + 25593744940652395086700970560658253564764671200 x - 44271959822073232588337476834309941911954845745332334416\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 192 \nu \)
\(\beta_{2}\)\(=\)\((\)\(-8104264459 \nu^{5} + 9642120199060495030 \nu^{4} + 126236081287745673418373496851 \nu^{3} - 106949459579029911739390268928063746990 \nu^{2} - 294136612685125663327920281815481415762311160172 \nu + 163304584067188222131014536083084269833355524978043256760\)\()/ \)\(18\!\cdots\!28\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-15406206736559 \nu^{5} + 18329670498414001052030 \nu^{4} + 239974790528004525168328017513751 \nu^{3} - 103951316688088910206759538881180972499734 \nu^{2} - 604622998011778477343174759352818594552930098917116 \nu - 277809164478093255030643853680584812361963496136214188267176\)\()/ \)\(26\!\cdots\!04\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-93048030767998127 \nu^{5} + 160224634309993103953457534 \nu^{4} + 1371124304204137778308284322826500375 \nu^{3} - 1918307183548659512975076546265468425108830998 \nu^{2} - 2724414459321415275488647375004634176252328285650280188 \nu + 3067155275934657273183312532303177718181013416044086712657234264\)\()/ \)\(78\!\cdots\!72\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-75779219167136606165 \nu^{5} + 137831567005917646627023429386 \nu^{4} + 1177211917287937830688521455354966290381 \nu^{3} - 1705187497977128892212824866543077461787976155410 \nu^{2} - 2868181449606997543965540779984442786486776928091482247700 \nu + 2719142611864231379739607468653334900790029404319710967216654414664\)\()/ \)\(67\!\cdots\!76\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/192\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 13307 \beta_{2} + 87863724858 \beta_{1} + 218250578218837967990784\)\()/36864\)
\(\nu^{3}\)\(=\)\((\)\(4131 \beta_{5} - 4639708 \beta_{4} + 6739710962 \beta_{3} + 107241347382062 \beta_{2} + 24417931701886985374437 \beta_{1} + 1198519366934203075696537283788800\)\()/442368\)
\(\nu^{4}\)\(=\)\((\)\(543893760081 \beta_{5} - 25654290487476 \beta_{4} + 14828846464893523799 \beta_{3} - 332657902425455306643601 \beta_{2} + 1909280277074058311388515397729 \beta_{1} + 2313032857572934116718473763038198554210304\)\()/36864\)
\(\nu^{5}\)\(=\)\((\)\(72111753371191048768899 \beta_{5} - 72636684087436959238921052 \beta_{4} + 158333854075625444882068546258 \beta_{3} - 2001500789982317072339343307927538 \beta_{2} + 310069303970187363553676491072993003877061 \beta_{1} + 26043847022394721292473370169085024603769408004096000\)\()/442368\)

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
3.81583e9
1.98644e9
7.14285e8
−1.47555e9
−1.61977e9
−3.42124e9
−6.88520e11 1.25704e18 3.22944e23 7.49066e26 −8.65494e29 4.37098e32 −1.18307e35 −3.89426e36 −5.15747e38
1.2 −3.37277e11 −2.07041e18 −3.73599e22 −2.41864e26 6.98302e29 −4.13826e32 6.35685e34 −1.18781e36 8.15750e37
1.3 −9.30224e10 3.16609e18 −1.42463e23 −6.28590e26 −2.94518e29 2.92216e32 2.73094e34 4.54975e36 5.84729e37
1.4 3.27426e11 −3.10225e18 −4.39082e22 −2.60453e25 −1.01576e30 6.63145e32 −6.38558e34 4.14958e36 −8.52790e36
1.5 3.55117e11 1.22969e18 −2.50080e22 1.38608e27 4.36684e29 −5.00002e32 −6.25444e34 −3.96226e36 4.92219e38
1.6 7.00999e11 9.61452e17 3.40284e23 −1.50068e27 6.73977e29 −2.07773e32 1.32607e35 −4.55001e36 −1.05197e39
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{78}^{\mathrm{new}}(\Gamma_0(1))\).