Properties

Label 495.2.a.d.1.2
Level $495$
Weight $2$
Character 495.1
Self dual yes
Analytic conductor $3.953$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.95259490005\)
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 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 495.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} +0.828427 q^{7} +4.41421 q^{8} +O(q^{10})\) \(q+2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} +0.828427 q^{7} +4.41421 q^{8} +2.41421 q^{10} +1.00000 q^{11} -5.65685 q^{13} +2.00000 q^{14} +3.00000 q^{16} +1.17157 q^{17} -6.82843 q^{19} +3.82843 q^{20} +2.41421 q^{22} +4.00000 q^{23} +1.00000 q^{25} -13.6569 q^{26} +3.17157 q^{28} +4.82843 q^{29} -1.58579 q^{32} +2.82843 q^{34} +0.828427 q^{35} +11.6569 q^{37} -16.4853 q^{38} +4.41421 q^{40} -4.82843 q^{41} -8.82843 q^{43} +3.82843 q^{44} +9.65685 q^{46} +4.00000 q^{47} -6.31371 q^{49} +2.41421 q^{50} -21.6569 q^{52} -9.31371 q^{53} +1.00000 q^{55} +3.65685 q^{56} +11.6569 q^{58} +4.00000 q^{59} -11.6569 q^{61} -9.82843 q^{64} -5.65685 q^{65} -5.65685 q^{67} +4.48528 q^{68} +2.00000 q^{70} -2.34315 q^{71} +11.3137 q^{73} +28.1421 q^{74} -26.1421 q^{76} +0.828427 q^{77} +8.48528 q^{79} +3.00000 q^{80} -11.6569 q^{82} +10.0000 q^{83} +1.17157 q^{85} -21.3137 q^{86} +4.41421 q^{88} -3.65685 q^{89} -4.68629 q^{91} +15.3137 q^{92} +9.65685 q^{94} -6.82843 q^{95} +11.6569 q^{97} -15.2426 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} - 4q^{7} + 6q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} - 4q^{7} + 6q^{8} + 2q^{10} + 2q^{11} + 4q^{14} + 6q^{16} + 8q^{17} - 8q^{19} + 2q^{20} + 2q^{22} + 8q^{23} + 2q^{25} - 16q^{26} + 12q^{28} + 4q^{29} - 6q^{32} - 4q^{35} + 12q^{37} - 16q^{38} + 6q^{40} - 4q^{41} - 12q^{43} + 2q^{44} + 8q^{46} + 8q^{47} + 10q^{49} + 2q^{50} - 32q^{52} + 4q^{53} + 2q^{55} - 4q^{56} + 12q^{58} + 8q^{59} - 12q^{61} - 14q^{64} - 8q^{68} + 4q^{70} - 16q^{71} + 28q^{74} - 24q^{76} - 4q^{77} + 6q^{80} - 12q^{82} + 20q^{83} + 8q^{85} - 20q^{86} + 6q^{88} + 4q^{89} - 32q^{91} + 8q^{92} + 8q^{94} - 8q^{95} + 12q^{97} - 22q^{98} + O(q^{100}) \)

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 \(-0.174443\pi\)
0.853553 + 0.521005i \(0.174443\pi\)
\(3\) 0 0
\(4\) 3.82843 1.91421
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0.828427 0.313116 0.156558 0.987669i \(-0.449960\pi\)
0.156558 + 0.987669i \(0.449960\pi\)
\(8\) 4.41421 1.56066
\(9\) 0 0
\(10\) 2.41421 0.763441
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −5.65685 −1.56893 −0.784465 0.620174i \(-0.787062\pi\)
−0.784465 + 0.620174i \(0.787062\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 1.17157 0.284148 0.142074 0.989856i \(-0.454623\pi\)
0.142074 + 0.989856i \(0.454623\pi\)
\(18\) 0 0
\(19\) −6.82843 −1.56655 −0.783274 0.621676i \(-0.786452\pi\)
−0.783274 + 0.621676i \(0.786452\pi\)
\(20\) 3.82843 0.856062
\(21\) 0 0
\(22\) 2.41421 0.514712
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −13.6569 −2.67833
\(27\) 0 0
\(28\) 3.17157 0.599371
\(29\) 4.82843 0.896616 0.448308 0.893879i \(-0.352027\pi\)
0.448308 + 0.893879i \(0.352027\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.58579 −0.280330
\(33\) 0 0
\(34\) 2.82843 0.485071
\(35\) 0.828427 0.140030
\(36\) 0 0
\(37\) 11.6569 1.91638 0.958188 0.286141i \(-0.0923726\pi\)
0.958188 + 0.286141i \(0.0923726\pi\)
\(38\) −16.4853 −2.67427
\(39\) 0 0
\(40\) 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
\(45\) 0 0
\(46\) 9.65685 1.42383
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) −6.31371 −0.901958
\(50\) 2.41421 0.341421
\(51\) 0 0
\(52\) −21.6569 −3.00327
\(53\) −9.31371 −1.27934 −0.639668 0.768651i \(-0.720928\pi\)
−0.639668 + 0.768651i \(0.720928\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 3.65685 0.488668
\(57\) 0 0
\(58\) 11.6569 1.53062
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −11.6569 −1.49251 −0.746254 0.665662i \(-0.768149\pi\)
−0.746254 + 0.665662i \(0.768149\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −9.82843 −1.22855
\(65\) −5.65685 −0.701646
\(66\) 0 0
\(67\) −5.65685 −0.691095 −0.345547 0.938401i \(-0.612307\pi\)
−0.345547 + 0.938401i \(0.612307\pi\)
\(68\) 4.48528 0.543920
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) −2.34315 −0.278080 −0.139040 0.990287i \(-0.544402\pi\)
−0.139040 + 0.990287i \(0.544402\pi\)
\(72\) 0 0
\(73\) 11.3137 1.32417 0.662085 0.749429i \(-0.269672\pi\)
0.662085 + 0.749429i \(0.269672\pi\)
\(74\) 28.1421 3.27146
\(75\) 0 0
\(76\) −26.1421 −2.99871
\(77\) 0.828427 0.0944080
\(78\) 0 0
\(79\) 8.48528 0.954669 0.477334 0.878722i \(-0.341603\pi\)
0.477334 + 0.878722i \(0.341603\pi\)
\(80\) 3.00000 0.335410
\(81\) 0 0
\(82\) −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
\(87\) 0 0
\(88\) 4.41421 0.470557
\(89\) −3.65685 −0.387626 −0.193813 0.981039i \(-0.562085\pi\)
−0.193813 + 0.981039i \(0.562085\pi\)
\(90\) 0 0
\(91\) −4.68629 −0.491257
\(92\) 15.3137 1.59656
\(93\) 0 0
\(94\) 9.65685 0.996028
\(95\) −6.82843 −0.700582
\(96\) 0 0
\(97\) 11.6569 1.18357 0.591787 0.806094i \(-0.298423\pi\)
0.591787 + 0.806094i \(0.298423\pi\)
\(98\) −15.2426 −1.53974
\(99\) 0 0
\(100\) 3.82843 0.382843
\(101\) 0.828427 0.0824316 0.0412158 0.999150i \(-0.486877\pi\)
0.0412158 + 0.999150i \(0.486877\pi\)
\(102\) 0 0
\(103\) −3.31371 −0.326509 −0.163255 0.986584i \(-0.552199\pi\)
−0.163255 + 0.986584i \(0.552199\pi\)
\(104\) −24.9706 −2.44857
\(105\) 0 0
\(106\) −22.4853 −2.18396
\(107\) −17.3137 −1.67378 −0.836890 0.547372i \(-0.815628\pi\)
−0.836890 + 0.547372i \(0.815628\pi\)
\(108\) 0 0
\(109\) 17.3137 1.65835 0.829176 0.558987i \(-0.188810\pi\)
0.829176 + 0.558987i \(0.188810\pi\)
\(110\) 2.41421 0.230186
\(111\) 0 0
\(112\) 2.48528 0.234837
\(113\) 18.9706 1.78460 0.892300 0.451442i \(-0.149090\pi\)
0.892300 + 0.451442i \(0.149090\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 18.4853 1.71632
\(117\) 0 0
\(118\) 9.65685 0.888985
\(119\) 0.970563 0.0889713
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −28.1421 −2.54787
\(123\) 0 0
\(124\) 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
\(129\) 0 0
\(130\) −13.6569 −1.19779
\(131\) −3.31371 −0.289520 −0.144760 0.989467i \(-0.546241\pi\)
−0.144760 + 0.989467i \(0.546241\pi\)
\(132\) 0 0
\(133\) −5.65685 −0.490511
\(134\) −13.6569 −1.17977
\(135\) 0 0
\(136\) 5.17157 0.443459
\(137\) 13.3137 1.13747 0.568733 0.822522i \(-0.307434\pi\)
0.568733 + 0.822522i \(0.307434\pi\)
\(138\) 0 0
\(139\) 0.485281 0.0411610 0.0205805 0.999788i \(-0.493449\pi\)
0.0205805 + 0.999788i \(0.493449\pi\)
\(140\) 3.17157 0.268047
\(141\) 0 0
\(142\) −5.65685 −0.474713
\(143\) −5.65685 −0.473050
\(144\) 0 0
\(145\) 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 \(-0.519762\pi\)
−0.0620453 + 0.998073i \(0.519762\pi\)
\(150\) 0 0
\(151\) 16.4853 1.34155 0.670777 0.741659i \(-0.265961\pi\)
0.670777 + 0.741659i \(0.265961\pi\)
\(152\) −30.1421 −2.44485
\(153\) 0 0
\(154\) 2.00000 0.161165
\(155\) 0 0
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 20.4853 1.62972
\(159\) 0 0
\(160\) −1.58579 −0.125367
\(161\) 3.31371 0.261157
\(162\) 0 0
\(163\) −7.31371 −0.572854 −0.286427 0.958102i \(-0.592468\pi\)
−0.286427 + 0.958102i \(0.592468\pi\)
\(164\) −18.4853 −1.44346
\(165\) 0 0
\(166\) 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
\(171\) 0 0
\(172\) −33.7990 −2.57715
\(173\) −2.82843 −0.215041 −0.107521 0.994203i \(-0.534291\pi\)
−0.107521 + 0.994203i \(0.534291\pi\)
\(174\) 0 0
\(175\) 0.828427 0.0626232
\(176\) 3.00000 0.226134
\(177\) 0 0
\(178\) −8.82843 −0.661719
\(179\) 17.6569 1.31974 0.659868 0.751382i \(-0.270612\pi\)
0.659868 + 0.751382i \(0.270612\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) −11.3137 −0.838628
\(183\) 0 0
\(184\) 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 \(-0.565608\pi\)
−0.204658 + 0.978834i \(0.565608\pi\)
\(192\) 0 0
\(193\) 13.6569 0.983042 0.491521 0.870866i \(-0.336441\pi\)
0.491521 + 0.870866i \(0.336441\pi\)
\(194\) 28.1421 2.02049
\(195\) 0 0
\(196\) −24.1716 −1.72654
\(197\) 8.48528 0.604551 0.302276 0.953221i \(-0.402254\pi\)
0.302276 + 0.953221i \(0.402254\pi\)
\(198\) 0 0
\(199\) −21.6569 −1.53521 −0.767607 0.640921i \(-0.778553\pi\)
−0.767607 + 0.640921i \(0.778553\pi\)
\(200\) 4.41421 0.312132
\(201\) 0 0
\(202\) 2.00000 0.140720
\(203\) 4.00000 0.280745
\(204\) 0 0
\(205\) −4.82843 −0.337232
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) −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
\(213\) 0 0
\(214\) −41.7990 −2.85732
\(215\) −8.82843 −0.602094
\(216\) 0 0
\(217\) 0 0
\(218\) 41.7990 2.83098
\(219\) 0 0
\(220\) 3.82843 0.258113
\(221\) −6.62742 −0.445808
\(222\) 0 0
\(223\) −6.34315 −0.424768 −0.212384 0.977186i \(-0.568123\pi\)
−0.212384 + 0.977186i \(0.568123\pi\)
\(224\) −1.31371 −0.0877758
\(225\) 0 0
\(226\) 45.7990 3.04650
\(227\) 14.0000 0.929213 0.464606 0.885517i \(-0.346196\pi\)
0.464606 + 0.885517i \(0.346196\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 9.65685 0.636754
\(231\) 0 0
\(232\) 21.3137 1.39931
\(233\) 18.8284 1.23349 0.616746 0.787163i \(-0.288451\pi\)
0.616746 + 0.787163i \(0.288451\pi\)
\(234\) 0 0
\(235\) 4.00000 0.260931
\(236\) 15.3137 0.996838
\(237\) 0 0
\(238\) 2.34315 0.151884
\(239\) −17.6569 −1.14213 −0.571063 0.820906i \(-0.693469\pi\)
−0.571063 + 0.820906i \(0.693469\pi\)
\(240\) 0 0
\(241\) −12.3431 −0.795092 −0.397546 0.917582i \(-0.630138\pi\)
−0.397546 + 0.917582i \(0.630138\pi\)
\(242\) 2.41421 0.155192
\(243\) 0 0
\(244\) −44.6274 −2.85698
\(245\) −6.31371 −0.403368
\(246\) 0 0
\(247\) 38.6274 2.45780
\(248\) 0 0
\(249\) 0 0
\(250\) 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
\(255\) 0 0
\(256\) −29.9706 −1.87316
\(257\) 16.3431 1.01946 0.509729 0.860335i \(-0.329746\pi\)
0.509729 + 0.860335i \(0.329746\pi\)
\(258\) 0 0
\(259\) 9.65685 0.600048
\(260\) −21.6569 −1.34310
\(261\) 0 0
\(262\) −8.00000 −0.494242
\(263\) −18.0000 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) 0 0
\(265\) −9.31371 −0.572137
\(266\) −13.6569 −0.837355
\(267\) 0 0
\(268\) −21.6569 −1.32290
\(269\) −20.6274 −1.25768 −0.628838 0.777536i \(-0.716469\pi\)
−0.628838 + 0.777536i \(0.716469\pi\)
\(270\) 0 0
\(271\) −11.7990 −0.716738 −0.358369 0.933580i \(-0.616667\pi\)
−0.358369 + 0.933580i \(0.616667\pi\)
\(272\) 3.51472 0.213111
\(273\) 0 0
\(274\) 32.1421 1.94178
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −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
\(280\) 3.65685 0.218539
\(281\) 11.1716 0.666440 0.333220 0.942849i \(-0.391865\pi\)
0.333220 + 0.942849i \(0.391865\pi\)
\(282\) 0 0
\(283\) −8.82843 −0.524796 −0.262398 0.964960i \(-0.584513\pi\)
−0.262398 + 0.964960i \(0.584513\pi\)
\(284\) −8.97056 −0.532305
\(285\) 0 0
\(286\) −13.6569 −0.807547
\(287\) −4.00000 −0.236113
\(288\) 0 0
\(289\) −15.6274 −0.919260
\(290\) 11.6569 0.684514
\(291\) 0 0
\(292\) 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
\(297\) 0 0
\(298\) −3.65685 −0.211836
\(299\) −22.6274 −1.30858
\(300\) 0 0
\(301\) −7.31371 −0.421555
\(302\) 39.7990 2.29017
\(303\) 0 0
\(304\) −20.4853 −1.17491
\(305\) −11.6569 −0.667470
\(306\) 0 0
\(307\) −3.17157 −0.181011 −0.0905056 0.995896i \(-0.528848\pi\)
−0.0905056 + 0.995896i \(0.528848\pi\)
\(308\) 3.17157 0.180717
\(309\) 0 0
\(310\) 0 0
\(311\) 3.31371 0.187903 0.0939516 0.995577i \(-0.470050\pi\)
0.0939516 + 0.995577i \(0.470050\pi\)
\(312\) 0 0
\(313\) 15.6569 0.884978 0.442489 0.896774i \(-0.354096\pi\)
0.442489 + 0.896774i \(0.354096\pi\)
\(314\) 43.4558 2.45236
\(315\) 0 0
\(316\) 32.4853 1.82744
\(317\) 26.2843 1.47627 0.738136 0.674652i \(-0.235706\pi\)
0.738136 + 0.674652i \(0.235706\pi\)
\(318\) 0 0
\(319\) 4.82843 0.270340
\(320\) −9.82843 −0.549426
\(321\) 0 0
\(322\) 8.00000 0.445823
\(323\) −8.00000 −0.445132
\(324\) 0 0
\(325\) −5.65685 −0.313786
\(326\) −17.6569 −0.977923
\(327\) 0 0
\(328\) −21.3137 −1.17685
\(329\) 3.31371 0.182691
\(330\) 0 0
\(331\) 6.34315 0.348651 0.174325 0.984688i \(-0.444226\pi\)
0.174325 + 0.984688i \(0.444226\pi\)
\(332\) 38.2843 2.10112
\(333\) 0 0
\(334\) 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
\(339\) 0 0
\(340\) 4.48528 0.243249
\(341\) 0 0
\(342\) 0 0
\(343\) −11.0294 −0.595534
\(344\) −38.9706 −2.10115
\(345\) 0 0
\(346\) −6.82843 −0.367099
\(347\) 29.3137 1.57364 0.786821 0.617181i \(-0.211725\pi\)
0.786821 + 0.617181i \(0.211725\pi\)
\(348\) 0 0
\(349\) 10.9706 0.587241 0.293620 0.955922i \(-0.405140\pi\)
0.293620 + 0.955922i \(0.405140\pi\)
\(350\) 2.00000 0.106904
\(351\) 0 0
\(352\) −1.58579 −0.0845227
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 0 0
\(355\) −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 \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) 27.6274 1.45407
\(362\) −33.7990 −1.77644
\(363\) 0 0
\(364\) −17.9411 −0.940370
\(365\) 11.3137 0.592187
\(366\) 0 0
\(367\) 9.65685 0.504084 0.252042 0.967716i \(-0.418898\pi\)
0.252042 + 0.967716i \(0.418898\pi\)
\(368\) 12.0000 0.625543
\(369\) 0 0
\(370\) 28.1421 1.46304
\(371\) −7.71573 −0.400581
\(372\) 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
\(375\) 0 0
\(376\) 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
\(381\) 0 0
\(382\) −13.6569 −0.698745
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) 0 0
\(385\) 0.828427 0.0422206
\(386\) 32.9706 1.67816
\(387\) 0 0
\(388\) 44.6274 2.26561
\(389\) 23.6569 1.19945 0.599725 0.800206i \(-0.295277\pi\)
0.599725 + 0.800206i \(0.295277\pi\)
\(390\) 0 0
\(391\) 4.68629 0.236996
\(392\) −27.8701 −1.40765
\(393\) 0 0
\(394\) 20.4853 1.03203
\(395\) 8.48528 0.426941
\(396\) 0 0
\(397\) −14.9706 −0.751351 −0.375676 0.926751i \(-0.622589\pi\)
−0.375676 + 0.926751i \(0.622589\pi\)
\(398\) −52.2843 −2.62077
\(399\) 0 0
\(400\) 3.00000 0.150000
\(401\) 6.68629 0.333897 0.166949 0.985966i \(-0.446609\pi\)
0.166949 + 0.985966i \(0.446609\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 3.17157 0.157792
\(405\) 0 0
\(406\) 9.65685 0.479262
\(407\) 11.6569 0.577809
\(408\) 0 0
\(409\) −19.6569 −0.971969 −0.485984 0.873967i \(-0.661539\pi\)
−0.485984 + 0.873967i \(0.661539\pi\)
\(410\) −11.6569 −0.575691
\(411\) 0 0
\(412\) −12.6863 −0.625009
\(413\) 3.31371 0.163057
\(414\) 0 0
\(415\) 10.0000 0.490881
\(416\) 8.97056 0.439818
\(417\) 0 0
\(418\) −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
\(423\) 0 0
\(424\) −41.1127 −1.99661
\(425\) 1.17157 0.0568296
\(426\) 0 0
\(427\) −9.65685 −0.467328
\(428\) −66.2843 −3.20397
\(429\) 0 0
\(430\) −21.3137 −1.02784
\(431\) −21.6569 −1.04317 −0.521587 0.853198i \(-0.674660\pi\)
−0.521587 + 0.853198i \(0.674660\pi\)
\(432\) 0 0
\(433\) −15.6569 −0.752420 −0.376210 0.926534i \(-0.622773\pi\)
−0.376210 + 0.926534i \(0.622773\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 66.2843 3.17444
\(437\) −27.3137 −1.30659
\(438\) 0 0
\(439\) −20.4853 −0.977709 −0.488855 0.872365i \(-0.662585\pi\)
−0.488855 + 0.872365i \(0.662585\pi\)
\(440\) 4.41421 0.210439
\(441\) 0 0
\(442\) −16.0000 −0.761042
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) −3.65685 −0.173352
\(446\) −15.3137 −0.725125
\(447\) 0 0
\(448\) −8.14214 −0.384680
\(449\) −30.9706 −1.46159 −0.730796 0.682596i \(-0.760851\pi\)
−0.730796 + 0.682596i \(0.760851\pi\)
\(450\) 0 0
\(451\) −4.82843 −0.227362
\(452\) 72.6274 3.41611
\(453\) 0 0
\(454\) 33.7990 1.58627
\(455\) −4.68629 −0.219697
\(456\) 0 0
\(457\) −23.3137 −1.09057 −0.545285 0.838251i \(-0.683578\pi\)
−0.545285 + 0.838251i \(0.683578\pi\)
\(458\) −4.82843 −0.225618
\(459\) 0 0
\(460\) 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 \(0.324613\pi\)
\(468\) 0 0
\(469\) −4.68629 −0.216393
\(470\) 9.65685 0.445437
\(471\) 0 0
\(472\) 17.6569 0.812723
\(473\) −8.82843 −0.405932
\(474\) 0 0
\(475\) −6.82843 −0.313310
\(476\) 3.71573 0.170310
\(477\) 0 0
\(478\) −42.6274 −1.94973
\(479\) −36.9706 −1.68923 −0.844614 0.535376i \(-0.820170\pi\)
−0.844614 + 0.535376i \(0.820170\pi\)
\(480\) 0 0
\(481\) −65.9411 −3.00666
\(482\) −29.7990 −1.35731
\(483\) 0 0
\(484\) 3.82843 0.174019
\(485\) 11.6569 0.529310
\(486\) 0 0
\(487\) −12.9706 −0.587752 −0.293876 0.955844i \(-0.594945\pi\)
−0.293876 + 0.955844i \(0.594945\pi\)
\(488\) −51.4558 −2.32930
\(489\) 0 0
\(490\) −15.2426 −0.688592
\(491\) −14.3431 −0.647297 −0.323649 0.946177i \(-0.604910\pi\)
−0.323649 + 0.946177i \(0.604910\pi\)
\(492\) 0 0
\(493\) 5.65685 0.254772
\(494\) 93.2548 4.19573
\(495\) 0 0
\(496\) 0 0
\(497\) −1.94113 −0.0870714
\(498\) 0 0
\(499\) −22.3431 −1.00022 −0.500108 0.865963i \(-0.666706\pi\)
−0.500108 + 0.865963i \(0.666706\pi\)
\(500\) 3.82843 0.171212
\(501\) 0 0
\(502\) −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
\(507\) 0 0
\(508\) −55.4558 −2.46046
\(509\) −18.6863 −0.828255 −0.414128 0.910219i \(-0.635913\pi\)
−0.414128 + 0.910219i \(0.635913\pi\)
\(510\) 0 0
\(511\) 9.37258 0.414619
\(512\) −31.2426 −1.38074
\(513\) 0 0
\(514\) 39.4558 1.74032
\(515\) −3.31371 −0.146019
\(516\) 0 0
\(517\) 4.00000 0.175920
\(518\) 23.3137 1.02435
\(519\) 0 0
\(520\) −24.9706 −1.09503
\(521\) 32.6274 1.42943 0.714717 0.699414i \(-0.246556\pi\)
0.714717 + 0.699414i \(0.246556\pi\)
\(522\) 0 0
\(523\) −9.51472 −0.416050 −0.208025 0.978124i \(-0.566703\pi\)
−0.208025 + 0.978124i \(0.566703\pi\)
\(524\) −12.6863 −0.554203
\(525\) 0 0
\(526\) −43.4558 −1.89476
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −22.4853 −0.976698
\(531\) 0 0
\(532\) −21.6569 −0.938944
\(533\) 27.3137 1.18309
\(534\) 0 0
\(535\) −17.3137 −0.748537
\(536\) −24.9706 −1.07856
\(537\) 0 0
\(538\) −49.7990 −2.14699
\(539\) −6.31371 −0.271951
\(540\) 0 0
\(541\) 17.3137 0.744374 0.372187 0.928158i \(-0.378608\pi\)
0.372187 + 0.928158i \(0.378608\pi\)
\(542\) −28.4853 −1.22355
\(543\) 0 0
\(544\) −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
\(549\) 0 0
\(550\) 2.41421 0.102942
\(551\) −32.9706 −1.40459
\(552\) 0 0
\(553\) 7.02944 0.298922
\(554\) −5.65685 −0.240337
\(555\) 0 0
\(556\) 1.85786 0.0787910
\(557\) −5.17157 −0.219127 −0.109563 0.993980i \(-0.534945\pi\)
−0.109563 + 0.993980i \(0.534945\pi\)
\(558\) 0 0
\(559\) 49.9411 2.11228
\(560\) 2.48528 0.105022
\(561\) 0 0
\(562\) 26.9706 1.13768
\(563\) −31.6569 −1.33418 −0.667089 0.744978i \(-0.732460\pi\)
−0.667089 + 0.744978i \(0.732460\pi\)
\(564\) 0 0
\(565\) 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 \(-0.233310\pi\)
0.743193 + 0.669077i \(0.233310\pi\)
\(570\) 0 0
\(571\) −16.4853 −0.689888 −0.344944 0.938623i \(-0.612102\pi\)
−0.344944 + 0.938623i \(0.612102\pi\)
\(572\) −21.6569 −0.905519
\(573\) 0 0
\(574\) −9.65685 −0.403069
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) −37.7279 −1.56927
\(579\) 0 0
\(580\) 18.4853 0.767560
\(581\) 8.28427 0.343689
\(582\) 0 0
\(583\) −9.31371 −0.385734
\(584\) 49.9411 2.06658
\(585\) 0 0
\(586\) 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
\(591\) 0 0
\(592\) 34.9706 1.43728
\(593\) 22.8284 0.937451 0.468726 0.883344i \(-0.344713\pi\)
0.468726 + 0.883344i \(0.344713\pi\)
\(594\) 0 0
\(595\) 0.970563 0.0397892
\(596\) −5.79899 −0.237536
\(597\) 0 0
\(598\) −54.6274 −2.23388
\(599\) −27.3137 −1.11601 −0.558004 0.829838i \(-0.688433\pi\)
−0.558004 + 0.829838i \(0.688433\pi\)
\(600\) 0 0
\(601\) −5.31371 −0.216751 −0.108375 0.994110i \(-0.534565\pi\)
−0.108375 + 0.994110i \(0.534565\pi\)
\(602\) −17.6569 −0.719640
\(603\) 0 0
\(604\) 63.1127 2.56802
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) 1.51472 0.0614805 0.0307403 0.999527i \(-0.490214\pi\)
0.0307403 + 0.999527i \(0.490214\pi\)
\(608\) 10.8284 0.439151
\(609\) 0 0
\(610\) −28.1421 −1.13944
\(611\) −22.6274 −0.915407
\(612\) 0 0
\(613\) −45.9411 −1.85554 −0.927772 0.373147i \(-0.878279\pi\)
−0.927772 + 0.373147i \(0.878279\pi\)
\(614\) −7.65685 −0.309005
\(615\) 0 0
\(616\) 3.65685 0.147339
\(617\) −0.343146 −0.0138145 −0.00690726 0.999976i \(-0.502199\pi\)
−0.00690726 + 0.999976i \(0.502199\pi\)
\(618\) 0 0
\(619\) −14.3431 −0.576500 −0.288250 0.957555i \(-0.593073\pi\)
−0.288250 + 0.957555i \(0.593073\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 8.00000 0.320771
\(623\) −3.02944 −0.121372
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 37.7990 1.51075
\(627\) 0 0
\(628\) 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
\(633\) 0 0
\(634\) 63.4558 2.52015
\(635\) −14.4853 −0.574831
\(636\) 0 0
\(637\) 35.7157 1.41511
\(638\) 11.6569 0.461499
\(639\) 0 0
\(640\) −20.5563 −0.812561
\(641\) −6.97056 −0.275321 −0.137660 0.990479i \(-0.543958\pi\)
−0.137660 + 0.990479i \(0.543958\pi\)
\(642\) 0 0
\(643\) 37.9411 1.49625 0.748126 0.663557i \(-0.230954\pi\)
0.748126 + 0.663557i \(0.230954\pi\)
\(644\) 12.6863 0.499910
\(645\) 0 0
\(646\) −19.3137 −0.759888
\(647\) −4.68629 −0.184237 −0.0921186 0.995748i \(-0.529364\pi\)
−0.0921186 + 0.995748i \(0.529364\pi\)
\(648\) 0 0
\(649\) 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 \(-0.543550\pi\)
−0.136390 + 0.990655i \(0.543550\pi\)
\(654\) 0 0
\(655\) −3.31371 −0.129477
\(656\) −14.4853 −0.565555
\(657\) 0 0
\(658\) 8.00000 0.311872
\(659\) 15.3137 0.596537 0.298269 0.954482i \(-0.403591\pi\)
0.298269 + 0.954482i \(0.403591\pi\)
\(660\) 0 0
\(661\) 9.31371 0.362261 0.181131 0.983459i \(-0.442024\pi\)
0.181131 + 0.983459i \(0.442024\pi\)
\(662\) 15.3137 0.595184
\(663\) 0 0
\(664\) 44.1421 1.71305
\(665\) −5.65685 −0.219363
\(666\) 0 0
\(667\) 19.3137 0.747830
\(668\) 50.9706 1.97211
\(669\) 0 0
\(670\) −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
\(675\) 0 0
\(676\) 72.7401 2.79770
\(677\) −29.4558 −1.13208 −0.566040 0.824378i \(-0.691525\pi\)
−0.566040 + 0.824378i \(0.691525\pi\)
\(678\) 0 0
\(679\) 9.65685 0.370596
\(680\) 5.17157 0.198321
\(681\) 0 0
\(682\) 0 0
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) 0 0
\(685\) 13.3137 0.508691
\(686\) −26.6274 −1.01664
\(687\) 0 0
\(688\) −26.4853 −1.00974
\(689\) 52.6863 2.00719
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) −10.8284 −0.411635
\(693\) 0 0
\(694\) 70.7696 2.68638
\(695\) 0.485281 0.0184078
\(696\) 0 0
\(697\) −5.65685 −0.214269
\(698\) 26.4853 1.00248
\(699\) 0 0
\(700\) 3.17157 0.119874
\(701\) −36.1421 −1.36507 −0.682535 0.730853i \(-0.739122\pi\)
−0.682535 + 0.730853i \(0.739122\pi\)
\(702\) 0 0
\(703\) −79.5980 −3.00209
\(704\) −9.82843 −0.370423
\(705\) 0 0
\(706\) −62.7696 −2.36236
\(707\) 0.686292 0.0258106
\(708\) 0 0
\(709\) 6.68629 0.251109 0.125554 0.992087i \(-0.459929\pi\)
0.125554 + 0.992087i \(0.459929\pi\)
\(710\) −5.65685 −0.212298
\(711\) 0 0
\(712\) −16.1421 −0.604952
\(713\) 0 0
\(714\) 0 0
\(715\) −5.65685 −0.211554
\(716\) 67.5980 2.52626
\(717\) 0 0
\(718\) −28.9706 −1.08117
\(719\) 47.5980 1.77511 0.887553 0.460706i \(-0.152404\pi\)
0.887553 + 0.460706i \(0.152404\pi\)
\(720\) 0 0
\(721\) −2.74517 −0.102235
\(722\) 66.6985 2.48226
\(723\) 0 0
\(724\) −53.5980 −1.99195
\(725\) 4.82843 0.179323
\(726\) 0 0
\(727\) 33.9411 1.25881 0.629403 0.777079i \(-0.283299\pi\)
0.629403 + 0.777079i \(0.283299\pi\)
\(728\) −20.6863 −0.766685
\(729\) 0 0
\(730\) 27.3137 1.01093
\(731\) −10.3431 −0.382555
\(732\) 0 0
\(733\) −6.34315 −0.234289 −0.117145 0.993115i \(-0.537374\pi\)
−0.117145 + 0.993115i \(0.537374\pi\)
\(734\) 23.3137 0.860525
\(735\) 0 0
\(736\) −6.34315 −0.233811
\(737\) −5.65685 −0.208373
\(738\) 0 0
\(739\) −15.1127 −0.555930 −0.277965 0.960591i \(-0.589660\pi\)
−0.277965 + 0.960591i \(0.589660\pi\)
\(740\) 44.6274 1.64054
\(741\) 0 0
\(742\) −18.6274 −0.683834
\(743\) −36.3431 −1.33330 −0.666650 0.745371i \(-0.732273\pi\)
−0.666650 + 0.745371i \(0.732273\pi\)
\(744\) 0 0
\(745\) −1.51472 −0.0554950
\(746\) 25.6569 0.939364
\(747\) 0 0
\(748\) 4.48528 0.163998
\(749\) −14.3431 −0.524087
\(750\) 0 0
\(751\) 20.2843 0.740184 0.370092 0.928995i \(-0.379326\pi\)
0.370092 + 0.928995i \(0.379326\pi\)
\(752\) 12.0000 0.437595
\(753\) 0 0
\(754\) −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
\(759\) 0 0
\(760\) −30.1421 −1.09337
\(761\) 28.8284 1.04503 0.522515 0.852630i \(-0.324994\pi\)
0.522515 + 0.852630i \(0.324994\pi\)
\(762\) 0 0
\(763\) 14.3431 0.519257
\(764\) −21.6569 −0.783517
\(765\) 0 0
\(766\) 19.3137 0.697833
\(767\) −22.6274 −0.817029
\(768\) 0 0
\(769\) 10.6863 0.385358 0.192679 0.981262i \(-0.438282\pi\)
0.192679 + 0.981262i \(0.438282\pi\)
\(770\) 2.00000 0.0720750
\(771\) 0 0
\(772\) 52.2843 1.88175
\(773\) −3.65685 −0.131528 −0.0657640 0.997835i \(-0.520948\pi\)
−0.0657640 + 0.997835i \(0.520948\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 51.4558 1.84716
\(777\) 0 0
\(778\) 57.1127 2.04759
\(779\) 32.9706 1.18129
\(780\) 0 0
\(781\) −2.34315 −0.0838443
\(782\) 11.3137 0.404577
\(783\) 0 0
\(784\) −18.9411 −0.676469
\(785\) 18.0000 0.642448
\(786\) 0 0
\(787\) −20.1421 −0.717990 −0.358995 0.933340i \(-0.616880\pi\)
−0.358995 + 0.933340i \(0.616880\pi\)
\(788\) 32.4853 1.15724
\(789\) 0 0
\(790\) 20.4853 0.728834
\(791\) 15.7157 0.558787
\(792\) 0 0
\(793\) 65.9411 2.34164
\(794\) −36.1421 −1.28264
\(795\) 0 0
\(796\) −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
\(801\) 0 0
\(802\) 16.1421 0.569999
\(803\) 11.3137 0.399252
\(804\) 0 0
\(805\) 3.31371 0.116793
\(806\) 0 0
\(807\) 0 0
\(808\) 3.65685 0.128648
\(809\) −28.4264 −0.999419 −0.499710 0.866193i \(-0.666560\pi\)
−0.499710 + 0.866193i \(0.666560\pi\)
\(810\) 0 0
\(811\) 0.485281 0.0170405 0.00852027 0.999964i \(-0.497288\pi\)
0.00852027 + 0.999964i \(0.497288\pi\)
\(812\) 15.3137 0.537406
\(813\) 0 0
\(814\) 28.1421 0.986381
\(815\) −7.31371 −0.256188
\(816\) 0 0
\(817\) 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 \(-0.428135\pi\)
0.223858 + 0.974622i \(0.428135\pi\)
\(822\) 0 0
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) −14.6274 −0.509570
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) −41.3137 −1.43662 −0.718309 0.695724i \(-0.755084\pi\)
−0.718309 + 0.695724i \(0.755084\pi\)
\(828\) 0 0
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 24.1421 0.837986
\(831\) 0 0
\(832\) 55.5980 1.92751
\(833\) −7.39697 −0.256290
\(834\) 0 0
\(835\) 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
\(843\) 0 0
\(844\) 4.48528 0.154390
\(845\) 19.0000 0.653620
\(846\) 0 0
\(847\) 0.828427 0.0284651
\(848\) −27.9411 −0.959502
\(849\) 0 0
\(850\) 2.82843 0.0970143
\(851\) 46.6274 1.59837
\(852\) 0 0
\(853\) −8.68629 −0.297413 −0.148706 0.988881i \(-0.547511\pi\)
−0.148706 + 0.988881i \(0.547511\pi\)
\(854\) −23.3137 −0.797779
\(855\) 0 0
\(856\) −76.4264 −2.61220
\(857\) 28.4853 0.973039 0.486519 0.873670i \(-0.338266\pi\)
0.486519 + 0.873670i \(0.338266\pi\)
\(858\) 0 0
\(859\) −52.9706 −1.80733 −0.903666 0.428238i \(-0.859135\pi\)
−0.903666 + 0.428238i \(0.859135\pi\)
\(860\) −33.7990 −1.15254
\(861\) 0 0
\(862\) −52.2843 −1.78081
\(863\) 20.6863 0.704170 0.352085 0.935968i \(-0.385473\pi\)
0.352085 + 0.935968i \(0.385473\pi\)
\(864\) 0 0
\(865\) −2.82843 −0.0961694
\(866\) −37.7990 −1.28446
\(867\) 0 0
\(868\) 0 0
\(869\) 8.48528 0.287843
\(870\) 0 0
\(871\) 32.0000 1.08428
\(872\) 76.4264 2.58812
\(873\) 0 0
\(874\) −65.9411 −2.23049
\(875\) 0.828427 0.0280059
\(876\) 0 0
\(877\) 2.62742 0.0887216 0.0443608 0.999016i \(-0.485875\pi\)
0.0443608 + 0.999016i \(0.485875\pi\)
\(878\) −49.4558 −1.66905
\(879\) 0 0
\(880\) 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
\(885\) 0 0
\(886\) −28.9706 −0.973285
\(887\) 15.6569 0.525706 0.262853 0.964836i \(-0.415337\pi\)
0.262853 + 0.964836i \(0.415337\pi\)
\(888\) 0 0
\(889\) −12.0000 −0.402467
\(890\) −8.82843 −0.295930
\(891\) 0 0
\(892\) −24.2843 −0.813098
\(893\) −27.3137 −0.914018
\(894\) 0 0
\(895\) 17.6569 0.590204
\(896\) −17.0294 −0.568914
\(897\) 0 0
\(898\) −74.7696 −2.49509
\(899\) 0 0
\(900\) 0 0
\(901\) −10.9117 −0.363521
\(902\) −11.6569 −0.388131
\(903\) 0 0
\(904\) 83.7401 2.78515
\(905\) −14.0000 −0.465376
\(906\) 0 0
\(907\) 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 0
\(910\) −11.3137 −0.375046
\(911\) 48.9706 1.62247 0.811234 0.584722i \(-0.198797\pi\)
0.811234 + 0.584722i \(0.198797\pi\)
\(912\) 0 0
\(913\) 10.0000 0.330952
\(914\) −56.2843 −1.86172
\(915\) 0 0
\(916\) −7.65685 −0.252990
\(917\) −2.74517 −0.0906534
\(918\) 0 0
\(919\) 11.5147 0.379836 0.189918 0.981800i \(-0.439178\pi\)
0.189918 + 0.981800i \(0.439178\pi\)
\(920\) 17.6569 0.582129
\(921\) 0 0
\(922\) −0.343146 −0.0113009
\(923\) 13.2548 0.436288
\(924\) 0 0
\(925\) 11.6569 0.383275
\(926\) 12.0000 0.394344
\(927\) 0 0
\(928\) −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
\(931\) 43.1127 1.41296
\(932\) 72.0833 2.36117
\(933\) 0 0
\(934\) 54.6274 1.78746
\(935\) 1.17157 0.0383145
\(936\) 0 0
\(937\) 11.0294 0.360316 0.180158 0.983638i \(-0.442339\pi\)
0.180158 + 0.983638i \(0.442339\pi\)
\(938\) −11.3137 −0.369406
\(939\) 0 0
\(940\) 15.3137 0.499478
\(941\) 34.7696 1.13346 0.566728 0.823905i \(-0.308209\pi\)
0.566728 + 0.823905i \(0.308209\pi\)
\(942\) 0 0
\(943\) −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 \(-0.534343\pi\)
−0.107681 + 0.994185i \(0.534343\pi\)
\(948\) 0 0
\(949\) −64.0000 −2.07753
\(950\) −16.4853 −0.534853
\(951\) 0 0
\(952\) 4.28427 0.138854
\(953\) 11.7990 0.382207 0.191103 0.981570i \(-0.438793\pi\)
0.191103 + 0.981570i \(0.438793\pi\)
\(954\) 0 0
\(955\) −5.65685 −0.183052
\(956\) −67.5980 −2.18627
\(957\) 0 0
\(958\) −89.2548 −2.88369
\(959\) 11.0294 0.356159
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −159.196 −5.13268
\(963\) 0 0
\(964\) −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
\(969\) 0 0
\(970\) 28.1421 0.903590
\(971\) 34.6274 1.11125 0.555623 0.831434i \(-0.312480\pi\)
0.555623 + 0.831434i \(0.312480\pi\)
\(972\) 0 0
\(973\) 0.402020 0.0128882
\(974\) −31.3137 −1.00336
\(975\) 0 0
\(976\) −34.9706 −1.11938
\(977\) −2.68629 −0.0859421 −0.0429710 0.999076i \(-0.513682\pi\)
−0.0429710 + 0.999076i \(0.513682\pi\)
\(978\) 0 0
\(979\) −3.65685 −0.116874
\(980\) −24.1716 −0.772133
\(981\) 0 0
\(982\) −34.6274 −1.10501
\(983\) 30.6274 0.976863 0.488431 0.872602i \(-0.337569\pi\)
0.488431 + 0.872602i \(0.337569\pi\)
\(984\) 0 0
\(985\) 8.48528 0.270364
\(986\) 13.6569 0.434923
\(987\) 0 0
\(988\) 147.882 4.70476
\(989\) −35.3137 −1.12291
\(990\) 0 0
\(991\) −30.6274 −0.972912 −0.486456 0.873705i \(-0.661711\pi\)
−0.486456 + 0.873705i \(0.661711\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −4.68629 −0.148640
\(995\) −21.6569 −0.686568
\(996\) 0 0
\(997\) −39.3137 −1.24508 −0.622539 0.782589i \(-0.713899\pi\)
−0.622539 + 0.782589i \(0.713899\pi\)
\(998\) −53.9411 −1.70748
\(999\) 0 0
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 495.2.a.d.1.2 2
3.2 odd 2 165.2.a.a.1.1 2
4.3 odd 2 7920.2.a.cg.1.1 2
5.2 odd 4 2475.2.c.m.199.4 4
5.3 odd 4 2475.2.c.m.199.1 4
5.4 even 2 2475.2.a.m.1.1 2
11.10 odd 2 5445.2.a.m.1.1 2
12.11 even 2 2640.2.a.bb.1.1 2
15.2 even 4 825.2.c.e.199.1 4
15.8 even 4 825.2.c.e.199.4 4
15.14 odd 2 825.2.a.g.1.2 2
21.20 even 2 8085.2.a.ba.1.1 2
33.32 even 2 1815.2.a.k.1.2 2
165.164 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 3.2 odd 2
495.2.a.d.1.2 2 1.1 even 1 trivial
825.2.a.g.1.2 2 15.14 odd 2
825.2.c.e.199.1 4 15.2 even 4
825.2.c.e.199.4 4 15.8 even 4
1815.2.a.k.1.2 2 33.32 even 2
2475.2.a.m.1.1 2 5.4 even 2
2475.2.c.m.199.1 4 5.3 odd 4
2475.2.c.m.199.4 4 5.2 odd 4
2640.2.a.bb.1.1 2 12.11 even 2
5445.2.a.m.1.1 2 11.10 odd 2
7920.2.a.cg.1.1 2 4.3 odd 2
8085.2.a.ba.1.1 2 21.20 even 2
9075.2.a.v.1.1 2 165.164 even 2