Properties

Label 4400.2.b.y.4049.3
Level $4400$
Weight $2$
Character 4400.4049
Analytic conductor $35.134$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(35.1341768894\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 275)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.3
Root \(1.30278i\) of defining polynomial
Character \(\chi\) \(=\) 4400.4049
Dual form 4400.2.b.y.4049.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.30278i q^{3} +4.30278i q^{7} +1.30278 q^{9} +O(q^{10})\) \(q+1.30278i q^{3} +4.30278i q^{7} +1.30278 q^{9} +1.00000 q^{11} +5.00000i q^{13} +3.90833i q^{17} -1.00000 q^{19} -5.60555 q^{21} +3.69722i q^{23} +5.60555i q^{27} +9.90833 q^{29} +4.21110 q^{31} +1.30278i q^{33} -9.60555i q^{37} -6.51388 q^{39} +1.60555 q^{41} +7.21110i q^{43} -3.00000i q^{47} -11.5139 q^{49} -5.09167 q^{51} +2.30278i q^{53} -1.30278i q^{57} +0.211103 q^{59} +2.90833 q^{61} +5.60555i q^{63} -4.00000i q^{67} -4.81665 q^{69} -4.60555 q^{71} +2.90833i q^{73} +4.30278i q^{77} -0.0916731 q^{79} -3.39445 q^{81} -14.5139i q^{83} +12.9083i q^{87} -5.30278 q^{89} -21.5139 q^{91} +5.48612i q^{93} -11.6972i q^{97} +1.30278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{9} + O(q^{10}) \) \( 4 q - 2 q^{9} + 4 q^{11} - 4 q^{19} - 8 q^{21} + 18 q^{29} - 12 q^{31} + 10 q^{39} - 8 q^{41} - 10 q^{49} - 42 q^{51} - 28 q^{59} - 10 q^{61} + 24 q^{69} - 4 q^{71} - 22 q^{79} - 28 q^{81} - 14 q^{89} - 50 q^{91} - 2 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times\).

\(n\) \(177\) \(1201\) \(2751\) \(3301\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 1.30278i 0.752158i 0.926588 + 0.376079i \(0.122728\pi\)
−0.926588 + 0.376079i \(0.877272\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.30278i 1.62630i 0.582057 + 0.813148i \(0.302248\pi\)
−0.582057 + 0.813148i \(0.697752\pi\)
\(8\) 0 0
\(9\) 1.30278 0.434259
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 5.00000i 1.38675i 0.720577 + 0.693375i \(0.243877\pi\)
−0.720577 + 0.693375i \(0.756123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.90833i 0.947909i 0.880549 + 0.473954i \(0.157174\pi\)
−0.880549 + 0.473954i \(0.842826\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) −5.60555 −1.22323
\(22\) 0 0
\(23\) 3.69722i 0.770925i 0.922724 + 0.385462i \(0.125958\pi\)
−0.922724 + 0.385462i \(0.874042\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.60555i 1.07879i
\(28\) 0 0
\(29\) 9.90833 1.83993 0.919965 0.392000i \(-0.128217\pi\)
0.919965 + 0.392000i \(0.128217\pi\)
\(30\) 0 0
\(31\) 4.21110 0.756336 0.378168 0.925737i \(-0.376554\pi\)
0.378168 + 0.925737i \(0.376554\pi\)
\(32\) 0 0
\(33\) 1.30278i 0.226784i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 9.60555i − 1.57914i −0.613659 0.789571i \(-0.710303\pi\)
0.613659 0.789571i \(-0.289697\pi\)
\(38\) 0 0
\(39\) −6.51388 −1.04306
\(40\) 0 0
\(41\) 1.60555 0.250745 0.125372 0.992110i \(-0.459987\pi\)
0.125372 + 0.992110i \(0.459987\pi\)
\(42\) 0 0
\(43\) 7.21110i 1.09968i 0.835269 + 0.549841i \(0.185312\pi\)
−0.835269 + 0.549841i \(0.814688\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 3.00000i − 0.437595i −0.975770 0.218797i \(-0.929787\pi\)
0.975770 0.218797i \(-0.0702134\pi\)
\(48\) 0 0
\(49\) −11.5139 −1.64484
\(50\) 0 0
\(51\) −5.09167 −0.712977
\(52\) 0 0
\(53\) 2.30278i 0.316311i 0.987414 + 0.158155i \(0.0505547\pi\)
−0.987414 + 0.158155i \(0.949445\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.30278i − 0.172557i
\(58\) 0 0
\(59\) 0.211103 0.0274832 0.0137416 0.999906i \(-0.495626\pi\)
0.0137416 + 0.999906i \(0.495626\pi\)
\(60\) 0 0
\(61\) 2.90833 0.372373 0.186187 0.982514i \(-0.440387\pi\)
0.186187 + 0.982514i \(0.440387\pi\)
\(62\) 0 0
\(63\) 5.60555i 0.706233i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) −4.81665 −0.579857
\(70\) 0 0
\(71\) −4.60555 −0.546578 −0.273289 0.961932i \(-0.588112\pi\)
−0.273289 + 0.961932i \(0.588112\pi\)
\(72\) 0 0
\(73\) 2.90833i 0.340394i 0.985410 + 0.170197i \(0.0544404\pi\)
−0.985410 + 0.170197i \(0.945560\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.30278i 0.490347i
\(78\) 0 0
\(79\) −0.0916731 −0.0103140 −0.00515701 0.999987i \(-0.501642\pi\)
−0.00515701 + 0.999987i \(0.501642\pi\)
\(80\) 0 0
\(81\) −3.39445 −0.377161
\(82\) 0 0
\(83\) − 14.5139i − 1.59311i −0.604569 0.796553i \(-0.706655\pi\)
0.604569 0.796553i \(-0.293345\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 12.9083i 1.38392i
\(88\) 0 0
\(89\) −5.30278 −0.562093 −0.281047 0.959694i \(-0.590682\pi\)
−0.281047 + 0.959694i \(0.590682\pi\)
\(90\) 0 0
\(91\) −21.5139 −2.25527
\(92\) 0 0
\(93\) 5.48612i 0.568884i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 11.6972i − 1.18767i −0.804586 0.593837i \(-0.797613\pi\)
0.804586 0.593837i \(-0.202387\pi\)
\(98\) 0 0
\(99\) 1.30278 0.130934
\(100\) 0 0
\(101\) 17.5139 1.74270 0.871348 0.490666i \(-0.163246\pi\)
0.871348 + 0.490666i \(0.163246\pi\)
\(102\) 0 0
\(103\) 7.90833i 0.779231i 0.920978 + 0.389615i \(0.127392\pi\)
−0.920978 + 0.389615i \(0.872608\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 3.00000i − 0.290021i −0.989430 0.145010i \(-0.953678\pi\)
0.989430 0.145010i \(-0.0463216\pi\)
\(108\) 0 0
\(109\) 6.51388 0.623916 0.311958 0.950096i \(-0.399015\pi\)
0.311958 + 0.950096i \(0.399015\pi\)
\(110\) 0 0
\(111\) 12.5139 1.18776
\(112\) 0 0
\(113\) 10.8167i 1.01755i 0.860901 + 0.508773i \(0.169901\pi\)
−0.860901 + 0.508773i \(0.830099\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 6.51388i 0.602208i
\(118\) 0 0
\(119\) −16.8167 −1.54158
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 2.09167i 0.188600i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 17.1194i − 1.51910i −0.650447 0.759552i \(-0.725418\pi\)
0.650447 0.759552i \(-0.274582\pi\)
\(128\) 0 0
\(129\) −9.39445 −0.827135
\(130\) 0 0
\(131\) −0.908327 −0.0793609 −0.0396804 0.999212i \(-0.512634\pi\)
−0.0396804 + 0.999212i \(0.512634\pi\)
\(132\) 0 0
\(133\) − 4.30278i − 0.373098i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.09167i 0.178704i 0.996000 + 0.0893518i \(0.0284796\pi\)
−0.996000 + 0.0893518i \(0.971520\pi\)
\(138\) 0 0
\(139\) 8.21110 0.696457 0.348228 0.937410i \(-0.386783\pi\)
0.348228 + 0.937410i \(0.386783\pi\)
\(140\) 0 0
\(141\) 3.90833 0.329141
\(142\) 0 0
\(143\) 5.00000i 0.418121i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 15.0000i − 1.23718i
\(148\) 0 0
\(149\) −2.78890 −0.228475 −0.114238 0.993453i \(-0.536443\pi\)
−0.114238 + 0.993453i \(0.536443\pi\)
\(150\) 0 0
\(151\) 20.8167 1.69404 0.847018 0.531565i \(-0.178396\pi\)
0.847018 + 0.531565i \(0.178396\pi\)
\(152\) 0 0
\(153\) 5.09167i 0.411637i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 4.78890i − 0.382196i −0.981571 0.191098i \(-0.938795\pi\)
0.981571 0.191098i \(-0.0612048\pi\)
\(158\) 0 0
\(159\) −3.00000 −0.237915
\(160\) 0 0
\(161\) −15.9083 −1.25375
\(162\) 0 0
\(163\) − 5.69722i − 0.446241i −0.974791 0.223121i \(-0.928376\pi\)
0.974791 0.223121i \(-0.0716243\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 15.4222i − 1.19341i −0.802462 0.596703i \(-0.796477\pi\)
0.802462 0.596703i \(-0.203523\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −1.30278 −0.0996257
\(172\) 0 0
\(173\) 16.8167i 1.27855i 0.768980 + 0.639273i \(0.220765\pi\)
−0.768980 + 0.639273i \(0.779235\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0.275019i 0.0206717i
\(178\) 0 0
\(179\) −5.51388 −0.412127 −0.206063 0.978539i \(-0.566065\pi\)
−0.206063 + 0.978539i \(0.566065\pi\)
\(180\) 0 0
\(181\) −9.09167 −0.675779 −0.337889 0.941186i \(-0.609713\pi\)
−0.337889 + 0.941186i \(0.609713\pi\)
\(182\) 0 0
\(183\) 3.78890i 0.280083i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 3.90833i 0.285805i
\(188\) 0 0
\(189\) −24.1194 −1.75443
\(190\) 0 0
\(191\) −6.69722 −0.484594 −0.242297 0.970202i \(-0.577901\pi\)
−0.242297 + 0.970202i \(0.577901\pi\)
\(192\) 0 0
\(193\) − 1.21110i − 0.0871771i −0.999050 0.0435885i \(-0.986121\pi\)
0.999050 0.0435885i \(-0.0138791\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 9.69722i − 0.690899i −0.938437 0.345449i \(-0.887727\pi\)
0.938437 0.345449i \(-0.112273\pi\)
\(198\) 0 0
\(199\) −24.5139 −1.73774 −0.868871 0.495038i \(-0.835154\pi\)
−0.868871 + 0.495038i \(0.835154\pi\)
\(200\) 0 0
\(201\) 5.21110 0.367563
\(202\) 0 0
\(203\) 42.6333i 2.99227i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 4.81665i 0.334781i
\(208\) 0 0
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) −25.2389 −1.73751 −0.868757 0.495238i \(-0.835081\pi\)
−0.868757 + 0.495238i \(0.835081\pi\)
\(212\) 0 0
\(213\) − 6.00000i − 0.411113i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 18.1194i 1.23003i
\(218\) 0 0
\(219\) −3.78890 −0.256030
\(220\) 0 0
\(221\) −19.5416 −1.31451
\(222\) 0 0
\(223\) − 20.6333i − 1.38171i −0.722994 0.690854i \(-0.757235\pi\)
0.722994 0.690854i \(-0.242765\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.30278i 0.351958i 0.984394 + 0.175979i \(0.0563090\pi\)
−0.984394 + 0.175979i \(0.943691\pi\)
\(228\) 0 0
\(229\) −13.7250 −0.906972 −0.453486 0.891263i \(-0.649820\pi\)
−0.453486 + 0.891263i \(0.649820\pi\)
\(230\) 0 0
\(231\) −5.60555 −0.368818
\(232\) 0 0
\(233\) 5.09167i 0.333567i 0.985994 + 0.166783i \(0.0533380\pi\)
−0.985994 + 0.166783i \(0.946662\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 0.119429i − 0.00775778i
\(238\) 0 0
\(239\) −4.11943 −0.266464 −0.133232 0.991085i \(-0.542535\pi\)
−0.133232 + 0.991085i \(0.542535\pi\)
\(240\) 0 0
\(241\) −24.9361 −1.60627 −0.803137 0.595794i \(-0.796837\pi\)
−0.803137 + 0.595794i \(0.796837\pi\)
\(242\) 0 0
\(243\) 12.3944i 0.795104i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 5.00000i − 0.318142i
\(248\) 0 0
\(249\) 18.9083 1.19827
\(250\) 0 0
\(251\) 3.90833 0.246691 0.123346 0.992364i \(-0.460638\pi\)
0.123346 + 0.992364i \(0.460638\pi\)
\(252\) 0 0
\(253\) 3.69722i 0.232443i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 18.0000i 1.12281i 0.827541 + 0.561405i \(0.189739\pi\)
−0.827541 + 0.561405i \(0.810261\pi\)
\(258\) 0 0
\(259\) 41.3305 2.56815
\(260\) 0 0
\(261\) 12.9083 0.799005
\(262\) 0 0
\(263\) 1.18335i 0.0729683i 0.999334 + 0.0364841i \(0.0116158\pi\)
−0.999334 + 0.0364841i \(0.988384\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 6.90833i − 0.422783i
\(268\) 0 0
\(269\) 23.7250 1.44654 0.723269 0.690567i \(-0.242639\pi\)
0.723269 + 0.690567i \(0.242639\pi\)
\(270\) 0 0
\(271\) −14.2111 −0.863263 −0.431632 0.902050i \(-0.642062\pi\)
−0.431632 + 0.902050i \(0.642062\pi\)
\(272\) 0 0
\(273\) − 28.0278i − 1.69632i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 21.6056i − 1.29815i −0.760724 0.649076i \(-0.775156\pi\)
0.760724 0.649076i \(-0.224844\pi\)
\(278\) 0 0
\(279\) 5.48612 0.328446
\(280\) 0 0
\(281\) −22.8167 −1.36113 −0.680564 0.732689i \(-0.738265\pi\)
−0.680564 + 0.732689i \(0.738265\pi\)
\(282\) 0 0
\(283\) − 2.69722i − 0.160333i −0.996781 0.0801667i \(-0.974455\pi\)
0.996781 0.0801667i \(-0.0255453\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.90833i 0.407786i
\(288\) 0 0
\(289\) 1.72498 0.101469
\(290\) 0 0
\(291\) 15.2389 0.893318
\(292\) 0 0
\(293\) 15.2111i 0.888642i 0.895868 + 0.444321i \(0.146555\pi\)
−0.895868 + 0.444321i \(0.853445\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.60555i 0.325267i
\(298\) 0 0
\(299\) −18.4861 −1.06908
\(300\) 0 0
\(301\) −31.0278 −1.78841
\(302\) 0 0
\(303\) 22.8167i 1.31078i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 6.09167i − 0.347670i −0.984775 0.173835i \(-0.944384\pi\)
0.984775 0.173835i \(-0.0556160\pi\)
\(308\) 0 0
\(309\) −10.3028 −0.586104
\(310\) 0 0
\(311\) 16.8167 0.953585 0.476792 0.879016i \(-0.341799\pi\)
0.476792 + 0.879016i \(0.341799\pi\)
\(312\) 0 0
\(313\) 21.8167i 1.23315i 0.787296 + 0.616575i \(0.211480\pi\)
−0.787296 + 0.616575i \(0.788520\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 9.90833i − 0.556507i −0.960508 0.278254i \(-0.910244\pi\)
0.960508 0.278254i \(-0.0897555\pi\)
\(318\) 0 0
\(319\) 9.90833 0.554760
\(320\) 0 0
\(321\) 3.90833 0.218142
\(322\) 0 0
\(323\) − 3.90833i − 0.217465i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 8.48612i 0.469284i
\(328\) 0 0
\(329\) 12.9083 0.711659
\(330\) 0 0
\(331\) 14.3944 0.791190 0.395595 0.918425i \(-0.370538\pi\)
0.395595 + 0.918425i \(0.370538\pi\)
\(332\) 0 0
\(333\) − 12.5139i − 0.685756i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 26.8444i − 1.46231i −0.682212 0.731154i \(-0.738982\pi\)
0.682212 0.731154i \(-0.261018\pi\)
\(338\) 0 0
\(339\) −14.0917 −0.765355
\(340\) 0 0
\(341\) 4.21110 0.228044
\(342\) 0 0
\(343\) − 19.4222i − 1.04870i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 5.51388i 0.296000i 0.988987 + 0.148000i \(0.0472836\pi\)
−0.988987 + 0.148000i \(0.952716\pi\)
\(348\) 0 0
\(349\) 26.8167 1.43546 0.717731 0.696320i \(-0.245181\pi\)
0.717731 + 0.696320i \(0.245181\pi\)
\(350\) 0 0
\(351\) −28.0278 −1.49601
\(352\) 0 0
\(353\) − 24.6333i − 1.31110i −0.755152 0.655549i \(-0.772437\pi\)
0.755152 0.655549i \(-0.227563\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 21.9083i − 1.15951i
\(358\) 0 0
\(359\) 15.2111 0.802811 0.401406 0.915900i \(-0.368522\pi\)
0.401406 + 0.915900i \(0.368522\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 1.30278i 0.0683780i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 24.3028i − 1.26859i −0.773089 0.634297i \(-0.781290\pi\)
0.773089 0.634297i \(-0.218710\pi\)
\(368\) 0 0
\(369\) 2.09167 0.108888
\(370\) 0 0
\(371\) −9.90833 −0.514415
\(372\) 0 0
\(373\) − 1.42221i − 0.0736390i −0.999322 0.0368195i \(-0.988277\pi\)
0.999322 0.0368195i \(-0.0117227\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 49.5416i 2.55152i
\(378\) 0 0
\(379\) 24.8167 1.27475 0.637373 0.770555i \(-0.280021\pi\)
0.637373 + 0.770555i \(0.280021\pi\)
\(380\) 0 0
\(381\) 22.3028 1.14261
\(382\) 0 0
\(383\) 21.6333i 1.10541i 0.833377 + 0.552705i \(0.186404\pi\)
−0.833377 + 0.552705i \(0.813596\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 9.39445i 0.477547i
\(388\) 0 0
\(389\) −12.0000 −0.608424 −0.304212 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 0 0
\(391\) −14.4500 −0.730766
\(392\) 0 0
\(393\) − 1.18335i − 0.0596919i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 25.3028i − 1.26991i −0.772549 0.634955i \(-0.781019\pi\)
0.772549 0.634955i \(-0.218981\pi\)
\(398\) 0 0
\(399\) 5.60555 0.280629
\(400\) 0 0
\(401\) −27.2111 −1.35886 −0.679429 0.733741i \(-0.737772\pi\)
−0.679429 + 0.733741i \(0.737772\pi\)
\(402\) 0 0
\(403\) 21.0555i 1.04885i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 9.60555i − 0.476129i
\(408\) 0 0
\(409\) −8.21110 −0.406013 −0.203006 0.979177i \(-0.565071\pi\)
−0.203006 + 0.979177i \(0.565071\pi\)
\(410\) 0 0
\(411\) −2.72498 −0.134413
\(412\) 0 0
\(413\) 0.908327i 0.0446958i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 10.6972i 0.523845i
\(418\) 0 0
\(419\) 13.6056 0.664675 0.332337 0.943161i \(-0.392163\pi\)
0.332337 + 0.943161i \(0.392163\pi\)
\(420\) 0 0
\(421\) 4.30278 0.209704 0.104852 0.994488i \(-0.466563\pi\)
0.104852 + 0.994488i \(0.466563\pi\)
\(422\) 0 0
\(423\) − 3.90833i − 0.190029i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 12.5139i 0.605589i
\(428\) 0 0
\(429\) −6.51388 −0.314493
\(430\) 0 0
\(431\) −33.0000 −1.58955 −0.794777 0.606902i \(-0.792412\pi\)
−0.794777 + 0.606902i \(0.792412\pi\)
\(432\) 0 0
\(433\) 5.00000i 0.240285i 0.992757 + 0.120142i \(0.0383351\pi\)
−0.992757 + 0.120142i \(0.961665\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 3.69722i − 0.176862i
\(438\) 0 0
\(439\) 20.6972 0.987825 0.493912 0.869512i \(-0.335566\pi\)
0.493912 + 0.869512i \(0.335566\pi\)
\(440\) 0 0
\(441\) −15.0000 −0.714286
\(442\) 0 0
\(443\) 1.39445i 0.0662523i 0.999451 + 0.0331261i \(0.0105463\pi\)
−0.999451 + 0.0331261i \(0.989454\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 3.63331i − 0.171850i
\(448\) 0 0
\(449\) −41.5139 −1.95916 −0.979581 0.201052i \(-0.935564\pi\)
−0.979581 + 0.201052i \(0.935564\pi\)
\(450\) 0 0
\(451\) 1.60555 0.0756025
\(452\) 0 0
\(453\) 27.1194i 1.27418i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 24.3028i 1.13684i 0.822740 + 0.568418i \(0.192444\pi\)
−0.822740 + 0.568418i \(0.807556\pi\)
\(458\) 0 0
\(459\) −21.9083 −1.02259
\(460\) 0 0
\(461\) 17.7889 0.828512 0.414256 0.910161i \(-0.364042\pi\)
0.414256 + 0.910161i \(0.364042\pi\)
\(462\) 0 0
\(463\) − 26.2111i − 1.21813i −0.793119 0.609067i \(-0.791544\pi\)
0.793119 0.609067i \(-0.208456\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 24.6333i 1.13989i 0.821682 + 0.569947i \(0.193036\pi\)
−0.821682 + 0.569947i \(0.806964\pi\)
\(468\) 0 0
\(469\) 17.2111 0.794735
\(470\) 0 0
\(471\) 6.23886 0.287471
\(472\) 0 0
\(473\) 7.21110i 0.331567i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 3.00000i 0.137361i
\(478\) 0 0
\(479\) −13.1833 −0.602362 −0.301181 0.953567i \(-0.597381\pi\)
−0.301181 + 0.953567i \(0.597381\pi\)
\(480\) 0 0
\(481\) 48.0278 2.18988
\(482\) 0 0
\(483\) − 20.7250i − 0.943019i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 10.2111i − 0.462709i −0.972869 0.231355i \(-0.925684\pi\)
0.972869 0.231355i \(-0.0743158\pi\)
\(488\) 0 0
\(489\) 7.42221 0.335644
\(490\) 0 0
\(491\) 24.2111 1.09263 0.546316 0.837579i \(-0.316030\pi\)
0.546316 + 0.837579i \(0.316030\pi\)
\(492\) 0 0
\(493\) 38.7250i 1.74409i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 19.8167i − 0.888898i
\(498\) 0 0
\(499\) −21.5139 −0.963093 −0.481547 0.876420i \(-0.659925\pi\)
−0.481547 + 0.876420i \(0.659925\pi\)
\(500\) 0 0
\(501\) 20.0917 0.897630
\(502\) 0 0
\(503\) − 16.6056i − 0.740405i −0.928951 0.370202i \(-0.879288\pi\)
0.928951 0.370202i \(-0.120712\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 15.6333i − 0.694300i
\(508\) 0 0
\(509\) 26.3028 1.16585 0.582925 0.812526i \(-0.301908\pi\)
0.582925 + 0.812526i \(0.301908\pi\)
\(510\) 0 0
\(511\) −12.5139 −0.553581
\(512\) 0 0
\(513\) − 5.60555i − 0.247491i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 3.00000i − 0.131940i
\(518\) 0 0
\(519\) −21.9083 −0.961669
\(520\) 0 0
\(521\) 23.4500 1.02736 0.513681 0.857981i \(-0.328282\pi\)
0.513681 + 0.857981i \(0.328282\pi\)
\(522\) 0 0
\(523\) 3.57779i 0.156446i 0.996936 + 0.0782230i \(0.0249246\pi\)
−0.996936 + 0.0782230i \(0.975075\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 16.4584i 0.716938i
\(528\) 0 0
\(529\) 9.33053 0.405675
\(530\) 0 0
\(531\) 0.275019 0.0119348
\(532\) 0 0
\(533\) 8.02776i 0.347721i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 7.18335i − 0.309984i
\(538\) 0 0
\(539\) −11.5139 −0.495938
\(540\) 0 0
\(541\) −6.72498 −0.289130 −0.144565 0.989495i \(-0.546178\pi\)
−0.144565 + 0.989495i \(0.546178\pi\)
\(542\) 0 0
\(543\) − 11.8444i − 0.508292i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 18.1194i 0.774731i 0.921926 + 0.387365i \(0.126615\pi\)
−0.921926 + 0.387365i \(0.873385\pi\)
\(548\) 0 0
\(549\) 3.78890 0.161706
\(550\) 0 0
\(551\) −9.90833 −0.422109
\(552\) 0 0
\(553\) − 0.394449i − 0.0167737i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9.42221i 0.399232i 0.979874 + 0.199616i \(0.0639694\pi\)
−0.979874 + 0.199616i \(0.936031\pi\)
\(558\) 0 0
\(559\) −36.0555 −1.52499
\(560\) 0 0
\(561\) −5.09167 −0.214971
\(562\) 0 0
\(563\) − 18.9083i − 0.796891i −0.917192 0.398445i \(-0.869550\pi\)
0.917192 0.398445i \(-0.130450\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 14.6056i − 0.613375i
\(568\) 0 0
\(569\) −15.1472 −0.635003 −0.317502 0.948258i \(-0.602844\pi\)
−0.317502 + 0.948258i \(0.602844\pi\)
\(570\) 0 0
\(571\) 17.3305 0.725260 0.362630 0.931933i \(-0.381879\pi\)
0.362630 + 0.931933i \(0.381879\pi\)
\(572\) 0 0
\(573\) − 8.72498i − 0.364491i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 31.3583i 1.30546i 0.757589 + 0.652731i \(0.226377\pi\)
−0.757589 + 0.652731i \(0.773623\pi\)
\(578\) 0 0
\(579\) 1.57779 0.0655709
\(580\) 0 0
\(581\) 62.4500 2.59086
\(582\) 0 0
\(583\) 2.30278i 0.0953712i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 37.5416i − 1.54951i −0.632262 0.774755i \(-0.717873\pi\)
0.632262 0.774755i \(-0.282127\pi\)
\(588\) 0 0
\(589\) −4.21110 −0.173515
\(590\) 0 0
\(591\) 12.6333 0.519665
\(592\) 0 0
\(593\) − 13.6056i − 0.558713i −0.960187 0.279357i \(-0.909879\pi\)
0.960187 0.279357i \(-0.0901211\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 31.9361i − 1.30706i
\(598\) 0 0
\(599\) −14.0917 −0.575770 −0.287885 0.957665i \(-0.592952\pi\)
−0.287885 + 0.957665i \(0.592952\pi\)
\(600\) 0 0
\(601\) 8.90833 0.363378 0.181689 0.983356i \(-0.441844\pi\)
0.181689 + 0.983356i \(0.441844\pi\)
\(602\) 0 0
\(603\) − 5.21110i − 0.212213i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 7.21110i − 0.292690i −0.989234 0.146345i \(-0.953249\pi\)
0.989234 0.146345i \(-0.0467509\pi\)
\(608\) 0 0
\(609\) −55.5416 −2.25066
\(610\) 0 0
\(611\) 15.0000 0.606835
\(612\) 0 0
\(613\) − 41.1194i − 1.66080i −0.557169 0.830399i \(-0.688112\pi\)
0.557169 0.830399i \(-0.311888\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 10.6056i − 0.426963i −0.976947 0.213482i \(-0.931520\pi\)
0.976947 0.213482i \(-0.0684804\pi\)
\(618\) 0 0
\(619\) 17.4222 0.700258 0.350129 0.936702i \(-0.386138\pi\)
0.350129 + 0.936702i \(0.386138\pi\)
\(620\) 0 0
\(621\) −20.7250 −0.831665
\(622\) 0 0
\(623\) − 22.8167i − 0.914130i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 1.30278i − 0.0520278i
\(628\) 0 0
\(629\) 37.5416 1.49688
\(630\) 0 0
\(631\) 39.9361 1.58983 0.794915 0.606721i \(-0.207515\pi\)
0.794915 + 0.606721i \(0.207515\pi\)
\(632\) 0 0
\(633\) − 32.8806i − 1.30689i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 57.5694i − 2.28098i
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 42.2111 1.66724 0.833619 0.552340i \(-0.186265\pi\)
0.833619 + 0.552340i \(0.186265\pi\)
\(642\) 0 0
\(643\) 22.0000i 0.867595i 0.901010 + 0.433798i \(0.142827\pi\)
−0.901010 + 0.433798i \(0.857173\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 17.2389i 0.677729i 0.940835 + 0.338865i \(0.110043\pi\)
−0.940835 + 0.338865i \(0.889957\pi\)
\(648\) 0 0
\(649\) 0.211103 0.00828650
\(650\) 0 0
\(651\) −23.6056 −0.925174
\(652\) 0 0
\(653\) − 19.1194i − 0.748201i −0.927388 0.374101i \(-0.877951\pi\)
0.927388 0.374101i \(-0.122049\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 3.78890i 0.147819i
\(658\) 0 0
\(659\) −20.0917 −0.782660 −0.391330 0.920250i \(-0.627985\pi\)
−0.391330 + 0.920250i \(0.627985\pi\)
\(660\) 0 0
\(661\) 12.8167 0.498510 0.249255 0.968438i \(-0.419814\pi\)
0.249255 + 0.968438i \(0.419814\pi\)
\(662\) 0 0
\(663\) − 25.4584i − 0.988721i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 36.6333i 1.41845i
\(668\) 0 0
\(669\) 26.8806 1.03926
\(670\) 0 0
\(671\) 2.90833 0.112275
\(672\) 0 0
\(673\) − 6.02776i − 0.232353i −0.993229 0.116176i \(-0.962936\pi\)
0.993229 0.116176i \(-0.0370638\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 26.2389i 1.00844i 0.863575 + 0.504221i \(0.168220\pi\)
−0.863575 + 0.504221i \(0.831780\pi\)
\(678\) 0 0
\(679\) 50.3305 1.93151
\(680\) 0 0
\(681\) −6.90833 −0.264728
\(682\) 0 0
\(683\) 9.84441i 0.376686i 0.982103 + 0.188343i \(0.0603116\pi\)
−0.982103 + 0.188343i \(0.939688\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 17.8806i − 0.682186i
\(688\) 0 0
\(689\) −11.5139 −0.438644
\(690\) 0 0
\(691\) 26.5416 1.00969 0.504846 0.863210i \(-0.331549\pi\)
0.504846 + 0.863210i \(0.331549\pi\)
\(692\) 0 0
\(693\) 5.60555i 0.212937i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 6.27502i 0.237683i
\(698\) 0 0
\(699\) −6.63331 −0.250895
\(700\) 0 0
\(701\) −26.7889 −1.01180 −0.505901 0.862591i \(-0.668840\pi\)
−0.505901 + 0.862591i \(0.668840\pi\)
\(702\) 0 0
\(703\) 9.60555i 0.362280i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 75.3583i 2.83414i
\(708\) 0 0
\(709\) −11.6333 −0.436898 −0.218449 0.975848i \(-0.570100\pi\)
−0.218449 + 0.975848i \(0.570100\pi\)
\(710\) 0 0
\(711\) −0.119429 −0.00447895
\(712\) 0 0
\(713\) 15.5694i 0.583078i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 5.36669i − 0.200423i
\(718\) 0 0
\(719\) 28.8167 1.07468 0.537340 0.843366i \(-0.319429\pi\)
0.537340 + 0.843366i \(0.319429\pi\)
\(720\) 0 0
\(721\) −34.0278 −1.26726
\(722\) 0 0
\(723\) − 32.4861i − 1.20817i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0.330532i 0.0122588i 0.999981 + 0.00612938i \(0.00195105\pi\)
−0.999981 + 0.00612938i \(0.998049\pi\)
\(728\) 0 0
\(729\) −26.3305 −0.975205
\(730\) 0 0
\(731\) −28.1833 −1.04240
\(732\) 0 0
\(733\) 12.3944i 0.457799i 0.973450 + 0.228900i \(0.0735128\pi\)
−0.973450 + 0.228900i \(0.926487\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 4.00000i − 0.147342i
\(738\) 0 0
\(739\) 9.88057 0.363463 0.181731 0.983348i \(-0.441830\pi\)
0.181731 + 0.983348i \(0.441830\pi\)
\(740\) 0 0
\(741\) 6.51388 0.239293
\(742\) 0 0
\(743\) − 44.3028i − 1.62531i −0.582744 0.812656i \(-0.698021\pi\)
0.582744 0.812656i \(-0.301979\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 18.9083i − 0.691820i
\(748\) 0 0
\(749\) 12.9083 0.471660
\(750\) 0 0
\(751\) 5.66947 0.206882 0.103441 0.994636i \(-0.467015\pi\)
0.103441 + 0.994636i \(0.467015\pi\)
\(752\) 0 0
\(753\) 5.09167i 0.185551i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 23.0555i 0.837967i 0.907994 + 0.418983i \(0.137613\pi\)
−0.907994 + 0.418983i \(0.862387\pi\)
\(758\) 0 0
\(759\) −4.81665 −0.174833
\(760\) 0 0
\(761\) −42.4222 −1.53780 −0.768902 0.639367i \(-0.779197\pi\)
−0.768902 + 0.639367i \(0.779197\pi\)
\(762\) 0 0
\(763\) 28.0278i 1.01467i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.05551i 0.0381124i
\(768\) 0 0
\(769\) 26.8167 0.967033 0.483517 0.875335i \(-0.339359\pi\)
0.483517 + 0.875335i \(0.339359\pi\)
\(770\) 0 0
\(771\) −23.4500 −0.844530
\(772\) 0 0
\(773\) − 22.1194i − 0.795581i −0.917476 0.397790i \(-0.869777\pi\)
0.917476 0.397790i \(-0.130223\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 53.8444i 1.93166i
\(778\) 0 0
\(779\) −1.60555 −0.0575248
\(780\) 0 0
\(781\) −4.60555 −0.164800
\(782\) 0 0
\(783\) 55.5416i 1.98490i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 4.21110i − 0.150110i −0.997179 0.0750548i \(-0.976087\pi\)
0.997179 0.0750548i \(-0.0239132\pi\)
\(788\) 0 0
\(789\) −1.54163 −0.0548836
\(790\) 0 0
\(791\) −46.5416 −1.65483
\(792\) 0 0
\(793\) 14.5416i 0.516389i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 14.5139i 0.514108i 0.966397 + 0.257054i \(0.0827518\pi\)
−0.966397 + 0.257054i \(0.917248\pi\)
\(798\) 0 0
\(799\) 11.7250 0.414800
\(800\) 0 0
\(801\) −6.90833 −0.244094
\(802\) 0 0
\(803\) 2.90833i 0.102633i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 30.9083i 1.08802i
\(808\) 0 0
\(809\) 3.63331 0.127740 0.0638701 0.997958i \(-0.479656\pi\)
0.0638701 + 0.997958i \(0.479656\pi\)
\(810\) 0 0
\(811\) 54.8722 1.92682 0.963411 0.268028i \(-0.0863719\pi\)
0.963411 + 0.268028i \(0.0863719\pi\)
\(812\) 0 0
\(813\) − 18.5139i − 0.649310i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 7.21110i − 0.252285i
\(818\) 0 0
\(819\) −28.0278 −0.979369
\(820\) 0 0
\(821\) 12.0000 0.418803 0.209401 0.977830i \(-0.432848\pi\)
0.209401 + 0.977830i \(0.432848\pi\)
\(822\) 0 0
\(823\) 10.4222i 0.363295i 0.983364 + 0.181648i \(0.0581430\pi\)
−0.983364 + 0.181648i \(0.941857\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 7.81665i − 0.271812i −0.990722 0.135906i \(-0.956606\pi\)
0.990722 0.135906i \(-0.0433945\pi\)
\(828\) 0 0
\(829\) 38.7527 1.34594 0.672969 0.739671i \(-0.265019\pi\)
0.672969 + 0.739671i \(0.265019\pi\)
\(830\) 0 0
\(831\) 28.1472 0.976415
\(832\) 0 0
\(833\) − 45.0000i − 1.55916i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 23.6056i 0.815927i
\(838\) 0 0
\(839\) 16.1194 0.556505 0.278252 0.960508i \(-0.410245\pi\)
0.278252 + 0.960508i \(0.410245\pi\)
\(840\) 0 0
\(841\) 69.1749 2.38534
\(842\) 0 0
\(843\) − 29.7250i − 1.02378i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 4.30278i 0.147845i
\(848\) 0 0
\(849\) 3.51388 0.120596
\(850\) 0 0
\(851\) 35.5139 1.21740
\(852\) 0 0
\(853\) 19.7250i 0.675370i 0.941259 + 0.337685i \(0.109644\pi\)
−0.941259 + 0.337685i \(0.890356\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3.00000i 0.102478i 0.998686 + 0.0512390i \(0.0163170\pi\)
−0.998686 + 0.0512390i \(0.983683\pi\)
\(858\) 0 0
\(859\) 48.6056 1.65840 0.829200 0.558952i \(-0.188796\pi\)
0.829200 + 0.558952i \(0.188796\pi\)
\(860\) 0 0
\(861\) −9.00000 −0.306719
\(862\) 0 0
\(863\) 19.6056i 0.667381i 0.942683 + 0.333690i \(0.108294\pi\)
−0.942683 + 0.333690i \(0.891706\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 2.24726i 0.0763210i
\(868\) 0 0
\(869\) −0.0916731 −0.00310980
\(870\) 0 0
\(871\) 20.0000 0.677674
\(872\) 0 0
\(873\) − 15.2389i − 0.515757i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 20.0000i − 0.675352i −0.941262 0.337676i \(-0.890359\pi\)
0.941262 0.337676i \(-0.109641\pi\)
\(878\) 0 0
\(879\) −19.8167 −0.668399
\(880\) 0 0
\(881\) −34.5416 −1.16374 −0.581869 0.813283i \(-0.697678\pi\)
−0.581869 + 0.813283i \(0.697678\pi\)
\(882\) 0 0
\(883\) 12.4500i 0.418975i 0.977811 + 0.209487i \(0.0671795\pi\)
−0.977811 + 0.209487i \(0.932821\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 47.2389i 1.58613i 0.609140 + 0.793063i \(0.291515\pi\)
−0.609140 + 0.793063i \(0.708485\pi\)
\(888\) 0 0
\(889\) 73.6611 2.47051
\(890\) 0 0
\(891\) −3.39445 −0.113718
\(892\) 0 0
\(893\) 3.00000i 0.100391i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 24.0833i − 0.804117i
\(898\) 0 0
\(899\) 41.7250 1.39161
\(900\) 0 0
\(901\) −9.00000 −0.299833
\(902\) 0 0
\(903\) − 40.4222i − 1.34517i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 4.00000i − 0.132818i −0.997792 0.0664089i \(-0.978846\pi\)
0.997792 0.0664089i \(-0.0211542\pi\)
\(908\) 0 0
\(909\) 22.8167 0.756781
\(910\) 0 0
\(911\) 39.2111 1.29912 0.649561 0.760310i \(-0.274953\pi\)
0.649561 + 0.760310i \(0.274953\pi\)
\(912\) 0 0
\(913\) − 14.5139i − 0.480339i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 3.90833i − 0.129064i
\(918\) 0 0
\(919\) 41.2111 1.35943 0.679714 0.733477i \(-0.262104\pi\)
0.679714 + 0.733477i \(0.262104\pi\)
\(920\) 0 0
\(921\) 7.93608 0.261503
\(922\) 0 0
\(923\) − 23.0278i − 0.757968i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 10.3028i 0.338388i
\(928\) 0 0
\(929\) −46.3944 −1.52215 −0.761076 0.648662i \(-0.775329\pi\)
−0.761076 + 0.648662i \(0.775329\pi\)
\(930\) 0 0
\(931\) 11.5139 0.377352
\(932\) 0 0
\(933\) 21.9083i 0.717246i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 5.21110i − 0.170239i −0.996371 0.0851196i \(-0.972873\pi\)
0.996371 0.0851196i \(-0.0271273\pi\)
\(938\) 0 0
\(939\) −28.4222 −0.927524
\(940\) 0 0
\(941\) 52.3944 1.70801 0.854005 0.520265i \(-0.174167\pi\)
0.854005 + 0.520265i \(0.174167\pi\)
\(942\) 0 0
\(943\) 5.93608i 0.193305i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 36.6333i − 1.19042i −0.803569 0.595211i \(-0.797068\pi\)
0.803569 0.595211i \(-0.202932\pi\)
\(948\) 0 0
\(949\) −14.5416 −0.472041
\(950\) 0 0
\(951\) 12.9083 0.418581
\(952\) 0 0
\(953\) − 49.2666i − 1.59590i −0.602722 0.797951i \(-0.705917\pi\)
0.602722 0.797951i \(-0.294083\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 12.9083i 0.417267i
\(958\) 0 0
\(959\) −9.00000 −0.290625
\(960\) 0 0
\(961\) −13.2666 −0.427955
\(962\) 0 0
\(963\) − 3.90833i − 0.125944i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 4.09167i 0.131579i 0.997834 + 0.0657897i \(0.0209566\pi\)
−0.997834 + 0.0657897i \(0.979043\pi\)
\(968\) 0 0
\(969\) 5.09167 0.163568
\(970\) 0 0
\(971\) 30.3583 0.974244 0.487122 0.873334i \(-0.338047\pi\)
0.487122 + 0.873334i \(0.338047\pi\)
\(972\) 0 0
\(973\) 35.3305i 1.13264i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 15.9722i 0.510997i 0.966809 + 0.255499i \(0.0822396\pi\)
−0.966809 + 0.255499i \(0.917760\pi\)
\(978\) 0 0
\(979\) −5.30278 −0.169477
\(980\) 0 0
\(981\) 8.48612 0.270941
\(982\) 0 0
\(983\) 48.8444i 1.55789i 0.627089 + 0.778947i \(0.284246\pi\)
−0.627089 + 0.778947i \(0.715754\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 16.8167i 0.535280i
\(988\) 0 0
\(989\) −26.6611 −0.847773
\(990\) 0 0
\(991\) 6.09167 0.193508 0.0967542 0.995308i \(-0.469154\pi\)
0.0967542 + 0.995308i \(0.469154\pi\)
\(992\) 0 0
\(993\) 18.7527i 0.595100i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 14.2750i − 0.452094i −0.974116 0.226047i \(-0.927420\pi\)
0.974116 0.226047i \(-0.0725804\pi\)
\(998\) 0 0
\(999\) 53.8444 1.70356
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 4400.2.b.y.4049.3 4
4.3 odd 2 275.2.b.c.199.1 4
5.2 odd 4 4400.2.a.bh.1.2 2
5.3 odd 4 4400.2.a.bs.1.1 2
5.4 even 2 inner 4400.2.b.y.4049.2 4
12.11 even 2 2475.2.c.k.199.4 4
20.3 even 4 275.2.a.e.1.1 2
20.7 even 4 275.2.a.f.1.2 yes 2
20.19 odd 2 275.2.b.c.199.4 4
60.23 odd 4 2475.2.a.t.1.2 2
60.47 odd 4 2475.2.a.o.1.1 2
60.59 even 2 2475.2.c.k.199.1 4
220.43 odd 4 3025.2.a.n.1.2 2
220.87 odd 4 3025.2.a.h.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
275.2.a.e.1.1 2 20.3 even 4
275.2.a.f.1.2 yes 2 20.7 even 4
275.2.b.c.199.1 4 4.3 odd 2
275.2.b.c.199.4 4 20.19 odd 2
2475.2.a.o.1.1 2 60.47 odd 4
2475.2.a.t.1.2 2 60.23 odd 4
2475.2.c.k.199.1 4 60.59 even 2
2475.2.c.k.199.4 4 12.11 even 2
3025.2.a.h.1.1 2 220.87 odd 4
3025.2.a.n.1.2 2 220.43 odd 4
4400.2.a.bh.1.2 2 5.2 odd 4
4400.2.a.bs.1.1 2 5.3 odd 4
4400.2.b.y.4049.2 4 5.4 even 2 inner
4400.2.b.y.4049.3 4 1.1 even 1 trivial