Properties

Label 3822.2.a.bv.1.2
Level $3822$
Weight $2$
Character 3822.1
Self dual yes
Analytic conductor $30.519$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3822,2,Mod(1,3822)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3822, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3822.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(30.5188236525\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.2700.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 15x - 20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.61323\) of defining polynomial
Character \(\chi\) \(=\) 3822.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.61323 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.61323 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -2.61323 q^{10} -3.55780 q^{11} -1.00000 q^{12} +1.00000 q^{13} -2.61323 q^{15} +1.00000 q^{16} -7.95529 q^{17} -1.00000 q^{18} -2.17103 q^{19} +2.61323 q^{20} +3.55780 q^{22} -2.55780 q^{23} +1.00000 q^{24} +1.82897 q^{25} -1.00000 q^{26} -1.00000 q^{27} -3.00000 q^{29} +2.61323 q^{30} -0.828970 q^{31} -1.00000 q^{32} +3.55780 q^{33} +7.95529 q^{34} +1.00000 q^{36} +7.78426 q^{37} +2.17103 q^{38} -1.00000 q^{39} -2.61323 q^{40} +11.7843 q^{41} -0.773540 q^{43} -3.55780 q^{44} +2.61323 q^{45} +2.55780 q^{46} -2.94457 q^{47} -1.00000 q^{48} -1.82897 q^{50} +7.95529 q^{51} +1.00000 q^{52} +12.5685 q^{53} +1.00000 q^{54} -9.29735 q^{55} +2.17103 q^{57} +3.00000 q^{58} +4.78426 q^{59} -2.61323 q^{60} +6.72883 q^{61} +0.828970 q^{62} +1.00000 q^{64} +2.61323 q^{65} -3.55780 q^{66} +14.5131 q^{67} -7.95529 q^{68} +2.55780 q^{69} -7.82897 q^{71} -1.00000 q^{72} +10.5578 q^{73} -7.78426 q^{74} -1.82897 q^{75} -2.17103 q^{76} +1.00000 q^{78} -2.39749 q^{79} +2.61323 q^{80} +1.00000 q^{81} -11.7843 q^{82} +15.5131 q^{83} -20.7890 q^{85} +0.773540 q^{86} +3.00000 q^{87} +3.55780 q^{88} +4.21574 q^{89} -2.61323 q^{90} -2.55780 q^{92} +0.828970 q^{93} +2.94457 q^{94} -5.67340 q^{95} +1.00000 q^{96} +10.9553 q^{97} -3.55780 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} - 3 q^{8} + 3 q^{9} - 3 q^{10} - 3 q^{11} - 3 q^{12} + 3 q^{13} - 3 q^{15} + 3 q^{16} + 6 q^{17} - 3 q^{18} + 6 q^{19} + 3 q^{20} + 3 q^{22} + 3 q^{24} + 18 q^{25} - 3 q^{26} - 3 q^{27} - 9 q^{29} + 3 q^{30} - 15 q^{31} - 3 q^{32} + 3 q^{33} - 6 q^{34} + 3 q^{36} + 6 q^{37} - 6 q^{38} - 3 q^{39} - 3 q^{40} + 18 q^{41} - 12 q^{43} - 3 q^{44} + 3 q^{45} - 6 q^{47} - 3 q^{48} - 18 q^{50} - 6 q^{51} + 3 q^{52} + 3 q^{53} + 3 q^{54} + 27 q^{55} - 6 q^{57} + 9 q^{58} - 3 q^{59} - 3 q^{60} + 15 q^{62} + 3 q^{64} + 3 q^{65} - 3 q^{66} + 6 q^{67} + 6 q^{68} - 36 q^{71} - 3 q^{72} + 24 q^{73} - 6 q^{74} - 18 q^{75} + 6 q^{76} + 3 q^{78} + 15 q^{79} + 3 q^{80} + 3 q^{81} - 18 q^{82} + 9 q^{83} - 24 q^{85} + 12 q^{86} + 9 q^{87} + 3 q^{88} + 30 q^{89} - 3 q^{90} + 15 q^{93} + 6 q^{94} + 6 q^{95} + 3 q^{96} + 3 q^{97} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 2.61323 1.16867 0.584336 0.811512i \(-0.301355\pi\)
0.584336 + 0.811512i \(0.301355\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −2.61323 −0.826376
\(11\) −3.55780 −1.07272 −0.536359 0.843990i \(-0.680201\pi\)
−0.536359 + 0.843990i \(0.680201\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −2.61323 −0.674733
\(16\) 1.00000 0.250000
\(17\) −7.95529 −1.92944 −0.964721 0.263276i \(-0.915197\pi\)
−0.964721 + 0.263276i \(0.915197\pi\)
\(18\) −1.00000 −0.235702
\(19\) −2.17103 −0.498068 −0.249034 0.968495i \(-0.580113\pi\)
−0.249034 + 0.968495i \(0.580113\pi\)
\(20\) 2.61323 0.584336
\(21\) 0 0
\(22\) 3.55780 0.758525
\(23\) −2.55780 −0.533338 −0.266669 0.963788i \(-0.585923\pi\)
−0.266669 + 0.963788i \(0.585923\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.82897 0.365794
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 2.61323 0.477108
\(31\) −0.828970 −0.148887 −0.0744437 0.997225i \(-0.523718\pi\)
−0.0744437 + 0.997225i \(0.523718\pi\)
\(32\) −1.00000 −0.176777
\(33\) 3.55780 0.619333
\(34\) 7.95529 1.36432
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 7.78426 1.27972 0.639862 0.768490i \(-0.278991\pi\)
0.639862 + 0.768490i \(0.278991\pi\)
\(38\) 2.17103 0.352188
\(39\) −1.00000 −0.160128
\(40\) −2.61323 −0.413188
\(41\) 11.7843 1.84039 0.920196 0.391458i \(-0.128029\pi\)
0.920196 + 0.391458i \(0.128029\pi\)
\(42\) 0 0
\(43\) −0.773540 −0.117964 −0.0589819 0.998259i \(-0.518785\pi\)
−0.0589819 + 0.998259i \(0.518785\pi\)
\(44\) −3.55780 −0.536359
\(45\) 2.61323 0.389557
\(46\) 2.55780 0.377127
\(47\) −2.94457 −0.429510 −0.214755 0.976668i \(-0.568895\pi\)
−0.214755 + 0.976668i \(0.568895\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −1.82897 −0.258655
\(51\) 7.95529 1.11396
\(52\) 1.00000 0.138675
\(53\) 12.5685 1.72642 0.863209 0.504846i \(-0.168451\pi\)
0.863209 + 0.504846i \(0.168451\pi\)
\(54\) 1.00000 0.136083
\(55\) −9.29735 −1.25365
\(56\) 0 0
\(57\) 2.17103 0.287560
\(58\) 3.00000 0.393919
\(59\) 4.78426 0.622858 0.311429 0.950269i \(-0.399192\pi\)
0.311429 + 0.950269i \(0.399192\pi\)
\(60\) −2.61323 −0.337367
\(61\) 6.72883 0.861538 0.430769 0.902462i \(-0.358242\pi\)
0.430769 + 0.902462i \(0.358242\pi\)
\(62\) 0.828970 0.105279
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.61323 0.324131
\(66\) −3.55780 −0.437935
\(67\) 14.5131 1.77306 0.886528 0.462675i \(-0.153110\pi\)
0.886528 + 0.462675i \(0.153110\pi\)
\(68\) −7.95529 −0.964721
\(69\) 2.55780 0.307923
\(70\) 0 0
\(71\) −7.82897 −0.929128 −0.464564 0.885540i \(-0.653789\pi\)
−0.464564 + 0.885540i \(0.653789\pi\)
\(72\) −1.00000 −0.117851
\(73\) 10.5578 1.23570 0.617849 0.786297i \(-0.288004\pi\)
0.617849 + 0.786297i \(0.288004\pi\)
\(74\) −7.78426 −0.904902
\(75\) −1.82897 −0.211191
\(76\) −2.17103 −0.249034
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) −2.39749 −0.269739 −0.134869 0.990863i \(-0.543061\pi\)
−0.134869 + 0.990863i \(0.543061\pi\)
\(80\) 2.61323 0.292168
\(81\) 1.00000 0.111111
\(82\) −11.7843 −1.30135
\(83\) 15.5131 1.70278 0.851391 0.524531i \(-0.175759\pi\)
0.851391 + 0.524531i \(0.175759\pi\)
\(84\) 0 0
\(85\) −20.7890 −2.25488
\(86\) 0.773540 0.0834130
\(87\) 3.00000 0.321634
\(88\) 3.55780 0.379263
\(89\) 4.21574 0.446868 0.223434 0.974719i \(-0.428273\pi\)
0.223434 + 0.974719i \(0.428273\pi\)
\(90\) −2.61323 −0.275459
\(91\) 0 0
\(92\) −2.55780 −0.266669
\(93\) 0.828970 0.0859602
\(94\) 2.94457 0.303709
\(95\) −5.67340 −0.582079
\(96\) 1.00000 0.102062
\(97\) 10.9553 1.11234 0.556171 0.831068i \(-0.312270\pi\)
0.556171 + 0.831068i \(0.312270\pi\)
\(98\) 0 0
\(99\) −3.55780 −0.357572
\(100\) 1.82897 0.182897
\(101\) 10.3421 1.02907 0.514537 0.857468i \(-0.327964\pi\)
0.514537 + 0.857468i \(0.327964\pi\)
\(102\) −7.95529 −0.787691
\(103\) −3.22646 −0.317913 −0.158956 0.987286i \(-0.550813\pi\)
−0.158956 + 0.987286i \(0.550813\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −12.5685 −1.22076
\(107\) 8.28663 0.801099 0.400549 0.916275i \(-0.368819\pi\)
0.400549 + 0.916275i \(0.368819\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 9.29735 0.886467
\(111\) −7.78426 −0.738849
\(112\) 0 0
\(113\) −17.1817 −1.61632 −0.808161 0.588961i \(-0.799537\pi\)
−0.808161 + 0.588961i \(0.799537\pi\)
\(114\) −2.17103 −0.203336
\(115\) −6.68412 −0.623297
\(116\) −3.00000 −0.278543
\(117\) 1.00000 0.0924500
\(118\) −4.78426 −0.440427
\(119\) 0 0
\(120\) 2.61323 0.238554
\(121\) 1.65794 0.150722
\(122\) −6.72883 −0.609200
\(123\) −11.7843 −1.06255
\(124\) −0.828970 −0.0744437
\(125\) −8.28663 −0.741179
\(126\) 0 0
\(127\) 19.2973 1.71236 0.856181 0.516675i \(-0.172831\pi\)
0.856181 + 0.516675i \(0.172831\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0.773540 0.0681064
\(130\) −2.61323 −0.229195
\(131\) 19.4082 1.69570 0.847852 0.530234i \(-0.177896\pi\)
0.847852 + 0.530234i \(0.177896\pi\)
\(132\) 3.55780 0.309667
\(133\) 0 0
\(134\) −14.5131 −1.25374
\(135\) −2.61323 −0.224911
\(136\) 7.95529 0.682160
\(137\) −0.215740 −0.0184319 −0.00921597 0.999958i \(-0.502934\pi\)
−0.00921597 + 0.999958i \(0.502934\pi\)
\(138\) −2.55780 −0.217734
\(139\) −3.44220 −0.291964 −0.145982 0.989287i \(-0.546634\pi\)
−0.145982 + 0.989287i \(0.546634\pi\)
\(140\) 0 0
\(141\) 2.94457 0.247978
\(142\) 7.82897 0.656993
\(143\) −3.55780 −0.297518
\(144\) 1.00000 0.0833333
\(145\) −7.83969 −0.651051
\(146\) −10.5578 −0.873770
\(147\) 0 0
\(148\) 7.78426 0.639862
\(149\) −9.44220 −0.773535 −0.386768 0.922177i \(-0.626408\pi\)
−0.386768 + 0.922177i \(0.626408\pi\)
\(150\) 1.82897 0.149335
\(151\) −6.44220 −0.524259 −0.262129 0.965033i \(-0.584425\pi\)
−0.262129 + 0.965033i \(0.584425\pi\)
\(152\) 2.17103 0.176094
\(153\) −7.95529 −0.643147
\(154\) 0 0
\(155\) −2.16629 −0.174001
\(156\) −1.00000 −0.0800641
\(157\) 6.83969 0.545867 0.272933 0.962033i \(-0.412006\pi\)
0.272933 + 0.962033i \(0.412006\pi\)
\(158\) 2.39749 0.190734
\(159\) −12.5685 −0.996748
\(160\) −2.61323 −0.206594
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −12.6239 −0.988784 −0.494392 0.869239i \(-0.664609\pi\)
−0.494392 + 0.869239i \(0.664609\pi\)
\(164\) 11.7843 0.920196
\(165\) 9.29735 0.723798
\(166\) −15.5131 −1.20405
\(167\) 3.71811 0.287716 0.143858 0.989598i \(-0.454049\pi\)
0.143858 + 0.989598i \(0.454049\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 20.7890 1.59444
\(171\) −2.17103 −0.166023
\(172\) −0.773540 −0.0589819
\(173\) 13.2866 1.01016 0.505082 0.863071i \(-0.331462\pi\)
0.505082 + 0.863071i \(0.331462\pi\)
\(174\) −3.00000 −0.227429
\(175\) 0 0
\(176\) −3.55780 −0.268179
\(177\) −4.78426 −0.359607
\(178\) −4.21574 −0.315983
\(179\) −14.7950 −1.10583 −0.552914 0.833238i \(-0.686484\pi\)
−0.552914 + 0.833238i \(0.686484\pi\)
\(180\) 2.61323 0.194779
\(181\) −1.95529 −0.145336 −0.0726678 0.997356i \(-0.523151\pi\)
−0.0726678 + 0.997356i \(0.523151\pi\)
\(182\) 0 0
\(183\) −6.72883 −0.497409
\(184\) 2.55780 0.188564
\(185\) 20.3421 1.49558
\(186\) −0.828970 −0.0607830
\(187\) 28.3033 2.06974
\(188\) −2.94457 −0.214755
\(189\) 0 0
\(190\) 5.67340 0.411592
\(191\) −4.11086 −0.297451 −0.148726 0.988878i \(-0.547517\pi\)
−0.148726 + 0.988878i \(0.547517\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −10.1817 −0.732898 −0.366449 0.930438i \(-0.619427\pi\)
−0.366449 + 0.930438i \(0.619427\pi\)
\(194\) −10.9553 −0.786544
\(195\) −2.61323 −0.187137
\(196\) 0 0
\(197\) −9.44220 −0.672729 −0.336364 0.941732i \(-0.609197\pi\)
−0.336364 + 0.941732i \(0.609197\pi\)
\(198\) 3.55780 0.252842
\(199\) 13.2419 0.938695 0.469347 0.883014i \(-0.344489\pi\)
0.469347 + 0.883014i \(0.344489\pi\)
\(200\) −1.82897 −0.129328
\(201\) −14.5131 −1.02367
\(202\) −10.3421 −0.727665
\(203\) 0 0
\(204\) 7.95529 0.556982
\(205\) 30.7950 2.15081
\(206\) 3.22646 0.224798
\(207\) −2.55780 −0.177779
\(208\) 1.00000 0.0693375
\(209\) 7.72409 0.534286
\(210\) 0 0
\(211\) 1.33134 0.0916532 0.0458266 0.998949i \(-0.485408\pi\)
0.0458266 + 0.998949i \(0.485408\pi\)
\(212\) 12.5685 0.863209
\(213\) 7.82897 0.536432
\(214\) −8.28663 −0.566462
\(215\) −2.02144 −0.137861
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −4.00000 −0.270914
\(219\) −10.5578 −0.713430
\(220\) −9.29735 −0.626827
\(221\) −7.95529 −0.535131
\(222\) 7.78426 0.522445
\(223\) 22.3528 1.49685 0.748426 0.663218i \(-0.230810\pi\)
0.748426 + 0.663218i \(0.230810\pi\)
\(224\) 0 0
\(225\) 1.82897 0.121931
\(226\) 17.1817 1.14291
\(227\) −19.8551 −1.31783 −0.658916 0.752216i \(-0.728985\pi\)
−0.658916 + 0.752216i \(0.728985\pi\)
\(228\) 2.17103 0.143780
\(229\) −19.1156 −1.26319 −0.631597 0.775297i \(-0.717600\pi\)
−0.631597 + 0.775297i \(0.717600\pi\)
\(230\) 6.68412 0.440738
\(231\) 0 0
\(232\) 3.00000 0.196960
\(233\) −21.0709 −1.38040 −0.690200 0.723619i \(-0.742477\pi\)
−0.690200 + 0.723619i \(0.742477\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −7.69484 −0.501956
\(236\) 4.78426 0.311429
\(237\) 2.39749 0.155734
\(238\) 0 0
\(239\) 7.39749 0.478504 0.239252 0.970958i \(-0.423098\pi\)
0.239252 + 0.970958i \(0.423098\pi\)
\(240\) −2.61323 −0.168683
\(241\) 21.6239 1.39292 0.696461 0.717595i \(-0.254757\pi\)
0.696461 + 0.717595i \(0.254757\pi\)
\(242\) −1.65794 −0.106576
\(243\) −1.00000 −0.0641500
\(244\) 6.72883 0.430769
\(245\) 0 0
\(246\) 11.7843 0.751337
\(247\) −2.17103 −0.138139
\(248\) 0.828970 0.0526397
\(249\) −15.5131 −0.983102
\(250\) 8.28663 0.524092
\(251\) −25.9660 −1.63896 −0.819480 0.573108i \(-0.805738\pi\)
−0.819480 + 0.573108i \(0.805738\pi\)
\(252\) 0 0
\(253\) 9.10014 0.572121
\(254\) −19.2973 −1.21082
\(255\) 20.7890 1.30186
\(256\) 1.00000 0.0625000
\(257\) −11.1156 −0.693372 −0.346686 0.937981i \(-0.612693\pi\)
−0.346686 + 0.937981i \(0.612693\pi\)
\(258\) −0.773540 −0.0481585
\(259\) 0 0
\(260\) 2.61323 0.162066
\(261\) −3.00000 −0.185695
\(262\) −19.4082 −1.19904
\(263\) 8.44694 0.520861 0.260430 0.965493i \(-0.416136\pi\)
0.260430 + 0.965493i \(0.416136\pi\)
\(264\) −3.55780 −0.218967
\(265\) 32.8444 2.01762
\(266\) 0 0
\(267\) −4.21574 −0.257999
\(268\) 14.5131 0.886528
\(269\) 11.2312 0.684778 0.342389 0.939558i \(-0.388764\pi\)
0.342389 + 0.939558i \(0.388764\pi\)
\(270\) 2.61323 0.159036
\(271\) −13.7890 −0.837622 −0.418811 0.908073i \(-0.637553\pi\)
−0.418811 + 0.908073i \(0.637553\pi\)
\(272\) −7.95529 −0.482360
\(273\) 0 0
\(274\) 0.215740 0.0130334
\(275\) −6.50711 −0.392394
\(276\) 2.55780 0.153961
\(277\) 20.2973 1.21955 0.609775 0.792575i \(-0.291260\pi\)
0.609775 + 0.792575i \(0.291260\pi\)
\(278\) 3.44220 0.206449
\(279\) −0.828970 −0.0496291
\(280\) 0 0
\(281\) 19.1370 1.14162 0.570810 0.821082i \(-0.306629\pi\)
0.570810 + 0.821082i \(0.306629\pi\)
\(282\) −2.94457 −0.175347
\(283\) 0.326600 0.0194144 0.00970718 0.999953i \(-0.496910\pi\)
0.00970718 + 0.999953i \(0.496910\pi\)
\(284\) −7.82897 −0.464564
\(285\) 5.67340 0.336063
\(286\) 3.55780 0.210377
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 46.2866 2.72274
\(290\) 7.83969 0.460362
\(291\) −10.9553 −0.642210
\(292\) 10.5578 0.617849
\(293\) 8.61323 0.503190 0.251595 0.967833i \(-0.419045\pi\)
0.251595 + 0.967833i \(0.419045\pi\)
\(294\) 0 0
\(295\) 12.5024 0.727916
\(296\) −7.78426 −0.452451
\(297\) 3.55780 0.206444
\(298\) 9.44220 0.546972
\(299\) −2.55780 −0.147921
\(300\) −1.82897 −0.105596
\(301\) 0 0
\(302\) 6.44220 0.370707
\(303\) −10.3421 −0.594136
\(304\) −2.17103 −0.124517
\(305\) 17.5840 1.00686
\(306\) 7.95529 0.454774
\(307\) 28.1710 1.60781 0.803903 0.594761i \(-0.202753\pi\)
0.803903 + 0.594761i \(0.202753\pi\)
\(308\) 0 0
\(309\) 3.22646 0.183547
\(310\) 2.16629 0.123037
\(311\) −24.2372 −1.37436 −0.687182 0.726485i \(-0.741153\pi\)
−0.687182 + 0.726485i \(0.741153\pi\)
\(312\) 1.00000 0.0566139
\(313\) 22.0554 1.24665 0.623323 0.781964i \(-0.285782\pi\)
0.623323 + 0.781964i \(0.285782\pi\)
\(314\) −6.83969 −0.385986
\(315\) 0 0
\(316\) −2.39749 −0.134869
\(317\) 8.48691 0.476672 0.238336 0.971183i \(-0.423398\pi\)
0.238336 + 0.971183i \(0.423398\pi\)
\(318\) 12.5685 0.704808
\(319\) 10.6734 0.597596
\(320\) 2.61323 0.146084
\(321\) −8.28663 −0.462515
\(322\) 0 0
\(323\) 17.2712 0.960994
\(324\) 1.00000 0.0555556
\(325\) 1.82897 0.101453
\(326\) 12.6239 0.699176
\(327\) −4.00000 −0.221201
\(328\) −11.7843 −0.650677
\(329\) 0 0
\(330\) −9.29735 −0.511802
\(331\) −29.0262 −1.59542 −0.797712 0.603039i \(-0.793956\pi\)
−0.797712 + 0.603039i \(0.793956\pi\)
\(332\) 15.5131 0.851391
\(333\) 7.78426 0.426575
\(334\) −3.71811 −0.203446
\(335\) 37.9260 2.07212
\(336\) 0 0
\(337\) −17.7950 −0.969354 −0.484677 0.874693i \(-0.661063\pi\)
−0.484677 + 0.874693i \(0.661063\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 17.1817 0.933185
\(340\) −20.7890 −1.12744
\(341\) 2.94931 0.159714
\(342\) 2.17103 0.117396
\(343\) 0 0
\(344\) 0.773540 0.0417065
\(345\) 6.68412 0.359861
\(346\) −13.2866 −0.714294
\(347\) −11.3528 −0.609449 −0.304725 0.952441i \(-0.598564\pi\)
−0.304725 + 0.952441i \(0.598564\pi\)
\(348\) 3.00000 0.160817
\(349\) 14.8999 0.797571 0.398786 0.917044i \(-0.369432\pi\)
0.398786 + 0.917044i \(0.369432\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 3.55780 0.189631
\(353\) −13.1370 −0.699214 −0.349607 0.936896i \(-0.613685\pi\)
−0.349607 + 0.936896i \(0.613685\pi\)
\(354\) 4.78426 0.254281
\(355\) −20.4589 −1.08585
\(356\) 4.21574 0.223434
\(357\) 0 0
\(358\) 14.7950 0.781939
\(359\) −18.0000 −0.950004 −0.475002 0.879985i \(-0.657553\pi\)
−0.475002 + 0.879985i \(0.657553\pi\)
\(360\) −2.61323 −0.137729
\(361\) −14.2866 −0.751928
\(362\) 1.95529 0.102768
\(363\) −1.65794 −0.0870193
\(364\) 0 0
\(365\) 27.5900 1.44412
\(366\) 6.72883 0.351722
\(367\) 1.71337 0.0894372 0.0447186 0.999000i \(-0.485761\pi\)
0.0447186 + 0.999000i \(0.485761\pi\)
\(368\) −2.55780 −0.133335
\(369\) 11.7843 0.613464
\(370\) −20.3421 −1.05753
\(371\) 0 0
\(372\) 0.828970 0.0429801
\(373\) 25.9767 1.34502 0.672512 0.740086i \(-0.265215\pi\)
0.672512 + 0.740086i \(0.265215\pi\)
\(374\) −28.3033 −1.46353
\(375\) 8.28663 0.427920
\(376\) 2.94457 0.151855
\(377\) −3.00000 −0.154508
\(378\) 0 0
\(379\) −14.4529 −0.742397 −0.371198 0.928554i \(-0.621053\pi\)
−0.371198 + 0.928554i \(0.621053\pi\)
\(380\) −5.67340 −0.291039
\(381\) −19.2973 −0.988633
\(382\) 4.11086 0.210330
\(383\) 20.4529 1.04510 0.522548 0.852610i \(-0.324982\pi\)
0.522548 + 0.852610i \(0.324982\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 10.1817 0.518237
\(387\) −0.773540 −0.0393213
\(388\) 10.9553 0.556171
\(389\) 23.3975 1.18630 0.593150 0.805092i \(-0.297884\pi\)
0.593150 + 0.805092i \(0.297884\pi\)
\(390\) 2.61323 0.132326
\(391\) 20.3480 1.02904
\(392\) 0 0
\(393\) −19.4082 −0.979015
\(394\) 9.44220 0.475691
\(395\) −6.26519 −0.315236
\(396\) −3.55780 −0.178786
\(397\) 22.4684 1.12766 0.563828 0.825892i \(-0.309328\pi\)
0.563828 + 0.825892i \(0.309328\pi\)
\(398\) −13.2419 −0.663757
\(399\) 0 0
\(400\) 1.82897 0.0914485
\(401\) 2.89986 0.144812 0.0724060 0.997375i \(-0.476932\pi\)
0.0724060 + 0.997375i \(0.476932\pi\)
\(402\) 14.5131 0.723847
\(403\) −0.828970 −0.0412939
\(404\) 10.3421 0.514537
\(405\) 2.61323 0.129852
\(406\) 0 0
\(407\) −27.6948 −1.37278
\(408\) −7.95529 −0.393846
\(409\) 5.27591 0.260877 0.130438 0.991456i \(-0.458361\pi\)
0.130438 + 0.991456i \(0.458361\pi\)
\(410\) −30.7950 −1.52086
\(411\) 0.215740 0.0106417
\(412\) −3.22646 −0.158956
\(413\) 0 0
\(414\) 2.55780 0.125709
\(415\) 40.5393 1.98999
\(416\) −1.00000 −0.0490290
\(417\) 3.44220 0.168565
\(418\) −7.72409 −0.377798
\(419\) −3.22646 −0.157623 −0.0788114 0.996890i \(-0.525113\pi\)
−0.0788114 + 0.996890i \(0.525113\pi\)
\(420\) 0 0
\(421\) 34.5792 1.68529 0.842644 0.538470i \(-0.180998\pi\)
0.842644 + 0.538470i \(0.180998\pi\)
\(422\) −1.33134 −0.0648086
\(423\) −2.94457 −0.143170
\(424\) −12.5685 −0.610381
\(425\) −14.5500 −0.705778
\(426\) −7.82897 −0.379315
\(427\) 0 0
\(428\) 8.28663 0.400549
\(429\) 3.55780 0.171772
\(430\) 2.02144 0.0974824
\(431\) 12.4529 0.599836 0.299918 0.953965i \(-0.403041\pi\)
0.299918 + 0.953965i \(0.403041\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −5.62869 −0.270498 −0.135249 0.990812i \(-0.543183\pi\)
−0.135249 + 0.990812i \(0.543183\pi\)
\(434\) 0 0
\(435\) 7.83969 0.375884
\(436\) 4.00000 0.191565
\(437\) 5.55306 0.265639
\(438\) 10.5578 0.504471
\(439\) 12.5238 0.597729 0.298864 0.954296i \(-0.403392\pi\)
0.298864 + 0.954296i \(0.403392\pi\)
\(440\) 9.29735 0.443234
\(441\) 0 0
\(442\) 7.95529 0.378395
\(443\) −8.50237 −0.403960 −0.201980 0.979390i \(-0.564738\pi\)
−0.201980 + 0.979390i \(0.564738\pi\)
\(444\) −7.78426 −0.369425
\(445\) 11.0167 0.522242
\(446\) −22.3528 −1.05843
\(447\) 9.44220 0.446601
\(448\) 0 0
\(449\) −34.9153 −1.64776 −0.823878 0.566767i \(-0.808194\pi\)
−0.823878 + 0.566767i \(0.808194\pi\)
\(450\) −1.82897 −0.0862185
\(451\) −41.9260 −1.97422
\(452\) −17.1817 −0.808161
\(453\) 6.44220 0.302681
\(454\) 19.8551 0.931848
\(455\) 0 0
\(456\) −2.17103 −0.101668
\(457\) 1.83969 0.0860570 0.0430285 0.999074i \(-0.486299\pi\)
0.0430285 + 0.999074i \(0.486299\pi\)
\(458\) 19.1156 0.893213
\(459\) 7.95529 0.371321
\(460\) −6.68412 −0.311649
\(461\) −23.4577 −1.09253 −0.546266 0.837612i \(-0.683951\pi\)
−0.546266 + 0.837612i \(0.683951\pi\)
\(462\) 0 0
\(463\) −22.4529 −1.04348 −0.521738 0.853106i \(-0.674716\pi\)
−0.521738 + 0.853106i \(0.674716\pi\)
\(464\) −3.00000 −0.139272
\(465\) 2.16629 0.100459
\(466\) 21.0709 0.976090
\(467\) 8.98928 0.415974 0.207987 0.978132i \(-0.433309\pi\)
0.207987 + 0.978132i \(0.433309\pi\)
\(468\) 1.00000 0.0462250
\(469\) 0 0
\(470\) 7.69484 0.354936
\(471\) −6.83969 −0.315156
\(472\) −4.78426 −0.220213
\(473\) 2.75210 0.126542
\(474\) −2.39749 −0.110120
\(475\) −3.97075 −0.182190
\(476\) 0 0
\(477\) 12.5685 0.575473
\(478\) −7.39749 −0.338353
\(479\) 35.5178 1.62285 0.811426 0.584456i \(-0.198692\pi\)
0.811426 + 0.584456i \(0.198692\pi\)
\(480\) 2.61323 0.119277
\(481\) 7.78426 0.354932
\(482\) −21.6239 −0.984944
\(483\) 0 0
\(484\) 1.65794 0.0753609
\(485\) 28.6287 1.29996
\(486\) 1.00000 0.0453609
\(487\) 18.0107 0.816144 0.408072 0.912950i \(-0.366201\pi\)
0.408072 + 0.912950i \(0.366201\pi\)
\(488\) −6.72883 −0.304600
\(489\) 12.6239 0.570875
\(490\) 0 0
\(491\) −12.8504 −0.579931 −0.289965 0.957037i \(-0.593644\pi\)
−0.289965 + 0.957037i \(0.593644\pi\)
\(492\) −11.7843 −0.531275
\(493\) 23.8659 1.07486
\(494\) 2.17103 0.0976792
\(495\) −9.29735 −0.417885
\(496\) −0.828970 −0.0372219
\(497\) 0 0
\(498\) 15.5131 0.695158
\(499\) 3.00474 0.134511 0.0672553 0.997736i \(-0.478576\pi\)
0.0672553 + 0.997736i \(0.478576\pi\)
\(500\) −8.28663 −0.370589
\(501\) −3.71811 −0.166113
\(502\) 25.9660 1.15892
\(503\) −33.9106 −1.51200 −0.755999 0.654573i \(-0.772849\pi\)
−0.755999 + 0.654573i \(0.772849\pi\)
\(504\) 0 0
\(505\) 27.0262 1.20265
\(506\) −9.10014 −0.404551
\(507\) −1.00000 −0.0444116
\(508\) 19.2973 0.856181
\(509\) −25.1925 −1.11664 −0.558318 0.829627i \(-0.688553\pi\)
−0.558318 + 0.829627i \(0.688553\pi\)
\(510\) −20.7890 −0.920552
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 2.17103 0.0958533
\(514\) 11.1156 0.490288
\(515\) −8.43148 −0.371535
\(516\) 0.773540 0.0340532
\(517\) 10.4762 0.460742
\(518\) 0 0
\(519\) −13.2866 −0.583218
\(520\) −2.61323 −0.114598
\(521\) 39.2574 1.71990 0.859948 0.510381i \(-0.170496\pi\)
0.859948 + 0.510381i \(0.170496\pi\)
\(522\) 3.00000 0.131306
\(523\) 13.6794 0.598157 0.299079 0.954228i \(-0.403321\pi\)
0.299079 + 0.954228i \(0.403321\pi\)
\(524\) 19.4082 0.847852
\(525\) 0 0
\(526\) −8.44694 −0.368304
\(527\) 6.59470 0.287270
\(528\) 3.55780 0.154833
\(529\) −16.4577 −0.715550
\(530\) −32.8444 −1.42667
\(531\) 4.78426 0.207619
\(532\) 0 0
\(533\) 11.7843 0.510433
\(534\) 4.21574 0.182433
\(535\) 21.6549 0.936222
\(536\) −14.5131 −0.626870
\(537\) 14.7950 0.638450
\(538\) −11.2312 −0.484211
\(539\) 0 0
\(540\) −2.61323 −0.112456
\(541\) −3.35278 −0.144147 −0.0720736 0.997399i \(-0.522962\pi\)
−0.0720736 + 0.997399i \(0.522962\pi\)
\(542\) 13.7890 0.592288
\(543\) 1.95529 0.0839095
\(544\) 7.95529 0.341080
\(545\) 10.4529 0.447754
\(546\) 0 0
\(547\) −0.899860 −0.0384752 −0.0192376 0.999815i \(-0.506124\pi\)
−0.0192376 + 0.999815i \(0.506124\pi\)
\(548\) −0.215740 −0.00921597
\(549\) 6.72883 0.287179
\(550\) 6.50711 0.277464
\(551\) 6.51309 0.277467
\(552\) −2.55780 −0.108867
\(553\) 0 0
\(554\) −20.2973 −0.862352
\(555\) −20.3421 −0.863472
\(556\) −3.44220 −0.145982
\(557\) 9.71337 0.411569 0.205784 0.978597i \(-0.434025\pi\)
0.205784 + 0.978597i \(0.434025\pi\)
\(558\) 0.828970 0.0350931
\(559\) −0.773540 −0.0327173
\(560\) 0 0
\(561\) −28.3033 −1.19497
\(562\) −19.1370 −0.807247
\(563\) −35.1710 −1.48228 −0.741141 0.671349i \(-0.765715\pi\)
−0.741141 + 0.671349i \(0.765715\pi\)
\(564\) 2.94457 0.123989
\(565\) −44.8999 −1.88895
\(566\) −0.326600 −0.0137280
\(567\) 0 0
\(568\) 7.82897 0.328496
\(569\) −12.0661 −0.505839 −0.252920 0.967487i \(-0.581391\pi\)
−0.252920 + 0.967487i \(0.581391\pi\)
\(570\) −5.67340 −0.237633
\(571\) 31.1370 1.30304 0.651522 0.758630i \(-0.274131\pi\)
0.651522 + 0.758630i \(0.274131\pi\)
\(572\) −3.55780 −0.148759
\(573\) 4.11086 0.171734
\(574\) 0 0
\(575\) −4.67814 −0.195092
\(576\) 1.00000 0.0416667
\(577\) −28.7395 −1.19644 −0.598222 0.801331i \(-0.704126\pi\)
−0.598222 + 0.801331i \(0.704126\pi\)
\(578\) −46.2866 −1.92527
\(579\) 10.1817 0.423139
\(580\) −7.83969 −0.325525
\(581\) 0 0
\(582\) 10.9553 0.454111
\(583\) −44.7163 −1.85196
\(584\) −10.5578 −0.436885
\(585\) 2.61323 0.108044
\(586\) −8.61323 −0.355809
\(587\) 12.7628 0.526778 0.263389 0.964690i \(-0.415160\pi\)
0.263389 + 0.964690i \(0.415160\pi\)
\(588\) 0 0
\(589\) 1.79972 0.0741561
\(590\) −12.5024 −0.514714
\(591\) 9.44220 0.388400
\(592\) 7.78426 0.319931
\(593\) 23.3528 0.958984 0.479492 0.877546i \(-0.340821\pi\)
0.479492 + 0.877546i \(0.340821\pi\)
\(594\) −3.55780 −0.145978
\(595\) 0 0
\(596\) −9.44220 −0.386768
\(597\) −13.2419 −0.541956
\(598\) 2.55780 0.104596
\(599\) 2.89986 0.118485 0.0592425 0.998244i \(-0.481131\pi\)
0.0592425 + 0.998244i \(0.481131\pi\)
\(600\) 1.82897 0.0746674
\(601\) 31.5947 1.28877 0.644387 0.764699i \(-0.277112\pi\)
0.644387 + 0.764699i \(0.277112\pi\)
\(602\) 0 0
\(603\) 14.5131 0.591019
\(604\) −6.44220 −0.262129
\(605\) 4.33258 0.176144
\(606\) 10.3421 0.420117
\(607\) 41.9600 1.70311 0.851553 0.524269i \(-0.175661\pi\)
0.851553 + 0.524269i \(0.175661\pi\)
\(608\) 2.17103 0.0880469
\(609\) 0 0
\(610\) −17.5840 −0.711954
\(611\) −2.94457 −0.119125
\(612\) −7.95529 −0.321574
\(613\) −18.0214 −0.727879 −0.363940 0.931423i \(-0.618569\pi\)
−0.363940 + 0.931423i \(0.618569\pi\)
\(614\) −28.1710 −1.13689
\(615\) −30.7950 −1.24177
\(616\) 0 0
\(617\) 0.551821 0.0222155 0.0111077 0.999938i \(-0.496464\pi\)
0.0111077 + 0.999938i \(0.496464\pi\)
\(618\) −3.22646 −0.129787
\(619\) −40.3421 −1.62148 −0.810742 0.585403i \(-0.800936\pi\)
−0.810742 + 0.585403i \(0.800936\pi\)
\(620\) −2.16629 −0.0870003
\(621\) 2.55780 0.102641
\(622\) 24.2372 0.971822
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) −30.7997 −1.23199
\(626\) −22.0554 −0.881512
\(627\) −7.72409 −0.308470
\(628\) 6.83969 0.272933
\(629\) −61.9260 −2.46915
\(630\) 0 0
\(631\) −27.5131 −1.09528 −0.547639 0.836714i \(-0.684473\pi\)
−0.547639 + 0.836714i \(0.684473\pi\)
\(632\) 2.39749 0.0953670
\(633\) −1.33134 −0.0529160
\(634\) −8.48691 −0.337058
\(635\) 50.4284 2.00119
\(636\) −12.5685 −0.498374
\(637\) 0 0
\(638\) −10.6734 −0.422564
\(639\) −7.82897 −0.309709
\(640\) −2.61323 −0.103297
\(641\) −2.23120 −0.0881271 −0.0440635 0.999029i \(-0.514030\pi\)
−0.0440635 + 0.999029i \(0.514030\pi\)
\(642\) 8.28663 0.327047
\(643\) −16.7134 −0.659111 −0.329555 0.944136i \(-0.606899\pi\)
−0.329555 + 0.944136i \(0.606899\pi\)
\(644\) 0 0
\(645\) 2.02144 0.0795941
\(646\) −17.2712 −0.679525
\(647\) −26.7950 −1.05342 −0.526710 0.850045i \(-0.676574\pi\)
−0.526710 + 0.850045i \(0.676574\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −17.0214 −0.668150
\(650\) −1.82897 −0.0717381
\(651\) 0 0
\(652\) −12.6239 −0.494392
\(653\) −24.9707 −0.977181 −0.488590 0.872513i \(-0.662489\pi\)
−0.488590 + 0.872513i \(0.662489\pi\)
\(654\) 4.00000 0.156412
\(655\) 50.7181 1.98172
\(656\) 11.7843 0.460098
\(657\) 10.5578 0.411899
\(658\) 0 0
\(659\) −20.8999 −0.814143 −0.407071 0.913396i \(-0.633450\pi\)
−0.407071 + 0.913396i \(0.633450\pi\)
\(660\) 9.29735 0.361899
\(661\) −47.7163 −1.85595 −0.927974 0.372645i \(-0.878451\pi\)
−0.927974 + 0.372645i \(0.878451\pi\)
\(662\) 29.0262 1.12813
\(663\) 7.95529 0.308958
\(664\) −15.5131 −0.602025
\(665\) 0 0
\(666\) −7.78426 −0.301634
\(667\) 7.67340 0.297115
\(668\) 3.71811 0.143858
\(669\) −22.3528 −0.864208
\(670\) −37.9260 −1.46521
\(671\) −23.9398 −0.924187
\(672\) 0 0
\(673\) 11.6025 0.447244 0.223622 0.974676i \(-0.428212\pi\)
0.223622 + 0.974676i \(0.428212\pi\)
\(674\) 17.7950 0.685437
\(675\) −1.82897 −0.0703971
\(676\) 1.00000 0.0384615
\(677\) 8.33732 0.320429 0.160215 0.987082i \(-0.448781\pi\)
0.160215 + 0.987082i \(0.448781\pi\)
\(678\) −17.1817 −0.659861
\(679\) 0 0
\(680\) 20.7890 0.797222
\(681\) 19.8551 0.760851
\(682\) −2.94931 −0.112935
\(683\) −32.5393 −1.24508 −0.622540 0.782588i \(-0.713899\pi\)
−0.622540 + 0.782588i \(0.713899\pi\)
\(684\) −2.17103 −0.0830114
\(685\) −0.563779 −0.0215409
\(686\) 0 0
\(687\) 19.1156 0.729306
\(688\) −0.773540 −0.0294909
\(689\) 12.5685 0.478822
\(690\) −6.68412 −0.254460
\(691\) −32.3128 −1.22924 −0.614619 0.788824i \(-0.710690\pi\)
−0.614619 + 0.788824i \(0.710690\pi\)
\(692\) 13.2866 0.505082
\(693\) 0 0
\(694\) 11.3528 0.430946
\(695\) −8.99526 −0.341210
\(696\) −3.00000 −0.113715
\(697\) −93.7472 −3.55093
\(698\) −14.8999 −0.563968
\(699\) 21.0709 0.796974
\(700\) 0 0
\(701\) −25.7657 −0.973158 −0.486579 0.873637i \(-0.661755\pi\)
−0.486579 + 0.873637i \(0.661755\pi\)
\(702\) 1.00000 0.0377426
\(703\) −16.8999 −0.637390
\(704\) −3.55780 −0.134090
\(705\) 7.69484 0.289804
\(706\) 13.1370 0.494419
\(707\) 0 0
\(708\) −4.78426 −0.179803
\(709\) −7.12158 −0.267457 −0.133728 0.991018i \(-0.542695\pi\)
−0.133728 + 0.991018i \(0.542695\pi\)
\(710\) 20.4589 0.767809
\(711\) −2.39749 −0.0899129
\(712\) −4.21574 −0.157992
\(713\) 2.12034 0.0794074
\(714\) 0 0
\(715\) −9.29735 −0.347701
\(716\) −14.7950 −0.552914
\(717\) −7.39749 −0.276264
\(718\) 18.0000 0.671754
\(719\) 13.7783 0.513843 0.256922 0.966432i \(-0.417292\pi\)
0.256922 + 0.966432i \(0.417292\pi\)
\(720\) 2.61323 0.0973893
\(721\) 0 0
\(722\) 14.2866 0.531693
\(723\) −21.6239 −0.804203
\(724\) −1.95529 −0.0726678
\(725\) −5.48691 −0.203779
\(726\) 1.65794 0.0615319
\(727\) −9.62395 −0.356933 −0.178466 0.983946i \(-0.557114\pi\)
−0.178466 + 0.983946i \(0.557114\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −27.5900 −1.02115
\(731\) 6.15374 0.227604
\(732\) −6.72883 −0.248705
\(733\) −26.0369 −0.961695 −0.480847 0.876804i \(-0.659671\pi\)
−0.480847 + 0.876804i \(0.659671\pi\)
\(734\) −1.71337 −0.0632417
\(735\) 0 0
\(736\) 2.55780 0.0942818
\(737\) −51.6347 −1.90199
\(738\) −11.7843 −0.433785
\(739\) 44.7056 1.64452 0.822260 0.569112i \(-0.192713\pi\)
0.822260 + 0.569112i \(0.192713\pi\)
\(740\) 20.3421 0.747789
\(741\) 2.17103 0.0797548
\(742\) 0 0
\(743\) 31.1972 1.14451 0.572257 0.820074i \(-0.306068\pi\)
0.572257 + 0.820074i \(0.306068\pi\)
\(744\) −0.828970 −0.0303915
\(745\) −24.6746 −0.904009
\(746\) −25.9767 −0.951076
\(747\) 15.5131 0.567594
\(748\) 28.3033 1.03487
\(749\) 0 0
\(750\) −8.28663 −0.302585
\(751\) −4.17577 −0.152376 −0.0761880 0.997093i \(-0.524275\pi\)
−0.0761880 + 0.997093i \(0.524275\pi\)
\(752\) −2.94457 −0.107377
\(753\) 25.9660 0.946254
\(754\) 3.00000 0.109254
\(755\) −16.8349 −0.612687
\(756\) 0 0
\(757\) −5.38203 −0.195613 −0.0978066 0.995205i \(-0.531183\pi\)
−0.0978066 + 0.995205i \(0.531183\pi\)
\(758\) 14.4529 0.524954
\(759\) −9.10014 −0.330314
\(760\) 5.67340 0.205796
\(761\) 53.1585 1.92699 0.963497 0.267720i \(-0.0862703\pi\)
0.963497 + 0.267720i \(0.0862703\pi\)
\(762\) 19.2973 0.699069
\(763\) 0 0
\(764\) −4.11086 −0.148726
\(765\) −20.7890 −0.751628
\(766\) −20.4529 −0.738994
\(767\) 4.78426 0.172750
\(768\) −1.00000 −0.0360844
\(769\) −38.7764 −1.39831 −0.699157 0.714968i \(-0.746441\pi\)
−0.699157 + 0.714968i \(0.746441\pi\)
\(770\) 0 0
\(771\) 11.1156 0.400319
\(772\) −10.1817 −0.366449
\(773\) −10.2217 −0.367650 −0.183825 0.982959i \(-0.558848\pi\)
−0.183825 + 0.982959i \(0.558848\pi\)
\(774\) 0.773540 0.0278043
\(775\) −1.51616 −0.0544621
\(776\) −10.9553 −0.393272
\(777\) 0 0
\(778\) −23.3975 −0.838841
\(779\) −25.5840 −0.916641
\(780\) −2.61323 −0.0935686
\(781\) 27.8539 0.996691
\(782\) −20.3480 −0.727644
\(783\) 3.00000 0.107211
\(784\) 0 0
\(785\) 17.8737 0.637939
\(786\) 19.4082 0.692268
\(787\) 3.28663 0.117156 0.0585778 0.998283i \(-0.481343\pi\)
0.0585778 + 0.998283i \(0.481343\pi\)
\(788\) −9.44220 −0.336364
\(789\) −8.44694 −0.300719
\(790\) 6.26519 0.222905
\(791\) 0 0
\(792\) 3.55780 0.126421
\(793\) 6.72883 0.238948
\(794\) −22.4684 −0.797373
\(795\) −32.8444 −1.16487
\(796\) 13.2419 0.469347
\(797\) 36.4284 1.29036 0.645180 0.764030i \(-0.276782\pi\)
0.645180 + 0.764030i \(0.276782\pi\)
\(798\) 0 0
\(799\) 23.4249 0.828714
\(800\) −1.82897 −0.0646639
\(801\) 4.21574 0.148956
\(802\) −2.89986 −0.102398
\(803\) −37.5625 −1.32555
\(804\) −14.5131 −0.511837
\(805\) 0 0
\(806\) 0.828970 0.0291992
\(807\) −11.2312 −0.395357
\(808\) −10.3421 −0.363832
\(809\) 43.6656 1.53520 0.767600 0.640929i \(-0.221451\pi\)
0.767600 + 0.640929i \(0.221451\pi\)
\(810\) −2.61323 −0.0918195
\(811\) 21.8891 0.768632 0.384316 0.923202i \(-0.374437\pi\)
0.384316 + 0.923202i \(0.374437\pi\)
\(812\) 0 0
\(813\) 13.7890 0.483601
\(814\) 27.6948 0.970704
\(815\) −32.9893 −1.15556
\(816\) 7.95529 0.278491
\(817\) 1.67938 0.0587540
\(818\) −5.27591 −0.184468
\(819\) 0 0
\(820\) 30.7950 1.07541
\(821\) 48.3450 1.68725 0.843625 0.536932i \(-0.180417\pi\)
0.843625 + 0.536932i \(0.180417\pi\)
\(822\) −0.215740 −0.00752481
\(823\) 28.2157 0.983539 0.491769 0.870725i \(-0.336350\pi\)
0.491769 + 0.870725i \(0.336350\pi\)
\(824\) 3.22646 0.112399
\(825\) 6.50711 0.226548
\(826\) 0 0
\(827\) 29.2586 1.01742 0.508711 0.860937i \(-0.330122\pi\)
0.508711 + 0.860937i \(0.330122\pi\)
\(828\) −2.55780 −0.0888897
\(829\) 44.0876 1.53123 0.765613 0.643302i \(-0.222436\pi\)
0.765613 + 0.643302i \(0.222436\pi\)
\(830\) −40.5393 −1.40714
\(831\) −20.2973 −0.704107
\(832\) 1.00000 0.0346688
\(833\) 0 0
\(834\) −3.44220 −0.119194
\(835\) 9.71628 0.336246
\(836\) 7.72409 0.267143
\(837\) 0.828970 0.0286534
\(838\) 3.22646 0.111456
\(839\) −39.9707 −1.37994 −0.689972 0.723836i \(-0.742377\pi\)
−0.689972 + 0.723836i \(0.742377\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −34.5792 −1.19168
\(843\) −19.1370 −0.659115
\(844\) 1.33134 0.0458266
\(845\) 2.61323 0.0898978
\(846\) 2.94457 0.101236
\(847\) 0 0
\(848\) 12.5685 0.431605
\(849\) −0.326600 −0.0112089
\(850\) 14.5500 0.499060
\(851\) −19.9106 −0.682526
\(852\) 7.82897 0.268216
\(853\) 7.10962 0.243429 0.121714 0.992565i \(-0.461161\pi\)
0.121714 + 0.992565i \(0.461161\pi\)
\(854\) 0 0
\(855\) −5.67340 −0.194026
\(856\) −8.28663 −0.283231
\(857\) −31.4344 −1.07378 −0.536889 0.843653i \(-0.680401\pi\)
−0.536889 + 0.843653i \(0.680401\pi\)
\(858\) −3.55780 −0.121461
\(859\) −27.5685 −0.940626 −0.470313 0.882500i \(-0.655859\pi\)
−0.470313 + 0.882500i \(0.655859\pi\)
\(860\) −2.02144 −0.0689305
\(861\) 0 0
\(862\) −12.4529 −0.424148
\(863\) 14.2312 0.484436 0.242218 0.970222i \(-0.422125\pi\)
0.242218 + 0.970222i \(0.422125\pi\)
\(864\) 1.00000 0.0340207
\(865\) 34.7210 1.18055
\(866\) 5.62869 0.191271
\(867\) −46.2866 −1.57198
\(868\) 0 0
\(869\) 8.52979 0.289353
\(870\) −7.83969 −0.265790
\(871\) 14.5131 0.491757
\(872\) −4.00000 −0.135457
\(873\) 10.9553 0.370780
\(874\) −5.55306 −0.187835
\(875\) 0 0
\(876\) −10.5578 −0.356715
\(877\) 36.1478 1.22062 0.610312 0.792161i \(-0.291044\pi\)
0.610312 + 0.792161i \(0.291044\pi\)
\(878\) −12.5238 −0.422658
\(879\) −8.61323 −0.290517
\(880\) −9.29735 −0.313414
\(881\) −10.8939 −0.367024 −0.183512 0.983017i \(-0.558747\pi\)
−0.183512 + 0.983017i \(0.558747\pi\)
\(882\) 0 0
\(883\) 56.2681 1.89357 0.946786 0.321863i \(-0.104309\pi\)
0.946786 + 0.321863i \(0.104309\pi\)
\(884\) −7.95529 −0.267565
\(885\) −12.5024 −0.420263
\(886\) 8.50237 0.285643
\(887\) −28.1632 −0.945628 −0.472814 0.881162i \(-0.656762\pi\)
−0.472814 + 0.881162i \(0.656762\pi\)
\(888\) 7.78426 0.261223
\(889\) 0 0
\(890\) −11.0167 −0.369281
\(891\) −3.55780 −0.119191
\(892\) 22.3528 0.748426
\(893\) 6.39275 0.213925
\(894\) −9.44220 −0.315794
\(895\) −38.6627 −1.29235
\(896\) 0 0
\(897\) 2.55780 0.0854025
\(898\) 34.9153 1.16514
\(899\) 2.48691 0.0829431
\(900\) 1.82897 0.0609657
\(901\) −99.9862 −3.33102
\(902\) 41.9260 1.39598
\(903\) 0 0
\(904\) 17.1817 0.571456
\(905\) −5.10962 −0.169850
\(906\) −6.44220 −0.214028
\(907\) −26.4469 −0.878156 −0.439078 0.898449i \(-0.644695\pi\)
−0.439078 + 0.898449i \(0.644695\pi\)
\(908\) −19.8551 −0.658916
\(909\) 10.3421 0.343024
\(910\) 0 0
\(911\) 46.0000 1.52405 0.762024 0.647549i \(-0.224206\pi\)
0.762024 + 0.647549i \(0.224206\pi\)
\(912\) 2.17103 0.0718900
\(913\) −55.1925 −1.82660
\(914\) −1.83969 −0.0608515
\(915\) −17.5840 −0.581308
\(916\) −19.1156 −0.631597
\(917\) 0 0
\(918\) −7.95529 −0.262564
\(919\) 2.54234 0.0838641 0.0419320 0.999120i \(-0.486649\pi\)
0.0419320 + 0.999120i \(0.486649\pi\)
\(920\) 6.68412 0.220369
\(921\) −28.1710 −0.928267
\(922\) 23.4577 0.772537
\(923\) −7.82897 −0.257694
\(924\) 0 0
\(925\) 14.2372 0.468116
\(926\) 22.4529 0.737849
\(927\) −3.22646 −0.105971
\(928\) 3.00000 0.0984798
\(929\) −5.88914 −0.193216 −0.0966082 0.995322i \(-0.530799\pi\)
−0.0966082 + 0.995322i \(0.530799\pi\)
\(930\) −2.16629 −0.0710354
\(931\) 0 0
\(932\) −21.0709 −0.690200
\(933\) 24.2372 0.793490
\(934\) −8.98928 −0.294138
\(935\) 73.9631 2.41885
\(936\) −1.00000 −0.0326860
\(937\) 21.3635 0.697915 0.348958 0.937139i \(-0.386536\pi\)
0.348958 + 0.937139i \(0.386536\pi\)
\(938\) 0 0
\(939\) −22.0554 −0.719752
\(940\) −7.69484 −0.250978
\(941\) −17.9446 −0.584976 −0.292488 0.956269i \(-0.594483\pi\)
−0.292488 + 0.956269i \(0.594483\pi\)
\(942\) 6.83969 0.222849
\(943\) −30.1418 −0.981551
\(944\) 4.78426 0.155714
\(945\) 0 0
\(946\) −2.75210 −0.0894785
\(947\) 45.6441 1.48324 0.741618 0.670823i \(-0.234059\pi\)
0.741618 + 0.670823i \(0.234059\pi\)
\(948\) 2.39749 0.0778668
\(949\) 10.5578 0.342721
\(950\) 3.97075 0.128828
\(951\) −8.48691 −0.275207
\(952\) 0 0
\(953\) 55.8444 1.80898 0.904489 0.426496i \(-0.140252\pi\)
0.904489 + 0.426496i \(0.140252\pi\)
\(954\) −12.5685 −0.406921
\(955\) −10.7426 −0.347623
\(956\) 7.39749 0.239252
\(957\) −10.6734 −0.345022
\(958\) −35.5178 −1.14753
\(959\) 0 0
\(960\) −2.61323 −0.0843416
\(961\) −30.3128 −0.977833
\(962\) −7.78426 −0.250975
\(963\) 8.28663 0.267033
\(964\) 21.6239 0.696461
\(965\) −26.6073 −0.856518
\(966\) 0 0
\(967\) −32.2419 −1.03683 −0.518415 0.855129i \(-0.673478\pi\)
−0.518415 + 0.855129i \(0.673478\pi\)
\(968\) −1.65794 −0.0532882
\(969\) −17.2712 −0.554830
\(970\) −28.6287 −0.919212
\(971\) 42.7764 1.37276 0.686381 0.727242i \(-0.259198\pi\)
0.686381 + 0.727242i \(0.259198\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −18.0107 −0.577101
\(975\) −1.82897 −0.0585739
\(976\) 6.72883 0.215385
\(977\) 11.1156 0.355620 0.177810 0.984065i \(-0.443099\pi\)
0.177810 + 0.984065i \(0.443099\pi\)
\(978\) −12.6239 −0.403669
\(979\) −14.9988 −0.479362
\(980\) 0 0
\(981\) 4.00000 0.127710
\(982\) 12.8504 0.410073
\(983\) −12.5131 −0.399106 −0.199553 0.979887i \(-0.563949\pi\)
−0.199553 + 0.979887i \(0.563949\pi\)
\(984\) 11.7843 0.375668
\(985\) −24.6746 −0.786199
\(986\) −23.8659 −0.760044
\(987\) 0 0
\(988\) −2.17103 −0.0690697
\(989\) 1.97856 0.0629146
\(990\) 9.29735 0.295489
\(991\) −44.9707 −1.42854 −0.714271 0.699869i \(-0.753242\pi\)
−0.714271 + 0.699869i \(0.753242\pi\)
\(992\) 0.828970 0.0263198
\(993\) 29.0262 0.921118
\(994\) 0 0
\(995\) 34.6042 1.09703
\(996\) −15.5131 −0.491551
\(997\) 18.1770 0.575672 0.287836 0.957680i \(-0.407064\pi\)
0.287836 + 0.957680i \(0.407064\pi\)
\(998\) −3.00474 −0.0951134
\(999\) −7.78426 −0.246283
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3822.2.a.bv.1.2 3
7.2 even 3 546.2.i.k.235.2 yes 6
7.4 even 3 546.2.i.k.79.2 6
7.6 odd 2 3822.2.a.bw.1.2 3
21.2 odd 6 1638.2.j.q.235.2 6
21.11 odd 6 1638.2.j.q.1171.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.i.k.79.2 6 7.4 even 3
546.2.i.k.235.2 yes 6 7.2 even 3
1638.2.j.q.235.2 6 21.2 odd 6
1638.2.j.q.1171.2 6 21.11 odd 6
3822.2.a.bv.1.2 3 1.1 even 1 trivial
3822.2.a.bw.1.2 3 7.6 odd 2