Properties

Label 4600.2.e.v.4049.4
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 4049.4
Root \(3.11721i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.v.4049.7

$q$-expansion

\(f(q)\) \(=\) \(q-1.33689i q^{3} +3.16736i q^{7} +1.21273 q^{9} +O(q^{10})\) \(q-1.33689i q^{3} +3.16736i q^{7} +1.21273 q^{9} -0.0955192 q^{11} +1.44343i q^{13} +2.29775i q^{17} -7.00371 q^{19} +4.23441 q^{21} -1.00000i q^{23} -5.63195i q^{27} -5.39076 q^{29} -0.584488 q^{31} +0.127699i q^{33} +9.29985i q^{37} +1.92970 q^{39} -2.86534 q^{41} +9.50208i q^{43} -7.09353i q^{47} -3.03218 q^{49} +3.07184 q^{51} -7.73922i q^{53} +9.36319i q^{57} -13.6426 q^{59} +0.234413 q^{61} +3.84114i q^{63} -7.49729i q^{67} -1.33689 q^{69} -5.18426 q^{71} -1.52384i q^{73} -0.302544i q^{77} +3.04068 q^{79} -3.89112 q^{81} -15.9909i q^{83} +7.20685i q^{87} -5.53325 q^{89} -4.57186 q^{91} +0.781396i q^{93} -2.58305i q^{97} -0.115839 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 8 q^{9} + O(q^{10}) \) \( 10 q - 8 q^{9} - 8 q^{11} - 8 q^{19} - 12 q^{21} + 22 q^{29} + 8 q^{31} - 62 q^{39} - 16 q^{41} + 4 q^{49} - 10 q^{51} - 46 q^{59} - 52 q^{61} - 6 q^{69} - 4 q^{71} - 86 q^{79} - 6 q^{81} - 30 q^{89} - 38 q^{91} - 74 q^{99} + O(q^{100}) \)

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
\(3\) − 1.33689i − 0.771854i −0.922529 0.385927i \(-0.873882\pi\)
0.922529 0.385927i \(-0.126118\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.16736i 1.19715i 0.801067 + 0.598575i \(0.204266\pi\)
−0.801067 + 0.598575i \(0.795734\pi\)
\(8\) 0 0
\(9\) 1.21273 0.404242
\(10\) 0 0
\(11\) −0.0955192 −0.0288001 −0.0144001 0.999896i \(-0.504584\pi\)
−0.0144001 + 0.999896i \(0.504584\pi\)
\(12\) 0 0
\(13\) 1.44343i 0.400335i 0.979762 + 0.200167i \(0.0641486\pi\)
−0.979762 + 0.200167i \(0.935851\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.29775i 0.557287i 0.960395 + 0.278643i \(0.0898848\pi\)
−0.960395 + 0.278643i \(0.910115\pi\)
\(18\) 0 0
\(19\) −7.00371 −1.60676 −0.803381 0.595466i \(-0.796968\pi\)
−0.803381 + 0.595466i \(0.796968\pi\)
\(20\) 0 0
\(21\) 4.23441 0.924025
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 5.63195i − 1.08387i
\(28\) 0 0
\(29\) −5.39076 −1.00104 −0.500519 0.865725i \(-0.666858\pi\)
−0.500519 + 0.865725i \(0.666858\pi\)
\(30\) 0 0
\(31\) −0.584488 −0.104977 −0.0524886 0.998622i \(-0.516715\pi\)
−0.0524886 + 0.998622i \(0.516715\pi\)
\(32\) 0 0
\(33\) 0.127699i 0.0222295i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 9.29985i 1.52889i 0.644691 + 0.764443i \(0.276986\pi\)
−0.644691 + 0.764443i \(0.723014\pi\)
\(38\) 0 0
\(39\) 1.92970 0.309000
\(40\) 0 0
\(41\) −2.86534 −0.447492 −0.223746 0.974648i \(-0.571829\pi\)
−0.223746 + 0.974648i \(0.571829\pi\)
\(42\) 0 0
\(43\) 9.50208i 1.44905i 0.689246 + 0.724527i \(0.257942\pi\)
−0.689246 + 0.724527i \(0.742058\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 7.09353i − 1.03470i −0.855775 0.517349i \(-0.826919\pi\)
0.855775 0.517349i \(-0.173081\pi\)
\(48\) 0 0
\(49\) −3.03218 −0.433168
\(50\) 0 0
\(51\) 3.07184 0.430144
\(52\) 0 0
\(53\) − 7.73922i − 1.06306i −0.847038 0.531532i \(-0.821617\pi\)
0.847038 0.531532i \(-0.178383\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 9.36319i 1.24018i
\(58\) 0 0
\(59\) −13.6426 −1.77612 −0.888059 0.459730i \(-0.847946\pi\)
−0.888059 + 0.459730i \(0.847946\pi\)
\(60\) 0 0
\(61\) 0.234413 0.0300135 0.0150068 0.999887i \(-0.495223\pi\)
0.0150068 + 0.999887i \(0.495223\pi\)
\(62\) 0 0
\(63\) 3.84114i 0.483938i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 7.49729i − 0.915940i −0.888968 0.457970i \(-0.848577\pi\)
0.888968 0.457970i \(-0.151423\pi\)
\(68\) 0 0
\(69\) −1.33689 −0.160943
\(70\) 0 0
\(71\) −5.18426 −0.615258 −0.307629 0.951506i \(-0.599536\pi\)
−0.307629 + 0.951506i \(0.599536\pi\)
\(72\) 0 0
\(73\) − 1.52384i − 0.178352i −0.996016 0.0891760i \(-0.971577\pi\)
0.996016 0.0891760i \(-0.0284234\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 0.302544i − 0.0344781i
\(78\) 0 0
\(79\) 3.04068 0.342103 0.171052 0.985262i \(-0.445283\pi\)
0.171052 + 0.985262i \(0.445283\pi\)
\(80\) 0 0
\(81\) −3.89112 −0.432347
\(82\) 0 0
\(83\) − 15.9909i − 1.75523i −0.479369 0.877613i \(-0.659135\pi\)
0.479369 0.877613i \(-0.340865\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 7.20685i 0.772655i
\(88\) 0 0
\(89\) −5.53325 −0.586523 −0.293261 0.956032i \(-0.594741\pi\)
−0.293261 + 0.956032i \(0.594741\pi\)
\(90\) 0 0
\(91\) −4.57186 −0.479261
\(92\) 0 0
\(93\) 0.781396i 0.0810270i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 2.58305i − 0.262269i −0.991365 0.131135i \(-0.958138\pi\)
0.991365 0.131135i \(-0.0418620\pi\)
\(98\) 0 0
\(99\) −0.115839 −0.0116422
\(100\) 0 0
\(101\) −6.65615 −0.662312 −0.331156 0.943576i \(-0.607439\pi\)
−0.331156 + 0.943576i \(0.607439\pi\)
\(102\) 0 0
\(103\) − 17.6668i − 1.74076i −0.492383 0.870378i \(-0.663874\pi\)
0.492383 0.870378i \(-0.336126\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.83635i 0.564221i 0.959382 + 0.282111i \(0.0910345\pi\)
−0.959382 + 0.282111i \(0.908965\pi\)
\(108\) 0 0
\(109\) −7.02096 −0.672486 −0.336243 0.941775i \(-0.609156\pi\)
−0.336243 + 0.941775i \(0.609156\pi\)
\(110\) 0 0
\(111\) 12.4329 1.18008
\(112\) 0 0
\(113\) − 17.1003i − 1.60866i −0.594185 0.804328i \(-0.702525\pi\)
0.594185 0.804328i \(-0.297475\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.75048i 0.161832i
\(118\) 0 0
\(119\) −7.27781 −0.667156
\(120\) 0 0
\(121\) −10.9909 −0.999171
\(122\) 0 0
\(123\) 3.83065i 0.345398i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 10.8050i 0.958787i 0.877600 + 0.479394i \(0.159143\pi\)
−0.877600 + 0.479394i \(0.840857\pi\)
\(128\) 0 0
\(129\) 12.7032 1.11846
\(130\) 0 0
\(131\) −8.45211 −0.738464 −0.369232 0.929337i \(-0.620379\pi\)
−0.369232 + 0.929337i \(0.620379\pi\)
\(132\) 0 0
\(133\) − 22.1833i − 1.92354i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 18.5322i 1.58331i 0.610969 + 0.791655i \(0.290780\pi\)
−0.610969 + 0.791655i \(0.709220\pi\)
\(138\) 0 0
\(139\) −7.42618 −0.629880 −0.314940 0.949112i \(-0.601984\pi\)
−0.314940 + 0.949112i \(0.601984\pi\)
\(140\) 0 0
\(141\) −9.48327 −0.798635
\(142\) 0 0
\(143\) − 0.137875i − 0.0115297i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 4.05369i 0.334343i
\(148\) 0 0
\(149\) 4.53968 0.371905 0.185952 0.982559i \(-0.440463\pi\)
0.185952 + 0.982559i \(0.440463\pi\)
\(150\) 0 0
\(151\) 0.00675366 0.000549605 0 0.000274803 1.00000i \(-0.499913\pi\)
0.000274803 1.00000i \(0.499913\pi\)
\(152\) 0 0
\(153\) 2.78654i 0.225279i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 12.1721i − 0.971439i −0.874115 0.485719i \(-0.838558\pi\)
0.874115 0.485719i \(-0.161442\pi\)
\(158\) 0 0
\(159\) −10.3465 −0.820529
\(160\) 0 0
\(161\) 3.16736 0.249623
\(162\) 0 0
\(163\) 20.2201i 1.58376i 0.610677 + 0.791879i \(0.290897\pi\)
−0.610677 + 0.791879i \(0.709103\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 8.04709i − 0.622702i −0.950295 0.311351i \(-0.899218\pi\)
0.950295 0.311351i \(-0.100782\pi\)
\(168\) 0 0
\(169\) 10.9165 0.839732
\(170\) 0 0
\(171\) −8.49358 −0.649520
\(172\) 0 0
\(173\) 16.8394i 1.28028i 0.768259 + 0.640139i \(0.221123\pi\)
−0.768259 + 0.640139i \(0.778877\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 18.2387i 1.37090i
\(178\) 0 0
\(179\) −23.7641 −1.77621 −0.888105 0.459641i \(-0.847978\pi\)
−0.888105 + 0.459641i \(0.847978\pi\)
\(180\) 0 0
\(181\) −3.13518 −0.233036 −0.116518 0.993189i \(-0.537173\pi\)
−0.116518 + 0.993189i \(0.537173\pi\)
\(182\) 0 0
\(183\) − 0.313385i − 0.0231661i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 0.219479i − 0.0160499i
\(188\) 0 0
\(189\) 17.8384 1.29755
\(190\) 0 0
\(191\) 6.17586 0.446870 0.223435 0.974719i \(-0.428273\pi\)
0.223435 + 0.974719i \(0.428273\pi\)
\(192\) 0 0
\(193\) 7.22267i 0.519899i 0.965622 + 0.259949i \(0.0837059\pi\)
−0.965622 + 0.259949i \(0.916294\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 13.0019i 0.926349i 0.886267 + 0.463174i \(0.153290\pi\)
−0.886267 + 0.463174i \(0.846710\pi\)
\(198\) 0 0
\(199\) 1.84114 0.130515 0.0652575 0.997868i \(-0.479213\pi\)
0.0652575 + 0.997868i \(0.479213\pi\)
\(200\) 0 0
\(201\) −10.0231 −0.706972
\(202\) 0 0
\(203\) − 17.0745i − 1.19839i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 1.21273i − 0.0842903i
\(208\) 0 0
\(209\) 0.668989 0.0462749
\(210\) 0 0
\(211\) −24.8791 −1.71275 −0.856375 0.516354i \(-0.827289\pi\)
−0.856375 + 0.516354i \(0.827289\pi\)
\(212\) 0 0
\(213\) 6.93078i 0.474889i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 1.85128i − 0.125673i
\(218\) 0 0
\(219\) −2.03721 −0.137662
\(220\) 0 0
\(221\) −3.31664 −0.223101
\(222\) 0 0
\(223\) − 4.04924i − 0.271157i −0.990767 0.135579i \(-0.956711\pi\)
0.990767 0.135579i \(-0.0432894\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 17.3916i 1.15432i 0.816630 + 0.577162i \(0.195840\pi\)
−0.816630 + 0.577162i \(0.804160\pi\)
\(228\) 0 0
\(229\) 3.51761 0.232450 0.116225 0.993223i \(-0.462921\pi\)
0.116225 + 0.993223i \(0.462921\pi\)
\(230\) 0 0
\(231\) −0.404468 −0.0266120
\(232\) 0 0
\(233\) 10.7910i 0.706941i 0.935446 + 0.353471i \(0.114999\pi\)
−0.935446 + 0.353471i \(0.885001\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 4.06506i − 0.264054i
\(238\) 0 0
\(239\) 2.86944 0.185608 0.0928042 0.995684i \(-0.470417\pi\)
0.0928042 + 0.995684i \(0.470417\pi\)
\(240\) 0 0
\(241\) 27.0486 1.74235 0.871176 0.490972i \(-0.163358\pi\)
0.871176 + 0.490972i \(0.163358\pi\)
\(242\) 0 0
\(243\) − 11.6939i − 0.750161i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 10.1093i − 0.643242i
\(248\) 0 0
\(249\) −21.3780 −1.35478
\(250\) 0 0
\(251\) −11.7794 −0.743510 −0.371755 0.928331i \(-0.621244\pi\)
−0.371755 + 0.928331i \(0.621244\pi\)
\(252\) 0 0
\(253\) 0.0955192i 0.00600524i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 8.47742i 0.528807i 0.964412 + 0.264404i \(0.0851751\pi\)
−0.964412 + 0.264404i \(0.914825\pi\)
\(258\) 0 0
\(259\) −29.4560 −1.83031
\(260\) 0 0
\(261\) −6.53751 −0.404662
\(262\) 0 0
\(263\) − 18.5118i − 1.14149i −0.821128 0.570745i \(-0.806655\pi\)
0.821128 0.570745i \(-0.193345\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 7.39734i 0.452710i
\(268\) 0 0
\(269\) 25.4583 1.55222 0.776109 0.630599i \(-0.217191\pi\)
0.776109 + 0.630599i \(0.217191\pi\)
\(270\) 0 0
\(271\) −14.1812 −0.861446 −0.430723 0.902484i \(-0.641741\pi\)
−0.430723 + 0.902484i \(0.641741\pi\)
\(272\) 0 0
\(273\) 6.11207i 0.369919i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 21.4281i 1.28749i 0.765241 + 0.643744i \(0.222620\pi\)
−0.765241 + 0.643744i \(0.777380\pi\)
\(278\) 0 0
\(279\) −0.708823 −0.0424361
\(280\) 0 0
\(281\) −8.90281 −0.531097 −0.265548 0.964098i \(-0.585553\pi\)
−0.265548 + 0.964098i \(0.585553\pi\)
\(282\) 0 0
\(283\) 13.0369i 0.774966i 0.921877 + 0.387483i \(0.126655\pi\)
−0.921877 + 0.387483i \(0.873345\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 9.07558i − 0.535715i
\(288\) 0 0
\(289\) 11.7203 0.689431
\(290\) 0 0
\(291\) −3.45325 −0.202433
\(292\) 0 0
\(293\) 6.74772i 0.394206i 0.980383 + 0.197103i \(0.0631533\pi\)
−0.980383 + 0.197103i \(0.936847\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0.537959i 0.0312156i
\(298\) 0 0
\(299\) 1.44343 0.0834755
\(300\) 0 0
\(301\) −30.0965 −1.73474
\(302\) 0 0
\(303\) 8.89854i 0.511208i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 15.6667i 0.894143i 0.894498 + 0.447071i \(0.147533\pi\)
−0.894498 + 0.447071i \(0.852467\pi\)
\(308\) 0 0
\(309\) −23.6185 −1.34361
\(310\) 0 0
\(311\) 8.27922 0.469472 0.234736 0.972059i \(-0.424577\pi\)
0.234736 + 0.972059i \(0.424577\pi\)
\(312\) 0 0
\(313\) − 15.4898i − 0.875534i −0.899088 0.437767i \(-0.855769\pi\)
0.899088 0.437767i \(-0.144231\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 10.5803i − 0.594251i −0.954838 0.297125i \(-0.903972\pi\)
0.954838 0.297125i \(-0.0960279\pi\)
\(318\) 0 0
\(319\) 0.514921 0.0288300
\(320\) 0 0
\(321\) 7.80256 0.435496
\(322\) 0 0
\(323\) − 16.0928i − 0.895427i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 9.38625i 0.519061i
\(328\) 0 0
\(329\) 22.4678 1.23869
\(330\) 0 0
\(331\) −1.32836 −0.0730133 −0.0365067 0.999333i \(-0.511623\pi\)
−0.0365067 + 0.999333i \(0.511623\pi\)
\(332\) 0 0
\(333\) 11.2782i 0.618040i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 33.1297i 1.80469i 0.431014 + 0.902345i \(0.358156\pi\)
−0.431014 + 0.902345i \(0.641844\pi\)
\(338\) 0 0
\(339\) −22.8611 −1.24165
\(340\) 0 0
\(341\) 0.0558298 0.00302335
\(342\) 0 0
\(343\) 12.5675i 0.678582i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 15.7579i 0.845926i 0.906147 + 0.422963i \(0.139010\pi\)
−0.906147 + 0.422963i \(0.860990\pi\)
\(348\) 0 0
\(349\) 14.5904 0.781004 0.390502 0.920602i \(-0.372301\pi\)
0.390502 + 0.920602i \(0.372301\pi\)
\(350\) 0 0
\(351\) 8.12931 0.433910
\(352\) 0 0
\(353\) 28.0097i 1.49080i 0.666615 + 0.745402i \(0.267742\pi\)
−0.666615 + 0.745402i \(0.732258\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 9.72964i 0.514947i
\(358\) 0 0
\(359\) −0.214689 −0.0113308 −0.00566542 0.999984i \(-0.501803\pi\)
−0.00566542 + 0.999984i \(0.501803\pi\)
\(360\) 0 0
\(361\) 30.0520 1.58168
\(362\) 0 0
\(363\) 14.6936i 0.771213i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 21.0197i 1.09722i 0.836079 + 0.548610i \(0.184843\pi\)
−0.836079 + 0.548610i \(0.815157\pi\)
\(368\) 0 0
\(369\) −3.47488 −0.180895
\(370\) 0 0
\(371\) 24.5129 1.27265
\(372\) 0 0
\(373\) 10.2558i 0.531023i 0.964108 + 0.265511i \(0.0855408\pi\)
−0.964108 + 0.265511i \(0.914459\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 7.78116i − 0.400750i
\(378\) 0 0
\(379\) 20.6274 1.05956 0.529780 0.848135i \(-0.322274\pi\)
0.529780 + 0.848135i \(0.322274\pi\)
\(380\) 0 0
\(381\) 14.4451 0.740044
\(382\) 0 0
\(383\) − 18.7419i − 0.957667i −0.877906 0.478833i \(-0.841060\pi\)
0.877906 0.478833i \(-0.158940\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 11.5234i 0.585769i
\(388\) 0 0
\(389\) 3.07719 0.156020 0.0780100 0.996953i \(-0.475143\pi\)
0.0780100 + 0.996953i \(0.475143\pi\)
\(390\) 0 0
\(391\) 2.29775 0.116202
\(392\) 0 0
\(393\) 11.2995i 0.569986i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 11.4670i 0.575513i 0.957704 + 0.287756i \(0.0929093\pi\)
−0.957704 + 0.287756i \(0.907091\pi\)
\(398\) 0 0
\(399\) −29.6566 −1.48469
\(400\) 0 0
\(401\) 14.1227 0.705252 0.352626 0.935764i \(-0.385289\pi\)
0.352626 + 0.935764i \(0.385289\pi\)
\(402\) 0 0
\(403\) − 0.843666i − 0.0420260i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 0.888314i − 0.0440321i
\(408\) 0 0
\(409\) 0.851429 0.0421005 0.0210502 0.999778i \(-0.493299\pi\)
0.0210502 + 0.999778i \(0.493299\pi\)
\(410\) 0 0
\(411\) 24.7755 1.22208
\(412\) 0 0
\(413\) − 43.2111i − 2.12628i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 9.92799i 0.486176i
\(418\) 0 0
\(419\) −0.680728 −0.0332557 −0.0166279 0.999862i \(-0.505293\pi\)
−0.0166279 + 0.999862i \(0.505293\pi\)
\(420\) 0 0
\(421\) −35.9671 −1.75293 −0.876466 0.481465i \(-0.840105\pi\)
−0.876466 + 0.481465i \(0.840105\pi\)
\(422\) 0 0
\(423\) − 8.60251i − 0.418268i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0.742472i 0.0359307i
\(428\) 0 0
\(429\) −0.184324 −0.00889923
\(430\) 0 0
\(431\) 15.2881 0.736402 0.368201 0.929746i \(-0.379974\pi\)
0.368201 + 0.929746i \(0.379974\pi\)
\(432\) 0 0
\(433\) 13.2632i 0.637390i 0.947857 + 0.318695i \(0.103245\pi\)
−0.947857 + 0.318695i \(0.896755\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.00371i 0.335033i
\(438\) 0 0
\(439\) −18.7740 −0.896035 −0.448018 0.894025i \(-0.647870\pi\)
−0.448018 + 0.894025i \(0.647870\pi\)
\(440\) 0 0
\(441\) −3.67720 −0.175105
\(442\) 0 0
\(443\) − 5.49270i − 0.260966i −0.991451 0.130483i \(-0.958347\pi\)
0.991451 0.130483i \(-0.0416528\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 6.06905i − 0.287056i
\(448\) 0 0
\(449\) −13.1705 −0.621553 −0.310777 0.950483i \(-0.600589\pi\)
−0.310777 + 0.950483i \(0.600589\pi\)
\(450\) 0 0
\(451\) 0.273695 0.0128878
\(452\) 0 0
\(453\) − 0.00902890i 0 0.000424215i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 20.9676i − 0.980821i −0.871492 0.490410i \(-0.836847\pi\)
0.871492 0.490410i \(-0.163153\pi\)
\(458\) 0 0
\(459\) 12.9408 0.604026
\(460\) 0 0
\(461\) 6.27335 0.292179 0.146089 0.989271i \(-0.453331\pi\)
0.146089 + 0.989271i \(0.453331\pi\)
\(462\) 0 0
\(463\) 9.49565i 0.441300i 0.975353 + 0.220650i \(0.0708179\pi\)
−0.975353 + 0.220650i \(0.929182\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 13.9066i − 0.643518i −0.946822 0.321759i \(-0.895726\pi\)
0.946822 0.321759i \(-0.104274\pi\)
\(468\) 0 0
\(469\) 23.7466 1.09652
\(470\) 0 0
\(471\) −16.2727 −0.749809
\(472\) 0 0
\(473\) − 0.907631i − 0.0417329i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 9.38555i − 0.429735i
\(478\) 0 0
\(479\) −3.66220 −0.167330 −0.0836652 0.996494i \(-0.526663\pi\)
−0.0836652 + 0.996494i \(0.526663\pi\)
\(480\) 0 0
\(481\) −13.4237 −0.612066
\(482\) 0 0
\(483\) − 4.23441i − 0.192672i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 28.9654i − 1.31255i −0.754523 0.656274i \(-0.772132\pi\)
0.754523 0.656274i \(-0.227868\pi\)
\(488\) 0 0
\(489\) 27.0320 1.22243
\(490\) 0 0
\(491\) −37.8991 −1.71036 −0.855182 0.518328i \(-0.826554\pi\)
−0.855182 + 0.518328i \(0.826554\pi\)
\(492\) 0 0
\(493\) − 12.3866i − 0.557866i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 16.4204i − 0.736557i
\(498\) 0 0
\(499\) 26.6914 1.19487 0.597436 0.801917i \(-0.296186\pi\)
0.597436 + 0.801917i \(0.296186\pi\)
\(500\) 0 0
\(501\) −10.7581 −0.480635
\(502\) 0 0
\(503\) − 18.2182i − 0.812309i −0.913804 0.406154i \(-0.866870\pi\)
0.913804 0.406154i \(-0.133130\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 14.5942i − 0.648150i
\(508\) 0 0
\(509\) 31.0138 1.37466 0.687332 0.726344i \(-0.258782\pi\)
0.687332 + 0.726344i \(0.258782\pi\)
\(510\) 0 0
\(511\) 4.82655 0.213514
\(512\) 0 0
\(513\) 39.4446i 1.74152i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0.677568i 0.0297994i
\(518\) 0 0
\(519\) 22.5125 0.988188
\(520\) 0 0
\(521\) −39.9824 −1.75166 −0.875831 0.482618i \(-0.839686\pi\)
−0.875831 + 0.482618i \(0.839686\pi\)
\(522\) 0 0
\(523\) 10.3111i 0.450873i 0.974258 + 0.225437i \(0.0723808\pi\)
−0.974258 + 0.225437i \(0.927619\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 1.34301i − 0.0585024i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −16.5448 −0.717981
\(532\) 0 0
\(533\) − 4.13592i − 0.179146i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 31.7699i 1.37097i
\(538\) 0 0
\(539\) 0.289631 0.0124753
\(540\) 0 0
\(541\) −42.5472 −1.82925 −0.914623 0.404307i \(-0.867513\pi\)
−0.914623 + 0.404307i \(0.867513\pi\)
\(542\) 0 0
\(543\) 4.19139i 0.179870i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 2.47986i − 0.106031i −0.998594 0.0530156i \(-0.983117\pi\)
0.998594 0.0530156i \(-0.0168833\pi\)
\(548\) 0 0
\(549\) 0.284279 0.0121327
\(550\) 0 0
\(551\) 37.7553 1.60843
\(552\) 0 0
\(553\) 9.63094i 0.409549i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 22.9814i − 0.973755i −0.873470 0.486877i \(-0.838136\pi\)
0.873470 0.486877i \(-0.161864\pi\)
\(558\) 0 0
\(559\) −13.7156 −0.580107
\(560\) 0 0
\(561\) −0.293420 −0.0123882
\(562\) 0 0
\(563\) − 18.4939i − 0.779427i −0.920936 0.389713i \(-0.872574\pi\)
0.920936 0.389713i \(-0.127426\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 12.3246i − 0.517584i
\(568\) 0 0
\(569\) −35.6081 −1.49277 −0.746385 0.665514i \(-0.768212\pi\)
−0.746385 + 0.665514i \(0.768212\pi\)
\(570\) 0 0
\(571\) 20.5827 0.861359 0.430680 0.902505i \(-0.358274\pi\)
0.430680 + 0.902505i \(0.358274\pi\)
\(572\) 0 0
\(573\) − 8.25645i − 0.344918i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 19.0378i 0.792554i 0.918131 + 0.396277i \(0.129698\pi\)
−0.918131 + 0.396277i \(0.870302\pi\)
\(578\) 0 0
\(579\) 9.65591 0.401286
\(580\) 0 0
\(581\) 50.6489 2.10127
\(582\) 0 0
\(583\) 0.739244i 0.0306163i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 7.29062i − 0.300916i −0.988616 0.150458i \(-0.951925\pi\)
0.988616 0.150458i \(-0.0480748\pi\)
\(588\) 0 0
\(589\) 4.09358 0.168673
\(590\) 0 0
\(591\) 17.3821 0.715006
\(592\) 0 0
\(593\) 10.5832i 0.434598i 0.976105 + 0.217299i \(0.0697247\pi\)
−0.976105 + 0.217299i \(0.930275\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 2.46140i − 0.100739i
\(598\) 0 0
\(599\) 5.63520 0.230248 0.115124 0.993351i \(-0.463273\pi\)
0.115124 + 0.993351i \(0.463273\pi\)
\(600\) 0 0
\(601\) −17.7093 −0.722377 −0.361188 0.932493i \(-0.617629\pi\)
−0.361188 + 0.932493i \(0.617629\pi\)
\(602\) 0 0
\(603\) − 9.09216i − 0.370261i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 22.0996i − 0.896996i −0.893784 0.448498i \(-0.851959\pi\)
0.893784 0.448498i \(-0.148041\pi\)
\(608\) 0 0
\(609\) −22.8267 −0.924984
\(610\) 0 0
\(611\) 10.2390 0.414225
\(612\) 0 0
\(613\) 5.08532i 0.205394i 0.994713 + 0.102697i \(0.0327472\pi\)
−0.994713 + 0.102697i \(0.967253\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 7.63073i − 0.307202i −0.988133 0.153601i \(-0.950913\pi\)
0.988133 0.153601i \(-0.0490870\pi\)
\(618\) 0 0
\(619\) −38.5591 −1.54982 −0.774910 0.632072i \(-0.782205\pi\)
−0.774910 + 0.632072i \(0.782205\pi\)
\(620\) 0 0
\(621\) −5.63195 −0.226002
\(622\) 0 0
\(623\) − 17.5258i − 0.702156i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 0.894364i − 0.0357175i
\(628\) 0 0
\(629\) −21.3688 −0.852028
\(630\) 0 0
\(631\) −1.86751 −0.0743444 −0.0371722 0.999309i \(-0.511835\pi\)
−0.0371722 + 0.999309i \(0.511835\pi\)
\(632\) 0 0
\(633\) 33.2607i 1.32199i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 4.37673i − 0.173412i
\(638\) 0 0
\(639\) −6.28708 −0.248713
\(640\) 0 0
\(641\) −1.91236 −0.0755338 −0.0377669 0.999287i \(-0.512024\pi\)
−0.0377669 + 0.999287i \(0.512024\pi\)
\(642\) 0 0
\(643\) 5.06670i 0.199811i 0.994997 + 0.0999055i \(0.0318541\pi\)
−0.994997 + 0.0999055i \(0.968146\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 37.0429i − 1.45631i −0.685414 0.728153i \(-0.740379\pi\)
0.685414 0.728153i \(-0.259621\pi\)
\(648\) 0 0
\(649\) 1.30313 0.0511524
\(650\) 0 0
\(651\) −2.47496 −0.0970014
\(652\) 0 0
\(653\) − 11.9884i − 0.469143i −0.972099 0.234572i \(-0.924631\pi\)
0.972099 0.234572i \(-0.0753687\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 1.84800i − 0.0720974i
\(658\) 0 0
\(659\) 2.82513 0.110051 0.0550257 0.998485i \(-0.482476\pi\)
0.0550257 + 0.998485i \(0.482476\pi\)
\(660\) 0 0
\(661\) −26.5004 −1.03075 −0.515374 0.856965i \(-0.672347\pi\)
−0.515374 + 0.856965i \(0.672347\pi\)
\(662\) 0 0
\(663\) 4.43398i 0.172202i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 5.39076i 0.208731i
\(668\) 0 0
\(669\) −5.41339 −0.209294
\(670\) 0 0
\(671\) −0.0223910 −0.000864393 0
\(672\) 0 0
\(673\) − 19.0620i − 0.734784i −0.930066 0.367392i \(-0.880251\pi\)
0.930066 0.367392i \(-0.119749\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 2.54346i 0.0977533i 0.998805 + 0.0488766i \(0.0155641\pi\)
−0.998805 + 0.0488766i \(0.984436\pi\)
\(678\) 0 0
\(679\) 8.18146 0.313975
\(680\) 0 0
\(681\) 23.2507 0.890969
\(682\) 0 0
\(683\) 0.377109i 0.0144297i 0.999974 + 0.00721483i \(0.00229657\pi\)
−0.999974 + 0.00721483i \(0.997703\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 4.70266i − 0.179418i
\(688\) 0 0
\(689\) 11.1710 0.425581
\(690\) 0 0
\(691\) 7.24743 0.275705 0.137853 0.990453i \(-0.455980\pi\)
0.137853 + 0.990453i \(0.455980\pi\)
\(692\) 0 0
\(693\) − 0.366903i − 0.0139375i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 6.58385i − 0.249381i
\(698\) 0 0
\(699\) 14.4264 0.545655
\(700\) 0 0
\(701\) −15.7381 −0.594420 −0.297210 0.954812i \(-0.596056\pi\)
−0.297210 + 0.954812i \(0.596056\pi\)
\(702\) 0 0
\(703\) − 65.1335i − 2.45656i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 21.0824i − 0.792887i
\(708\) 0 0
\(709\) 45.9062 1.72404 0.862022 0.506870i \(-0.169198\pi\)
0.862022 + 0.506870i \(0.169198\pi\)
\(710\) 0 0
\(711\) 3.68751 0.138293
\(712\) 0 0
\(713\) 0.584488i 0.0218892i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 3.83612i − 0.143263i
\(718\) 0 0
\(719\) 50.4975 1.88324 0.941619 0.336682i \(-0.109305\pi\)
0.941619 + 0.336682i \(0.109305\pi\)
\(720\) 0 0
\(721\) 55.9570 2.08395
\(722\) 0 0
\(723\) − 36.1609i − 1.34484i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 19.0273i − 0.705685i −0.935683 0.352842i \(-0.885215\pi\)
0.935683 0.352842i \(-0.114785\pi\)
\(728\) 0 0
\(729\) −27.3067 −1.01136
\(730\) 0 0
\(731\) −21.8334 −0.807539
\(732\) 0 0
\(733\) 42.1027i 1.55510i 0.628822 + 0.777549i \(0.283537\pi\)
−0.628822 + 0.777549i \(0.716463\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0.716135i 0.0263792i
\(738\) 0 0
\(739\) −47.9212 −1.76281 −0.881405 0.472361i \(-0.843402\pi\)
−0.881405 + 0.472361i \(0.843402\pi\)
\(740\) 0 0
\(741\) −13.5151 −0.496489
\(742\) 0 0
\(743\) − 21.3492i − 0.783225i −0.920130 0.391613i \(-0.871917\pi\)
0.920130 0.391613i \(-0.128083\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 19.3925i − 0.709536i
\(748\) 0 0
\(749\) −18.4858 −0.675458
\(750\) 0 0
\(751\) −27.8218 −1.01523 −0.507617 0.861583i \(-0.669473\pi\)
−0.507617 + 0.861583i \(0.669473\pi\)
\(752\) 0 0
\(753\) 15.7478i 0.573881i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 8.74990i 0.318020i 0.987277 + 0.159010i \(0.0508303\pi\)
−0.987277 + 0.159010i \(0.949170\pi\)
\(758\) 0 0
\(759\) 0.127699 0.00463517
\(760\) 0 0
\(761\) 21.9923 0.797221 0.398610 0.917120i \(-0.369493\pi\)
0.398610 + 0.917120i \(0.369493\pi\)
\(762\) 0 0
\(763\) − 22.2379i − 0.805066i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 19.6921i − 0.711041i
\(768\) 0 0
\(769\) 31.3229 1.12953 0.564766 0.825251i \(-0.308966\pi\)
0.564766 + 0.825251i \(0.308966\pi\)
\(770\) 0 0
\(771\) 11.3334 0.408162
\(772\) 0 0
\(773\) − 32.3308i − 1.16286i −0.813597 0.581429i \(-0.802494\pi\)
0.813597 0.581429i \(-0.197506\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 39.3794i 1.41273i
\(778\) 0 0
\(779\) 20.0680 0.719012
\(780\) 0 0
\(781\) 0.495196 0.0177195
\(782\) 0 0
\(783\) 30.3605i 1.08499i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 18.5712i − 0.661990i −0.943632 0.330995i \(-0.892616\pi\)
0.943632 0.330995i \(-0.107384\pi\)
\(788\) 0 0
\(789\) −24.7483 −0.881063
\(790\) 0 0
\(791\) 54.1627 1.92580
\(792\) 0 0
\(793\) 0.338358i 0.0120155i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 24.1985i − 0.857154i −0.903505 0.428577i \(-0.859015\pi\)
0.903505 0.428577i \(-0.140985\pi\)
\(798\) 0 0
\(799\) 16.2992 0.576624
\(800\) 0 0
\(801\) −6.71031 −0.237097
\(802\) 0 0
\(803\) 0.145556i 0.00513656i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 34.0349i − 1.19809i
\(808\) 0 0
\(809\) 30.4574 1.07082 0.535412 0.844591i \(-0.320156\pi\)
0.535412 + 0.844591i \(0.320156\pi\)
\(810\) 0 0
\(811\) 4.05626 0.142435 0.0712174 0.997461i \(-0.477312\pi\)
0.0712174 + 0.997461i \(0.477312\pi\)
\(812\) 0 0
\(813\) 18.9587i 0.664910i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 66.5499i − 2.32829i
\(818\) 0 0
\(819\) −5.54441 −0.193737
\(820\) 0 0
\(821\) 19.0615 0.665252 0.332626 0.943059i \(-0.392065\pi\)
0.332626 + 0.943059i \(0.392065\pi\)
\(822\) 0 0
\(823\) 42.9810i 1.49822i 0.662444 + 0.749111i \(0.269519\pi\)
−0.662444 + 0.749111i \(0.730481\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 51.6777i 1.79701i 0.438963 + 0.898505i \(0.355346\pi\)
−0.438963 + 0.898505i \(0.644654\pi\)
\(828\) 0 0
\(829\) −25.6228 −0.889915 −0.444958 0.895552i \(-0.646781\pi\)
−0.444958 + 0.895552i \(0.646781\pi\)
\(830\) 0 0
\(831\) 28.6470 0.993753
\(832\) 0 0
\(833\) − 6.96720i − 0.241399i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 3.29181i 0.113781i
\(838\) 0 0
\(839\) 53.0313 1.83084 0.915422 0.402495i \(-0.131857\pi\)
0.915422 + 0.402495i \(0.131857\pi\)
\(840\) 0 0
\(841\) 0.0602571 0.00207783
\(842\) 0 0
\(843\) 11.9021i 0.409929i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 34.8121i − 1.19616i
\(848\) 0 0
\(849\) 17.4290 0.598160
\(850\) 0 0
\(851\) 9.29985 0.318795
\(852\) 0 0
\(853\) − 13.8668i − 0.474791i −0.971413 0.237395i \(-0.923706\pi\)
0.971413 0.237395i \(-0.0762937\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 16.3518i 0.558567i 0.960209 + 0.279283i \(0.0900969\pi\)
−0.960209 + 0.279283i \(0.909903\pi\)
\(858\) 0 0
\(859\) −42.1443 −1.43795 −0.718973 0.695038i \(-0.755388\pi\)
−0.718973 + 0.695038i \(0.755388\pi\)
\(860\) 0 0
\(861\) −12.1331 −0.413493
\(862\) 0 0
\(863\) 15.2130i 0.517856i 0.965897 + 0.258928i \(0.0833692\pi\)
−0.965897 + 0.258928i \(0.916631\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 15.6688i − 0.532140i
\(868\) 0 0
\(869\) −0.290443 −0.00985262
\(870\) 0 0
\(871\) 10.8218 0.366683
\(872\) 0 0
\(873\) − 3.13253i − 0.106020i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 53.6934i − 1.81310i −0.422101 0.906549i \(-0.638707\pi\)
0.422101 0.906549i \(-0.361293\pi\)
\(878\) 0 0
\(879\) 9.02096 0.304269
\(880\) 0 0
\(881\) 46.5787 1.56928 0.784638 0.619954i \(-0.212849\pi\)
0.784638 + 0.619954i \(0.212849\pi\)
\(882\) 0 0
\(883\) 19.9787i 0.672337i 0.941802 + 0.336169i \(0.109131\pi\)
−0.941802 + 0.336169i \(0.890869\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 51.9251i 1.74347i 0.489974 + 0.871737i \(0.337006\pi\)
−0.489974 + 0.871737i \(0.662994\pi\)
\(888\) 0 0
\(889\) −34.2233 −1.14781
\(890\) 0 0
\(891\) 0.371676 0.0124516
\(892\) 0 0
\(893\) 49.6810i 1.66251i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 1.92970i − 0.0644309i
\(898\) 0 0
\(899\) 3.15083 0.105086
\(900\) 0 0
\(901\) 17.7828 0.592431
\(902\) 0 0
\(903\) 40.2358i 1.33896i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 6.64335i − 0.220589i −0.993899 0.110294i \(-0.964821\pi\)
0.993899 0.110294i \(-0.0351794\pi\)
\(908\) 0 0
\(909\) −8.07209 −0.267734
\(910\) 0 0
\(911\) 15.2064 0.503811 0.251905 0.967752i \(-0.418943\pi\)
0.251905 + 0.967752i \(0.418943\pi\)
\(912\) 0 0
\(913\) 1.52744i 0.0505507i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 26.7709i − 0.884052i
\(918\) 0 0
\(919\) −41.2068 −1.35929 −0.679644 0.733542i \(-0.737866\pi\)
−0.679644 + 0.733542i \(0.737866\pi\)
\(920\) 0 0
\(921\) 20.9446 0.690148
\(922\) 0 0
\(923\) − 7.48310i − 0.246309i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 21.4249i − 0.703687i
\(928\) 0 0
\(929\) 4.86715 0.159686 0.0798431 0.996807i \(-0.474558\pi\)
0.0798431 + 0.996807i \(0.474558\pi\)
\(930\) 0 0
\(931\) 21.2365 0.695999
\(932\) 0 0
\(933\) − 11.0684i − 0.362364i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 3.44469i − 0.112533i −0.998416 0.0562666i \(-0.982080\pi\)
0.998416 0.0562666i \(-0.0179197\pi\)
\(938\) 0 0
\(939\) −20.7081 −0.675784
\(940\) 0 0
\(941\) −34.8591 −1.13637 −0.568187 0.822900i \(-0.692355\pi\)
−0.568187 + 0.822900i \(0.692355\pi\)
\(942\) 0 0
\(943\) 2.86534i 0.0933084i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 5.17298i − 0.168099i −0.996462 0.0840496i \(-0.973215\pi\)
0.996462 0.0840496i \(-0.0267854\pi\)
\(948\) 0 0
\(949\) 2.19955 0.0714005
\(950\) 0 0
\(951\) −14.1447 −0.458675
\(952\) 0 0
\(953\) − 3.41090i − 0.110490i −0.998473 0.0552449i \(-0.982406\pi\)
0.998473 0.0552449i \(-0.0175940\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 0.688392i − 0.0222526i
\(958\) 0 0
\(959\) −58.6981 −1.89546
\(960\) 0 0
\(961\) −30.6584 −0.988980
\(962\) 0 0
\(963\) 7.07789i 0.228082i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 17.5047i 0.562914i 0.959574 + 0.281457i \(0.0908176\pi\)
−0.959574 + 0.281457i \(0.909182\pi\)
\(968\) 0 0
\(969\) −21.5143 −0.691139
\(970\) 0 0
\(971\) 35.9954 1.15515 0.577574 0.816338i \(-0.303999\pi\)
0.577574 + 0.816338i \(0.303999\pi\)
\(972\) 0 0
\(973\) − 23.5214i − 0.754062i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 31.3634i − 1.00340i −0.865040 0.501702i \(-0.832707\pi\)
0.865040 0.501702i \(-0.167293\pi\)
\(978\) 0 0
\(979\) 0.528531 0.0168919
\(980\) 0 0
\(981\) −8.51450 −0.271847
\(982\) 0 0
\(983\) 36.4262i 1.16182i 0.813969 + 0.580908i \(0.197302\pi\)
−0.813969 + 0.580908i \(0.802698\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 30.0369i − 0.956086i
\(988\) 0 0
\(989\) 9.50208 0.302149
\(990\) 0 0
\(991\) 6.22316 0.197685 0.0988426 0.995103i \(-0.468486\pi\)
0.0988426 + 0.995103i \(0.468486\pi\)
\(992\) 0 0
\(993\) 1.77587i 0.0563556i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 24.0896i − 0.762925i −0.924384 0.381463i \(-0.875420\pi\)
0.924384 0.381463i \(-0.124580\pi\)
\(998\) 0 0
\(999\) 52.3763 1.65711
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.4049.4 10
5.2 odd 4 4600.2.a.bc.1.3 5
5.3 odd 4 4600.2.a.bg.1.3 yes 5
5.4 even 2 inner 4600.2.e.v.4049.7 10
20.3 even 4 9200.2.a.cs.1.3 5
20.7 even 4 9200.2.a.cw.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bc.1.3 5 5.2 odd 4
4600.2.a.bg.1.3 yes 5 5.3 odd 4
4600.2.e.v.4049.4 10 1.1 even 1 trivial
4600.2.e.v.4049.7 10 5.4 even 2 inner
9200.2.a.cs.1.3 5 20.3 even 4
9200.2.a.cw.1.3 5 20.7 even 4