Properties

Label 8085.2.a.bn.1.3
Level 8085
Weight 2
Character 8085.1
Self dual yes
Analytic conductor 64.559
Analytic rank 1
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8085.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.5590500342\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.13448.1
Defining polynomial: \(x^{4} - 7 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1155)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.546295\) of defining polynomial
Character \(\chi\) \(=\) 8085.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.546295 q^{2} -1.00000 q^{3} -1.70156 q^{4} +1.00000 q^{5} -0.546295 q^{6} -2.02214 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.546295 q^{2} -1.00000 q^{3} -1.70156 q^{4} +1.00000 q^{5} -0.546295 q^{6} -2.02214 q^{8} +1.00000 q^{9} +0.546295 q^{10} -1.00000 q^{11} +1.70156 q^{12} +0.568438 q^{13} -1.00000 q^{15} +2.29844 q^{16} -2.70156 q^{17} +0.546295 q^{18} -6.62049 q^{19} -1.70156 q^{20} -0.546295 q^{22} +8.93103 q^{23} +2.02214 q^{24} +1.00000 q^{25} +0.310535 q^{26} -1.00000 q^{27} +5.27000 q^{29} -0.546295 q^{30} -9.66103 q^{31} +5.29991 q^{32} +1.00000 q^{33} -1.47585 q^{34} -1.70156 q^{36} -2.75362 q^{37} -3.61674 q^{38} -0.568438 q^{39} -2.02214 q^{40} +10.9831 q^{41} -0.0520550 q^{43} +1.70156 q^{44} +1.00000 q^{45} +4.87897 q^{46} +10.1567 q^{47} -2.29844 q^{48} +0.546295 q^{50} +2.70156 q^{51} -0.967233 q^{52} +1.56469 q^{53} -0.546295 q^{54} -1.00000 q^{55} +6.62049 q^{57} +2.87897 q^{58} -6.36259 q^{59} +1.70156 q^{60} +3.71308 q^{61} -5.27777 q^{62} -1.70156 q^{64} +0.568438 q^{65} +0.546295 q^{66} -10.4146 q^{67} +4.59688 q^{68} -8.93103 q^{69} +2.56844 q^{71} -2.02214 q^{72} -2.26625 q^{73} -1.50429 q^{74} -1.00000 q^{75} +11.2652 q^{76} -0.310535 q^{78} -2.56844 q^{79} +2.29844 q^{80} +1.00000 q^{81} +6.00000 q^{82} -17.1605 q^{83} -2.70156 q^{85} -0.0284374 q^{86} -5.27000 q^{87} +2.02214 q^{88} +10.7772 q^{89} +0.546295 q^{90} -15.1967 q^{92} +9.66103 q^{93} +5.54857 q^{94} -6.62049 q^{95} -5.29991 q^{96} -17.9952 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + 6q^{4} + 4q^{5} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} + 6q^{4} + 4q^{5} + 4q^{9} - 4q^{11} - 6q^{12} - 8q^{13} - 4q^{15} + 22q^{16} + 2q^{17} - 10q^{19} + 6q^{20} - 2q^{23} + 4q^{25} - 20q^{26} - 4q^{27} - 2q^{29} - 24q^{31} + 4q^{33} + 6q^{36} + 8q^{37} - 16q^{38} + 8q^{39} + 6q^{43} - 6q^{44} + 4q^{45} - 12q^{46} - 4q^{47} - 22q^{48} - 2q^{51} - 12q^{52} + 14q^{53} - 4q^{55} + 10q^{57} - 20q^{58} + 2q^{59} - 6q^{60} - 6q^{61} - 8q^{62} + 6q^{64} - 8q^{65} - 8q^{67} + 44q^{68} + 2q^{69} - 4q^{73} - 36q^{74} - 4q^{75} - 56q^{76} + 20q^{78} + 22q^{80} + 4q^{81} + 24q^{82} - 6q^{83} + 2q^{85} - 36q^{86} + 2q^{87} - 18q^{89} - 44q^{92} + 24q^{93} + 36q^{94} - 10q^{95} + 6q^{97} - 4q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.546295 0.386289 0.193144 0.981170i \(-0.438131\pi\)
0.193144 + 0.981170i \(0.438131\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.70156 −0.850781
\(5\) 1.00000 0.447214
\(6\) −0.546295 −0.223024
\(7\) 0 0
\(8\) −2.02214 −0.714936
\(9\) 1.00000 0.333333
\(10\) 0.546295 0.172754
\(11\) −1.00000 −0.301511
\(12\) 1.70156 0.491199
\(13\) 0.568438 0.157656 0.0788282 0.996888i \(-0.474882\pi\)
0.0788282 + 0.996888i \(0.474882\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 2.29844 0.574609
\(17\) −2.70156 −0.655225 −0.327613 0.944812i \(-0.606244\pi\)
−0.327613 + 0.944812i \(0.606244\pi\)
\(18\) 0.546295 0.128763
\(19\) −6.62049 −1.51885 −0.759423 0.650598i \(-0.774518\pi\)
−0.759423 + 0.650598i \(0.774518\pi\)
\(20\) −1.70156 −0.380481
\(21\) 0 0
\(22\) −0.546295 −0.116470
\(23\) 8.93103 1.86225 0.931124 0.364703i \(-0.118829\pi\)
0.931124 + 0.364703i \(0.118829\pi\)
\(24\) 2.02214 0.412768
\(25\) 1.00000 0.200000
\(26\) 0.310535 0.0609009
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 5.27000 0.978615 0.489307 0.872111i \(-0.337250\pi\)
0.489307 + 0.872111i \(0.337250\pi\)
\(30\) −0.546295 −0.0997393
\(31\) −9.66103 −1.73517 −0.867586 0.497287i \(-0.834329\pi\)
−0.867586 + 0.497287i \(0.834329\pi\)
\(32\) 5.29991 0.936901
\(33\) 1.00000 0.174078
\(34\) −1.47585 −0.253106
\(35\) 0 0
\(36\) −1.70156 −0.283594
\(37\) −2.75362 −0.452692 −0.226346 0.974047i \(-0.572678\pi\)
−0.226346 + 0.974047i \(0.572678\pi\)
\(38\) −3.61674 −0.586713
\(39\) −0.568438 −0.0910230
\(40\) −2.02214 −0.319729
\(41\) 10.9831 1.71527 0.857635 0.514259i \(-0.171933\pi\)
0.857635 + 0.514259i \(0.171933\pi\)
\(42\) 0 0
\(43\) −0.0520550 −0.00793831 −0.00396916 0.999992i \(-0.501263\pi\)
−0.00396916 + 0.999992i \(0.501263\pi\)
\(44\) 1.70156 0.256520
\(45\) 1.00000 0.149071
\(46\) 4.87897 0.719365
\(47\) 10.1567 1.48151 0.740756 0.671774i \(-0.234467\pi\)
0.740756 + 0.671774i \(0.234467\pi\)
\(48\) −2.29844 −0.331751
\(49\) 0 0
\(50\) 0.546295 0.0772577
\(51\) 2.70156 0.378294
\(52\) −0.967233 −0.134131
\(53\) 1.56469 0.214926 0.107463 0.994209i \(-0.465727\pi\)
0.107463 + 0.994209i \(0.465727\pi\)
\(54\) −0.546295 −0.0743413
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 6.62049 0.876906
\(58\) 2.87897 0.378028
\(59\) −6.36259 −0.828339 −0.414169 0.910200i \(-0.635928\pi\)
−0.414169 + 0.910200i \(0.635928\pi\)
\(60\) 1.70156 0.219671
\(61\) 3.71308 0.475412 0.237706 0.971337i \(-0.423605\pi\)
0.237706 + 0.971337i \(0.423605\pi\)
\(62\) −5.27777 −0.670277
\(63\) 0 0
\(64\) −1.70156 −0.212695
\(65\) 0.568438 0.0705061
\(66\) 0.546295 0.0672442
\(67\) −10.4146 −1.27235 −0.636176 0.771544i \(-0.719485\pi\)
−0.636176 + 0.771544i \(0.719485\pi\)
\(68\) 4.59688 0.557453
\(69\) −8.93103 −1.07517
\(70\) 0 0
\(71\) 2.56844 0.304818 0.152409 0.988318i \(-0.451297\pi\)
0.152409 + 0.988318i \(0.451297\pi\)
\(72\) −2.02214 −0.238312
\(73\) −2.26625 −0.265244 −0.132622 0.991167i \(-0.542340\pi\)
−0.132622 + 0.991167i \(0.542340\pi\)
\(74\) −1.50429 −0.174870
\(75\) −1.00000 −0.115470
\(76\) 11.2652 1.29220
\(77\) 0 0
\(78\) −0.310535 −0.0351612
\(79\) −2.56844 −0.288972 −0.144486 0.989507i \(-0.546153\pi\)
−0.144486 + 0.989507i \(0.546153\pi\)
\(80\) 2.29844 0.256973
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −17.1605 −1.88361 −0.941804 0.336161i \(-0.890871\pi\)
−0.941804 + 0.336161i \(0.890871\pi\)
\(84\) 0 0
\(85\) −2.70156 −0.293026
\(86\) −0.0284374 −0.00306648
\(87\) −5.27000 −0.565003
\(88\) 2.02214 0.215561
\(89\) 10.7772 1.14238 0.571192 0.820816i \(-0.306481\pi\)
0.571192 + 0.820816i \(0.306481\pi\)
\(90\) 0.546295 0.0575845
\(91\) 0 0
\(92\) −15.1967 −1.58437
\(93\) 9.66103 1.00180
\(94\) 5.54857 0.572292
\(95\) −6.62049 −0.679248
\(96\) −5.29991 −0.540920
\(97\) −17.9952 −1.82713 −0.913567 0.406689i \(-0.866683\pi\)
−0.913567 + 0.406689i \(0.866683\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) −1.70156 −0.170156
\(101\) −0.826342 −0.0822241 −0.0411120 0.999155i \(-0.513090\pi\)
−0.0411120 + 0.999155i \(0.513090\pi\)
\(102\) 1.47585 0.146131
\(103\) −10.5921 −1.04367 −0.521833 0.853048i \(-0.674752\pi\)
−0.521833 + 0.853048i \(0.674752\pi\)
\(104\) −1.14946 −0.112714
\(105\) 0 0
\(106\) 0.854779 0.0830235
\(107\) −13.7864 −1.33278 −0.666390 0.745603i \(-0.732161\pi\)
−0.666390 + 0.745603i \(0.732161\pi\)
\(108\) 1.70156 0.163733
\(109\) 19.8905 1.90516 0.952582 0.304282i \(-0.0984166\pi\)
0.952582 + 0.304282i \(0.0984166\pi\)
\(110\) −0.546295 −0.0520872
\(111\) 2.75362 0.261362
\(112\) 0 0
\(113\) 6.43531 0.605383 0.302692 0.953089i \(-0.402115\pi\)
0.302692 + 0.953089i \(0.402115\pi\)
\(114\) 3.61674 0.338739
\(115\) 8.93103 0.832823
\(116\) −8.96723 −0.832587
\(117\) 0.568438 0.0525521
\(118\) −3.47585 −0.319978
\(119\) 0 0
\(120\) 2.02214 0.184596
\(121\) 1.00000 0.0909091
\(122\) 2.02844 0.183646
\(123\) −10.9831 −0.990311
\(124\) 16.4388 1.47625
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 4.05206 0.359562 0.179781 0.983707i \(-0.442461\pi\)
0.179781 + 0.983707i \(0.442461\pi\)
\(128\) −11.5294 −1.01906
\(129\) 0.0520550 0.00458319
\(130\) 0.310535 0.0272357
\(131\) 5.69781 0.497820 0.248910 0.968527i \(-0.419928\pi\)
0.248910 + 0.968527i \(0.419928\pi\)
\(132\) −1.70156 −0.148102
\(133\) 0 0
\(134\) −5.68947 −0.491495
\(135\) −1.00000 −0.0860663
\(136\) 5.46295 0.468444
\(137\) −3.40312 −0.290749 −0.145374 0.989377i \(-0.546439\pi\)
−0.145374 + 0.989377i \(0.546439\pi\)
\(138\) −4.87897 −0.415326
\(139\) 8.04724 0.682558 0.341279 0.939962i \(-0.389140\pi\)
0.341279 + 0.939962i \(0.389140\pi\)
\(140\) 0 0
\(141\) −10.1567 −0.855352
\(142\) 1.40312 0.117748
\(143\) −0.568438 −0.0475352
\(144\) 2.29844 0.191536
\(145\) 5.27000 0.437650
\(146\) −1.23804 −0.102461
\(147\) 0 0
\(148\) 4.68545 0.385142
\(149\) 7.50723 0.615017 0.307508 0.951545i \(-0.400505\pi\)
0.307508 + 0.951545i \(0.400505\pi\)
\(150\) −0.546295 −0.0446048
\(151\) 3.21795 0.261873 0.130936 0.991391i \(-0.458202\pi\)
0.130936 + 0.991391i \(0.458202\pi\)
\(152\) 13.3876 1.08588
\(153\) −2.70156 −0.218408
\(154\) 0 0
\(155\) −9.66103 −0.775992
\(156\) 0.967233 0.0774406
\(157\) 3.35107 0.267444 0.133722 0.991019i \(-0.457307\pi\)
0.133722 + 0.991019i \(0.457307\pi\)
\(158\) −1.40312 −0.111627
\(159\) −1.56469 −0.124088
\(160\) 5.29991 0.418995
\(161\) 0 0
\(162\) 0.546295 0.0429210
\(163\) 10.9600 0.858457 0.429228 0.903196i \(-0.358786\pi\)
0.429228 + 0.903196i \(0.358786\pi\)
\(164\) −18.6884 −1.45932
\(165\) 1.00000 0.0778499
\(166\) −9.37469 −0.727617
\(167\) 7.58830 0.587201 0.293600 0.955928i \(-0.405147\pi\)
0.293600 + 0.955928i \(0.405147\pi\)
\(168\) 0 0
\(169\) −12.6769 −0.975144
\(170\) −1.47585 −0.113192
\(171\) −6.62049 −0.506282
\(172\) 0.0885748 0.00675377
\(173\) 18.2094 1.38443 0.692216 0.721690i \(-0.256634\pi\)
0.692216 + 0.721690i \(0.256634\pi\)
\(174\) −2.87897 −0.218254
\(175\) 0 0
\(176\) −2.29844 −0.173251
\(177\) 6.36259 0.478242
\(178\) 5.88755 0.441290
\(179\) 4.75362 0.355302 0.177651 0.984094i \(-0.443150\pi\)
0.177651 + 0.984094i \(0.443150\pi\)
\(180\) −1.70156 −0.126827
\(181\) −25.2652 −1.87795 −0.938973 0.343991i \(-0.888221\pi\)
−0.938973 + 0.343991i \(0.888221\pi\)
\(182\) 0 0
\(183\) −3.71308 −0.274479
\(184\) −18.0598 −1.33139
\(185\) −2.75362 −0.202450
\(186\) 5.27777 0.386985
\(187\) 2.70156 0.197558
\(188\) −17.2823 −1.26044
\(189\) 0 0
\(190\) −3.61674 −0.262386
\(191\) −17.4504 −1.26266 −0.631332 0.775513i \(-0.717491\pi\)
−0.631332 + 0.775513i \(0.717491\pi\)
\(192\) 1.70156 0.122800
\(193\) 22.9546 1.65231 0.826156 0.563442i \(-0.190523\pi\)
0.826156 + 0.563442i \(0.190523\pi\)
\(194\) −9.83067 −0.705801
\(195\) −0.568438 −0.0407067
\(196\) 0 0
\(197\) −3.36634 −0.239842 −0.119921 0.992783i \(-0.538264\pi\)
−0.119921 + 0.992783i \(0.538264\pi\)
\(198\) −0.546295 −0.0388235
\(199\) −9.54402 −0.676557 −0.338279 0.941046i \(-0.609845\pi\)
−0.338279 + 0.941046i \(0.609845\pi\)
\(200\) −2.02214 −0.142987
\(201\) 10.4146 0.734592
\(202\) −0.451426 −0.0317622
\(203\) 0 0
\(204\) −4.59688 −0.321846
\(205\) 10.9831 0.767092
\(206\) −5.78638 −0.403156
\(207\) 8.93103 0.620749
\(208\) 1.30652 0.0905909
\(209\) 6.62049 0.457949
\(210\) 0 0
\(211\) −7.73375 −0.532413 −0.266207 0.963916i \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(212\) −2.66241 −0.182855
\(213\) −2.56844 −0.175986
\(214\) −7.53143 −0.514838
\(215\) −0.0520550 −0.00355012
\(216\) 2.02214 0.137589
\(217\) 0 0
\(218\) 10.8661 0.735943
\(219\) 2.26625 0.153139
\(220\) 1.70156 0.114719
\(221\) −1.53567 −0.103300
\(222\) 1.50429 0.100961
\(223\) 13.9141 0.931758 0.465879 0.884848i \(-0.345738\pi\)
0.465879 + 0.884848i \(0.345738\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 3.51558 0.233853
\(227\) −0.597452 −0.0396543 −0.0198271 0.999803i \(-0.506312\pi\)
−0.0198271 + 0.999803i \(0.506312\pi\)
\(228\) −11.2652 −0.746055
\(229\) −21.1895 −1.40024 −0.700121 0.714024i \(-0.746871\pi\)
−0.700121 + 0.714024i \(0.746871\pi\)
\(230\) 4.87897 0.321710
\(231\) 0 0
\(232\) −10.6567 −0.699647
\(233\) −20.2821 −1.32872 −0.664362 0.747411i \(-0.731297\pi\)
−0.664362 + 0.747411i \(0.731297\pi\)
\(234\) 0.310535 0.0203003
\(235\) 10.1567 0.662553
\(236\) 10.8263 0.704735
\(237\) 2.56844 0.166838
\(238\) 0 0
\(239\) 2.23723 0.144715 0.0723573 0.997379i \(-0.476948\pi\)
0.0723573 + 0.997379i \(0.476948\pi\)
\(240\) −2.29844 −0.148364
\(241\) −8.93045 −0.575261 −0.287630 0.957741i \(-0.592867\pi\)
−0.287630 + 0.957741i \(0.592867\pi\)
\(242\) 0.546295 0.0351172
\(243\) −1.00000 −0.0641500
\(244\) −6.31804 −0.404471
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) −3.76334 −0.239456
\(248\) 19.5360 1.24054
\(249\) 17.1605 1.08750
\(250\) 0.546295 0.0345507
\(251\) −3.35884 −0.212008 −0.106004 0.994366i \(-0.533806\pi\)
−0.106004 + 0.994366i \(0.533806\pi\)
\(252\) 0 0
\(253\) −8.93103 −0.561489
\(254\) 2.21362 0.138895
\(255\) 2.70156 0.169178
\(256\) −2.89531 −0.180957
\(257\) −5.80943 −0.362382 −0.181191 0.983448i \(-0.557995\pi\)
−0.181191 + 0.983448i \(0.557995\pi\)
\(258\) 0.0284374 0.00177043
\(259\) 0 0
\(260\) −0.967233 −0.0599853
\(261\) 5.27000 0.326205
\(262\) 3.11268 0.192302
\(263\) −23.3221 −1.43810 −0.719050 0.694959i \(-0.755423\pi\)
−0.719050 + 0.694959i \(0.755423\pi\)
\(264\) −2.02214 −0.124454
\(265\) 1.56469 0.0961179
\(266\) 0 0
\(267\) −10.7772 −0.659556
\(268\) 17.7212 1.08249
\(269\) −29.1246 −1.77576 −0.887878 0.460080i \(-0.847821\pi\)
−0.887878 + 0.460080i \(0.847821\pi\)
\(270\) −0.546295 −0.0332464
\(271\) −31.2416 −1.89779 −0.948895 0.315592i \(-0.897797\pi\)
−0.948895 + 0.315592i \(0.897797\pi\)
\(272\) −6.20937 −0.376499
\(273\) 0 0
\(274\) −1.85911 −0.112313
\(275\) −1.00000 −0.0603023
\(276\) 15.1967 0.914734
\(277\) −15.8988 −0.955269 −0.477634 0.878559i \(-0.658506\pi\)
−0.477634 + 0.878559i \(0.658506\pi\)
\(278\) 4.39616 0.263664
\(279\) −9.66103 −0.578391
\(280\) 0 0
\(281\) −14.1041 −0.841381 −0.420690 0.907204i \(-0.638212\pi\)
−0.420690 + 0.907204i \(0.638212\pi\)
\(282\) −5.54857 −0.330413
\(283\) 1.32745 0.0789088 0.0394544 0.999221i \(-0.487438\pi\)
0.0394544 + 0.999221i \(0.487438\pi\)
\(284\) −4.37036 −0.259333
\(285\) 6.62049 0.392164
\(286\) −0.310535 −0.0183623
\(287\) 0 0
\(288\) 5.29991 0.312300
\(289\) −9.70156 −0.570680
\(290\) 2.87897 0.169059
\(291\) 17.9952 1.05490
\(292\) 3.85616 0.225665
\(293\) −18.3784 −1.07368 −0.536840 0.843684i \(-0.680382\pi\)
−0.536840 + 0.843684i \(0.680382\pi\)
\(294\) 0 0
\(295\) −6.36259 −0.370444
\(296\) 5.56821 0.323646
\(297\) 1.00000 0.0580259
\(298\) 4.10116 0.237574
\(299\) 5.07674 0.293595
\(300\) 1.70156 0.0982397
\(301\) 0 0
\(302\) 1.75795 0.101158
\(303\) 0.826342 0.0474721
\(304\) −15.2168 −0.872743
\(305\) 3.71308 0.212611
\(306\) −1.47585 −0.0843687
\(307\) −6.45143 −0.368202 −0.184101 0.982907i \(-0.558937\pi\)
−0.184101 + 0.982907i \(0.558937\pi\)
\(308\) 0 0
\(309\) 10.5921 0.602561
\(310\) −5.27777 −0.299757
\(311\) 29.4136 1.66789 0.833946 0.551847i \(-0.186077\pi\)
0.833946 + 0.551847i \(0.186077\pi\)
\(312\) 1.14946 0.0650756
\(313\) 24.7542 1.39919 0.699595 0.714540i \(-0.253364\pi\)
0.699595 + 0.714540i \(0.253364\pi\)
\(314\) 1.83067 0.103311
\(315\) 0 0
\(316\) 4.37036 0.245852
\(317\) −15.0955 −0.847850 −0.423925 0.905697i \(-0.639348\pi\)
−0.423925 + 0.905697i \(0.639348\pi\)
\(318\) −0.854779 −0.0479336
\(319\) −5.27000 −0.295063
\(320\) −1.70156 −0.0951202
\(321\) 13.7864 0.769481
\(322\) 0 0
\(323\) 17.8857 0.995186
\(324\) −1.70156 −0.0945312
\(325\) 0.568438 0.0315313
\(326\) 5.98741 0.331612
\(327\) −19.8905 −1.09995
\(328\) −22.2094 −1.22631
\(329\) 0 0
\(330\) 0.546295 0.0300725
\(331\) 6.78263 0.372807 0.186404 0.982473i \(-0.440317\pi\)
0.186404 + 0.982473i \(0.440317\pi\)
\(332\) 29.1996 1.60254
\(333\) −2.75362 −0.150897
\(334\) 4.14545 0.226829
\(335\) −10.4146 −0.569013
\(336\) 0 0
\(337\) −31.9509 −1.74048 −0.870238 0.492631i \(-0.836035\pi\)
−0.870238 + 0.492631i \(0.836035\pi\)
\(338\) −6.92531 −0.376687
\(339\) −6.43531 −0.349518
\(340\) 4.59688 0.249301
\(341\) 9.66103 0.523174
\(342\) −3.61674 −0.195571
\(343\) 0 0
\(344\) 0.105263 0.00567538
\(345\) −8.93103 −0.480830
\(346\) 9.94768 0.534791
\(347\) −13.6694 −0.733810 −0.366905 0.930258i \(-0.619583\pi\)
−0.366905 + 0.930258i \(0.619583\pi\)
\(348\) 8.96723 0.480694
\(349\) −16.1720 −0.865668 −0.432834 0.901474i \(-0.642486\pi\)
−0.432834 + 0.901474i \(0.642486\pi\)
\(350\) 0 0
\(351\) −0.568438 −0.0303410
\(352\) −5.29991 −0.282486
\(353\) 9.94313 0.529219 0.264610 0.964356i \(-0.414757\pi\)
0.264610 + 0.964356i \(0.414757\pi\)
\(354\) 3.47585 0.184739
\(355\) 2.56844 0.136319
\(356\) −18.3381 −0.971919
\(357\) 0 0
\(358\) 2.59688 0.137249
\(359\) −12.5845 −0.664187 −0.332094 0.943246i \(-0.607755\pi\)
−0.332094 + 0.943246i \(0.607755\pi\)
\(360\) −2.02214 −0.106576
\(361\) 24.8309 1.30689
\(362\) −13.8022 −0.725429
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) −2.26625 −0.118621
\(366\) −2.02844 −0.106028
\(367\) −36.9699 −1.92981 −0.964907 0.262592i \(-0.915423\pi\)
−0.964907 + 0.262592i \(0.915423\pi\)
\(368\) 20.5274 1.07007
\(369\) 10.9831 0.571756
\(370\) −1.50429 −0.0782041
\(371\) 0 0
\(372\) −16.4388 −0.852314
\(373\) −13.4109 −0.694390 −0.347195 0.937793i \(-0.612866\pi\)
−0.347195 + 0.937793i \(0.612866\pi\)
\(374\) 1.47585 0.0763143
\(375\) −1.00000 −0.0516398
\(376\) −20.5384 −1.05919
\(377\) 2.99567 0.154285
\(378\) 0 0
\(379\) −0.805672 −0.0413846 −0.0206923 0.999786i \(-0.506587\pi\)
−0.0206923 + 0.999786i \(0.506587\pi\)
\(380\) 11.2652 0.577892
\(381\) −4.05206 −0.207593
\(382\) −9.53304 −0.487753
\(383\) 22.3494 1.14200 0.571001 0.820949i \(-0.306555\pi\)
0.571001 + 0.820949i \(0.306555\pi\)
\(384\) 11.5294 0.588356
\(385\) 0 0
\(386\) 12.5400 0.638269
\(387\) −0.0520550 −0.00264610
\(388\) 30.6199 1.55449
\(389\) −29.4863 −1.49501 −0.747507 0.664253i \(-0.768750\pi\)
−0.747507 + 0.664253i \(0.768750\pi\)
\(390\) −0.310535 −0.0157245
\(391\) −24.1277 −1.22019
\(392\) 0 0
\(393\) −5.69781 −0.287416
\(394\) −1.83902 −0.0926482
\(395\) −2.56844 −0.129232
\(396\) 1.70156 0.0855067
\(397\) −4.45143 −0.223411 −0.111705 0.993741i \(-0.535631\pi\)
−0.111705 + 0.993741i \(0.535631\pi\)
\(398\) −5.21384 −0.261346
\(399\) 0 0
\(400\) 2.29844 0.114922
\(401\) 22.7123 1.13420 0.567099 0.823650i \(-0.308066\pi\)
0.567099 + 0.823650i \(0.308066\pi\)
\(402\) 5.68947 0.283765
\(403\) −5.49170 −0.273561
\(404\) 1.40607 0.0699547
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 2.75362 0.136492
\(408\) −5.46295 −0.270456
\(409\) −5.09259 −0.251812 −0.125906 0.992042i \(-0.540184\pi\)
−0.125906 + 0.992042i \(0.540184\pi\)
\(410\) 6.00000 0.296319
\(411\) 3.40312 0.167864
\(412\) 18.0230 0.887932
\(413\) 0 0
\(414\) 4.87897 0.239788
\(415\) −17.1605 −0.842376
\(416\) 3.01267 0.147708
\(417\) −8.04724 −0.394075
\(418\) 3.61674 0.176901
\(419\) 2.86286 0.139860 0.0699300 0.997552i \(-0.477722\pi\)
0.0699300 + 0.997552i \(0.477722\pi\)
\(420\) 0 0
\(421\) −16.1047 −0.784894 −0.392447 0.919775i \(-0.628371\pi\)
−0.392447 + 0.919775i \(0.628371\pi\)
\(422\) −4.22491 −0.205665
\(423\) 10.1567 0.493838
\(424\) −3.16402 −0.153658
\(425\) −2.70156 −0.131045
\(426\) −1.40312 −0.0679816
\(427\) 0 0
\(428\) 23.4584 1.13390
\(429\) 0.568438 0.0274445
\(430\) −0.0284374 −0.00137137
\(431\) −4.67794 −0.225329 −0.112664 0.993633i \(-0.535938\pi\)
−0.112664 + 0.993633i \(0.535938\pi\)
\(432\) −2.29844 −0.110584
\(433\) 27.4337 1.31838 0.659189 0.751977i \(-0.270900\pi\)
0.659189 + 0.751977i \(0.270900\pi\)
\(434\) 0 0
\(435\) −5.27000 −0.252677
\(436\) −33.8449 −1.62088
\(437\) −59.1278 −2.82847
\(438\) 1.23804 0.0591558
\(439\) −5.73433 −0.273685 −0.136842 0.990593i \(-0.543695\pi\)
−0.136842 + 0.990593i \(0.543695\pi\)
\(440\) 2.02214 0.0964019
\(441\) 0 0
\(442\) −0.838929 −0.0399038
\(443\) −9.58536 −0.455414 −0.227707 0.973730i \(-0.573123\pi\)
−0.227707 + 0.973730i \(0.573123\pi\)
\(444\) −4.68545 −0.222362
\(445\) 10.7772 0.510890
\(446\) 7.60121 0.359927
\(447\) −7.50723 −0.355080
\(448\) 0 0
\(449\) 30.9463 1.46045 0.730223 0.683209i \(-0.239416\pi\)
0.730223 + 0.683209i \(0.239416\pi\)
\(450\) 0.546295 0.0257526
\(451\) −10.9831 −0.517173
\(452\) −10.9501 −0.515049
\(453\) −3.21795 −0.151192
\(454\) −0.326385 −0.0153180
\(455\) 0 0
\(456\) −13.3876 −0.626931
\(457\) −42.2246 −1.97519 −0.987593 0.157036i \(-0.949806\pi\)
−0.987593 + 0.157036i \(0.949806\pi\)
\(458\) −11.5757 −0.540898
\(459\) 2.70156 0.126098
\(460\) −15.1967 −0.708550
\(461\) −9.87464 −0.459908 −0.229954 0.973201i \(-0.573858\pi\)
−0.229954 + 0.973201i \(0.573858\pi\)
\(462\) 0 0
\(463\) 2.56009 0.118978 0.0594888 0.998229i \(-0.481053\pi\)
0.0594888 + 0.998229i \(0.481053\pi\)
\(464\) 12.1128 0.562321
\(465\) 9.66103 0.448019
\(466\) −11.0800 −0.513271
\(467\) 19.6924 0.911256 0.455628 0.890170i \(-0.349415\pi\)
0.455628 + 0.890170i \(0.349415\pi\)
\(468\) −0.967233 −0.0447104
\(469\) 0 0
\(470\) 5.54857 0.255937
\(471\) −3.35107 −0.154409
\(472\) 12.8661 0.592209
\(473\) 0.0520550 0.00239349
\(474\) 1.40312 0.0644476
\(475\) −6.62049 −0.303769
\(476\) 0 0
\(477\) 1.56469 0.0716420
\(478\) 1.22219 0.0559016
\(479\) −15.3672 −0.702144 −0.351072 0.936348i \(-0.614183\pi\)
−0.351072 + 0.936348i \(0.614183\pi\)
\(480\) −5.29991 −0.241907
\(481\) −1.56526 −0.0713698
\(482\) −4.87866 −0.222217
\(483\) 0 0
\(484\) −1.70156 −0.0773437
\(485\) −17.9952 −0.817119
\(486\) −0.546295 −0.0247804
\(487\) −1.52848 −0.0692621 −0.0346310 0.999400i \(-0.511026\pi\)
−0.0346310 + 0.999400i \(0.511026\pi\)
\(488\) −7.50839 −0.339889
\(489\) −10.9600 −0.495630
\(490\) 0 0
\(491\) −40.7359 −1.83839 −0.919193 0.393808i \(-0.871157\pi\)
−0.919193 + 0.393808i \(0.871157\pi\)
\(492\) 18.6884 0.842538
\(493\) −14.2372 −0.641213
\(494\) −2.05589 −0.0924990
\(495\) −1.00000 −0.0449467
\(496\) −22.2053 −0.997046
\(497\) 0 0
\(498\) 9.37469 0.420090
\(499\) −20.3312 −0.910150 −0.455075 0.890453i \(-0.650388\pi\)
−0.455075 + 0.890453i \(0.650388\pi\)
\(500\) −1.70156 −0.0760962
\(501\) −7.58830 −0.339020
\(502\) −1.83491 −0.0818963
\(503\) 23.5952 1.05206 0.526030 0.850466i \(-0.323680\pi\)
0.526030 + 0.850466i \(0.323680\pi\)
\(504\) 0 0
\(505\) −0.826342 −0.0367717
\(506\) −4.87897 −0.216897
\(507\) 12.6769 0.563000
\(508\) −6.89482 −0.305908
\(509\) −38.0993 −1.68872 −0.844361 0.535775i \(-0.820019\pi\)
−0.844361 + 0.535775i \(0.820019\pi\)
\(510\) 1.47585 0.0653517
\(511\) 0 0
\(512\) 21.4771 0.949161
\(513\) 6.62049 0.292302
\(514\) −3.17366 −0.139984
\(515\) −10.5921 −0.466742
\(516\) −0.0885748 −0.00389929
\(517\) −10.1567 −0.446693
\(518\) 0 0
\(519\) −18.2094 −0.799303
\(520\) −1.14946 −0.0504073
\(521\) −33.9952 −1.48936 −0.744678 0.667424i \(-0.767397\pi\)
−0.744678 + 0.667424i \(0.767397\pi\)
\(522\) 2.87897 0.126009
\(523\) −10.9517 −0.478884 −0.239442 0.970911i \(-0.576965\pi\)
−0.239442 + 0.970911i \(0.576965\pi\)
\(524\) −9.69518 −0.423536
\(525\) 0 0
\(526\) −12.7407 −0.555522
\(527\) 26.0999 1.13693
\(528\) 2.29844 0.100027
\(529\) 56.7633 2.46797
\(530\) 0.854779 0.0371292
\(531\) −6.36259 −0.276113
\(532\) 0 0
\(533\) 6.24321 0.270423
\(534\) −5.88755 −0.254779
\(535\) −13.7864 −0.596037
\(536\) 21.0599 0.909650
\(537\) −4.75362 −0.205134
\(538\) −15.9106 −0.685954
\(539\) 0 0
\(540\) 1.70156 0.0732236
\(541\) 13.4971 0.580285 0.290143 0.956983i \(-0.406297\pi\)
0.290143 + 0.956983i \(0.406297\pi\)
\(542\) −17.0671 −0.733095
\(543\) 25.2652 1.08423
\(544\) −14.3180 −0.613881
\(545\) 19.8905 0.852015
\(546\) 0 0
\(547\) −42.0182 −1.79657 −0.898285 0.439414i \(-0.855186\pi\)
−0.898285 + 0.439414i \(0.855186\pi\)
\(548\) 5.79063 0.247363
\(549\) 3.71308 0.158471
\(550\) −0.546295 −0.0232941
\(551\) −34.8900 −1.48636
\(552\) 18.0598 0.768677
\(553\) 0 0
\(554\) −8.68545 −0.369009
\(555\) 2.75362 0.116885
\(556\) −13.6929 −0.580707
\(557\) −27.9461 −1.18411 −0.592057 0.805896i \(-0.701684\pi\)
−0.592057 + 0.805896i \(0.701684\pi\)
\(558\) −5.27777 −0.223426
\(559\) −0.0295901 −0.00125153
\(560\) 0 0
\(561\) −2.70156 −0.114060
\(562\) −7.70500 −0.325016
\(563\) 0.266247 0.0112210 0.00561050 0.999984i \(-0.498214\pi\)
0.00561050 + 0.999984i \(0.498214\pi\)
\(564\) 17.2823 0.727717
\(565\) 6.43531 0.270736
\(566\) 0.725180 0.0304816
\(567\) 0 0
\(568\) −5.19375 −0.217925
\(569\) −35.3983 −1.48397 −0.741987 0.670414i \(-0.766116\pi\)
−0.741987 + 0.670414i \(0.766116\pi\)
\(570\) 3.61674 0.151489
\(571\) −0.440134 −0.0184191 −0.00920953 0.999958i \(-0.502932\pi\)
−0.00920953 + 0.999958i \(0.502932\pi\)
\(572\) 0.967233 0.0404421
\(573\) 17.4504 0.728999
\(574\) 0 0
\(575\) 8.93103 0.372450
\(576\) −1.70156 −0.0708984
\(577\) 25.1030 1.04505 0.522527 0.852623i \(-0.324990\pi\)
0.522527 + 0.852623i \(0.324990\pi\)
\(578\) −5.29991 −0.220447
\(579\) −22.9546 −0.953963
\(580\) −8.96723 −0.372344
\(581\) 0 0
\(582\) 9.83067 0.407494
\(583\) −1.56469 −0.0648026
\(584\) 4.58268 0.189633
\(585\) 0.568438 0.0235020
\(586\) −10.0400 −0.414750
\(587\) −3.24638 −0.133993 −0.0669963 0.997753i \(-0.521342\pi\)
−0.0669963 + 0.997753i \(0.521342\pi\)
\(588\) 0 0
\(589\) 63.9608 2.63546
\(590\) −3.47585 −0.143098
\(591\) 3.36634 0.138473
\(592\) −6.32902 −0.260121
\(593\) −10.7252 −0.440430 −0.220215 0.975451i \(-0.570676\pi\)
−0.220215 + 0.975451i \(0.570676\pi\)
\(594\) 0.546295 0.0224147
\(595\) 0 0
\(596\) −12.7740 −0.523244
\(597\) 9.54402 0.390611
\(598\) 2.77340 0.113413
\(599\) 36.3607 1.48566 0.742829 0.669481i \(-0.233483\pi\)
0.742829 + 0.669481i \(0.233483\pi\)
\(600\) 2.02214 0.0825537
\(601\) 39.7034 1.61954 0.809769 0.586749i \(-0.199593\pi\)
0.809769 + 0.586749i \(0.199593\pi\)
\(602\) 0 0
\(603\) −10.4146 −0.424117
\(604\) −5.47553 −0.222796
\(605\) 1.00000 0.0406558
\(606\) 0.451426 0.0183379
\(607\) 9.91893 0.402597 0.201299 0.979530i \(-0.435484\pi\)
0.201299 + 0.979530i \(0.435484\pi\)
\(608\) −35.0880 −1.42301
\(609\) 0 0
\(610\) 2.02844 0.0821290
\(611\) 5.77348 0.233570
\(612\) 4.59688 0.185818
\(613\) 39.2681 1.58602 0.793012 0.609206i \(-0.208512\pi\)
0.793012 + 0.609206i \(0.208512\pi\)
\(614\) −3.52438 −0.142232
\(615\) −10.9831 −0.442881
\(616\) 0 0
\(617\) 31.3462 1.26195 0.630976 0.775802i \(-0.282655\pi\)
0.630976 + 0.775802i \(0.282655\pi\)
\(618\) 5.78638 0.232762
\(619\) 12.3916 0.498061 0.249030 0.968496i \(-0.419888\pi\)
0.249030 + 0.968496i \(0.419888\pi\)
\(620\) 16.4388 0.660200
\(621\) −8.93103 −0.358390
\(622\) 16.0685 0.644287
\(623\) 0 0
\(624\) −1.30652 −0.0523027
\(625\) 1.00000 0.0400000
\(626\) 13.5231 0.540491
\(627\) −6.62049 −0.264397
\(628\) −5.70205 −0.227537
\(629\) 7.43907 0.296615
\(630\) 0 0
\(631\) 33.2233 1.32260 0.661299 0.750123i \(-0.270006\pi\)
0.661299 + 0.750123i \(0.270006\pi\)
\(632\) 5.19375 0.206596
\(633\) 7.73375 0.307389
\(634\) −8.24661 −0.327515
\(635\) 4.05206 0.160801
\(636\) 2.66241 0.105571
\(637\) 0 0
\(638\) −2.87897 −0.113980
\(639\) 2.56844 0.101606
\(640\) −11.5294 −0.455739
\(641\) 18.9179 0.747211 0.373605 0.927588i \(-0.378121\pi\)
0.373605 + 0.927588i \(0.378121\pi\)
\(642\) 7.53143 0.297242
\(643\) 5.82554 0.229737 0.114868 0.993381i \(-0.463355\pi\)
0.114868 + 0.993381i \(0.463355\pi\)
\(644\) 0 0
\(645\) 0.0520550 0.00204966
\(646\) 9.77085 0.384429
\(647\) −8.30759 −0.326605 −0.163302 0.986576i \(-0.552215\pi\)
−0.163302 + 0.986576i \(0.552215\pi\)
\(648\) −2.02214 −0.0794373
\(649\) 6.36259 0.249753
\(650\) 0.310535 0.0121802
\(651\) 0 0
\(652\) −18.6492 −0.730359
\(653\) 44.4182 1.73822 0.869109 0.494621i \(-0.164693\pi\)
0.869109 + 0.494621i \(0.164693\pi\)
\(654\) −10.8661 −0.424897
\(655\) 5.69781 0.222632
\(656\) 25.2439 0.985610
\(657\) −2.26625 −0.0884147
\(658\) 0 0
\(659\) −13.4965 −0.525750 −0.262875 0.964830i \(-0.584671\pi\)
−0.262875 + 0.964830i \(0.584671\pi\)
\(660\) −1.70156 −0.0662332
\(661\) −31.7450 −1.23474 −0.617370 0.786673i \(-0.711802\pi\)
−0.617370 + 0.786673i \(0.711802\pi\)
\(662\) 3.70532 0.144011
\(663\) 1.53567 0.0596405
\(664\) 34.7010 1.34666
\(665\) 0 0
\(666\) −1.50429 −0.0582899
\(667\) 47.0665 1.82242
\(668\) −12.9120 −0.499579
\(669\) −13.9141 −0.537951
\(670\) −5.68947 −0.219803
\(671\) −3.71308 −0.143342
\(672\) 0 0
\(673\) 5.81295 0.224073 0.112036 0.993704i \(-0.464263\pi\)
0.112036 + 0.993704i \(0.464263\pi\)
\(674\) −17.4546 −0.672326
\(675\) −1.00000 −0.0384900
\(676\) 21.5705 0.829634
\(677\) 23.3699 0.898177 0.449088 0.893487i \(-0.351749\pi\)
0.449088 + 0.893487i \(0.351749\pi\)
\(678\) −3.51558 −0.135015
\(679\) 0 0
\(680\) 5.46295 0.209494
\(681\) 0.597452 0.0228944
\(682\) 5.27777 0.202096
\(683\) 13.4705 0.515433 0.257716 0.966221i \(-0.417030\pi\)
0.257716 + 0.966221i \(0.417030\pi\)
\(684\) 11.2652 0.430735
\(685\) −3.40312 −0.130027
\(686\) 0 0
\(687\) 21.1895 0.808430
\(688\) −0.119645 −0.00456143
\(689\) 0.889427 0.0338845
\(690\) −4.87897 −0.185739
\(691\) −9.79358 −0.372565 −0.186283 0.982496i \(-0.559644\pi\)
−0.186283 + 0.982496i \(0.559644\pi\)
\(692\) −30.9844 −1.17785
\(693\) 0 0
\(694\) −7.46751 −0.283463
\(695\) 8.04724 0.305249
\(696\) 10.6567 0.403941
\(697\) −29.6715 −1.12389
\(698\) −8.83469 −0.334398
\(699\) 20.2821 0.767139
\(700\) 0 0
\(701\) −9.27000 −0.350123 −0.175062 0.984557i \(-0.556012\pi\)
−0.175062 + 0.984557i \(0.556012\pi\)
\(702\) −0.310535 −0.0117204
\(703\) 18.2303 0.687569
\(704\) 1.70156 0.0641300
\(705\) −10.1567 −0.382525
\(706\) 5.43188 0.204431
\(707\) 0 0
\(708\) −10.8263 −0.406879
\(709\) −19.9898 −0.750732 −0.375366 0.926877i \(-0.622483\pi\)
−0.375366 + 0.926877i \(0.622483\pi\)
\(710\) 1.40312 0.0526583
\(711\) −2.56844 −0.0963240
\(712\) −21.7931 −0.816732
\(713\) −86.2829 −3.23132
\(714\) 0 0
\(715\) −0.568438 −0.0212584
\(716\) −8.08857 −0.302284
\(717\) −2.23723 −0.0835510
\(718\) −6.87487 −0.256568
\(719\) 26.7174 0.996391 0.498196 0.867065i \(-0.333996\pi\)
0.498196 + 0.867065i \(0.333996\pi\)
\(720\) 2.29844 0.0856577
\(721\) 0 0
\(722\) 13.5650 0.504837
\(723\) 8.93045 0.332127
\(724\) 42.9903 1.59772
\(725\) 5.27000 0.195723
\(726\) −0.546295 −0.0202749
\(727\) −16.4955 −0.611782 −0.305891 0.952066i \(-0.598954\pi\)
−0.305891 + 0.952066i \(0.598954\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −1.23804 −0.0458219
\(731\) 0.140630 0.00520138
\(732\) 6.31804 0.233522
\(733\) 22.6838 0.837847 0.418923 0.908022i \(-0.362408\pi\)
0.418923 + 0.908022i \(0.362408\pi\)
\(734\) −20.1965 −0.745465
\(735\) 0 0
\(736\) 47.3337 1.74474
\(737\) 10.4146 0.383628
\(738\) 6.00000 0.220863
\(739\) −41.7166 −1.53457 −0.767285 0.641306i \(-0.778393\pi\)
−0.767285 + 0.641306i \(0.778393\pi\)
\(740\) 4.68545 0.172241
\(741\) 3.76334 0.138250
\(742\) 0 0
\(743\) −26.4664 −0.970959 −0.485480 0.874248i \(-0.661355\pi\)
−0.485480 + 0.874248i \(0.661355\pi\)
\(744\) −19.5360 −0.716224
\(745\) 7.50723 0.275044
\(746\) −7.32630 −0.268235
\(747\) −17.1605 −0.627870
\(748\) −4.59688 −0.168078
\(749\) 0 0
\(750\) −0.546295 −0.0199479
\(751\) −36.2802 −1.32388 −0.661942 0.749555i \(-0.730268\pi\)
−0.661942 + 0.749555i \(0.730268\pi\)
\(752\) 23.3446 0.851291
\(753\) 3.35884 0.122403
\(754\) 1.63652 0.0595985
\(755\) 3.21795 0.117113
\(756\) 0 0
\(757\) 27.3451 0.993874 0.496937 0.867786i \(-0.334458\pi\)
0.496937 + 0.867786i \(0.334458\pi\)
\(758\) −0.440134 −0.0159864
\(759\) 8.93103 0.324176
\(760\) 13.3876 0.485619
\(761\) 16.2537 0.589195 0.294597 0.955621i \(-0.404814\pi\)
0.294597 + 0.955621i \(0.404814\pi\)
\(762\) −2.21362 −0.0801909
\(763\) 0 0
\(764\) 29.6929 1.07425
\(765\) −2.70156 −0.0976752
\(766\) 12.2094 0.441143
\(767\) −3.61674 −0.130593
\(768\) 2.89531 0.104476
\(769\) 38.2698 1.38004 0.690022 0.723789i \(-0.257601\pi\)
0.690022 + 0.723789i \(0.257601\pi\)
\(770\) 0 0
\(771\) 5.80943 0.209221
\(772\) −39.0588 −1.40576
\(773\) −6.60438 −0.237543 −0.118772 0.992922i \(-0.537896\pi\)
−0.118772 + 0.992922i \(0.537896\pi\)
\(774\) −0.0284374 −0.00102216
\(775\) −9.66103 −0.347034
\(776\) 36.3888 1.30628
\(777\) 0 0
\(778\) −16.1082 −0.577507
\(779\) −72.7134 −2.60523
\(780\) 0.967233 0.0346325
\(781\) −2.56844 −0.0919060
\(782\) −13.1808 −0.471346
\(783\) −5.27000 −0.188334
\(784\) 0 0
\(785\) 3.35107 0.119605
\(786\) −3.11268 −0.111026
\(787\) 23.1916 0.826692 0.413346 0.910574i \(-0.364360\pi\)
0.413346 + 0.910574i \(0.364360\pi\)
\(788\) 5.72804 0.204053
\(789\) 23.3221 0.830287
\(790\) −1.40312 −0.0499209
\(791\) 0 0
\(792\) 2.02214 0.0718537
\(793\) 2.11066 0.0749517
\(794\) −2.43179 −0.0863010
\(795\) −1.56469 −0.0554937
\(796\) 16.2397 0.575602
\(797\) −32.5872 −1.15430 −0.577150 0.816638i \(-0.695835\pi\)
−0.577150 + 0.816638i \(0.695835\pi\)
\(798\) 0 0
\(799\) −27.4391 −0.970724
\(800\) 5.29991 0.187380
\(801\) 10.7772 0.380795
\(802\) 12.4076 0.438127
\(803\) 2.26625 0.0799741
\(804\) −17.7212 −0.624977
\(805\) 0 0
\(806\) −3.00009 −0.105674
\(807\) 29.1246 1.02523
\(808\) 1.67098 0.0587849
\(809\) 13.5303 0.475699 0.237850 0.971302i \(-0.423557\pi\)
0.237850 + 0.971302i \(0.423557\pi\)
\(810\) 0.546295 0.0191948
\(811\) −27.7241 −0.973525 −0.486763 0.873534i \(-0.661822\pi\)
−0.486763 + 0.873534i \(0.661822\pi\)
\(812\) 0 0
\(813\) 31.2416 1.09569
\(814\) 1.50429 0.0527252
\(815\) 10.9600 0.383914
\(816\) 6.20937 0.217372
\(817\) 0.344630 0.0120571
\(818\) −2.78205 −0.0972723
\(819\) 0 0
\(820\) −18.6884 −0.652627
\(821\) 40.9774 1.43012 0.715061 0.699062i \(-0.246399\pi\)
0.715061 + 0.699062i \(0.246399\pi\)
\(822\) 1.85911 0.0648439
\(823\) 3.27352 0.114108 0.0570539 0.998371i \(-0.481829\pi\)
0.0570539 + 0.998371i \(0.481829\pi\)
\(824\) 21.4187 0.746154
\(825\) 1.00000 0.0348155
\(826\) 0 0
\(827\) −36.5217 −1.26998 −0.634992 0.772519i \(-0.718997\pi\)
−0.634992 + 0.772519i \(0.718997\pi\)
\(828\) −15.1967 −0.528122
\(829\) −21.2335 −0.737469 −0.368735 0.929535i \(-0.620209\pi\)
−0.368735 + 0.929535i \(0.620209\pi\)
\(830\) −9.37469 −0.325400
\(831\) 15.8988 0.551525
\(832\) −0.967233 −0.0335328
\(833\) 0 0
\(834\) −4.39616 −0.152227
\(835\) 7.58830 0.262604
\(836\) −11.2652 −0.389614
\(837\) 9.66103 0.333934
\(838\) 1.56397 0.0540263
\(839\) −20.7571 −0.716616 −0.358308 0.933603i \(-0.616646\pi\)
−0.358308 + 0.933603i \(0.616646\pi\)
\(840\) 0 0
\(841\) −1.22709 −0.0423136
\(842\) −8.79790 −0.303196
\(843\) 14.1041 0.485771
\(844\) 13.1595 0.452967
\(845\) −12.6769 −0.436098
\(846\) 5.54857 0.190764
\(847\) 0 0
\(848\) 3.59633 0.123499
\(849\) −1.32745 −0.0455580
\(850\) −1.47585 −0.0506212
\(851\) −24.5926 −0.843025
\(852\) 4.37036 0.149726
\(853\) 31.3333 1.07283 0.536417 0.843953i \(-0.319778\pi\)
0.536417 + 0.843953i \(0.319778\pi\)
\(854\) 0 0
\(855\) −6.62049 −0.226416
\(856\) 27.8780 0.952852
\(857\) −13.5411 −0.462554 −0.231277 0.972888i \(-0.574290\pi\)
−0.231277 + 0.972888i \(0.574290\pi\)
\(858\) 0.310535 0.0106015
\(859\) 19.5644 0.667530 0.333765 0.942656i \(-0.391681\pi\)
0.333765 + 0.942656i \(0.391681\pi\)
\(860\) 0.0885748 0.00302038
\(861\) 0 0
\(862\) −2.55554 −0.0870419
\(863\) −8.91434 −0.303448 −0.151724 0.988423i \(-0.548482\pi\)
−0.151724 + 0.988423i \(0.548482\pi\)
\(864\) −5.29991 −0.180307
\(865\) 18.2094 0.619137
\(866\) 14.9869 0.509275
\(867\) 9.70156 0.329482
\(868\) 0 0
\(869\) 2.56844 0.0871283
\(870\) −2.87897 −0.0976063
\(871\) −5.92008 −0.200594
\(872\) −40.2214 −1.36207
\(873\) −17.9952 −0.609045
\(874\) −32.3012 −1.09260
\(875\) 0 0
\(876\) −3.85616 −0.130288
\(877\) 6.17626 0.208557 0.104279 0.994548i \(-0.466747\pi\)
0.104279 + 0.994548i \(0.466747\pi\)
\(878\) −3.13263 −0.105721
\(879\) 18.3784 0.619889
\(880\) −2.29844 −0.0774803
\(881\) −19.3511 −0.651954 −0.325977 0.945378i \(-0.605693\pi\)
−0.325977 + 0.945378i \(0.605693\pi\)
\(882\) 0 0
\(883\) 28.7925 0.968945 0.484473 0.874806i \(-0.339012\pi\)
0.484473 + 0.874806i \(0.339012\pi\)
\(884\) 2.61304 0.0878861
\(885\) 6.36259 0.213876
\(886\) −5.23643 −0.175921
\(887\) 43.4970 1.46049 0.730243 0.683187i \(-0.239407\pi\)
0.730243 + 0.683187i \(0.239407\pi\)
\(888\) −5.56821 −0.186857
\(889\) 0 0
\(890\) 5.88755 0.197351
\(891\) −1.00000 −0.0335013
\(892\) −23.6757 −0.792722
\(893\) −67.2426 −2.25019
\(894\) −4.10116 −0.137163
\(895\) 4.75362 0.158896
\(896\) 0 0
\(897\) −5.07674 −0.169507
\(898\) 16.9058 0.564154
\(899\) −50.9136 −1.69806
\(900\) −1.70156 −0.0567187
\(901\) −4.22709 −0.140825
\(902\) −6.00000 −0.199778
\(903\) 0 0
\(904\) −13.0131 −0.432810
\(905\) −25.2652 −0.839843
\(906\) −1.75795 −0.0584039
\(907\) −9.20210 −0.305551 −0.152775 0.988261i \(-0.548821\pi\)
−0.152775 + 0.988261i \(0.548821\pi\)
\(908\) 1.01660 0.0337371
\(909\) −0.826342 −0.0274080
\(910\) 0 0
\(911\) 7.98331 0.264499 0.132249 0.991216i \(-0.457780\pi\)
0.132249 + 0.991216i \(0.457780\pi\)
\(912\) 15.2168 0.503878
\(913\) 17.1605 0.567929
\(914\) −23.0671 −0.762992
\(915\) −3.71308 −0.122751
\(916\) 36.0553 1.19130
\(917\) 0 0
\(918\) 1.47585 0.0487103
\(919\) −36.4176 −1.20131 −0.600653 0.799510i \(-0.705093\pi\)
−0.600653 + 0.799510i \(0.705093\pi\)
\(920\) −18.0598 −0.595415
\(921\) 6.45143 0.212582
\(922\) −5.39447 −0.177657
\(923\) 1.46000 0.0480565
\(924\) 0 0
\(925\) −2.75362 −0.0905384
\(926\) 1.39857 0.0459597
\(927\) −10.5921 −0.347889
\(928\) 27.9305 0.916865
\(929\) −1.06660 −0.0349940 −0.0174970 0.999847i \(-0.505570\pi\)
−0.0174970 + 0.999847i \(0.505570\pi\)
\(930\) 5.27777 0.173065
\(931\) 0 0
\(932\) 34.5112 1.13045
\(933\) −29.4136 −0.962957
\(934\) 10.7579 0.352008
\(935\) 2.70156 0.0883505
\(936\) −1.14946 −0.0375714
\(937\) 4.74611 0.155049 0.0775243 0.996990i \(-0.475298\pi\)
0.0775243 + 0.996990i \(0.475298\pi\)
\(938\) 0 0
\(939\) −24.7542 −0.807823
\(940\) −17.2823 −0.563687
\(941\) −33.1988 −1.08225 −0.541125 0.840942i \(-0.682001\pi\)
−0.541125 + 0.840942i \(0.682001\pi\)
\(942\) −1.83067 −0.0596465
\(943\) 98.0902 3.19426
\(944\) −14.6240 −0.475971
\(945\) 0 0
\(946\) 0.0284374 0.000924579 0
\(947\) 6.30361 0.204840 0.102420 0.994741i \(-0.467341\pi\)
0.102420 + 0.994741i \(0.467341\pi\)
\(948\) −4.37036 −0.141943
\(949\) −1.28822 −0.0418175
\(950\) −3.61674 −0.117343
\(951\) 15.0955 0.489506
\(952\) 0 0
\(953\) −39.4705 −1.27857 −0.639287 0.768968i \(-0.720770\pi\)
−0.639287 + 0.768968i \(0.720770\pi\)
\(954\) 0.854779 0.0276745
\(955\) −17.4504 −0.564680
\(956\) −3.80679 −0.123120
\(957\) 5.27000 0.170355
\(958\) −8.39501 −0.271230
\(959\) 0 0
\(960\) 1.70156 0.0549177
\(961\) 62.3355 2.01082
\(962\) −0.855094 −0.0275693
\(963\) −13.7864 −0.444260
\(964\) 15.1957 0.489421
\(965\) 22.9546 0.738936
\(966\) 0 0
\(967\) −20.7192 −0.666285 −0.333142 0.942877i \(-0.608109\pi\)
−0.333142 + 0.942877i \(0.608109\pi\)
\(968\) −2.02214 −0.0649942
\(969\) −17.8857 −0.574571
\(970\) −9.83067 −0.315644
\(971\) −0.281521 −0.00903444 −0.00451722 0.999990i \(-0.501438\pi\)
−0.00451722 + 0.999990i \(0.501438\pi\)
\(972\) 1.70156 0.0545776
\(973\) 0 0
\(974\) −0.835001 −0.0267551
\(975\) −0.568438 −0.0182046
\(976\) 8.53429 0.273176
\(977\) 53.1267 1.69967 0.849836 0.527047i \(-0.176701\pi\)
0.849836 + 0.527047i \(0.176701\pi\)
\(978\) −5.98741 −0.191456
\(979\) −10.7772 −0.344442
\(980\) 0 0
\(981\) 19.8905 0.635055
\(982\) −22.2538 −0.710147
\(983\) 47.2072 1.50567 0.752837 0.658207i \(-0.228685\pi\)
0.752837 + 0.658207i \(0.228685\pi\)
\(984\) 22.2094 0.708009
\(985\) −3.36634 −0.107261
\(986\) −7.77773 −0.247693
\(987\) 0 0
\(988\) 6.40356 0.203724
\(989\) −0.464905 −0.0147831
\(990\) −0.546295 −0.0173624
\(991\) −30.4901 −0.968549 −0.484274 0.874916i \(-0.660916\pi\)
−0.484274 + 0.874916i \(0.660916\pi\)
\(992\) −51.2026 −1.62568
\(993\) −6.78263 −0.215240
\(994\) 0 0
\(995\) −9.54402 −0.302566
\(996\) −29.1996 −0.925226
\(997\) −6.62107 −0.209691 −0.104846 0.994489i \(-0.533435\pi\)
−0.104846 + 0.994489i \(0.533435\pi\)
\(998\) −11.1068 −0.351581
\(999\) 2.75362 0.0871206
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 8085.2.a.bn.1.3 4
7.6 odd 2 1155.2.a.u.1.3 4
21.20 even 2 3465.2.a.bl.1.2 4
35.34 odd 2 5775.2.a.bz.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1155.2.a.u.1.3 4 7.6 odd 2
3465.2.a.bl.1.2 4 21.20 even 2
5775.2.a.bz.1.2 4 35.34 odd 2
8085.2.a.bn.1.3 4 1.1 even 1 trivial