Properties

Label 1.62.a.a
Level 1
Weight 62
Character orbit 1.a
Self dual yes
Analytic conductor 23.566
Analytic rank 1
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 62 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.5656183265\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{28}\cdot 3^{8}\cdot 5^{2}\cdot 7 \)
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.

\(f(q)\) \(=\) \( q +(286578000 - \beta_{1}) q^{2} +(-143430899755500 + 48580 \beta_{1} + \beta_{2}) q^{3} +(1100749418957151232 - 838426474 \beta_{1} + 343 \beta_{2} + \beta_{3}) q^{4} +(-\)\(13\!\cdots\!50\)\( + 100252966240 \beta_{1} - 48540 \beta_{2} - 120 \beta_{3}) q^{5} +(-\)\(20\!\cdots\!88\)\( + 157027925895036 \beta_{1} - 109667352 \beta_{2} - 189864 \beta_{3}) q^{6} +(-\)\(15\!\cdots\!00\)\( + 14513866966047176 \beta_{1} - 46993043006 \beta_{2} - 19628000 \beta_{3}) q^{7} +(\)\(24\!\cdots\!00\)\( - 1373007096556182656 \beta_{1} - 7496719202112 \beta_{2} + 1190856000 \beta_{3}) q^{8} +(\)\(29\!\cdots\!13\)\( + 1326939277665425472 \beta_{1} - 362685230082504 \beta_{2} - 19284989328 \beta_{3}) q^{9} +O(q^{10})\) \( q +(286578000 - \beta_{1}) q^{2} +(-143430899755500 + 48580 \beta_{1} + \beta_{2}) q^{3} +(1100749418957151232 - 838426474 \beta_{1} + 343 \beta_{2} + \beta_{3}) q^{4} +(-\)\(13\!\cdots\!50\)\( + 100252966240 \beta_{1} - 48540 \beta_{2} - 120 \beta_{3}) q^{5} +(-\)\(20\!\cdots\!88\)\( + 157027925895036 \beta_{1} - 109667352 \beta_{2} - 189864 \beta_{3}) q^{6} +(-\)\(15\!\cdots\!00\)\( + 14513866966047176 \beta_{1} - 46993043006 \beta_{2} - 19628000 \beta_{3}) q^{7} +(\)\(24\!\cdots\!00\)\( - 1373007096556182656 \beta_{1} - 7496719202112 \beta_{2} + 1190856000 \beta_{3}) q^{8} +(\)\(29\!\cdots\!13\)\( + 1326939277665425472 \beta_{1} - 362685230082504 \beta_{2} - 19284989328 \beta_{3}) q^{9} +(-\)\(37\!\cdots\!00\)\( + \)\(44\!\cdots\!30\)\( \beta_{1} + 900415543063520 \beta_{2} - 141501833440 \beta_{3}) q^{10} +(-\)\(10\!\cdots\!88\)\( + \)\(12\!\cdots\!80\)\( \beta_{1} + 130667557581888915 \beta_{2} + 11035029161280 \beta_{3}) q^{11} +(-\)\(24\!\cdots\!00\)\( + \)\(57\!\cdots\!92\)\( \beta_{1} - 877602562285349684 \beta_{2} - 217648250923500 \beta_{3}) q^{12} +(\)\(26\!\cdots\!50\)\( + \)\(39\!\cdots\!32\)\( \beta_{1} - 12482522028380172860 \beta_{2} + 2435574015901000 \beta_{3}) q^{13} +(-\)\(52\!\cdots\!76\)\( + \)\(65\!\cdots\!28\)\( \beta_{1} + \)\(15\!\cdots\!04\)\( \beta_{2} - 15743169581051472 \beta_{3}) q^{14} +(\)\(28\!\cdots\!00\)\( - \)\(69\!\cdots\!60\)\( \beta_{1} + \)\(10\!\cdots\!10\)\( \beta_{2} + 26260172144333280 \beta_{3}) q^{15} +(\)\(27\!\cdots\!56\)\( - \)\(36\!\cdots\!88\)\( \beta_{1} - \)\(88\!\cdots\!84\)\( \beta_{2} + 603732751100608512 \beta_{3}) q^{16} +(-\)\(10\!\cdots\!50\)\( - \)\(38\!\cdots\!60\)\( \beta_{1} + \)\(33\!\cdots\!52\)\( \beta_{2} - 7263830520990282000 \beta_{3}) q^{17} +(\)\(39\!\cdots\!00\)\( + \)\(17\!\cdots\!15\)\( \beta_{1} + \)\(18\!\cdots\!36\)\( \beta_{2} + 42183522593278488000 \beta_{3}) q^{18} +(-\)\(89\!\cdots\!80\)\( + \)\(48\!\cdots\!64\)\( \beta_{1} - \)\(16\!\cdots\!23\)\( \beta_{2} - \)\(10\!\cdots\!36\)\( \beta_{3}) q^{19} +(-\)\(12\!\cdots\!00\)\( + \)\(66\!\cdots\!80\)\( \beta_{1} + \)\(97\!\cdots\!70\)\( \beta_{2} - \)\(34\!\cdots\!90\)\( \beta_{3}) q^{20} +(-\)\(13\!\cdots\!68\)\( - \)\(11\!\cdots\!48\)\( \beta_{1} + \)\(31\!\cdots\!36\)\( \beta_{2} + \)\(50\!\cdots\!52\)\( \beta_{3}) q^{21} +(-\)\(43\!\cdots\!00\)\( - \)\(79\!\cdots\!92\)\( \beta_{1} - \)\(10\!\cdots\!60\)\( \beta_{2} - \)\(26\!\cdots\!00\)\( \beta_{3}) q^{22} +(-\)\(10\!\cdots\!00\)\( + \)\(24\!\cdots\!24\)\( \beta_{1} - \)\(24\!\cdots\!66\)\( \beta_{2} + \)\(75\!\cdots\!00\)\( \beta_{3}) q^{23} +(-\)\(15\!\cdots\!40\)\( + \)\(67\!\cdots\!92\)\( \beta_{1} + \)\(18\!\cdots\!56\)\( \beta_{2} - \)\(10\!\cdots\!08\)\( \beta_{3}) q^{24} +(-\)\(41\!\cdots\!25\)\( - \)\(79\!\cdots\!00\)\( \beta_{1} - \)\(11\!\cdots\!00\)\( \beta_{2} + \)\(41\!\cdots\!00\)\( \beta_{3}) q^{25} +(-\)\(12\!\cdots\!08\)\( - \)\(54\!\cdots\!42\)\( \beta_{1} - \)\(19\!\cdots\!56\)\( \beta_{2} - \)\(11\!\cdots\!92\)\( \beta_{3}) q^{26} +(-\)\(32\!\cdots\!00\)\( - \)\(18\!\cdots\!96\)\( \beta_{1} + \)\(19\!\cdots\!58\)\( \beta_{2} + \)\(11\!\cdots\!00\)\( \beta_{3}) q^{27} +(-\)\(19\!\cdots\!00\)\( + \)\(86\!\cdots\!96\)\( \beta_{1} + \)\(19\!\cdots\!76\)\( \beta_{2} - \)\(48\!\cdots\!00\)\( \beta_{3}) q^{28} +(\)\(16\!\cdots\!30\)\( + \)\(32\!\cdots\!76\)\( \beta_{1} - \)\(54\!\cdots\!32\)\( \beta_{2} + \)\(93\!\cdots\!76\)\( \beta_{3}) q^{29} +(\)\(23\!\cdots\!00\)\( - \)\(12\!\cdots\!20\)\( \beta_{1} - \)\(18\!\cdots\!80\)\( \beta_{2} + \)\(64\!\cdots\!60\)\( \beta_{3}) q^{30} +(\)\(81\!\cdots\!32\)\( - \)\(32\!\cdots\!60\)\( \beta_{1} + \)\(47\!\cdots\!20\)\( \beta_{2} - \)\(92\!\cdots\!60\)\( \beta_{3}) q^{31} +(\)\(73\!\cdots\!00\)\( - \)\(27\!\cdots\!56\)\( \beta_{1} + \)\(14\!\cdots\!80\)\( \beta_{2} + \)\(24\!\cdots\!00\)\( \beta_{3}) q^{32} +(\)\(20\!\cdots\!00\)\( + \)\(41\!\cdots\!20\)\( \beta_{1} - \)\(54\!\cdots\!68\)\( \beta_{2} - \)\(19\!\cdots\!00\)\( \beta_{3}) q^{33} +(\)\(99\!\cdots\!04\)\( + \)\(22\!\cdots\!42\)\( \beta_{1} + \)\(54\!\cdots\!56\)\( \beta_{2} - \)\(38\!\cdots\!08\)\( \beta_{3}) q^{34} +(\)\(23\!\cdots\!00\)\( - \)\(10\!\cdots\!20\)\( \beta_{1} - \)\(23\!\cdots\!80\)\( \beta_{2} + \)\(57\!\cdots\!60\)\( \beta_{3}) q^{35} +(-\)\(12\!\cdots\!84\)\( - \)\(89\!\cdots\!58\)\( \beta_{1} + \)\(48\!\cdots\!31\)\( \beta_{2} + \)\(18\!\cdots\!17\)\( \beta_{3}) q^{36} +(-\)\(17\!\cdots\!50\)\( - \)\(22\!\cdots\!72\)\( \beta_{1} - \)\(16\!\cdots\!12\)\( \beta_{2} - \)\(11\!\cdots\!00\)\( \beta_{3}) q^{37} +(-\)\(16\!\cdots\!00\)\( + \)\(61\!\cdots\!16\)\( \beta_{1} + \)\(83\!\cdots\!32\)\( \beta_{2} - \)\(29\!\cdots\!00\)\( \beta_{3}) q^{38} +(-\)\(13\!\cdots\!44\)\( + \)\(35\!\cdots\!28\)\( \beta_{1} + \)\(34\!\cdots\!54\)\( \beta_{2} + \)\(98\!\cdots\!28\)\( \beta_{3}) q^{39} +(-\)\(17\!\cdots\!00\)\( + \)\(13\!\cdots\!00\)\( \beta_{1} + \)\(29\!\cdots\!00\)\( \beta_{2} - \)\(62\!\cdots\!00\)\( \beta_{3}) q^{40} +(\)\(40\!\cdots\!42\)\( - \)\(27\!\cdots\!60\)\( \beta_{1} - \)\(10\!\cdots\!80\)\( \beta_{2} - \)\(34\!\cdots\!60\)\( \beta_{3}) q^{41} +(\)\(37\!\cdots\!00\)\( - \)\(15\!\cdots\!84\)\( \beta_{1} - \)\(38\!\cdots\!24\)\( \beta_{2} + \)\(89\!\cdots\!00\)\( \beta_{3}) q^{42} +(\)\(18\!\cdots\!00\)\( + \)\(70\!\cdots\!68\)\( \beta_{1} + \)\(15\!\cdots\!67\)\( \beta_{2} - \)\(33\!\cdots\!00\)\( \beta_{3}) q^{43} +(\)\(38\!\cdots\!84\)\( + \)\(67\!\cdots\!72\)\( \beta_{1} - \)\(86\!\cdots\!04\)\( \beta_{2} - \)\(13\!\cdots\!28\)\( \beta_{3}) q^{44} +(\)\(15\!\cdots\!50\)\( + \)\(10\!\cdots\!20\)\( \beta_{1} - \)\(59\!\cdots\!20\)\( \beta_{2} - \)\(23\!\cdots\!60\)\( \beta_{3}) q^{45} +(-\)\(37\!\cdots\!28\)\( - \)\(52\!\cdots\!80\)\( \beta_{1} - \)\(56\!\cdots\!40\)\( \beta_{2} + \)\(62\!\cdots\!20\)\( \beta_{3}) q^{46} +(-\)\(54\!\cdots\!00\)\( - \)\(53\!\cdots\!48\)\( \beta_{1} + \)\(90\!\cdots\!16\)\( \beta_{2} + \)\(28\!\cdots\!00\)\( \beta_{3}) q^{47} +(-\)\(21\!\cdots\!00\)\( + \)\(76\!\cdots\!64\)\( \beta_{1} + \)\(24\!\cdots\!64\)\( \beta_{2} - \)\(47\!\cdots\!00\)\( \beta_{3}) q^{48} +(\)\(38\!\cdots\!57\)\( - \)\(12\!\cdots\!80\)\( \beta_{1} - \)\(50\!\cdots\!40\)\( \beta_{2} + \)\(74\!\cdots\!20\)\( \beta_{3}) q^{49} +(-\)\(93\!\cdots\!00\)\( + \)\(40\!\cdots\!25\)\( \beta_{1} - \)\(28\!\cdots\!00\)\( \beta_{2} + \)\(11\!\cdots\!00\)\( \beta_{3}) q^{50} +(\)\(51\!\cdots\!72\)\( - \)\(23\!\cdots\!36\)\( \beta_{1} - \)\(84\!\cdots\!98\)\( \beta_{2} - \)\(13\!\cdots\!36\)\( \beta_{3}) q^{51} +(\)\(85\!\cdots\!00\)\( + \)\(29\!\cdots\!36\)\( \beta_{1} + \)\(41\!\cdots\!74\)\( \beta_{2} + \)\(20\!\cdots\!50\)\( \beta_{3}) q^{52} +(\)\(21\!\cdots\!50\)\( - \)\(18\!\cdots\!80\)\( \beta_{1} - \)\(98\!\cdots\!44\)\( \beta_{2} - \)\(81\!\cdots\!00\)\( \beta_{3}) q^{53} +(\)\(51\!\cdots\!20\)\( - \)\(25\!\cdots\!76\)\( \beta_{1} - \)\(84\!\cdots\!68\)\( \beta_{2} + \)\(20\!\cdots\!24\)\( \beta_{3}) q^{54} +(-\)\(47\!\cdots\!00\)\( - \)\(80\!\cdots\!20\)\( \beta_{1} + \)\(10\!\cdots\!70\)\( \beta_{2} + \)\(16\!\cdots\!60\)\( \beta_{3}) q^{55} +(-\)\(22\!\cdots\!80\)\( + \)\(19\!\cdots\!64\)\( \beta_{1} - \)\(22\!\cdots\!48\)\( \beta_{2} - \)\(97\!\cdots\!36\)\( \beta_{3}) q^{56} +(-\)\(12\!\cdots\!00\)\( - \)\(14\!\cdots\!52\)\( \beta_{1} + \)\(67\!\cdots\!96\)\( \beta_{2} + \)\(14\!\cdots\!00\)\( \beta_{3}) q^{57} +(-\)\(10\!\cdots\!00\)\( - \)\(17\!\cdots\!06\)\( \beta_{1} - \)\(68\!\cdots\!12\)\( \beta_{2} + \)\(82\!\cdots\!00\)\( \beta_{3}) q^{58} +(-\)\(53\!\cdots\!40\)\( - \)\(31\!\cdots\!88\)\( \beta_{1} - \)\(66\!\cdots\!09\)\( \beta_{2} - \)\(38\!\cdots\!88\)\( \beta_{3}) q^{59} +(\)\(41\!\cdots\!00\)\( - \)\(28\!\cdots\!20\)\( \beta_{1} - \)\(69\!\cdots\!80\)\( \beta_{2} + \)\(11\!\cdots\!60\)\( \beta_{3}) q^{60} +(\)\(10\!\cdots\!62\)\( + \)\(63\!\cdots\!00\)\( \beta_{1} + \)\(25\!\cdots\!00\)\( \beta_{2} + \)\(50\!\cdots\!00\)\( \beta_{3}) q^{61} +(\)\(12\!\cdots\!00\)\( + \)\(94\!\cdots\!28\)\( \beta_{1} + \)\(72\!\cdots\!20\)\( \beta_{2} - \)\(11\!\cdots\!00\)\( \beta_{3}) q^{62} +(\)\(44\!\cdots\!00\)\( + \)\(22\!\cdots\!96\)\( \beta_{1} - \)\(15\!\cdots\!82\)\( \beta_{2} - \)\(82\!\cdots\!00\)\( \beta_{3}) q^{63} +(\)\(49\!\cdots\!52\)\( - \)\(57\!\cdots\!88\)\( \beta_{1} + \)\(11\!\cdots\!16\)\( \beta_{2} + \)\(25\!\cdots\!12\)\( \beta_{3}) q^{64} +(-\)\(10\!\cdots\!00\)\( - \)\(36\!\cdots\!60\)\( \beta_{1} - \)\(50\!\cdots\!40\)\( \beta_{2} - \)\(25\!\cdots\!20\)\( \beta_{3}) q^{65} +(-\)\(81\!\cdots\!56\)\( - \)\(12\!\cdots\!08\)\( \beta_{1} + \)\(18\!\cdots\!56\)\( \beta_{2} + \)\(26\!\cdots\!92\)\( \beta_{3}) q^{66} +(-\)\(37\!\cdots\!00\)\( + \)\(19\!\cdots\!60\)\( \beta_{1} + \)\(59\!\cdots\!17\)\( \beta_{2} + \)\(26\!\cdots\!00\)\( \beta_{3}) q^{67} +(-\)\(49\!\cdots\!00\)\( + \)\(18\!\cdots\!24\)\( \beta_{1} - \)\(61\!\cdots\!58\)\( \beta_{2} - \)\(15\!\cdots\!50\)\( \beta_{3}) q^{68} +(-\)\(16\!\cdots\!04\)\( + \)\(21\!\cdots\!04\)\( \beta_{1} - \)\(16\!\cdots\!28\)\( \beta_{2} + \)\(37\!\cdots\!04\)\( \beta_{3}) q^{69} +(\)\(41\!\cdots\!00\)\( - \)\(41\!\cdots\!40\)\( \beta_{1} - \)\(38\!\cdots\!60\)\( \beta_{2} + \)\(15\!\cdots\!20\)\( \beta_{3}) q^{70} +(\)\(66\!\cdots\!72\)\( - \)\(92\!\cdots\!00\)\( \beta_{1} + \)\(71\!\cdots\!50\)\( \beta_{2} + \)\(40\!\cdots\!00\)\( \beta_{3}) q^{71} +(\)\(25\!\cdots\!00\)\( - \)\(11\!\cdots\!88\)\( \beta_{1} - \)\(57\!\cdots\!76\)\( \beta_{2} - \)\(69\!\cdots\!00\)\( \beta_{3}) q^{72} +(\)\(10\!\cdots\!50\)\( + \)\(79\!\cdots\!44\)\( \beta_{1} + \)\(17\!\cdots\!24\)\( \beta_{2} - \)\(19\!\cdots\!00\)\( \beta_{3}) q^{73} +(\)\(68\!\cdots\!64\)\( + \)\(10\!\cdots\!30\)\( \beta_{1} + \)\(32\!\cdots\!40\)\( \beta_{2} + \)\(45\!\cdots\!80\)\( \beta_{3}) q^{74} +(\)\(57\!\cdots\!00\)\( - \)\(17\!\cdots\!00\)\( \beta_{1} - \)\(42\!\cdots\!25\)\( \beta_{2} - \)\(13\!\cdots\!00\)\( \beta_{3}) q^{75} +(-\)\(22\!\cdots\!60\)\( + \)\(14\!\cdots\!68\)\( \beta_{1} + \)\(58\!\cdots\!24\)\( \beta_{2} - \)\(59\!\cdots\!32\)\( \beta_{3}) q^{76} +(-\)\(15\!\cdots\!00\)\( - \)\(14\!\cdots\!68\)\( \beta_{1} + \)\(38\!\cdots\!68\)\( \beta_{2} + \)\(34\!\cdots\!00\)\( \beta_{3}) q^{77} +(-\)\(15\!\cdots\!00\)\( - \)\(58\!\cdots\!84\)\( \beta_{1} - \)\(80\!\cdots\!36\)\( \beta_{2} - \)\(44\!\cdots\!00\)\( \beta_{3}) q^{78} +(-\)\(54\!\cdots\!20\)\( - \)\(32\!\cdots\!24\)\( \beta_{1} - \)\(14\!\cdots\!32\)\( \beta_{2} + \)\(16\!\cdots\!76\)\( \beta_{3}) q^{79} +(-\)\(20\!\cdots\!00\)\( + \)\(22\!\cdots\!40\)\( \beta_{1} + \)\(20\!\cdots\!60\)\( \beta_{2} - \)\(83\!\cdots\!20\)\( \beta_{3}) q^{80} +(\)\(41\!\cdots\!21\)\( + \)\(57\!\cdots\!56\)\( \beta_{1} - \)\(15\!\cdots\!92\)\( \beta_{2} - \)\(21\!\cdots\!44\)\( \beta_{3}) q^{81} +(\)\(10\!\cdots\!00\)\( + \)\(18\!\cdots\!18\)\( \beta_{1} + \)\(29\!\cdots\!20\)\( \beta_{2} + \)\(29\!\cdots\!00\)\( \beta_{3}) q^{82} +(-\)\(48\!\cdots\!00\)\( + \)\(14\!\cdots\!56\)\( \beta_{1} + \)\(64\!\cdots\!33\)\( \beta_{2} - \)\(66\!\cdots\!00\)\( \beta_{3}) q^{83} +(\)\(67\!\cdots\!24\)\( - \)\(38\!\cdots\!04\)\( \beta_{1} - \)\(13\!\cdots\!72\)\( \beta_{2} + \)\(13\!\cdots\!96\)\( \beta_{3}) q^{84} +(\)\(59\!\cdots\!00\)\( - \)\(22\!\cdots\!20\)\( \beta_{1} + \)\(74\!\cdots\!20\)\( \beta_{2} + \)\(18\!\cdots\!60\)\( \beta_{3}) q^{85} +(-\)\(17\!\cdots\!68\)\( - \)\(99\!\cdots\!96\)\( \beta_{1} + \)\(95\!\cdots\!72\)\( \beta_{2} - \)\(29\!\cdots\!96\)\( \beta_{3}) q^{86} +(-\)\(64\!\cdots\!00\)\( + \)\(24\!\cdots\!32\)\( \beta_{1} - \)\(16\!\cdots\!86\)\( \beta_{2} + \)\(17\!\cdots\!00\)\( \beta_{3}) q^{87} +(-\)\(11\!\cdots\!00\)\( + \)\(47\!\cdots\!28\)\( \beta_{1} + \)\(32\!\cdots\!56\)\( \beta_{2} - \)\(24\!\cdots\!00\)\( \beta_{3}) q^{88} +(-\)\(15\!\cdots\!10\)\( + \)\(39\!\cdots\!28\)\( \beta_{1} - \)\(20\!\cdots\!96\)\( \beta_{2} + \)\(35\!\cdots\!28\)\( \beta_{3}) q^{89} +(-\)\(31\!\cdots\!00\)\( - \)\(35\!\cdots\!10\)\( \beta_{1} + \)\(19\!\cdots\!60\)\( \beta_{2} - \)\(32\!\cdots\!20\)\( \beta_{3}) q^{90} +(-\)\(11\!\cdots\!88\)\( - \)\(72\!\cdots\!16\)\( \beta_{1} - \)\(95\!\cdots\!88\)\( \beta_{2} - \)\(80\!\cdots\!16\)\( \beta_{3}) q^{91} +(\)\(40\!\cdots\!00\)\( - \)\(12\!\cdots\!40\)\( \beta_{1} + \)\(15\!\cdots\!92\)\( \beta_{2} - \)\(16\!\cdots\!00\)\( \beta_{3}) q^{92} +(-\)\(10\!\cdots\!00\)\( - \)\(27\!\cdots\!60\)\( \beta_{1} + \)\(20\!\cdots\!92\)\( \beta_{2} - \)\(29\!\cdots\!00\)\( \beta_{3}) q^{93} +(\)\(16\!\cdots\!44\)\( + \)\(17\!\cdots\!04\)\( \beta_{1} - \)\(12\!\cdots\!28\)\( \beta_{2} + \)\(41\!\cdots\!04\)\( \beta_{3}) q^{94} +(\)\(27\!\cdots\!00\)\( - \)\(17\!\cdots\!00\)\( \beta_{1} - \)\(70\!\cdots\!50\)\( \beta_{2} + \)\(70\!\cdots\!00\)\( \beta_{3}) q^{95} +(\)\(37\!\cdots\!52\)\( + \)\(19\!\cdots\!76\)\( \beta_{1} - \)\(10\!\cdots\!32\)\( \beta_{2} - \)\(10\!\cdots\!24\)\( \beta_{3}) q^{96} +(-\)\(20\!\cdots\!50\)\( + \)\(47\!\cdots\!52\)\( \beta_{1} - \)\(30\!\cdots\!84\)\( \beta_{2} - \)\(42\!\cdots\!00\)\( \beta_{3}) q^{97} +(\)\(42\!\cdots\!00\)\( - \)\(25\!\cdots\!77\)\( \beta_{1} - \)\(48\!\cdots\!40\)\( \beta_{2} + \)\(22\!\cdots\!00\)\( \beta_{3}) q^{98} +(-\)\(78\!\cdots\!44\)\( - \)\(27\!\cdots\!96\)\( \beta_{1} + \)\(21\!\cdots\!47\)\( \beta_{2} + \)\(10\!\cdots\!04\)\( \beta_{3}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 1146312000q^{2} - 573723599022000q^{3} + 4402997675828604928q^{4} - \)\(52\!\cdots\!00\)\(q^{5} - \)\(81\!\cdots\!52\)\(q^{6} - \)\(63\!\cdots\!00\)\(q^{7} + \)\(97\!\cdots\!00\)\(q^{8} + \)\(11\!\cdots\!52\)\(q^{9} + O(q^{10}) \) \( 4q + 1146312000q^{2} - 573723599022000q^{3} + 4402997675828604928q^{4} - \)\(52\!\cdots\!00\)\(q^{5} - \)\(81\!\cdots\!52\)\(q^{6} - \)\(63\!\cdots\!00\)\(q^{7} + \)\(97\!\cdots\!00\)\(q^{8} + \)\(11\!\cdots\!52\)\(q^{9} - \)\(14\!\cdots\!00\)\(q^{10} - \)\(41\!\cdots\!52\)\(q^{11} - \)\(99\!\cdots\!00\)\(q^{12} + \)\(10\!\cdots\!00\)\(q^{13} - \)\(21\!\cdots\!04\)\(q^{14} + \)\(11\!\cdots\!00\)\(q^{15} + \)\(10\!\cdots\!24\)\(q^{16} - \)\(40\!\cdots\!00\)\(q^{17} + \)\(15\!\cdots\!00\)\(q^{18} - \)\(35\!\cdots\!20\)\(q^{19} - \)\(50\!\cdots\!00\)\(q^{20} - \)\(55\!\cdots\!72\)\(q^{21} - \)\(17\!\cdots\!00\)\(q^{22} - \)\(41\!\cdots\!00\)\(q^{23} - \)\(61\!\cdots\!60\)\(q^{24} - \)\(16\!\cdots\!00\)\(q^{25} - \)\(49\!\cdots\!32\)\(q^{26} - \)\(12\!\cdots\!00\)\(q^{27} - \)\(79\!\cdots\!00\)\(q^{28} + \)\(66\!\cdots\!20\)\(q^{29} + \)\(95\!\cdots\!00\)\(q^{30} + \)\(32\!\cdots\!28\)\(q^{31} + \)\(29\!\cdots\!00\)\(q^{32} + \)\(80\!\cdots\!00\)\(q^{33} + \)\(39\!\cdots\!16\)\(q^{34} + \)\(94\!\cdots\!00\)\(q^{35} - \)\(49\!\cdots\!36\)\(q^{36} - \)\(71\!\cdots\!00\)\(q^{37} - \)\(65\!\cdots\!00\)\(q^{38} - \)\(53\!\cdots\!76\)\(q^{39} - \)\(69\!\cdots\!00\)\(q^{40} + \)\(16\!\cdots\!68\)\(q^{41} + \)\(15\!\cdots\!00\)\(q^{42} + \)\(75\!\cdots\!00\)\(q^{43} + \)\(15\!\cdots\!36\)\(q^{44} + \)\(60\!\cdots\!00\)\(q^{45} - \)\(15\!\cdots\!12\)\(q^{46} - \)\(21\!\cdots\!00\)\(q^{47} - \)\(84\!\cdots\!00\)\(q^{48} + \)\(15\!\cdots\!28\)\(q^{49} - \)\(37\!\cdots\!00\)\(q^{50} + \)\(20\!\cdots\!88\)\(q^{51} + \)\(34\!\cdots\!00\)\(q^{52} + \)\(84\!\cdots\!00\)\(q^{53} + \)\(20\!\cdots\!80\)\(q^{54} - \)\(18\!\cdots\!00\)\(q^{55} - \)\(89\!\cdots\!20\)\(q^{56} - \)\(48\!\cdots\!00\)\(q^{57} - \)\(42\!\cdots\!00\)\(q^{58} - \)\(21\!\cdots\!60\)\(q^{59} + \)\(16\!\cdots\!00\)\(q^{60} + \)\(42\!\cdots\!48\)\(q^{61} + \)\(51\!\cdots\!00\)\(q^{62} + \)\(17\!\cdots\!00\)\(q^{63} + \)\(19\!\cdots\!08\)\(q^{64} - \)\(40\!\cdots\!00\)\(q^{65} - \)\(32\!\cdots\!24\)\(q^{66} - \)\(15\!\cdots\!00\)\(q^{67} - \)\(19\!\cdots\!00\)\(q^{68} - \)\(65\!\cdots\!16\)\(q^{69} + \)\(16\!\cdots\!00\)\(q^{70} + \)\(26\!\cdots\!88\)\(q^{71} + \)\(10\!\cdots\!00\)\(q^{72} + \)\(43\!\cdots\!00\)\(q^{73} + \)\(27\!\cdots\!56\)\(q^{74} + \)\(22\!\cdots\!00\)\(q^{75} - \)\(91\!\cdots\!40\)\(q^{76} - \)\(62\!\cdots\!00\)\(q^{77} - \)\(62\!\cdots\!00\)\(q^{78} - \)\(21\!\cdots\!80\)\(q^{79} - \)\(81\!\cdots\!00\)\(q^{80} + \)\(16\!\cdots\!84\)\(q^{81} + \)\(41\!\cdots\!00\)\(q^{82} - \)\(19\!\cdots\!00\)\(q^{83} + \)\(26\!\cdots\!96\)\(q^{84} + \)\(23\!\cdots\!00\)\(q^{85} - \)\(71\!\cdots\!72\)\(q^{86} - \)\(25\!\cdots\!00\)\(q^{87} - \)\(45\!\cdots\!00\)\(q^{88} - \)\(60\!\cdots\!40\)\(q^{89} - \)\(12\!\cdots\!00\)\(q^{90} - \)\(45\!\cdots\!52\)\(q^{91} + \)\(16\!\cdots\!00\)\(q^{92} - \)\(43\!\cdots\!00\)\(q^{93} + \)\(64\!\cdots\!76\)\(q^{94} + \)\(11\!\cdots\!00\)\(q^{95} + \)\(15\!\cdots\!08\)\(q^{96} - \)\(80\!\cdots\!00\)\(q^{97} + \)\(17\!\cdots\!00\)\(q^{98} - \)\(31\!\cdots\!76\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 180363795469121 x^{2} + 166129321978984507920 x + 2785609847439483545242446300\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 192 \nu - 48 \)
\(\beta_{2}\)\(=\)\((\)\( 12 \nu^{3} + 20695116 \nu^{2} - 1765108515928884 \nu - 371162118903041598888 \)\()/13402613\)
\(\beta_{3}\)\(=\)\((\)\( -588 \nu^{3} + 69567928692 \nu^{2} + 184007562081366900 \nu - 6347030828361347486393976 \)\()/1914659\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 48\)\()/192\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 343 \beta_{2} - 265270378 \beta_{1} + 3324465478086847488\)\()/36864\)
\(\nu^{3}\)\(=\)\((\)\(-1724593 \beta_{3} + 40581291737 \beta_{2} + 28699219691868298 \beta_{1} - 4593138507376746544705536\)\()/36864\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.21599e7
4.76281e6
−3.61406e6
−1.33086e7
−2.04812e9 1.78996e14 1.88894e18 −2.27994e20 −3.66606e23 −4.42559e25 8.53860e26 −9.51338e28 4.66958e29
1.2 −6.27881e8 −6.22194e14 −1.91161e18 2.33985e20 3.90664e23 6.25792e25 2.64806e27 2.59952e29 −1.46915e29
1.3 9.80478e8 2.49037e14 −1.34451e18 1.59503e20 2.44176e23 1.63507e25 −3.57909e27 −6.51539e28 1.56389e29
1.4 2.84183e9 −3.79563e14 5.77017e18 −6.89620e20 −1.07865e24 −9.80127e25 9.84504e27 1.68945e28 −1.95979e30
\(n\): 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 1.62.a.a 4
3.b odd 2 1 9.62.a.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.62.a.a 4 1.a even 1 1 trivial
9.62.a.a 4 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{62}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 1146312000 T + 3067202781185085440 T^{2} - \)\(53\!\cdots\!00\)\( T^{3} + \)\(70\!\cdots\!08\)\( T^{4} - \)\(12\!\cdots\!00\)\( T^{5} + \)\(16\!\cdots\!60\)\( T^{6} - \)\(14\!\cdots\!00\)\( T^{7} + \)\(28\!\cdots\!16\)\( T^{8} \)
$3$ \( 1 + 573723599022000 T + \)\(36\!\cdots\!80\)\( T^{2} + \)\(16\!\cdots\!00\)\( T^{3} + \)\(69\!\cdots\!18\)\( T^{4} + \)\(20\!\cdots\!00\)\( T^{5} + \)\(58\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!00\)\( T^{7} + \)\(26\!\cdots\!81\)\( T^{8} \)
$5$ \( 1 + \)\(52\!\cdots\!00\)\( T + \)\(17\!\cdots\!00\)\( T^{2} + \)\(67\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!50\)\( T^{4} + \)\(29\!\cdots\!00\)\( T^{5} + \)\(32\!\cdots\!00\)\( T^{6} + \)\(42\!\cdots\!00\)\( T^{7} + \)\(35\!\cdots\!25\)\( T^{8} \)
$7$ \( 1 + \)\(63\!\cdots\!00\)\( T + \)\(83\!\cdots\!00\)\( T^{2} + \)\(47\!\cdots\!00\)\( T^{3} + \)\(38\!\cdots\!98\)\( T^{4} + \)\(17\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!00\)\( T^{6} + \)\(28\!\cdots\!00\)\( T^{7} + \)\(15\!\cdots\!01\)\( T^{8} \)
$11$ \( 1 + \)\(41\!\cdots\!52\)\( T + \)\(70\!\cdots\!08\)\( T^{2} + \)\(24\!\cdots\!04\)\( T^{3} + \)\(24\!\cdots\!70\)\( T^{4} + \)\(81\!\cdots\!44\)\( T^{5} + \)\(78\!\cdots\!68\)\( T^{6} + \)\(15\!\cdots\!12\)\( T^{7} + \)\(12\!\cdots\!41\)\( T^{8} \)
$13$ \( 1 - \)\(10\!\cdots\!00\)\( T + \)\(17\!\cdots\!60\)\( T^{2} - \)\(12\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!38\)\( T^{4} - \)\(10\!\cdots\!00\)\( T^{5} + \)\(13\!\cdots\!40\)\( T^{6} - \)\(76\!\cdots\!00\)\( T^{7} + \)\(63\!\cdots\!61\)\( T^{8} \)
$17$ \( 1 + \)\(40\!\cdots\!00\)\( T + \)\(40\!\cdots\!20\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(63\!\cdots\!78\)\( T^{4} + \)\(11\!\cdots\!00\)\( T^{5} + \)\(52\!\cdots\!80\)\( T^{6} + \)\(60\!\cdots\!00\)\( T^{7} + \)\(16\!\cdots\!21\)\( T^{8} \)
$19$ \( 1 + \)\(35\!\cdots\!20\)\( T + \)\(16\!\cdots\!76\)\( T^{2} + \)\(29\!\cdots\!40\)\( T^{3} + \)\(19\!\cdots\!66\)\( T^{4} + \)\(29\!\cdots\!60\)\( T^{5} + \)\(16\!\cdots\!36\)\( T^{6} + \)\(36\!\cdots\!80\)\( T^{7} + \)\(10\!\cdots\!21\)\( T^{8} \)
$23$ \( 1 + \)\(41\!\cdots\!00\)\( T + \)\(43\!\cdots\!40\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(73\!\cdots\!58\)\( T^{4} + \)\(15\!\cdots\!00\)\( T^{5} + \)\(58\!\cdots\!60\)\( T^{6} + \)\(64\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!41\)\( T^{8} \)
$29$ \( 1 - \)\(66\!\cdots\!20\)\( T + \)\(43\!\cdots\!16\)\( T^{2} + \)\(46\!\cdots\!60\)\( T^{3} + \)\(85\!\cdots\!46\)\( T^{4} + \)\(74\!\cdots\!40\)\( T^{5} + \)\(11\!\cdots\!56\)\( T^{6} - \)\(27\!\cdots\!80\)\( T^{7} + \)\(66\!\cdots\!81\)\( T^{8} \)
$31$ \( 1 - \)\(32\!\cdots\!28\)\( T + \)\(28\!\cdots\!68\)\( T^{2} - \)\(75\!\cdots\!76\)\( T^{3} + \)\(38\!\cdots\!70\)\( T^{4} - \)\(70\!\cdots\!56\)\( T^{5} + \)\(25\!\cdots\!48\)\( T^{6} - \)\(27\!\cdots\!48\)\( T^{7} + \)\(78\!\cdots\!21\)\( T^{8} \)
$37$ \( 1 + \)\(71\!\cdots\!00\)\( T + \)\(98\!\cdots\!60\)\( T^{2} + \)\(68\!\cdots\!00\)\( T^{3} + \)\(69\!\cdots\!38\)\( T^{4} + \)\(31\!\cdots\!00\)\( T^{5} + \)\(20\!\cdots\!40\)\( T^{6} + \)\(67\!\cdots\!00\)\( T^{7} + \)\(43\!\cdots\!61\)\( T^{8} \)
$41$ \( 1 - \)\(16\!\cdots\!68\)\( T + \)\(80\!\cdots\!48\)\( T^{2} - \)\(85\!\cdots\!16\)\( T^{3} + \)\(26\!\cdots\!70\)\( T^{4} - \)\(20\!\cdots\!56\)\( T^{5} + \)\(46\!\cdots\!88\)\( T^{6} - \)\(22\!\cdots\!28\)\( T^{7} + \)\(33\!\cdots\!61\)\( T^{8} \)
$43$ \( 1 - \)\(75\!\cdots\!00\)\( T + \)\(13\!\cdots\!00\)\( T^{2} - \)\(79\!\cdots\!00\)\( T^{3} + \)\(84\!\cdots\!98\)\( T^{4} - \)\(34\!\cdots\!00\)\( T^{5} + \)\(25\!\cdots\!00\)\( T^{6} - \)\(63\!\cdots\!00\)\( T^{7} + \)\(36\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 + \)\(21\!\cdots\!00\)\( T + \)\(36\!\cdots\!80\)\( T^{2} + \)\(46\!\cdots\!00\)\( T^{3} + \)\(54\!\cdots\!18\)\( T^{4} + \)\(45\!\cdots\!00\)\( T^{5} + \)\(36\!\cdots\!20\)\( T^{6} + \)\(21\!\cdots\!00\)\( T^{7} + \)\(98\!\cdots\!81\)\( T^{8} \)
$53$ \( 1 - \)\(84\!\cdots\!00\)\( T + \)\(55\!\cdots\!80\)\( T^{2} - \)\(20\!\cdots\!00\)\( T^{3} + \)\(89\!\cdots\!18\)\( T^{4} - \)\(31\!\cdots\!00\)\( T^{5} + \)\(12\!\cdots\!20\)\( T^{6} - \)\(29\!\cdots\!00\)\( T^{7} + \)\(52\!\cdots\!81\)\( T^{8} \)
$59$ \( 1 + \)\(21\!\cdots\!60\)\( T + \)\(37\!\cdots\!36\)\( T^{2} + \)\(50\!\cdots\!20\)\( T^{3} + \)\(62\!\cdots\!86\)\( T^{4} + \)\(53\!\cdots\!80\)\( T^{5} + \)\(41\!\cdots\!16\)\( T^{6} + \)\(24\!\cdots\!40\)\( T^{7} + \)\(12\!\cdots\!61\)\( T^{8} \)
$61$ \( 1 - \)\(42\!\cdots\!48\)\( T + \)\(31\!\cdots\!08\)\( T^{2} - \)\(10\!\cdots\!96\)\( T^{3} + \)\(37\!\cdots\!70\)\( T^{4} - \)\(81\!\cdots\!56\)\( T^{5} + \)\(20\!\cdots\!68\)\( T^{6} - \)\(22\!\cdots\!88\)\( T^{7} + \)\(41\!\cdots\!41\)\( T^{8} \)
$67$ \( 1 + \)\(15\!\cdots\!00\)\( T + \)\(14\!\cdots\!20\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(56\!\cdots\!78\)\( T^{4} + \)\(24\!\cdots\!00\)\( T^{5} + \)\(89\!\cdots\!80\)\( T^{6} + \)\(22\!\cdots\!00\)\( T^{7} + \)\(36\!\cdots\!21\)\( T^{8} \)
$71$ \( 1 - \)\(26\!\cdots\!88\)\( T + \)\(15\!\cdots\!88\)\( T^{2} + \)\(22\!\cdots\!64\)\( T^{3} + \)\(64\!\cdots\!70\)\( T^{4} + \)\(18\!\cdots\!44\)\( T^{5} + \)\(10\!\cdots\!08\)\( T^{6} - \)\(16\!\cdots\!68\)\( T^{7} + \)\(50\!\cdots\!81\)\( T^{8} \)
$73$ \( 1 - \)\(43\!\cdots\!00\)\( T + \)\(13\!\cdots\!40\)\( T^{2} - \)\(45\!\cdots\!00\)\( T^{3} + \)\(84\!\cdots\!58\)\( T^{4} - \)\(20\!\cdots\!00\)\( T^{5} + \)\(28\!\cdots\!60\)\( T^{6} - \)\(42\!\cdots\!00\)\( T^{7} + \)\(44\!\cdots\!41\)\( T^{8} \)
$79$ \( 1 + \)\(21\!\cdots\!80\)\( T + \)\(11\!\cdots\!16\)\( T^{2} - \)\(36\!\cdots\!40\)\( T^{3} + \)\(56\!\cdots\!46\)\( T^{4} - \)\(20\!\cdots\!60\)\( T^{5} + \)\(38\!\cdots\!56\)\( T^{6} + \)\(40\!\cdots\!20\)\( T^{7} + \)\(10\!\cdots\!81\)\( T^{8} \)
$83$ \( 1 + \)\(19\!\cdots\!00\)\( T + \)\(16\!\cdots\!20\)\( T^{2} + \)\(86\!\cdots\!00\)\( T^{3} + \)\(18\!\cdots\!78\)\( T^{4} + \)\(10\!\cdots\!00\)\( T^{5} + \)\(22\!\cdots\!80\)\( T^{6} + \)\(29\!\cdots\!00\)\( T^{7} + \)\(17\!\cdots\!21\)\( T^{8} \)
$89$ \( 1 + \)\(60\!\cdots\!40\)\( T + \)\(42\!\cdots\!56\)\( T^{2} + \)\(15\!\cdots\!80\)\( T^{3} + \)\(56\!\cdots\!26\)\( T^{4} + \)\(12\!\cdots\!20\)\( T^{5} + \)\(28\!\cdots\!76\)\( T^{6} + \)\(33\!\cdots\!60\)\( T^{7} + \)\(44\!\cdots\!41\)\( T^{8} \)
$97$ \( 1 + \)\(80\!\cdots\!00\)\( T + \)\(80\!\cdots\!80\)\( T^{2} + \)\(38\!\cdots\!00\)\( T^{3} + \)\(20\!\cdots\!18\)\( T^{4} + \)\(59\!\cdots\!00\)\( T^{5} + \)\(19\!\cdots\!20\)\( T^{6} + \)\(30\!\cdots\!00\)\( T^{7} + \)\(59\!\cdots\!81\)\( T^{8} \)
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