Properties

Label 1110.2.a.s.1.3
Level $1110$
Weight $2$
Character 1110.1
Self dual yes
Analytic conductor $8.863$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.54764.1
Defining polynomial: \(x^{4} - x^{3} - 9 x^{2} + 3 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.339102\) of defining polynomial
Character \(\chi\) \(=\) 1110.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.84556 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.84556 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +0.832640 q^{11} +1.00000 q^{12} +3.21973 q^{13} +1.84556 q^{14} +1.00000 q^{15} +1.00000 q^{16} -7.95030 q^{17} +1.00000 q^{18} +4.20681 q^{19} +1.00000 q^{20} +1.84556 q^{21} +0.832640 q^{22} -6.57614 q^{23} +1.00000 q^{24} +1.00000 q^{25} +3.21973 q^{26} +1.00000 q^{27} +1.84556 q^{28} +3.03945 q^{29} +1.00000 q^{30} -4.73057 q^{31} +1.00000 q^{32} +0.832640 q^{33} -7.95030 q^{34} +1.84556 q^{35} +1.00000 q^{36} -1.00000 q^{37} +4.20681 q^{38} +3.21973 q^{39} +1.00000 q^{40} +2.73057 q^{41} +1.84556 q^{42} -5.03945 q^{43} +0.832640 q^{44} +1.00000 q^{45} -6.57614 q^{46} +9.11766 q^{47} +1.00000 q^{48} -3.59390 q^{49} +1.00000 q^{50} -7.95030 q^{51} +3.21973 q^{52} +3.95030 q^{53} +1.00000 q^{54} +0.832640 q^{55} +1.84556 q^{56} +4.20681 q^{57} +3.03945 q^{58} -3.69113 q^{59} +1.00000 q^{60} +5.40878 q^{61} -4.73057 q^{62} +1.84556 q^{63} +1.00000 q^{64} +3.21973 q^{65} +0.832640 q^{66} +11.7959 q^{67} -7.95030 q^{68} -6.57614 q^{69} +1.84556 q^{70} -13.7700 q^{71} +1.00000 q^{72} -3.52860 q^{73} -1.00000 q^{74} +1.00000 q^{75} +4.20681 q^{76} +1.53669 q^{77} +3.21973 q^{78} +4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.73057 q^{82} -3.55445 q^{83} +1.84556 q^{84} -7.95030 q^{85} -5.03945 q^{86} +3.03945 q^{87} +0.832640 q^{88} -6.55029 q^{89} +1.00000 q^{90} +5.94222 q^{91} -6.57614 q^{92} -4.73057 q^{93} +9.11766 q^{94} +4.20681 q^{95} +1.00000 q^{96} -10.4918 q^{97} -3.59390 q^{98} +0.832640 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 4q^{3} + 4q^{4} + 4q^{5} + 4q^{6} + 4q^{7} + 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{2} + 4q^{3} + 4q^{4} + 4q^{5} + 4q^{6} + 4q^{7} + 4q^{8} + 4q^{9} + 4q^{10} + 2q^{11} + 4q^{12} - 3q^{13} + 4q^{14} + 4q^{15} + 4q^{16} + 6q^{17} + 4q^{18} + 3q^{19} + 4q^{20} + 4q^{21} + 2q^{22} - q^{23} + 4q^{24} + 4q^{25} - 3q^{26} + 4q^{27} + 4q^{28} - 3q^{29} + 4q^{30} + 3q^{31} + 4q^{32} + 2q^{33} + 6q^{34} + 4q^{35} + 4q^{36} - 4q^{37} + 3q^{38} - 3q^{39} + 4q^{40} - 11q^{41} + 4q^{42} - 5q^{43} + 2q^{44} + 4q^{45} - q^{46} + 4q^{48} + 14q^{49} + 4q^{50} + 6q^{51} - 3q^{52} - 22q^{53} + 4q^{54} + 2q^{55} + 4q^{56} + 3q^{57} - 3q^{58} - 8q^{59} + 4q^{60} - 5q^{61} + 3q^{62} + 4q^{63} + 4q^{64} - 3q^{65} + 2q^{66} + 6q^{67} + 6q^{68} - q^{69} + 4q^{70} - 18q^{71} + 4q^{72} - 5q^{73} - 4q^{74} + 4q^{75} + 3q^{76} - 4q^{77} - 3q^{78} + 16q^{79} + 4q^{80} + 4q^{81} - 11q^{82} - q^{83} + 4q^{84} + 6q^{85} - 5q^{86} - 3q^{87} + 2q^{88} - 5q^{89} + 4q^{90} - 13q^{91} - q^{92} + 3q^{93} + 3q^{95} + 4q^{96} + 7q^{97} + 14q^{98} + 2q^{99} + O(q^{100}) \)

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
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) 1.84556 0.697557 0.348779 0.937205i \(-0.386596\pi\)
0.348779 + 0.937205i \(0.386596\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 0.832640 0.251050 0.125525 0.992090i \(-0.459938\pi\)
0.125525 + 0.992090i \(0.459938\pi\)
\(12\) 1.00000 0.288675
\(13\) 3.21973 0.892992 0.446496 0.894786i \(-0.352672\pi\)
0.446496 + 0.894786i \(0.352672\pi\)
\(14\) 1.84556 0.493248
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −7.95030 −1.92823 −0.964116 0.265482i \(-0.914469\pi\)
−0.964116 + 0.265482i \(0.914469\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.20681 0.965108 0.482554 0.875866i \(-0.339709\pi\)
0.482554 + 0.875866i \(0.339709\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.84556 0.402735
\(22\) 0.832640 0.177519
\(23\) −6.57614 −1.37122 −0.685610 0.727969i \(-0.740464\pi\)
−0.685610 + 0.727969i \(0.740464\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 3.21973 0.631441
\(27\) 1.00000 0.192450
\(28\) 1.84556 0.348779
\(29\) 3.03945 0.564411 0.282206 0.959354i \(-0.408934\pi\)
0.282206 + 0.959354i \(0.408934\pi\)
\(30\) 1.00000 0.182574
\(31\) −4.73057 −0.849636 −0.424818 0.905279i \(-0.639662\pi\)
−0.424818 + 0.905279i \(0.639662\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.832640 0.144944
\(34\) −7.95030 −1.36347
\(35\) 1.84556 0.311957
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399
\(38\) 4.20681 0.682434
\(39\) 3.21973 0.515569
\(40\) 1.00000 0.158114
\(41\) 2.73057 0.426444 0.213222 0.977004i \(-0.431604\pi\)
0.213222 + 0.977004i \(0.431604\pi\)
\(42\) 1.84556 0.284777
\(43\) −5.03945 −0.768508 −0.384254 0.923227i \(-0.625541\pi\)
−0.384254 + 0.923227i \(0.625541\pi\)
\(44\) 0.832640 0.125525
\(45\) 1.00000 0.149071
\(46\) −6.57614 −0.969598
\(47\) 9.11766 1.32995 0.664974 0.746867i \(-0.268443\pi\)
0.664974 + 0.746867i \(0.268443\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.59390 −0.513414
\(50\) 1.00000 0.141421
\(51\) −7.95030 −1.11327
\(52\) 3.21973 0.446496
\(53\) 3.95030 0.542616 0.271308 0.962493i \(-0.412544\pi\)
0.271308 + 0.962493i \(0.412544\pi\)
\(54\) 1.00000 0.136083
\(55\) 0.832640 0.112273
\(56\) 1.84556 0.246624
\(57\) 4.20681 0.557205
\(58\) 3.03945 0.399099
\(59\) −3.69113 −0.480544 −0.240272 0.970706i \(-0.577237\pi\)
−0.240272 + 0.970706i \(0.577237\pi\)
\(60\) 1.00000 0.129099
\(61\) 5.40878 0.692523 0.346261 0.938138i \(-0.387451\pi\)
0.346261 + 0.938138i \(0.387451\pi\)
\(62\) −4.73057 −0.600783
\(63\) 1.84556 0.232519
\(64\) 1.00000 0.125000
\(65\) 3.21973 0.399358
\(66\) 0.832640 0.102491
\(67\) 11.7959 1.44109 0.720547 0.693406i \(-0.243891\pi\)
0.720547 + 0.693406i \(0.243891\pi\)
\(68\) −7.95030 −0.964116
\(69\) −6.57614 −0.791674
\(70\) 1.84556 0.220587
\(71\) −13.7700 −1.63420 −0.817100 0.576495i \(-0.804420\pi\)
−0.817100 + 0.576495i \(0.804420\pi\)
\(72\) 1.00000 0.117851
\(73\) −3.52860 −0.412992 −0.206496 0.978447i \(-0.566206\pi\)
−0.206496 + 0.978447i \(0.566206\pi\)
\(74\) −1.00000 −0.116248
\(75\) 1.00000 0.115470
\(76\) 4.20681 0.482554
\(77\) 1.53669 0.175122
\(78\) 3.21973 0.364563
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 2.73057 0.301541
\(83\) −3.55445 −0.390151 −0.195076 0.980788i \(-0.562495\pi\)
−0.195076 + 0.980788i \(0.562495\pi\)
\(84\) 1.84556 0.201367
\(85\) −7.95030 −0.862331
\(86\) −5.03945 −0.543417
\(87\) 3.03945 0.325863
\(88\) 0.832640 0.0887597
\(89\) −6.55029 −0.694329 −0.347165 0.937804i \(-0.612856\pi\)
−0.347165 + 0.937804i \(0.612856\pi\)
\(90\) 1.00000 0.105409
\(91\) 5.94222 0.622913
\(92\) −6.57614 −0.685610
\(93\) −4.73057 −0.490538
\(94\) 9.11766 0.940415
\(95\) 4.20681 0.431609
\(96\) 1.00000 0.102062
\(97\) −10.4918 −1.06528 −0.532642 0.846341i \(-0.678801\pi\)
−0.532642 + 0.846341i \(0.678801\pi\)
\(98\) −3.59390 −0.363038
\(99\) 0.832640 0.0836835
\(100\) 1.00000 0.100000
\(101\) −3.63067 −0.361265 −0.180633 0.983551i \(-0.557814\pi\)
−0.180633 + 0.983551i \(0.557814\pi\)
\(102\) −7.95030 −0.787197
\(103\) −9.11766 −0.898390 −0.449195 0.893434i \(-0.648289\pi\)
−0.449195 + 0.893434i \(0.648289\pi\)
\(104\) 3.21973 0.315720
\(105\) 1.84556 0.180109
\(106\) 3.95030 0.383687
\(107\) 8.91086 0.861445 0.430723 0.902484i \(-0.358259\pi\)
0.430723 + 0.902484i \(0.358259\pi\)
\(108\) 1.00000 0.0962250
\(109\) 11.2979 1.08215 0.541073 0.840975i \(-0.318018\pi\)
0.541073 + 0.840975i \(0.318018\pi\)
\(110\) 0.832640 0.0793891
\(111\) −1.00000 −0.0949158
\(112\) 1.84556 0.174389
\(113\) −11.0394 −1.03850 −0.519252 0.854621i \(-0.673789\pi\)
−0.519252 + 0.854621i \(0.673789\pi\)
\(114\) 4.20681 0.394004
\(115\) −6.57614 −0.613228
\(116\) 3.03945 0.282206
\(117\) 3.21973 0.297664
\(118\) −3.69113 −0.339796
\(119\) −14.6728 −1.34505
\(120\) 1.00000 0.0912871
\(121\) −10.3067 −0.936974
\(122\) 5.40878 0.489688
\(123\) 2.73057 0.246208
\(124\) −4.73057 −0.424818
\(125\) 1.00000 0.0894427
\(126\) 1.84556 0.164416
\(127\) −0.136678 −0.0121282 −0.00606410 0.999982i \(-0.501930\pi\)
−0.00606410 + 0.999982i \(0.501930\pi\)
\(128\) 1.00000 0.0883883
\(129\) −5.03945 −0.443699
\(130\) 3.21973 0.282389
\(131\) 2.07889 0.181634 0.0908169 0.995868i \(-0.471052\pi\)
0.0908169 + 0.995868i \(0.471052\pi\)
\(132\) 0.832640 0.0724720
\(133\) 7.76393 0.673218
\(134\) 11.7959 1.01901
\(135\) 1.00000 0.0860663
\(136\) −7.95030 −0.681733
\(137\) 3.06597 0.261943 0.130972 0.991386i \(-0.458190\pi\)
0.130972 + 0.991386i \(0.458190\pi\)
\(138\) −6.57614 −0.559798
\(139\) 22.1917 1.88228 0.941139 0.338021i \(-0.109757\pi\)
0.941139 + 0.338021i \(0.109757\pi\)
\(140\) 1.84556 0.155979
\(141\) 9.11766 0.767846
\(142\) −13.7700 −1.15555
\(143\) 2.68088 0.224186
\(144\) 1.00000 0.0833333
\(145\) 3.03945 0.252412
\(146\) −3.52860 −0.292029
\(147\) −3.59390 −0.296420
\(148\) −1.00000 −0.0821995
\(149\) −14.4999 −1.18788 −0.593940 0.804510i \(-0.702428\pi\)
−0.593940 + 0.804510i \(0.702428\pi\)
\(150\) 1.00000 0.0816497
\(151\) 8.24142 0.670677 0.335339 0.942098i \(-0.391149\pi\)
0.335339 + 0.942098i \(0.391149\pi\)
\(152\) 4.20681 0.341217
\(153\) −7.95030 −0.642744
\(154\) 1.53669 0.123830
\(155\) −4.73057 −0.379969
\(156\) 3.21973 0.257785
\(157\) −6.11283 −0.487857 −0.243928 0.969793i \(-0.578436\pi\)
−0.243928 + 0.969793i \(0.578436\pi\)
\(158\) 4.00000 0.318223
\(159\) 3.95030 0.313279
\(160\) 1.00000 0.0790569
\(161\) −12.1367 −0.956504
\(162\) 1.00000 0.0785674
\(163\) −21.4373 −1.67910 −0.839549 0.543283i \(-0.817181\pi\)
−0.839549 + 0.543283i \(0.817181\pi\)
\(164\) 2.73057 0.213222
\(165\) 0.832640 0.0648210
\(166\) −3.55445 −0.275879
\(167\) −22.5815 −1.74741 −0.873704 0.486458i \(-0.838289\pi\)
−0.873704 + 0.486458i \(0.838289\pi\)
\(168\) 1.84556 0.142388
\(169\) −2.63334 −0.202565
\(170\) −7.95030 −0.609760
\(171\) 4.20681 0.321703
\(172\) −5.03945 −0.384254
\(173\) −6.48916 −0.493361 −0.246681 0.969097i \(-0.579340\pi\)
−0.246681 + 0.969097i \(0.579340\pi\)
\(174\) 3.03945 0.230420
\(175\) 1.84556 0.139511
\(176\) 0.832640 0.0627626
\(177\) −3.69113 −0.277442
\(178\) −6.55029 −0.490965
\(179\) 16.2095 1.21155 0.605777 0.795635i \(-0.292862\pi\)
0.605777 + 0.795635i \(0.292862\pi\)
\(180\) 1.00000 0.0745356
\(181\) 5.38225 0.400060 0.200030 0.979790i \(-0.435896\pi\)
0.200030 + 0.979790i \(0.435896\pi\)
\(182\) 5.94222 0.440466
\(183\) 5.40878 0.399828
\(184\) −6.57614 −0.484799
\(185\) −1.00000 −0.0735215
\(186\) −4.73057 −0.346862
\(187\) −6.61974 −0.484083
\(188\) 9.11766 0.664974
\(189\) 1.84556 0.134245
\(190\) 4.20681 0.305194
\(191\) −10.5680 −0.764677 −0.382339 0.924022i \(-0.624881\pi\)
−0.382339 + 0.924022i \(0.624881\pi\)
\(192\) 1.00000 0.0721688
\(193\) 3.11766 0.224414 0.112207 0.993685i \(-0.464208\pi\)
0.112207 + 0.993685i \(0.464208\pi\)
\(194\) −10.4918 −0.753269
\(195\) 3.21973 0.230570
\(196\) −3.59390 −0.256707
\(197\) 0.576137 0.0410481 0.0205240 0.999789i \(-0.493467\pi\)
0.0205240 + 0.999789i \(0.493467\pi\)
\(198\) 0.832640 0.0591732
\(199\) 6.64359 0.470952 0.235476 0.971880i \(-0.424335\pi\)
0.235476 + 0.971880i \(0.424335\pi\)
\(200\) 1.00000 0.0707107
\(201\) 11.7959 0.832016
\(202\) −3.63067 −0.255453
\(203\) 5.60949 0.393709
\(204\) −7.95030 −0.556633
\(205\) 2.73057 0.190712
\(206\) −9.11766 −0.635258
\(207\) −6.57614 −0.457073
\(208\) 3.21973 0.223248
\(209\) 3.50276 0.242291
\(210\) 1.84556 0.127356
\(211\) −8.83531 −0.608248 −0.304124 0.952632i \(-0.598364\pi\)
−0.304124 + 0.952632i \(0.598364\pi\)
\(212\) 3.95030 0.271308
\(213\) −13.7700 −0.943506
\(214\) 8.91086 0.609134
\(215\) −5.03945 −0.343687
\(216\) 1.00000 0.0680414
\(217\) −8.73057 −0.592670
\(218\) 11.2979 0.765193
\(219\) −3.52860 −0.238441
\(220\) 0.832640 0.0561366
\(221\) −25.5978 −1.72190
\(222\) −1.00000 −0.0671156
\(223\) 17.8828 1.19752 0.598762 0.800927i \(-0.295660\pi\)
0.598762 + 0.800927i \(0.295660\pi\)
\(224\) 1.84556 0.123312
\(225\) 1.00000 0.0666667
\(226\) −11.0394 −0.734333
\(227\) 25.1959 1.67231 0.836155 0.548494i \(-0.184799\pi\)
0.836155 + 0.548494i \(0.184799\pi\)
\(228\) 4.20681 0.278603
\(229\) −5.48699 −0.362591 −0.181295 0.983429i \(-0.558029\pi\)
−0.181295 + 0.983429i \(0.558029\pi\)
\(230\) −6.57614 −0.433618
\(231\) 1.53669 0.101107
\(232\) 3.03945 0.199549
\(233\) −28.5788 −1.87226 −0.936130 0.351654i \(-0.885619\pi\)
−0.936130 + 0.351654i \(0.885619\pi\)
\(234\) 3.21973 0.210480
\(235\) 9.11766 0.594771
\(236\) −3.69113 −0.240272
\(237\) 4.00000 0.259828
\(238\) −14.6728 −0.951096
\(239\) 1.77811 0.115016 0.0575081 0.998345i \(-0.481684\pi\)
0.0575081 + 0.998345i \(0.481684\pi\)
\(240\) 1.00000 0.0645497
\(241\) 22.2612 1.43397 0.716984 0.697090i \(-0.245522\pi\)
0.716984 + 0.697090i \(0.245522\pi\)
\(242\) −10.3067 −0.662540
\(243\) 1.00000 0.0641500
\(244\) 5.40878 0.346261
\(245\) −3.59390 −0.229606
\(246\) 2.73057 0.174095
\(247\) 13.5448 0.861834
\(248\) −4.73057 −0.300392
\(249\) −3.55445 −0.225254
\(250\) 1.00000 0.0632456
\(251\) −28.6748 −1.80994 −0.904968 0.425479i \(-0.860106\pi\)
−0.904968 + 0.425479i \(0.860106\pi\)
\(252\) 1.84556 0.116260
\(253\) −5.47556 −0.344245
\(254\) −0.136678 −0.00857593
\(255\) −7.95030 −0.497867
\(256\) 1.00000 0.0625000
\(257\) 9.19388 0.573499 0.286749 0.958006i \(-0.407425\pi\)
0.286749 + 0.958006i \(0.407425\pi\)
\(258\) −5.03945 −0.313742
\(259\) −1.84556 −0.114678
\(260\) 3.21973 0.199679
\(261\) 3.03945 0.188137
\(262\) 2.07889 0.128434
\(263\) −8.02652 −0.494937 −0.247468 0.968896i \(-0.579599\pi\)
−0.247468 + 0.968896i \(0.579599\pi\)
\(264\) 0.832640 0.0512455
\(265\) 3.95030 0.242665
\(266\) 7.76393 0.476037
\(267\) −6.55029 −0.400871
\(268\) 11.7959 0.720547
\(269\) 21.3849 1.30386 0.651931 0.758278i \(-0.273959\pi\)
0.651931 + 0.758278i \(0.273959\pi\)
\(270\) 1.00000 0.0608581
\(271\) 2.64359 0.160587 0.0802934 0.996771i \(-0.474414\pi\)
0.0802934 + 0.996771i \(0.474414\pi\)
\(272\) −7.95030 −0.482058
\(273\) 5.94222 0.359639
\(274\) 3.06597 0.185222
\(275\) 0.832640 0.0502101
\(276\) −6.57614 −0.395837
\(277\) −14.4456 −0.867949 −0.433975 0.900925i \(-0.642889\pi\)
−0.433975 + 0.900925i \(0.642889\pi\)
\(278\) 22.1917 1.33097
\(279\) −4.73057 −0.283212
\(280\) 1.84556 0.110294
\(281\) −11.2197 −0.669313 −0.334656 0.942340i \(-0.608620\pi\)
−0.334656 + 0.942340i \(0.608620\pi\)
\(282\) 9.11766 0.542949
\(283\) 3.86332 0.229651 0.114825 0.993386i \(-0.463369\pi\)
0.114825 + 0.993386i \(0.463369\pi\)
\(284\) −13.7700 −0.817100
\(285\) 4.20681 0.249190
\(286\) 2.68088 0.158524
\(287\) 5.03945 0.297469
\(288\) 1.00000 0.0589256
\(289\) 46.2073 2.71808
\(290\) 3.03945 0.178482
\(291\) −10.4918 −0.615042
\(292\) −3.52860 −0.206496
\(293\) −16.1544 −0.943752 −0.471876 0.881665i \(-0.656423\pi\)
−0.471876 + 0.881665i \(0.656423\pi\)
\(294\) −3.59390 −0.209600
\(295\) −3.69113 −0.214906
\(296\) −1.00000 −0.0581238
\(297\) 0.832640 0.0483147
\(298\) −14.4999 −0.839958
\(299\) −21.1734 −1.22449
\(300\) 1.00000 0.0577350
\(301\) −9.30062 −0.536079
\(302\) 8.24142 0.474240
\(303\) −3.63067 −0.208577
\(304\) 4.20681 0.241277
\(305\) 5.40878 0.309706
\(306\) −7.95030 −0.454489
\(307\) 6.28303 0.358591 0.179296 0.983795i \(-0.442618\pi\)
0.179296 + 0.983795i \(0.442618\pi\)
\(308\) 1.53669 0.0875611
\(309\) −9.11766 −0.518686
\(310\) −4.73057 −0.268679
\(311\) 20.6312 1.16989 0.584943 0.811074i \(-0.301117\pi\)
0.584943 + 0.811074i \(0.301117\pi\)
\(312\) 3.21973 0.182281
\(313\) −7.32180 −0.413852 −0.206926 0.978357i \(-0.566346\pi\)
−0.206926 + 0.978357i \(0.566346\pi\)
\(314\) −6.11283 −0.344967
\(315\) 1.84556 0.103986
\(316\) 4.00000 0.225018
\(317\) 19.1442 1.07524 0.537622 0.843186i \(-0.319323\pi\)
0.537622 + 0.843186i \(0.319323\pi\)
\(318\) 3.95030 0.221522
\(319\) 2.53077 0.141696
\(320\) 1.00000 0.0559017
\(321\) 8.91086 0.497356
\(322\) −12.1367 −0.676351
\(323\) −33.4454 −1.86095
\(324\) 1.00000 0.0555556
\(325\) 3.21973 0.178598
\(326\) −21.4373 −1.18730
\(327\) 11.2979 0.624778
\(328\) 2.73057 0.150771
\(329\) 16.8272 0.927715
\(330\) 0.832640 0.0458353
\(331\) 0.908184 0.0499183 0.0249591 0.999688i \(-0.492054\pi\)
0.0249591 + 0.999688i \(0.492054\pi\)
\(332\) −3.55445 −0.195076
\(333\) −1.00000 −0.0547997
\(334\) −22.5815 −1.23560
\(335\) 11.7959 0.644477
\(336\) 1.84556 0.100684
\(337\) −8.11083 −0.441825 −0.220913 0.975294i \(-0.570904\pi\)
−0.220913 + 0.975294i \(0.570904\pi\)
\(338\) −2.63334 −0.143235
\(339\) −11.0394 −0.599580
\(340\) −7.95030 −0.431166
\(341\) −3.93887 −0.213302
\(342\) 4.20681 0.227478
\(343\) −19.5517 −1.05569
\(344\) −5.03945 −0.271709
\(345\) −6.57614 −0.354047
\(346\) −6.48916 −0.348859
\(347\) 9.35641 0.502278 0.251139 0.967951i \(-0.419195\pi\)
0.251139 + 0.967951i \(0.419195\pi\)
\(348\) 3.03945 0.162931
\(349\) −35.4665 −1.89848 −0.949239 0.314556i \(-0.898144\pi\)
−0.949239 + 0.314556i \(0.898144\pi\)
\(350\) 1.84556 0.0986495
\(351\) 3.21973 0.171856
\(352\) 0.832640 0.0443799
\(353\) −36.0923 −1.92100 −0.960500 0.278279i \(-0.910236\pi\)
−0.960500 + 0.278279i \(0.910236\pi\)
\(354\) −3.69113 −0.196181
\(355\) −13.7700 −0.730837
\(356\) −6.55029 −0.347165
\(357\) −14.6728 −0.776566
\(358\) 16.2095 0.856698
\(359\) 2.07889 0.109720 0.0548599 0.998494i \(-0.482529\pi\)
0.0548599 + 0.998494i \(0.482529\pi\)
\(360\) 1.00000 0.0527046
\(361\) −1.30278 −0.0685674
\(362\) 5.38225 0.282885
\(363\) −10.3067 −0.540962
\(364\) 5.94222 0.311457
\(365\) −3.52860 −0.184696
\(366\) 5.40878 0.282721
\(367\) 34.2645 1.78859 0.894297 0.447474i \(-0.147676\pi\)
0.894297 + 0.447474i \(0.147676\pi\)
\(368\) −6.57614 −0.342805
\(369\) 2.73057 0.142148
\(370\) −1.00000 −0.0519875
\(371\) 7.29053 0.378506
\(372\) −4.73057 −0.245269
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) −6.61974 −0.342299
\(375\) 1.00000 0.0516398
\(376\) 9.11766 0.470208
\(377\) 9.78620 0.504015
\(378\) 1.84556 0.0949255
\(379\) 29.8217 1.53184 0.765919 0.642937i \(-0.222284\pi\)
0.765919 + 0.642937i \(0.222284\pi\)
\(380\) 4.20681 0.215805
\(381\) −0.136678 −0.00700222
\(382\) −10.5680 −0.540708
\(383\) −6.78027 −0.346456 −0.173228 0.984882i \(-0.555420\pi\)
−0.173228 + 0.984882i \(0.555420\pi\)
\(384\) 1.00000 0.0510310
\(385\) 1.53669 0.0783170
\(386\) 3.11766 0.158685
\(387\) −5.03945 −0.256169
\(388\) −10.4918 −0.532642
\(389\) 35.1959 1.78450 0.892251 0.451540i \(-0.149125\pi\)
0.892251 + 0.451540i \(0.149125\pi\)
\(390\) 3.21973 0.163037
\(391\) 52.2823 2.64403
\(392\) −3.59390 −0.181519
\(393\) 2.07889 0.104866
\(394\) 0.576137 0.0290254
\(395\) 4.00000 0.201262
\(396\) 0.832640 0.0418417
\(397\) −3.56054 −0.178698 −0.0893492 0.996000i \(-0.528479\pi\)
−0.0893492 + 0.996000i \(0.528479\pi\)
\(398\) 6.64359 0.333013
\(399\) 7.76393 0.388683
\(400\) 1.00000 0.0500000
\(401\) 1.44971 0.0723950 0.0361975 0.999345i \(-0.488475\pi\)
0.0361975 + 0.999345i \(0.488475\pi\)
\(402\) 11.7959 0.588324
\(403\) −15.2312 −0.758718
\(404\) −3.63067 −0.180633
\(405\) 1.00000 0.0496904
\(406\) 5.60949 0.278394
\(407\) −0.832640 −0.0412724
\(408\) −7.95030 −0.393599
\(409\) 25.1781 1.24498 0.622489 0.782629i \(-0.286122\pi\)
0.622489 + 0.782629i \(0.286122\pi\)
\(410\) 2.73057 0.134853
\(411\) 3.06597 0.151233
\(412\) −9.11766 −0.449195
\(413\) −6.81221 −0.335207
\(414\) −6.57614 −0.323199
\(415\) −3.55445 −0.174481
\(416\) 3.21973 0.157860
\(417\) 22.1917 1.08673
\(418\) 3.50276 0.171325
\(419\) 31.6938 1.54834 0.774172 0.632976i \(-0.218167\pi\)
0.774172 + 0.632976i \(0.218167\pi\)
\(420\) 1.84556 0.0900543
\(421\) −5.14351 −0.250679 −0.125340 0.992114i \(-0.540002\pi\)
−0.125340 + 0.992114i \(0.540002\pi\)
\(422\) −8.83531 −0.430096
\(423\) 9.11766 0.443316
\(424\) 3.95030 0.191844
\(425\) −7.95030 −0.385646
\(426\) −13.7700 −0.667160
\(427\) 9.98224 0.483075
\(428\) 8.91086 0.430723
\(429\) 2.68088 0.129434
\(430\) −5.03945 −0.243024
\(431\) 2.72448 0.131234 0.0656168 0.997845i \(-0.479098\pi\)
0.0656168 + 0.997845i \(0.479098\pi\)
\(432\) 1.00000 0.0481125
\(433\) −8.68088 −0.417176 −0.208588 0.978004i \(-0.566887\pi\)
−0.208588 + 0.978004i \(0.566887\pi\)
\(434\) −8.73057 −0.419081
\(435\) 3.03945 0.145730
\(436\) 11.2979 0.541073
\(437\) −27.6645 −1.32337
\(438\) −3.52860 −0.168603
\(439\) 7.64752 0.364996 0.182498 0.983206i \(-0.441582\pi\)
0.182498 + 0.983206i \(0.441582\pi\)
\(440\) 0.832640 0.0396946
\(441\) −3.59390 −0.171138
\(442\) −25.5978 −1.21756
\(443\) 16.7917 0.797798 0.398899 0.916995i \(-0.369392\pi\)
0.398899 + 0.916995i \(0.369392\pi\)
\(444\) −1.00000 −0.0474579
\(445\) −6.55029 −0.310514
\(446\) 17.8828 0.846777
\(447\) −14.4999 −0.685823
\(448\) 1.84556 0.0871947
\(449\) 26.1047 1.23196 0.615979 0.787762i \(-0.288760\pi\)
0.615979 + 0.787762i \(0.288760\pi\)
\(450\) 1.00000 0.0471405
\(451\) 2.27359 0.107059
\(452\) −11.0394 −0.519252
\(453\) 8.24142 0.387216
\(454\) 25.1959 1.18250
\(455\) 5.94222 0.278575
\(456\) 4.20681 0.197002
\(457\) −17.6646 −0.826315 −0.413158 0.910660i \(-0.635574\pi\)
−0.413158 + 0.910660i \(0.635574\pi\)
\(458\) −5.48699 −0.256390
\(459\) −7.95030 −0.371088
\(460\) −6.57614 −0.306614
\(461\) −21.2231 −0.988457 −0.494229 0.869332i \(-0.664549\pi\)
−0.494229 + 0.869332i \(0.664549\pi\)
\(462\) 1.53669 0.0714933
\(463\) 32.5530 1.51286 0.756432 0.654072i \(-0.226941\pi\)
0.756432 + 0.654072i \(0.226941\pi\)
\(464\) 3.03945 0.141103
\(465\) −4.73057 −0.219375
\(466\) −28.5788 −1.32389
\(467\) 15.1170 0.699531 0.349765 0.936837i \(-0.386261\pi\)
0.349765 + 0.936837i \(0.386261\pi\)
\(468\) 3.21973 0.148832
\(469\) 21.7700 1.00525
\(470\) 9.11766 0.420566
\(471\) −6.11283 −0.281664
\(472\) −3.69113 −0.169898
\(473\) −4.19605 −0.192934
\(474\) 4.00000 0.183726
\(475\) 4.20681 0.193022
\(476\) −14.6728 −0.672526
\(477\) 3.95030 0.180872
\(478\) 1.77811 0.0813288
\(479\) 3.44971 0.157621 0.0788106 0.996890i \(-0.474888\pi\)
0.0788106 + 0.996890i \(0.474888\pi\)
\(480\) 1.00000 0.0456435
\(481\) −3.21973 −0.146807
\(482\) 22.2612 1.01397
\(483\) −12.1367 −0.552238
\(484\) −10.3067 −0.468487
\(485\) −10.4918 −0.476409
\(486\) 1.00000 0.0453609
\(487\) −15.6619 −0.709707 −0.354853 0.934922i \(-0.615469\pi\)
−0.354853 + 0.934922i \(0.615469\pi\)
\(488\) 5.40878 0.244844
\(489\) −21.4373 −0.969428
\(490\) −3.59390 −0.162356
\(491\) 7.89793 0.356429 0.178214 0.983992i \(-0.442968\pi\)
0.178214 + 0.983992i \(0.442968\pi\)
\(492\) 2.73057 0.123104
\(493\) −24.1645 −1.08832
\(494\) 13.5448 0.609408
\(495\) 0.832640 0.0374244
\(496\) −4.73057 −0.212409
\(497\) −25.4134 −1.13995
\(498\) −3.55445 −0.159279
\(499\) 5.87209 0.262871 0.131435 0.991325i \(-0.458041\pi\)
0.131435 + 0.991325i \(0.458041\pi\)
\(500\) 1.00000 0.0447214
\(501\) −22.5815 −1.00887
\(502\) −28.6748 −1.27982
\(503\) 3.33056 0.148502 0.0742512 0.997240i \(-0.476343\pi\)
0.0742512 + 0.997240i \(0.476343\pi\)
\(504\) 1.84556 0.0822079
\(505\) −3.63067 −0.161563
\(506\) −5.47556 −0.243418
\(507\) −2.63334 −0.116951
\(508\) −0.136678 −0.00606410
\(509\) −30.9508 −1.37187 −0.685935 0.727663i \(-0.740607\pi\)
−0.685935 + 0.727663i \(0.740607\pi\)
\(510\) −7.95030 −0.352045
\(511\) −6.51226 −0.288085
\(512\) 1.00000 0.0441942
\(513\) 4.20681 0.185735
\(514\) 9.19388 0.405525
\(515\) −9.11766 −0.401772
\(516\) −5.03945 −0.221849
\(517\) 7.59173 0.333884
\(518\) −1.84556 −0.0810894
\(519\) −6.48916 −0.284842
\(520\) 3.21973 0.141194
\(521\) 7.45306 0.326524 0.163262 0.986583i \(-0.447798\pi\)
0.163262 + 0.986583i \(0.447798\pi\)
\(522\) 3.03945 0.133033
\(523\) 31.5917 1.38141 0.690705 0.723137i \(-0.257300\pi\)
0.690705 + 0.723137i \(0.257300\pi\)
\(524\) 2.07889 0.0908169
\(525\) 1.84556 0.0805470
\(526\) −8.02652 −0.349973
\(527\) 37.6095 1.63830
\(528\) 0.832640 0.0362360
\(529\) 20.2456 0.880242
\(530\) 3.95030 0.171590
\(531\) −3.69113 −0.160181
\(532\) 7.76393 0.336609
\(533\) 8.79171 0.380811
\(534\) −6.55029 −0.283459
\(535\) 8.91086 0.385250
\(536\) 11.7959 0.509504
\(537\) 16.2095 0.699491
\(538\) 21.3849 0.921970
\(539\) −2.99242 −0.128893
\(540\) 1.00000 0.0430331
\(541\) 36.0816 1.55127 0.775634 0.631183i \(-0.217430\pi\)
0.775634 + 0.631183i \(0.217430\pi\)
\(542\) 2.64359 0.113552
\(543\) 5.38225 0.230975
\(544\) −7.95030 −0.340866
\(545\) 11.2979 0.483951
\(546\) 5.94222 0.254303
\(547\) −38.7420 −1.65649 −0.828244 0.560367i \(-0.810660\pi\)
−0.828244 + 0.560367i \(0.810660\pi\)
\(548\) 3.06597 0.130972
\(549\) 5.40878 0.230841
\(550\) 0.832640 0.0355039
\(551\) 12.7864 0.544717
\(552\) −6.57614 −0.279899
\(553\) 7.38225 0.313925
\(554\) −14.4456 −0.613733
\(555\) −1.00000 −0.0424476
\(556\) 22.1917 0.941139
\(557\) −23.3618 −0.989869 −0.494935 0.868930i \(-0.664808\pi\)
−0.494935 + 0.868930i \(0.664808\pi\)
\(558\) −4.73057 −0.200261
\(559\) −16.2257 −0.686272
\(560\) 1.84556 0.0779893
\(561\) −6.61974 −0.279486
\(562\) −11.2197 −0.473276
\(563\) 39.0351 1.64513 0.822567 0.568668i \(-0.192541\pi\)
0.822567 + 0.568668i \(0.192541\pi\)
\(564\) 9.11766 0.383923
\(565\) −11.0394 −0.464433
\(566\) 3.86332 0.162388
\(567\) 1.84556 0.0775064
\(568\) −13.7700 −0.577777
\(569\) 35.4809 1.48744 0.743718 0.668493i \(-0.233060\pi\)
0.743718 + 0.668493i \(0.233060\pi\)
\(570\) 4.20681 0.176204
\(571\) 37.0093 1.54879 0.774395 0.632702i \(-0.218054\pi\)
0.774395 + 0.632702i \(0.218054\pi\)
\(572\) 2.68088 0.112093
\(573\) −10.5680 −0.441487
\(574\) 5.03945 0.210342
\(575\) −6.57614 −0.274244
\(576\) 1.00000 0.0416667
\(577\) −15.7354 −0.655074 −0.327537 0.944838i \(-0.606219\pi\)
−0.327537 + 0.944838i \(0.606219\pi\)
\(578\) 46.2073 1.92197
\(579\) 3.11766 0.129566
\(580\) 3.03945 0.126206
\(581\) −6.55996 −0.272153
\(582\) −10.4918 −0.434900
\(583\) 3.28918 0.136224
\(584\) −3.52860 −0.146015
\(585\) 3.21973 0.133119
\(586\) −16.1544 −0.667334
\(587\) 37.3006 1.53956 0.769781 0.638309i \(-0.220366\pi\)
0.769781 + 0.638309i \(0.220366\pi\)
\(588\) −3.59390 −0.148210
\(589\) −19.9006 −0.819990
\(590\) −3.69113 −0.151961
\(591\) 0.576137 0.0236991
\(592\) −1.00000 −0.0410997
\(593\) −17.9829 −0.738470 −0.369235 0.929336i \(-0.620380\pi\)
−0.369235 + 0.929336i \(0.620380\pi\)
\(594\) 0.832640 0.0341636
\(595\) −14.6728 −0.601526
\(596\) −14.4999 −0.593940
\(597\) 6.64359 0.271904
\(598\) −21.1734 −0.865844
\(599\) −32.9998 −1.34834 −0.674168 0.738578i \(-0.735498\pi\)
−0.674168 + 0.738578i \(0.735498\pi\)
\(600\) 1.00000 0.0408248
\(601\) 33.0334 1.34746 0.673729 0.738978i \(-0.264691\pi\)
0.673729 + 0.738978i \(0.264691\pi\)
\(602\) −9.30062 −0.379065
\(603\) 11.7959 0.480365
\(604\) 8.24142 0.335339
\(605\) −10.3067 −0.419027
\(606\) −3.63067 −0.147486
\(607\) 27.6102 1.12066 0.560331 0.828269i \(-0.310674\pi\)
0.560331 + 0.828269i \(0.310674\pi\)
\(608\) 4.20681 0.170609
\(609\) 5.60949 0.227308
\(610\) 5.40878 0.218995
\(611\) 29.3564 1.18763
\(612\) −7.95030 −0.321372
\(613\) 1.62999 0.0658348 0.0329174 0.999458i \(-0.489520\pi\)
0.0329174 + 0.999458i \(0.489520\pi\)
\(614\) 6.28303 0.253562
\(615\) 2.73057 0.110107
\(616\) 1.53669 0.0619150
\(617\) 28.6143 1.15197 0.575985 0.817460i \(-0.304619\pi\)
0.575985 + 0.817460i \(0.304619\pi\)
\(618\) −9.11766 −0.366766
\(619\) 19.6830 0.791128 0.395564 0.918438i \(-0.370549\pi\)
0.395564 + 0.918438i \(0.370549\pi\)
\(620\) −4.73057 −0.189984
\(621\) −6.57614 −0.263891
\(622\) 20.6312 0.827235
\(623\) −12.0890 −0.484335
\(624\) 3.21973 0.128892
\(625\) 1.00000 0.0400000
\(626\) −7.32180 −0.292638
\(627\) 3.50276 0.139887
\(628\) −6.11283 −0.243928
\(629\) 7.95030 0.316999
\(630\) 1.84556 0.0735290
\(631\) −5.76193 −0.229379 −0.114689 0.993401i \(-0.536587\pi\)
−0.114689 + 0.993401i \(0.536587\pi\)
\(632\) 4.00000 0.159111
\(633\) −8.83531 −0.351172
\(634\) 19.1442 0.760313
\(635\) −0.136678 −0.00542390
\(636\) 3.95030 0.156640
\(637\) −11.5714 −0.458474
\(638\) 2.53077 0.100194
\(639\) −13.7700 −0.544734
\(640\) 1.00000 0.0395285
\(641\) 20.8967 0.825369 0.412685 0.910874i \(-0.364591\pi\)
0.412685 + 0.910874i \(0.364591\pi\)
\(642\) 8.91086 0.351683
\(643\) 30.6197 1.20752 0.603762 0.797164i \(-0.293668\pi\)
0.603762 + 0.797164i \(0.293668\pi\)
\(644\) −12.1367 −0.478252
\(645\) −5.03945 −0.198428
\(646\) −33.4454 −1.31589
\(647\) −10.6278 −0.417823 −0.208912 0.977935i \(-0.566992\pi\)
−0.208912 + 0.977935i \(0.566992\pi\)
\(648\) 1.00000 0.0392837
\(649\) −3.07338 −0.120641
\(650\) 3.21973 0.126288
\(651\) −8.73057 −0.342178
\(652\) −21.4373 −0.839549
\(653\) −5.90061 −0.230909 −0.115454 0.993313i \(-0.536832\pi\)
−0.115454 + 0.993313i \(0.536832\pi\)
\(654\) 11.2979 0.441785
\(655\) 2.07889 0.0812291
\(656\) 2.73057 0.106611
\(657\) −3.52860 −0.137664
\(658\) 16.8272 0.655994
\(659\) 19.8305 0.772486 0.386243 0.922397i \(-0.373773\pi\)
0.386243 + 0.922397i \(0.373773\pi\)
\(660\) 0.832640 0.0324105
\(661\) −9.32379 −0.362653 −0.181327 0.983423i \(-0.558039\pi\)
−0.181327 + 0.983423i \(0.558039\pi\)
\(662\) 0.908184 0.0352976
\(663\) −25.5978 −0.994137
\(664\) −3.55445 −0.137939
\(665\) 7.76393 0.301072
\(666\) −1.00000 −0.0387492
\(667\) −19.9878 −0.773931
\(668\) −22.5815 −0.873704
\(669\) 17.8828 0.691391
\(670\) 11.7959 0.455714
\(671\) 4.50357 0.173858
\(672\) 1.84556 0.0711942
\(673\) 33.0113 1.27249 0.636245 0.771487i \(-0.280487\pi\)
0.636245 + 0.771487i \(0.280487\pi\)
\(674\) −8.11083 −0.312418
\(675\) 1.00000 0.0384900
\(676\) −2.63334 −0.101282
\(677\) 26.1245 1.00405 0.502023 0.864854i \(-0.332589\pi\)
0.502023 + 0.864854i \(0.332589\pi\)
\(678\) −11.0394 −0.423967
\(679\) −19.3633 −0.743097
\(680\) −7.95030 −0.304880
\(681\) 25.1959 0.965508
\(682\) −3.93887 −0.150827
\(683\) −39.3957 −1.50743 −0.753717 0.657199i \(-0.771741\pi\)
−0.753717 + 0.657199i \(0.771741\pi\)
\(684\) 4.20681 0.160851
\(685\) 3.06597 0.117145
\(686\) −19.5517 −0.746488
\(687\) −5.48699 −0.209342
\(688\) −5.03945 −0.192127
\(689\) 12.7189 0.484552
\(690\) −6.57614 −0.250349
\(691\) −9.45306 −0.359611 −0.179806 0.983702i \(-0.557547\pi\)
−0.179806 + 0.983702i \(0.557547\pi\)
\(692\) −6.48916 −0.246681
\(693\) 1.53669 0.0583740
\(694\) 9.35641 0.355164
\(695\) 22.1917 0.841780
\(696\) 3.03945 0.115210
\(697\) −21.7089 −0.822283
\(698\) −35.4665 −1.34243
\(699\) −28.5788 −1.08095
\(700\) 1.84556 0.0697557
\(701\) 37.2842 1.40821 0.704103 0.710098i \(-0.251349\pi\)
0.704103 + 0.710098i \(0.251349\pi\)
\(702\) 3.21973 0.121521
\(703\) −4.20681 −0.158663
\(704\) 0.832640 0.0313813
\(705\) 9.11766 0.343391
\(706\) −36.0923 −1.35835
\(707\) −6.70063 −0.252003
\(708\) −3.69113 −0.138721
\(709\) −19.8110 −0.744016 −0.372008 0.928230i \(-0.621331\pi\)
−0.372008 + 0.928230i \(0.621331\pi\)
\(710\) −13.7700 −0.516780
\(711\) 4.00000 0.150012
\(712\) −6.55029 −0.245483
\(713\) 31.1089 1.16504
\(714\) −14.6728 −0.549115
\(715\) 2.68088 0.100259
\(716\) 16.2095 0.605777
\(717\) 1.77811 0.0664046
\(718\) 2.07889 0.0775836
\(719\) −19.2136 −0.716548 −0.358274 0.933617i \(-0.616635\pi\)
−0.358274 + 0.933617i \(0.616635\pi\)
\(720\) 1.00000 0.0372678
\(721\) −16.8272 −0.626679
\(722\) −1.30278 −0.0484845
\(723\) 22.2612 0.827902
\(724\) 5.38225 0.200030
\(725\) 3.03945 0.112882
\(726\) −10.3067 −0.382518
\(727\) −45.0183 −1.66964 −0.834818 0.550527i \(-0.814427\pi\)
−0.834818 + 0.550527i \(0.814427\pi\)
\(728\) 5.94222 0.220233
\(729\) 1.00000 0.0370370
\(730\) −3.52860 −0.130599
\(731\) 40.0651 1.48186
\(732\) 5.40878 0.199914
\(733\) −20.7262 −0.765541 −0.382771 0.923843i \(-0.625030\pi\)
−0.382771 + 0.923843i \(0.625030\pi\)
\(734\) 34.2645 1.26473
\(735\) −3.59390 −0.132563
\(736\) −6.57614 −0.242400
\(737\) 9.82171 0.361787
\(738\) 2.73057 0.100514
\(739\) 24.2176 0.890858 0.445429 0.895317i \(-0.353051\pi\)
0.445429 + 0.895317i \(0.353051\pi\)
\(740\) −1.00000 −0.0367607
\(741\) 13.5448 0.497580
\(742\) 7.29053 0.267644
\(743\) 13.9530 0.511885 0.255943 0.966692i \(-0.417614\pi\)
0.255943 + 0.966692i \(0.417614\pi\)
\(744\) −4.73057 −0.173431
\(745\) −14.4999 −0.531236
\(746\) −6.00000 −0.219676
\(747\) −3.55445 −0.130050
\(748\) −6.61974 −0.242042
\(749\) 16.4456 0.600907
\(750\) 1.00000 0.0365148
\(751\) 41.6176 1.51865 0.759324 0.650713i \(-0.225530\pi\)
0.759324 + 0.650713i \(0.225530\pi\)
\(752\) 9.11766 0.332487
\(753\) −28.6748 −1.04497
\(754\) 9.78620 0.356392
\(755\) 8.24142 0.299936
\(756\) 1.84556 0.0671225
\(757\) −45.4034 −1.65021 −0.825107 0.564977i \(-0.808885\pi\)
−0.825107 + 0.564977i \(0.808885\pi\)
\(758\) 29.8217 1.08317
\(759\) −5.47556 −0.198750
\(760\) 4.20681 0.152597
\(761\) 18.9347 0.686383 0.343191 0.939266i \(-0.388492\pi\)
0.343191 + 0.939266i \(0.388492\pi\)
\(762\) −0.136678 −0.00495132
\(763\) 20.8511 0.754860
\(764\) −10.5680 −0.382339
\(765\) −7.95030 −0.287444
\(766\) −6.78027 −0.244981
\(767\) −11.8844 −0.429122
\(768\) 1.00000 0.0360844
\(769\) 31.0884 1.12108 0.560538 0.828129i \(-0.310594\pi\)
0.560538 + 0.828129i \(0.310594\pi\)
\(770\) 1.53669 0.0553785
\(771\) 9.19388 0.331110
\(772\) 3.11766 0.112207
\(773\) 34.3542 1.23564 0.617818 0.786321i \(-0.288017\pi\)
0.617818 + 0.786321i \(0.288017\pi\)
\(774\) −5.03945 −0.181139
\(775\) −4.73057 −0.169927
\(776\) −10.4918 −0.376635
\(777\) −1.84556 −0.0662092
\(778\) 35.1959 1.26183
\(779\) 11.4870 0.411564
\(780\) 3.21973 0.115285
\(781\) −11.4655 −0.410267
\(782\) 52.2823 1.86961
\(783\) 3.03945 0.108621
\(784\) −3.59390 −0.128353
\(785\) −6.11283 −0.218176
\(786\) 2.07889 0.0741517
\(787\) 36.0053 1.28345 0.641726 0.766934i \(-0.278219\pi\)
0.641726 + 0.766934i \(0.278219\pi\)
\(788\) 0.576137 0.0205240
\(789\) −8.02652 −0.285752
\(790\) 4.00000 0.142314
\(791\) −20.3740 −0.724416
\(792\) 0.832640 0.0295866
\(793\) 17.4148 0.618418
\(794\) −3.56054 −0.126359
\(795\) 3.95030 0.140103
\(796\) 6.64359 0.235476
\(797\) 43.2107 1.53060 0.765300 0.643674i \(-0.222591\pi\)
0.765300 + 0.643674i \(0.222591\pi\)
\(798\) 7.76393 0.274840
\(799\) −72.4882 −2.56445
\(800\) 1.00000 0.0353553
\(801\) −6.55029 −0.231443
\(802\) 1.44971 0.0511910
\(803\) −2.93806 −0.103682
\(804\) 11.7959 0.416008
\(805\) −12.1367 −0.427762
\(806\) −15.2312 −0.536495
\(807\) 21.3849 0.752785
\(808\) −3.63067 −0.127727
\(809\) −32.2209 −1.13283 −0.566414 0.824121i \(-0.691669\pi\)
−0.566414 + 0.824121i \(0.691669\pi\)
\(810\) 1.00000 0.0351364
\(811\) 8.87892 0.311781 0.155890 0.987774i \(-0.450175\pi\)
0.155890 + 0.987774i \(0.450175\pi\)
\(812\) 5.60949 0.196855
\(813\) 2.64359 0.0927148
\(814\) −0.832640 −0.0291840
\(815\) −21.4373 −0.750916
\(816\) −7.95030 −0.278316
\(817\) −21.2000 −0.741693
\(818\) 25.1781 0.880332
\(819\) 5.94222 0.207638
\(820\) 2.73057 0.0953558
\(821\) 2.31155 0.0806735 0.0403368 0.999186i \(-0.487157\pi\)
0.0403368 + 0.999186i \(0.487157\pi\)
\(822\) 3.06597 0.106938
\(823\) 13.6280 0.475042 0.237521 0.971382i \(-0.423665\pi\)
0.237521 + 0.971382i \(0.423665\pi\)
\(824\) −9.11766 −0.317629
\(825\) 0.832640 0.0289888
\(826\) −6.81221 −0.237027
\(827\) −17.8298 −0.620003 −0.310001 0.950736i \(-0.600330\pi\)
−0.310001 + 0.950736i \(0.600330\pi\)
\(828\) −6.57614 −0.228537
\(829\) −29.7388 −1.03287 −0.516435 0.856326i \(-0.672741\pi\)
−0.516435 + 0.856326i \(0.672741\pi\)
\(830\) −3.55445 −0.123377
\(831\) −14.4456 −0.501111
\(832\) 3.21973 0.111624
\(833\) 28.5726 0.989980
\(834\) 22.1917 0.768436
\(835\) −22.5815 −0.781464
\(836\) 3.50276 0.121145
\(837\) −4.73057 −0.163513
\(838\) 31.6938 1.09484
\(839\) −10.4556 −0.360969 −0.180484 0.983578i \(-0.557766\pi\)
−0.180484 + 0.983578i \(0.557766\pi\)
\(840\) 1.84556 0.0636780
\(841\) −19.7618 −0.681440
\(842\) −5.14351 −0.177257
\(843\) −11.2197 −0.386428
\(844\) −8.83531 −0.304124
\(845\) −2.63334 −0.0905897
\(846\) 9.11766 0.313472
\(847\) −19.0217 −0.653593
\(848\) 3.95030 0.135654
\(849\) 3.86332 0.132589
\(850\) −7.95030 −0.272693
\(851\) 6.57614 0.225427
\(852\) −13.7700 −0.471753
\(853\) −15.6333 −0.535275 −0.267638 0.963520i \(-0.586243\pi\)
−0.267638 + 0.963520i \(0.586243\pi\)
\(854\) 9.98224 0.341585
\(855\) 4.20681 0.143870
\(856\) 8.91086 0.304567
\(857\) −14.4945 −0.495123 −0.247561 0.968872i \(-0.579629\pi\)
−0.247561 + 0.968872i \(0.579629\pi\)
\(858\) 2.68088 0.0915236
\(859\) 3.38493 0.115492 0.0577461 0.998331i \(-0.481609\pi\)
0.0577461 + 0.998331i \(0.481609\pi\)
\(860\) −5.03945 −0.171844
\(861\) 5.03945 0.171744
\(862\) 2.72448 0.0927962
\(863\) −56.3924 −1.91962 −0.959810 0.280649i \(-0.909450\pi\)
−0.959810 + 0.280649i \(0.909450\pi\)
\(864\) 1.00000 0.0340207
\(865\) −6.48916 −0.220638
\(866\) −8.68088 −0.294988
\(867\) 46.2073 1.56928
\(868\) −8.73057 −0.296335
\(869\) 3.33056 0.112982
\(870\) 3.03945 0.103047
\(871\) 37.9795 1.28689
\(872\) 11.2979 0.382597
\(873\) −10.4918 −0.355095
\(874\) −27.6645 −0.935767
\(875\) 1.84556 0.0623914
\(876\) −3.52860 −0.119220
\(877\) −32.1755 −1.08649 −0.543245 0.839574i \(-0.682805\pi\)
−0.543245 + 0.839574i \(0.682805\pi\)
\(878\) 7.64752 0.258091
\(879\) −16.1544 −0.544876
\(880\) 0.832640 0.0280683
\(881\) 14.6097 0.492212 0.246106 0.969243i \(-0.420849\pi\)
0.246106 + 0.969243i \(0.420849\pi\)
\(882\) −3.59390 −0.121013
\(883\) 18.5203 0.623259 0.311630 0.950204i \(-0.399125\pi\)
0.311630 + 0.950204i \(0.399125\pi\)
\(884\) −25.5978 −0.860948
\(885\) −3.69113 −0.124076
\(886\) 16.7917 0.564128
\(887\) −28.5611 −0.958986 −0.479493 0.877546i \(-0.659179\pi\)
−0.479493 + 0.877546i \(0.659179\pi\)
\(888\) −1.00000 −0.0335578
\(889\) −0.252248 −0.00846012
\(890\) −6.55029 −0.219566
\(891\) 0.832640 0.0278945
\(892\) 17.8828 0.598762
\(893\) 38.3562 1.28354
\(894\) −14.4999 −0.484950
\(895\) 16.2095 0.541823
\(896\) 1.84556 0.0616559
\(897\) −21.1734 −0.706959
\(898\) 26.1047 0.871126
\(899\) −14.3783 −0.479544
\(900\) 1.00000 0.0333333
\(901\) −31.4061 −1.04629
\(902\) 2.27359 0.0757021
\(903\) −9.30062 −0.309505
\(904\) −11.0394 −0.367167
\(905\) 5.38225 0.178912
\(906\) 8.24142 0.273803
\(907\) −25.8428 −0.858097 −0.429048 0.903282i \(-0.641151\pi\)
−0.429048 + 0.903282i \(0.641151\pi\)
\(908\) 25.1959 0.836155
\(909\) −3.63067 −0.120422
\(910\) 5.94222 0.196983
\(911\) 24.9998 0.828281 0.414141 0.910213i \(-0.364082\pi\)
0.414141 + 0.910213i \(0.364082\pi\)
\(912\) 4.20681 0.139301
\(913\) −2.95958 −0.0979477
\(914\) −17.6646 −0.584293
\(915\) 5.40878 0.178809
\(916\) −5.48699 −0.181295
\(917\) 3.83673 0.126700
\(918\) −7.95030 −0.262399
\(919\) −34.3931 −1.13452 −0.567262 0.823537i \(-0.691997\pi\)
−0.567262 + 0.823537i \(0.691997\pi\)
\(920\) −6.57614 −0.216809
\(921\) 6.28303 0.207033
\(922\) −21.2231 −0.698945
\(923\) −44.3357 −1.45933
\(924\) 1.53669 0.0505534
\(925\) −1.00000 −0.0328798
\(926\) 32.5530 1.06976
\(927\) −9.11766 −0.299463
\(928\) 3.03945 0.0997747
\(929\) 55.6310 1.82519 0.912597 0.408860i \(-0.134074\pi\)
0.912597 + 0.408860i \(0.134074\pi\)
\(930\) −4.73057 −0.155122
\(931\) −15.1188 −0.495499
\(932\) −28.5788 −0.936130
\(933\) 20.6312 0.675434
\(934\) 15.1170 0.494643
\(935\) −6.61974 −0.216489
\(936\) 3.21973 0.105240
\(937\) −53.9576 −1.76272 −0.881360 0.472446i \(-0.843371\pi\)
−0.881360 + 0.472446i \(0.843371\pi\)
\(938\) 21.7700 0.710816
\(939\) −7.32180 −0.238938
\(940\) 9.11766 0.297385
\(941\) −50.1092 −1.63351 −0.816756 0.576983i \(-0.804230\pi\)
−0.816756 + 0.576983i \(0.804230\pi\)
\(942\) −6.11283 −0.199167
\(943\) −17.9566 −0.584748
\(944\) −3.69113 −0.120136
\(945\) 1.84556 0.0600362
\(946\) −4.19605 −0.136425
\(947\) 39.3795 1.27966 0.639831 0.768516i \(-0.279004\pi\)
0.639831 + 0.768516i \(0.279004\pi\)
\(948\) 4.00000 0.129914
\(949\) −11.3611 −0.368798
\(950\) 4.20681 0.136487
\(951\) 19.1442 0.620793
\(952\) −14.6728 −0.475548
\(953\) −20.5096 −0.664371 −0.332185 0.943214i \(-0.607786\pi\)
−0.332185 + 0.943214i \(0.607786\pi\)
\(954\) 3.95030 0.127896
\(955\) −10.5680 −0.341974
\(956\) 1.77811 0.0575081
\(957\) 2.53077 0.0818080
\(958\) 3.44971 0.111455
\(959\) 5.65844 0.182721
\(960\) 1.00000 0.0322749
\(961\) −8.62168 −0.278119
\(962\) −3.21973 −0.103808
\(963\) 8.91086 0.287148
\(964\) 22.2612 0.716984
\(965\) 3.11766 0.100361
\(966\) −12.1367 −0.390491
\(967\) 18.6265 0.598988 0.299494 0.954098i \(-0.403182\pi\)
0.299494 + 0.954098i \(0.403182\pi\)
\(968\) −10.3067 −0.331270
\(969\) −33.4454 −1.07442
\(970\) −10.4918 −0.336872
\(971\) −3.56656 −0.114456 −0.0572282 0.998361i \(-0.518226\pi\)
−0.0572282 + 0.998361i \(0.518226\pi\)
\(972\) 1.00000 0.0320750
\(973\) 40.9562 1.31300
\(974\) −15.6619 −0.501838
\(975\) 3.21973 0.103114
\(976\) 5.40878 0.173131
\(977\) −43.6998 −1.39808 −0.699041 0.715082i \(-0.746389\pi\)
−0.699041 + 0.715082i \(0.746389\pi\)
\(978\) −21.4373 −0.685489
\(979\) −5.45404 −0.174312
\(980\) −3.59390 −0.114803
\(981\) 11.2979 0.360716
\(982\) 7.89793 0.252033
\(983\) 53.1096 1.69393 0.846966 0.531647i \(-0.178427\pi\)
0.846966 + 0.531647i \(0.178427\pi\)
\(984\) 2.73057 0.0870475
\(985\) 0.576137 0.0183572
\(986\) −24.1645 −0.769555
\(987\) 16.8272 0.535616
\(988\) 13.5448 0.430917
\(989\) 33.1401 1.05379
\(990\) 0.832640 0.0264630
\(991\) −9.14283 −0.290432 −0.145216 0.989400i \(-0.546388\pi\)
−0.145216 + 0.989400i \(0.546388\pi\)
\(992\) −4.73057 −0.150196
\(993\) 0.908184 0.0288203
\(994\) −25.4134 −0.806066
\(995\) 6.64359 0.210616
\(996\) −3.55445 −0.112627
\(997\) −15.6280 −0.494944 −0.247472 0.968895i \(-0.579600\pi\)
−0.247472 + 0.968895i \(0.579600\pi\)
\(998\) 5.87209 0.185878
\(999\) −1.00000 −0.0316386
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 1110.2.a.s.1.3 4
3.2 odd 2 3330.2.a.bj.1.3 4
4.3 odd 2 8880.2.a.cg.1.2 4
5.4 even 2 5550.2.a.cj.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.s.1.3 4 1.1 even 1 trivial
3330.2.a.bj.1.3 4 3.2 odd 2
5550.2.a.cj.1.2 4 5.4 even 2
8880.2.a.cg.1.2 4 4.3 odd 2