Properties

Label 4600.2.e.r.4049.2
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.24681024.1
Defining polynomial: \( x^{6} + 12x^{4} + 36x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(-2.14510i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.r.4049.5

$q$-expansion

\(f(q)\) \(=\) \(q-2.14510i q^{3} +1.14510i q^{7} -1.60147 q^{9} +O(q^{10})\) \(q-2.14510i q^{3} +1.14510i q^{7} -1.60147 q^{9} +5.89167 q^{11} -4.89167i q^{13} +5.89167i q^{17} +2.34803 q^{19} +2.45636 q^{21} +1.00000i q^{23} -3.00000i q^{27} -3.74657 q^{29} +5.68874 q^{31} -12.6382i q^{33} -4.00000i q^{37} -10.4931 q^{39} -1.05783 q^{41} +11.4931i q^{43} -7.74657i q^{47} +5.68874 q^{49} +12.6382 q^{51} +12.9863i q^{53} -5.03677i q^{57} -0.797069 q^{59} +13.8917 q^{61} -1.83384i q^{63} +15.5667i q^{67} +2.14510 q^{69} +2.94217 q^{71} +6.32698i q^{73} +6.74657i q^{77} -11.2397 q^{81} -0.912726i q^{83} +8.03677i q^{87} +15.4931 q^{89} +5.60147 q^{91} -12.2029i q^{93} -13.3848i q^{97} -9.43531 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{9} + 6 q^{11} - 6 q^{19} + 24 q^{21} - 6 q^{29} + 12 q^{31} - 30 q^{39} - 12 q^{41} + 12 q^{49} + 30 q^{51} - 12 q^{59} + 54 q^{61} + 12 q^{71} - 18 q^{81} + 60 q^{89} + 30 q^{91} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) − 2.14510i − 1.23848i −0.785204 0.619238i \(-0.787442\pi\)
0.785204 0.619238i \(-0.212558\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.14510i 0.432808i 0.976304 + 0.216404i \(0.0694329\pi\)
−0.976304 + 0.216404i \(0.930567\pi\)
\(8\) 0 0
\(9\) −1.60147 −0.533822
\(10\) 0 0
\(11\) 5.89167 1.77641 0.888203 0.459452i \(-0.151954\pi\)
0.888203 + 0.459452i \(0.151954\pi\)
\(12\) 0 0
\(13\) − 4.89167i − 1.35671i −0.734736 0.678353i \(-0.762694\pi\)
0.734736 0.678353i \(-0.237306\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.89167i 1.42894i 0.699666 + 0.714470i \(0.253332\pi\)
−0.699666 + 0.714470i \(0.746668\pi\)
\(18\) 0 0
\(19\) 2.34803 0.538676 0.269338 0.963046i \(-0.413195\pi\)
0.269338 + 0.963046i \(0.413195\pi\)
\(20\) 0 0
\(21\) 2.45636 0.536022
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 3.00000i − 0.577350i
\(28\) 0 0
\(29\) −3.74657 −0.695720 −0.347860 0.937546i \(-0.613092\pi\)
−0.347860 + 0.937546i \(0.613092\pi\)
\(30\) 0 0
\(31\) 5.68874 1.02173 0.510864 0.859662i \(-0.329326\pi\)
0.510864 + 0.859662i \(0.329326\pi\)
\(32\) 0 0
\(33\) − 12.6382i − 2.20004i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 4.00000i − 0.657596i −0.944400 0.328798i \(-0.893356\pi\)
0.944400 0.328798i \(-0.106644\pi\)
\(38\) 0 0
\(39\) −10.4931 −1.68025
\(40\) 0 0
\(41\) −1.05783 −0.165205 −0.0826025 0.996583i \(-0.526323\pi\)
−0.0826025 + 0.996583i \(0.526323\pi\)
\(42\) 0 0
\(43\) 11.4931i 1.75269i 0.481687 + 0.876343i \(0.340024\pi\)
−0.481687 + 0.876343i \(0.659976\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 7.74657i − 1.12995i −0.825107 0.564977i \(-0.808885\pi\)
0.825107 0.564977i \(-0.191115\pi\)
\(48\) 0 0
\(49\) 5.68874 0.812677
\(50\) 0 0
\(51\) 12.6382 1.76971
\(52\) 0 0
\(53\) 12.9863i 1.78380i 0.452231 + 0.891901i \(0.350628\pi\)
−0.452231 + 0.891901i \(0.649372\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 5.03677i − 0.667137i
\(58\) 0 0
\(59\) −0.797069 −0.103770 −0.0518848 0.998653i \(-0.516523\pi\)
−0.0518848 + 0.998653i \(0.516523\pi\)
\(60\) 0 0
\(61\) 13.8917 1.77865 0.889323 0.457279i \(-0.151176\pi\)
0.889323 + 0.457279i \(0.151176\pi\)
\(62\) 0 0
\(63\) − 1.83384i − 0.231042i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 15.5667i 1.90177i 0.309540 + 0.950887i \(0.399825\pi\)
−0.309540 + 0.950887i \(0.600175\pi\)
\(68\) 0 0
\(69\) 2.14510 0.258240
\(70\) 0 0
\(71\) 2.94217 0.349172 0.174586 0.984642i \(-0.444141\pi\)
0.174586 + 0.984642i \(0.444141\pi\)
\(72\) 0 0
\(73\) 6.32698i 0.740517i 0.928929 + 0.370258i \(0.120731\pi\)
−0.928929 + 0.370258i \(0.879269\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.74657i 0.768843i
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −11.2397 −1.24886
\(82\) 0 0
\(83\) − 0.912726i − 0.100185i −0.998745 0.0500923i \(-0.984048\pi\)
0.998745 0.0500923i \(-0.0159516\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 8.03677i 0.861633i
\(88\) 0 0
\(89\) 15.4931 1.64227 0.821135 0.570735i \(-0.193341\pi\)
0.821135 + 0.570735i \(0.193341\pi\)
\(90\) 0 0
\(91\) 5.60147 0.587193
\(92\) 0 0
\(93\) − 12.2029i − 1.26539i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 13.3848i − 1.35902i −0.733666 0.679511i \(-0.762192\pi\)
0.733666 0.679511i \(-0.237808\pi\)
\(98\) 0 0
\(99\) −9.43531 −0.948284
\(100\) 0 0
\(101\) 2.79707 0.278319 0.139159 0.990270i \(-0.455560\pi\)
0.139159 + 0.990270i \(0.455560\pi\)
\(102\) 0 0
\(103\) − 9.89167i − 0.974655i −0.873219 0.487328i \(-0.837972\pi\)
0.873219 0.487328i \(-0.162028\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 12.6961i − 1.22738i −0.789549 0.613688i \(-0.789685\pi\)
0.789549 0.613688i \(-0.210315\pi\)
\(108\) 0 0
\(109\) −3.65197 −0.349795 −0.174897 0.984587i \(-0.555959\pi\)
−0.174897 + 0.984587i \(0.555959\pi\)
\(110\) 0 0
\(111\) −8.58041 −0.814417
\(112\) 0 0
\(113\) − 10.0000i − 0.940721i −0.882474 0.470360i \(-0.844124\pi\)
0.882474 0.470360i \(-0.155876\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 7.83384i 0.724239i
\(118\) 0 0
\(119\) −6.74657 −0.618457
\(120\) 0 0
\(121\) 23.7118 2.15562
\(122\) 0 0
\(123\) 2.26915i 0.204602i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 12.1524i − 1.07835i −0.842193 0.539177i \(-0.818735\pi\)
0.842193 0.539177i \(-0.181265\pi\)
\(128\) 0 0
\(129\) 24.6540 2.17066
\(130\) 0 0
\(131\) −6.94950 −0.607181 −0.303590 0.952803i \(-0.598185\pi\)
−0.303590 + 0.952803i \(0.598185\pi\)
\(132\) 0 0
\(133\) 2.68874i 0.233143i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.44904i 0.209235i 0.994513 + 0.104618i \(0.0333619\pi\)
−0.994513 + 0.104618i \(0.966638\pi\)
\(138\) 0 0
\(139\) −9.23970 −0.783702 −0.391851 0.920029i \(-0.628165\pi\)
−0.391851 + 0.920029i \(0.628165\pi\)
\(140\) 0 0
\(141\) −16.6172 −1.39942
\(142\) 0 0
\(143\) − 28.8201i − 2.41006i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 12.2029i − 1.00648i
\(148\) 0 0
\(149\) 2.05783 0.168584 0.0842919 0.996441i \(-0.473137\pi\)
0.0842919 + 0.996441i \(0.473137\pi\)
\(150\) 0 0
\(151\) 9.34803 0.760732 0.380366 0.924836i \(-0.375798\pi\)
0.380366 + 0.924836i \(0.375798\pi\)
\(152\) 0 0
\(153\) − 9.43531i − 0.762799i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3.60879i 0.288013i 0.989577 + 0.144007i \(0.0459986\pi\)
−0.989577 + 0.144007i \(0.954001\pi\)
\(158\) 0 0
\(159\) 27.8569 2.20920
\(160\) 0 0
\(161\) −1.14510 −0.0902467
\(162\) 0 0
\(163\) 21.2554i 1.66485i 0.554135 + 0.832427i \(0.313049\pi\)
−0.554135 + 0.832427i \(0.686951\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 10.9863i 0.850143i 0.905160 + 0.425072i \(0.139751\pi\)
−0.905160 + 0.425072i \(0.860249\pi\)
\(168\) 0 0
\(169\) −10.9284 −0.840650
\(170\) 0 0
\(171\) −3.76030 −0.287557
\(172\) 0 0
\(173\) − 6.28288i − 0.477678i −0.971059 0.238839i \(-0.923233\pi\)
0.971059 0.238839i \(-0.0767669\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1.70979i 0.128516i
\(178\) 0 0
\(179\) −10.1524 −0.758828 −0.379414 0.925227i \(-0.623874\pi\)
−0.379414 + 0.925227i \(0.623874\pi\)
\(180\) 0 0
\(181\) 11.1451 0.828409 0.414204 0.910184i \(-0.364060\pi\)
0.414204 + 0.910184i \(0.364060\pi\)
\(182\) 0 0
\(183\) − 29.7991i − 2.20281i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 34.7118i 2.53838i
\(188\) 0 0
\(189\) 3.43531 0.249882
\(190\) 0 0
\(191\) −14.6961 −1.06337 −0.531685 0.846942i \(-0.678441\pi\)
−0.531685 + 0.846942i \(0.678441\pi\)
\(192\) 0 0
\(193\) − 26.4005i − 1.90035i −0.311715 0.950176i \(-0.600903\pi\)
0.311715 0.950176i \(-0.399097\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4.26076i 0.303567i 0.988414 + 0.151783i \(0.0485016\pi\)
−0.988414 + 0.151783i \(0.951498\pi\)
\(198\) 0 0
\(199\) −18.0735 −1.28120 −0.640600 0.767875i \(-0.721314\pi\)
−0.640600 + 0.767875i \(0.721314\pi\)
\(200\) 0 0
\(201\) 33.3921 2.35530
\(202\) 0 0
\(203\) − 4.29021i − 0.301113i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 1.60147i − 0.111310i
\(208\) 0 0
\(209\) 13.8338 0.956907
\(210\) 0 0
\(211\) −11.8990 −0.819161 −0.409580 0.912274i \(-0.634325\pi\)
−0.409580 + 0.912274i \(0.634325\pi\)
\(212\) 0 0
\(213\) − 6.31126i − 0.432440i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 6.51419i 0.442212i
\(218\) 0 0
\(219\) 13.5720 0.917112
\(220\) 0 0
\(221\) 28.8201 1.93865
\(222\) 0 0
\(223\) 4.58041i 0.306727i 0.988170 + 0.153363i \(0.0490105\pi\)
−0.988170 + 0.153363i \(0.950989\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 2.21666i − 0.147125i −0.997291 0.0735624i \(-0.976563\pi\)
0.997291 0.0735624i \(-0.0234368\pi\)
\(228\) 0 0
\(229\) 19.1608 1.26618 0.633091 0.774077i \(-0.281786\pi\)
0.633091 + 0.774077i \(0.281786\pi\)
\(230\) 0 0
\(231\) 14.4721 0.952193
\(232\) 0 0
\(233\) − 21.7466i − 1.42467i −0.701842 0.712333i \(-0.747639\pi\)
0.701842 0.712333i \(-0.252361\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −20.9495 −1.35511 −0.677555 0.735472i \(-0.736961\pi\)
−0.677555 + 0.735472i \(0.736961\pi\)
\(240\) 0 0
\(241\) 24.0735 1.55071 0.775357 0.631523i \(-0.217570\pi\)
0.775357 + 0.631523i \(0.217570\pi\)
\(242\) 0 0
\(243\) 15.1103i 0.969328i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 11.4858i − 0.730825i
\(248\) 0 0
\(249\) −1.95789 −0.124076
\(250\) 0 0
\(251\) −7.82651 −0.494005 −0.247003 0.969015i \(-0.579446\pi\)
−0.247003 + 0.969015i \(0.579446\pi\)
\(252\) 0 0
\(253\) 5.89167i 0.370406i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 9.82012i − 0.612562i −0.951941 0.306281i \(-0.900915\pi\)
0.951941 0.306281i \(-0.0990847\pi\)
\(258\) 0 0
\(259\) 4.58041 0.284613
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) − 14.6887i − 0.905746i −0.891575 0.452873i \(-0.850399\pi\)
0.891575 0.452873i \(-0.149601\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 33.2344i − 2.03391i
\(268\) 0 0
\(269\) 0.253432 0.0154520 0.00772600 0.999970i \(-0.497541\pi\)
0.00772600 + 0.999970i \(0.497541\pi\)
\(270\) 0 0
\(271\) −19.6099 −1.19121 −0.595607 0.803276i \(-0.703088\pi\)
−0.595607 + 0.803276i \(0.703088\pi\)
\(272\) 0 0
\(273\) − 12.0157i − 0.727224i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 29.1976i − 1.75431i −0.480204 0.877157i \(-0.659437\pi\)
0.480204 0.877157i \(-0.340563\pi\)
\(278\) 0 0
\(279\) −9.11032 −0.545421
\(280\) 0 0
\(281\) −27.1755 −1.62115 −0.810577 0.585633i \(-0.800846\pi\)
−0.810577 + 0.585633i \(0.800846\pi\)
\(282\) 0 0
\(283\) 18.1471i 1.07873i 0.842071 + 0.539366i \(0.181336\pi\)
−0.842071 + 0.539366i \(0.818664\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 1.21132i − 0.0715021i
\(288\) 0 0
\(289\) −17.7118 −1.04187
\(290\) 0 0
\(291\) −28.7118 −1.68311
\(292\) 0 0
\(293\) 0.681412i 0.0398085i 0.999802 + 0.0199043i \(0.00633614\pi\)
−0.999802 + 0.0199043i \(0.993664\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 17.6750i − 1.02561i
\(298\) 0 0
\(299\) 4.89167 0.282893
\(300\) 0 0
\(301\) −13.1608 −0.758577
\(302\) 0 0
\(303\) − 6.00000i − 0.344691i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 26.5877i − 1.51744i −0.651415 0.758721i \(-0.725824\pi\)
0.651415 0.758721i \(-0.274176\pi\)
\(308\) 0 0
\(309\) −21.2186 −1.20709
\(310\) 0 0
\(311\) 13.5299 0.767211 0.383605 0.923497i \(-0.374682\pi\)
0.383605 + 0.923497i \(0.374682\pi\)
\(312\) 0 0
\(313\) 19.8412i 1.12149i 0.827989 + 0.560745i \(0.189485\pi\)
−0.827989 + 0.560745i \(0.810515\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 3.55096i − 0.199442i −0.995015 0.0997210i \(-0.968205\pi\)
0.995015 0.0997210i \(-0.0317950\pi\)
\(318\) 0 0
\(319\) −22.0735 −1.23588
\(320\) 0 0
\(321\) −27.2344 −1.52007
\(322\) 0 0
\(323\) 13.8338i 0.769736i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 7.83384i 0.433212i
\(328\) 0 0
\(329\) 8.87062 0.489053
\(330\) 0 0
\(331\) −13.3133 −0.731763 −0.365881 0.930662i \(-0.619232\pi\)
−0.365881 + 0.930662i \(0.619232\pi\)
\(332\) 0 0
\(333\) 6.40586i 0.351039i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 20.4951i 1.11644i 0.829693 + 0.558220i \(0.188516\pi\)
−0.829693 + 0.558220i \(0.811484\pi\)
\(338\) 0 0
\(339\) −21.4510 −1.16506
\(340\) 0 0
\(341\) 33.5162 1.81500
\(342\) 0 0
\(343\) 14.5299i 0.784541i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7.60147i 0.408068i 0.978964 + 0.204034i \(0.0654053\pi\)
−0.978964 + 0.204034i \(0.934595\pi\)
\(348\) 0 0
\(349\) −5.55736 −0.297479 −0.148739 0.988876i \(-0.547522\pi\)
−0.148739 + 0.988876i \(0.547522\pi\)
\(350\) 0 0
\(351\) −14.6750 −0.783294
\(352\) 0 0
\(353\) 1.96323i 0.104492i 0.998634 + 0.0522460i \(0.0166380\pi\)
−0.998634 + 0.0522460i \(0.983362\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 14.4721i 0.765944i
\(358\) 0 0
\(359\) −10.5069 −0.554531 −0.277266 0.960793i \(-0.589428\pi\)
−0.277266 + 0.960793i \(0.589428\pi\)
\(360\) 0 0
\(361\) −13.4867 −0.709828
\(362\) 0 0
\(363\) − 50.8642i − 2.66968i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 33.7412i − 1.76128i −0.473788 0.880639i \(-0.657114\pi\)
0.473788 0.880639i \(-0.342886\pi\)
\(368\) 0 0
\(369\) 1.69408 0.0881901
\(370\) 0 0
\(371\) −14.8706 −0.772044
\(372\) 0 0
\(373\) − 17.4510i − 0.903580i −0.892124 0.451790i \(-0.850786\pi\)
0.892124 0.451790i \(-0.149214\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 18.3270i 0.943887i
\(378\) 0 0
\(379\) 33.4584 1.71864 0.859320 0.511438i \(-0.170887\pi\)
0.859320 + 0.511438i \(0.170887\pi\)
\(380\) 0 0
\(381\) −26.0682 −1.33551
\(382\) 0 0
\(383\) 13.0873i 0.668728i 0.942444 + 0.334364i \(0.108522\pi\)
−0.942444 + 0.334364i \(0.891478\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 18.4059i − 0.935623i
\(388\) 0 0
\(389\) −14.1083 −0.715321 −0.357660 0.933852i \(-0.616425\pi\)
−0.357660 + 0.933852i \(0.616425\pi\)
\(390\) 0 0
\(391\) −5.89167 −0.297955
\(392\) 0 0
\(393\) 14.9074i 0.751978i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 16.6245i − 0.834360i −0.908824 0.417180i \(-0.863018\pi\)
0.908824 0.417180i \(-0.136982\pi\)
\(398\) 0 0
\(399\) 5.76762 0.288742
\(400\) 0 0
\(401\) 22.2628 1.11175 0.555874 0.831266i \(-0.312384\pi\)
0.555874 + 0.831266i \(0.312384\pi\)
\(402\) 0 0
\(403\) − 27.8274i − 1.38618i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 23.5667i − 1.16816i
\(408\) 0 0
\(409\) −15.6887 −0.775758 −0.387879 0.921710i \(-0.626792\pi\)
−0.387879 + 0.921710i \(0.626792\pi\)
\(410\) 0 0
\(411\) 5.25343 0.259133
\(412\) 0 0
\(413\) − 0.912726i − 0.0449123i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 19.8201i 0.970595i
\(418\) 0 0
\(419\) 5.66769 0.276885 0.138442 0.990371i \(-0.455790\pi\)
0.138442 + 0.990371i \(0.455790\pi\)
\(420\) 0 0
\(421\) 15.0716 0.734543 0.367271 0.930114i \(-0.380292\pi\)
0.367271 + 0.930114i \(0.380292\pi\)
\(422\) 0 0
\(423\) 12.4059i 0.603194i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 15.9074i 0.769813i
\(428\) 0 0
\(429\) −61.8221 −2.98480
\(430\) 0 0
\(431\) 29.3500 1.41374 0.706870 0.707343i \(-0.250106\pi\)
0.706870 + 0.707343i \(0.250106\pi\)
\(432\) 0 0
\(433\) 29.6245i 1.42366i 0.702350 + 0.711832i \(0.252134\pi\)
−0.702350 + 0.711832i \(0.747866\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.34803i 0.112322i
\(438\) 0 0
\(439\) 20.7907 0.992285 0.496142 0.868241i \(-0.334749\pi\)
0.496142 + 0.868241i \(0.334749\pi\)
\(440\) 0 0
\(441\) −9.11032 −0.433825
\(442\) 0 0
\(443\) 21.2975i 1.01188i 0.862570 + 0.505938i \(0.168854\pi\)
−0.862570 + 0.505938i \(0.831146\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 4.41425i − 0.208787i
\(448\) 0 0
\(449\) 19.3197 0.911751 0.455875 0.890044i \(-0.349326\pi\)
0.455875 + 0.890044i \(0.349326\pi\)
\(450\) 0 0
\(451\) −6.23238 −0.293471
\(452\) 0 0
\(453\) − 20.0525i − 0.942148i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 0.912726i − 0.0426955i −0.999772 0.0213478i \(-0.993204\pi\)
0.999772 0.0213478i \(-0.00679572\pi\)
\(458\) 0 0
\(459\) 17.6750 0.824999
\(460\) 0 0
\(461\) 2.83384 0.131985 0.0659926 0.997820i \(-0.478979\pi\)
0.0659926 + 0.997820i \(0.478979\pi\)
\(462\) 0 0
\(463\) − 17.2618i − 0.802225i −0.916029 0.401112i \(-0.868624\pi\)
0.916029 0.401112i \(-0.131376\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 21.4510i 0.992635i 0.868141 + 0.496318i \(0.165315\pi\)
−0.868141 + 0.496318i \(0.834685\pi\)
\(468\) 0 0
\(469\) −17.8255 −0.823103
\(470\) 0 0
\(471\) 7.74123 0.356697
\(472\) 0 0
\(473\) 67.7138i 3.11348i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 20.7971i − 0.952232i
\(478\) 0 0
\(479\) −9.49314 −0.433752 −0.216876 0.976199i \(-0.569587\pi\)
−0.216876 + 0.976199i \(0.569587\pi\)
\(480\) 0 0
\(481\) −19.5667 −0.892164
\(482\) 0 0
\(483\) 2.45636i 0.111768i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 38.3270i − 1.73676i −0.495898 0.868381i \(-0.665161\pi\)
0.495898 0.868381i \(-0.334839\pi\)
\(488\) 0 0
\(489\) 45.5951 2.06188
\(490\) 0 0
\(491\) −13.9947 −0.631570 −0.315785 0.948831i \(-0.602268\pi\)
−0.315785 + 0.948831i \(0.602268\pi\)
\(492\) 0 0
\(493\) − 22.0735i − 0.994143i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.36909i 0.151124i
\(498\) 0 0
\(499\) 0.0642277 0.00287522 0.00143761 0.999999i \(-0.499542\pi\)
0.00143761 + 0.999999i \(0.499542\pi\)
\(500\) 0 0
\(501\) 23.5667 1.05288
\(502\) 0 0
\(503\) − 24.2785i − 1.08252i −0.840854 0.541262i \(-0.817947\pi\)
0.840854 0.541262i \(-0.182053\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 23.4426i 1.04112i
\(508\) 0 0
\(509\) 2.65929 0.117871 0.0589356 0.998262i \(-0.481229\pi\)
0.0589356 + 0.998262i \(0.481229\pi\)
\(510\) 0 0
\(511\) −7.24504 −0.320502
\(512\) 0 0
\(513\) − 7.04410i − 0.311005i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 45.6402i − 2.00726i
\(518\) 0 0
\(519\) −13.4774 −0.591593
\(520\) 0 0
\(521\) −17.0598 −0.747404 −0.373702 0.927549i \(-0.621912\pi\)
−0.373702 + 0.927549i \(0.621912\pi\)
\(522\) 0 0
\(523\) − 17.5667i − 0.768137i −0.923305 0.384069i \(-0.874523\pi\)
0.923305 0.384069i \(-0.125477\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 33.5162i 1.45999i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 1.27648 0.0553944
\(532\) 0 0
\(533\) 5.17455i 0.224135i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 21.7780i 0.939790i
\(538\) 0 0
\(539\) 33.5162 1.44364
\(540\) 0 0
\(541\) −8.36909 −0.359815 −0.179908 0.983684i \(-0.557580\pi\)
−0.179908 + 0.983684i \(0.557580\pi\)
\(542\) 0 0
\(543\) − 23.9074i − 1.02596i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 16.7486i 0.716117i 0.933699 + 0.358058i \(0.116561\pi\)
−0.933699 + 0.358058i \(0.883439\pi\)
\(548\) 0 0
\(549\) −22.2470 −0.949480
\(550\) 0 0
\(551\) −8.79707 −0.374768
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 7.26182i 0.307693i 0.988095 + 0.153847i \(0.0491662\pi\)
−0.988095 + 0.153847i \(0.950834\pi\)
\(558\) 0 0
\(559\) 56.2206 2.37788
\(560\) 0 0
\(561\) 74.4603 3.14372
\(562\) 0 0
\(563\) 34.5530i 1.45623i 0.685453 + 0.728117i \(0.259604\pi\)
−0.685453 + 0.728117i \(0.740396\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 12.8706i − 0.540515i
\(568\) 0 0
\(569\) −34.7275 −1.45585 −0.727926 0.685655i \(-0.759516\pi\)
−0.727926 + 0.685655i \(0.759516\pi\)
\(570\) 0 0
\(571\) 14.7034 0.615318 0.307659 0.951497i \(-0.400454\pi\)
0.307659 + 0.951497i \(0.400454\pi\)
\(572\) 0 0
\(573\) 31.5246i 1.31696i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 6.13777i 0.255519i 0.991805 + 0.127759i \(0.0407785\pi\)
−0.991805 + 0.127759i \(0.959221\pi\)
\(578\) 0 0
\(579\) −56.6318 −2.35354
\(580\) 0 0
\(581\) 1.04516 0.0433607
\(582\) 0 0
\(583\) 76.5108i 3.16876i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.33338i 0.0550344i 0.999621 + 0.0275172i \(0.00876010\pi\)
−0.999621 + 0.0275172i \(0.991240\pi\)
\(588\) 0 0
\(589\) 13.3574 0.550380
\(590\) 0 0
\(591\) 9.13977 0.375960
\(592\) 0 0
\(593\) 1.88434i 0.0773807i 0.999251 + 0.0386903i \(0.0123186\pi\)
−0.999251 + 0.0386903i \(0.987681\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 38.7696i 1.58673i
\(598\) 0 0
\(599\) 43.7632 1.78812 0.894058 0.447951i \(-0.147846\pi\)
0.894058 + 0.447951i \(0.147846\pi\)
\(600\) 0 0
\(601\) 4.50046 0.183578 0.0917889 0.995778i \(-0.470742\pi\)
0.0917889 + 0.995778i \(0.470742\pi\)
\(602\) 0 0
\(603\) − 24.9295i − 1.01521i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 6.17455i − 0.250617i −0.992118 0.125309i \(-0.960008\pi\)
0.992118 0.125309i \(-0.0399921\pi\)
\(608\) 0 0
\(609\) −9.20293 −0.372922
\(610\) 0 0
\(611\) −37.8937 −1.53301
\(612\) 0 0
\(613\) − 9.72445i − 0.392767i −0.980527 0.196383i \(-0.937080\pi\)
0.980527 0.196383i \(-0.0629197\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0.730849i 0.0294229i 0.999892 + 0.0147114i \(0.00468297\pi\)
−0.999892 + 0.0147114i \(0.995317\pi\)
\(618\) 0 0
\(619\) −11.1598 −0.448549 −0.224274 0.974526i \(-0.572001\pi\)
−0.224274 + 0.974526i \(0.572001\pi\)
\(620\) 0 0
\(621\) 3.00000 0.120386
\(622\) 0 0
\(623\) 17.7412i 0.710787i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 29.6750i − 1.18511i
\(628\) 0 0
\(629\) 23.5667 0.939665
\(630\) 0 0
\(631\) −4.87062 −0.193896 −0.0969481 0.995289i \(-0.530908\pi\)
−0.0969481 + 0.995289i \(0.530908\pi\)
\(632\) 0 0
\(633\) 25.5246i 1.01451i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 27.8274i − 1.10256i
\(638\) 0 0
\(639\) −4.71179 −0.186395
\(640\) 0 0
\(641\) 32.6540 1.28975 0.644877 0.764286i \(-0.276909\pi\)
0.644877 + 0.764286i \(0.276909\pi\)
\(642\) 0 0
\(643\) 47.9579i 1.89127i 0.325223 + 0.945637i \(0.394561\pi\)
−0.325223 + 0.945637i \(0.605439\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 12.1103i − 0.476106i −0.971252 0.238053i \(-0.923491\pi\)
0.971252 0.238053i \(-0.0765092\pi\)
\(648\) 0 0
\(649\) −4.69607 −0.184337
\(650\) 0 0
\(651\) 13.9736 0.547669
\(652\) 0 0
\(653\) − 43.2133i − 1.69107i −0.533922 0.845534i \(-0.679282\pi\)
0.533922 0.845534i \(-0.320718\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 10.1324i − 0.395304i
\(658\) 0 0
\(659\) −20.4333 −0.795969 −0.397984 0.917392i \(-0.630290\pi\)
−0.397984 + 0.917392i \(0.630290\pi\)
\(660\) 0 0
\(661\) 48.1398 1.87242 0.936210 0.351441i \(-0.114308\pi\)
0.936210 + 0.351441i \(0.114308\pi\)
\(662\) 0 0
\(663\) − 61.8221i − 2.40097i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 3.74657i − 0.145068i
\(668\) 0 0
\(669\) 9.82545 0.379874
\(670\) 0 0
\(671\) 81.8452 3.15960
\(672\) 0 0
\(673\) 16.5436i 0.637710i 0.947803 + 0.318855i \(0.103298\pi\)
−0.947803 + 0.318855i \(0.896702\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 48.9863i 1.88270i 0.337438 + 0.941348i \(0.390440\pi\)
−0.337438 + 0.941348i \(0.609560\pi\)
\(678\) 0 0
\(679\) 15.3270 0.588195
\(680\) 0 0
\(681\) −4.75496 −0.182210
\(682\) 0 0
\(683\) − 21.8779i − 0.837136i −0.908185 0.418568i \(-0.862532\pi\)
0.908185 0.418568i \(-0.137468\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 41.1019i − 1.56814i
\(688\) 0 0
\(689\) 63.5246 2.42009
\(690\) 0 0
\(691\) −3.30393 −0.125688 −0.0628438 0.998023i \(-0.520017\pi\)
−0.0628438 + 0.998023i \(0.520017\pi\)
\(692\) 0 0
\(693\) − 10.8044i − 0.410425i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 6.23238i − 0.236068i
\(698\) 0 0
\(699\) −46.6486 −1.76441
\(700\) 0 0
\(701\) −4.46369 −0.168591 −0.0842956 0.996441i \(-0.526864\pi\)
−0.0842956 + 0.996441i \(0.526864\pi\)
\(702\) 0 0
\(703\) − 9.39214i − 0.354231i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 3.20293i 0.120459i
\(708\) 0 0
\(709\) 2.39853 0.0900789 0.0450394 0.998985i \(-0.485659\pi\)
0.0450394 + 0.998985i \(0.485659\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 5.68874i 0.213045i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 44.9388i 1.67827i
\(718\) 0 0
\(719\) −12.0725 −0.450228 −0.225114 0.974332i \(-0.572275\pi\)
−0.225114 + 0.974332i \(0.572275\pi\)
\(720\) 0 0
\(721\) 11.3270 0.421839
\(722\) 0 0
\(723\) − 51.6402i − 1.92052i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 11.4269i − 0.423801i −0.977291 0.211900i \(-0.932035\pi\)
0.977291 0.211900i \(-0.0679653\pi\)
\(728\) 0 0
\(729\) −1.30592 −0.0483676
\(730\) 0 0
\(731\) −67.7138 −2.50448
\(732\) 0 0
\(733\) − 3.08727i − 0.114031i −0.998373 0.0570155i \(-0.981842\pi\)
0.998373 0.0570155i \(-0.0181585\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 91.7138i 3.37832i
\(738\) 0 0
\(739\) 8.36909 0.307862 0.153931 0.988082i \(-0.450807\pi\)
0.153931 + 0.988082i \(0.450807\pi\)
\(740\) 0 0
\(741\) −24.6382 −0.905108
\(742\) 0 0
\(743\) 15.1598i 0.556158i 0.960558 + 0.278079i \(0.0896976\pi\)
−0.960558 + 0.278079i \(0.910302\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 1.46170i 0.0534808i
\(748\) 0 0
\(749\) 14.5383 0.531218
\(750\) 0 0
\(751\) −33.6823 −1.22909 −0.614543 0.788883i \(-0.710660\pi\)
−0.614543 + 0.788883i \(0.710660\pi\)
\(752\) 0 0
\(753\) 16.7887i 0.611813i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 29.1608i 1.05987i 0.848039 + 0.529934i \(0.177783\pi\)
−0.848039 + 0.529934i \(0.822217\pi\)
\(758\) 0 0
\(759\) 12.6382 0.458739
\(760\) 0 0
\(761\) −6.91912 −0.250818 −0.125409 0.992105i \(-0.540024\pi\)
−0.125409 + 0.992105i \(0.540024\pi\)
\(762\) 0 0
\(763\) − 4.18188i − 0.151394i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.89900i 0.140785i
\(768\) 0 0
\(769\) 14.3491 0.517442 0.258721 0.965952i \(-0.416699\pi\)
0.258721 + 0.965952i \(0.416699\pi\)
\(770\) 0 0
\(771\) −21.0652 −0.758643
\(772\) 0 0
\(773\) 50.6265i 1.82091i 0.413609 + 0.910454i \(0.364268\pi\)
−0.413609 + 0.910454i \(0.635732\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 9.82545i − 0.352486i
\(778\) 0 0
\(779\) −2.48382 −0.0889920
\(780\) 0 0
\(781\) 17.3343 0.620270
\(782\) 0 0
\(783\) 11.2397i 0.401674i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 20.9442i − 0.746579i −0.927715 0.373289i \(-0.878230\pi\)
0.927715 0.373289i \(-0.121770\pi\)
\(788\) 0 0
\(789\) −31.5089 −1.12174
\(790\) 0 0
\(791\) 11.4510 0.407152
\(792\) 0 0
\(793\) − 67.9535i − 2.41310i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 19.0009i − 0.673047i −0.941675 0.336524i \(-0.890749\pi\)
0.941675 0.336524i \(-0.109251\pi\)
\(798\) 0 0
\(799\) 45.6402 1.61464
\(800\) 0 0
\(801\) −24.8117 −0.876679
\(802\) 0 0
\(803\) 37.2765i 1.31546i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 0.543637i − 0.0191369i
\(808\) 0 0
\(809\) −18.4300 −0.647963 −0.323982 0.946063i \(-0.605022\pi\)
−0.323982 + 0.946063i \(0.605022\pi\)
\(810\) 0 0
\(811\) −21.1387 −0.742280 −0.371140 0.928577i \(-0.621033\pi\)
−0.371140 + 0.928577i \(0.621033\pi\)
\(812\) 0 0
\(813\) 42.0652i 1.47529i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 26.9863i 0.944130i
\(818\) 0 0
\(819\) −8.97055 −0.313457
\(820\) 0 0
\(821\) −16.7550 −0.584752 −0.292376 0.956303i \(-0.594446\pi\)
−0.292376 + 0.956303i \(0.594446\pi\)
\(822\) 0 0
\(823\) 24.9220i 0.868728i 0.900737 + 0.434364i \(0.143027\pi\)
−0.900737 + 0.434364i \(0.856973\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 6.62252i 0.230288i 0.993349 + 0.115144i \(0.0367329\pi\)
−0.993349 + 0.115144i \(0.963267\pi\)
\(828\) 0 0
\(829\) −52.7001 −1.83035 −0.915174 0.403058i \(-0.867947\pi\)
−0.915174 + 0.403058i \(0.867947\pi\)
\(830\) 0 0
\(831\) −62.6318 −2.17267
\(832\) 0 0
\(833\) 33.5162i 1.16127i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 17.0662i − 0.589895i
\(838\) 0 0
\(839\) −42.5951 −1.47054 −0.735272 0.677772i \(-0.762946\pi\)
−0.735272 + 0.677772i \(0.762946\pi\)
\(840\) 0 0
\(841\) −14.9632 −0.515973
\(842\) 0 0
\(843\) 58.2942i 2.00776i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 27.1524i 0.932969i
\(848\) 0 0
\(849\) 38.9274 1.33598
\(850\) 0 0
\(851\) 4.00000 0.137118
\(852\) 0 0
\(853\) − 57.7255i − 1.97648i −0.152898 0.988242i \(-0.548861\pi\)
0.152898 0.988242i \(-0.451139\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 55.3721i 1.89148i 0.324930 + 0.945738i \(0.394659\pi\)
−0.324930 + 0.945738i \(0.605341\pi\)
\(858\) 0 0
\(859\) 14.3544 0.489767 0.244883 0.969553i \(-0.421250\pi\)
0.244883 + 0.969553i \(0.421250\pi\)
\(860\) 0 0
\(861\) −2.59841 −0.0885536
\(862\) 0 0
\(863\) 15.9211i 0.541961i 0.962585 + 0.270981i \(0.0873479\pi\)
−0.962585 + 0.270981i \(0.912652\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 37.9936i 1.29033i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 76.1471 2.58015
\(872\) 0 0
\(873\) 21.4353i 0.725475i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 30.2975i 1.02308i 0.859261 + 0.511538i \(0.170924\pi\)
−0.859261 + 0.511538i \(0.829076\pi\)
\(878\) 0 0
\(879\) 1.46170 0.0493019
\(880\) 0 0
\(881\) −11.1883 −0.376943 −0.188471 0.982079i \(-0.560353\pi\)
−0.188471 + 0.982079i \(0.560353\pi\)
\(882\) 0 0
\(883\) 25.2882i 0.851016i 0.904955 + 0.425508i \(0.139905\pi\)
−0.904955 + 0.425508i \(0.860095\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 32.2849i 1.08402i 0.840372 + 0.542010i \(0.182336\pi\)
−0.840372 + 0.542010i \(0.817664\pi\)
\(888\) 0 0
\(889\) 13.9158 0.466720
\(890\) 0 0
\(891\) −66.2206 −2.21847
\(892\) 0 0
\(893\) − 18.1892i − 0.608679i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 10.4931i − 0.350356i
\(898\) 0 0
\(899\) −21.3133 −0.710837
\(900\) 0 0
\(901\) −76.5108 −2.54895
\(902\) 0 0
\(903\) 28.2313i 0.939479i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.47848i 0.0490922i 0.999699 + 0.0245461i \(0.00781404\pi\)
−0.999699 + 0.0245461i \(0.992186\pi\)
\(908\) 0 0
\(909\) −4.47941 −0.148573
\(910\) 0 0
\(911\) 5.70979 0.189174 0.0945870 0.995517i \(-0.469847\pi\)
0.0945870 + 0.995517i \(0.469847\pi\)
\(912\) 0 0
\(913\) − 5.37748i − 0.177969i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 7.95789i − 0.262793i
\(918\) 0 0
\(919\) −22.6265 −0.746379 −0.373190 0.927755i \(-0.621736\pi\)
−0.373190 + 0.927755i \(0.621736\pi\)
\(920\) 0 0
\(921\) −57.0334 −1.87932
\(922\) 0 0
\(923\) − 14.3921i − 0.473723i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 15.8412i 0.520292i
\(928\) 0 0
\(929\) 50.0829 1.64317 0.821583 0.570089i \(-0.193091\pi\)
0.821583 + 0.570089i \(0.193091\pi\)
\(930\) 0 0
\(931\) 13.3574 0.437770
\(932\) 0 0
\(933\) − 29.0230i − 0.950172i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 9.98428i − 0.326172i −0.986612 0.163086i \(-0.947855\pi\)
0.986612 0.163086i \(-0.0521448\pi\)
\(938\) 0 0
\(939\) 42.5613 1.38894
\(940\) 0 0
\(941\) −36.0809 −1.17620 −0.588101 0.808787i \(-0.700124\pi\)
−0.588101 + 0.808787i \(0.700124\pi\)
\(942\) 0 0
\(943\) − 1.05783i − 0.0344476i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 32.4730i 1.05523i 0.849483 + 0.527616i \(0.176914\pi\)
−0.849483 + 0.527616i \(0.823086\pi\)
\(948\) 0 0
\(949\) 30.9495 1.00466
\(950\) 0 0
\(951\) −7.61718 −0.247004
\(952\) 0 0
\(953\) − 39.3427i − 1.27443i −0.770684 0.637217i \(-0.780085\pi\)
0.770684 0.637217i \(-0.219915\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 47.3500i 1.53061i
\(958\) 0 0
\(959\) −2.80440 −0.0905587
\(960\) 0 0
\(961\) 1.36176 0.0439278
\(962\) 0 0
\(963\) 20.3323i 0.655200i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 23.5720i − 0.758025i −0.925392 0.379013i \(-0.876264\pi\)
0.925392 0.379013i \(-0.123736\pi\)
\(968\) 0 0
\(969\) 29.6750 0.953299
\(970\) 0 0
\(971\) 12.9368 0.415163 0.207581 0.978218i \(-0.433441\pi\)
0.207581 + 0.978218i \(0.433441\pi\)
\(972\) 0 0
\(973\) − 10.5804i − 0.339192i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 22.7118i 0.726614i 0.931669 + 0.363307i \(0.118352\pi\)
−0.931669 + 0.363307i \(0.881648\pi\)
\(978\) 0 0
\(979\) 91.2805 2.91734
\(980\) 0 0
\(981\) 5.84850 0.186728
\(982\) 0 0
\(983\) 26.1819i 0.835072i 0.908660 + 0.417536i \(0.137106\pi\)
−0.908660 + 0.417536i \(0.862894\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 19.0284i − 0.605680i
\(988\) 0 0
\(989\) −11.4931 −0.365460
\(990\) 0 0
\(991\) −56.3143 −1.78888 −0.894442 0.447185i \(-0.852427\pi\)
−0.894442 + 0.447185i \(0.852427\pi\)
\(992\) 0 0
\(993\) 28.5583i 0.906270i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 3.74123i 0.118486i 0.998244 + 0.0592430i \(0.0188687\pi\)
−0.998244 + 0.0592430i \(0.981131\pi\)
\(998\) 0 0
\(999\) −12.0000 −0.379663
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.r.4049.2 6
5.2 odd 4 920.2.a.g.1.1 3
5.3 odd 4 4600.2.a.y.1.3 3
5.4 even 2 inner 4600.2.e.r.4049.5 6
15.2 even 4 8280.2.a.bo.1.1 3
20.3 even 4 9200.2.a.cd.1.1 3
20.7 even 4 1840.2.a.t.1.3 3
40.27 even 4 7360.2.a.ca.1.1 3
40.37 odd 4 7360.2.a.cb.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.g.1.1 3 5.2 odd 4
1840.2.a.t.1.3 3 20.7 even 4
4600.2.a.y.1.3 3 5.3 odd 4
4600.2.e.r.4049.2 6 1.1 even 1 trivial
4600.2.e.r.4049.5 6 5.4 even 2 inner
7360.2.a.ca.1.1 3 40.27 even 4
7360.2.a.cb.1.3 3 40.37 odd 4
8280.2.a.bo.1.1 3 15.2 even 4
9200.2.a.cd.1.1 3 20.3 even 4