Properties

Label 4600.2.e.u.4049.2
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(x^{10} + 28 x^{8} + 260 x^{6} + 897 x^{4} + 1056 x^{2} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.2
Root \(-3.30649i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.u.4049.9

$q$-expansion

\(f(q)\) \(=\) \(q-3.30649i q^{3} -2.55040i q^{7} -7.93288 q^{9} +O(q^{10})\) \(q-3.30649i q^{3} -2.55040i q^{7} -7.93288 q^{9} -2.72314 q^{11} -7.12637i q^{13} +0.924010i q^{17} -7.51623 q^{19} -8.43286 q^{21} +1.00000i q^{23} +16.3105i q^{27} +2.38248 q^{29} +0.866248 q^{31} +9.00402i q^{33} +0.352855i q^{37} -23.5633 q^{39} +4.34066 q^{41} -13.3239i q^{47} +0.495474 q^{49} +3.05523 q^{51} -3.99262i q^{53} +24.8523i q^{57} +3.84064 q^{59} -9.14262 q^{61} +20.2320i q^{63} -3.15933i q^{67} +3.30649 q^{69} -6.07883 q^{71} +11.3239i q^{73} +6.94508i q^{77} +12.0593 q^{79} +30.1319 q^{81} +6.35285i q^{83} -7.87765i q^{87} +9.71377 q^{89} -18.1751 q^{91} -2.86424i q^{93} +8.76465i q^{97} +21.6023 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 26 q^{9} + O(q^{10}) \) \( 10 q - 26 q^{9} - 2 q^{11} - 14 q^{19} + 12 q^{21} - 8 q^{29} + 38 q^{31} - 38 q^{39} + 50 q^{41} - 50 q^{49} + 38 q^{51} + 2 q^{59} - 10 q^{61} + 2 q^{71} + 4 q^{79} + 114 q^{81} - 12 q^{89} + 22 q^{91} + 130 q^{99} + O(q^{100}) \)

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
\(3\) − 3.30649i − 1.90900i −0.298206 0.954501i \(-0.596388\pi\)
0.298206 0.954501i \(-0.403612\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.55040i − 0.963960i −0.876182 0.481980i \(-0.839918\pi\)
0.876182 0.481980i \(-0.160082\pi\)
\(8\) 0 0
\(9\) −7.93288 −2.64429
\(10\) 0 0
\(11\) −2.72314 −0.821056 −0.410528 0.911848i \(-0.634656\pi\)
−0.410528 + 0.911848i \(0.634656\pi\)
\(12\) 0 0
\(13\) − 7.12637i − 1.97650i −0.152845 0.988250i \(-0.548844\pi\)
0.152845 0.988250i \(-0.451156\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.924010i 0.224105i 0.993702 + 0.112053i \(0.0357426\pi\)
−0.993702 + 0.112053i \(0.964257\pi\)
\(18\) 0 0
\(19\) −7.51623 −1.72434 −0.862171 0.506617i \(-0.830896\pi\)
−0.862171 + 0.506617i \(0.830896\pi\)
\(20\) 0 0
\(21\) −8.43286 −1.84020
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 16.3105i 3.13896i
\(28\) 0 0
\(29\) 2.38248 0.442415 0.221208 0.975227i \(-0.429000\pi\)
0.221208 + 0.975227i \(0.429000\pi\)
\(30\) 0 0
\(31\) 0.866248 0.155583 0.0777913 0.996970i \(-0.475213\pi\)
0.0777913 + 0.996970i \(0.475213\pi\)
\(32\) 0 0
\(33\) 9.00402i 1.56740i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.352855i 0.0580089i 0.999579 + 0.0290045i \(0.00923370\pi\)
−0.999579 + 0.0290045i \(0.990766\pi\)
\(38\) 0 0
\(39\) −23.5633 −3.77315
\(40\) 0 0
\(41\) 4.34066 0.677896 0.338948 0.940805i \(-0.389929\pi\)
0.338948 + 0.940805i \(0.389929\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 13.3239i − 1.94349i −0.236027 0.971746i \(-0.575845\pi\)
0.236027 0.971746i \(-0.424155\pi\)
\(48\) 0 0
\(49\) 0.495474 0.0707819
\(50\) 0 0
\(51\) 3.05523 0.427818
\(52\) 0 0
\(53\) − 3.99262i − 0.548429i −0.961669 0.274214i \(-0.911582\pi\)
0.961669 0.274214i \(-0.0884178\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 24.8523i 3.29177i
\(58\) 0 0
\(59\) 3.84064 0.500009 0.250004 0.968245i \(-0.419568\pi\)
0.250004 + 0.968245i \(0.419568\pi\)
\(60\) 0 0
\(61\) −9.14262 −1.17059 −0.585296 0.810820i \(-0.699022\pi\)
−0.585296 + 0.810820i \(0.699022\pi\)
\(62\) 0 0
\(63\) 20.2320i 2.54899i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 3.15933i − 0.385974i −0.981201 0.192987i \(-0.938183\pi\)
0.981201 0.192987i \(-0.0618175\pi\)
\(68\) 0 0
\(69\) 3.30649 0.398055
\(70\) 0 0
\(71\) −6.07883 −0.721424 −0.360712 0.932677i \(-0.617466\pi\)
−0.360712 + 0.932677i \(0.617466\pi\)
\(72\) 0 0
\(73\) 11.3239i 1.32536i 0.748901 + 0.662682i \(0.230582\pi\)
−0.748901 + 0.662682i \(0.769418\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.94508i 0.791465i
\(78\) 0 0
\(79\) 12.0593 1.35677 0.678386 0.734706i \(-0.262680\pi\)
0.678386 + 0.734706i \(0.262680\pi\)
\(80\) 0 0
\(81\) 30.1319 3.34799
\(82\) 0 0
\(83\) 6.35285i 0.697316i 0.937250 + 0.348658i \(0.113363\pi\)
−0.937250 + 0.348658i \(0.886637\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 7.87765i − 0.844572i
\(88\) 0 0
\(89\) 9.71377 1.02966 0.514829 0.857293i \(-0.327855\pi\)
0.514829 + 0.857293i \(0.327855\pi\)
\(90\) 0 0
\(91\) −18.1751 −1.90527
\(92\) 0 0
\(93\) − 2.86424i − 0.297008i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.76465i 0.889916i 0.895552 + 0.444958i \(0.146781\pi\)
−0.895552 + 0.444958i \(0.853219\pi\)
\(98\) 0 0
\(99\) 21.6023 2.17111
\(100\) 0 0
\(101\) 4.74794 0.472438 0.236219 0.971700i \(-0.424092\pi\)
0.236219 + 0.971700i \(0.424092\pi\)
\(102\) 0 0
\(103\) − 12.6304i − 1.24451i −0.782814 0.622255i \(-0.786217\pi\)
0.782814 0.622255i \(-0.213783\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 12.9658i − 1.25345i −0.779239 0.626727i \(-0.784394\pi\)
0.779239 0.626727i \(-0.215606\pi\)
\(108\) 0 0
\(109\) 15.0040 1.43712 0.718562 0.695463i \(-0.244801\pi\)
0.718562 + 0.695463i \(0.244801\pi\)
\(110\) 0 0
\(111\) 1.16671 0.110739
\(112\) 0 0
\(113\) − 2.13496i − 0.200840i −0.994945 0.100420i \(-0.967981\pi\)
0.994945 0.100420i \(-0.0320187\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 56.5326i 5.22644i
\(118\) 0 0
\(119\) 2.35659 0.216029
\(120\) 0 0
\(121\) −3.58454 −0.325867
\(122\) 0 0
\(123\) − 14.3523i − 1.29411i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 3.54181i − 0.314285i −0.987576 0.157142i \(-0.949772\pi\)
0.987576 0.157142i \(-0.0502282\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 19.5499 1.70808 0.854042 0.520205i \(-0.174144\pi\)
0.854042 + 0.520205i \(0.174144\pi\)
\(132\) 0 0
\(133\) 19.1694i 1.66220i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.41544i 0.377236i 0.982051 + 0.188618i \(0.0604008\pi\)
−0.982051 + 0.188618i \(0.939599\pi\)
\(138\) 0 0
\(139\) −21.9954 −1.86563 −0.932814 0.360358i \(-0.882655\pi\)
−0.932814 + 0.360358i \(0.882655\pi\)
\(140\) 0 0
\(141\) −44.0554 −3.71013
\(142\) 0 0
\(143\) 19.4061i 1.62282i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 1.63828i − 0.135123i
\(148\) 0 0
\(149\) −19.0780 −1.56293 −0.781466 0.623947i \(-0.785528\pi\)
−0.781466 + 0.623947i \(0.785528\pi\)
\(150\) 0 0
\(151\) −7.75273 −0.630908 −0.315454 0.948941i \(-0.602157\pi\)
−0.315454 + 0.948941i \(0.602157\pi\)
\(152\) 0 0
\(153\) − 7.33006i − 0.592600i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 17.5129i 1.39768i 0.715277 + 0.698841i \(0.246300\pi\)
−0.715277 + 0.698841i \(0.753700\pi\)
\(158\) 0 0
\(159\) −13.2016 −1.04695
\(160\) 0 0
\(161\) 2.55040 0.200999
\(162\) 0 0
\(163\) 2.55491i 0.200116i 0.994982 + 0.100058i \(0.0319028\pi\)
−0.994982 + 0.100058i \(0.968097\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 3.61301i − 0.279583i −0.990181 0.139791i \(-0.955357\pi\)
0.990181 0.139791i \(-0.0446432\pi\)
\(168\) 0 0
\(169\) −37.7852 −2.90655
\(170\) 0 0
\(171\) 59.6253 4.55966
\(172\) 0 0
\(173\) − 12.7824i − 0.971827i −0.874007 0.485913i \(-0.838487\pi\)
0.874007 0.485913i \(-0.161513\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 12.6990i − 0.954519i
\(178\) 0 0
\(179\) −10.0149 −0.748546 −0.374273 0.927318i \(-0.622108\pi\)
−0.374273 + 0.927318i \(0.622108\pi\)
\(180\) 0 0
\(181\) 9.96248 0.740505 0.370252 0.928931i \(-0.379271\pi\)
0.370252 + 0.928931i \(0.379271\pi\)
\(182\) 0 0
\(183\) 30.2300i 2.23466i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 2.51621i − 0.184003i
\(188\) 0 0
\(189\) 41.5983 3.02583
\(190\) 0 0
\(191\) −6.61298 −0.478498 −0.239249 0.970958i \(-0.576901\pi\)
−0.239249 + 0.970958i \(0.576901\pi\)
\(192\) 0 0
\(193\) 19.0540i 1.37154i 0.727820 + 0.685768i \(0.240534\pi\)
−0.727820 + 0.685768i \(0.759466\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 11.0886i − 0.790030i −0.918675 0.395015i \(-0.870739\pi\)
0.918675 0.395015i \(-0.129261\pi\)
\(198\) 0 0
\(199\) −3.71377 −0.263263 −0.131631 0.991299i \(-0.542021\pi\)
−0.131631 + 0.991299i \(0.542021\pi\)
\(200\) 0 0
\(201\) −10.4463 −0.736825
\(202\) 0 0
\(203\) − 6.07627i − 0.426471i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 7.93288i − 0.551373i
\(208\) 0 0
\(209\) 20.4677 1.41578
\(210\) 0 0
\(211\) −21.7656 −1.49841 −0.749204 0.662339i \(-0.769564\pi\)
−0.749204 + 0.662339i \(0.769564\pi\)
\(212\) 0 0
\(213\) 20.0996i 1.37720i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 2.20928i − 0.149975i
\(218\) 0 0
\(219\) 37.4424 2.53012
\(220\) 0 0
\(221\) 6.58484 0.442944
\(222\) 0 0
\(223\) − 6.90730i − 0.462547i −0.972889 0.231273i \(-0.925711\pi\)
0.972889 0.231273i \(-0.0742892\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 23.0325i − 1.52872i −0.644791 0.764359i \(-0.723055\pi\)
0.644791 0.764359i \(-0.276945\pi\)
\(228\) 0 0
\(229\) −12.9665 −0.856854 −0.428427 0.903576i \(-0.640932\pi\)
−0.428427 + 0.903576i \(0.640932\pi\)
\(230\) 0 0
\(231\) 22.9638 1.51091
\(232\) 0 0
\(233\) 4.80934i 0.315071i 0.987513 + 0.157535i \(0.0503548\pi\)
−0.987513 + 0.157535i \(0.949645\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 39.8738i − 2.59008i
\(238\) 0 0
\(239\) 20.4661 1.32384 0.661922 0.749573i \(-0.269741\pi\)
0.661922 + 0.749573i \(0.269741\pi\)
\(240\) 0 0
\(241\) 26.8657 1.73057 0.865287 0.501277i \(-0.167136\pi\)
0.865287 + 0.501277i \(0.167136\pi\)
\(242\) 0 0
\(243\) − 50.6993i − 3.25236i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 53.5635i 3.40816i
\(248\) 0 0
\(249\) 21.0057 1.33118
\(250\) 0 0
\(251\) −15.9625 −1.00754 −0.503771 0.863837i \(-0.668055\pi\)
−0.503771 + 0.863837i \(0.668055\pi\)
\(252\) 0 0
\(253\) − 2.72314i − 0.171202i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.33867i 0.208261i 0.994564 + 0.104130i \(0.0332059\pi\)
−0.994564 + 0.104130i \(0.966794\pi\)
\(258\) 0 0
\(259\) 0.899919 0.0559183
\(260\) 0 0
\(261\) −18.8999 −1.16988
\(262\) 0 0
\(263\) 23.3880i 1.44217i 0.692849 + 0.721083i \(0.256355\pi\)
−0.692849 + 0.721083i \(0.743645\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 32.1185i − 1.96562i
\(268\) 0 0
\(269\) −14.1378 −0.861995 −0.430998 0.902353i \(-0.641838\pi\)
−0.430998 + 0.902353i \(0.641838\pi\)
\(270\) 0 0
\(271\) 25.9283 1.57503 0.787517 0.616293i \(-0.211366\pi\)
0.787517 + 0.616293i \(0.211366\pi\)
\(272\) 0 0
\(273\) 60.0957i 3.63716i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.07117i 0.0643603i 0.999482 + 0.0321802i \(0.0102450\pi\)
−0.999482 + 0.0321802i \(0.989755\pi\)
\(278\) 0 0
\(279\) −6.87184 −0.411406
\(280\) 0 0
\(281\) 12.2260 0.729341 0.364671 0.931137i \(-0.381182\pi\)
0.364671 + 0.931137i \(0.381182\pi\)
\(282\) 0 0
\(283\) 30.5454i 1.81573i 0.419259 + 0.907867i \(0.362290\pi\)
−0.419259 + 0.907867i \(0.637710\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 11.0704i − 0.653465i
\(288\) 0 0
\(289\) 16.1462 0.949777
\(290\) 0 0
\(291\) 28.9802 1.69885
\(292\) 0 0
\(293\) 8.46838i 0.494728i 0.968923 + 0.247364i \(0.0795644\pi\)
−0.968923 + 0.247364i \(0.920436\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 44.4157i − 2.57726i
\(298\) 0 0
\(299\) 7.12637 0.412129
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) − 15.6990i − 0.901885i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 1.32802i − 0.0757942i −0.999282 0.0378971i \(-0.987934\pi\)
0.999282 0.0378971i \(-0.0120659\pi\)
\(308\) 0 0
\(309\) −41.7623 −2.37578
\(310\) 0 0
\(311\) 24.2971 1.37776 0.688882 0.724874i \(-0.258102\pi\)
0.688882 + 0.724874i \(0.258102\pi\)
\(312\) 0 0
\(313\) − 20.3478i − 1.15013i −0.818109 0.575063i \(-0.804977\pi\)
0.818109 0.575063i \(-0.195023\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 8.57309i − 0.481513i −0.970586 0.240756i \(-0.922604\pi\)
0.970586 0.240756i \(-0.0773955\pi\)
\(318\) 0 0
\(319\) −6.48781 −0.363248
\(320\) 0 0
\(321\) −42.8714 −2.39285
\(322\) 0 0
\(323\) − 6.94508i − 0.386434i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 49.6106i − 2.74347i
\(328\) 0 0
\(329\) −33.9813 −1.87345
\(330\) 0 0
\(331\) −20.9767 −1.15299 −0.576493 0.817102i \(-0.695579\pi\)
−0.576493 + 0.817102i \(0.695579\pi\)
\(332\) 0 0
\(333\) − 2.79915i − 0.153393i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 2.17314i − 0.118379i −0.998247 0.0591894i \(-0.981148\pi\)
0.998247 0.0591894i \(-0.0188516\pi\)
\(338\) 0 0
\(339\) −7.05922 −0.383404
\(340\) 0 0
\(341\) −2.35891 −0.127742
\(342\) 0 0
\(343\) − 19.1164i − 1.03219i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 32.1335i − 1.72502i −0.506042 0.862509i \(-0.668892\pi\)
0.506042 0.862509i \(-0.331108\pi\)
\(348\) 0 0
\(349\) −13.5157 −0.723479 −0.361740 0.932279i \(-0.617817\pi\)
−0.361740 + 0.932279i \(0.617817\pi\)
\(350\) 0 0
\(351\) 116.235 6.20415
\(352\) 0 0
\(353\) 28.0645i 1.49372i 0.664981 + 0.746861i \(0.268440\pi\)
−0.664981 + 0.746861i \(0.731560\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 7.79205i − 0.412399i
\(358\) 0 0
\(359\) −20.2340 −1.06791 −0.533956 0.845513i \(-0.679295\pi\)
−0.533956 + 0.845513i \(0.679295\pi\)
\(360\) 0 0
\(361\) 37.4937 1.97336
\(362\) 0 0
\(363\) 11.8522i 0.622081i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 4.76666i 0.248818i 0.992231 + 0.124409i \(0.0397034\pi\)
−0.992231 + 0.124409i \(0.960297\pi\)
\(368\) 0 0
\(369\) −34.4339 −1.79256
\(370\) 0 0
\(371\) −10.1828 −0.528663
\(372\) 0 0
\(373\) 29.1510i 1.50938i 0.656082 + 0.754690i \(0.272213\pi\)
−0.656082 + 0.754690i \(0.727787\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 16.9784i − 0.874434i
\(378\) 0 0
\(379\) −4.20856 −0.216179 −0.108090 0.994141i \(-0.534473\pi\)
−0.108090 + 0.994141i \(0.534473\pi\)
\(380\) 0 0
\(381\) −11.7110 −0.599971
\(382\) 0 0
\(383\) − 3.55206i − 0.181502i −0.995874 0.0907508i \(-0.971073\pi\)
0.995874 0.0907508i \(-0.0289267\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 29.2515 1.48311 0.741554 0.670893i \(-0.234089\pi\)
0.741554 + 0.670893i \(0.234089\pi\)
\(390\) 0 0
\(391\) −0.924010 −0.0467292
\(392\) 0 0
\(393\) − 64.6416i − 3.26074i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 17.8000i − 0.893356i −0.894695 0.446678i \(-0.852607\pi\)
0.894695 0.446678i \(-0.147393\pi\)
\(398\) 0 0
\(399\) 63.3834 3.17314
\(400\) 0 0
\(401\) −11.6210 −0.580327 −0.290164 0.956977i \(-0.593710\pi\)
−0.290164 + 0.956977i \(0.593710\pi\)
\(402\) 0 0
\(403\) − 6.17320i − 0.307509i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 0.960871i − 0.0476286i
\(408\) 0 0
\(409\) 13.5710 0.671041 0.335521 0.942033i \(-0.391088\pi\)
0.335521 + 0.942033i \(0.391088\pi\)
\(410\) 0 0
\(411\) 14.5996 0.720145
\(412\) 0 0
\(413\) − 9.79516i − 0.481988i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 72.7277i 3.56149i
\(418\) 0 0
\(419\) −13.4959 −0.659317 −0.329658 0.944100i \(-0.606934\pi\)
−0.329658 + 0.944100i \(0.606934\pi\)
\(420\) 0 0
\(421\) 37.6495 1.83492 0.917462 0.397824i \(-0.130234\pi\)
0.917462 + 0.397824i \(0.130234\pi\)
\(422\) 0 0
\(423\) 105.697i 5.13916i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 23.3173i 1.12840i
\(428\) 0 0
\(429\) 64.1660 3.09796
\(430\) 0 0
\(431\) 10.0568 0.484421 0.242210 0.970224i \(-0.422128\pi\)
0.242210 + 0.970224i \(0.422128\pi\)
\(432\) 0 0
\(433\) − 2.30001i − 0.110531i −0.998472 0.0552657i \(-0.982399\pi\)
0.998472 0.0552657i \(-0.0176006\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 7.51623i − 0.359550i
\(438\) 0 0
\(439\) −12.4273 −0.593124 −0.296562 0.955014i \(-0.595840\pi\)
−0.296562 + 0.955014i \(0.595840\pi\)
\(440\) 0 0
\(441\) −3.93053 −0.187168
\(442\) 0 0
\(443\) − 2.18268i − 0.103702i −0.998655 0.0518510i \(-0.983488\pi\)
0.998655 0.0518510i \(-0.0165121\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 63.0813i 2.98364i
\(448\) 0 0
\(449\) −3.09603 −0.146111 −0.0730554 0.997328i \(-0.523275\pi\)
−0.0730554 + 0.997328i \(0.523275\pi\)
\(450\) 0 0
\(451\) −11.8202 −0.556591
\(452\) 0 0
\(453\) 25.6343i 1.20441i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 0.320363i − 0.0149860i −0.999972 0.00749298i \(-0.997615\pi\)
0.999972 0.00749298i \(-0.00238511\pi\)
\(458\) 0 0
\(459\) −15.0711 −0.703458
\(460\) 0 0
\(461\) −15.3239 −0.713706 −0.356853 0.934161i \(-0.616150\pi\)
−0.356853 + 0.934161i \(0.616150\pi\)
\(462\) 0 0
\(463\) − 6.81556i − 0.316746i −0.987379 0.158373i \(-0.949375\pi\)
0.987379 0.158373i \(-0.0506248\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 27.2444i 1.26072i 0.776303 + 0.630360i \(0.217093\pi\)
−0.776303 + 0.630360i \(0.782907\pi\)
\(468\) 0 0
\(469\) −8.05755 −0.372063
\(470\) 0 0
\(471\) 57.9062 2.66818
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 31.6730i 1.45021i
\(478\) 0 0
\(479\) −23.5031 −1.07388 −0.536942 0.843619i \(-0.680421\pi\)
−0.536942 + 0.843619i \(0.680421\pi\)
\(480\) 0 0
\(481\) 2.51457 0.114655
\(482\) 0 0
\(483\) − 8.43286i − 0.383709i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 7.63058i 0.345774i 0.984942 + 0.172887i \(0.0553096\pi\)
−0.984942 + 0.172887i \(0.944690\pi\)
\(488\) 0 0
\(489\) 8.44779 0.382022
\(490\) 0 0
\(491\) −19.1206 −0.862902 −0.431451 0.902136i \(-0.641998\pi\)
−0.431451 + 0.902136i \(0.641998\pi\)
\(492\) 0 0
\(493\) 2.20144i 0.0991477i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 15.5034i 0.695424i
\(498\) 0 0
\(499\) 25.6581 1.14861 0.574306 0.818641i \(-0.305272\pi\)
0.574306 + 0.818641i \(0.305272\pi\)
\(500\) 0 0
\(501\) −11.9464 −0.533725
\(502\) 0 0
\(503\) 8.42282i 0.375555i 0.982212 + 0.187777i \(0.0601284\pi\)
−0.982212 + 0.187777i \(0.939872\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 124.936i 5.54862i
\(508\) 0 0
\(509\) −18.2328 −0.808153 −0.404076 0.914725i \(-0.632407\pi\)
−0.404076 + 0.914725i \(0.632407\pi\)
\(510\) 0 0
\(511\) 28.8805 1.27760
\(512\) 0 0
\(513\) − 122.594i − 5.41264i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 36.2828i 1.59572i
\(518\) 0 0
\(519\) −42.2648 −1.85522
\(520\) 0 0
\(521\) 19.8738 0.870689 0.435345 0.900264i \(-0.356627\pi\)
0.435345 + 0.900264i \(0.356627\pi\)
\(522\) 0 0
\(523\) 19.7228i 0.862418i 0.902252 + 0.431209i \(0.141913\pi\)
−0.902252 + 0.431209i \(0.858087\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0.800422i 0.0348669i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −30.4673 −1.32217
\(532\) 0 0
\(533\) − 30.9331i − 1.33986i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 33.1141i 1.42898i
\(538\) 0 0
\(539\) −1.34924 −0.0581160
\(540\) 0 0
\(541\) −16.4662 −0.707938 −0.353969 0.935257i \(-0.615168\pi\)
−0.353969 + 0.935257i \(0.615168\pi\)
\(542\) 0 0
\(543\) − 32.9408i − 1.41363i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 38.3286i − 1.63881i −0.573215 0.819405i \(-0.694304\pi\)
0.573215 0.819405i \(-0.305696\pi\)
\(548\) 0 0
\(549\) 72.5273 3.09539
\(550\) 0 0
\(551\) −17.9073 −0.762875
\(552\) 0 0
\(553\) − 30.7559i − 1.30787i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 20.3973i − 0.864263i −0.901811 0.432132i \(-0.857762\pi\)
0.901811 0.432132i \(-0.142238\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −8.31981 −0.351263
\(562\) 0 0
\(563\) 1.72270i 0.0726032i 0.999341 + 0.0363016i \(0.0115577\pi\)
−0.999341 + 0.0363016i \(0.988442\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 76.8483i − 3.22733i
\(568\) 0 0
\(569\) −25.3283 −1.06182 −0.530909 0.847429i \(-0.678150\pi\)
−0.530909 + 0.847429i \(0.678150\pi\)
\(570\) 0 0
\(571\) −5.68968 −0.238106 −0.119053 0.992888i \(-0.537986\pi\)
−0.119053 + 0.992888i \(0.537986\pi\)
\(572\) 0 0
\(573\) 21.8658i 0.913455i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 18.0377i − 0.750918i −0.926839 0.375459i \(-0.877485\pi\)
0.926839 0.375459i \(-0.122515\pi\)
\(578\) 0 0
\(579\) 63.0019 2.61827
\(580\) 0 0
\(581\) 16.2023 0.672185
\(582\) 0 0
\(583\) 10.8724i 0.450291i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 13.1894i 0.544383i 0.962243 + 0.272192i \(0.0877485\pi\)
−0.962243 + 0.272192i \(0.912252\pi\)
\(588\) 0 0
\(589\) −6.51092 −0.268278
\(590\) 0 0
\(591\) −36.6643 −1.50817
\(592\) 0 0
\(593\) 0.326755i 0.0134182i 0.999977 + 0.00670911i \(0.00213559\pi\)
−0.999977 + 0.00670911i \(0.997864\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 12.2796i 0.502569i
\(598\) 0 0
\(599\) 7.49619 0.306286 0.153143 0.988204i \(-0.451061\pi\)
0.153143 + 0.988204i \(0.451061\pi\)
\(600\) 0 0
\(601\) 0.121724 0.00496523 0.00248262 0.999997i \(-0.499210\pi\)
0.00248262 + 0.999997i \(0.499210\pi\)
\(602\) 0 0
\(603\) 25.0626i 1.02063i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 1.52992i 0.0620975i 0.999518 + 0.0310488i \(0.00988471\pi\)
−0.999518 + 0.0310488i \(0.990115\pi\)
\(608\) 0 0
\(609\) −20.0911 −0.814134
\(610\) 0 0
\(611\) −94.9512 −3.84131
\(612\) 0 0
\(613\) − 24.3927i − 0.985211i −0.870253 0.492605i \(-0.836045\pi\)
0.870253 0.492605i \(-0.163955\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 27.8906i 1.12283i 0.827534 + 0.561416i \(0.189743\pi\)
−0.827534 + 0.561416i \(0.810257\pi\)
\(618\) 0 0
\(619\) −12.8617 −0.516957 −0.258478 0.966017i \(-0.583221\pi\)
−0.258478 + 0.966017i \(0.583221\pi\)
\(620\) 0 0
\(621\) −16.3105 −0.654518
\(622\) 0 0
\(623\) − 24.7740i − 0.992549i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 67.6763i − 2.70273i
\(628\) 0 0
\(629\) −0.326041 −0.0130001
\(630\) 0 0
\(631\) −11.9518 −0.475794 −0.237897 0.971290i \(-0.576458\pi\)
−0.237897 + 0.971290i \(0.576458\pi\)
\(632\) 0 0
\(633\) 71.9679i 2.86047i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 3.53093i − 0.139901i
\(638\) 0 0
\(639\) 48.2226 1.90766
\(640\) 0 0
\(641\) −23.9139 −0.944544 −0.472272 0.881453i \(-0.656566\pi\)
−0.472272 + 0.881453i \(0.656566\pi\)
\(642\) 0 0
\(643\) − 3.87313i − 0.152741i −0.997079 0.0763707i \(-0.975667\pi\)
0.997079 0.0763707i \(-0.0243333\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 10.3330i − 0.406231i −0.979155 0.203115i \(-0.934893\pi\)
0.979155 0.203115i \(-0.0651067\pi\)
\(648\) 0 0
\(649\) −10.4586 −0.410535
\(650\) 0 0
\(651\) −7.30495 −0.286303
\(652\) 0 0
\(653\) − 21.6319i − 0.846522i −0.906008 0.423261i \(-0.860885\pi\)
0.906008 0.423261i \(-0.139115\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 89.8312i − 3.50465i
\(658\) 0 0
\(659\) −15.7168 −0.612238 −0.306119 0.951993i \(-0.599030\pi\)
−0.306119 + 0.951993i \(0.599030\pi\)
\(660\) 0 0
\(661\) −40.3002 −1.56750 −0.783749 0.621078i \(-0.786695\pi\)
−0.783749 + 0.621078i \(0.786695\pi\)
\(662\) 0 0
\(663\) − 21.7727i − 0.845582i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 2.38248i 0.0922500i
\(668\) 0 0
\(669\) −22.8389 −0.883003
\(670\) 0 0
\(671\) 24.8966 0.961122
\(672\) 0 0
\(673\) 6.69888i 0.258223i 0.991630 + 0.129111i \(0.0412125\pi\)
−0.991630 + 0.129111i \(0.958788\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 22.5812i − 0.867866i −0.900945 0.433933i \(-0.857125\pi\)
0.900945 0.433933i \(-0.142875\pi\)
\(678\) 0 0
\(679\) 22.3533 0.857843
\(680\) 0 0
\(681\) −76.1566 −2.91833
\(682\) 0 0
\(683\) 1.42797i 0.0546398i 0.999627 + 0.0273199i \(0.00869728\pi\)
−0.999627 + 0.0273199i \(0.991303\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 42.8738i 1.63574i
\(688\) 0 0
\(689\) −28.4529 −1.08397
\(690\) 0 0
\(691\) −1.50255 −0.0571595 −0.0285798 0.999592i \(-0.509098\pi\)
−0.0285798 + 0.999592i \(0.509098\pi\)
\(692\) 0 0
\(693\) − 55.0944i − 2.09286i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 4.01081i 0.151920i
\(698\) 0 0
\(699\) 15.9020 0.601471
\(700\) 0 0
\(701\) 18.7341 0.707577 0.353789 0.935325i \(-0.384893\pi\)
0.353789 + 0.935325i \(0.384893\pi\)
\(702\) 0 0
\(703\) − 2.65214i − 0.100027i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 12.1091i − 0.455411i
\(708\) 0 0
\(709\) 13.5052 0.507199 0.253599 0.967309i \(-0.418385\pi\)
0.253599 + 0.967309i \(0.418385\pi\)
\(710\) 0 0
\(711\) −95.6646 −3.58770
\(712\) 0 0
\(713\) 0.866248i 0.0324412i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 67.6711i − 2.52722i
\(718\) 0 0
\(719\) −31.7496 −1.18406 −0.592030 0.805916i \(-0.701673\pi\)
−0.592030 + 0.805916i \(0.701673\pi\)
\(720\) 0 0
\(721\) −32.2126 −1.19966
\(722\) 0 0
\(723\) − 88.8313i − 3.30367i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 21.6531i 0.803070i 0.915844 + 0.401535i \(0.131523\pi\)
−0.915844 + 0.401535i \(0.868477\pi\)
\(728\) 0 0
\(729\) −77.2411 −2.86078
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) − 0.640507i − 0.0236577i −0.999930 0.0118288i \(-0.996235\pi\)
0.999930 0.0118288i \(-0.00376533\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.60329i 0.316906i
\(738\) 0 0
\(739\) −16.8191 −0.618700 −0.309350 0.950948i \(-0.600111\pi\)
−0.309350 + 0.950948i \(0.600111\pi\)
\(740\) 0 0
\(741\) 177.107 6.50619
\(742\) 0 0
\(743\) 15.9691i 0.585851i 0.956135 + 0.292925i \(0.0946288\pi\)
−0.956135 + 0.292925i \(0.905371\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 50.3964i − 1.84391i
\(748\) 0 0
\(749\) −33.0680 −1.20828
\(750\) 0 0
\(751\) −8.09841 −0.295515 −0.147758 0.989024i \(-0.547206\pi\)
−0.147758 + 0.989024i \(0.547206\pi\)
\(752\) 0 0
\(753\) 52.7798i 1.92340i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 20.5691i 0.747598i 0.927510 + 0.373799i \(0.121945\pi\)
−0.927510 + 0.373799i \(0.878055\pi\)
\(758\) 0 0
\(759\) −9.00402 −0.326825
\(760\) 0 0
\(761\) −17.1840 −0.622919 −0.311459 0.950259i \(-0.600818\pi\)
−0.311459 + 0.950259i \(0.600818\pi\)
\(762\) 0 0
\(763\) − 38.2662i − 1.38533i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 27.3698i − 0.988268i
\(768\) 0 0
\(769\) −41.8648 −1.50968 −0.754841 0.655908i \(-0.772286\pi\)
−0.754841 + 0.655908i \(0.772286\pi\)
\(770\) 0 0
\(771\) 11.0393 0.397570
\(772\) 0 0
\(773\) 35.7639i 1.28634i 0.765724 + 0.643170i \(0.222381\pi\)
−0.765724 + 0.643170i \(0.777619\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 2.97557i − 0.106748i
\(778\) 0 0
\(779\) −32.6254 −1.16893
\(780\) 0 0
\(781\) 16.5535 0.592330
\(782\) 0 0
\(783\) 38.8595i 1.38872i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 26.9835i 0.961858i 0.876759 + 0.480929i \(0.159701\pi\)
−0.876759 + 0.480929i \(0.840299\pi\)
\(788\) 0 0
\(789\) 77.3321 2.75310
\(790\) 0 0
\(791\) −5.44500 −0.193602
\(792\) 0 0
\(793\) 65.1537i 2.31368i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 26.7226i 0.946561i 0.880912 + 0.473281i \(0.156930\pi\)
−0.880912 + 0.473281i \(0.843070\pi\)
\(798\) 0 0
\(799\) 12.3114 0.435547
\(800\) 0 0
\(801\) −77.0582 −2.72272
\(802\) 0 0
\(803\) − 30.8366i − 1.08820i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 46.7464i 1.64555i
\(808\) 0 0
\(809\) −50.0319 −1.75903 −0.879513 0.475874i \(-0.842132\pi\)
−0.879513 + 0.475874i \(0.842132\pi\)
\(810\) 0 0
\(811\) 33.5774 1.17906 0.589531 0.807746i \(-0.299313\pi\)
0.589531 + 0.807746i \(0.299313\pi\)
\(812\) 0 0
\(813\) − 85.7318i − 3.00675i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 144.181 5.03808
\(820\) 0 0
\(821\) 27.1429 0.947295 0.473647 0.880715i \(-0.342937\pi\)
0.473647 + 0.880715i \(0.342937\pi\)
\(822\) 0 0
\(823\) − 32.6392i − 1.13773i −0.822431 0.568866i \(-0.807382\pi\)
0.822431 0.568866i \(-0.192618\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 37.0828i − 1.28949i −0.764396 0.644747i \(-0.776963\pi\)
0.764396 0.644747i \(-0.223037\pi\)
\(828\) 0 0
\(829\) −33.9405 −1.17880 −0.589401 0.807841i \(-0.700636\pi\)
−0.589401 + 0.807841i \(0.700636\pi\)
\(830\) 0 0
\(831\) 3.54181 0.122864
\(832\) 0 0
\(833\) 0.457823i 0.0158626i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 14.1289i 0.488368i
\(838\) 0 0
\(839\) 1.36929 0.0472730 0.0236365 0.999721i \(-0.492476\pi\)
0.0236365 + 0.999721i \(0.492476\pi\)
\(840\) 0 0
\(841\) −23.3238 −0.804269
\(842\) 0 0
\(843\) − 40.4251i − 1.39231i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 9.14199i 0.314122i
\(848\) 0 0
\(849\) 100.998 3.46624
\(850\) 0 0
\(851\) −0.352855 −0.0120957
\(852\) 0 0
\(853\) − 9.21382i − 0.315475i −0.987481 0.157738i \(-0.949580\pi\)
0.987481 0.157738i \(-0.0504200\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 14.3564i 0.490405i 0.969472 + 0.245202i \(0.0788544\pi\)
−0.969472 + 0.245202i \(0.921146\pi\)
\(858\) 0 0
\(859\) −31.3758 −1.07053 −0.535264 0.844685i \(-0.679788\pi\)
−0.535264 + 0.844685i \(0.679788\pi\)
\(860\) 0 0
\(861\) −36.6042 −1.24747
\(862\) 0 0
\(863\) − 26.1214i − 0.889182i −0.895734 0.444591i \(-0.853349\pi\)
0.895734 0.444591i \(-0.146651\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 53.3873i − 1.81313i
\(868\) 0 0
\(869\) −32.8390 −1.11399
\(870\) 0 0
\(871\) −22.5146 −0.762877
\(872\) 0 0
\(873\) − 69.5289i − 2.35320i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 11.5530i − 0.390118i −0.980792 0.195059i \(-0.937510\pi\)
0.980792 0.195059i \(-0.0624898\pi\)
\(878\) 0 0
\(879\) 28.0006 0.944437
\(880\) 0 0
\(881\) −34.3559 −1.15748 −0.578740 0.815512i \(-0.696456\pi\)
−0.578740 + 0.815512i \(0.696456\pi\)
\(882\) 0 0
\(883\) − 12.8316i − 0.431818i −0.976414 0.215909i \(-0.930729\pi\)
0.976414 0.215909i \(-0.0692714\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 1.46724i − 0.0492652i −0.999697 0.0246326i \(-0.992158\pi\)
0.999697 0.0246326i \(-0.00784159\pi\)
\(888\) 0 0
\(889\) −9.03303 −0.302958
\(890\) 0 0
\(891\) −82.0533 −2.74889
\(892\) 0 0
\(893\) 100.146i 3.35125i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 23.5633i − 0.786755i
\(898\) 0 0
\(899\) 2.06382 0.0688322
\(900\) 0 0
\(901\) 3.68922 0.122906
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 4.44558i 0.147613i 0.997273 + 0.0738066i \(0.0235147\pi\)
−0.997273 + 0.0738066i \(0.976485\pi\)
\(908\) 0 0
\(909\) −37.6648 −1.24926
\(910\) 0 0
\(911\) 40.1697 1.33088 0.665440 0.746451i \(-0.268244\pi\)
0.665440 + 0.746451i \(0.268244\pi\)
\(912\) 0 0
\(913\) − 17.2997i − 0.572536i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 49.8600i − 1.64652i
\(918\) 0 0
\(919\) −46.8063 −1.54400 −0.771999 0.635624i \(-0.780743\pi\)
−0.771999 + 0.635624i \(0.780743\pi\)
\(920\) 0 0
\(921\) −4.39109 −0.144691
\(922\) 0 0
\(923\) 43.3200i 1.42590i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 100.195i 3.29085i
\(928\) 0 0
\(929\) −8.92145 −0.292703 −0.146352 0.989233i \(-0.546753\pi\)
−0.146352 + 0.989233i \(0.546753\pi\)
\(930\) 0 0
\(931\) −3.72409 −0.122052
\(932\) 0 0
\(933\) − 80.3382i − 2.63016i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 7.70070i 0.251571i 0.992057 + 0.125785i \(0.0401451\pi\)
−0.992057 + 0.125785i \(0.959855\pi\)
\(938\) 0 0
\(939\) −67.2799 −2.19560
\(940\) 0 0
\(941\) −16.8979 −0.550856 −0.275428 0.961322i \(-0.588820\pi\)
−0.275428 + 0.961322i \(0.588820\pi\)
\(942\) 0 0
\(943\) 4.34066i 0.141351i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 43.7150i − 1.42055i −0.703927 0.710273i \(-0.748572\pi\)
0.703927 0.710273i \(-0.251428\pi\)
\(948\) 0 0
\(949\) 80.6985 2.61958
\(950\) 0 0
\(951\) −28.3469 −0.919210
\(952\) 0 0
\(953\) − 30.2353i − 0.979418i −0.871886 0.489709i \(-0.837103\pi\)
0.871886 0.489709i \(-0.162897\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 21.4519i 0.693441i
\(958\) 0 0
\(959\) 11.2611 0.363641
\(960\) 0 0
\(961\) −30.2496 −0.975794
\(962\) 0 0
\(963\) 102.856i 3.31450i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 16.2727i 0.523296i 0.965163 + 0.261648i \(0.0842659\pi\)
−0.965163 + 0.261648i \(0.915734\pi\)
\(968\) 0 0
\(969\) −22.9638 −0.737704
\(970\) 0 0
\(971\) 6.21424 0.199424 0.0997122 0.995016i \(-0.468208\pi\)
0.0997122 + 0.995016i \(0.468208\pi\)
\(972\) 0 0
\(973\) 56.0971i 1.79839i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 25.6705i 0.821271i 0.911800 + 0.410636i \(0.134693\pi\)
−0.911800 + 0.410636i \(0.865307\pi\)
\(978\) 0 0
\(979\) −26.4519 −0.845407
\(980\) 0 0
\(981\) −119.025 −3.80018
\(982\) 0 0
\(983\) 30.7124i 0.979574i 0.871842 + 0.489787i \(0.162925\pi\)
−0.871842 + 0.489787i \(0.837075\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 112.359i 3.57642i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 29.4052 0.934086 0.467043 0.884235i \(-0.345319\pi\)
0.467043 + 0.884235i \(0.345319\pi\)
\(992\) 0 0
\(993\) 69.3594i 2.20105i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 5.59825i 0.177298i 0.996063 + 0.0886492i \(0.0282550\pi\)
−0.996063 + 0.0886492i \(0.971745\pi\)
\(998\) 0 0
\(999\) −5.75524 −0.182088
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.u.4049.2 10
5.2 odd 4 4600.2.a.be.1.1 5
5.3 odd 4 920.2.a.j.1.5 5
5.4 even 2 inner 4600.2.e.u.4049.9 10
15.8 even 4 8280.2.a.bs.1.2 5
20.3 even 4 1840.2.a.v.1.1 5
20.7 even 4 9200.2.a.cu.1.5 5
40.3 even 4 7360.2.a.cp.1.5 5
40.13 odd 4 7360.2.a.co.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.j.1.5 5 5.3 odd 4
1840.2.a.v.1.1 5 20.3 even 4
4600.2.a.be.1.1 5 5.2 odd 4
4600.2.e.u.4049.2 10 1.1 even 1 trivial
4600.2.e.u.4049.9 10 5.4 even 2 inner
7360.2.a.co.1.1 5 40.13 odd 4
7360.2.a.cp.1.5 5 40.3 even 4
8280.2.a.bs.1.2 5 15.8 even 4
9200.2.a.cu.1.5 5 20.7 even 4