Properties

Label 4600.2.e.v.4049.10
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \(x^{10} + 18 x^{8} + 103 x^{6} + 239 x^{4} + 197 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.10
Root \(-0.144312i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.v.4049.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.08344i q^{3} +0.555022i q^{7} -6.50759 q^{9} +O(q^{10})\) \(q+3.08344i q^{3} +0.555022i q^{7} -6.50759 q^{9} +4.65190 q^{11} +5.02256i q^{13} -1.32867i q^{17} +0.196402 q^{19} -1.71138 q^{21} +1.00000i q^{23} -10.8154i q^{27} +0.812298 q^{29} -2.11145 q^{31} +14.3439i q^{33} +5.64564i q^{37} -15.4868 q^{39} -4.89714 q^{41} +1.66507i q^{43} +9.89310i q^{47} +6.69195 q^{49} +4.09688 q^{51} +2.23261i q^{53} +0.605593i q^{57} -2.43488 q^{59} -5.71138 q^{61} -3.61185i q^{63} -6.91830i q^{67} -3.08344 q^{69} +0.0120411 q^{71} +15.2989i q^{73} +2.58191i q^{77} -10.6351 q^{79} +13.8260 q^{81} -5.64020i q^{83} +2.50467i q^{87} -9.06693 q^{89} -2.78763 q^{91} -6.51051i q^{93} +14.0720i q^{97} -30.2727 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 8 q^{9} + O(q^{10}) \) \( 10 q - 8 q^{9} - 8 q^{11} - 8 q^{19} - 12 q^{21} + 22 q^{29} + 8 q^{31} - 62 q^{39} - 16 q^{41} + 4 q^{49} - 10 q^{51} - 46 q^{59} - 52 q^{61} - 6 q^{69} - 4 q^{71} - 86 q^{79} - 6 q^{81} - 30 q^{89} - 38 q^{91} - 74 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 0
\(3\) 3.08344i 1.78022i 0.455742 + 0.890112i \(0.349374\pi\)
−0.455742 + 0.890112i \(0.650626\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.555022i 0.209779i 0.994484 + 0.104889i \(0.0334488\pi\)
−0.994484 + 0.104889i \(0.966551\pi\)
\(8\) 0 0
\(9\) −6.50759 −2.16920
\(10\) 0 0
\(11\) 4.65190 1.40260 0.701301 0.712866i \(-0.252603\pi\)
0.701301 + 0.712866i \(0.252603\pi\)
\(12\) 0 0
\(13\) 5.02256i 1.39301i 0.717553 + 0.696504i \(0.245262\pi\)
−0.717553 + 0.696504i \(0.754738\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 1.32867i − 0.322251i −0.986934 0.161125i \(-0.948488\pi\)
0.986934 0.161125i \(-0.0515123\pi\)
\(18\) 0 0
\(19\) 0.196402 0.0450577 0.0225288 0.999746i \(-0.492828\pi\)
0.0225288 + 0.999746i \(0.492828\pi\)
\(20\) 0 0
\(21\) −1.71138 −0.373453
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 10.8154i − 2.08143i
\(28\) 0 0
\(29\) 0.812298 0.150840 0.0754200 0.997152i \(-0.475970\pi\)
0.0754200 + 0.997152i \(0.475970\pi\)
\(30\) 0 0
\(31\) −2.11145 −0.379227 −0.189613 0.981859i \(-0.560723\pi\)
−0.189613 + 0.981859i \(0.560723\pi\)
\(32\) 0 0
\(33\) 14.3439i 2.49694i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.64564i 0.928138i 0.885799 + 0.464069i \(0.153611\pi\)
−0.885799 + 0.464069i \(0.846389\pi\)
\(38\) 0 0
\(39\) −15.4868 −2.47987
\(40\) 0 0
\(41\) −4.89714 −0.764804 −0.382402 0.923996i \(-0.624903\pi\)
−0.382402 + 0.923996i \(0.624903\pi\)
\(42\) 0 0
\(43\) 1.66507i 0.253920i 0.991908 + 0.126960i \(0.0405220\pi\)
−0.991908 + 0.126960i \(0.959478\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.89310i 1.44306i 0.692385 + 0.721528i \(0.256560\pi\)
−0.692385 + 0.721528i \(0.743440\pi\)
\(48\) 0 0
\(49\) 6.69195 0.955993
\(50\) 0 0
\(51\) 4.09688 0.573678
\(52\) 0 0
\(53\) 2.23261i 0.306673i 0.988174 + 0.153336i \(0.0490018\pi\)
−0.988174 + 0.153336i \(0.950998\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.605593i 0.0802127i
\(58\) 0 0
\(59\) −2.43488 −0.316994 −0.158497 0.987359i \(-0.550665\pi\)
−0.158497 + 0.987359i \(0.550665\pi\)
\(60\) 0 0
\(61\) −5.71138 −0.731267 −0.365633 0.930759i \(-0.619148\pi\)
−0.365633 + 0.930759i \(0.619148\pi\)
\(62\) 0 0
\(63\) − 3.61185i − 0.455051i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 6.91830i − 0.845205i −0.906315 0.422602i \(-0.861117\pi\)
0.906315 0.422602i \(-0.138883\pi\)
\(68\) 0 0
\(69\) −3.08344 −0.371202
\(70\) 0 0
\(71\) 0.0120411 0.00142902 0.000714510 1.00000i \(-0.499773\pi\)
0.000714510 1.00000i \(0.499773\pi\)
\(72\) 0 0
\(73\) 15.2989i 1.79061i 0.445458 + 0.895303i \(0.353041\pi\)
−0.445458 + 0.895303i \(0.646959\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.58191i 0.294236i
\(78\) 0 0
\(79\) −10.6351 −1.19654 −0.598272 0.801293i \(-0.704146\pi\)
−0.598272 + 0.801293i \(0.704146\pi\)
\(80\) 0 0
\(81\) 13.8260 1.53622
\(82\) 0 0
\(83\) − 5.64020i − 0.619092i −0.950884 0.309546i \(-0.899823\pi\)
0.950884 0.309546i \(-0.100177\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.50467i 0.268529i
\(88\) 0 0
\(89\) −9.06693 −0.961093 −0.480547 0.876969i \(-0.659562\pi\)
−0.480547 + 0.876969i \(0.659562\pi\)
\(90\) 0 0
\(91\) −2.78763 −0.292223
\(92\) 0 0
\(93\) − 6.51051i − 0.675108i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.0720i 1.42880i 0.699739 + 0.714398i \(0.253299\pi\)
−0.699739 + 0.714398i \(0.746701\pi\)
\(98\) 0 0
\(99\) −30.2727 −3.04252
\(100\) 0 0
\(101\) 7.53015 0.749278 0.374639 0.927171i \(-0.377767\pi\)
0.374639 + 0.927171i \(0.377767\pi\)
\(102\) 0 0
\(103\) − 14.4476i − 1.42357i −0.702399 0.711784i \(-0.747888\pi\)
0.702399 0.711784i \(-0.252112\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 2.35862i − 0.228016i −0.993480 0.114008i \(-0.963631\pi\)
0.993480 0.114008i \(-0.0363690\pi\)
\(108\) 0 0
\(109\) 10.3578 0.992101 0.496050 0.868294i \(-0.334783\pi\)
0.496050 + 0.868294i \(0.334783\pi\)
\(110\) 0 0
\(111\) −17.4080 −1.65229
\(112\) 0 0
\(113\) − 19.2723i − 1.81298i −0.422226 0.906491i \(-0.638751\pi\)
0.422226 0.906491i \(-0.361249\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 32.6848i − 3.02171i
\(118\) 0 0
\(119\) 0.737442 0.0676012
\(120\) 0 0
\(121\) 10.6402 0.967291
\(122\) 0 0
\(123\) − 15.1000i − 1.36152i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 18.4934i 1.64102i 0.571632 + 0.820510i \(0.306311\pi\)
−0.571632 + 0.820510i \(0.693689\pi\)
\(128\) 0 0
\(129\) −5.13413 −0.452035
\(130\) 0 0
\(131\) −14.7727 −1.29070 −0.645351 0.763887i \(-0.723289\pi\)
−0.645351 + 0.763887i \(0.723289\pi\)
\(132\) 0 0
\(133\) 0.109007i 0.00945213i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 11.6173i − 0.992533i −0.868170 0.496266i \(-0.834704\pi\)
0.868170 0.496266i \(-0.165296\pi\)
\(138\) 0 0
\(139\) −11.1389 −0.944787 −0.472393 0.881388i \(-0.656610\pi\)
−0.472393 + 0.881388i \(0.656610\pi\)
\(140\) 0 0
\(141\) −30.5047 −2.56896
\(142\) 0 0
\(143\) 23.3645i 1.95384i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 20.6342i 1.70188i
\(148\) 0 0
\(149\) 12.4796 1.02237 0.511184 0.859472i \(-0.329207\pi\)
0.511184 + 0.859472i \(0.329207\pi\)
\(150\) 0 0
\(151\) 1.32673 0.107968 0.0539840 0.998542i \(-0.482808\pi\)
0.0539840 + 0.998542i \(0.482808\pi\)
\(152\) 0 0
\(153\) 8.64646i 0.699025i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 23.1754i − 1.84960i −0.380458 0.924798i \(-0.624234\pi\)
0.380458 0.924798i \(-0.375766\pi\)
\(158\) 0 0
\(159\) −6.88412 −0.545946
\(160\) 0 0
\(161\) −0.555022 −0.0437418
\(162\) 0 0
\(163\) 25.0682i 1.96349i 0.190196 + 0.981746i \(0.439088\pi\)
−0.190196 + 0.981746i \(0.560912\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.39603i 0.340175i 0.985429 + 0.170087i \(0.0544050\pi\)
−0.985429 + 0.170087i \(0.945595\pi\)
\(168\) 0 0
\(169\) −12.2261 −0.940473
\(170\) 0 0
\(171\) −1.27810 −0.0977389
\(172\) 0 0
\(173\) 19.4211i 1.47656i 0.674493 + 0.738281i \(0.264362\pi\)
−0.674493 + 0.738281i \(0.735638\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 7.50779i − 0.564320i
\(178\) 0 0
\(179\) −10.4169 −0.778592 −0.389296 0.921113i \(-0.627282\pi\)
−0.389296 + 0.921113i \(0.627282\pi\)
\(180\) 0 0
\(181\) −9.13693 −0.679143 −0.339571 0.940580i \(-0.610282\pi\)
−0.339571 + 0.940580i \(0.610282\pi\)
\(182\) 0 0
\(183\) − 17.6107i − 1.30182i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 6.18086i − 0.451989i
\(188\) 0 0
\(189\) 6.00280 0.436640
\(190\) 0 0
\(191\) −1.49819 −0.108405 −0.0542026 0.998530i \(-0.517262\pi\)
−0.0542026 + 0.998530i \(0.517262\pi\)
\(192\) 0 0
\(193\) 14.4113i 1.03735i 0.854972 + 0.518675i \(0.173575\pi\)
−0.854972 + 0.518675i \(0.826425\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 24.4686i − 1.74331i −0.490116 0.871657i \(-0.663046\pi\)
0.490116 0.871657i \(-0.336954\pi\)
\(198\) 0 0
\(199\) 1.61185 0.114261 0.0571307 0.998367i \(-0.481805\pi\)
0.0571307 + 0.998367i \(0.481805\pi\)
\(200\) 0 0
\(201\) 21.3321 1.50465
\(202\) 0 0
\(203\) 0.450843i 0.0316430i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 6.50759i − 0.452309i
\(208\) 0 0
\(209\) 0.913642 0.0631979
\(210\) 0 0
\(211\) 6.25081 0.430324 0.215162 0.976578i \(-0.430972\pi\)
0.215162 + 0.976578i \(0.430972\pi\)
\(212\) 0 0
\(213\) 0.0371281i 0.00254398i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 1.17190i − 0.0795536i
\(218\) 0 0
\(219\) −47.1733 −3.18768
\(220\) 0 0
\(221\) 6.67334 0.448898
\(222\) 0 0
\(223\) 21.2953i 1.42604i 0.701144 + 0.713019i \(0.252673\pi\)
−0.701144 + 0.713019i \(0.747327\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 10.5174i − 0.698061i −0.937111 0.349031i \(-0.886511\pi\)
0.937111 0.349031i \(-0.113489\pi\)
\(228\) 0 0
\(229\) 24.0063 1.58638 0.793190 0.608975i \(-0.208419\pi\)
0.793190 + 0.608975i \(0.208419\pi\)
\(230\) 0 0
\(231\) −7.96115 −0.523805
\(232\) 0 0
\(233\) − 11.3112i − 0.741021i −0.928828 0.370510i \(-0.879183\pi\)
0.928828 0.370510i \(-0.120817\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 32.7927i − 2.13012i
\(238\) 0 0
\(239\) 25.5592 1.65329 0.826644 0.562726i \(-0.190247\pi\)
0.826644 + 0.562726i \(0.190247\pi\)
\(240\) 0 0
\(241\) −16.1617 −1.04106 −0.520532 0.853842i \(-0.674266\pi\)
−0.520532 + 0.853842i \(0.674266\pi\)
\(242\) 0 0
\(243\) 10.1852i 0.653380i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0.986440i 0.0627657i
\(248\) 0 0
\(249\) 17.3912 1.10212
\(250\) 0 0
\(251\) −13.3497 −0.842627 −0.421313 0.906915i \(-0.638431\pi\)
−0.421313 + 0.906915i \(0.638431\pi\)
\(252\) 0 0
\(253\) 4.65190i 0.292463i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 10.1834i − 0.635222i −0.948221 0.317611i \(-0.897119\pi\)
0.948221 0.317611i \(-0.102881\pi\)
\(258\) 0 0
\(259\) −3.13345 −0.194703
\(260\) 0 0
\(261\) −5.28610 −0.327201
\(262\) 0 0
\(263\) − 23.3073i − 1.43719i −0.695430 0.718594i \(-0.744786\pi\)
0.695430 0.718594i \(-0.255214\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 27.9573i − 1.71096i
\(268\) 0 0
\(269\) −9.80817 −0.598015 −0.299007 0.954251i \(-0.596656\pi\)
−0.299007 + 0.954251i \(0.596656\pi\)
\(270\) 0 0
\(271\) −6.08332 −0.369535 −0.184768 0.982782i \(-0.559153\pi\)
−0.184768 + 0.982782i \(0.559153\pi\)
\(272\) 0 0
\(273\) − 8.59549i − 0.520223i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 0.0445722i − 0.00267809i −0.999999 0.00133904i \(-0.999574\pi\)
0.999999 0.00133904i \(-0.000426231\pi\)
\(278\) 0 0
\(279\) 13.7404 0.822617
\(280\) 0 0
\(281\) 27.9996 1.67032 0.835158 0.550010i \(-0.185376\pi\)
0.835158 + 0.550010i \(0.185376\pi\)
\(282\) 0 0
\(283\) − 12.1798i − 0.724013i −0.932176 0.362006i \(-0.882092\pi\)
0.932176 0.362006i \(-0.117908\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 2.71802i − 0.160440i
\(288\) 0 0
\(289\) 15.2346 0.896155
\(290\) 0 0
\(291\) −43.3902 −2.54358
\(292\) 0 0
\(293\) 2.71056i 0.158352i 0.996861 + 0.0791762i \(0.0252290\pi\)
−0.996861 + 0.0791762i \(0.974771\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 50.3124i − 2.91942i
\(298\) 0 0
\(299\) −5.02256 −0.290462
\(300\) 0 0
\(301\) −0.924148 −0.0532670
\(302\) 0 0
\(303\) 23.2188i 1.33388i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 32.1631i − 1.83564i −0.396994 0.917821i \(-0.629947\pi\)
0.396994 0.917821i \(-0.370053\pi\)
\(308\) 0 0
\(309\) 44.5484 2.53427
\(310\) 0 0
\(311\) −4.07951 −0.231328 −0.115664 0.993288i \(-0.536900\pi\)
−0.115664 + 0.993288i \(0.536900\pi\)
\(312\) 0 0
\(313\) − 13.7806i − 0.778925i −0.921042 0.389462i \(-0.872661\pi\)
0.921042 0.389462i \(-0.127339\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.4630i 0.587658i 0.955858 + 0.293829i \(0.0949297\pi\)
−0.955858 + 0.293829i \(0.905070\pi\)
\(318\) 0 0
\(319\) 3.77873 0.211568
\(320\) 0 0
\(321\) 7.27266 0.405920
\(322\) 0 0
\(323\) − 0.260954i − 0.0145199i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 31.9377i 1.76616i
\(328\) 0 0
\(329\) −5.49088 −0.302722
\(330\) 0 0
\(331\) −12.6451 −0.695038 −0.347519 0.937673i \(-0.612976\pi\)
−0.347519 + 0.937673i \(0.612976\pi\)
\(332\) 0 0
\(333\) − 36.7395i − 2.01331i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 24.5733i 1.33859i 0.742997 + 0.669295i \(0.233404\pi\)
−0.742997 + 0.669295i \(0.766596\pi\)
\(338\) 0 0
\(339\) 59.4248 3.22751
\(340\) 0 0
\(341\) −9.82224 −0.531904
\(342\) 0 0
\(343\) 7.59933i 0.410325i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 3.50569i − 0.188195i −0.995563 0.0940977i \(-0.970003\pi\)
0.995563 0.0940977i \(-0.0299966\pi\)
\(348\) 0 0
\(349\) 18.1085 0.969327 0.484664 0.874701i \(-0.338942\pi\)
0.484664 + 0.874701i \(0.338942\pi\)
\(350\) 0 0
\(351\) 54.3212 2.89945
\(352\) 0 0
\(353\) 20.1915i 1.07469i 0.843364 + 0.537343i \(0.180572\pi\)
−0.843364 + 0.537343i \(0.819428\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 2.27386i 0.120345i
\(358\) 0 0
\(359\) 9.43409 0.497912 0.248956 0.968515i \(-0.419912\pi\)
0.248956 + 0.968515i \(0.419912\pi\)
\(360\) 0 0
\(361\) −18.9614 −0.997970
\(362\) 0 0
\(363\) 32.8084i 1.72199i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 30.3412i − 1.58380i −0.610652 0.791899i \(-0.709092\pi\)
0.610652 0.791899i \(-0.290908\pi\)
\(368\) 0 0
\(369\) 31.8686 1.65901
\(370\) 0 0
\(371\) −1.23915 −0.0643333
\(372\) 0 0
\(373\) − 14.7893i − 0.765758i −0.923798 0.382879i \(-0.874933\pi\)
0.923798 0.382879i \(-0.125067\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.07982i 0.210121i
\(378\) 0 0
\(379\) 11.2405 0.577385 0.288693 0.957422i \(-0.406779\pi\)
0.288693 + 0.957422i \(0.406779\pi\)
\(380\) 0 0
\(381\) −57.0231 −2.92138
\(382\) 0 0
\(383\) 30.4602i 1.55644i 0.627991 + 0.778221i \(0.283878\pi\)
−0.627991 + 0.778221i \(0.716122\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 10.8356i − 0.550803i
\(388\) 0 0
\(389\) −1.94011 −0.0983673 −0.0491836 0.998790i \(-0.515662\pi\)
−0.0491836 + 0.998790i \(0.515662\pi\)
\(390\) 0 0
\(391\) 1.32867 0.0671939
\(392\) 0 0
\(393\) − 45.5509i − 2.29774i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 18.3206i 0.919487i 0.888052 + 0.459743i \(0.152059\pi\)
−0.888052 + 0.459743i \(0.847941\pi\)
\(398\) 0 0
\(399\) −0.336117 −0.0168269
\(400\) 0 0
\(401\) −1.32199 −0.0660170 −0.0330085 0.999455i \(-0.510509\pi\)
−0.0330085 + 0.999455i \(0.510509\pi\)
\(402\) 0 0
\(403\) − 10.6049i − 0.528266i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 26.2630i 1.30181i
\(408\) 0 0
\(409\) −21.9963 −1.08765 −0.543823 0.839200i \(-0.683024\pi\)
−0.543823 + 0.839200i \(0.683024\pi\)
\(410\) 0 0
\(411\) 35.8212 1.76693
\(412\) 0 0
\(413\) − 1.35141i − 0.0664985i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 34.3460i − 1.68193i
\(418\) 0 0
\(419\) −28.3194 −1.38349 −0.691746 0.722141i \(-0.743158\pi\)
−0.691746 + 0.722141i \(0.743158\pi\)
\(420\) 0 0
\(421\) 14.7650 0.719602 0.359801 0.933029i \(-0.382845\pi\)
0.359801 + 0.933029i \(0.382845\pi\)
\(422\) 0 0
\(423\) − 64.3802i − 3.13027i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 3.16994i − 0.153404i
\(428\) 0 0
\(429\) −72.0429 −3.47826
\(430\) 0 0
\(431\) −27.0514 −1.30302 −0.651510 0.758640i \(-0.725864\pi\)
−0.651510 + 0.758640i \(0.725864\pi\)
\(432\) 0 0
\(433\) − 38.2700i − 1.83914i −0.392926 0.919570i \(-0.628537\pi\)
0.392926 0.919570i \(-0.371463\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0.196402i 0.00939517i
\(438\) 0 0
\(439\) 2.39959 0.114526 0.0572630 0.998359i \(-0.481763\pi\)
0.0572630 + 0.998359i \(0.481763\pi\)
\(440\) 0 0
\(441\) −43.5485 −2.07374
\(442\) 0 0
\(443\) 10.6542i 0.506198i 0.967440 + 0.253099i \(0.0814499\pi\)
−0.967440 + 0.253099i \(0.918550\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 38.4800i 1.82004i
\(448\) 0 0
\(449\) 28.9533 1.36639 0.683195 0.730236i \(-0.260590\pi\)
0.683195 + 0.730236i \(0.260590\pi\)
\(450\) 0 0
\(451\) −22.7810 −1.07272
\(452\) 0 0
\(453\) 4.09090i 0.192207i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 38.5855i 1.80496i 0.430736 + 0.902478i \(0.358254\pi\)
−0.430736 + 0.902478i \(0.641746\pi\)
\(458\) 0 0
\(459\) −14.3702 −0.670743
\(460\) 0 0
\(461\) −8.07659 −0.376164 −0.188082 0.982153i \(-0.560227\pi\)
−0.188082 + 0.982153i \(0.560227\pi\)
\(462\) 0 0
\(463\) 6.07771i 0.282455i 0.989977 + 0.141228i \(0.0451050\pi\)
−0.989977 + 0.141228i \(0.954895\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 10.2961i 0.476446i 0.971210 + 0.238223i \(0.0765649\pi\)
−0.971210 + 0.238223i \(0.923435\pi\)
\(468\) 0 0
\(469\) 3.83981 0.177306
\(470\) 0 0
\(471\) 71.4598 3.29270
\(472\) 0 0
\(473\) 7.74572i 0.356149i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 14.5289i − 0.665233i
\(478\) 0 0
\(479\) 33.9760 1.55240 0.776201 0.630486i \(-0.217144\pi\)
0.776201 + 0.630486i \(0.217144\pi\)
\(480\) 0 0
\(481\) −28.3556 −1.29290
\(482\) 0 0
\(483\) − 1.71138i − 0.0778703i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 16.4951i − 0.747464i −0.927537 0.373732i \(-0.878078\pi\)
0.927537 0.373732i \(-0.121922\pi\)
\(488\) 0 0
\(489\) −77.2962 −3.49546
\(490\) 0 0
\(491\) 27.4876 1.24050 0.620249 0.784405i \(-0.287032\pi\)
0.620249 + 0.784405i \(0.287032\pi\)
\(492\) 0 0
\(493\) − 1.07928i − 0.0486082i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0.00668310i 0 0.000299778i
\(498\) 0 0
\(499\) −18.2961 −0.819045 −0.409523 0.912300i \(-0.634305\pi\)
−0.409523 + 0.912300i \(0.634305\pi\)
\(500\) 0 0
\(501\) −13.5549 −0.605587
\(502\) 0 0
\(503\) − 1.97389i − 0.0880115i −0.999031 0.0440058i \(-0.985988\pi\)
0.999031 0.0440058i \(-0.0140120\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 37.6986i − 1.67425i
\(508\) 0 0
\(509\) 38.3441 1.69957 0.849787 0.527126i \(-0.176731\pi\)
0.849787 + 0.527126i \(0.176731\pi\)
\(510\) 0 0
\(511\) −8.49125 −0.375631
\(512\) 0 0
\(513\) − 2.12417i − 0.0937844i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 46.0217i 2.02403i
\(518\) 0 0
\(519\) −59.8839 −2.62861
\(520\) 0 0
\(521\) −23.1814 −1.01560 −0.507799 0.861476i \(-0.669541\pi\)
−0.507799 + 0.861476i \(0.669541\pi\)
\(522\) 0 0
\(523\) − 43.5123i − 1.90266i −0.308172 0.951331i \(-0.599717\pi\)
0.308172 0.951331i \(-0.400283\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.80542i 0.122206i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 15.8452 0.687622
\(532\) 0 0
\(533\) − 24.5962i − 1.06538i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 32.1197i − 1.38607i
\(538\) 0 0
\(539\) 31.1303 1.34088
\(540\) 0 0
\(541\) 18.6112 0.800158 0.400079 0.916481i \(-0.368983\pi\)
0.400079 + 0.916481i \(0.368983\pi\)
\(542\) 0 0
\(543\) − 28.1732i − 1.20903i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 6.78956i − 0.290300i −0.989410 0.145150i \(-0.953633\pi\)
0.989410 0.145150i \(-0.0463665\pi\)
\(548\) 0 0
\(549\) 37.1673 1.58626
\(550\) 0 0
\(551\) 0.159537 0.00679649
\(552\) 0 0
\(553\) − 5.90272i − 0.251009i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 24.9869i 1.05873i 0.848395 + 0.529364i \(0.177570\pi\)
−0.848395 + 0.529364i \(0.822430\pi\)
\(558\) 0 0
\(559\) −8.36290 −0.353713
\(560\) 0 0
\(561\) 19.0583 0.804642
\(562\) 0 0
\(563\) 44.7551i 1.88620i 0.332508 + 0.943100i \(0.392105\pi\)
−0.332508 + 0.943100i \(0.607895\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 7.67371i 0.322265i
\(568\) 0 0
\(569\) 27.4895 1.15242 0.576211 0.817301i \(-0.304531\pi\)
0.576211 + 0.817301i \(0.304531\pi\)
\(570\) 0 0
\(571\) −1.40495 −0.0587952 −0.0293976 0.999568i \(-0.509359\pi\)
−0.0293976 + 0.999568i \(0.509359\pi\)
\(572\) 0 0
\(573\) − 4.61957i − 0.192985i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 19.1987i − 0.799253i −0.916678 0.399627i \(-0.869140\pi\)
0.916678 0.399627i \(-0.130860\pi\)
\(578\) 0 0
\(579\) −44.4364 −1.84671
\(580\) 0 0
\(581\) 3.13043 0.129872
\(582\) 0 0
\(583\) 10.3859i 0.430139i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 22.8565i − 0.943390i −0.881762 0.471695i \(-0.843642\pi\)
0.881762 0.471695i \(-0.156358\pi\)
\(588\) 0 0
\(589\) −0.414692 −0.0170871
\(590\) 0 0
\(591\) 75.4473 3.10349
\(592\) 0 0
\(593\) 26.5387i 1.08981i 0.838497 + 0.544906i \(0.183435\pi\)
−0.838497 + 0.544906i \(0.816565\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 4.97005i 0.203411i
\(598\) 0 0
\(599\) 8.82768 0.360689 0.180345 0.983603i \(-0.442279\pi\)
0.180345 + 0.983603i \(0.442279\pi\)
\(600\) 0 0
\(601\) −30.3072 −1.23626 −0.618129 0.786077i \(-0.712109\pi\)
−0.618129 + 0.786077i \(0.712109\pi\)
\(602\) 0 0
\(603\) 45.0215i 1.83342i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 38.0102i 1.54278i 0.636360 + 0.771392i \(0.280439\pi\)
−0.636360 + 0.771392i \(0.719561\pi\)
\(608\) 0 0
\(609\) −1.39015 −0.0563316
\(610\) 0 0
\(611\) −49.6887 −2.01019
\(612\) 0 0
\(613\) 31.7417i 1.28204i 0.767526 + 0.641018i \(0.221488\pi\)
−0.767526 + 0.641018i \(0.778512\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 46.6699i 1.87886i 0.342741 + 0.939430i \(0.388645\pi\)
−0.342741 + 0.939430i \(0.611355\pi\)
\(618\) 0 0
\(619\) −23.2223 −0.933383 −0.466692 0.884420i \(-0.654554\pi\)
−0.466692 + 0.884420i \(0.654554\pi\)
\(620\) 0 0
\(621\) 10.8154 0.434009
\(622\) 0 0
\(623\) − 5.03235i − 0.201617i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 2.81716i 0.112506i
\(628\) 0 0
\(629\) 7.50121 0.299093
\(630\) 0 0
\(631\) −13.0906 −0.521129 −0.260565 0.965456i \(-0.583909\pi\)
−0.260565 + 0.965456i \(0.583909\pi\)
\(632\) 0 0
\(633\) 19.2740i 0.766072i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 33.6107i 1.33171i
\(638\) 0 0
\(639\) −0.0783588 −0.00309983
\(640\) 0 0
\(641\) 22.8746 0.903494 0.451747 0.892146i \(-0.350801\pi\)
0.451747 + 0.892146i \(0.350801\pi\)
\(642\) 0 0
\(643\) 30.6333i 1.20806i 0.796962 + 0.604030i \(0.206439\pi\)
−0.796962 + 0.604030i \(0.793561\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 8.74454i − 0.343783i −0.985116 0.171892i \(-0.945012\pi\)
0.985116 0.171892i \(-0.0549879\pi\)
\(648\) 0 0
\(649\) −11.3268 −0.444616
\(650\) 0 0
\(651\) 3.61348 0.141623
\(652\) 0 0
\(653\) 24.7443i 0.968320i 0.874979 + 0.484160i \(0.160875\pi\)
−0.874979 + 0.484160i \(0.839125\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 99.5593i − 3.88418i
\(658\) 0 0
\(659\) −8.30726 −0.323605 −0.161803 0.986823i \(-0.551731\pi\)
−0.161803 + 0.986823i \(0.551731\pi\)
\(660\) 0 0
\(661\) −14.1755 −0.551364 −0.275682 0.961249i \(-0.588904\pi\)
−0.275682 + 0.961249i \(0.588904\pi\)
\(662\) 0 0
\(663\) 20.5768i 0.799138i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0.812298i 0.0314523i
\(668\) 0 0
\(669\) −65.6627 −2.53867
\(670\) 0 0
\(671\) −26.5688 −1.02568
\(672\) 0 0
\(673\) − 7.30229i − 0.281482i −0.990046 0.140741i \(-0.955051\pi\)
0.990046 0.140741i \(-0.0449486\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 39.7090i 1.52614i 0.646315 + 0.763071i \(0.276309\pi\)
−0.646315 + 0.763071i \(0.723691\pi\)
\(678\) 0 0
\(679\) −7.81027 −0.299731
\(680\) 0 0
\(681\) 32.4296 1.24271
\(682\) 0 0
\(683\) 38.5993i 1.47696i 0.674274 + 0.738481i \(0.264457\pi\)
−0.674274 + 0.738481i \(0.735543\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 74.0219i 2.82411i
\(688\) 0 0
\(689\) −11.2134 −0.427198
\(690\) 0 0
\(691\) −21.4163 −0.814713 −0.407357 0.913269i \(-0.633549\pi\)
−0.407357 + 0.913269i \(0.633549\pi\)
\(692\) 0 0
\(693\) − 16.8020i − 0.638255i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 6.50669i 0.246459i
\(698\) 0 0
\(699\) 34.8773 1.31918
\(700\) 0 0
\(701\) −5.40147 −0.204011 −0.102005 0.994784i \(-0.532526\pi\)
−0.102005 + 0.994784i \(0.532526\pi\)
\(702\) 0 0
\(703\) 1.10881i 0.0418197i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4.17940i 0.157182i
\(708\) 0 0
\(709\) 15.3161 0.575208 0.287604 0.957749i \(-0.407141\pi\)
0.287604 + 0.957749i \(0.407141\pi\)
\(710\) 0 0
\(711\) 69.2090 2.59554
\(712\) 0 0
\(713\) − 2.11145i − 0.0790742i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 78.8102i 2.94322i
\(718\) 0 0
\(719\) −11.6505 −0.434491 −0.217245 0.976117i \(-0.569707\pi\)
−0.217245 + 0.976117i \(0.569707\pi\)
\(720\) 0 0
\(721\) 8.01875 0.298634
\(722\) 0 0
\(723\) − 49.8334i − 1.85333i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 28.8188i − 1.06883i −0.845223 0.534415i \(-0.820532\pi\)
0.845223 0.534415i \(-0.179468\pi\)
\(728\) 0 0
\(729\) 10.0725 0.373056
\(730\) 0 0
\(731\) 2.21233 0.0818259
\(732\) 0 0
\(733\) − 25.2304i − 0.931906i −0.884810 0.465953i \(-0.845712\pi\)
0.884810 0.465953i \(-0.154288\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 32.1833i − 1.18549i
\(738\) 0 0
\(739\) 1.22800 0.0451728 0.0225864 0.999745i \(-0.492810\pi\)
0.0225864 + 0.999745i \(0.492810\pi\)
\(740\) 0 0
\(741\) −3.04163 −0.111737
\(742\) 0 0
\(743\) 35.1117i 1.28812i 0.764974 + 0.644061i \(0.222752\pi\)
−0.764974 + 0.644061i \(0.777248\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 36.7041i 1.34293i
\(748\) 0 0
\(749\) 1.30909 0.0478329
\(750\) 0 0
\(751\) 31.7367 1.15809 0.579044 0.815296i \(-0.303426\pi\)
0.579044 + 0.815296i \(0.303426\pi\)
\(752\) 0 0
\(753\) − 41.1630i − 1.50006i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 3.22287i 0.117137i 0.998283 + 0.0585685i \(0.0186536\pi\)
−0.998283 + 0.0585685i \(0.981346\pi\)
\(758\) 0 0
\(759\) −14.3439 −0.520649
\(760\) 0 0
\(761\) −9.60077 −0.348027 −0.174014 0.984743i \(-0.555674\pi\)
−0.174014 + 0.984743i \(0.555674\pi\)
\(762\) 0 0
\(763\) 5.74882i 0.208121i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 12.2293i − 0.441575i
\(768\) 0 0
\(769\) −14.9778 −0.540113 −0.270056 0.962844i \(-0.587042\pi\)
−0.270056 + 0.962844i \(0.587042\pi\)
\(770\) 0 0
\(771\) 31.3998 1.13084
\(772\) 0 0
\(773\) 28.7430i 1.03381i 0.856042 + 0.516906i \(0.172916\pi\)
−0.856042 + 0.516906i \(0.827084\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 9.66181i − 0.346616i
\(778\) 0 0
\(779\) −0.961806 −0.0344603
\(780\) 0 0
\(781\) 0.0560142 0.00200435
\(782\) 0 0
\(783\) − 8.78536i − 0.313963i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 34.1965i 1.21897i 0.792797 + 0.609486i \(0.208624\pi\)
−0.792797 + 0.609486i \(0.791376\pi\)
\(788\) 0 0
\(789\) 71.8666 2.55852
\(790\) 0 0
\(791\) 10.6965 0.380324
\(792\) 0 0
\(793\) − 28.6857i − 1.01866i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0.303391i 0.0107467i 0.999986 + 0.00537334i \(0.00171039\pi\)
−0.999986 + 0.00537334i \(0.998290\pi\)
\(798\) 0 0
\(799\) 13.1447 0.465026
\(800\) 0 0
\(801\) 59.0039 2.08480
\(802\) 0 0
\(803\) 71.1692i 2.51151i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 30.2429i − 1.06460i
\(808\) 0 0
\(809\) 13.1865 0.463611 0.231806 0.972762i \(-0.425537\pi\)
0.231806 + 0.972762i \(0.425537\pi\)
\(810\) 0 0
\(811\) 14.7284 0.517185 0.258593 0.965986i \(-0.416741\pi\)
0.258593 + 0.965986i \(0.416741\pi\)
\(812\) 0 0
\(813\) − 18.7575i − 0.657856i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0.327022i 0.0114410i
\(818\) 0 0
\(819\) 18.1408 0.633890
\(820\) 0 0
\(821\) 36.6177 1.27797 0.638984 0.769220i \(-0.279355\pi\)
0.638984 + 0.769220i \(0.279355\pi\)
\(822\) 0 0
\(823\) 0.420867i 0.0146705i 0.999973 + 0.00733525i \(0.00233490\pi\)
−0.999973 + 0.00733525i \(0.997665\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 35.1118i − 1.22096i −0.792032 0.610479i \(-0.790977\pi\)
0.792032 0.610479i \(-0.209023\pi\)
\(828\) 0 0
\(829\) 35.6234 1.23725 0.618626 0.785685i \(-0.287689\pi\)
0.618626 + 0.785685i \(0.287689\pi\)
\(830\) 0 0
\(831\) 0.137436 0.00476759
\(832\) 0 0
\(833\) − 8.89141i − 0.308069i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 22.8362i 0.789335i
\(838\) 0 0
\(839\) 19.9998 0.690469 0.345235 0.938516i \(-0.387799\pi\)
0.345235 + 0.938516i \(0.387799\pi\)
\(840\) 0 0
\(841\) −28.3402 −0.977247
\(842\) 0 0
\(843\) 86.3350i 2.97354i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 5.90554i 0.202917i
\(848\) 0 0
\(849\) 37.5556 1.28890
\(850\) 0 0
\(851\) −5.64564 −0.193530
\(852\) 0 0
\(853\) 42.4994i 1.45515i 0.686026 + 0.727577i \(0.259353\pi\)
−0.686026 + 0.727577i \(0.740647\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 18.0100i 0.615208i 0.951514 + 0.307604i \(0.0995273\pi\)
−0.951514 + 0.307604i \(0.900473\pi\)
\(858\) 0 0
\(859\) −22.3869 −0.763832 −0.381916 0.924197i \(-0.624736\pi\)
−0.381916 + 0.924197i \(0.624736\pi\)
\(860\) 0 0
\(861\) 8.38084 0.285618
\(862\) 0 0
\(863\) 18.7613i 0.638642i 0.947647 + 0.319321i \(0.103455\pi\)
−0.947647 + 0.319321i \(0.896545\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 46.9750i 1.59536i
\(868\) 0 0
\(869\) −49.4735 −1.67827
\(870\) 0 0
\(871\) 34.7476 1.17738
\(872\) 0 0
\(873\) − 91.5749i − 3.09934i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 31.6345i 1.06822i 0.845415 + 0.534111i \(0.179353\pi\)
−0.845415 + 0.534111i \(0.820647\pi\)
\(878\) 0 0
\(879\) −8.35783 −0.281903
\(880\) 0 0
\(881\) 39.6850 1.33702 0.668512 0.743702i \(-0.266932\pi\)
0.668512 + 0.743702i \(0.266932\pi\)
\(882\) 0 0
\(883\) − 10.3778i − 0.349242i −0.984636 0.174621i \(-0.944130\pi\)
0.984636 0.174621i \(-0.0558701\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 24.0199i − 0.806508i −0.915088 0.403254i \(-0.867879\pi\)
0.915088 0.403254i \(-0.132121\pi\)
\(888\) 0 0
\(889\) −10.2642 −0.344251
\(890\) 0 0
\(891\) 64.3170 2.15470
\(892\) 0 0
\(893\) 1.94302i 0.0650207i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 15.4868i − 0.517088i
\(898\) 0 0
\(899\) −1.71512 −0.0572025
\(900\) 0 0
\(901\) 2.96641 0.0988254
\(902\) 0 0
\(903\) − 2.84955i − 0.0948271i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 18.5181i 0.614885i 0.951567 + 0.307442i \(0.0994731\pi\)
−0.951567 + 0.307442i \(0.900527\pi\)
\(908\) 0 0
\(909\) −49.0032 −1.62533
\(910\) 0 0
\(911\) −3.34956 −0.110976 −0.0554879 0.998459i \(-0.517671\pi\)
−0.0554879 + 0.998459i \(0.517671\pi\)
\(912\) 0 0
\(913\) − 26.2376i − 0.868339i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 8.19920i − 0.270761i
\(918\) 0 0
\(919\) 32.9754 1.08776 0.543878 0.839164i \(-0.316955\pi\)
0.543878 + 0.839164i \(0.316955\pi\)
\(920\) 0 0
\(921\) 99.1728 3.26785
\(922\) 0 0
\(923\) 0.0604774i 0.00199064i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 94.0193i 3.08800i
\(928\) 0 0
\(929\) 36.5008 1.19755 0.598777 0.800916i \(-0.295654\pi\)
0.598777 + 0.800916i \(0.295654\pi\)
\(930\) 0 0
\(931\) 1.31431 0.0430748
\(932\) 0 0
\(933\) − 12.5789i − 0.411816i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 12.4598i 0.407043i 0.979071 + 0.203521i \(0.0652386\pi\)
−0.979071 + 0.203521i \(0.934761\pi\)
\(938\) 0 0
\(939\) 42.4916 1.38666
\(940\) 0 0
\(941\) −45.2938 −1.47653 −0.738267 0.674508i \(-0.764356\pi\)
−0.738267 + 0.674508i \(0.764356\pi\)
\(942\) 0 0
\(943\) − 4.89714i − 0.159473i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 41.4016i 1.34537i 0.739929 + 0.672685i \(0.234859\pi\)
−0.739929 + 0.672685i \(0.765141\pi\)
\(948\) 0 0
\(949\) −76.8399 −2.49433
\(950\) 0 0
\(951\) −32.2619 −1.04616
\(952\) 0 0
\(953\) 15.3738i 0.498006i 0.968503 + 0.249003i \(0.0801029\pi\)
−0.968503 + 0.249003i \(0.919897\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 11.6515i 0.376639i
\(958\) 0 0
\(959\) 6.44785 0.208212
\(960\) 0 0
\(961\) −26.5418 −0.856187
\(962\) 0 0
\(963\) 15.3489i 0.494612i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 53.4925i 1.72020i 0.510124 + 0.860101i \(0.329600\pi\)
−0.510124 + 0.860101i \(0.670400\pi\)
\(968\) 0 0
\(969\) 0.804635 0.0258486
\(970\) 0 0
\(971\) 19.8881 0.638241 0.319120 0.947714i \(-0.396613\pi\)
0.319120 + 0.947714i \(0.396613\pi\)
\(972\) 0 0
\(973\) − 6.18231i − 0.198196i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 14.8758i − 0.475919i −0.971275 0.237960i \(-0.923521\pi\)
0.971275 0.237960i \(-0.0764786\pi\)
\(978\) 0 0
\(979\) −42.1785 −1.34803
\(980\) 0 0
\(981\) −67.4045 −2.15206
\(982\) 0 0
\(983\) 1.16403i 0.0371269i 0.999828 + 0.0185634i \(0.00590926\pi\)
−0.999828 + 0.0185634i \(0.994091\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 16.9308i − 0.538913i
\(988\) 0 0
\(989\) −1.66507 −0.0529460
\(990\) 0 0
\(991\) −47.8693 −1.52062 −0.760310 0.649561i \(-0.774953\pi\)
−0.760310 + 0.649561i \(0.774953\pi\)
\(992\) 0 0
\(993\) − 38.9904i − 1.23732i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 11.4353i − 0.362159i −0.983468 0.181079i \(-0.942041\pi\)
0.983468 0.181079i \(-0.0579591\pi\)
\(998\) 0 0
\(999\) 61.0601 1.93186
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 4600.2.e.v.4049.10 10
5.2 odd 4 4600.2.a.bg.1.5 yes 5
5.3 odd 4 4600.2.a.bc.1.1 5
5.4 even 2 inner 4600.2.e.v.4049.1 10
20.3 even 4 9200.2.a.cw.1.5 5
20.7 even 4 9200.2.a.cs.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bc.1.1 5 5.3 odd 4
4600.2.a.bg.1.5 yes 5 5.2 odd 4
4600.2.e.v.4049.1 10 5.4 even 2 inner
4600.2.e.v.4049.10 10 1.1 even 1 trivial
9200.2.a.cs.1.1 5 20.7 even 4
9200.2.a.cw.1.5 5 20.3 even 4