Properties

Label 8085.2.a.ba.1.1
Level $8085$
Weight $2$
Character 8085.1
Self dual yes
Analytic conductor $64.559$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 8085.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} +1.00000 q^{5} -2.41421 q^{6} -4.41421 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} +1.00000 q^{5} -2.41421 q^{6} -4.41421 q^{8} +1.00000 q^{9} -2.41421 q^{10} -1.00000 q^{11} +3.82843 q^{12} +5.65685 q^{13} +1.00000 q^{15} +3.00000 q^{16} +1.17157 q^{17} -2.41421 q^{18} +6.82843 q^{19} +3.82843 q^{20} +2.41421 q^{22} -4.00000 q^{23} -4.41421 q^{24} +1.00000 q^{25} -13.6569 q^{26} +1.00000 q^{27} -4.82843 q^{29} -2.41421 q^{30} +1.58579 q^{32} -1.00000 q^{33} -2.82843 q^{34} +3.82843 q^{36} +11.6569 q^{37} -16.4853 q^{38} +5.65685 q^{39} -4.41421 q^{40} -4.82843 q^{41} -8.82843 q^{43} -3.82843 q^{44} +1.00000 q^{45} +9.65685 q^{46} +4.00000 q^{47} +3.00000 q^{48} -2.41421 q^{50} +1.17157 q^{51} +21.6569 q^{52} +9.31371 q^{53} -2.41421 q^{54} -1.00000 q^{55} +6.82843 q^{57} +11.6569 q^{58} +4.00000 q^{59} +3.82843 q^{60} +11.6569 q^{61} -9.82843 q^{64} +5.65685 q^{65} +2.41421 q^{66} -5.65685 q^{67} +4.48528 q^{68} -4.00000 q^{69} +2.34315 q^{71} -4.41421 q^{72} -11.3137 q^{73} -28.1421 q^{74} +1.00000 q^{75} +26.1421 q^{76} -13.6569 q^{78} +8.48528 q^{79} +3.00000 q^{80} +1.00000 q^{81} +11.6569 q^{82} +10.0000 q^{83} +1.17157 q^{85} +21.3137 q^{86} -4.82843 q^{87} +4.41421 q^{88} -3.65685 q^{89} -2.41421 q^{90} -15.3137 q^{92} -9.65685 q^{94} +6.82843 q^{95} +1.58579 q^{96} -11.6569 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 6q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 6q^{8} + 2q^{9} - 2q^{10} - 2q^{11} + 2q^{12} + 2q^{15} + 6q^{16} + 8q^{17} - 2q^{18} + 8q^{19} + 2q^{20} + 2q^{22} - 8q^{23} - 6q^{24} + 2q^{25} - 16q^{26} + 2q^{27} - 4q^{29} - 2q^{30} + 6q^{32} - 2q^{33} + 2q^{36} + 12q^{37} - 16q^{38} - 6q^{40} - 4q^{41} - 12q^{43} - 2q^{44} + 2q^{45} + 8q^{46} + 8q^{47} + 6q^{48} - 2q^{50} + 8q^{51} + 32q^{52} - 4q^{53} - 2q^{54} - 2q^{55} + 8q^{57} + 12q^{58} + 8q^{59} + 2q^{60} + 12q^{61} - 14q^{64} + 2q^{66} - 8q^{68} - 8q^{69} + 16q^{71} - 6q^{72} - 28q^{74} + 2q^{75} + 24q^{76} - 16q^{78} + 6q^{80} + 2q^{81} + 12q^{82} + 20q^{83} + 8q^{85} + 20q^{86} - 4q^{87} + 6q^{88} + 4q^{89} - 2q^{90} - 8q^{92} - 8q^{94} + 8q^{95} + 6q^{96} - 12q^{97} - 2q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.41421 −1.70711 −0.853553 0.521005i \(-0.825557\pi\)
−0.853553 + 0.521005i \(0.825557\pi\)
\(3\) 1.00000 0.577350
\(4\) 3.82843 1.91421
\(5\) 1.00000 0.447214
\(6\) −2.41421 −0.985599
\(7\) 0 0
\(8\) −4.41421 −1.56066
\(9\) 1.00000 0.333333
\(10\) −2.41421 −0.763441
\(11\) −1.00000 −0.301511
\(12\) 3.82843 1.10517
\(13\) 5.65685 1.56893 0.784465 0.620174i \(-0.212938\pi\)
0.784465 + 0.620174i \(0.212938\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 3.00000 0.750000
\(17\) 1.17157 0.284148 0.142074 0.989856i \(-0.454623\pi\)
0.142074 + 0.989856i \(0.454623\pi\)
\(18\) −2.41421 −0.569036
\(19\) 6.82843 1.56655 0.783274 0.621676i \(-0.213548\pi\)
0.783274 + 0.621676i \(0.213548\pi\)
\(20\) 3.82843 0.856062
\(21\) 0 0
\(22\) 2.41421 0.514712
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −4.41421 −0.901048
\(25\) 1.00000 0.200000
\(26\) −13.6569 −2.67833
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −4.82843 −0.896616 −0.448308 0.893879i \(-0.647973\pi\)
−0.448308 + 0.893879i \(0.647973\pi\)
\(30\) −2.41421 −0.440773
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.58579 0.280330
\(33\) −1.00000 −0.174078
\(34\) −2.82843 −0.485071
\(35\) 0 0
\(36\) 3.82843 0.638071
\(37\) 11.6569 1.91638 0.958188 0.286141i \(-0.0923726\pi\)
0.958188 + 0.286141i \(0.0923726\pi\)
\(38\) −16.4853 −2.67427
\(39\) 5.65685 0.905822
\(40\) −4.41421 −0.697948
\(41\) −4.82843 −0.754074 −0.377037 0.926198i \(-0.623057\pi\)
−0.377037 + 0.926198i \(0.623057\pi\)
\(42\) 0 0
\(43\) −8.82843 −1.34632 −0.673161 0.739496i \(-0.735064\pi\)
−0.673161 + 0.739496i \(0.735064\pi\)
\(44\) −3.82843 −0.577157
\(45\) 1.00000 0.149071
\(46\) 9.65685 1.42383
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 3.00000 0.433013
\(49\) 0 0
\(50\) −2.41421 −0.341421
\(51\) 1.17157 0.164053
\(52\) 21.6569 3.00327
\(53\) 9.31371 1.27934 0.639668 0.768651i \(-0.279072\pi\)
0.639668 + 0.768651i \(0.279072\pi\)
\(54\) −2.41421 −0.328533
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 6.82843 0.904447
\(58\) 11.6569 1.53062
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 3.82843 0.494248
\(61\) 11.6569 1.49251 0.746254 0.665662i \(-0.231851\pi\)
0.746254 + 0.665662i \(0.231851\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −9.82843 −1.22855
\(65\) 5.65685 0.701646
\(66\) 2.41421 0.297169
\(67\) −5.65685 −0.691095 −0.345547 0.938401i \(-0.612307\pi\)
−0.345547 + 0.938401i \(0.612307\pi\)
\(68\) 4.48528 0.543920
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 2.34315 0.278080 0.139040 0.990287i \(-0.455598\pi\)
0.139040 + 0.990287i \(0.455598\pi\)
\(72\) −4.41421 −0.520220
\(73\) −11.3137 −1.32417 −0.662085 0.749429i \(-0.730328\pi\)
−0.662085 + 0.749429i \(0.730328\pi\)
\(74\) −28.1421 −3.27146
\(75\) 1.00000 0.115470
\(76\) 26.1421 2.99871
\(77\) 0 0
\(78\) −13.6569 −1.54633
\(79\) 8.48528 0.954669 0.477334 0.878722i \(-0.341603\pi\)
0.477334 + 0.878722i \(0.341603\pi\)
\(80\) 3.00000 0.335410
\(81\) 1.00000 0.111111
\(82\) 11.6569 1.28728
\(83\) 10.0000 1.09764 0.548821 0.835940i \(-0.315077\pi\)
0.548821 + 0.835940i \(0.315077\pi\)
\(84\) 0 0
\(85\) 1.17157 0.127075
\(86\) 21.3137 2.29832
\(87\) −4.82843 −0.517662
\(88\) 4.41421 0.470557
\(89\) −3.65685 −0.387626 −0.193813 0.981039i \(-0.562085\pi\)
−0.193813 + 0.981039i \(0.562085\pi\)
\(90\) −2.41421 −0.254480
\(91\) 0 0
\(92\) −15.3137 −1.59656
\(93\) 0 0
\(94\) −9.65685 −0.996028
\(95\) 6.82843 0.700582
\(96\) 1.58579 0.161849
\(97\) −11.6569 −1.18357 −0.591787 0.806094i \(-0.701577\pi\)
−0.591787 + 0.806094i \(0.701577\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 3.82843 0.382843
\(101\) 0.828427 0.0824316 0.0412158 0.999150i \(-0.486877\pi\)
0.0412158 + 0.999150i \(0.486877\pi\)
\(102\) −2.82843 −0.280056
\(103\) 3.31371 0.326509 0.163255 0.986584i \(-0.447801\pi\)
0.163255 + 0.986584i \(0.447801\pi\)
\(104\) −24.9706 −2.44857
\(105\) 0 0
\(106\) −22.4853 −2.18396
\(107\) 17.3137 1.67378 0.836890 0.547372i \(-0.184372\pi\)
0.836890 + 0.547372i \(0.184372\pi\)
\(108\) 3.82843 0.368391
\(109\) 17.3137 1.65835 0.829176 0.558987i \(-0.188810\pi\)
0.829176 + 0.558987i \(0.188810\pi\)
\(110\) 2.41421 0.230186
\(111\) 11.6569 1.10642
\(112\) 0 0
\(113\) −18.9706 −1.78460 −0.892300 0.451442i \(-0.850910\pi\)
−0.892300 + 0.451442i \(0.850910\pi\)
\(114\) −16.4853 −1.54399
\(115\) −4.00000 −0.373002
\(116\) −18.4853 −1.71632
\(117\) 5.65685 0.522976
\(118\) −9.65685 −0.888985
\(119\) 0 0
\(120\) −4.41421 −0.402961
\(121\) 1.00000 0.0909091
\(122\) −28.1421 −2.54787
\(123\) −4.82843 −0.435365
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −14.4853 −1.28536 −0.642680 0.766134i \(-0.722178\pi\)
−0.642680 + 0.766134i \(0.722178\pi\)
\(128\) 20.5563 1.81694
\(129\) −8.82843 −0.777300
\(130\) −13.6569 −1.19779
\(131\) −3.31371 −0.289520 −0.144760 0.989467i \(-0.546241\pi\)
−0.144760 + 0.989467i \(0.546241\pi\)
\(132\) −3.82843 −0.333222
\(133\) 0 0
\(134\) 13.6569 1.17977
\(135\) 1.00000 0.0860663
\(136\) −5.17157 −0.443459
\(137\) −13.3137 −1.13747 −0.568733 0.822522i \(-0.692566\pi\)
−0.568733 + 0.822522i \(0.692566\pi\)
\(138\) 9.65685 0.822046
\(139\) −0.485281 −0.0411610 −0.0205805 0.999788i \(-0.506551\pi\)
−0.0205805 + 0.999788i \(0.506551\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) −5.65685 −0.474713
\(143\) −5.65685 −0.473050
\(144\) 3.00000 0.250000
\(145\) −4.82843 −0.400979
\(146\) 27.3137 2.26050
\(147\) 0 0
\(148\) 44.6274 3.66835
\(149\) 1.51472 0.124091 0.0620453 0.998073i \(-0.480238\pi\)
0.0620453 + 0.998073i \(0.480238\pi\)
\(150\) −2.41421 −0.197120
\(151\) 16.4853 1.34155 0.670777 0.741659i \(-0.265961\pi\)
0.670777 + 0.741659i \(0.265961\pi\)
\(152\) −30.1421 −2.44485
\(153\) 1.17157 0.0947161
\(154\) 0 0
\(155\) 0 0
\(156\) 21.6569 1.73394
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) −20.4853 −1.62972
\(159\) 9.31371 0.738625
\(160\) 1.58579 0.125367
\(161\) 0 0
\(162\) −2.41421 −0.189679
\(163\) −7.31371 −0.572854 −0.286427 0.958102i \(-0.592468\pi\)
−0.286427 + 0.958102i \(0.592468\pi\)
\(164\) −18.4853 −1.44346
\(165\) −1.00000 −0.0778499
\(166\) −24.1421 −1.87379
\(167\) 13.3137 1.03025 0.515123 0.857116i \(-0.327746\pi\)
0.515123 + 0.857116i \(0.327746\pi\)
\(168\) 0 0
\(169\) 19.0000 1.46154
\(170\) −2.82843 −0.216930
\(171\) 6.82843 0.522183
\(172\) −33.7990 −2.57715
\(173\) −2.82843 −0.215041 −0.107521 0.994203i \(-0.534291\pi\)
−0.107521 + 0.994203i \(0.534291\pi\)
\(174\) 11.6569 0.883704
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 4.00000 0.300658
\(178\) 8.82843 0.661719
\(179\) −17.6569 −1.31974 −0.659868 0.751382i \(-0.729388\pi\)
−0.659868 + 0.751382i \(0.729388\pi\)
\(180\) 3.82843 0.285354
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 11.6569 0.861699
\(184\) 17.6569 1.30168
\(185\) 11.6569 0.857029
\(186\) 0 0
\(187\) −1.17157 −0.0856739
\(188\) 15.3137 1.11687
\(189\) 0 0
\(190\) −16.4853 −1.19597
\(191\) 5.65685 0.409316 0.204658 0.978834i \(-0.434392\pi\)
0.204658 + 0.978834i \(0.434392\pi\)
\(192\) −9.82843 −0.709306
\(193\) 13.6569 0.983042 0.491521 0.870866i \(-0.336441\pi\)
0.491521 + 0.870866i \(0.336441\pi\)
\(194\) 28.1421 2.02049
\(195\) 5.65685 0.405096
\(196\) 0 0
\(197\) −8.48528 −0.604551 −0.302276 0.953221i \(-0.597746\pi\)
−0.302276 + 0.953221i \(0.597746\pi\)
\(198\) 2.41421 0.171571
\(199\) 21.6569 1.53521 0.767607 0.640921i \(-0.221447\pi\)
0.767607 + 0.640921i \(0.221447\pi\)
\(200\) −4.41421 −0.312132
\(201\) −5.65685 −0.399004
\(202\) −2.00000 −0.140720
\(203\) 0 0
\(204\) 4.48528 0.314033
\(205\) −4.82843 −0.337232
\(206\) −8.00000 −0.557386
\(207\) −4.00000 −0.278019
\(208\) 16.9706 1.17670
\(209\) −6.82843 −0.472332
\(210\) 0 0
\(211\) 1.17157 0.0806544 0.0403272 0.999187i \(-0.487160\pi\)
0.0403272 + 0.999187i \(0.487160\pi\)
\(212\) 35.6569 2.44892
\(213\) 2.34315 0.160550
\(214\) −41.7990 −2.85732
\(215\) −8.82843 −0.602094
\(216\) −4.41421 −0.300349
\(217\) 0 0
\(218\) −41.7990 −2.83098
\(219\) −11.3137 −0.764510
\(220\) −3.82843 −0.258113
\(221\) 6.62742 0.445808
\(222\) −28.1421 −1.88878
\(223\) 6.34315 0.424768 0.212384 0.977186i \(-0.431877\pi\)
0.212384 + 0.977186i \(0.431877\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 45.7990 3.04650
\(227\) 14.0000 0.929213 0.464606 0.885517i \(-0.346196\pi\)
0.464606 + 0.885517i \(0.346196\pi\)
\(228\) 26.1421 1.73131
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 9.65685 0.636754
\(231\) 0 0
\(232\) 21.3137 1.39931
\(233\) −18.8284 −1.23349 −0.616746 0.787163i \(-0.711549\pi\)
−0.616746 + 0.787163i \(0.711549\pi\)
\(234\) −13.6569 −0.892776
\(235\) 4.00000 0.260931
\(236\) 15.3137 0.996838
\(237\) 8.48528 0.551178
\(238\) 0 0
\(239\) 17.6569 1.14213 0.571063 0.820906i \(-0.306531\pi\)
0.571063 + 0.820906i \(0.306531\pi\)
\(240\) 3.00000 0.193649
\(241\) 12.3431 0.795092 0.397546 0.917582i \(-0.369862\pi\)
0.397546 + 0.917582i \(0.369862\pi\)
\(242\) −2.41421 −0.155192
\(243\) 1.00000 0.0641500
\(244\) 44.6274 2.85698
\(245\) 0 0
\(246\) 11.6569 0.743214
\(247\) 38.6274 2.45780
\(248\) 0 0
\(249\) 10.0000 0.633724
\(250\) −2.41421 −0.152688
\(251\) −20.9706 −1.32365 −0.661825 0.749658i \(-0.730218\pi\)
−0.661825 + 0.749658i \(0.730218\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 34.9706 2.19425
\(255\) 1.17157 0.0733667
\(256\) −29.9706 −1.87316
\(257\) 16.3431 1.01946 0.509729 0.860335i \(-0.329746\pi\)
0.509729 + 0.860335i \(0.329746\pi\)
\(258\) 21.3137 1.32693
\(259\) 0 0
\(260\) 21.6569 1.34310
\(261\) −4.82843 −0.298872
\(262\) 8.00000 0.494242
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 4.41421 0.271676
\(265\) 9.31371 0.572137
\(266\) 0 0
\(267\) −3.65685 −0.223796
\(268\) −21.6569 −1.32290
\(269\) −20.6274 −1.25768 −0.628838 0.777536i \(-0.716469\pi\)
−0.628838 + 0.777536i \(0.716469\pi\)
\(270\) −2.41421 −0.146924
\(271\) 11.7990 0.716738 0.358369 0.933580i \(-0.383333\pi\)
0.358369 + 0.933580i \(0.383333\pi\)
\(272\) 3.51472 0.213111
\(273\) 0 0
\(274\) 32.1421 1.94178
\(275\) −1.00000 −0.0603023
\(276\) −15.3137 −0.921777
\(277\) −2.34315 −0.140786 −0.0703930 0.997519i \(-0.522425\pi\)
−0.0703930 + 0.997519i \(0.522425\pi\)
\(278\) 1.17157 0.0702663
\(279\) 0 0
\(280\) 0 0
\(281\) −11.1716 −0.666440 −0.333220 0.942849i \(-0.608135\pi\)
−0.333220 + 0.942849i \(0.608135\pi\)
\(282\) −9.65685 −0.575057
\(283\) 8.82843 0.524796 0.262398 0.964960i \(-0.415487\pi\)
0.262398 + 0.964960i \(0.415487\pi\)
\(284\) 8.97056 0.532305
\(285\) 6.82843 0.404481
\(286\) 13.6569 0.807547
\(287\) 0 0
\(288\) 1.58579 0.0934434
\(289\) −15.6274 −0.919260
\(290\) 11.6569 0.684514
\(291\) −11.6569 −0.683337
\(292\) −43.3137 −2.53474
\(293\) 6.82843 0.398921 0.199460 0.979906i \(-0.436081\pi\)
0.199460 + 0.979906i \(0.436081\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) −51.4558 −2.99081
\(297\) −1.00000 −0.0580259
\(298\) −3.65685 −0.211836
\(299\) −22.6274 −1.30858
\(300\) 3.82843 0.221034
\(301\) 0 0
\(302\) −39.7990 −2.29017
\(303\) 0.828427 0.0475919
\(304\) 20.4853 1.17491
\(305\) 11.6569 0.667470
\(306\) −2.82843 −0.161690
\(307\) 3.17157 0.181011 0.0905056 0.995896i \(-0.471152\pi\)
0.0905056 + 0.995896i \(0.471152\pi\)
\(308\) 0 0
\(309\) 3.31371 0.188510
\(310\) 0 0
\(311\) 3.31371 0.187903 0.0939516 0.995577i \(-0.470050\pi\)
0.0939516 + 0.995577i \(0.470050\pi\)
\(312\) −24.9706 −1.41368
\(313\) −15.6569 −0.884978 −0.442489 0.896774i \(-0.645904\pi\)
−0.442489 + 0.896774i \(0.645904\pi\)
\(314\) 43.4558 2.45236
\(315\) 0 0
\(316\) 32.4853 1.82744
\(317\) −26.2843 −1.47627 −0.738136 0.674652i \(-0.764294\pi\)
−0.738136 + 0.674652i \(0.764294\pi\)
\(318\) −22.4853 −1.26091
\(319\) 4.82843 0.270340
\(320\) −9.82843 −0.549426
\(321\) 17.3137 0.966357
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) 3.82843 0.212690
\(325\) 5.65685 0.313786
\(326\) 17.6569 0.977923
\(327\) 17.3137 0.957450
\(328\) 21.3137 1.17685
\(329\) 0 0
\(330\) 2.41421 0.132898
\(331\) 6.34315 0.348651 0.174325 0.984688i \(-0.444226\pi\)
0.174325 + 0.984688i \(0.444226\pi\)
\(332\) 38.2843 2.10112
\(333\) 11.6569 0.638792
\(334\) −32.1421 −1.75874
\(335\) −5.65685 −0.309067
\(336\) 0 0
\(337\) 3.31371 0.180509 0.0902546 0.995919i \(-0.471232\pi\)
0.0902546 + 0.995919i \(0.471232\pi\)
\(338\) −45.8701 −2.49500
\(339\) −18.9706 −1.03034
\(340\) 4.48528 0.243249
\(341\) 0 0
\(342\) −16.4853 −0.891422
\(343\) 0 0
\(344\) 38.9706 2.10115
\(345\) −4.00000 −0.215353
\(346\) 6.82843 0.367099
\(347\) −29.3137 −1.57364 −0.786821 0.617181i \(-0.788275\pi\)
−0.786821 + 0.617181i \(0.788275\pi\)
\(348\) −18.4853 −0.990915
\(349\) −10.9706 −0.587241 −0.293620 0.955922i \(-0.594860\pi\)
−0.293620 + 0.955922i \(0.594860\pi\)
\(350\) 0 0
\(351\) 5.65685 0.301941
\(352\) −1.58579 −0.0845227
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) −9.65685 −0.513256
\(355\) 2.34315 0.124361
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) 42.6274 2.25293
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) −4.41421 −0.232649
\(361\) 27.6274 1.45407
\(362\) −33.7990 −1.77644
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) −11.3137 −0.592187
\(366\) −28.1421 −1.47101
\(367\) −9.65685 −0.504084 −0.252042 0.967716i \(-0.581102\pi\)
−0.252042 + 0.967716i \(0.581102\pi\)
\(368\) −12.0000 −0.625543
\(369\) −4.82843 −0.251358
\(370\) −28.1421 −1.46304
\(371\) 0 0
\(372\) 0 0
\(373\) 10.6274 0.550267 0.275133 0.961406i \(-0.411278\pi\)
0.275133 + 0.961406i \(0.411278\pi\)
\(374\) 2.82843 0.146254
\(375\) 1.00000 0.0516398
\(376\) −17.6569 −0.910583
\(377\) −27.3137 −1.40673
\(378\) 0 0
\(379\) −23.3137 −1.19754 −0.598772 0.800919i \(-0.704345\pi\)
−0.598772 + 0.800919i \(0.704345\pi\)
\(380\) 26.1421 1.34106
\(381\) −14.4853 −0.742103
\(382\) −13.6569 −0.698745
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) 20.5563 1.04901
\(385\) 0 0
\(386\) −32.9706 −1.67816
\(387\) −8.82843 −0.448774
\(388\) −44.6274 −2.26561
\(389\) −23.6569 −1.19945 −0.599725 0.800206i \(-0.704723\pi\)
−0.599725 + 0.800206i \(0.704723\pi\)
\(390\) −13.6569 −0.691542
\(391\) −4.68629 −0.236996
\(392\) 0 0
\(393\) −3.31371 −0.167154
\(394\) 20.4853 1.03203
\(395\) 8.48528 0.426941
\(396\) −3.82843 −0.192386
\(397\) 14.9706 0.751351 0.375676 0.926751i \(-0.377411\pi\)
0.375676 + 0.926751i \(0.377411\pi\)
\(398\) −52.2843 −2.62077
\(399\) 0 0
\(400\) 3.00000 0.150000
\(401\) −6.68629 −0.333897 −0.166949 0.985966i \(-0.553391\pi\)
−0.166949 + 0.985966i \(0.553391\pi\)
\(402\) 13.6569 0.681142
\(403\) 0 0
\(404\) 3.17157 0.157792
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −11.6569 −0.577809
\(408\) −5.17157 −0.256031
\(409\) 19.6569 0.971969 0.485984 0.873967i \(-0.338461\pi\)
0.485984 + 0.873967i \(0.338461\pi\)
\(410\) 11.6569 0.575691
\(411\) −13.3137 −0.656717
\(412\) 12.6863 0.625009
\(413\) 0 0
\(414\) 9.65685 0.474608
\(415\) 10.0000 0.490881
\(416\) 8.97056 0.439818
\(417\) −0.485281 −0.0237643
\(418\) 16.4853 0.806321
\(419\) 36.9706 1.80613 0.903065 0.429504i \(-0.141311\pi\)
0.903065 + 0.429504i \(0.141311\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) −2.82843 −0.137686
\(423\) 4.00000 0.194487
\(424\) −41.1127 −1.99661
\(425\) 1.17157 0.0568296
\(426\) −5.65685 −0.274075
\(427\) 0 0
\(428\) 66.2843 3.20397
\(429\) −5.65685 −0.273115
\(430\) 21.3137 1.02784
\(431\) 21.6569 1.04317 0.521587 0.853198i \(-0.325340\pi\)
0.521587 + 0.853198i \(0.325340\pi\)
\(432\) 3.00000 0.144338
\(433\) 15.6569 0.752420 0.376210 0.926534i \(-0.377227\pi\)
0.376210 + 0.926534i \(0.377227\pi\)
\(434\) 0 0
\(435\) −4.82843 −0.231505
\(436\) 66.2843 3.17444
\(437\) −27.3137 −1.30659
\(438\) 27.3137 1.30510
\(439\) 20.4853 0.977709 0.488855 0.872365i \(-0.337415\pi\)
0.488855 + 0.872365i \(0.337415\pi\)
\(440\) 4.41421 0.210439
\(441\) 0 0
\(442\) −16.0000 −0.761042
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 44.6274 2.11792
\(445\) −3.65685 −0.173352
\(446\) −15.3137 −0.725125
\(447\) 1.51472 0.0716437
\(448\) 0 0
\(449\) 30.9706 1.46159 0.730796 0.682596i \(-0.239149\pi\)
0.730796 + 0.682596i \(0.239149\pi\)
\(450\) −2.41421 −0.113807
\(451\) 4.82843 0.227362
\(452\) −72.6274 −3.41611
\(453\) 16.4853 0.774546
\(454\) −33.7990 −1.58627
\(455\) 0 0
\(456\) −30.1421 −1.41153
\(457\) −23.3137 −1.09057 −0.545285 0.838251i \(-0.683578\pi\)
−0.545285 + 0.838251i \(0.683578\pi\)
\(458\) −4.82843 −0.225618
\(459\) 1.17157 0.0546843
\(460\) −15.3137 −0.714005
\(461\) −0.142136 −0.00661992 −0.00330996 0.999995i \(-0.501054\pi\)
−0.00330996 + 0.999995i \(0.501054\pi\)
\(462\) 0 0
\(463\) 4.97056 0.231002 0.115501 0.993307i \(-0.463153\pi\)
0.115501 + 0.993307i \(0.463153\pi\)
\(464\) −14.4853 −0.672462
\(465\) 0 0
\(466\) 45.4558 2.10570
\(467\) 22.6274 1.04707 0.523536 0.852004i \(-0.324613\pi\)
0.523536 + 0.852004i \(0.324613\pi\)
\(468\) 21.6569 1.00109
\(469\) 0 0
\(470\) −9.65685 −0.445437
\(471\) −18.0000 −0.829396
\(472\) −17.6569 −0.812723
\(473\) 8.82843 0.405932
\(474\) −20.4853 −0.940920
\(475\) 6.82843 0.313310
\(476\) 0 0
\(477\) 9.31371 0.426445
\(478\) −42.6274 −1.94973
\(479\) −36.9706 −1.68923 −0.844614 0.535376i \(-0.820170\pi\)
−0.844614 + 0.535376i \(0.820170\pi\)
\(480\) 1.58579 0.0723809
\(481\) 65.9411 3.00666
\(482\) −29.7990 −1.35731
\(483\) 0 0
\(484\) 3.82843 0.174019
\(485\) −11.6569 −0.529310
\(486\) −2.41421 −0.109511
\(487\) −12.9706 −0.587752 −0.293876 0.955844i \(-0.594945\pi\)
−0.293876 + 0.955844i \(0.594945\pi\)
\(488\) −51.4558 −2.32930
\(489\) −7.31371 −0.330737
\(490\) 0 0
\(491\) 14.3431 0.647297 0.323649 0.946177i \(-0.395090\pi\)
0.323649 + 0.946177i \(0.395090\pi\)
\(492\) −18.4853 −0.833381
\(493\) −5.65685 −0.254772
\(494\) −93.2548 −4.19573
\(495\) −1.00000 −0.0449467
\(496\) 0 0
\(497\) 0 0
\(498\) −24.1421 −1.08183
\(499\) −22.3431 −1.00022 −0.500108 0.865963i \(-0.666706\pi\)
−0.500108 + 0.865963i \(0.666706\pi\)
\(500\) 3.82843 0.171212
\(501\) 13.3137 0.594813
\(502\) 50.6274 2.25961
\(503\) −17.3137 −0.771980 −0.385990 0.922503i \(-0.626140\pi\)
−0.385990 + 0.922503i \(0.626140\pi\)
\(504\) 0 0
\(505\) 0.828427 0.0368645
\(506\) −9.65685 −0.429300
\(507\) 19.0000 0.843820
\(508\) −55.4558 −2.46046
\(509\) −18.6863 −0.828255 −0.414128 0.910219i \(-0.635913\pi\)
−0.414128 + 0.910219i \(0.635913\pi\)
\(510\) −2.82843 −0.125245
\(511\) 0 0
\(512\) 31.2426 1.38074
\(513\) 6.82843 0.301482
\(514\) −39.4558 −1.74032
\(515\) 3.31371 0.146019
\(516\) −33.7990 −1.48792
\(517\) −4.00000 −0.175920
\(518\) 0 0
\(519\) −2.82843 −0.124154
\(520\) −24.9706 −1.09503
\(521\) 32.6274 1.42943 0.714717 0.699414i \(-0.246556\pi\)
0.714717 + 0.699414i \(0.246556\pi\)
\(522\) 11.6569 0.510207
\(523\) 9.51472 0.416050 0.208025 0.978124i \(-0.433297\pi\)
0.208025 + 0.978124i \(0.433297\pi\)
\(524\) −12.6863 −0.554203
\(525\) 0 0
\(526\) −43.4558 −1.89476
\(527\) 0 0
\(528\) −3.00000 −0.130558
\(529\) −7.00000 −0.304348
\(530\) −22.4853 −0.976698
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) −27.3137 −1.18309
\(534\) 8.82843 0.382043
\(535\) 17.3137 0.748537
\(536\) 24.9706 1.07856
\(537\) −17.6569 −0.761950
\(538\) 49.7990 2.14699
\(539\) 0 0
\(540\) 3.82843 0.164749
\(541\) 17.3137 0.744374 0.372187 0.928158i \(-0.378608\pi\)
0.372187 + 0.928158i \(0.378608\pi\)
\(542\) −28.4853 −1.22355
\(543\) 14.0000 0.600798
\(544\) 1.85786 0.0796553
\(545\) 17.3137 0.741638
\(546\) 0 0
\(547\) −8.14214 −0.348133 −0.174066 0.984734i \(-0.555691\pi\)
−0.174066 + 0.984734i \(0.555691\pi\)
\(548\) −50.9706 −2.17735
\(549\) 11.6569 0.497502
\(550\) 2.41421 0.102942
\(551\) −32.9706 −1.40459
\(552\) 17.6569 0.751526
\(553\) 0 0
\(554\) 5.65685 0.240337
\(555\) 11.6569 0.494806
\(556\) −1.85786 −0.0787910
\(557\) 5.17157 0.219127 0.109563 0.993980i \(-0.465055\pi\)
0.109563 + 0.993980i \(0.465055\pi\)
\(558\) 0 0
\(559\) −49.9411 −2.11228
\(560\) 0 0
\(561\) −1.17157 −0.0494638
\(562\) 26.9706 1.13768
\(563\) −31.6569 −1.33418 −0.667089 0.744978i \(-0.732460\pi\)
−0.667089 + 0.744978i \(0.732460\pi\)
\(564\) 15.3137 0.644823
\(565\) −18.9706 −0.798098
\(566\) −21.3137 −0.895882
\(567\) 0 0
\(568\) −10.3431 −0.433989
\(569\) −35.4558 −1.48639 −0.743193 0.669077i \(-0.766690\pi\)
−0.743193 + 0.669077i \(0.766690\pi\)
\(570\) −16.4853 −0.690492
\(571\) −16.4853 −0.689888 −0.344944 0.938623i \(-0.612102\pi\)
−0.344944 + 0.938623i \(0.612102\pi\)
\(572\) −21.6569 −0.905519
\(573\) 5.65685 0.236318
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) −9.82843 −0.409518
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 37.7279 1.56927
\(579\) 13.6569 0.567559
\(580\) −18.4853 −0.767560
\(581\) 0 0
\(582\) 28.1421 1.16653
\(583\) −9.31371 −0.385734
\(584\) 49.9411 2.06658
\(585\) 5.65685 0.233882
\(586\) −16.4853 −0.681001
\(587\) −14.6274 −0.603738 −0.301869 0.953349i \(-0.597611\pi\)
−0.301869 + 0.953349i \(0.597611\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −9.65685 −0.397566
\(591\) −8.48528 −0.349038
\(592\) 34.9706 1.43728
\(593\) 22.8284 0.937451 0.468726 0.883344i \(-0.344713\pi\)
0.468726 + 0.883344i \(0.344713\pi\)
\(594\) 2.41421 0.0990564
\(595\) 0 0
\(596\) 5.79899 0.237536
\(597\) 21.6569 0.886356
\(598\) 54.6274 2.23388
\(599\) 27.3137 1.11601 0.558004 0.829838i \(-0.311567\pi\)
0.558004 + 0.829838i \(0.311567\pi\)
\(600\) −4.41421 −0.180210
\(601\) 5.31371 0.216751 0.108375 0.994110i \(-0.465435\pi\)
0.108375 + 0.994110i \(0.465435\pi\)
\(602\) 0 0
\(603\) −5.65685 −0.230365
\(604\) 63.1127 2.56802
\(605\) 1.00000 0.0406558
\(606\) −2.00000 −0.0812444
\(607\) −1.51472 −0.0614805 −0.0307403 0.999527i \(-0.509786\pi\)
−0.0307403 + 0.999527i \(0.509786\pi\)
\(608\) 10.8284 0.439151
\(609\) 0 0
\(610\) −28.1421 −1.13944
\(611\) 22.6274 0.915407
\(612\) 4.48528 0.181307
\(613\) −45.9411 −1.85554 −0.927772 0.373147i \(-0.878279\pi\)
−0.927772 + 0.373147i \(0.878279\pi\)
\(614\) −7.65685 −0.309005
\(615\) −4.82843 −0.194701
\(616\) 0 0
\(617\) 0.343146 0.0138145 0.00690726 0.999976i \(-0.497801\pi\)
0.00690726 + 0.999976i \(0.497801\pi\)
\(618\) −8.00000 −0.321807
\(619\) 14.3431 0.576500 0.288250 0.957555i \(-0.406927\pi\)
0.288250 + 0.957555i \(0.406927\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) −8.00000 −0.320771
\(623\) 0 0
\(624\) 16.9706 0.679366
\(625\) 1.00000 0.0400000
\(626\) 37.7990 1.51075
\(627\) −6.82843 −0.272701
\(628\) −68.9117 −2.74988
\(629\) 13.6569 0.544534
\(630\) 0 0
\(631\) 45.6569 1.81757 0.908785 0.417264i \(-0.137011\pi\)
0.908785 + 0.417264i \(0.137011\pi\)
\(632\) −37.4558 −1.48991
\(633\) 1.17157 0.0465658
\(634\) 63.4558 2.52015
\(635\) −14.4853 −0.574831
\(636\) 35.6569 1.41389
\(637\) 0 0
\(638\) −11.6569 −0.461499
\(639\) 2.34315 0.0926934
\(640\) 20.5563 0.812561
\(641\) 6.97056 0.275321 0.137660 0.990479i \(-0.456042\pi\)
0.137660 + 0.990479i \(0.456042\pi\)
\(642\) −41.7990 −1.64967
\(643\) −37.9411 −1.49625 −0.748126 0.663557i \(-0.769046\pi\)
−0.748126 + 0.663557i \(0.769046\pi\)
\(644\) 0 0
\(645\) −8.82843 −0.347619
\(646\) −19.3137 −0.759888
\(647\) −4.68629 −0.184237 −0.0921186 0.995748i \(-0.529364\pi\)
−0.0921186 + 0.995748i \(0.529364\pi\)
\(648\) −4.41421 −0.173407
\(649\) −4.00000 −0.157014
\(650\) −13.6569 −0.535666
\(651\) 0 0
\(652\) −28.0000 −1.09656
\(653\) 6.97056 0.272779 0.136390 0.990655i \(-0.456450\pi\)
0.136390 + 0.990655i \(0.456450\pi\)
\(654\) −41.7990 −1.63447
\(655\) −3.31371 −0.129477
\(656\) −14.4853 −0.565555
\(657\) −11.3137 −0.441390
\(658\) 0 0
\(659\) −15.3137 −0.596537 −0.298269 0.954482i \(-0.596409\pi\)
−0.298269 + 0.954482i \(0.596409\pi\)
\(660\) −3.82843 −0.149021
\(661\) −9.31371 −0.362261 −0.181131 0.983459i \(-0.557976\pi\)
−0.181131 + 0.983459i \(0.557976\pi\)
\(662\) −15.3137 −0.595184
\(663\) 6.62742 0.257388
\(664\) −44.1421 −1.71305
\(665\) 0 0
\(666\) −28.1421 −1.09049
\(667\) 19.3137 0.747830
\(668\) 50.9706 1.97211
\(669\) 6.34315 0.245240
\(670\) 13.6569 0.527610
\(671\) −11.6569 −0.450008
\(672\) 0 0
\(673\) 18.3431 0.707076 0.353538 0.935420i \(-0.384978\pi\)
0.353538 + 0.935420i \(0.384978\pi\)
\(674\) −8.00000 −0.308148
\(675\) 1.00000 0.0384900
\(676\) 72.7401 2.79770
\(677\) −29.4558 −1.13208 −0.566040 0.824378i \(-0.691525\pi\)
−0.566040 + 0.824378i \(0.691525\pi\)
\(678\) 45.7990 1.75890
\(679\) 0 0
\(680\) −5.17157 −0.198321
\(681\) 14.0000 0.536481
\(682\) 0 0
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 26.1421 0.999570
\(685\) −13.3137 −0.508691
\(686\) 0 0
\(687\) 2.00000 0.0763048
\(688\) −26.4853 −1.00974
\(689\) 52.6863 2.00719
\(690\) 9.65685 0.367630
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) −10.8284 −0.411635
\(693\) 0 0
\(694\) 70.7696 2.68638
\(695\) −0.485281 −0.0184078
\(696\) 21.3137 0.807894
\(697\) −5.65685 −0.214269
\(698\) 26.4853 1.00248
\(699\) −18.8284 −0.712157
\(700\) 0 0
\(701\) 36.1421 1.36507 0.682535 0.730853i \(-0.260878\pi\)
0.682535 + 0.730853i \(0.260878\pi\)
\(702\) −13.6569 −0.515445
\(703\) 79.5980 3.00209
\(704\) 9.82843 0.370423
\(705\) 4.00000 0.150649
\(706\) 62.7696 2.36236
\(707\) 0 0
\(708\) 15.3137 0.575524
\(709\) 6.68629 0.251109 0.125554 0.992087i \(-0.459929\pi\)
0.125554 + 0.992087i \(0.459929\pi\)
\(710\) −5.65685 −0.212298
\(711\) 8.48528 0.318223
\(712\) 16.1421 0.604952
\(713\) 0 0
\(714\) 0 0
\(715\) −5.65685 −0.211554
\(716\) −67.5980 −2.52626
\(717\) 17.6569 0.659407
\(718\) −28.9706 −1.08117
\(719\) 47.5980 1.77511 0.887553 0.460706i \(-0.152404\pi\)
0.887553 + 0.460706i \(0.152404\pi\)
\(720\) 3.00000 0.111803
\(721\) 0 0
\(722\) −66.6985 −2.48226
\(723\) 12.3431 0.459047
\(724\) 53.5980 1.99195
\(725\) −4.82843 −0.179323
\(726\) −2.41421 −0.0895999
\(727\) −33.9411 −1.25881 −0.629403 0.777079i \(-0.716701\pi\)
−0.629403 + 0.777079i \(0.716701\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 27.3137 1.01093
\(731\) −10.3431 −0.382555
\(732\) 44.6274 1.64948
\(733\) 6.34315 0.234289 0.117145 0.993115i \(-0.462626\pi\)
0.117145 + 0.993115i \(0.462626\pi\)
\(734\) 23.3137 0.860525
\(735\) 0 0
\(736\) −6.34315 −0.233811
\(737\) 5.65685 0.208373
\(738\) 11.6569 0.429095
\(739\) −15.1127 −0.555930 −0.277965 0.960591i \(-0.589660\pi\)
−0.277965 + 0.960591i \(0.589660\pi\)
\(740\) 44.6274 1.64054
\(741\) 38.6274 1.41901
\(742\) 0 0
\(743\) 36.3431 1.33330 0.666650 0.745371i \(-0.267727\pi\)
0.666650 + 0.745371i \(0.267727\pi\)
\(744\) 0 0
\(745\) 1.51472 0.0554950
\(746\) −25.6569 −0.939364
\(747\) 10.0000 0.365881
\(748\) −4.48528 −0.163998
\(749\) 0 0
\(750\) −2.41421 −0.0881546
\(751\) 20.2843 0.740184 0.370092 0.928995i \(-0.379326\pi\)
0.370092 + 0.928995i \(0.379326\pi\)
\(752\) 12.0000 0.437595
\(753\) −20.9706 −0.764210
\(754\) 65.9411 2.40143
\(755\) 16.4853 0.599961
\(756\) 0 0
\(757\) −36.6274 −1.33125 −0.665623 0.746288i \(-0.731834\pi\)
−0.665623 + 0.746288i \(0.731834\pi\)
\(758\) 56.2843 2.04434
\(759\) 4.00000 0.145191
\(760\) −30.1421 −1.09337
\(761\) 28.8284 1.04503 0.522515 0.852630i \(-0.324994\pi\)
0.522515 + 0.852630i \(0.324994\pi\)
\(762\) 34.9706 1.26685
\(763\) 0 0
\(764\) 21.6569 0.783517
\(765\) 1.17157 0.0423583
\(766\) −19.3137 −0.697833
\(767\) 22.6274 0.817029
\(768\) −29.9706 −1.08147
\(769\) −10.6863 −0.385358 −0.192679 0.981262i \(-0.561718\pi\)
−0.192679 + 0.981262i \(0.561718\pi\)
\(770\) 0 0
\(771\) 16.3431 0.588584
\(772\) 52.2843 1.88175
\(773\) −3.65685 −0.131528 −0.0657640 0.997835i \(-0.520948\pi\)
−0.0657640 + 0.997835i \(0.520948\pi\)
\(774\) 21.3137 0.766105
\(775\) 0 0
\(776\) 51.4558 1.84716
\(777\) 0 0
\(778\) 57.1127 2.04759
\(779\) −32.9706 −1.18129
\(780\) 21.6569 0.775440
\(781\) −2.34315 −0.0838443
\(782\) 11.3137 0.404577
\(783\) −4.82843 −0.172554
\(784\) 0 0
\(785\) −18.0000 −0.642448
\(786\) 8.00000 0.285351
\(787\) 20.1421 0.717990 0.358995 0.933340i \(-0.383120\pi\)
0.358995 + 0.933340i \(0.383120\pi\)
\(788\) −32.4853 −1.15724
\(789\) 18.0000 0.640817
\(790\) −20.4853 −0.728834
\(791\) 0 0
\(792\) 4.41421 0.156852
\(793\) 65.9411 2.34164
\(794\) −36.1421 −1.28264
\(795\) 9.31371 0.330323
\(796\) 82.9117 2.93873
\(797\) −34.9706 −1.23872 −0.619360 0.785107i \(-0.712608\pi\)
−0.619360 + 0.785107i \(0.712608\pi\)
\(798\) 0 0
\(799\) 4.68629 0.165789
\(800\) 1.58579 0.0560660
\(801\) −3.65685 −0.129209
\(802\) 16.1421 0.569999
\(803\) 11.3137 0.399252
\(804\) −21.6569 −0.763778
\(805\) 0 0
\(806\) 0 0
\(807\) −20.6274 −0.726119
\(808\) −3.65685 −0.128648
\(809\) 28.4264 0.999419 0.499710 0.866193i \(-0.333440\pi\)
0.499710 + 0.866193i \(0.333440\pi\)
\(810\) −2.41421 −0.0848268
\(811\) −0.485281 −0.0170405 −0.00852027 0.999964i \(-0.502712\pi\)
−0.00852027 + 0.999964i \(0.502712\pi\)
\(812\) 0 0
\(813\) 11.7990 0.413809
\(814\) 28.1421 0.986381
\(815\) −7.31371 −0.256188
\(816\) 3.51472 0.123040
\(817\) −60.2843 −2.10908
\(818\) −47.4558 −1.65925
\(819\) 0 0
\(820\) −18.4853 −0.645534
\(821\) −12.8284 −0.447715 −0.223858 0.974622i \(-0.571865\pi\)
−0.223858 + 0.974622i \(0.571865\pi\)
\(822\) 32.1421 1.12109
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) −14.6274 −0.509570
\(825\) −1.00000 −0.0348155
\(826\) 0 0
\(827\) 41.3137 1.43662 0.718309 0.695724i \(-0.244916\pi\)
0.718309 + 0.695724i \(0.244916\pi\)
\(828\) −15.3137 −0.532188
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) −24.1421 −0.837986
\(831\) −2.34315 −0.0812828
\(832\) −55.5980 −1.92751
\(833\) 0 0
\(834\) 1.17157 0.0405683
\(835\) 13.3137 0.460740
\(836\) −26.1421 −0.904145
\(837\) 0 0
\(838\) −89.2548 −3.08326
\(839\) 22.6274 0.781185 0.390593 0.920564i \(-0.372270\pi\)
0.390593 + 0.920564i \(0.372270\pi\)
\(840\) 0 0
\(841\) −5.68629 −0.196079
\(842\) 14.4853 0.499196
\(843\) −11.1716 −0.384769
\(844\) 4.48528 0.154390
\(845\) 19.0000 0.653620
\(846\) −9.65685 −0.332009
\(847\) 0 0
\(848\) 27.9411 0.959502
\(849\) 8.82843 0.302991
\(850\) −2.82843 −0.0970143
\(851\) −46.6274 −1.59837
\(852\) 8.97056 0.307326
\(853\) 8.68629 0.297413 0.148706 0.988881i \(-0.452489\pi\)
0.148706 + 0.988881i \(0.452489\pi\)
\(854\) 0 0
\(855\) 6.82843 0.233527
\(856\) −76.4264 −2.61220
\(857\) 28.4853 0.973039 0.486519 0.873670i \(-0.338266\pi\)
0.486519 + 0.873670i \(0.338266\pi\)
\(858\) 13.6569 0.466237
\(859\) 52.9706 1.80733 0.903666 0.428238i \(-0.140865\pi\)
0.903666 + 0.428238i \(0.140865\pi\)
\(860\) −33.7990 −1.15254
\(861\) 0 0
\(862\) −52.2843 −1.78081
\(863\) −20.6863 −0.704170 −0.352085 0.935968i \(-0.614527\pi\)
−0.352085 + 0.935968i \(0.614527\pi\)
\(864\) 1.58579 0.0539496
\(865\) −2.82843 −0.0961694
\(866\) −37.7990 −1.28446
\(867\) −15.6274 −0.530735
\(868\) 0 0
\(869\) −8.48528 −0.287843
\(870\) 11.6569 0.395204
\(871\) −32.0000 −1.08428
\(872\) −76.4264 −2.58812
\(873\) −11.6569 −0.394525
\(874\) 65.9411 2.23049
\(875\) 0 0
\(876\) −43.3137 −1.46343
\(877\) 2.62742 0.0887216 0.0443608 0.999016i \(-0.485875\pi\)
0.0443608 + 0.999016i \(0.485875\pi\)
\(878\) −49.4558 −1.66905
\(879\) 6.82843 0.230317
\(880\) −3.00000 −0.101130
\(881\) 46.9706 1.58248 0.791239 0.611507i \(-0.209436\pi\)
0.791239 + 0.611507i \(0.209436\pi\)
\(882\) 0 0
\(883\) −5.37258 −0.180802 −0.0904009 0.995905i \(-0.528815\pi\)
−0.0904009 + 0.995905i \(0.528815\pi\)
\(884\) 25.3726 0.853372
\(885\) 4.00000 0.134459
\(886\) −28.9706 −0.973285
\(887\) 15.6569 0.525706 0.262853 0.964836i \(-0.415337\pi\)
0.262853 + 0.964836i \(0.415337\pi\)
\(888\) −51.4558 −1.72675
\(889\) 0 0
\(890\) 8.82843 0.295930
\(891\) −1.00000 −0.0335013
\(892\) 24.2843 0.813098
\(893\) 27.3137 0.914018
\(894\) −3.65685 −0.122304
\(895\) −17.6569 −0.590204
\(896\) 0 0
\(897\) −22.6274 −0.755507
\(898\) −74.7696 −2.49509
\(899\) 0 0
\(900\) 3.82843 0.127614
\(901\) 10.9117 0.363521
\(902\) −11.6569 −0.388131
\(903\) 0 0
\(904\) 83.7401 2.78515
\(905\) 14.0000 0.465376
\(906\) −39.7990 −1.32223
\(907\) 40.9706 1.36041 0.680203 0.733024i \(-0.261892\pi\)
0.680203 + 0.733024i \(0.261892\pi\)
\(908\) 53.5980 1.77871
\(909\) 0.828427 0.0274772
\(910\) 0 0
\(911\) −48.9706 −1.62247 −0.811234 0.584722i \(-0.801203\pi\)
−0.811234 + 0.584722i \(0.801203\pi\)
\(912\) 20.4853 0.678335
\(913\) −10.0000 −0.330952
\(914\) 56.2843 1.86172
\(915\) 11.6569 0.385364
\(916\) 7.65685 0.252990
\(917\) 0 0
\(918\) −2.82843 −0.0933520
\(919\) 11.5147 0.379836 0.189918 0.981800i \(-0.439178\pi\)
0.189918 + 0.981800i \(0.439178\pi\)
\(920\) 17.6569 0.582129
\(921\) 3.17157 0.104507
\(922\) 0.343146 0.0113009
\(923\) 13.2548 0.436288
\(924\) 0 0
\(925\) 11.6569 0.383275
\(926\) −12.0000 −0.394344
\(927\) 3.31371 0.108836
\(928\) −7.65685 −0.251349
\(929\) 45.5980 1.49602 0.748011 0.663687i \(-0.231009\pi\)
0.748011 + 0.663687i \(0.231009\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −72.0833 −2.36117
\(933\) 3.31371 0.108486
\(934\) −54.6274 −1.78746
\(935\) −1.17157 −0.0383145
\(936\) −24.9706 −0.816188
\(937\) −11.0294 −0.360316 −0.180158 0.983638i \(-0.557661\pi\)
−0.180158 + 0.983638i \(0.557661\pi\)
\(938\) 0 0
\(939\) −15.6569 −0.510942
\(940\) 15.3137 0.499478
\(941\) 34.7696 1.13346 0.566728 0.823905i \(-0.308209\pi\)
0.566728 + 0.823905i \(0.308209\pi\)
\(942\) 43.4558 1.41587
\(943\) 19.3137 0.628941
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) −21.3137 −0.692968
\(947\) 6.62742 0.215362 0.107681 0.994185i \(-0.465657\pi\)
0.107681 + 0.994185i \(0.465657\pi\)
\(948\) 32.4853 1.05507
\(949\) −64.0000 −2.07753
\(950\) −16.4853 −0.534853
\(951\) −26.2843 −0.852326
\(952\) 0 0
\(953\) −11.7990 −0.382207 −0.191103 0.981570i \(-0.561207\pi\)
−0.191103 + 0.981570i \(0.561207\pi\)
\(954\) −22.4853 −0.727988
\(955\) 5.65685 0.183052
\(956\) 67.5980 2.18627
\(957\) 4.82843 0.156081
\(958\) 89.2548 2.88369
\(959\) 0 0
\(960\) −9.82843 −0.317211
\(961\) −31.0000 −1.00000
\(962\) −159.196 −5.13268
\(963\) 17.3137 0.557926
\(964\) 47.2548 1.52198
\(965\) 13.6569 0.439630
\(966\) 0 0
\(967\) 11.4558 0.368395 0.184198 0.982889i \(-0.441031\pi\)
0.184198 + 0.982889i \(0.441031\pi\)
\(968\) −4.41421 −0.141878
\(969\) 8.00000 0.256997
\(970\) 28.1421 0.903590
\(971\) 34.6274 1.11125 0.555623 0.831434i \(-0.312480\pi\)
0.555623 + 0.831434i \(0.312480\pi\)
\(972\) 3.82843 0.122797
\(973\) 0 0
\(974\) 31.3137 1.00336
\(975\) 5.65685 0.181164
\(976\) 34.9706 1.11938
\(977\) 2.68629 0.0859421 0.0429710 0.999076i \(-0.486318\pi\)
0.0429710 + 0.999076i \(0.486318\pi\)
\(978\) 17.6569 0.564604
\(979\) 3.65685 0.116874
\(980\) 0 0
\(981\) 17.3137 0.552784
\(982\) −34.6274 −1.10501
\(983\) 30.6274 0.976863 0.488431 0.872602i \(-0.337569\pi\)
0.488431 + 0.872602i \(0.337569\pi\)
\(984\) 21.3137 0.679456
\(985\) −8.48528 −0.270364
\(986\) 13.6569 0.434923
\(987\) 0 0
\(988\) 147.882 4.70476
\(989\) 35.3137 1.12291
\(990\) 2.41421 0.0767287
\(991\) −30.6274 −0.972912 −0.486456 0.873705i \(-0.661711\pi\)
−0.486456 + 0.873705i \(0.661711\pi\)
\(992\) 0 0
\(993\) 6.34315 0.201294
\(994\) 0 0
\(995\) 21.6569 0.686568
\(996\) 38.2843 1.21308
\(997\) 39.3137 1.24508 0.622539 0.782589i \(-0.286101\pi\)
0.622539 + 0.782589i \(0.286101\pi\)
\(998\) 53.9411 1.70748
\(999\) 11.6569 0.368807
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.ba.1.1 2
7.6 odd 2 165.2.a.a.1.1 2
21.20 even 2 495.2.a.d.1.2 2
28.27 even 2 2640.2.a.bb.1.1 2
35.13 even 4 825.2.c.e.199.4 4
35.27 even 4 825.2.c.e.199.1 4
35.34 odd 2 825.2.a.g.1.2 2
77.76 even 2 1815.2.a.k.1.2 2
84.83 odd 2 7920.2.a.cg.1.1 2
105.62 odd 4 2475.2.c.m.199.4 4
105.83 odd 4 2475.2.c.m.199.1 4
105.104 even 2 2475.2.a.m.1.1 2
231.230 odd 2 5445.2.a.m.1.1 2
385.384 even 2 9075.2.a.v.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.a.1.1 2 7.6 odd 2
495.2.a.d.1.2 2 21.20 even 2
825.2.a.g.1.2 2 35.34 odd 2
825.2.c.e.199.1 4 35.27 even 4
825.2.c.e.199.4 4 35.13 even 4
1815.2.a.k.1.2 2 77.76 even 2
2475.2.a.m.1.1 2 105.104 even 2
2475.2.c.m.199.1 4 105.83 odd 4
2475.2.c.m.199.4 4 105.62 odd 4
2640.2.a.bb.1.1 2 28.27 even 2
5445.2.a.m.1.1 2 231.230 odd 2
7920.2.a.cg.1.1 2 84.83 odd 2
8085.2.a.ba.1.1 2 1.1 even 1 trivial
9075.2.a.v.1.1 2 385.384 even 2