Properties

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

$q$-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 \(-0.650626\pi\)
0.455742 0.890112i \(-0.349374\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 0.555022i − 0.209779i −0.994484 0.104889i \(-0.966551\pi\)
0.994484 0.104889i \(-0.0334488\pi\)
\(8\) 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 \(-0.754738\pi\)
0.717553 0.696504i \(-0.245262\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.32867i 0.322251i 0.986934 + 0.161125i \(0.0515123\pi\)
−0.986934 + 0.161125i \(0.948488\pi\)
\(18\) 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 \(-0.846389\pi\)
0.885799 0.464069i \(-0.153611\pi\)
\(38\) 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 \(-0.959478\pi\)
0.991908 0.126960i \(-0.0405220\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 9.89310i − 1.44306i −0.692385 0.721528i \(-0.743440\pi\)
0.692385 0.721528i \(-0.256560\pi\)
\(48\) 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 \(-0.950998\pi\)
0.988174 0.153336i \(-0.0490018\pi\)
\(54\) 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 \(0.138883\pi\)
−0.906315 + 0.422602i \(0.861117\pi\)
\(68\) 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 \(-0.646959\pi\)
0.445458 0.895303i \(-0.353041\pi\)
\(74\) 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 \(0.100177\pi\)
−0.950884 + 0.309546i \(0.899823\pi\)
\(84\) 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 \(-0.746701\pi\)
0.699739 0.714398i \(-0.253299\pi\)
\(98\) 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 \(0.252112\pi\)
−0.702399 + 0.711784i \(0.747888\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.35862i 0.228016i 0.993480 + 0.114008i \(0.0363690\pi\)
−0.993480 + 0.114008i \(0.963631\pi\)
\(108\) 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 \(0.361249\pi\)
−0.422226 + 0.906491i \(0.638751\pi\)
\(114\) 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 \(-0.693689\pi\)
0.571632 0.820510i \(-0.306311\pi\)
\(128\) 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 \(0.165296\pi\)
−0.868170 + 0.496266i \(0.834704\pi\)
\(138\) 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 \(0.375766\pi\)
−0.380458 + 0.924798i \(0.624234\pi\)
\(158\) 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 \(-0.560912\pi\)
0.190196 0.981746i \(-0.439088\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 4.39603i − 0.340175i −0.985429 0.170087i \(-0.945595\pi\)
0.985429 0.170087i \(-0.0544050\pi\)
\(168\) 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 \(-0.735638\pi\)
0.674493 0.738281i \(-0.264362\pi\)
\(174\) 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 \(-0.826425\pi\)
0.854972 0.518675i \(-0.173575\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 24.4686i 1.74331i 0.490116 + 0.871657i \(0.336954\pi\)
−0.490116 + 0.871657i \(0.663046\pi\)
\(198\) 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 \(-0.747327\pi\)
0.701144 0.713019i \(-0.252673\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 10.5174i 0.698061i 0.937111 + 0.349031i \(0.113489\pi\)
−0.937111 + 0.349031i \(0.886511\pi\)
\(228\) 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 \(0.120817\pi\)
−0.928828 + 0.370510i \(0.879183\pi\)
\(234\) 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 \(0.102881\pi\)
−0.948221 + 0.317611i \(0.897119\pi\)
\(258\) 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 \(0.255214\pi\)
−0.695430 + 0.718594i \(0.744786\pi\)
\(264\) 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 \(0.000426231\pi\)
−0.999999 + 0.00133904i \(0.999574\pi\)
\(278\) 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 \(0.117908\pi\)
−0.932176 + 0.362006i \(0.882092\pi\)
\(284\) 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 \(-0.974771\pi\)
0.996861 0.0791762i \(-0.0252290\pi\)
\(294\) 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 \(0.370053\pi\)
−0.396994 + 0.917821i \(0.629947\pi\)
\(308\) 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 \(0.127339\pi\)
−0.921042 + 0.389462i \(0.872661\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 10.4630i − 0.587658i −0.955858 0.293829i \(-0.905070\pi\)
0.955858 0.293829i \(-0.0949297\pi\)
\(318\) 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 \(-0.766596\pi\)
0.742997 0.669295i \(-0.233404\pi\)
\(338\) 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 \(0.0299966\pi\)
−0.995563 + 0.0940977i \(0.970003\pi\)
\(348\) 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 \(-0.819428\pi\)
0.843364 0.537343i \(-0.180572\pi\)
\(354\) 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 \(0.290908\pi\)
−0.610652 + 0.791899i \(0.709092\pi\)
\(368\) 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 \(0.125067\pi\)
−0.923798 + 0.382879i \(0.874933\pi\)
\(374\) 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 \(-0.716122\pi\)
0.627991 0.778221i \(-0.283878\pi\)
\(384\) 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 \(-0.847941\pi\)
0.888052 0.459743i \(-0.152059\pi\)
\(398\) 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 \(0.371463\pi\)
−0.392926 + 0.919570i \(0.628537\pi\)
\(434\) 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 \(-0.918550\pi\)
0.967440 0.253099i \(-0.0814499\pi\)
\(444\) 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 \(-0.641746\pi\)
0.430736 0.902478i \(-0.358254\pi\)
\(458\) 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 \(-0.954895\pi\)
0.989977 0.141228i \(-0.0451050\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 10.2961i − 0.476446i −0.971210 0.238223i \(-0.923435\pi\)
0.971210 0.238223i \(-0.0765649\pi\)
\(468\) 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 \(0.121922\pi\)
−0.927537 + 0.373732i \(0.878078\pi\)
\(488\) 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 \(0.0140120\pi\)
−0.999031 + 0.0440058i \(0.985988\pi\)
\(504\) 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 \(0.400283\pi\)
−0.308172 + 0.951331i \(0.599717\pi\)
\(524\) 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 \(0.0463665\pi\)
−0.989410 + 0.145150i \(0.953633\pi\)
\(548\) 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 \(-0.822430\pi\)
0.848395 0.529364i \(-0.177570\pi\)
\(558\) 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 \(-0.607895\pi\)
0.332508 0.943100i \(-0.392105\pi\)
\(564\) 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 \(0.130860\pi\)
−0.916678 + 0.399627i \(0.869140\pi\)
\(578\) 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 \(0.156358\pi\)
−0.881762 + 0.471695i \(0.843642\pi\)
\(588\) 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 \(-0.816565\pi\)
0.838497 0.544906i \(-0.183435\pi\)
\(594\) 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 \(-0.719561\pi\)
0.636360 0.771392i \(-0.280439\pi\)
\(608\) 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 \(-0.778512\pi\)
0.767526 0.641018i \(-0.221488\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 46.6699i − 1.87886i −0.342741 0.939430i \(-0.611355\pi\)
0.342741 0.939430i \(-0.388645\pi\)
\(618\) 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 \(-0.793561\pi\)
0.796962 0.604030i \(-0.206439\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 8.74454i 0.343783i 0.985116 + 0.171892i \(0.0549879\pi\)
−0.985116 + 0.171892i \(0.945012\pi\)
\(648\) 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 \(-0.839125\pi\)
0.874979 0.484160i \(-0.160875\pi\)
\(654\) 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 \(0.0449486\pi\)
−0.990046 + 0.140741i \(0.955051\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 39.7090i − 1.52614i −0.646315 0.763071i \(-0.723691\pi\)
0.646315 0.763071i \(-0.276309\pi\)
\(678\) 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 \(-0.735543\pi\)
0.674274 0.738481i \(-0.264457\pi\)
\(684\) 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 \(0.179468\pi\)
−0.845223 + 0.534415i \(0.820532\pi\)
\(728\) 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 \(0.154288\pi\)
−0.884810 + 0.465953i \(0.845712\pi\)
\(734\) 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 \(-0.777248\pi\)
0.764974 0.644061i \(-0.222752\pi\)
\(744\) 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 \(-0.981346\pi\)
0.998283 0.0585685i \(-0.0186536\pi\)
\(758\) 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 \(-0.827084\pi\)
0.856042 0.516906i \(-0.172916\pi\)
\(774\) 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 \(-0.791376\pi\)
0.792797 0.609486i \(-0.208624\pi\)
\(788\) 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 \(-0.998290\pi\)
0.999986 0.00537334i \(-0.00171039\pi\)
\(798\) 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 \(-0.997665\pi\)
0.999973 0.00733525i \(-0.00233490\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 35.1118i 1.22096i 0.792032 + 0.610479i \(0.209023\pi\)
−0.792032 + 0.610479i \(0.790977\pi\)
\(828\) 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 \(-0.740647\pi\)
0.686026 0.727577i \(-0.259353\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 18.0100i − 0.615208i −0.951514 0.307604i \(-0.900473\pi\)
0.951514 0.307604i \(-0.0995273\pi\)
\(858\) 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 \(-0.896545\pi\)
0.947647 0.319321i \(-0.103455\pi\)
\(864\) 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 \(-0.820647\pi\)
0.845415 0.534111i \(-0.179353\pi\)
\(878\) 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 \(0.0558701\pi\)
−0.984636 + 0.174621i \(0.944130\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 24.0199i 0.806508i 0.915088 + 0.403254i \(0.132121\pi\)
−0.915088 + 0.403254i \(0.867879\pi\)
\(888\) 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 \(-0.900527\pi\)
0.951567 0.307442i \(-0.0994731\pi\)
\(908\) 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 \(-0.934761\pi\)
0.979071 0.203521i \(-0.0652386\pi\)
\(938\) 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 \(-0.765141\pi\)
0.739929 0.672685i \(-0.234859\pi\)
\(948\) 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 \(-0.919897\pi\)
0.968503 0.249003i \(-0.0801029\pi\)
\(954\) 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 \(-0.670400\pi\)
0.510124 0.860101i \(-0.329600\pi\)
\(968\) 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 \(0.0764786\pi\)
−0.971275 + 0.237960i \(0.923521\pi\)
\(978\) 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 \(-0.994091\pi\)
0.999828 0.0185634i \(-0.00590926\pi\)
\(984\) 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 \(0.0579591\pi\)
−0.983468 + 0.181079i \(0.942041\pi\)
\(998\) 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.4049.1 10
5.2 odd 4 4600.2.a.bc.1.1 5
5.3 odd 4 4600.2.a.bg.1.5 yes 5
5.4 even 2 inner 4600.2.e.v.4049.10 10
20.3 even 4 9200.2.a.cs.1.1 5
20.7 even 4 9200.2.a.cw.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bc.1.1 5 5.2 odd 4
4600.2.a.bg.1.5 yes 5 5.3 odd 4
4600.2.e.v.4049.1 10 1.1 even 1 trivial
4600.2.e.v.4049.10 10 5.4 even 2 inner
9200.2.a.cs.1.1 5 20.3 even 4
9200.2.a.cw.1.5 5 20.7 even 4