Properties

Label 1.52.a.a
Level 1
Weight 52
Character orbit 1.a
Self dual yes
Analytic conductor 16.473
Analytic rank 0
Dimension 4
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 52 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.4731353414\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 2 x^{3} - 495735060514 x^{2} - 23954614981416598 x + 48979992255622025570313\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{23}\cdot 3^{10}\cdot 5^{3}\cdot 7^{2}\cdot 13\cdot 17 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q +(8189010 + \beta_{1}) q^{2} +(100965943260 - 6677 \beta_{1} - \beta_{2}) q^{3} +(1994525867718928 - 7825327 \beta_{1} + 511 \beta_{2} - \beta_{3}) q^{4} +(303528278241889470 - 2170290676 \beta_{1} - 86148 \beta_{2} - 144 \beta_{3}) q^{5} +(-27078920080376717448 - 186291114948 \beta_{1} - 61334496 \beta_{2} + 16416 \beta_{3}) q^{6} +(\)\(16\!\cdots\!00\)\( + 3775025640302 \beta_{1} - 224454986 \beta_{2} - 736960 \beta_{3}) q^{7} +(-\)\(34\!\cdots\!80\)\( + 1959718379828168 \beta_{1} + 49429856952 \beta_{2} + 20189880 \beta_{3}) q^{8} +(\)\(12\!\cdots\!57\)\( + 41221787348461896 \beta_{1} - 666471696408 \beta_{2} - 386651232 \beta_{3}) q^{9} +O(q^{10})\) \( q +(8189010 + \beta_{1}) q^{2} +(100965943260 - 6677 \beta_{1} - \beta_{2}) q^{3} +(1994525867718928 - 7825327 \beta_{1} + 511 \beta_{2} - \beta_{3}) q^{4} +(303528278241889470 - 2170290676 \beta_{1} - 86148 \beta_{2} - 144 \beta_{3}) q^{5} +(-27078920080376717448 - 186291114948 \beta_{1} - 61334496 \beta_{2} + 16416 \beta_{3}) q^{6} +(\)\(16\!\cdots\!00\)\( + 3775025640302 \beta_{1} - 224454986 \beta_{2} - 736960 \beta_{3}) q^{7} +(-\)\(34\!\cdots\!80\)\( + 1959718379828168 \beta_{1} + 49429856952 \beta_{2} + 20189880 \beta_{3}) q^{8} +(\)\(12\!\cdots\!57\)\( + 41221787348461896 \beta_{1} - 666471696408 \beta_{2} - 386651232 \beta_{3}) q^{9} +(-\)\(65\!\cdots\!80\)\( + 576521306120640574 \beta_{1} - 2667384869248 \beta_{2} + 5506416256 \beta_{3}) q^{10} +(\)\(88\!\cdots\!12\)\( - 2258744929721205575 \beta_{1} + 155930790349845 \beta_{2} - 59943070080 \beta_{3}) q^{11} +(-\)\(12\!\cdots\!80\)\( - 64267735385758560740 \beta_{1} - 1787241473591644 \beta_{2} + 498954927780 \beta_{3}) q^{12} +(\)\(76\!\cdots\!70\)\( + 75259059392487711692 \beta_{1} + 8796829725936700 \beta_{2} - 3042585439120 \beta_{3}) q^{13} +(\)\(29\!\cdots\!56\)\( + \)\(28\!\cdots\!44\)\( \beta_{1} + 6489920058124608 \beta_{2} + 11190698955072 \beta_{3}) q^{14} +(\)\(36\!\cdots\!40\)\( - \)\(26\!\cdots\!02\)\( \beta_{1} - 365235659009187246 \beta_{2} + 10467930808512 \beta_{3}) q^{15} +(\)\(34\!\cdots\!36\)\( - \)\(67\!\cdots\!96\)\( \beta_{1} + 2232721294548523968 \beta_{2} - 539432326636608 \beta_{3}) q^{16} +(\)\(12\!\cdots\!30\)\( + \)\(12\!\cdots\!04\)\( \beta_{1} - 5095661844231778968 \beta_{2} + 4845742691513760 \beta_{3}) q^{17} +(\)\(18\!\cdots\!10\)\( + \)\(10\!\cdots\!33\)\( \beta_{1} - 8325439050872514816 \beta_{2} - 28026030112508160 \beta_{3}) q^{18} +(\)\(20\!\cdots\!20\)\( + \)\(21\!\cdots\!67\)\( \beta_{1} + 56753234257291972779 \beta_{2} + 115731936015363456 \beta_{3}) q^{19} +(\)\(16\!\cdots\!60\)\( - \)\(22\!\cdots\!78\)\( \beta_{1} + \)\(20\!\cdots\!06\)\( \beta_{2} - 321772238171413182 \beta_{3}) q^{20} +(\)\(78\!\cdots\!12\)\( - \)\(25\!\cdots\!96\)\( \beta_{1} - \)\(25\!\cdots\!72\)\( \beta_{2} + 357837433134495552 \beta_{3}) q^{21} +(-\)\(87\!\cdots\!80\)\( + \)\(29\!\cdots\!12\)\( \beta_{1} + \)\(93\!\cdots\!80\)\( \beta_{2} + 1779618650400099680 \beta_{3}) q^{22} +(-\)\(13\!\cdots\!20\)\( - \)\(14\!\cdots\!14\)\( \beta_{1} - \)\(12\!\cdots\!34\)\( \beta_{2} - 12037007883164961600 \beta_{3}) q^{23} +(-\)\(21\!\cdots\!40\)\( - \)\(14\!\cdots\!64\)\( \beta_{1} - \)\(10\!\cdots\!88\)\( \beta_{2} + 36055666143180714528 \beta_{3}) q^{24} +(-\)\(14\!\cdots\!25\)\( - \)\(36\!\cdots\!80\)\( \beta_{1} - \)\(29\!\cdots\!40\)\( \beta_{2} - 44536012526878198720 \beta_{3}) q^{25} +(\)\(37\!\cdots\!12\)\( + \)\(15\!\cdots\!06\)\( \beta_{1} + \)\(62\!\cdots\!12\)\( \beta_{2} - \)\(10\!\cdots\!52\)\( \beta_{3}) q^{26} +(\)\(91\!\cdots\!80\)\( + \)\(15\!\cdots\!62\)\( \beta_{1} - \)\(21\!\cdots\!82\)\( \beta_{2} + \)\(67\!\cdots\!20\)\( \beta_{3}) q^{27} +(\)\(86\!\cdots\!20\)\( - \)\(44\!\cdots\!76\)\( \beta_{1} + \)\(20\!\cdots\!84\)\( \beta_{2} - \)\(14\!\cdots\!80\)\( \beta_{3}) q^{28} +(\)\(61\!\cdots\!30\)\( - \)\(20\!\cdots\!12\)\( \beta_{1} + \)\(68\!\cdots\!36\)\( \beta_{2} + \)\(50\!\cdots\!64\)\( \beta_{3}) q^{29} +(-\)\(81\!\cdots\!60\)\( + \)\(24\!\cdots\!48\)\( \beta_{1} - \)\(22\!\cdots\!96\)\( \beta_{2} + \)\(60\!\cdots\!12\)\( \beta_{3}) q^{30} +(-\)\(18\!\cdots\!68\)\( + \)\(61\!\cdots\!00\)\( \beta_{1} + \)\(18\!\cdots\!60\)\( \beta_{2} - \)\(16\!\cdots\!40\)\( \beta_{3}) q^{31} +(-\)\(17\!\cdots\!40\)\( + \)\(19\!\cdots\!96\)\( \beta_{1} - \)\(34\!\cdots\!40\)\( \beta_{2} + \)\(93\!\cdots\!60\)\( \beta_{3}) q^{32} +(-\)\(43\!\cdots\!80\)\( - \)\(85\!\cdots\!24\)\( \beta_{1} + \)\(13\!\cdots\!48\)\( \beta_{2} + \)\(35\!\cdots\!60\)\( \beta_{3}) q^{33} +(\)\(14\!\cdots\!16\)\( + \)\(67\!\cdots\!86\)\( \beta_{1} - \)\(40\!\cdots\!88\)\( \beta_{2} - \)\(46\!\cdots\!72\)\( \beta_{3}) q^{34} +(\)\(13\!\cdots\!20\)\( - \)\(10\!\cdots\!56\)\( \beta_{1} - \)\(10\!\cdots\!88\)\( \beta_{2} - \)\(15\!\cdots\!64\)\( \beta_{3}) q^{35} +(\)\(31\!\cdots\!96\)\( + \)\(12\!\cdots\!89\)\( \beta_{1} + \)\(22\!\cdots\!63\)\( \beta_{2} + \)\(36\!\cdots\!47\)\( \beta_{3}) q^{36} +(\)\(23\!\cdots\!90\)\( - \)\(15\!\cdots\!92\)\( \beta_{1} - \)\(34\!\cdots\!92\)\( \beta_{2} + \)\(87\!\cdots\!40\)\( \beta_{3}) q^{37} +(\)\(10\!\cdots\!80\)\( - \)\(26\!\cdots\!28\)\( \beta_{1} + \)\(16\!\cdots\!48\)\( \beta_{2} - \)\(46\!\cdots\!40\)\( \beta_{3}) q^{38} +(-\)\(29\!\cdots\!76\)\( - \)\(51\!\cdots\!66\)\( \beta_{1} - \)\(56\!\cdots\!42\)\( \beta_{2} + \)\(51\!\cdots\!12\)\( \beta_{3}) q^{39} +(-\)\(65\!\cdots\!00\)\( + \)\(14\!\cdots\!20\)\( \beta_{1} + \)\(13\!\cdots\!60\)\( \beta_{2} + \)\(13\!\cdots\!80\)\( \beta_{3}) q^{40} +(\)\(36\!\cdots\!42\)\( + \)\(88\!\cdots\!00\)\( \beta_{1} + \)\(36\!\cdots\!60\)\( \beta_{2} - \)\(51\!\cdots\!40\)\( \beta_{3}) q^{41} +(-\)\(98\!\cdots\!20\)\( - \)\(49\!\cdots\!64\)\( \beta_{1} - \)\(16\!\cdots\!04\)\( \beta_{2} + \)\(43\!\cdots\!40\)\( \beta_{3}) q^{42} +(-\)\(95\!\cdots\!00\)\( - \)\(24\!\cdots\!51\)\( \beta_{1} + \)\(12\!\cdots\!33\)\( \beta_{2} + \)\(99\!\cdots\!00\)\( \beta_{3}) q^{43} +(\)\(97\!\cdots\!36\)\( - \)\(81\!\cdots\!24\)\( \beta_{1} + \)\(29\!\cdots\!92\)\( \beta_{2} - \)\(28\!\cdots\!52\)\( \beta_{3}) q^{44} +(\)\(63\!\cdots\!90\)\( + \)\(17\!\cdots\!68\)\( \beta_{1} - \)\(37\!\cdots\!36\)\( \beta_{2} + \)\(15\!\cdots\!92\)\( \beta_{3}) q^{45} +(-\)\(17\!\cdots\!28\)\( + \)\(44\!\cdots\!60\)\( \beta_{1} - \)\(43\!\cdots\!20\)\( \beta_{2} + \)\(34\!\cdots\!40\)\( \beta_{3}) q^{46} +(\)\(17\!\cdots\!20\)\( + \)\(37\!\cdots\!60\)\( \beta_{1} + \)\(13\!\cdots\!16\)\( \beta_{2} - \)\(42\!\cdots\!00\)\( \beta_{3}) q^{47} +(-\)\(49\!\cdots\!20\)\( - \)\(12\!\cdots\!04\)\( \beta_{1} + \)\(18\!\cdots\!16\)\( \beta_{2} - \)\(22\!\cdots\!40\)\( \beta_{3}) q^{48} +(-\)\(53\!\cdots\!07\)\( - \)\(84\!\cdots\!20\)\( \beta_{1} + \)\(95\!\cdots\!20\)\( \beta_{2} - \)\(72\!\cdots\!00\)\( \beta_{3}) q^{49} +(-\)\(16\!\cdots\!50\)\( - \)\(13\!\cdots\!05\)\( \beta_{1} - \)\(24\!\cdots\!40\)\( \beta_{2} + \)\(46\!\cdots\!80\)\( \beta_{3}) q^{50} +(\)\(17\!\cdots\!32\)\( + \)\(27\!\cdots\!58\)\( \beta_{1} - \)\(14\!\cdots\!54\)\( \beta_{2} - \)\(35\!\cdots\!56\)\( \beta_{3}) q^{51} +(\)\(51\!\cdots\!00\)\( + \)\(40\!\cdots\!82\)\( \beta_{1} + \)\(26\!\cdots\!74\)\( \beta_{2} - \)\(13\!\cdots\!50\)\( \beta_{3}) q^{52} +(-\)\(11\!\cdots\!90\)\( - \)\(26\!\cdots\!32\)\( \beta_{1} + \)\(39\!\cdots\!84\)\( \beta_{2} + \)\(28\!\cdots\!80\)\( \beta_{3}) q^{53} +(\)\(73\!\cdots\!20\)\( - \)\(14\!\cdots\!28\)\( \beta_{1} - \)\(13\!\cdots\!36\)\( \beta_{2} - \)\(63\!\cdots\!04\)\( \beta_{3}) q^{54} +(\)\(72\!\cdots\!40\)\( - \)\(25\!\cdots\!62\)\( \beta_{1} + \)\(64\!\cdots\!74\)\( \beta_{2} - \)\(35\!\cdots\!28\)\( \beta_{3}) q^{55} +(-\)\(17\!\cdots\!20\)\( + \)\(64\!\cdots\!68\)\( \beta_{1} + \)\(11\!\cdots\!96\)\( \beta_{2} + \)\(24\!\cdots\!04\)\( \beta_{3}) q^{56} +(-\)\(22\!\cdots\!40\)\( - \)\(38\!\cdots\!76\)\( \beta_{1} - \)\(18\!\cdots\!04\)\( \beta_{2} - \)\(56\!\cdots\!00\)\( \beta_{3}) q^{57} +(-\)\(81\!\cdots\!80\)\( + \)\(11\!\cdots\!58\)\( \beta_{1} + \)\(27\!\cdots\!92\)\( \beta_{2} + \)\(13\!\cdots\!60\)\( \beta_{3}) q^{58} +(\)\(27\!\cdots\!60\)\( - \)\(19\!\cdots\!99\)\( \beta_{1} - \)\(25\!\cdots\!43\)\( \beta_{2} - \)\(22\!\cdots\!12\)\( \beta_{3}) q^{59} +(\)\(13\!\cdots\!20\)\( - \)\(26\!\cdots\!56\)\( \beta_{1} - \)\(51\!\cdots\!88\)\( \beta_{2} - \)\(15\!\cdots\!64\)\( \beta_{3}) q^{60} +(\)\(85\!\cdots\!62\)\( + \)\(22\!\cdots\!00\)\( \beta_{1} + \)\(50\!\cdots\!00\)\( \beta_{2} + \)\(69\!\cdots\!00\)\( \beta_{3}) q^{61} +(\)\(24\!\cdots\!20\)\( + \)\(95\!\cdots\!32\)\( \beta_{1} + \)\(17\!\cdots\!40\)\( \beta_{2} - \)\(50\!\cdots\!60\)\( \beta_{3}) q^{62} +(\)\(54\!\cdots\!40\)\( + \)\(10\!\cdots\!50\)\( \beta_{1} - \)\(10\!\cdots\!38\)\( \beta_{2} + \)\(29\!\cdots\!60\)\( \beta_{3}) q^{63} +(-\)\(85\!\cdots\!32\)\( - \)\(74\!\cdots\!84\)\( \beta_{1} - \)\(44\!\cdots\!68\)\( \beta_{2} - \)\(89\!\cdots\!72\)\( \beta_{3}) q^{64} +(\)\(26\!\cdots\!40\)\( - \)\(23\!\cdots\!12\)\( \beta_{1} + \)\(29\!\cdots\!24\)\( \beta_{2} - \)\(10\!\cdots\!28\)\( \beta_{3}) q^{65} +(-\)\(39\!\cdots\!76\)\( - \)\(30\!\cdots\!76\)\( \beta_{1} + \)\(25\!\cdots\!88\)\( \beta_{2} + \)\(66\!\cdots\!32\)\( \beta_{3}) q^{66} +(\)\(77\!\cdots\!80\)\( + \)\(20\!\cdots\!39\)\( \beta_{1} + \)\(69\!\cdots\!47\)\( \beta_{2} - \)\(47\!\cdots\!40\)\( \beta_{3}) q^{67} +(-\)\(23\!\cdots\!40\)\( + \)\(40\!\cdots\!90\)\( \beta_{1} - \)\(10\!\cdots\!82\)\( \beta_{2} - \)\(68\!\cdots\!30\)\( \beta_{3}) q^{68} +(\)\(39\!\cdots\!44\)\( + \)\(22\!\cdots\!72\)\( \beta_{1} - \)\(51\!\cdots\!56\)\( \beta_{2} + \)\(21\!\cdots\!76\)\( \beta_{3}) q^{69} +(-\)\(31\!\cdots\!80\)\( + \)\(17\!\cdots\!44\)\( \beta_{1} - \)\(76\!\cdots\!88\)\( \beta_{2} + \)\(14\!\cdots\!36\)\( \beta_{3}) q^{70} +(\)\(99\!\cdots\!72\)\( - \)\(24\!\cdots\!50\)\( \beta_{1} - \)\(30\!\cdots\!50\)\( \beta_{2} + \)\(25\!\cdots\!00\)\( \beta_{3}) q^{71} +(\)\(12\!\cdots\!40\)\( - \)\(10\!\cdots\!04\)\( \beta_{1} + \)\(20\!\cdots\!64\)\( \beta_{2} - \)\(87\!\cdots\!20\)\( \beta_{3}) q^{72} +(\)\(27\!\cdots\!30\)\( - \)\(13\!\cdots\!44\)\( \beta_{1} - \)\(13\!\cdots\!44\)\( \beta_{2} - \)\(37\!\cdots\!40\)\( \beta_{3}) q^{73} +(-\)\(61\!\cdots\!64\)\( + \)\(16\!\cdots\!70\)\( \beta_{1} - \)\(30\!\cdots\!20\)\( \beta_{2} + \)\(16\!\cdots\!00\)\( \beta_{3}) q^{74} +(\)\(18\!\cdots\!00\)\( + \)\(12\!\cdots\!65\)\( \beta_{1} + \)\(28\!\cdots\!45\)\( \beta_{2} - \)\(46\!\cdots\!40\)\( \beta_{3}) q^{75} +(-\)\(48\!\cdots\!40\)\( + \)\(15\!\cdots\!16\)\( \beta_{1} + \)\(64\!\cdots\!52\)\( \beta_{2} - \)\(16\!\cdots\!52\)\( \beta_{3}) q^{76} +(\)\(34\!\cdots\!00\)\( - \)\(70\!\cdots\!76\)\( \beta_{1} + \)\(42\!\cdots\!48\)\( \beta_{2} - \)\(16\!\cdots\!40\)\( \beta_{3}) q^{77} +(-\)\(24\!\cdots\!00\)\( - \)\(33\!\cdots\!72\)\( \beta_{1} - \)\(71\!\cdots\!04\)\( \beta_{2} + \)\(48\!\cdots\!20\)\( \beta_{3}) q^{78} +(\)\(10\!\cdots\!80\)\( + \)\(34\!\cdots\!68\)\( \beta_{1} - \)\(25\!\cdots\!64\)\( \beta_{2} + \)\(32\!\cdots\!44\)\( \beta_{3}) q^{79} +(\)\(16\!\cdots\!20\)\( - \)\(57\!\cdots\!36\)\( \beta_{1} + \)\(75\!\cdots\!72\)\( \beta_{2} - \)\(10\!\cdots\!84\)\( \beta_{3}) q^{80} +(\)\(39\!\cdots\!61\)\( + \)\(13\!\cdots\!12\)\( \beta_{1} - \)\(18\!\cdots\!16\)\( \beta_{2} + \)\(15\!\cdots\!56\)\( \beta_{3}) q^{81} +(\)\(66\!\cdots\!20\)\( + \)\(14\!\cdots\!42\)\( \beta_{1} + \)\(34\!\cdots\!40\)\( \beta_{2} + \)\(44\!\cdots\!40\)\( \beta_{3}) q^{82} +(\)\(26\!\cdots\!40\)\( + \)\(51\!\cdots\!75\)\( \beta_{1} + \)\(14\!\cdots\!67\)\( \beta_{2} + \)\(30\!\cdots\!00\)\( \beta_{3}) q^{83} +(-\)\(46\!\cdots\!64\)\( - \)\(18\!\cdots\!52\)\( \beta_{1} - \)\(53\!\cdots\!44\)\( \beta_{2} + \)\(57\!\cdots\!44\)\( \beta_{3}) q^{84} +(-\)\(55\!\cdots\!80\)\( + \)\(50\!\cdots\!84\)\( \beta_{1} - \)\(37\!\cdots\!68\)\( \beta_{2} - \)\(15\!\cdots\!04\)\( \beta_{3}) q^{85} +(-\)\(10\!\cdots\!08\)\( - \)\(19\!\cdots\!32\)\( \beta_{1} + \)\(34\!\cdots\!56\)\( \beta_{2} - \)\(47\!\cdots\!36\)\( \beta_{3}) q^{86} +(-\)\(15\!\cdots\!60\)\( - \)\(27\!\cdots\!14\)\( \beta_{1} + \)\(11\!\cdots\!34\)\( \beta_{2} - \)\(17\!\cdots\!60\)\( \beta_{3}) q^{87} +(-\)\(62\!\cdots\!60\)\( + \)\(10\!\cdots\!16\)\( \beta_{1} - \)\(10\!\cdots\!76\)\( \beta_{2} + \)\(61\!\cdots\!60\)\( \beta_{3}) q^{88} +(-\)\(22\!\cdots\!10\)\( - \)\(25\!\cdots\!36\)\( \beta_{1} - \)\(11\!\cdots\!92\)\( \beta_{2} + \)\(60\!\cdots\!92\)\( \beta_{3}) q^{89} +(\)\(79\!\cdots\!40\)\( - \)\(98\!\cdots\!82\)\( \beta_{1} - \)\(16\!\cdots\!36\)\( \beta_{2} - \)\(17\!\cdots\!08\)\( \beta_{3}) q^{90} +(\)\(25\!\cdots\!72\)\( + \)\(20\!\cdots\!08\)\( \beta_{1} + \)\(24\!\cdots\!76\)\( \beta_{2} - \)\(79\!\cdots\!76\)\( \beta_{3}) q^{91} +(\)\(48\!\cdots\!80\)\( - \)\(10\!\cdots\!96\)\( \beta_{1} - \)\(33\!\cdots\!88\)\( \beta_{2} + \)\(20\!\cdots\!40\)\( \beta_{3}) q^{92} +(-\)\(77\!\cdots\!80\)\( - \)\(12\!\cdots\!64\)\( \beta_{1} - \)\(83\!\cdots\!52\)\( \beta_{2} + \)\(23\!\cdots\!80\)\( \beta_{3}) q^{93} +(\)\(15\!\cdots\!96\)\( + \)\(43\!\cdots\!12\)\( \beta_{1} + \)\(37\!\cdots\!44\)\( \beta_{2} - \)\(31\!\cdots\!84\)\( \beta_{3}) q^{94} +(-\)\(10\!\cdots\!00\)\( + \)\(21\!\cdots\!30\)\( \beta_{1} - \)\(20\!\cdots\!10\)\( \beta_{2} - \)\(25\!\cdots\!80\)\( \beta_{3}) q^{95} +(-\)\(61\!\cdots\!08\)\( + \)\(13\!\cdots\!92\)\( \beta_{1} + \)\(72\!\cdots\!64\)\( \beta_{2} + \)\(26\!\cdots\!16\)\( \beta_{3}) q^{96} +(-\)\(33\!\cdots\!30\)\( + \)\(19\!\cdots\!20\)\( \beta_{1} - \)\(31\!\cdots\!24\)\( \beta_{2} + \)\(34\!\cdots\!20\)\( \beta_{3}) q^{97} +(-\)\(79\!\cdots\!70\)\( - \)\(38\!\cdots\!27\)\( \beta_{1} + \)\(18\!\cdots\!60\)\( \beta_{2} + \)\(20\!\cdots\!40\)\( \beta_{3}) q^{98} +(-\)\(42\!\cdots\!16\)\( + \)\(44\!\cdots\!77\)\( \beta_{1} + \)\(69\!\cdots\!69\)\( \beta_{2} + \)\(90\!\cdots\!56\)\( \beta_{3}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 32756040q^{2} + 403863773040q^{3} + 7978103470875712q^{4} + 1214113112967557880q^{5} - \)\(10\!\cdots\!92\)\(q^{6} + \)\(65\!\cdots\!00\)\(q^{7} - \)\(13\!\cdots\!20\)\(q^{8} + \)\(50\!\cdots\!28\)\(q^{9} + O(q^{10}) \) \( 4q + 32756040q^{2} + 403863773040q^{3} + 7978103470875712q^{4} + 1214113112967557880q^{5} - \)\(10\!\cdots\!92\)\(q^{6} + \)\(65\!\cdots\!00\)\(q^{7} - \)\(13\!\cdots\!20\)\(q^{8} + \)\(50\!\cdots\!28\)\(q^{9} - \)\(26\!\cdots\!20\)\(q^{10} + \)\(35\!\cdots\!48\)\(q^{11} - \)\(49\!\cdots\!20\)\(q^{12} + \)\(30\!\cdots\!80\)\(q^{13} + \)\(11\!\cdots\!24\)\(q^{14} + \)\(14\!\cdots\!60\)\(q^{15} + \)\(13\!\cdots\!44\)\(q^{16} + \)\(48\!\cdots\!20\)\(q^{17} + \)\(73\!\cdots\!40\)\(q^{18} + \)\(81\!\cdots\!80\)\(q^{19} + \)\(66\!\cdots\!40\)\(q^{20} + \)\(31\!\cdots\!48\)\(q^{21} - \)\(34\!\cdots\!20\)\(q^{22} - \)\(54\!\cdots\!80\)\(q^{23} - \)\(87\!\cdots\!60\)\(q^{24} - \)\(57\!\cdots\!00\)\(q^{25} + \)\(15\!\cdots\!48\)\(q^{26} + \)\(36\!\cdots\!20\)\(q^{27} + \)\(34\!\cdots\!80\)\(q^{28} + \)\(24\!\cdots\!20\)\(q^{29} - \)\(32\!\cdots\!40\)\(q^{30} - \)\(74\!\cdots\!72\)\(q^{31} - \)\(69\!\cdots\!60\)\(q^{32} - \)\(17\!\cdots\!20\)\(q^{33} + \)\(59\!\cdots\!64\)\(q^{34} + \)\(54\!\cdots\!80\)\(q^{35} + \)\(12\!\cdots\!84\)\(q^{36} + \)\(92\!\cdots\!60\)\(q^{37} + \)\(42\!\cdots\!20\)\(q^{38} - \)\(11\!\cdots\!04\)\(q^{39} - \)\(26\!\cdots\!00\)\(q^{40} + \)\(14\!\cdots\!68\)\(q^{41} - \)\(39\!\cdots\!80\)\(q^{42} - \)\(38\!\cdots\!00\)\(q^{43} + \)\(39\!\cdots\!44\)\(q^{44} + \)\(25\!\cdots\!60\)\(q^{45} - \)\(68\!\cdots\!12\)\(q^{46} + \)\(70\!\cdots\!80\)\(q^{47} - \)\(19\!\cdots\!80\)\(q^{48} - \)\(21\!\cdots\!28\)\(q^{49} - \)\(65\!\cdots\!00\)\(q^{50} + \)\(69\!\cdots\!28\)\(q^{51} + \)\(20\!\cdots\!00\)\(q^{52} - \)\(46\!\cdots\!60\)\(q^{53} + \)\(29\!\cdots\!80\)\(q^{54} + \)\(28\!\cdots\!60\)\(q^{55} - \)\(71\!\cdots\!80\)\(q^{56} - \)\(88\!\cdots\!60\)\(q^{57} - \)\(32\!\cdots\!20\)\(q^{58} + \)\(11\!\cdots\!40\)\(q^{59} + \)\(55\!\cdots\!80\)\(q^{60} + \)\(34\!\cdots\!48\)\(q^{61} + \)\(96\!\cdots\!80\)\(q^{62} + \)\(21\!\cdots\!60\)\(q^{63} - \)\(34\!\cdots\!28\)\(q^{64} + \)\(10\!\cdots\!60\)\(q^{65} - \)\(15\!\cdots\!04\)\(q^{66} + \)\(30\!\cdots\!20\)\(q^{67} - \)\(92\!\cdots\!60\)\(q^{68} + \)\(15\!\cdots\!76\)\(q^{69} - \)\(12\!\cdots\!20\)\(q^{70} + \)\(39\!\cdots\!88\)\(q^{71} + \)\(50\!\cdots\!60\)\(q^{72} + \)\(10\!\cdots\!20\)\(q^{73} - \)\(24\!\cdots\!56\)\(q^{74} + \)\(72\!\cdots\!00\)\(q^{75} - \)\(19\!\cdots\!60\)\(q^{76} + \)\(13\!\cdots\!00\)\(q^{77} - \)\(96\!\cdots\!00\)\(q^{78} + \)\(40\!\cdots\!20\)\(q^{79} + \)\(65\!\cdots\!80\)\(q^{80} + \)\(15\!\cdots\!44\)\(q^{81} + \)\(26\!\cdots\!80\)\(q^{82} + \)\(10\!\cdots\!60\)\(q^{83} - \)\(18\!\cdots\!56\)\(q^{84} - \)\(22\!\cdots\!20\)\(q^{85} - \)\(43\!\cdots\!32\)\(q^{86} - \)\(62\!\cdots\!40\)\(q^{87} - \)\(25\!\cdots\!40\)\(q^{88} - \)\(90\!\cdots\!40\)\(q^{89} + \)\(31\!\cdots\!60\)\(q^{90} + \)\(10\!\cdots\!88\)\(q^{91} + \)\(19\!\cdots\!20\)\(q^{92} - \)\(31\!\cdots\!20\)\(q^{93} + \)\(63\!\cdots\!84\)\(q^{94} - \)\(41\!\cdots\!00\)\(q^{95} - \)\(24\!\cdots\!32\)\(q^{96} - \)\(13\!\cdots\!20\)\(q^{97} - \)\(31\!\cdots\!80\)\(q^{98} - \)\(17\!\cdots\!64\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 495735060514 x^{2} - 23954614981416598 x + 48979992255622025570313\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -252006 \nu^{3} + 93063836718 \nu^{2} + 70872137420108946 \nu - 18539821380123502730388 \)\()/ 77619756584545 \)
\(\beta_{2}\)\(=\)\((\)\( 3755577054 \nu^{3} - 2376866094373062 \nu^{2} - 745977194049162568314 \nu + 521672957126789442477814692 \)\()/ 77619756584545 \)
\(\beta_{3}\)\(=\)\((\)\(-334830962412 \nu^{3} + 368681755057133436 \nu^{2} + 113767409889319937410692 \nu - 85368483888402915793673001576\)\()/ 11088536654935 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(31 \beta_{3} + 53711 \beta_{2} + 512120665 \beta_{1} + 134730086400\)\()/ 269460172800 \)
\(\nu^{2}\)\(=\)\((\)\(4857031 \beta_{3} - 2148411769 \beta_{2} - 77190687886655 \beta_{1} + 33395213767414954291200\)\()/ 134730086400 \)
\(\nu^{3}\)\(=\)\((\)\(251133001843 \beta_{3} + 275887056853763 \beta_{2} + 81981363281812165 \beta_{1} + 98802273274899860911564800\)\()/ 5499187200 \)

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
−457245.
644100.
−511801.
324949.
−8.71140e7 6.49653e11 5.33705e15 8.70295e17 −5.65939e19 3.14283e21 −2.68769e23 −1.73165e24 −7.58149e25
1.2 −1.27049e7 −5.15198e11 −2.09038e15 −3.83602e17 6.54557e18 −2.02239e21 5.51672e22 −1.88826e24 4.87365e24
1.3 5.13381e7 2.68083e12 3.83803e14 4.84667e17 1.37629e20 2.59004e21 −9.58994e22 5.03313e24 2.48819e25
1.4 8.12369e7 −2.41142e12 4.34763e15 2.42754e17 −1.95896e20 2.85586e21 1.70259e23 3.66124e24 1.97206e25
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1.52.a.a 4
3.b odd 2 1 9.52.a.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.52.a.a 4 1.a even 1 1 trivial
9.52.a.b 4 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{52}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 32756040 T + 1051026970173440 T^{2} + \)\(48\!\cdots\!80\)\( T^{3} - \)\(79\!\cdots\!92\)\( T^{4} + \)\(10\!\cdots\!40\)\( T^{5} + \)\(53\!\cdots\!60\)\( T^{6} - \)\(37\!\cdots\!80\)\( T^{7} + \)\(25\!\cdots\!16\)\( T^{8} \)
$3$ \( 1 - 403863773040 T + \)\(18\!\cdots\!80\)\( T^{2} - \)\(16\!\cdots\!80\)\( T^{3} + \)\(86\!\cdots\!18\)\( T^{4} - \)\(35\!\cdots\!60\)\( T^{5} + \)\(85\!\cdots\!20\)\( T^{6} - \)\(40\!\cdots\!20\)\( T^{7} + \)\(21\!\cdots\!81\)\( T^{8} \)
$5$ \( 1 - 1214113112967557880 T + \)\(19\!\cdots\!00\)\( T^{2} - \)\(14\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!50\)\( T^{4} - \)\(63\!\cdots\!00\)\( T^{5} + \)\(37\!\cdots\!00\)\( T^{6} - \)\(10\!\cdots\!00\)\( T^{7} + \)\(38\!\cdots\!25\)\( T^{8} \)
$7$ \( 1 - \)\(65\!\cdots\!00\)\( T + \)\(57\!\cdots\!00\)\( T^{2} - \)\(22\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!98\)\( T^{4} - \)\(27\!\cdots\!00\)\( T^{5} + \)\(91\!\cdots\!00\)\( T^{6} - \)\(13\!\cdots\!00\)\( T^{7} + \)\(25\!\cdots\!01\)\( T^{8} \)
$11$ \( 1 - \)\(35\!\cdots\!48\)\( T + \)\(30\!\cdots\!08\)\( T^{2} - \)\(67\!\cdots\!96\)\( T^{3} + \)\(47\!\cdots\!70\)\( T^{4} - \)\(87\!\cdots\!56\)\( T^{5} + \)\(51\!\cdots\!68\)\( T^{6} - \)\(75\!\cdots\!88\)\( T^{7} + \)\(27\!\cdots\!41\)\( T^{8} \)
$13$ \( 1 - \)\(30\!\cdots\!80\)\( T + \)\(22\!\cdots\!60\)\( T^{2} - \)\(55\!\cdots\!60\)\( T^{3} + \)\(20\!\cdots\!38\)\( T^{4} - \)\(35\!\cdots\!20\)\( T^{5} + \)\(94\!\cdots\!40\)\( T^{6} - \)\(83\!\cdots\!40\)\( T^{7} + \)\(17\!\cdots\!61\)\( T^{8} \)
$17$ \( 1 - \)\(48\!\cdots\!20\)\( T + \)\(25\!\cdots\!20\)\( T^{2} - \)\(75\!\cdots\!60\)\( T^{3} + \)\(23\!\cdots\!78\)\( T^{4} - \)\(42\!\cdots\!80\)\( T^{5} + \)\(83\!\cdots\!80\)\( T^{6} - \)\(87\!\cdots\!40\)\( T^{7} + \)\(10\!\cdots\!21\)\( T^{8} \)
$19$ \( 1 - \)\(81\!\cdots\!80\)\( T + \)\(63\!\cdots\!76\)\( T^{2} - \)\(33\!\cdots\!60\)\( T^{3} + \)\(15\!\cdots\!66\)\( T^{4} - \)\(55\!\cdots\!40\)\( T^{5} + \)\(17\!\cdots\!36\)\( T^{6} - \)\(36\!\cdots\!20\)\( T^{7} + \)\(73\!\cdots\!21\)\( T^{8} \)
$23$ \( 1 + \)\(54\!\cdots\!80\)\( T + \)\(90\!\cdots\!40\)\( T^{2} + \)\(44\!\cdots\!60\)\( T^{3} + \)\(35\!\cdots\!58\)\( T^{4} + \)\(12\!\cdots\!20\)\( T^{5} + \)\(71\!\cdots\!60\)\( T^{6} + \)\(12\!\cdots\!40\)\( T^{7} + \)\(62\!\cdots\!41\)\( T^{8} \)
$29$ \( 1 - \)\(24\!\cdots\!20\)\( T + \)\(10\!\cdots\!16\)\( T^{2} - \)\(15\!\cdots\!40\)\( T^{3} + \)\(49\!\cdots\!46\)\( T^{4} - \)\(60\!\cdots\!60\)\( T^{5} + \)\(15\!\cdots\!56\)\( T^{6} - \)\(13\!\cdots\!80\)\( T^{7} + \)\(21\!\cdots\!81\)\( T^{8} \)
$31$ \( 1 + \)\(74\!\cdots\!72\)\( T + \)\(38\!\cdots\!68\)\( T^{2} + \)\(18\!\cdots\!24\)\( T^{3} + \)\(60\!\cdots\!70\)\( T^{4} + \)\(21\!\cdots\!44\)\( T^{5} + \)\(50\!\cdots\!48\)\( T^{6} + \)\(11\!\cdots\!52\)\( T^{7} + \)\(17\!\cdots\!21\)\( T^{8} \)
$37$ \( 1 - \)\(92\!\cdots\!60\)\( T + \)\(13\!\cdots\!60\)\( T^{2} + \)\(58\!\cdots\!20\)\( T^{3} - \)\(26\!\cdots\!62\)\( T^{4} + \)\(55\!\cdots\!60\)\( T^{5} + \)\(11\!\cdots\!40\)\( T^{6} - \)\(79\!\cdots\!20\)\( T^{7} + \)\(81\!\cdots\!61\)\( T^{8} \)
$41$ \( 1 - \)\(14\!\cdots\!68\)\( T + \)\(29\!\cdots\!48\)\( T^{2} - \)\(34\!\cdots\!16\)\( T^{3} + \)\(74\!\cdots\!70\)\( T^{4} - \)\(62\!\cdots\!56\)\( T^{5} + \)\(92\!\cdots\!88\)\( T^{6} - \)\(83\!\cdots\!28\)\( T^{7} + \)\(10\!\cdots\!61\)\( T^{8} \)
$43$ \( 1 + \)\(38\!\cdots\!00\)\( T + \)\(56\!\cdots\!00\)\( T^{2} + \)\(14\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!98\)\( T^{4} + \)\(28\!\cdots\!00\)\( T^{5} + \)\(23\!\cdots\!00\)\( T^{6} + \)\(31\!\cdots\!00\)\( T^{7} + \)\(16\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 - \)\(70\!\cdots\!80\)\( T + \)\(60\!\cdots\!80\)\( T^{2} - \)\(43\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!18\)\( T^{4} - \)\(81\!\cdots\!20\)\( T^{5} + \)\(21\!\cdots\!20\)\( T^{6} - \)\(47\!\cdots\!60\)\( T^{7} + \)\(12\!\cdots\!81\)\( T^{8} \)
$53$ \( 1 + \)\(46\!\cdots\!60\)\( T + \)\(11\!\cdots\!80\)\( T^{2} - \)\(14\!\cdots\!80\)\( T^{3} + \)\(49\!\cdots\!18\)\( T^{4} - \)\(12\!\cdots\!60\)\( T^{5} + \)\(89\!\cdots\!20\)\( T^{6} + \)\(30\!\cdots\!80\)\( T^{7} + \)\(56\!\cdots\!81\)\( T^{8} \)
$59$ \( 1 - \)\(11\!\cdots\!40\)\( T + \)\(43\!\cdots\!36\)\( T^{2} - \)\(78\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!86\)\( T^{4} - \)\(16\!\cdots\!20\)\( T^{5} + \)\(18\!\cdots\!16\)\( T^{6} - \)\(97\!\cdots\!60\)\( T^{7} + \)\(17\!\cdots\!61\)\( T^{8} \)
$61$ \( 1 - \)\(34\!\cdots\!48\)\( T + \)\(36\!\cdots\!08\)\( T^{2} - \)\(87\!\cdots\!96\)\( T^{3} + \)\(55\!\cdots\!70\)\( T^{4} - \)\(98\!\cdots\!56\)\( T^{5} + \)\(45\!\cdots\!68\)\( T^{6} - \)\(49\!\cdots\!88\)\( T^{7} + \)\(16\!\cdots\!41\)\( T^{8} \)
$67$ \( 1 - \)\(30\!\cdots\!20\)\( T + \)\(47\!\cdots\!20\)\( T^{2} - \)\(12\!\cdots\!60\)\( T^{3} + \)\(91\!\cdots\!78\)\( T^{4} - \)\(16\!\cdots\!80\)\( T^{5} + \)\(86\!\cdots\!80\)\( T^{6} - \)\(75\!\cdots\!40\)\( T^{7} + \)\(33\!\cdots\!21\)\( T^{8} \)
$71$ \( 1 - \)\(39\!\cdots\!88\)\( T + \)\(95\!\cdots\!88\)\( T^{2} - \)\(15\!\cdots\!36\)\( T^{3} + \)\(26\!\cdots\!70\)\( T^{4} - \)\(41\!\cdots\!56\)\( T^{5} + \)\(64\!\cdots\!08\)\( T^{6} - \)\(69\!\cdots\!68\)\( T^{7} + \)\(45\!\cdots\!81\)\( T^{8} \)
$73$ \( 1 - \)\(10\!\cdots\!20\)\( T + \)\(74\!\cdots\!40\)\( T^{2} - \)\(35\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!58\)\( T^{4} - \)\(38\!\cdots\!80\)\( T^{5} + \)\(85\!\cdots\!60\)\( T^{6} - \)\(13\!\cdots\!60\)\( T^{7} + \)\(13\!\cdots\!41\)\( T^{8} \)
$79$ \( 1 - \)\(40\!\cdots\!20\)\( T + \)\(18\!\cdots\!16\)\( T^{2} - \)\(34\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!46\)\( T^{4} - \)\(20\!\cdots\!60\)\( T^{5} + \)\(66\!\cdots\!56\)\( T^{6} - \)\(88\!\cdots\!80\)\( T^{7} + \)\(13\!\cdots\!81\)\( T^{8} \)
$83$ \( 1 - \)\(10\!\cdots\!60\)\( T + \)\(32\!\cdots\!20\)\( T^{2} - \)\(23\!\cdots\!20\)\( T^{3} + \)\(37\!\cdots\!78\)\( T^{4} - \)\(17\!\cdots\!40\)\( T^{5} + \)\(18\!\cdots\!80\)\( T^{6} - \)\(43\!\cdots\!80\)\( T^{7} + \)\(31\!\cdots\!21\)\( T^{8} \)
$89$ \( 1 + \)\(90\!\cdots\!40\)\( T + \)\(11\!\cdots\!56\)\( T^{2} + \)\(71\!\cdots\!80\)\( T^{3} + \)\(47\!\cdots\!26\)\( T^{4} + \)\(18\!\cdots\!20\)\( T^{5} + \)\(80\!\cdots\!76\)\( T^{6} + \)\(16\!\cdots\!60\)\( T^{7} + \)\(47\!\cdots\!41\)\( T^{8} \)
$97$ \( 1 + \)\(13\!\cdots\!20\)\( T + \)\(81\!\cdots\!80\)\( T^{2} + \)\(24\!\cdots\!60\)\( T^{3} + \)\(68\!\cdots\!18\)\( T^{4} + \)\(52\!\cdots\!80\)\( T^{5} + \)\(36\!\cdots\!20\)\( T^{6} + \)\(12\!\cdots\!40\)\( T^{7} + \)\(20\!\cdots\!81\)\( T^{8} \)
show more
show less