Properties

Label 3822.2.a.bo.1.1
Level $3822$
Weight $2$
Character 3822.1
Self dual yes
Analytic conductor $30.519$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.5188236525\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
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 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 3822.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.56155 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.56155 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -3.56155 q^{10} -1.56155 q^{11} -1.00000 q^{12} -1.00000 q^{13} +3.56155 q^{15} +1.00000 q^{16} +6.68466 q^{17} +1.00000 q^{18} +4.68466 q^{19} -3.56155 q^{20} -1.56155 q^{22} -5.56155 q^{23} -1.00000 q^{24} +7.68466 q^{25} -1.00000 q^{26} -1.00000 q^{27} +6.68466 q^{29} +3.56155 q^{30} -6.24621 q^{31} +1.00000 q^{32} +1.56155 q^{33} +6.68466 q^{34} +1.00000 q^{36} -7.56155 q^{37} +4.68466 q^{38} +1.00000 q^{39} -3.56155 q^{40} +1.12311 q^{41} -6.43845 q^{43} -1.56155 q^{44} -3.56155 q^{45} -5.56155 q^{46} -1.00000 q^{48} +7.68466 q^{50} -6.68466 q^{51} -1.00000 q^{52} +12.2462 q^{53} -1.00000 q^{54} +5.56155 q^{55} -4.68466 q^{57} +6.68466 q^{58} -2.24621 q^{59} +3.56155 q^{60} -6.68466 q^{61} -6.24621 q^{62} +1.00000 q^{64} +3.56155 q^{65} +1.56155 q^{66} -7.12311 q^{67} +6.68466 q^{68} +5.56155 q^{69} +8.00000 q^{71} +1.00000 q^{72} +3.56155 q^{73} -7.56155 q^{74} -7.68466 q^{75} +4.68466 q^{76} +1.00000 q^{78} -11.1231 q^{79} -3.56155 q^{80} +1.00000 q^{81} +1.12311 q^{82} -8.87689 q^{83} -23.8078 q^{85} -6.43845 q^{86} -6.68466 q^{87} -1.56155 q^{88} -10.0000 q^{89} -3.56155 q^{90} -5.56155 q^{92} +6.24621 q^{93} -16.6847 q^{95} -1.00000 q^{96} -14.4924 q^{97} -1.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 3q^{5} - 2q^{6} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 3q^{5} - 2q^{6} + 2q^{8} + 2q^{9} - 3q^{10} + q^{11} - 2q^{12} - 2q^{13} + 3q^{15} + 2q^{16} + q^{17} + 2q^{18} - 3q^{19} - 3q^{20} + q^{22} - 7q^{23} - 2q^{24} + 3q^{25} - 2q^{26} - 2q^{27} + q^{29} + 3q^{30} + 4q^{31} + 2q^{32} - q^{33} + q^{34} + 2q^{36} - 11q^{37} - 3q^{38} + 2q^{39} - 3q^{40} - 6q^{41} - 17q^{43} + q^{44} - 3q^{45} - 7q^{46} - 2q^{48} + 3q^{50} - q^{51} - 2q^{52} + 8q^{53} - 2q^{54} + 7q^{55} + 3q^{57} + q^{58} + 12q^{59} + 3q^{60} - q^{61} + 4q^{62} + 2q^{64} + 3q^{65} - q^{66} - 6q^{67} + q^{68} + 7q^{69} + 16q^{71} + 2q^{72} + 3q^{73} - 11q^{74} - 3q^{75} - 3q^{76} + 2q^{78} - 14q^{79} - 3q^{80} + 2q^{81} - 6q^{82} - 26q^{83} - 27q^{85} - 17q^{86} - q^{87} + q^{88} - 20q^{89} - 3q^{90} - 7q^{92} - 4q^{93} - 21q^{95} - 2q^{96} + 4q^{97} + q^{99} + O(q^{100}) \)

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
\(5\) −3.56155 −1.59277 −0.796387 0.604787i \(-0.793258\pi\)
−0.796387 + 0.604787i \(0.793258\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −3.56155 −1.12626
\(11\) −1.56155 −0.470826 −0.235413 0.971895i \(-0.575644\pi\)
−0.235413 + 0.971895i \(0.575644\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 3.56155 0.919589
\(16\) 1.00000 0.250000
\(17\) 6.68466 1.62127 0.810634 0.585553i \(-0.199123\pi\)
0.810634 + 0.585553i \(0.199123\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.68466 1.07473 0.537367 0.843348i \(-0.319419\pi\)
0.537367 + 0.843348i \(0.319419\pi\)
\(20\) −3.56155 −0.796387
\(21\) 0 0
\(22\) −1.56155 −0.332924
\(23\) −5.56155 −1.15966 −0.579832 0.814736i \(-0.696882\pi\)
−0.579832 + 0.814736i \(0.696882\pi\)
\(24\) −1.00000 −0.204124
\(25\) 7.68466 1.53693
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 6.68466 1.24131 0.620655 0.784084i \(-0.286867\pi\)
0.620655 + 0.784084i \(0.286867\pi\)
\(30\) 3.56155 0.650248
\(31\) −6.24621 −1.12185 −0.560926 0.827866i \(-0.689555\pi\)
−0.560926 + 0.827866i \(0.689555\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.56155 0.271831
\(34\) 6.68466 1.14641
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −7.56155 −1.24311 −0.621556 0.783370i \(-0.713499\pi\)
−0.621556 + 0.783370i \(0.713499\pi\)
\(38\) 4.68466 0.759952
\(39\) 1.00000 0.160128
\(40\) −3.56155 −0.563131
\(41\) 1.12311 0.175400 0.0876998 0.996147i \(-0.472048\pi\)
0.0876998 + 0.996147i \(0.472048\pi\)
\(42\) 0 0
\(43\) −6.43845 −0.981854 −0.490927 0.871201i \(-0.663342\pi\)
−0.490927 + 0.871201i \(0.663342\pi\)
\(44\) −1.56155 −0.235413
\(45\) −3.56155 −0.530925
\(46\) −5.56155 −0.820006
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 7.68466 1.08677
\(51\) −6.68466 −0.936039
\(52\) −1.00000 −0.138675
\(53\) 12.2462 1.68215 0.841073 0.540921i \(-0.181924\pi\)
0.841073 + 0.540921i \(0.181924\pi\)
\(54\) −1.00000 −0.136083
\(55\) 5.56155 0.749920
\(56\) 0 0
\(57\) −4.68466 −0.620498
\(58\) 6.68466 0.877739
\(59\) −2.24621 −0.292432 −0.146216 0.989253i \(-0.546709\pi\)
−0.146216 + 0.989253i \(0.546709\pi\)
\(60\) 3.56155 0.459794
\(61\) −6.68466 −0.855883 −0.427941 0.903806i \(-0.640761\pi\)
−0.427941 + 0.903806i \(0.640761\pi\)
\(62\) −6.24621 −0.793270
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.56155 0.441756
\(66\) 1.56155 0.192214
\(67\) −7.12311 −0.870226 −0.435113 0.900376i \(-0.643292\pi\)
−0.435113 + 0.900376i \(0.643292\pi\)
\(68\) 6.68466 0.810634
\(69\) 5.56155 0.669532
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 1.00000 0.117851
\(73\) 3.56155 0.416848 0.208424 0.978039i \(-0.433167\pi\)
0.208424 + 0.978039i \(0.433167\pi\)
\(74\) −7.56155 −0.879013
\(75\) −7.68466 −0.887348
\(76\) 4.68466 0.537367
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) −11.1231 −1.25145 −0.625724 0.780045i \(-0.715196\pi\)
−0.625724 + 0.780045i \(0.715196\pi\)
\(80\) −3.56155 −0.398194
\(81\) 1.00000 0.111111
\(82\) 1.12311 0.124026
\(83\) −8.87689 −0.974366 −0.487183 0.873300i \(-0.661975\pi\)
−0.487183 + 0.873300i \(0.661975\pi\)
\(84\) 0 0
\(85\) −23.8078 −2.58231
\(86\) −6.43845 −0.694276
\(87\) −6.68466 −0.716671
\(88\) −1.56155 −0.166462
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) −3.56155 −0.375421
\(91\) 0 0
\(92\) −5.56155 −0.579832
\(93\) 6.24621 0.647702
\(94\) 0 0
\(95\) −16.6847 −1.71181
\(96\) −1.00000 −0.102062
\(97\) −14.4924 −1.47148 −0.735741 0.677263i \(-0.763166\pi\)
−0.735741 + 0.677263i \(0.763166\pi\)
\(98\) 0 0
\(99\) −1.56155 −0.156942
\(100\) 7.68466 0.768466
\(101\) 5.12311 0.509768 0.254884 0.966972i \(-0.417963\pi\)
0.254884 + 0.966972i \(0.417963\pi\)
\(102\) −6.68466 −0.661880
\(103\) 0.684658 0.0674614 0.0337307 0.999431i \(-0.489261\pi\)
0.0337307 + 0.999431i \(0.489261\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 12.2462 1.18946
\(107\) −8.87689 −0.858162 −0.429081 0.903266i \(-0.641162\pi\)
−0.429081 + 0.903266i \(0.641162\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −16.9309 −1.62168 −0.810842 0.585266i \(-0.800990\pi\)
−0.810842 + 0.585266i \(0.800990\pi\)
\(110\) 5.56155 0.530273
\(111\) 7.56155 0.717711
\(112\) 0 0
\(113\) −4.24621 −0.399450 −0.199725 0.979852i \(-0.564005\pi\)
−0.199725 + 0.979852i \(0.564005\pi\)
\(114\) −4.68466 −0.438758
\(115\) 19.8078 1.84708
\(116\) 6.68466 0.620655
\(117\) −1.00000 −0.0924500
\(118\) −2.24621 −0.206781
\(119\) 0 0
\(120\) 3.56155 0.325124
\(121\) −8.56155 −0.778323
\(122\) −6.68466 −0.605201
\(123\) −1.12311 −0.101267
\(124\) −6.24621 −0.560926
\(125\) −9.56155 −0.855211
\(126\) 0 0
\(127\) −4.87689 −0.432754 −0.216377 0.976310i \(-0.569424\pi\)
−0.216377 + 0.976310i \(0.569424\pi\)
\(128\) 1.00000 0.0883883
\(129\) 6.43845 0.566874
\(130\) 3.56155 0.312369
\(131\) 9.56155 0.835397 0.417698 0.908586i \(-0.362837\pi\)
0.417698 + 0.908586i \(0.362837\pi\)
\(132\) 1.56155 0.135916
\(133\) 0 0
\(134\) −7.12311 −0.615343
\(135\) 3.56155 0.306530
\(136\) 6.68466 0.573205
\(137\) −3.56155 −0.304284 −0.152142 0.988359i \(-0.548617\pi\)
−0.152142 + 0.988359i \(0.548617\pi\)
\(138\) 5.56155 0.473431
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) 1.56155 0.130584
\(144\) 1.00000 0.0833333
\(145\) −23.8078 −1.97713
\(146\) 3.56155 0.294756
\(147\) 0 0
\(148\) −7.56155 −0.621556
\(149\) −17.6155 −1.44312 −0.721560 0.692352i \(-0.756575\pi\)
−0.721560 + 0.692352i \(0.756575\pi\)
\(150\) −7.68466 −0.627450
\(151\) 11.8078 0.960902 0.480451 0.877022i \(-0.340473\pi\)
0.480451 + 0.877022i \(0.340473\pi\)
\(152\) 4.68466 0.379976
\(153\) 6.68466 0.540423
\(154\) 0 0
\(155\) 22.2462 1.78686
\(156\) 1.00000 0.0800641
\(157\) 15.5616 1.24195 0.620974 0.783832i \(-0.286737\pi\)
0.620974 + 0.783832i \(0.286737\pi\)
\(158\) −11.1231 −0.884907
\(159\) −12.2462 −0.971188
\(160\) −3.56155 −0.281565
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −16.8769 −1.32190 −0.660950 0.750430i \(-0.729847\pi\)
−0.660950 + 0.750430i \(0.729847\pi\)
\(164\) 1.12311 0.0876998
\(165\) −5.56155 −0.432966
\(166\) −8.87689 −0.688981
\(167\) −22.9309 −1.77444 −0.887222 0.461343i \(-0.847368\pi\)
−0.887222 + 0.461343i \(0.847368\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −23.8078 −1.82597
\(171\) 4.68466 0.358245
\(172\) −6.43845 −0.490927
\(173\) −20.2462 −1.53929 −0.769645 0.638471i \(-0.779567\pi\)
−0.769645 + 0.638471i \(0.779567\pi\)
\(174\) −6.68466 −0.506763
\(175\) 0 0
\(176\) −1.56155 −0.117706
\(177\) 2.24621 0.168836
\(178\) −10.0000 −0.749532
\(179\) 16.4924 1.23270 0.616351 0.787472i \(-0.288610\pi\)
0.616351 + 0.787472i \(0.288610\pi\)
\(180\) −3.56155 −0.265462
\(181\) 0.246211 0.0183007 0.00915037 0.999958i \(-0.497087\pi\)
0.00915037 + 0.999958i \(0.497087\pi\)
\(182\) 0 0
\(183\) 6.68466 0.494144
\(184\) −5.56155 −0.410003
\(185\) 26.9309 1.98000
\(186\) 6.24621 0.457994
\(187\) −10.4384 −0.763335
\(188\) 0 0
\(189\) 0 0
\(190\) −16.6847 −1.21043
\(191\) 14.9309 1.08036 0.540180 0.841550i \(-0.318356\pi\)
0.540180 + 0.841550i \(0.318356\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 19.3693 1.39423 0.697117 0.716957i \(-0.254466\pi\)
0.697117 + 0.716957i \(0.254466\pi\)
\(194\) −14.4924 −1.04050
\(195\) −3.56155 −0.255048
\(196\) 0 0
\(197\) 10.8769 0.774947 0.387473 0.921881i \(-0.373348\pi\)
0.387473 + 0.921881i \(0.373348\pi\)
\(198\) −1.56155 −0.110975
\(199\) −6.93087 −0.491316 −0.245658 0.969357i \(-0.579004\pi\)
−0.245658 + 0.969357i \(0.579004\pi\)
\(200\) 7.68466 0.543387
\(201\) 7.12311 0.502425
\(202\) 5.12311 0.360460
\(203\) 0 0
\(204\) −6.68466 −0.468020
\(205\) −4.00000 −0.279372
\(206\) 0.684658 0.0477024
\(207\) −5.56155 −0.386555
\(208\) −1.00000 −0.0693375
\(209\) −7.31534 −0.506013
\(210\) 0 0
\(211\) −15.8078 −1.08825 −0.544126 0.839004i \(-0.683139\pi\)
−0.544126 + 0.839004i \(0.683139\pi\)
\(212\) 12.2462 0.841073
\(213\) −8.00000 −0.548151
\(214\) −8.87689 −0.606812
\(215\) 22.9309 1.56387
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −16.9309 −1.14670
\(219\) −3.56155 −0.240667
\(220\) 5.56155 0.374960
\(221\) −6.68466 −0.449659
\(222\) 7.56155 0.507498
\(223\) 23.6155 1.58141 0.790706 0.612196i \(-0.209713\pi\)
0.790706 + 0.612196i \(0.209713\pi\)
\(224\) 0 0
\(225\) 7.68466 0.512311
\(226\) −4.24621 −0.282454
\(227\) 7.12311 0.472777 0.236389 0.971659i \(-0.424036\pi\)
0.236389 + 0.971659i \(0.424036\pi\)
\(228\) −4.68466 −0.310249
\(229\) −28.2462 −1.86656 −0.933281 0.359147i \(-0.883068\pi\)
−0.933281 + 0.359147i \(0.883068\pi\)
\(230\) 19.8078 1.30609
\(231\) 0 0
\(232\) 6.68466 0.438869
\(233\) 27.3693 1.79302 0.896512 0.443020i \(-0.146093\pi\)
0.896512 + 0.443020i \(0.146093\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) −2.24621 −0.146216
\(237\) 11.1231 0.722523
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 3.56155 0.229897
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) −8.56155 −0.550357
\(243\) −1.00000 −0.0641500
\(244\) −6.68466 −0.427941
\(245\) 0 0
\(246\) −1.12311 −0.0716066
\(247\) −4.68466 −0.298078
\(248\) −6.24621 −0.396635
\(249\) 8.87689 0.562550
\(250\) −9.56155 −0.604726
\(251\) 7.80776 0.492822 0.246411 0.969165i \(-0.420749\pi\)
0.246411 + 0.969165i \(0.420749\pi\)
\(252\) 0 0
\(253\) 8.68466 0.546000
\(254\) −4.87689 −0.306004
\(255\) 23.8078 1.49090
\(256\) 1.00000 0.0625000
\(257\) 4.24621 0.264871 0.132436 0.991192i \(-0.457720\pi\)
0.132436 + 0.991192i \(0.457720\pi\)
\(258\) 6.43845 0.400840
\(259\) 0 0
\(260\) 3.56155 0.220878
\(261\) 6.68466 0.413770
\(262\) 9.56155 0.590715
\(263\) −30.2462 −1.86506 −0.932531 0.361091i \(-0.882404\pi\)
−0.932531 + 0.361091i \(0.882404\pi\)
\(264\) 1.56155 0.0961069
\(265\) −43.6155 −2.67928
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) −7.12311 −0.435113
\(269\) −20.2462 −1.23443 −0.617217 0.786793i \(-0.711740\pi\)
−0.617217 + 0.786793i \(0.711740\pi\)
\(270\) 3.56155 0.216749
\(271\) −4.87689 −0.296250 −0.148125 0.988969i \(-0.547324\pi\)
−0.148125 + 0.988969i \(0.547324\pi\)
\(272\) 6.68466 0.405317
\(273\) 0 0
\(274\) −3.56155 −0.215161
\(275\) −12.0000 −0.723627
\(276\) 5.56155 0.334766
\(277\) −22.4924 −1.35144 −0.675719 0.737159i \(-0.736167\pi\)
−0.675719 + 0.737159i \(0.736167\pi\)
\(278\) −12.0000 −0.719712
\(279\) −6.24621 −0.373951
\(280\) 0 0
\(281\) 16.2462 0.969168 0.484584 0.874745i \(-0.338971\pi\)
0.484584 + 0.874745i \(0.338971\pi\)
\(282\) 0 0
\(283\) −18.2462 −1.08462 −0.542312 0.840177i \(-0.682451\pi\)
−0.542312 + 0.840177i \(0.682451\pi\)
\(284\) 8.00000 0.474713
\(285\) 16.6847 0.988314
\(286\) 1.56155 0.0923366
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 27.6847 1.62851
\(290\) −23.8078 −1.39804
\(291\) 14.4924 0.849561
\(292\) 3.56155 0.208424
\(293\) −24.7386 −1.44525 −0.722623 0.691242i \(-0.757064\pi\)
−0.722623 + 0.691242i \(0.757064\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) −7.56155 −0.439506
\(297\) 1.56155 0.0906105
\(298\) −17.6155 −1.02044
\(299\) 5.56155 0.321633
\(300\) −7.68466 −0.443674
\(301\) 0 0
\(302\) 11.8078 0.679460
\(303\) −5.12311 −0.294315
\(304\) 4.68466 0.268684
\(305\) 23.8078 1.36323
\(306\) 6.68466 0.382136
\(307\) −26.2462 −1.49795 −0.748975 0.662598i \(-0.769454\pi\)
−0.748975 + 0.662598i \(0.769454\pi\)
\(308\) 0 0
\(309\) −0.684658 −0.0389489
\(310\) 22.2462 1.26350
\(311\) 12.8769 0.730182 0.365091 0.930972i \(-0.381038\pi\)
0.365091 + 0.930972i \(0.381038\pi\)
\(312\) 1.00000 0.0566139
\(313\) 13.6155 0.769595 0.384798 0.923001i \(-0.374271\pi\)
0.384798 + 0.923001i \(0.374271\pi\)
\(314\) 15.5616 0.878189
\(315\) 0 0
\(316\) −11.1231 −0.625724
\(317\) 2.87689 0.161582 0.0807912 0.996731i \(-0.474255\pi\)
0.0807912 + 0.996731i \(0.474255\pi\)
\(318\) −12.2462 −0.686733
\(319\) −10.4384 −0.584441
\(320\) −3.56155 −0.199097
\(321\) 8.87689 0.495460
\(322\) 0 0
\(323\) 31.3153 1.74243
\(324\) 1.00000 0.0555556
\(325\) −7.68466 −0.426268
\(326\) −16.8769 −0.934725
\(327\) 16.9309 0.936279
\(328\) 1.12311 0.0620131
\(329\) 0 0
\(330\) −5.56155 −0.306153
\(331\) 13.3693 0.734844 0.367422 0.930054i \(-0.380240\pi\)
0.367422 + 0.930054i \(0.380240\pi\)
\(332\) −8.87689 −0.487183
\(333\) −7.56155 −0.414371
\(334\) −22.9309 −1.25472
\(335\) 25.3693 1.38607
\(336\) 0 0
\(337\) 4.43845 0.241778 0.120889 0.992666i \(-0.461426\pi\)
0.120889 + 0.992666i \(0.461426\pi\)
\(338\) 1.00000 0.0543928
\(339\) 4.24621 0.230623
\(340\) −23.8078 −1.29116
\(341\) 9.75379 0.528197
\(342\) 4.68466 0.253317
\(343\) 0 0
\(344\) −6.43845 −0.347138
\(345\) −19.8078 −1.06641
\(346\) −20.2462 −1.08844
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) −6.68466 −0.358335
\(349\) −7.75379 −0.415051 −0.207525 0.978230i \(-0.566541\pi\)
−0.207525 + 0.978230i \(0.566541\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) −1.56155 −0.0832310
\(353\) −30.4924 −1.62295 −0.811474 0.584389i \(-0.801334\pi\)
−0.811474 + 0.584389i \(0.801334\pi\)
\(354\) 2.24621 0.119385
\(355\) −28.4924 −1.51222
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) 16.4924 0.871652
\(359\) −26.7386 −1.41121 −0.705606 0.708605i \(-0.749325\pi\)
−0.705606 + 0.708605i \(0.749325\pi\)
\(360\) −3.56155 −0.187710
\(361\) 2.94602 0.155054
\(362\) 0.246211 0.0129406
\(363\) 8.56155 0.449365
\(364\) 0 0
\(365\) −12.6847 −0.663945
\(366\) 6.68466 0.349413
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) −5.56155 −0.289916
\(369\) 1.12311 0.0584665
\(370\) 26.9309 1.40007
\(371\) 0 0
\(372\) 6.24621 0.323851
\(373\) −17.6155 −0.912097 −0.456049 0.889955i \(-0.650736\pi\)
−0.456049 + 0.889955i \(0.650736\pi\)
\(374\) −10.4384 −0.539759
\(375\) 9.56155 0.493756
\(376\) 0 0
\(377\) −6.68466 −0.344277
\(378\) 0 0
\(379\) 34.2462 1.75911 0.879555 0.475797i \(-0.157840\pi\)
0.879555 + 0.475797i \(0.157840\pi\)
\(380\) −16.6847 −0.855905
\(381\) 4.87689 0.249851
\(382\) 14.9309 0.763930
\(383\) −27.4233 −1.40126 −0.700632 0.713522i \(-0.747099\pi\)
−0.700632 + 0.713522i \(0.747099\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 19.3693 0.985872
\(387\) −6.43845 −0.327285
\(388\) −14.4924 −0.735741
\(389\) −28.7386 −1.45711 −0.728553 0.684989i \(-0.759807\pi\)
−0.728553 + 0.684989i \(0.759807\pi\)
\(390\) −3.56155 −0.180346
\(391\) −37.1771 −1.88013
\(392\) 0 0
\(393\) −9.56155 −0.482317
\(394\) 10.8769 0.547970
\(395\) 39.6155 1.99327
\(396\) −1.56155 −0.0784710
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) −6.93087 −0.347413
\(399\) 0 0
\(400\) 7.68466 0.384233
\(401\) 8.24621 0.411796 0.205898 0.978573i \(-0.433988\pi\)
0.205898 + 0.978573i \(0.433988\pi\)
\(402\) 7.12311 0.355268
\(403\) 6.24621 0.311146
\(404\) 5.12311 0.254884
\(405\) −3.56155 −0.176975
\(406\) 0 0
\(407\) 11.8078 0.585289
\(408\) −6.68466 −0.330940
\(409\) 4.93087 0.243816 0.121908 0.992541i \(-0.461099\pi\)
0.121908 + 0.992541i \(0.461099\pi\)
\(410\) −4.00000 −0.197546
\(411\) 3.56155 0.175678
\(412\) 0.684658 0.0337307
\(413\) 0 0
\(414\) −5.56155 −0.273335
\(415\) 31.6155 1.55195
\(416\) −1.00000 −0.0490290
\(417\) 12.0000 0.587643
\(418\) −7.31534 −0.357805
\(419\) 36.6847 1.79216 0.896081 0.443890i \(-0.146402\pi\)
0.896081 + 0.443890i \(0.146402\pi\)
\(420\) 0 0
\(421\) −3.75379 −0.182948 −0.0914742 0.995807i \(-0.529158\pi\)
−0.0914742 + 0.995807i \(0.529158\pi\)
\(422\) −15.8078 −0.769510
\(423\) 0 0
\(424\) 12.2462 0.594729
\(425\) 51.3693 2.49178
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) −8.87689 −0.429081
\(429\) −1.56155 −0.0753925
\(430\) 22.9309 1.10582
\(431\) −33.3693 −1.60734 −0.803672 0.595073i \(-0.797123\pi\)
−0.803672 + 0.595073i \(0.797123\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 31.3693 1.50751 0.753757 0.657154i \(-0.228240\pi\)
0.753757 + 0.657154i \(0.228240\pi\)
\(434\) 0 0
\(435\) 23.8078 1.14149
\(436\) −16.9309 −0.810842
\(437\) −26.0540 −1.24633
\(438\) −3.56155 −0.170178
\(439\) 6.93087 0.330792 0.165396 0.986227i \(-0.447110\pi\)
0.165396 + 0.986227i \(0.447110\pi\)
\(440\) 5.56155 0.265137
\(441\) 0 0
\(442\) −6.68466 −0.317957
\(443\) −5.36932 −0.255104 −0.127552 0.991832i \(-0.540712\pi\)
−0.127552 + 0.991832i \(0.540712\pi\)
\(444\) 7.56155 0.358855
\(445\) 35.6155 1.68834
\(446\) 23.6155 1.11823
\(447\) 17.6155 0.833186
\(448\) 0 0
\(449\) 10.6847 0.504240 0.252120 0.967696i \(-0.418872\pi\)
0.252120 + 0.967696i \(0.418872\pi\)
\(450\) 7.68466 0.362258
\(451\) −1.75379 −0.0825827
\(452\) −4.24621 −0.199725
\(453\) −11.8078 −0.554777
\(454\) 7.12311 0.334304
\(455\) 0 0
\(456\) −4.68466 −0.219379
\(457\) 27.3693 1.28028 0.640141 0.768257i \(-0.278876\pi\)
0.640141 + 0.768257i \(0.278876\pi\)
\(458\) −28.2462 −1.31986
\(459\) −6.68466 −0.312013
\(460\) 19.8078 0.923542
\(461\) 28.0540 1.30660 0.653302 0.757097i \(-0.273383\pi\)
0.653302 + 0.757097i \(0.273383\pi\)
\(462\) 0 0
\(463\) −8.68466 −0.403610 −0.201805 0.979426i \(-0.564681\pi\)
−0.201805 + 0.979426i \(0.564681\pi\)
\(464\) 6.68466 0.310327
\(465\) −22.2462 −1.03164
\(466\) 27.3693 1.26786
\(467\) 3.31534 0.153416 0.0767079 0.997054i \(-0.475559\pi\)
0.0767079 + 0.997054i \(0.475559\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 0 0
\(470\) 0 0
\(471\) −15.5616 −0.717039
\(472\) −2.24621 −0.103390
\(473\) 10.0540 0.462282
\(474\) 11.1231 0.510901
\(475\) 36.0000 1.65179
\(476\) 0 0
\(477\) 12.2462 0.560715
\(478\) 16.0000 0.731823
\(479\) −32.3002 −1.47583 −0.737917 0.674892i \(-0.764190\pi\)
−0.737917 + 0.674892i \(0.764190\pi\)
\(480\) 3.56155 0.162562
\(481\) 7.56155 0.344777
\(482\) 14.0000 0.637683
\(483\) 0 0
\(484\) −8.56155 −0.389161
\(485\) 51.6155 2.34374
\(486\) −1.00000 −0.0453609
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) −6.68466 −0.302600
\(489\) 16.8769 0.763200
\(490\) 0 0
\(491\) −8.87689 −0.400609 −0.200304 0.979734i \(-0.564193\pi\)
−0.200304 + 0.979734i \(0.564193\pi\)
\(492\) −1.12311 −0.0506335
\(493\) 44.6847 2.01250
\(494\) −4.68466 −0.210773
\(495\) 5.56155 0.249973
\(496\) −6.24621 −0.280463
\(497\) 0 0
\(498\) 8.87689 0.397783
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) −9.56155 −0.427606
\(501\) 22.9309 1.02448
\(502\) 7.80776 0.348478
\(503\) 3.12311 0.139252 0.0696262 0.997573i \(-0.477819\pi\)
0.0696262 + 0.997573i \(0.477819\pi\)
\(504\) 0 0
\(505\) −18.2462 −0.811946
\(506\) 8.68466 0.386080
\(507\) −1.00000 −0.0444116
\(508\) −4.87689 −0.216377
\(509\) 12.0540 0.534283 0.267142 0.963657i \(-0.413921\pi\)
0.267142 + 0.963657i \(0.413921\pi\)
\(510\) 23.8078 1.05423
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −4.68466 −0.206833
\(514\) 4.24621 0.187292
\(515\) −2.43845 −0.107451
\(516\) 6.43845 0.283437
\(517\) 0 0
\(518\) 0 0
\(519\) 20.2462 0.888710
\(520\) 3.56155 0.156184
\(521\) −2.68466 −0.117617 −0.0588085 0.998269i \(-0.518730\pi\)
−0.0588085 + 0.998269i \(0.518730\pi\)
\(522\) 6.68466 0.292580
\(523\) −21.7538 −0.951227 −0.475613 0.879654i \(-0.657774\pi\)
−0.475613 + 0.879654i \(0.657774\pi\)
\(524\) 9.56155 0.417698
\(525\) 0 0
\(526\) −30.2462 −1.31880
\(527\) −41.7538 −1.81882
\(528\) 1.56155 0.0679579
\(529\) 7.93087 0.344820
\(530\) −43.6155 −1.89454
\(531\) −2.24621 −0.0974773
\(532\) 0 0
\(533\) −1.12311 −0.0486471
\(534\) 10.0000 0.432742
\(535\) 31.6155 1.36686
\(536\) −7.12311 −0.307671
\(537\) −16.4924 −0.711701
\(538\) −20.2462 −0.872876
\(539\) 0 0
\(540\) 3.56155 0.153265
\(541\) −32.9309 −1.41581 −0.707904 0.706308i \(-0.750359\pi\)
−0.707904 + 0.706308i \(0.750359\pi\)
\(542\) −4.87689 −0.209481
\(543\) −0.246211 −0.0105659
\(544\) 6.68466 0.286602
\(545\) 60.3002 2.58298
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −3.56155 −0.152142
\(549\) −6.68466 −0.285294
\(550\) −12.0000 −0.511682
\(551\) 31.3153 1.33408
\(552\) 5.56155 0.236715
\(553\) 0 0
\(554\) −22.4924 −0.955611
\(555\) −26.9309 −1.14315
\(556\) −12.0000 −0.508913
\(557\) −3.36932 −0.142763 −0.0713813 0.997449i \(-0.522741\pi\)
−0.0713813 + 0.997449i \(0.522741\pi\)
\(558\) −6.24621 −0.264423
\(559\) 6.43845 0.272317
\(560\) 0 0
\(561\) 10.4384 0.440712
\(562\) 16.2462 0.685305
\(563\) 6.05398 0.255145 0.127572 0.991829i \(-0.459282\pi\)
0.127572 + 0.991829i \(0.459282\pi\)
\(564\) 0 0
\(565\) 15.1231 0.636234
\(566\) −18.2462 −0.766945
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) 41.2311 1.72850 0.864248 0.503066i \(-0.167795\pi\)
0.864248 + 0.503066i \(0.167795\pi\)
\(570\) 16.6847 0.698843
\(571\) 18.2462 0.763580 0.381790 0.924249i \(-0.375308\pi\)
0.381790 + 0.924249i \(0.375308\pi\)
\(572\) 1.56155 0.0652918
\(573\) −14.9309 −0.623746
\(574\) 0 0
\(575\) −42.7386 −1.78232
\(576\) 1.00000 0.0416667
\(577\) 30.0000 1.24892 0.624458 0.781058i \(-0.285320\pi\)
0.624458 + 0.781058i \(0.285320\pi\)
\(578\) 27.6847 1.15153
\(579\) −19.3693 −0.804961
\(580\) −23.8078 −0.988564
\(581\) 0 0
\(582\) 14.4924 0.600730
\(583\) −19.1231 −0.791998
\(584\) 3.56155 0.147378
\(585\) 3.56155 0.147252
\(586\) −24.7386 −1.02194
\(587\) −13.3693 −0.551811 −0.275905 0.961185i \(-0.588978\pi\)
−0.275905 + 0.961185i \(0.588978\pi\)
\(588\) 0 0
\(589\) −29.2614 −1.20569
\(590\) 8.00000 0.329355
\(591\) −10.8769 −0.447416
\(592\) −7.56155 −0.310778
\(593\) 20.2462 0.831412 0.415706 0.909499i \(-0.363534\pi\)
0.415706 + 0.909499i \(0.363534\pi\)
\(594\) 1.56155 0.0640713
\(595\) 0 0
\(596\) −17.6155 −0.721560
\(597\) 6.93087 0.283662
\(598\) 5.56155 0.227429
\(599\) 21.5616 0.880981 0.440491 0.897757i \(-0.354805\pi\)
0.440491 + 0.897757i \(0.354805\pi\)
\(600\) −7.68466 −0.313725
\(601\) 10.8769 0.443678 0.221839 0.975083i \(-0.428794\pi\)
0.221839 + 0.975083i \(0.428794\pi\)
\(602\) 0 0
\(603\) −7.12311 −0.290075
\(604\) 11.8078 0.480451
\(605\) 30.4924 1.23969
\(606\) −5.12311 −0.208112
\(607\) 18.4384 0.748393 0.374197 0.927349i \(-0.377918\pi\)
0.374197 + 0.927349i \(0.377918\pi\)
\(608\) 4.68466 0.189988
\(609\) 0 0
\(610\) 23.8078 0.963948
\(611\) 0 0
\(612\) 6.68466 0.270211
\(613\) 44.5464 1.79921 0.899606 0.436702i \(-0.143854\pi\)
0.899606 + 0.436702i \(0.143854\pi\)
\(614\) −26.2462 −1.05921
\(615\) 4.00000 0.161296
\(616\) 0 0
\(617\) −33.4233 −1.34557 −0.672786 0.739838i \(-0.734902\pi\)
−0.672786 + 0.739838i \(0.734902\pi\)
\(618\) −0.684658 −0.0275410
\(619\) −19.3153 −0.776349 −0.388175 0.921586i \(-0.626894\pi\)
−0.388175 + 0.921586i \(0.626894\pi\)
\(620\) 22.2462 0.893429
\(621\) 5.56155 0.223177
\(622\) 12.8769 0.516316
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) −4.36932 −0.174773
\(626\) 13.6155 0.544186
\(627\) 7.31534 0.292147
\(628\) 15.5616 0.620974
\(629\) −50.5464 −2.01542
\(630\) 0 0
\(631\) −22.9309 −0.912864 −0.456432 0.889758i \(-0.650873\pi\)
−0.456432 + 0.889758i \(0.650873\pi\)
\(632\) −11.1231 −0.442453
\(633\) 15.8078 0.628302
\(634\) 2.87689 0.114256
\(635\) 17.3693 0.689280
\(636\) −12.2462 −0.485594
\(637\) 0 0
\(638\) −10.4384 −0.413262
\(639\) 8.00000 0.316475
\(640\) −3.56155 −0.140783
\(641\) −20.2462 −0.799677 −0.399839 0.916586i \(-0.630934\pi\)
−0.399839 + 0.916586i \(0.630934\pi\)
\(642\) 8.87689 0.350343
\(643\) −16.1922 −0.638559 −0.319280 0.947661i \(-0.603441\pi\)
−0.319280 + 0.947661i \(0.603441\pi\)
\(644\) 0 0
\(645\) −22.9309 −0.902902
\(646\) 31.3153 1.23209
\(647\) 14.2462 0.560076 0.280038 0.959989i \(-0.409653\pi\)
0.280038 + 0.959989i \(0.409653\pi\)
\(648\) 1.00000 0.0392837
\(649\) 3.50758 0.137684
\(650\) −7.68466 −0.301417
\(651\) 0 0
\(652\) −16.8769 −0.660950
\(653\) −16.9309 −0.662556 −0.331278 0.943533i \(-0.607480\pi\)
−0.331278 + 0.943533i \(0.607480\pi\)
\(654\) 16.9309 0.662049
\(655\) −34.0540 −1.33060
\(656\) 1.12311 0.0438499
\(657\) 3.56155 0.138949
\(658\) 0 0
\(659\) 21.3693 0.832430 0.416215 0.909266i \(-0.363356\pi\)
0.416215 + 0.909266i \(0.363356\pi\)
\(660\) −5.56155 −0.216483
\(661\) 0.246211 0.00957651 0.00478825 0.999989i \(-0.498476\pi\)
0.00478825 + 0.999989i \(0.498476\pi\)
\(662\) 13.3693 0.519613
\(663\) 6.68466 0.259611
\(664\) −8.87689 −0.344490
\(665\) 0 0
\(666\) −7.56155 −0.293004
\(667\) −37.1771 −1.43950
\(668\) −22.9309 −0.887222
\(669\) −23.6155 −0.913029
\(670\) 25.3693 0.980102
\(671\) 10.4384 0.402972
\(672\) 0 0
\(673\) −33.8078 −1.30319 −0.651597 0.758566i \(-0.725901\pi\)
−0.651597 + 0.758566i \(0.725901\pi\)
\(674\) 4.43845 0.170963
\(675\) −7.68466 −0.295783
\(676\) 1.00000 0.0384615
\(677\) −2.49242 −0.0957916 −0.0478958 0.998852i \(-0.515252\pi\)
−0.0478958 + 0.998852i \(0.515252\pi\)
\(678\) 4.24621 0.163075
\(679\) 0 0
\(680\) −23.8078 −0.912986
\(681\) −7.12311 −0.272958
\(682\) 9.75379 0.373492
\(683\) 35.3153 1.35130 0.675652 0.737221i \(-0.263862\pi\)
0.675652 + 0.737221i \(0.263862\pi\)
\(684\) 4.68466 0.179122
\(685\) 12.6847 0.484656
\(686\) 0 0
\(687\) 28.2462 1.07766
\(688\) −6.43845 −0.245463
\(689\) −12.2462 −0.466543
\(690\) −19.8078 −0.754069
\(691\) −16.4924 −0.627401 −0.313701 0.949522i \(-0.601569\pi\)
−0.313701 + 0.949522i \(0.601569\pi\)
\(692\) −20.2462 −0.769645
\(693\) 0 0
\(694\) −20.0000 −0.759190
\(695\) 42.7386 1.62117
\(696\) −6.68466 −0.253381
\(697\) 7.50758 0.284370
\(698\) −7.75379 −0.293485
\(699\) −27.3693 −1.03520
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 1.00000 0.0377426
\(703\) −35.4233 −1.33601
\(704\) −1.56155 −0.0588532
\(705\) 0 0
\(706\) −30.4924 −1.14760
\(707\) 0 0
\(708\) 2.24621 0.0844178
\(709\) −16.2462 −0.610139 −0.305070 0.952330i \(-0.598680\pi\)
−0.305070 + 0.952330i \(0.598680\pi\)
\(710\) −28.4924 −1.06930
\(711\) −11.1231 −0.417149
\(712\) −10.0000 −0.374766
\(713\) 34.7386 1.30097
\(714\) 0 0
\(715\) −5.56155 −0.207990
\(716\) 16.4924 0.616351
\(717\) −16.0000 −0.597531
\(718\) −26.7386 −0.997877
\(719\) −43.1231 −1.60822 −0.804110 0.594480i \(-0.797358\pi\)
−0.804110 + 0.594480i \(0.797358\pi\)
\(720\) −3.56155 −0.132731
\(721\) 0 0
\(722\) 2.94602 0.109640
\(723\) −14.0000 −0.520666
\(724\) 0.246211 0.00915037
\(725\) 51.3693 1.90781
\(726\) 8.56155 0.317749
\(727\) 6.93087 0.257052 0.128526 0.991706i \(-0.458975\pi\)
0.128526 + 0.991706i \(0.458975\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −12.6847 −0.469480
\(731\) −43.0388 −1.59185
\(732\) 6.68466 0.247072
\(733\) 41.6155 1.53710 0.768552 0.639787i \(-0.220977\pi\)
0.768552 + 0.639787i \(0.220977\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) −5.56155 −0.205002
\(737\) 11.1231 0.409725
\(738\) 1.12311 0.0413421
\(739\) −32.1080 −1.18111 −0.590555 0.806997i \(-0.701091\pi\)
−0.590555 + 0.806997i \(0.701091\pi\)
\(740\) 26.9309 0.989998
\(741\) 4.68466 0.172095
\(742\) 0 0
\(743\) 28.1080 1.03118 0.515590 0.856835i \(-0.327573\pi\)
0.515590 + 0.856835i \(0.327573\pi\)
\(744\) 6.24621 0.228997
\(745\) 62.7386 2.29857
\(746\) −17.6155 −0.644950
\(747\) −8.87689 −0.324789
\(748\) −10.4384 −0.381667
\(749\) 0 0
\(750\) 9.56155 0.349139
\(751\) −6.24621 −0.227927 −0.113964 0.993485i \(-0.536355\pi\)
−0.113964 + 0.993485i \(0.536355\pi\)
\(752\) 0 0
\(753\) −7.80776 −0.284531
\(754\) −6.68466 −0.243441
\(755\) −42.0540 −1.53050
\(756\) 0 0
\(757\) 34.4924 1.25365 0.626824 0.779161i \(-0.284354\pi\)
0.626824 + 0.779161i \(0.284354\pi\)
\(758\) 34.2462 1.24388
\(759\) −8.68466 −0.315233
\(760\) −16.6847 −0.605216
\(761\) −5.12311 −0.185712 −0.0928562 0.995680i \(-0.529600\pi\)
−0.0928562 + 0.995680i \(0.529600\pi\)
\(762\) 4.87689 0.176671
\(763\) 0 0
\(764\) 14.9309 0.540180
\(765\) −23.8078 −0.860772
\(766\) −27.4233 −0.990844
\(767\) 2.24621 0.0811060
\(768\) −1.00000 −0.0360844
\(769\) 16.4384 0.592786 0.296393 0.955066i \(-0.404216\pi\)
0.296393 + 0.955066i \(0.404216\pi\)
\(770\) 0 0
\(771\) −4.24621 −0.152924
\(772\) 19.3693 0.697117
\(773\) −52.9309 −1.90379 −0.951896 0.306423i \(-0.900868\pi\)
−0.951896 + 0.306423i \(0.900868\pi\)
\(774\) −6.43845 −0.231425
\(775\) −48.0000 −1.72421
\(776\) −14.4924 −0.520248
\(777\) 0 0
\(778\) −28.7386 −1.03033
\(779\) 5.26137 0.188508
\(780\) −3.56155 −0.127524
\(781\) −12.4924 −0.447014
\(782\) −37.1771 −1.32945
\(783\) −6.68466 −0.238890
\(784\) 0 0
\(785\) −55.4233 −1.97814
\(786\) −9.56155 −0.341049
\(787\) −20.3002 −0.723624 −0.361812 0.932251i \(-0.617842\pi\)
−0.361812 + 0.932251i \(0.617842\pi\)
\(788\) 10.8769 0.387473
\(789\) 30.2462 1.07679
\(790\) 39.6155 1.40946
\(791\) 0 0
\(792\) −1.56155 −0.0554874
\(793\) 6.68466 0.237379
\(794\) 18.0000 0.638796
\(795\) 43.6155 1.54688
\(796\) −6.93087 −0.245658
\(797\) −31.3693 −1.11116 −0.555579 0.831464i \(-0.687503\pi\)
−0.555579 + 0.831464i \(0.687503\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 7.68466 0.271694
\(801\) −10.0000 −0.353333
\(802\) 8.24621 0.291184
\(803\) −5.56155 −0.196263
\(804\) 7.12311 0.251213
\(805\) 0 0
\(806\) 6.24621 0.220013
\(807\) 20.2462 0.712700
\(808\) 5.12311 0.180230
\(809\) −2.49242 −0.0876289 −0.0438145 0.999040i \(-0.513951\pi\)
−0.0438145 + 0.999040i \(0.513951\pi\)
\(810\) −3.56155 −0.125140
\(811\) −41.5616 −1.45942 −0.729712 0.683755i \(-0.760346\pi\)
−0.729712 + 0.683755i \(0.760346\pi\)
\(812\) 0 0
\(813\) 4.87689 0.171040
\(814\) 11.8078 0.413862
\(815\) 60.1080 2.10549
\(816\) −6.68466 −0.234010
\(817\) −30.1619 −1.05523
\(818\) 4.93087 0.172404
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) 14.3845 0.502022 0.251011 0.967984i \(-0.419237\pi\)
0.251011 + 0.967984i \(0.419237\pi\)
\(822\) 3.56155 0.124223
\(823\) 4.49242 0.156596 0.0782980 0.996930i \(-0.475051\pi\)
0.0782980 + 0.996930i \(0.475051\pi\)
\(824\) 0.684658 0.0238512
\(825\) 12.0000 0.417786
\(826\) 0 0
\(827\) 26.9309 0.936478 0.468239 0.883602i \(-0.344889\pi\)
0.468239 + 0.883602i \(0.344889\pi\)
\(828\) −5.56155 −0.193277
\(829\) −38.6847 −1.34357 −0.671787 0.740744i \(-0.734473\pi\)
−0.671787 + 0.740744i \(0.734473\pi\)
\(830\) 31.6155 1.09739
\(831\) 22.4924 0.780253
\(832\) −1.00000 −0.0346688
\(833\) 0 0
\(834\) 12.0000 0.415526
\(835\) 81.6695 2.82629
\(836\) −7.31534 −0.253006
\(837\) 6.24621 0.215901
\(838\) 36.6847 1.26725
\(839\) 10.7386 0.370739 0.185369 0.982669i \(-0.440652\pi\)
0.185369 + 0.982669i \(0.440652\pi\)
\(840\) 0 0
\(841\) 15.6847 0.540850
\(842\) −3.75379 −0.129364
\(843\) −16.2462 −0.559549
\(844\) −15.8078 −0.544126
\(845\) −3.56155 −0.122521
\(846\) 0 0
\(847\) 0 0
\(848\) 12.2462 0.420537
\(849\) 18.2462 0.626208
\(850\) 51.3693 1.76195
\(851\) 42.0540 1.44159
\(852\) −8.00000 −0.274075
\(853\) 27.3693 0.937108 0.468554 0.883435i \(-0.344775\pi\)
0.468554 + 0.883435i \(0.344775\pi\)
\(854\) 0 0
\(855\) −16.6847 −0.570603
\(856\) −8.87689 −0.303406
\(857\) 34.4924 1.17824 0.589119 0.808046i \(-0.299475\pi\)
0.589119 + 0.808046i \(0.299475\pi\)
\(858\) −1.56155 −0.0533105
\(859\) −5.75379 −0.196317 −0.0981584 0.995171i \(-0.531295\pi\)
−0.0981584 + 0.995171i \(0.531295\pi\)
\(860\) 22.9309 0.781936
\(861\) 0 0
\(862\) −33.3693 −1.13656
\(863\) −17.3693 −0.591258 −0.295629 0.955303i \(-0.595529\pi\)
−0.295629 + 0.955303i \(0.595529\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 72.1080 2.45174
\(866\) 31.3693 1.06597
\(867\) −27.6847 −0.940220
\(868\) 0 0
\(869\) 17.3693 0.589214
\(870\) 23.8078 0.807159
\(871\) 7.12311 0.241357
\(872\) −16.9309 −0.573352
\(873\) −14.4924 −0.490494
\(874\) −26.0540 −0.881289
\(875\) 0 0
\(876\) −3.56155 −0.120334
\(877\) −40.2462 −1.35902 −0.679509 0.733667i \(-0.737807\pi\)
−0.679509 + 0.733667i \(0.737807\pi\)
\(878\) 6.93087 0.233906
\(879\) 24.7386 0.834413
\(880\) 5.56155 0.187480
\(881\) 19.1771 0.646092 0.323046 0.946383i \(-0.395293\pi\)
0.323046 + 0.946383i \(0.395293\pi\)
\(882\) 0 0
\(883\) 1.56155 0.0525504 0.0262752 0.999655i \(-0.491635\pi\)
0.0262752 + 0.999655i \(0.491635\pi\)
\(884\) −6.68466 −0.224829
\(885\) −8.00000 −0.268917
\(886\) −5.36932 −0.180386
\(887\) 52.4924 1.76252 0.881262 0.472629i \(-0.156695\pi\)
0.881262 + 0.472629i \(0.156695\pi\)
\(888\) 7.56155 0.253749
\(889\) 0 0
\(890\) 35.6155 1.19384
\(891\) −1.56155 −0.0523140
\(892\) 23.6155 0.790706
\(893\) 0 0
\(894\) 17.6155 0.589151
\(895\) −58.7386 −1.96342
\(896\) 0 0
\(897\) −5.56155 −0.185695
\(898\) 10.6847 0.356552
\(899\) −41.7538 −1.39257
\(900\) 7.68466 0.256155
\(901\) 81.8617 2.72721
\(902\) −1.75379 −0.0583948
\(903\) 0 0
\(904\) −4.24621 −0.141227
\(905\) −0.876894 −0.0291490
\(906\) −11.8078 −0.392287
\(907\) −7.50758 −0.249285 −0.124643 0.992202i \(-0.539778\pi\)
−0.124643 + 0.992202i \(0.539778\pi\)
\(908\) 7.12311 0.236389
\(909\) 5.12311 0.169923
\(910\) 0 0
\(911\) −11.4233 −0.378471 −0.189235 0.981932i \(-0.560601\pi\)
−0.189235 + 0.981932i \(0.560601\pi\)
\(912\) −4.68466 −0.155125
\(913\) 13.8617 0.458757
\(914\) 27.3693 0.905297
\(915\) −23.8078 −0.787060
\(916\) −28.2462 −0.933281
\(917\) 0 0
\(918\) −6.68466 −0.220627
\(919\) 19.1231 0.630813 0.315407 0.948957i \(-0.397859\pi\)
0.315407 + 0.948957i \(0.397859\pi\)
\(920\) 19.8078 0.653043
\(921\) 26.2462 0.864842
\(922\) 28.0540 0.923908
\(923\) −8.00000 −0.263323
\(924\) 0 0
\(925\) −58.1080 −1.91058
\(926\) −8.68466 −0.285396
\(927\) 0.684658 0.0224871
\(928\) 6.68466 0.219435
\(929\) −25.6155 −0.840418 −0.420209 0.907427i \(-0.638043\pi\)
−0.420209 + 0.907427i \(0.638043\pi\)
\(930\) −22.2462 −0.729482
\(931\) 0 0
\(932\) 27.3693 0.896512
\(933\) −12.8769 −0.421571
\(934\) 3.31534 0.108481
\(935\) 37.1771 1.21582
\(936\) −1.00000 −0.0326860
\(937\) 46.9848 1.53493 0.767464 0.641092i \(-0.221518\pi\)
0.767464 + 0.641092i \(0.221518\pi\)
\(938\) 0 0
\(939\) −13.6155 −0.444326
\(940\) 0 0
\(941\) 34.0000 1.10837 0.554184 0.832394i \(-0.313030\pi\)
0.554184 + 0.832394i \(0.313030\pi\)
\(942\) −15.5616 −0.507023
\(943\) −6.24621 −0.203405
\(944\) −2.24621 −0.0731079
\(945\) 0 0
\(946\) 10.0540 0.326883
\(947\) −40.7926 −1.32558 −0.662791 0.748805i \(-0.730628\pi\)
−0.662791 + 0.748805i \(0.730628\pi\)
\(948\) 11.1231 0.361262
\(949\) −3.56155 −0.115613
\(950\) 36.0000 1.16799
\(951\) −2.87689 −0.0932897
\(952\) 0 0
\(953\) 11.3693 0.368288 0.184144 0.982899i \(-0.441049\pi\)
0.184144 + 0.982899i \(0.441049\pi\)
\(954\) 12.2462 0.396486
\(955\) −53.1771 −1.72077
\(956\) 16.0000 0.517477
\(957\) 10.4384 0.337427
\(958\) −32.3002 −1.04357
\(959\) 0 0
\(960\) 3.56155 0.114949
\(961\) 8.01515 0.258553
\(962\) 7.56155 0.243794
\(963\) −8.87689 −0.286054
\(964\) 14.0000 0.450910
\(965\) −68.9848 −2.22070
\(966\) 0 0
\(967\) −21.5616 −0.693373 −0.346686 0.937981i \(-0.612693\pi\)
−0.346686 + 0.937981i \(0.612693\pi\)
\(968\) −8.56155 −0.275179
\(969\) −31.3153 −1.00599
\(970\) 51.6155 1.65727
\(971\) −2.24621 −0.0720843 −0.0360422 0.999350i \(-0.511475\pi\)
−0.0360422 + 0.999350i \(0.511475\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −8.00000 −0.256337
\(975\) 7.68466 0.246106
\(976\) −6.68466 −0.213971
\(977\) −22.6847 −0.725747 −0.362873 0.931839i \(-0.618204\pi\)
−0.362873 + 0.931839i \(0.618204\pi\)
\(978\) 16.8769 0.539664
\(979\) 15.6155 0.499074
\(980\) 0 0
\(981\) −16.9309 −0.540561
\(982\) −8.87689 −0.283273
\(983\) 13.1771 0.420284 0.210142 0.977671i \(-0.432607\pi\)
0.210142 + 0.977671i \(0.432607\pi\)
\(984\) −1.12311 −0.0358033
\(985\) −38.7386 −1.23432
\(986\) 44.6847 1.42305
\(987\) 0 0
\(988\) −4.68466 −0.149039
\(989\) 35.8078 1.13862
\(990\) 5.56155 0.176758
\(991\) −7.61553 −0.241915 −0.120958 0.992658i \(-0.538597\pi\)
−0.120958 + 0.992658i \(0.538597\pi\)
\(992\) −6.24621 −0.198317
\(993\) −13.3693 −0.424262
\(994\) 0 0
\(995\) 24.6847 0.782556
\(996\) 8.87689 0.281275
\(997\) −40.7386 −1.29021 −0.645103 0.764096i \(-0.723185\pi\)
−0.645103 + 0.764096i \(0.723185\pi\)
\(998\) 36.0000 1.13956
\(999\) 7.56155 0.239237
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.bo.1.1 2
7.6 odd 2 546.2.a.j.1.2 2
21.20 even 2 1638.2.a.u.1.1 2
28.27 even 2 4368.2.a.be.1.2 2
91.90 odd 2 7098.2.a.bl.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.j.1.2 2 7.6 odd 2
1638.2.a.u.1.1 2 21.20 even 2
3822.2.a.bo.1.1 2 1.1 even 1 trivial
4368.2.a.be.1.2 2 28.27 even 2
7098.2.a.bl.1.1 2 91.90 odd 2