Properties

Label 2800.2.g.i.449.2
Level 2800
Weight 2
Character 2800.449
Analytic conductor 22.358
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 2800.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.3581125660\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 350)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.2
Root \(1.00000i\)
Character \(\chi\) = 2800.449
Dual form 2800.2.g.i.449.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} -1.00000i q^{7} +2.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} -1.00000i q^{7} +2.00000 q^{9} -3.00000 q^{11} -2.00000i q^{13} +3.00000i q^{17} -7.00000 q^{19} +1.00000 q^{21} +5.00000i q^{27} +6.00000 q^{29} +4.00000 q^{31} -3.00000i q^{33} +8.00000i q^{37} +2.00000 q^{39} -9.00000 q^{41} +8.00000i q^{43} +6.00000i q^{47} -1.00000 q^{49} -3.00000 q^{51} +12.0000i q^{53} -7.00000i q^{57} +12.0000 q^{59} -10.0000 q^{61} -2.00000i q^{63} +7.00000i q^{67} -6.00000 q^{71} -5.00000i q^{73} +3.00000i q^{77} +14.0000 q^{79} +1.00000 q^{81} -9.00000i q^{83} +6.00000i q^{87} +15.0000 q^{89} -2.00000 q^{91} +4.00000i q^{93} -10.0000i q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{9} + O(q^{10}) \) \( 2q + 4q^{9} - 6q^{11} - 14q^{19} + 2q^{21} + 12q^{29} + 8q^{31} + 4q^{39} - 18q^{41} - 2q^{49} - 6q^{51} + 24q^{59} - 20q^{61} - 12q^{71} + 28q^{79} + 2q^{81} + 30q^{89} - 4q^{91} - 12q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(801\) \(2101\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000i 0.577350i 0.957427 + 0.288675i \(0.0932147\pi\)
−0.957427 + 0.288675i \(0.906785\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000i 0.727607i 0.931476 + 0.363803i \(0.118522\pi\)
−0.931476 + 0.363803i \(0.881478\pi\)
\(18\) 0 0
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.00000i 0.962250i
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) − 3.00000i − 0.522233i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000i 0.875190i 0.899172 + 0.437595i \(0.144170\pi\)
−0.899172 + 0.437595i \(0.855830\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) 12.0000i 1.64833i 0.566352 + 0.824163i \(0.308354\pi\)
−0.566352 + 0.824163i \(0.691646\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 7.00000i − 0.927173i
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) − 2.00000i − 0.251976i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.00000i 0.855186i 0.903971 + 0.427593i \(0.140638\pi\)
−0.903971 + 0.427593i \(0.859362\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) − 5.00000i − 0.585206i −0.956234 0.292603i \(-0.905479\pi\)
0.956234 0.292603i \(-0.0945214\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.00000i 0.341882i
\(78\) 0 0
\(79\) 14.0000 1.57512 0.787562 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 9.00000i − 0.987878i −0.869496 0.493939i \(-0.835557\pi\)
0.869496 0.493939i \(-0.164443\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.00000i 0.643268i
\(88\) 0 0
\(89\) 15.0000 1.59000 0.794998 0.606612i \(-0.207472\pi\)
0.794998 + 0.606612i \(0.207472\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 4.00000i 0.414781i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 20.0000i 1.97066i 0.170664 + 0.985329i \(0.445409\pi\)
−0.170664 + 0.985329i \(0.554591\pi\)
\(104\) 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
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) 9.00000i 0.846649i 0.905978 + 0.423324i \(0.139137\pi\)
−0.905978 + 0.423324i \(0.860863\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 4.00000i − 0.369800i
\(118\) 0 0
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) − 9.00000i − 0.811503i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 2.00000i − 0.177471i −0.996055 0.0887357i \(-0.971717\pi\)
0.996055 0.0887357i \(-0.0282826\pi\)
\(128\) 0 0
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 7.00000i 0.606977i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 21.0000i 1.79415i 0.441877 + 0.897076i \(0.354313\pi\)
−0.441877 + 0.897076i \(0.645687\pi\)
\(138\) 0 0
\(139\) −7.00000 −0.593732 −0.296866 0.954919i \(-0.595942\pi\)
−0.296866 + 0.954919i \(0.595942\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) 6.00000i 0.501745i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 1.00000i − 0.0824786i
\(148\) 0 0
\(149\) −12.0000 −0.983078 −0.491539 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 20.0000i 1.59617i 0.602542 + 0.798087i \(0.294154\pi\)
−0.602542 + 0.798087i \(0.705846\pi\)
\(158\) 0 0
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 5.00000i 0.391630i 0.980641 + 0.195815i \(0.0627352\pi\)
−0.980641 + 0.195815i \(0.937265\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 12.0000i − 0.928588i −0.885681 0.464294i \(-0.846308\pi\)
0.885681 0.464294i \(-0.153692\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) −14.0000 −1.07061
\(172\) 0 0
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 12.0000i 0.901975i
\(178\) 0 0
\(179\) 3.00000 0.224231 0.112115 0.993695i \(-0.464237\pi\)
0.112115 + 0.993695i \(0.464237\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) − 10.0000i − 0.739221i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 9.00000i − 0.658145i
\(188\) 0 0
\(189\) 5.00000 0.363696
\(190\) 0 0
\(191\) 6.00000 0.434145 0.217072 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) 0 0
\(193\) − 5.00000i − 0.359908i −0.983675 0.179954i \(-0.942405\pi\)
0.983675 0.179954i \(-0.0575949\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 6.00000i − 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) 0 0
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 0 0
\(201\) −7.00000 −0.493742
\(202\) 0 0
\(203\) − 6.00000i − 0.421117i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 21.0000 1.45260
\(210\) 0 0
\(211\) −17.0000 −1.17033 −0.585164 0.810915i \(-0.698970\pi\)
−0.585164 + 0.810915i \(0.698970\pi\)
\(212\) 0 0
\(213\) − 6.00000i − 0.411113i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 4.00000i − 0.271538i
\(218\) 0 0
\(219\) 5.00000 0.337869
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 0 0
\(223\) 14.0000i 0.937509i 0.883328 + 0.468755i \(0.155297\pi\)
−0.883328 + 0.468755i \(0.844703\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) 0 0
\(229\) −26.0000 −1.71813 −0.859064 0.511868i \(-0.828954\pi\)
−0.859064 + 0.511868i \(0.828954\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) 0 0
\(233\) − 6.00000i − 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 14.0000i 0.909398i
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −25.0000 −1.61039 −0.805196 0.593009i \(-0.797940\pi\)
−0.805196 + 0.593009i \(0.797940\pi\)
\(242\) 0 0
\(243\) 16.0000i 1.02640i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 14.0000i 0.890799i
\(248\) 0 0
\(249\) 9.00000 0.570352
\(250\) 0 0
\(251\) −15.0000 −0.946792 −0.473396 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 30.0000i − 1.87135i −0.352865 0.935674i \(-0.614792\pi\)
0.352865 0.935674i \(-0.385208\pi\)
\(258\) 0 0
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) 12.0000 0.742781
\(262\) 0 0
\(263\) 6.00000i 0.369976i 0.982741 + 0.184988i \(0.0592246\pi\)
−0.982741 + 0.184988i \(0.940775\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 15.0000i 0.917985i
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −2.00000 −0.121491 −0.0607457 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) 0 0
\(273\) − 2.00000i − 0.121046i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.00000i 0.120168i 0.998193 + 0.0600842i \(0.0191369\pi\)
−0.998193 + 0.0600842i \(0.980863\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) − 1.00000i − 0.0594438i −0.999558 0.0297219i \(-0.990538\pi\)
0.999558 0.0297219i \(-0.00946217\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 9.00000i 0.531253i
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) 0 0
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 15.0000i − 0.870388i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 7.00000i 0.399511i 0.979846 + 0.199756i \(0.0640148\pi\)
−0.979846 + 0.199756i \(0.935985\pi\)
\(308\) 0 0
\(309\) −20.0000 −1.13776
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) 10.0000i 0.565233i 0.959233 + 0.282617i \(0.0912024\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 12.0000i − 0.673987i −0.941507 0.336994i \(-0.890590\pi\)
0.941507 0.336994i \(-0.109410\pi\)
\(318\) 0 0
\(319\) −18.0000 −1.00781
\(320\) 0 0
\(321\) 3.00000 0.167444
\(322\) 0 0
\(323\) − 21.0000i − 1.16847i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 14.0000i − 0.774202i
\(328\) 0 0
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) 25.0000 1.37412 0.687062 0.726599i \(-0.258900\pi\)
0.687062 + 0.726599i \(0.258900\pi\)
\(332\) 0 0
\(333\) 16.0000i 0.876795i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 13.0000i − 0.708155i −0.935216 0.354078i \(-0.884795\pi\)
0.935216 0.354078i \(-0.115205\pi\)
\(338\) 0 0
\(339\) −9.00000 −0.488813
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 21.0000i 1.12734i 0.826000 + 0.563670i \(0.190611\pi\)
−0.826000 + 0.563670i \(0.809389\pi\)
\(348\) 0 0
\(349\) −8.00000 −0.428230 −0.214115 0.976808i \(-0.568687\pi\)
−0.214115 + 0.976808i \(0.568687\pi\)
\(350\) 0 0
\(351\) 10.0000 0.533761
\(352\) 0 0
\(353\) − 30.0000i − 1.59674i −0.602168 0.798369i \(-0.705696\pi\)
0.602168 0.798369i \(-0.294304\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 3.00000i 0.158777i
\(358\) 0 0
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 0 0
\(363\) − 2.00000i − 0.104973i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 8.00000i − 0.417597i −0.977959 0.208798i \(-0.933045\pi\)
0.977959 0.208798i \(-0.0669552\pi\)
\(368\) 0 0
\(369\) −18.0000 −0.937043
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) 4.00000i 0.207112i 0.994624 + 0.103556i \(0.0330221\pi\)
−0.994624 + 0.103556i \(0.966978\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 12.0000i − 0.618031i
\(378\) 0 0
\(379\) 17.0000 0.873231 0.436616 0.899648i \(-0.356177\pi\)
0.436616 + 0.899648i \(0.356177\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) 0 0
\(383\) − 30.0000i − 1.53293i −0.642287 0.766464i \(-0.722014\pi\)
0.642287 0.766464i \(-0.277986\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 16.0000i 0.813326i
\(388\) 0 0
\(389\) 24.0000 1.21685 0.608424 0.793612i \(-0.291802\pi\)
0.608424 + 0.793612i \(0.291802\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 2.00000i 0.100377i 0.998740 + 0.0501886i \(0.0159822\pi\)
−0.998740 + 0.0501886i \(0.984018\pi\)
\(398\) 0 0
\(399\) −7.00000 −0.350438
\(400\) 0 0
\(401\) −27.0000 −1.34832 −0.674158 0.738587i \(-0.735493\pi\)
−0.674158 + 0.738587i \(0.735493\pi\)
\(402\) 0 0
\(403\) − 8.00000i − 0.398508i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 24.0000i − 1.18964i
\(408\) 0 0
\(409\) 25.0000 1.23617 0.618085 0.786111i \(-0.287909\pi\)
0.618085 + 0.786111i \(0.287909\pi\)
\(410\) 0 0
\(411\) −21.0000 −1.03585
\(412\) 0 0
\(413\) − 12.0000i − 0.590481i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 7.00000i − 0.342791i
\(418\) 0 0
\(419\) −3.00000 −0.146560 −0.0732798 0.997311i \(-0.523347\pi\)
−0.0732798 + 0.997311i \(0.523347\pi\)
\(420\) 0 0
\(421\) 20.0000 0.974740 0.487370 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) 0 0
\(423\) 12.0000i 0.583460i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 10.0000i 0.483934i
\(428\) 0 0
\(429\) −6.00000 −0.289683
\(430\) 0 0
\(431\) 36.0000 1.73406 0.867029 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) 0 0
\(433\) − 11.0000i − 0.528626i −0.964437 0.264313i \(-0.914855\pi\)
0.964437 0.264313i \(-0.0851452\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −4.00000 −0.190910 −0.0954548 0.995434i \(-0.530431\pi\)
−0.0954548 + 0.995434i \(0.530431\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) 0 0
\(443\) 21.0000i 0.997740i 0.866677 + 0.498870i \(0.166252\pi\)
−0.866677 + 0.498870i \(0.833748\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 12.0000i − 0.567581i
\(448\) 0 0
\(449\) 15.0000 0.707894 0.353947 0.935266i \(-0.384839\pi\)
0.353947 + 0.935266i \(0.384839\pi\)
\(450\) 0 0
\(451\) 27.0000 1.27138
\(452\) 0 0
\(453\) − 8.00000i − 0.375873i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 17.0000i 0.795226i 0.917553 + 0.397613i \(0.130161\pi\)
−0.917553 + 0.397613i \(0.869839\pi\)
\(458\) 0 0
\(459\) −15.0000 −0.700140
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) 8.00000i 0.371792i 0.982569 + 0.185896i \(0.0595187\pi\)
−0.982569 + 0.185896i \(0.940481\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 36.0000i − 1.66588i −0.553362 0.832941i \(-0.686655\pi\)
0.553362 0.832941i \(-0.313345\pi\)
\(468\) 0 0
\(469\) 7.00000 0.323230
\(470\) 0 0
\(471\) −20.0000 −0.921551
\(472\) 0 0
\(473\) − 24.0000i − 1.10352i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 24.0000i 1.09888i
\(478\) 0 0
\(479\) 18.0000 0.822441 0.411220 0.911536i \(-0.365103\pi\)
0.411220 + 0.911536i \(0.365103\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 34.0000i 1.54069i 0.637629 + 0.770344i \(0.279915\pi\)
−0.637629 + 0.770344i \(0.720085\pi\)
\(488\) 0 0
\(489\) −5.00000 −0.226108
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 18.0000i 0.810679i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.00000i 0.269137i
\(498\) 0 0
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) − 6.00000i − 0.267527i −0.991013 0.133763i \(-0.957294\pi\)
0.991013 0.133763i \(-0.0427062\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 9.00000i 0.399704i
\(508\) 0 0
\(509\) −42.0000 −1.86162 −0.930809 0.365507i \(-0.880896\pi\)
−0.930809 + 0.365507i \(0.880896\pi\)
\(510\) 0 0
\(511\) −5.00000 −0.221187
\(512\) 0 0
\(513\) − 35.0000i − 1.54529i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 18.0000i − 0.791639i
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) 39.0000 1.70862 0.854311 0.519763i \(-0.173980\pi\)
0.854311 + 0.519763i \(0.173980\pi\)
\(522\) 0 0
\(523\) − 7.00000i − 0.306089i −0.988219 0.153044i \(-0.951092\pi\)
0.988219 0.153044i \(-0.0489077\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.0000i 0.522728i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 24.0000 1.04151
\(532\) 0 0
\(533\) 18.0000i 0.779667i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 3.00000i 0.129460i
\(538\) 0 0
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 0 0
\(543\) 2.00000i 0.0858282i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 35.0000i − 1.49649i −0.663421 0.748246i \(-0.730896\pi\)
0.663421 0.748246i \(-0.269104\pi\)
\(548\) 0 0
\(549\) −20.0000 −0.853579
\(550\) 0 0
\(551\) −42.0000 −1.78926
\(552\) 0 0
\(553\) − 14.0000i − 0.595341i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 36.0000i 1.52537i 0.646771 + 0.762684i \(0.276119\pi\)
−0.646771 + 0.762684i \(0.723881\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 9.00000 0.379980
\(562\) 0 0
\(563\) 12.0000i 0.505740i 0.967500 + 0.252870i \(0.0813744\pi\)
−0.967500 + 0.252870i \(0.918626\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 1.00000i − 0.0419961i
\(568\) 0 0
\(569\) 27.0000 1.13190 0.565949 0.824440i \(-0.308510\pi\)
0.565949 + 0.824440i \(0.308510\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) 6.00000i 0.250654i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 7.00000i − 0.291414i −0.989328 0.145707i \(-0.953454\pi\)
0.989328 0.145707i \(-0.0465456\pi\)
\(578\) 0 0
\(579\) 5.00000 0.207793
\(580\) 0 0
\(581\) −9.00000 −0.373383
\(582\) 0 0
\(583\) − 36.0000i − 1.49097i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 39.0000i − 1.60970i −0.593477 0.804851i \(-0.702245\pi\)
0.593477 0.804851i \(-0.297755\pi\)
\(588\) 0 0
\(589\) −28.0000 −1.15372
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 0 0
\(593\) 27.0000i 1.10876i 0.832265 + 0.554379i \(0.187044\pi\)
−0.832265 + 0.554379i \(0.812956\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 14.0000i 0.572982i
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −7.00000 −0.285536 −0.142768 0.989756i \(-0.545600\pi\)
−0.142768 + 0.989756i \(0.545600\pi\)
\(602\) 0 0
\(603\) 14.0000i 0.570124i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 44.0000i − 1.78590i −0.450151 0.892952i \(-0.648630\pi\)
0.450151 0.892952i \(-0.351370\pi\)
\(608\) 0 0
\(609\) 6.00000 0.243132
\(610\) 0 0
\(611\) 12.0000 0.485468
\(612\) 0 0
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 18.0000i − 0.724653i −0.932051 0.362326i \(-0.881983\pi\)
0.932051 0.362326i \(-0.118017\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 15.0000i − 0.600962i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 21.0000i 0.838659i
\(628\) 0 0
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) 0 0
\(633\) − 17.0000i − 0.675689i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2.00000i 0.0792429i
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) − 40.0000i − 1.57745i −0.614749 0.788723i \(-0.710743\pi\)
0.614749 0.788723i \(-0.289257\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.0000i 0.943537i 0.881722 + 0.471769i \(0.156384\pi\)
−0.881722 + 0.471769i \(0.843616\pi\)
\(648\) 0 0
\(649\) −36.0000 −1.41312
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) 0 0
\(653\) − 36.0000i − 1.40879i −0.709809 0.704394i \(-0.751219\pi\)
0.709809 0.704394i \(-0.248781\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 10.0000i − 0.390137i
\(658\) 0 0
\(659\) 9.00000 0.350590 0.175295 0.984516i \(-0.443912\pi\)
0.175295 + 0.984516i \(0.443912\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 0 0
\(663\) 6.00000i 0.233021i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −14.0000 −0.541271
\(670\) 0 0
\(671\) 30.0000 1.15814
\(672\) 0 0
\(673\) − 2.00000i − 0.0770943i −0.999257 0.0385472i \(-0.987727\pi\)
0.999257 0.0385472i \(-0.0122730\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 3.00000i 0.114792i 0.998351 + 0.0573959i \(0.0182797\pi\)
−0.998351 + 0.0573959i \(0.981720\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 26.0000i − 0.991962i
\(688\) 0 0
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) 19.0000 0.722794 0.361397 0.932412i \(-0.382300\pi\)
0.361397 + 0.932412i \(0.382300\pi\)
\(692\) 0 0
\(693\) 6.00000i 0.227921i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 27.0000i − 1.02270i
\(698\) 0 0
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) − 56.0000i − 2.11208i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 28.0000 1.05156 0.525781 0.850620i \(-0.323773\pi\)
0.525781 + 0.850620i \(0.323773\pi\)
\(710\) 0 0
\(711\) 28.0000 1.05008
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 12.0000i − 0.448148i
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 20.0000 0.744839
\(722\) 0 0
\(723\) − 25.0000i − 0.929760i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 34.0000i 1.26099i 0.776193 + 0.630495i \(0.217148\pi\)
−0.776193 + 0.630495i \(0.782852\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) 0 0
\(733\) 40.0000i 1.47743i 0.674016 + 0.738717i \(0.264568\pi\)
−0.674016 + 0.738717i \(0.735432\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 21.0000i − 0.773545i
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) −14.0000 −0.514303
\(742\) 0 0
\(743\) − 24.0000i − 0.880475i −0.897881 0.440237i \(-0.854894\pi\)
0.897881 0.440237i \(-0.145106\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 18.0000i − 0.658586i
\(748\) 0 0
\(749\) −3.00000 −0.109618
\(750\) 0 0
\(751\) 46.0000 1.67856 0.839282 0.543696i \(-0.182976\pi\)
0.839282 + 0.543696i \(0.182976\pi\)
\(752\) 0 0
\(753\) − 15.0000i − 0.546630i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 34.0000i − 1.23575i −0.786276 0.617876i \(-0.787994\pi\)
0.786276 0.617876i \(-0.212006\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −9.00000 −0.326250 −0.163125 0.986605i \(-0.552157\pi\)
−0.163125 + 0.986605i \(0.552157\pi\)
\(762\) 0 0
\(763\) 14.0000i 0.506834i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 24.0000i − 0.866590i
\(768\) 0 0
\(769\) −23.0000 −0.829401 −0.414701 0.909958i \(-0.636114\pi\)
−0.414701 + 0.909958i \(0.636114\pi\)
\(770\) 0 0
\(771\) 30.0000 1.08042
\(772\) 0 0
\(773\) 12.0000i 0.431610i 0.976436 + 0.215805i \(0.0692376\pi\)
−0.976436 + 0.215805i \(0.930762\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 8.00000i 0.286998i
\(778\) 0 0
\(779\) 63.0000 2.25721
\(780\) 0 0
\(781\) 18.0000 0.644091
\(782\) 0 0
\(783\) 30.0000i 1.07211i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 4.00000i 0.142585i 0.997455 + 0.0712923i \(0.0227123\pi\)
−0.997455 + 0.0712923i \(0.977288\pi\)
\(788\) 0 0
\(789\) −6.00000 −0.213606
\(790\) 0 0
\(791\) 9.00000 0.320003
\(792\) 0 0
\(793\) 20.0000i 0.710221i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 30.0000i − 1.06265i −0.847167 0.531327i \(-0.821693\pi\)
0.847167 0.531327i \(-0.178307\pi\)
\(798\) 0 0
\(799\) −18.0000 −0.636794
\(800\) 0 0
\(801\) 30.0000 1.06000
\(802\) 0 0
\(803\) 15.0000i 0.529339i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 6.00000i 0.211210i
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) −44.0000 −1.54505 −0.772524 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) 0 0
\(813\) − 2.00000i − 0.0701431i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 56.0000i − 1.95919i
\(818\) 0 0
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) 24.0000 0.837606 0.418803 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) 0 0
\(823\) 26.0000i 0.906303i 0.891434 + 0.453152i \(0.149700\pi\)
−0.891434 + 0.453152i \(0.850300\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 9.00000i − 0.312961i −0.987681 0.156480i \(-0.949985\pi\)
0.987681 0.156480i \(-0.0500148\pi\)
\(828\) 0 0
\(829\) 4.00000 0.138926 0.0694629 0.997585i \(-0.477871\pi\)
0.0694629 + 0.997585i \(0.477871\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 0 0
\(833\) − 3.00000i − 0.103944i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 20.0000i 0.691301i
\(838\) 0 0
\(839\) −6.00000 −0.207143 −0.103572 0.994622i \(-0.533027\pi\)
−0.103572 + 0.994622i \(0.533027\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) − 18.0000i − 0.619953i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 2.00000i 0.0687208i
\(848\) 0 0
\(849\) 1.00000 0.0343199
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 10.0000i 0.342393i 0.985237 + 0.171197i \(0.0547634\pi\)
−0.985237 + 0.171197i \(0.945237\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 15.0000i 0.512390i 0.966625 + 0.256195i \(0.0824690\pi\)
−0.966625 + 0.256195i \(0.917531\pi\)
\(858\) 0 0
\(859\) −31.0000 −1.05771 −0.528853 0.848713i \(-0.677378\pi\)
−0.528853 + 0.848713i \(0.677378\pi\)
\(860\) 0 0
\(861\) −9.00000 −0.306719
\(862\) 0 0
\(863\) − 12.0000i − 0.408485i −0.978920 0.204242i \(-0.934527\pi\)
0.978920 0.204242i \(-0.0654731\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 8.00000i 0.271694i
\(868\) 0 0
\(869\) −42.0000 −1.42475
\(870\) 0 0
\(871\) 14.0000 0.474372
\(872\) 0 0
\(873\) − 20.0000i − 0.676897i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 32.0000i 1.08056i 0.841484 + 0.540282i \(0.181682\pi\)
−0.841484 + 0.540282i \(0.818318\pi\)
\(878\) 0 0
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 0 0
\(883\) 47.0000i 1.58168i 0.612026 + 0.790838i \(0.290355\pi\)
−0.612026 + 0.790838i \(0.709645\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 6.00000i − 0.201460i −0.994914 0.100730i \(-0.967882\pi\)
0.994914 0.100730i \(-0.0321179\pi\)
\(888\) 0 0
\(889\) −2.00000 −0.0670778
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 0 0
\(893\) − 42.0000i − 1.40548i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 8.00000i 0.266223i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 4.00000i 0.132818i 0.997792 + 0.0664089i \(0.0211542\pi\)
−0.997792 + 0.0664089i \(0.978846\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 0 0
\(913\) 27.0000i 0.893570i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −34.0000 −1.12156 −0.560778 0.827966i \(-0.689498\pi\)
−0.560778 + 0.827966i \(0.689498\pi\)
\(920\) 0 0
\(921\) −7.00000 −0.230658
\(922\) 0 0
\(923\) 12.0000i 0.394985i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 40.0000i 1.31377i
\(928\) 0 0
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) 7.00000 0.229416
\(932\) 0 0
\(933\) 18.0000i 0.589294i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 29.0000i 0.947389i 0.880689 + 0.473694i \(0.157080\pi\)
−0.880689 + 0.473694i \(0.842920\pi\)
\(938\) 0 0
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) 24.0000 0.782378 0.391189 0.920310i \(-0.372064\pi\)
0.391189 + 0.920310i \(0.372064\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 12.0000i − 0.389948i −0.980808 0.194974i \(-0.937538\pi\)
0.980808 0.194974i \(-0.0624622\pi\)
\(948\) 0 0
\(949\) −10.0000 −0.324614
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) 0 0
\(953\) − 57.0000i − 1.84641i −0.384307 0.923206i \(-0.625559\pi\)
0.384307 0.923206i \(-0.374441\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 18.0000i − 0.581857i
\(958\) 0 0
\(959\) 21.0000 0.678125
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) − 6.00000i − 0.193347i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 34.0000i 1.09337i 0.837340 + 0.546683i \(0.184110\pi\)
−0.837340 + 0.546683i \(0.815890\pi\)
\(968\) 0 0
\(969\) 21.0000 0.674617
\(970\) 0 0
\(971\) −9.00000 −0.288824 −0.144412 0.989518i \(-0.546129\pi\)
−0.144412 + 0.989518i \(0.546129\pi\)
\(972\) 0 0
\(973\) 7.00000i 0.224410i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 3.00000i − 0.0959785i −0.998848 0.0479893i \(-0.984719\pi\)
0.998848 0.0479893i \(-0.0152813\pi\)
\(978\) 0 0
\(979\) −45.0000 −1.43821
\(980\) 0 0
\(981\) −28.0000 −0.893971
\(982\) 0 0
\(983\) 12.0000i 0.382741i 0.981518 + 0.191370i \(0.0612931\pi\)
−0.981518 + 0.191370i \(0.938707\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 6.00000i 0.190982i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −20.0000 −0.635321 −0.317660 0.948205i \(-0.602897\pi\)
−0.317660 + 0.948205i \(0.602897\pi\)
\(992\) 0 0
\(993\) 25.0000i 0.793351i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 62.0000i 1.96356i 0.190022 + 0.981780i \(0.439144\pi\)
−0.190022 + 0.981780i \(0.560856\pi\)
\(998\) 0 0
\(999\) −40.0000 −1.26554
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 2800.2.g.i.449.2 2
4.3 odd 2 350.2.c.c.99.2 2
5.2 odd 4 2800.2.a.x.1.1 1
5.3 odd 4 2800.2.a.h.1.1 1
5.4 even 2 inner 2800.2.g.i.449.1 2
12.11 even 2 3150.2.g.f.2899.1 2
20.3 even 4 350.2.a.e.1.1 yes 1
20.7 even 4 350.2.a.a.1.1 1
20.19 odd 2 350.2.c.c.99.1 2
28.27 even 2 2450.2.c.h.99.2 2
60.23 odd 4 3150.2.a.m.1.1 1
60.47 odd 4 3150.2.a.x.1.1 1
60.59 even 2 3150.2.g.f.2899.2 2
140.27 odd 4 2450.2.a.m.1.1 1
140.83 odd 4 2450.2.a.x.1.1 1
140.139 even 2 2450.2.c.h.99.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
350.2.a.a.1.1 1 20.7 even 4
350.2.a.e.1.1 yes 1 20.3 even 4
350.2.c.c.99.1 2 20.19 odd 2
350.2.c.c.99.2 2 4.3 odd 2
2450.2.a.m.1.1 1 140.27 odd 4
2450.2.a.x.1.1 1 140.83 odd 4
2450.2.c.h.99.1 2 140.139 even 2
2450.2.c.h.99.2 2 28.27 even 2
2800.2.a.h.1.1 1 5.3 odd 4
2800.2.a.x.1.1 1 5.2 odd 4
2800.2.g.i.449.1 2 5.4 even 2 inner
2800.2.g.i.449.2 2 1.1 even 1 trivial
3150.2.a.m.1.1 1 60.23 odd 4
3150.2.a.x.1.1 1 60.47 odd 4
3150.2.g.f.2899.1 2 12.11 even 2
3150.2.g.f.2899.2 2 60.59 even 2