Properties

Label 2800.2.g.t.449.1
Level $2800$
Weight $2$
Character 2800.449
Analytic conductor $22.358$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \(x^{4} + 9 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.1
Root \(-2.56155i\) of defining polynomial
Character \(\chi\) \(=\) 2800.449
Dual form 2800.2.g.t.449.4

$q$-expansion

\(f(q)\) \(=\) \(q-2.56155i q^{3} +1.00000i q^{7} -3.56155 q^{9} +O(q^{10})\) \(q-2.56155i q^{3} +1.00000i q^{7} -3.56155 q^{9} -2.56155 q^{11} -4.56155i q^{13} -4.56155i q^{17} +1.12311 q^{19} +2.56155 q^{21} -5.12311i q^{23} +1.43845i q^{27} +5.68466 q^{29} +6.56155i q^{33} +6.00000i q^{37} -11.6847 q^{39} -3.12311 q^{41} +9.12311i q^{43} -3.68466i q^{47} -1.00000 q^{49} -11.6847 q^{51} -3.12311i q^{53} -2.87689i q^{57} -4.00000 q^{59} -9.36932 q^{61} -3.56155i q^{63} +6.24621i q^{67} -13.1231 q^{69} -8.00000 q^{71} -4.24621i q^{73} -2.56155i q^{77} -6.56155 q^{79} -7.00000 q^{81} +4.00000i q^{83} -14.5616i q^{87} -7.12311 q^{89} +4.56155 q^{91} -14.8078i q^{97} +9.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 6q^{9} + O(q^{10}) \) \( 4q - 6q^{9} - 2q^{11} - 12q^{19} + 2q^{21} - 2q^{29} - 22q^{39} + 4q^{41} - 4q^{49} - 22q^{51} - 16q^{59} + 12q^{61} - 36q^{69} - 32q^{71} - 18q^{79} - 28q^{81} - 12q^{89} + 10q^{91} + 20q^{99} + O(q^{100}) \)

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
\(3\) − 2.56155i − 1.47891i −0.673204 0.739457i \(-0.735083\pi\)
0.673204 0.739457i \(-0.264917\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −3.56155 −1.18718
\(10\) 0 0
\(11\) −2.56155 −0.772337 −0.386169 0.922428i \(-0.626202\pi\)
−0.386169 + 0.922428i \(0.626202\pi\)
\(12\) 0 0
\(13\) − 4.56155i − 1.26515i −0.774500 0.632574i \(-0.781999\pi\)
0.774500 0.632574i \(-0.218001\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 4.56155i − 1.10634i −0.833069 0.553170i \(-0.813418\pi\)
0.833069 0.553170i \(-0.186582\pi\)
\(18\) 0 0
\(19\) 1.12311 0.257658 0.128829 0.991667i \(-0.458878\pi\)
0.128829 + 0.991667i \(0.458878\pi\)
\(20\) 0 0
\(21\) 2.56155 0.558977
\(22\) 0 0
\(23\) − 5.12311i − 1.06824i −0.845408 0.534121i \(-0.820643\pi\)
0.845408 0.534121i \(-0.179357\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.43845i 0.276829i
\(28\) 0 0
\(29\) 5.68466 1.05561 0.527807 0.849364i \(-0.323014\pi\)
0.527807 + 0.849364i \(0.323014\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 6.56155i 1.14222i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) −11.6847 −1.87104
\(40\) 0 0
\(41\) −3.12311 −0.487747 −0.243874 0.969807i \(-0.578418\pi\)
−0.243874 + 0.969807i \(0.578418\pi\)
\(42\) 0 0
\(43\) 9.12311i 1.39126i 0.718400 + 0.695630i \(0.244875\pi\)
−0.718400 + 0.695630i \(0.755125\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 3.68466i − 0.537463i −0.963215 0.268731i \(-0.913396\pi\)
0.963215 0.268731i \(-0.0866044\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −11.6847 −1.63618
\(52\) 0 0
\(53\) − 3.12311i − 0.428992i −0.976725 0.214496i \(-0.931189\pi\)
0.976725 0.214496i \(-0.0688108\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 2.87689i − 0.381054i
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −9.36932 −1.19962 −0.599809 0.800143i \(-0.704757\pi\)
−0.599809 + 0.800143i \(0.704757\pi\)
\(62\) 0 0
\(63\) − 3.56155i − 0.448713i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.24621i 0.763096i 0.924349 + 0.381548i \(0.124609\pi\)
−0.924349 + 0.381548i \(0.875391\pi\)
\(68\) 0 0
\(69\) −13.1231 −1.57984
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) − 4.24621i − 0.496981i −0.968634 0.248491i \(-0.920065\pi\)
0.968634 0.248491i \(-0.0799345\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 2.56155i − 0.291916i
\(78\) 0 0
\(79\) −6.56155 −0.738232 −0.369116 0.929383i \(-0.620340\pi\)
−0.369116 + 0.929383i \(0.620340\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 14.5616i − 1.56116i
\(88\) 0 0
\(89\) −7.12311 −0.755048 −0.377524 0.926000i \(-0.623224\pi\)
−0.377524 + 0.926000i \(0.623224\pi\)
\(90\) 0 0
\(91\) 4.56155 0.478181
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 14.8078i − 1.50350i −0.659448 0.751750i \(-0.729210\pi\)
0.659448 0.751750i \(-0.270790\pi\)
\(98\) 0 0
\(99\) 9.12311 0.916907
\(100\) 0 0
\(101\) 0.246211 0.0244989 0.0122495 0.999925i \(-0.496101\pi\)
0.0122495 + 0.999925i \(0.496101\pi\)
\(102\) 0 0
\(103\) 1.43845i 0.141734i 0.997486 + 0.0708672i \(0.0225767\pi\)
−0.997486 + 0.0708672i \(0.977423\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.3693i 1.09911i 0.835456 + 0.549557i \(0.185203\pi\)
−0.835456 + 0.549557i \(0.814797\pi\)
\(108\) 0 0
\(109\) −17.6847 −1.69388 −0.846942 0.531686i \(-0.821559\pi\)
−0.846942 + 0.531686i \(0.821559\pi\)
\(110\) 0 0
\(111\) 15.3693 1.45879
\(112\) 0 0
\(113\) 14.0000i 1.31701i 0.752577 + 0.658505i \(0.228811\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 16.2462i 1.50196i
\(118\) 0 0
\(119\) 4.56155 0.418157
\(120\) 0 0
\(121\) −4.43845 −0.403495
\(122\) 0 0
\(123\) 8.00000i 0.721336i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 10.2462i − 0.909204i −0.890695 0.454602i \(-0.849781\pi\)
0.890695 0.454602i \(-0.150219\pi\)
\(128\) 0 0
\(129\) 23.3693 2.05755
\(130\) 0 0
\(131\) 9.12311 0.797089 0.398545 0.917149i \(-0.369515\pi\)
0.398545 + 0.917149i \(0.369515\pi\)
\(132\) 0 0
\(133\) 1.12311i 0.0973856i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 8.87689i − 0.758404i −0.925314 0.379202i \(-0.876199\pi\)
0.925314 0.379202i \(-0.123801\pi\)
\(138\) 0 0
\(139\) −6.87689 −0.583291 −0.291645 0.956527i \(-0.594203\pi\)
−0.291645 + 0.956527i \(0.594203\pi\)
\(140\) 0 0
\(141\) −9.43845 −0.794861
\(142\) 0 0
\(143\) 11.6847i 0.977120i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 2.56155i 0.211273i
\(148\) 0 0
\(149\) 4.24621 0.347863 0.173932 0.984758i \(-0.444353\pi\)
0.173932 + 0.984758i \(0.444353\pi\)
\(150\) 0 0
\(151\) −21.9309 −1.78471 −0.892354 0.451335i \(-0.850948\pi\)
−0.892354 + 0.451335i \(0.850948\pi\)
\(152\) 0 0
\(153\) 16.2462i 1.31343i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3.75379i 0.299585i 0.988717 + 0.149792i \(0.0478606\pi\)
−0.988717 + 0.149792i \(0.952139\pi\)
\(158\) 0 0
\(159\) −8.00000 −0.634441
\(160\) 0 0
\(161\) 5.12311 0.403757
\(162\) 0 0
\(163\) 1.12311i 0.0879684i 0.999032 + 0.0439842i \(0.0140051\pi\)
−0.999032 + 0.0439842i \(0.985995\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 21.9309i − 1.69706i −0.529146 0.848531i \(-0.677488\pi\)
0.529146 0.848531i \(-0.322512\pi\)
\(168\) 0 0
\(169\) −7.80776 −0.600597
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) 8.56155i 0.650923i 0.945555 + 0.325461i \(0.105520\pi\)
−0.945555 + 0.325461i \(0.894480\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 10.2462i 0.770152i
\(178\) 0 0
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) 23.6155 1.75533 0.877664 0.479276i \(-0.159101\pi\)
0.877664 + 0.479276i \(0.159101\pi\)
\(182\) 0 0
\(183\) 24.0000i 1.77413i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 11.6847i 0.854467i
\(188\) 0 0
\(189\) −1.43845 −0.104632
\(190\) 0 0
\(191\) 9.43845 0.682942 0.341471 0.939892i \(-0.389075\pi\)
0.341471 + 0.939892i \(0.389075\pi\)
\(192\) 0 0
\(193\) 5.36932i 0.386492i 0.981150 + 0.193246i \(0.0619015\pi\)
−0.981150 + 0.193246i \(0.938098\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 7.12311i − 0.507500i −0.967270 0.253750i \(-0.918336\pi\)
0.967270 0.253750i \(-0.0816641\pi\)
\(198\) 0 0
\(199\) −18.2462 −1.29344 −0.646720 0.762728i \(-0.723860\pi\)
−0.646720 + 0.762728i \(0.723860\pi\)
\(200\) 0 0
\(201\) 16.0000 1.12855
\(202\) 0 0
\(203\) 5.68466i 0.398985i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 18.2462i 1.26820i
\(208\) 0 0
\(209\) −2.87689 −0.198999
\(210\) 0 0
\(211\) 23.0540 1.58710 0.793551 0.608504i \(-0.208230\pi\)
0.793551 + 0.608504i \(0.208230\pi\)
\(212\) 0 0
\(213\) 20.4924i 1.40412i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −10.8769 −0.734992
\(220\) 0 0
\(221\) −20.8078 −1.39968
\(222\) 0 0
\(223\) − 6.56155i − 0.439394i −0.975568 0.219697i \(-0.929493\pi\)
0.975568 0.219697i \(-0.0705069\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 23.6847i − 1.57201i −0.618223 0.786003i \(-0.712147\pi\)
0.618223 0.786003i \(-0.287853\pi\)
\(228\) 0 0
\(229\) −19.1231 −1.26369 −0.631845 0.775095i \(-0.717702\pi\)
−0.631845 + 0.775095i \(0.717702\pi\)
\(230\) 0 0
\(231\) −6.56155 −0.431718
\(232\) 0 0
\(233\) 3.12311i 0.204601i 0.994754 + 0.102301i \(0.0326204\pi\)
−0.994754 + 0.102301i \(0.967380\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 16.8078i 1.09178i
\(238\) 0 0
\(239\) −0.807764 −0.0522499 −0.0261250 0.999659i \(-0.508317\pi\)
−0.0261250 + 0.999659i \(0.508317\pi\)
\(240\) 0 0
\(241\) 12.2462 0.788848 0.394424 0.918929i \(-0.370944\pi\)
0.394424 + 0.918929i \(0.370944\pi\)
\(242\) 0 0
\(243\) 22.2462i 1.42710i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 5.12311i − 0.325975i
\(248\) 0 0
\(249\) 10.2462 0.649327
\(250\) 0 0
\(251\) 17.1231 1.08080 0.540400 0.841408i \(-0.318273\pi\)
0.540400 + 0.841408i \(0.318273\pi\)
\(252\) 0 0
\(253\) 13.1231i 0.825043i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 22.4924i 1.40304i 0.712650 + 0.701519i \(0.247495\pi\)
−0.712650 + 0.701519i \(0.752505\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) −20.2462 −1.25321
\(262\) 0 0
\(263\) − 21.1231i − 1.30251i −0.758860 0.651253i \(-0.774244\pi\)
0.758860 0.651253i \(-0.225756\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 18.2462i 1.11665i
\(268\) 0 0
\(269\) −28.7386 −1.75223 −0.876113 0.482106i \(-0.839872\pi\)
−0.876113 + 0.482106i \(0.839872\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 0 0
\(273\) − 11.6847i − 0.707188i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 16.2462i 0.976140i 0.872804 + 0.488070i \(0.162299\pi\)
−0.872804 + 0.488070i \(0.837701\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 16.5616 0.987979 0.493990 0.869468i \(-0.335538\pi\)
0.493990 + 0.869468i \(0.335538\pi\)
\(282\) 0 0
\(283\) − 23.6847i − 1.40791i −0.710246 0.703953i \(-0.751416\pi\)
0.710246 0.703953i \(-0.248584\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 3.12311i − 0.184351i
\(288\) 0 0
\(289\) −3.80776 −0.223986
\(290\) 0 0
\(291\) −37.9309 −2.22355
\(292\) 0 0
\(293\) − 9.68466i − 0.565784i −0.959152 0.282892i \(-0.908706\pi\)
0.959152 0.282892i \(-0.0912938\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 3.68466i − 0.213806i
\(298\) 0 0
\(299\) −23.3693 −1.35148
\(300\) 0 0
\(301\) −9.12311 −0.525847
\(302\) 0 0
\(303\) − 0.630683i − 0.0362318i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 31.6847i 1.80834i 0.427174 + 0.904169i \(0.359509\pi\)
−0.427174 + 0.904169i \(0.640491\pi\)
\(308\) 0 0
\(309\) 3.68466 0.209613
\(310\) 0 0
\(311\) 9.61553 0.545247 0.272623 0.962121i \(-0.412109\pi\)
0.272623 + 0.962121i \(0.412109\pi\)
\(312\) 0 0
\(313\) − 31.3002i − 1.76919i −0.466359 0.884596i \(-0.654434\pi\)
0.466359 0.884596i \(-0.345566\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 22.4924i − 1.26330i −0.775254 0.631650i \(-0.782378\pi\)
0.775254 0.631650i \(-0.217622\pi\)
\(318\) 0 0
\(319\) −14.5616 −0.815290
\(320\) 0 0
\(321\) 29.1231 1.62549
\(322\) 0 0
\(323\) − 5.12311i − 0.285057i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 45.3002i 2.50511i
\(328\) 0 0
\(329\) 3.68466 0.203142
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 0 0
\(333\) − 21.3693i − 1.17103i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 34.4924i − 1.87892i −0.342656 0.939461i \(-0.611326\pi\)
0.342656 0.939461i \(-0.388674\pi\)
\(338\) 0 0
\(339\) 35.8617 1.94774
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.12311i 0.0602915i 0.999546 + 0.0301457i \(0.00959714\pi\)
−0.999546 + 0.0301457i \(0.990403\pi\)
\(348\) 0 0
\(349\) 22.4924 1.20399 0.601996 0.798499i \(-0.294372\pi\)
0.601996 + 0.798499i \(0.294372\pi\)
\(350\) 0 0
\(351\) 6.56155 0.350230
\(352\) 0 0
\(353\) 14.8078i 0.788138i 0.919081 + 0.394069i \(0.128933\pi\)
−0.919081 + 0.394069i \(0.871067\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 11.6847i − 0.618418i
\(358\) 0 0
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) −17.7386 −0.933612
\(362\) 0 0
\(363\) 11.3693i 0.596734i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 3.68466i − 0.192338i −0.995365 0.0961688i \(-0.969341\pi\)
0.995365 0.0961688i \(-0.0306589\pi\)
\(368\) 0 0
\(369\) 11.1231 0.579046
\(370\) 0 0
\(371\) 3.12311 0.162144
\(372\) 0 0
\(373\) − 29.3693i − 1.52069i −0.649522 0.760343i \(-0.725031\pi\)
0.649522 0.760343i \(-0.274969\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 25.9309i − 1.33551i
\(378\) 0 0
\(379\) 16.4924 0.847159 0.423579 0.905859i \(-0.360773\pi\)
0.423579 + 0.905859i \(0.360773\pi\)
\(380\) 0 0
\(381\) −26.2462 −1.34463
\(382\) 0 0
\(383\) − 10.2462i − 0.523557i −0.965128 0.261778i \(-0.915691\pi\)
0.965128 0.261778i \(-0.0843090\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 32.4924i − 1.65168i
\(388\) 0 0
\(389\) −3.93087 −0.199303 −0.0996515 0.995022i \(-0.531773\pi\)
−0.0996515 + 0.995022i \(0.531773\pi\)
\(390\) 0 0
\(391\) −23.3693 −1.18184
\(392\) 0 0
\(393\) − 23.3693i − 1.17883i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 23.4384i 1.17634i 0.808737 + 0.588171i \(0.200152\pi\)
−0.808737 + 0.588171i \(0.799848\pi\)
\(398\) 0 0
\(399\) 2.87689 0.144025
\(400\) 0 0
\(401\) 27.4384 1.37021 0.685105 0.728444i \(-0.259756\pi\)
0.685105 + 0.728444i \(0.259756\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 15.3693i − 0.761829i
\(408\) 0 0
\(409\) 26.4924 1.30997 0.654983 0.755644i \(-0.272676\pi\)
0.654983 + 0.755644i \(0.272676\pi\)
\(410\) 0 0
\(411\) −22.7386 −1.12161
\(412\) 0 0
\(413\) − 4.00000i − 0.196827i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 17.6155i 0.862636i
\(418\) 0 0
\(419\) 9.75379 0.476504 0.238252 0.971203i \(-0.423426\pi\)
0.238252 + 0.971203i \(0.423426\pi\)
\(420\) 0 0
\(421\) 9.68466 0.472001 0.236001 0.971753i \(-0.424163\pi\)
0.236001 + 0.971753i \(0.424163\pi\)
\(422\) 0 0
\(423\) 13.1231i 0.638067i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 9.36932i − 0.453413i
\(428\) 0 0
\(429\) 29.9309 1.44508
\(430\) 0 0
\(431\) −0.807764 −0.0389086 −0.0194543 0.999811i \(-0.506193\pi\)
−0.0194543 + 0.999811i \(0.506193\pi\)
\(432\) 0 0
\(433\) 8.24621i 0.396288i 0.980173 + 0.198144i \(0.0634913\pi\)
−0.980173 + 0.198144i \(0.936509\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 5.75379i − 0.275241i
\(438\) 0 0
\(439\) −15.3693 −0.733537 −0.366769 0.930312i \(-0.619536\pi\)
−0.366769 + 0.930312i \(0.619536\pi\)
\(440\) 0 0
\(441\) 3.56155 0.169598
\(442\) 0 0
\(443\) − 27.3693i − 1.30036i −0.759782 0.650178i \(-0.774694\pi\)
0.759782 0.650178i \(-0.225306\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 10.8769i − 0.514459i
\(448\) 0 0
\(449\) −18.8078 −0.887593 −0.443797 0.896128i \(-0.646369\pi\)
−0.443797 + 0.896128i \(0.646369\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) 0 0
\(453\) 56.1771i 2.63943i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 8.87689i − 0.415244i −0.978209 0.207622i \(-0.933428\pi\)
0.978209 0.207622i \(-0.0665723\pi\)
\(458\) 0 0
\(459\) 6.56155 0.306267
\(460\) 0 0
\(461\) −4.87689 −0.227140 −0.113570 0.993530i \(-0.536229\pi\)
−0.113570 + 0.993530i \(0.536229\pi\)
\(462\) 0 0
\(463\) − 20.4924i − 0.952364i −0.879347 0.476182i \(-0.842020\pi\)
0.879347 0.476182i \(-0.157980\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 26.5616i − 1.22912i −0.788869 0.614561i \(-0.789333\pi\)
0.788869 0.614561i \(-0.210667\pi\)
\(468\) 0 0
\(469\) −6.24621 −0.288423
\(470\) 0 0
\(471\) 9.61553 0.443060
\(472\) 0 0
\(473\) − 23.3693i − 1.07452i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 11.1231i 0.509292i
\(478\) 0 0
\(479\) 13.1231 0.599610 0.299805 0.954001i \(-0.403078\pi\)
0.299805 + 0.954001i \(0.403078\pi\)
\(480\) 0 0
\(481\) 27.3693 1.24793
\(482\) 0 0
\(483\) − 13.1231i − 0.597122i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 5.12311i − 0.232150i −0.993240 0.116075i \(-0.962969\pi\)
0.993240 0.116075i \(-0.0370313\pi\)
\(488\) 0 0
\(489\) 2.87689 0.130098
\(490\) 0 0
\(491\) −4.17708 −0.188509 −0.0942545 0.995548i \(-0.530047\pi\)
−0.0942545 + 0.995548i \(0.530047\pi\)
\(492\) 0 0
\(493\) − 25.9309i − 1.16787i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 8.00000i − 0.358849i
\(498\) 0 0
\(499\) −4.17708 −0.186992 −0.0934959 0.995620i \(-0.529804\pi\)
−0.0934959 + 0.995620i \(0.529804\pi\)
\(500\) 0 0
\(501\) −56.1771 −2.50981
\(502\) 0 0
\(503\) 10.0691i 0.448960i 0.974479 + 0.224480i \(0.0720684\pi\)
−0.974479 + 0.224480i \(0.927932\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 20.0000i 0.888231i
\(508\) 0 0
\(509\) 28.2462 1.25199 0.625996 0.779827i \(-0.284693\pi\)
0.625996 + 0.779827i \(0.284693\pi\)
\(510\) 0 0
\(511\) 4.24621 0.187841
\(512\) 0 0
\(513\) 1.61553i 0.0713273i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 9.43845i 0.415102i
\(518\) 0 0
\(519\) 21.9309 0.962658
\(520\) 0 0
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) 0 0
\(523\) 7.50758i 0.328283i 0.986437 + 0.164142i \(0.0524854\pi\)
−0.986437 + 0.164142i \(0.947515\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −3.24621 −0.141140
\(530\) 0 0
\(531\) 14.2462 0.618233
\(532\) 0 0
\(533\) 14.2462i 0.617072i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 51.2311i − 2.21078i
\(538\) 0 0
\(539\) 2.56155 0.110334
\(540\) 0 0
\(541\) −17.1922 −0.739152 −0.369576 0.929201i \(-0.620497\pi\)
−0.369576 + 0.929201i \(0.620497\pi\)
\(542\) 0 0
\(543\) − 60.4924i − 2.59598i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 14.2462i − 0.609124i −0.952493 0.304562i \(-0.901490\pi\)
0.952493 0.304562i \(-0.0985101\pi\)
\(548\) 0 0
\(549\) 33.3693 1.42417
\(550\) 0 0
\(551\) 6.38447 0.271988
\(552\) 0 0
\(553\) − 6.56155i − 0.279026i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 4.87689i − 0.206641i −0.994648 0.103320i \(-0.967053\pi\)
0.994648 0.103320i \(-0.0329467\pi\)
\(558\) 0 0
\(559\) 41.6155 1.76015
\(560\) 0 0
\(561\) 29.9309 1.26368
\(562\) 0 0
\(563\) − 28.0000i − 1.18006i −0.807382 0.590030i \(-0.799116\pi\)
0.807382 0.590030i \(-0.200884\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 7.00000i − 0.293972i
\(568\) 0 0
\(569\) −34.9848 −1.46664 −0.733320 0.679883i \(-0.762031\pi\)
−0.733320 + 0.679883i \(0.762031\pi\)
\(570\) 0 0
\(571\) −7.50758 −0.314182 −0.157091 0.987584i \(-0.550212\pi\)
−0.157091 + 0.987584i \(0.550212\pi\)
\(572\) 0 0
\(573\) − 24.1771i − 1.01001i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 13.0540i 0.543444i 0.962376 + 0.271722i \(0.0875931\pi\)
−0.962376 + 0.271722i \(0.912407\pi\)
\(578\) 0 0
\(579\) 13.7538 0.571588
\(580\) 0 0
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) 8.00000i 0.331326i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 9.75379i 0.402582i 0.979531 + 0.201291i \(0.0645137\pi\)
−0.979531 + 0.201291i \(0.935486\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −18.2462 −0.750549
\(592\) 0 0
\(593\) 23.4384i 0.962502i 0.876583 + 0.481251i \(0.159817\pi\)
−0.876583 + 0.481251i \(0.840183\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 46.7386i 1.91288i
\(598\) 0 0
\(599\) 8.80776 0.359875 0.179938 0.983678i \(-0.442410\pi\)
0.179938 + 0.983678i \(0.442410\pi\)
\(600\) 0 0
\(601\) −26.4924 −1.08065 −0.540324 0.841457i \(-0.681698\pi\)
−0.540324 + 0.841457i \(0.681698\pi\)
\(602\) 0 0
\(603\) − 22.2462i − 0.905936i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 4.94602i 0.200753i 0.994950 + 0.100376i \(0.0320047\pi\)
−0.994950 + 0.100376i \(0.967995\pi\)
\(608\) 0 0
\(609\) 14.5616 0.590064
\(610\) 0 0
\(611\) −16.8078 −0.679969
\(612\) 0 0
\(613\) 8.73863i 0.352950i 0.984305 + 0.176475i \(0.0564695\pi\)
−0.984305 + 0.176475i \(0.943531\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 15.7538i 0.634224i 0.948388 + 0.317112i \(0.102713\pi\)
−0.948388 + 0.317112i \(0.897287\pi\)
\(618\) 0 0
\(619\) −42.1080 −1.69246 −0.846231 0.532817i \(-0.821134\pi\)
−0.846231 + 0.532817i \(0.821134\pi\)
\(620\) 0 0
\(621\) 7.36932 0.295720
\(622\) 0 0
\(623\) − 7.12311i − 0.285381i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 7.36932i 0.294302i
\(628\) 0 0
\(629\) 27.3693 1.09129
\(630\) 0 0
\(631\) −8.80776 −0.350632 −0.175316 0.984512i \(-0.556095\pi\)
−0.175316 + 0.984512i \(0.556095\pi\)
\(632\) 0 0
\(633\) − 59.0540i − 2.34718i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 4.56155i 0.180735i
\(638\) 0 0
\(639\) 28.4924 1.12714
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) − 2.56155i − 0.101018i −0.998724 0.0505089i \(-0.983916\pi\)
0.998724 0.0505089i \(-0.0160843\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 3.50758i − 0.137897i −0.997620 0.0689486i \(-0.978036\pi\)
0.997620 0.0689486i \(-0.0219644\pi\)
\(648\) 0 0
\(649\) 10.2462 0.402199
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 49.2311i − 1.92656i −0.268499 0.963280i \(-0.586527\pi\)
0.268499 0.963280i \(-0.413473\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 15.1231i 0.590009i
\(658\) 0 0
\(659\) −36.1771 −1.40926 −0.704629 0.709575i \(-0.748887\pi\)
−0.704629 + 0.709575i \(0.748887\pi\)
\(660\) 0 0
\(661\) 3.12311 0.121475 0.0607374 0.998154i \(-0.480655\pi\)
0.0607374 + 0.998154i \(0.480655\pi\)
\(662\) 0 0
\(663\) 53.3002i 2.07001i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 29.1231i − 1.12765i
\(668\) 0 0
\(669\) −16.8078 −0.649826
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) 25.8617i 0.996897i 0.866919 + 0.498448i \(0.166097\pi\)
−0.866919 + 0.498448i \(0.833903\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 23.9309i − 0.919738i −0.887987 0.459869i \(-0.847896\pi\)
0.887987 0.459869i \(-0.152104\pi\)
\(678\) 0 0
\(679\) 14.8078 0.568270
\(680\) 0 0
\(681\) −60.6695 −2.32486
\(682\) 0 0
\(683\) 42.7386i 1.63535i 0.575681 + 0.817674i \(0.304737\pi\)
−0.575681 + 0.817674i \(0.695263\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 48.9848i 1.86889i
\(688\) 0 0
\(689\) −14.2462 −0.542737
\(690\) 0 0
\(691\) −8.49242 −0.323067 −0.161533 0.986867i \(-0.551644\pi\)
−0.161533 + 0.986867i \(0.551644\pi\)
\(692\) 0 0
\(693\) 9.12311i 0.346558i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 14.2462i 0.539614i
\(698\) 0 0
\(699\) 8.00000 0.302588
\(700\) 0 0
\(701\) 0.0691303 0.00261102 0.00130551 0.999999i \(-0.499584\pi\)
0.00130551 + 0.999999i \(0.499584\pi\)
\(702\) 0 0
\(703\) 6.73863i 0.254152i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0.246211i 0.00925973i
\(708\) 0 0
\(709\) 18.1771 0.682655 0.341327 0.939945i \(-0.389124\pi\)
0.341327 + 0.939945i \(0.389124\pi\)
\(710\) 0 0
\(711\) 23.3693 0.876418
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 2.06913i 0.0772731i
\(718\) 0 0
\(719\) 49.6155 1.85035 0.925173 0.379544i \(-0.123919\pi\)
0.925173 + 0.379544i \(0.123919\pi\)
\(720\) 0 0
\(721\) −1.43845 −0.0535706
\(722\) 0 0
\(723\) − 31.3693i − 1.16664i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 19.5076i − 0.723496i −0.932276 0.361748i \(-0.882180\pi\)
0.932276 0.361748i \(-0.117820\pi\)
\(728\) 0 0
\(729\) 35.9848 1.33277
\(730\) 0 0
\(731\) 41.6155 1.53921
\(732\) 0 0
\(733\) 5.68466i 0.209968i 0.994474 + 0.104984i \(0.0334791\pi\)
−0.994474 + 0.104984i \(0.966521\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 16.0000i − 0.589368i
\(738\) 0 0
\(739\) 6.06913 0.223257 0.111628 0.993750i \(-0.464393\pi\)
0.111628 + 0.993750i \(0.464393\pi\)
\(740\) 0 0
\(741\) −13.1231 −0.482089
\(742\) 0 0
\(743\) 32.9848i 1.21010i 0.796189 + 0.605048i \(0.206846\pi\)
−0.796189 + 0.605048i \(0.793154\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 14.2462i − 0.521242i
\(748\) 0 0
\(749\) −11.3693 −0.415426
\(750\) 0 0
\(751\) −45.9309 −1.67604 −0.838021 0.545639i \(-0.816287\pi\)
−0.838021 + 0.545639i \(0.816287\pi\)
\(752\) 0 0
\(753\) − 43.8617i − 1.59841i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 14.6307i 0.531761i 0.964006 + 0.265881i \(0.0856627\pi\)
−0.964006 + 0.265881i \(0.914337\pi\)
\(758\) 0 0
\(759\) 33.6155 1.22017
\(760\) 0 0
\(761\) 31.7538 1.15107 0.575537 0.817776i \(-0.304793\pi\)
0.575537 + 0.817776i \(0.304793\pi\)
\(762\) 0 0
\(763\) − 17.6847i − 0.640228i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 18.2462i 0.658833i
\(768\) 0 0
\(769\) 9.50758 0.342852 0.171426 0.985197i \(-0.445163\pi\)
0.171426 + 0.985197i \(0.445163\pi\)
\(770\) 0 0
\(771\) 57.6155 2.07497
\(772\) 0 0
\(773\) − 8.06913i − 0.290226i −0.989415 0.145113i \(-0.953645\pi\)
0.989415 0.145113i \(-0.0463546\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 15.3693i 0.551371i
\(778\) 0 0
\(779\) −3.50758 −0.125672
\(780\) 0 0
\(781\) 20.4924 0.733277
\(782\) 0 0
\(783\) 8.17708i 0.292225i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 3.82292i 0.136272i 0.997676 + 0.0681362i \(0.0217052\pi\)
−0.997676 + 0.0681362i \(0.978295\pi\)
\(788\) 0 0
\(789\) −54.1080 −1.92629
\(790\) 0 0
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) 42.7386i 1.51769i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 13.0540i − 0.462396i −0.972907 0.231198i \(-0.925736\pi\)
0.972907 0.231198i \(-0.0742644\pi\)
\(798\) 0 0
\(799\) −16.8078 −0.594616
\(800\) 0 0
\(801\) 25.3693 0.896381
\(802\) 0 0
\(803\) 10.8769i 0.383837i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 73.6155i 2.59139i
\(808\) 0 0
\(809\) 53.5464 1.88259 0.941296 0.337584i \(-0.109610\pi\)
0.941296 + 0.337584i \(0.109610\pi\)
\(810\) 0 0
\(811\) 21.6155 0.759024 0.379512 0.925187i \(-0.376092\pi\)
0.379512 + 0.925187i \(0.376092\pi\)
\(812\) 0 0
\(813\) − 40.9848i − 1.43740i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 10.2462i 0.358470i
\(818\) 0 0
\(819\) −16.2462 −0.567689
\(820\) 0 0
\(821\) 40.4233 1.41078 0.705391 0.708818i \(-0.250771\pi\)
0.705391 + 0.708818i \(0.250771\pi\)
\(822\) 0 0
\(823\) − 3.50758i − 0.122266i −0.998130 0.0611332i \(-0.980529\pi\)
0.998130 0.0611332i \(-0.0194715\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 19.3693i − 0.673537i −0.941587 0.336769i \(-0.890666\pi\)
0.941587 0.336769i \(-0.109334\pi\)
\(828\) 0 0
\(829\) −43.1231 −1.49773 −0.748864 0.662724i \(-0.769400\pi\)
−0.748864 + 0.662724i \(0.769400\pi\)
\(830\) 0 0
\(831\) 41.6155 1.44363
\(832\) 0 0
\(833\) 4.56155i 0.158048i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −37.1231 −1.28163 −0.640816 0.767695i \(-0.721404\pi\)
−0.640816 + 0.767695i \(0.721404\pi\)
\(840\) 0 0
\(841\) 3.31534 0.114322
\(842\) 0 0
\(843\) − 42.4233i − 1.46114i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 4.43845i − 0.152507i
\(848\) 0 0
\(849\) −60.6695 −2.08217
\(850\) 0 0
\(851\) 30.7386 1.05371
\(852\) 0 0
\(853\) 56.7386i 1.94269i 0.237666 + 0.971347i \(0.423618\pi\)
−0.237666 + 0.971347i \(0.576382\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 32.2462i − 1.10151i −0.834667 0.550755i \(-0.814340\pi\)
0.834667 0.550755i \(-0.185660\pi\)
\(858\) 0 0
\(859\) 16.4924 0.562714 0.281357 0.959603i \(-0.409215\pi\)
0.281357 + 0.959603i \(0.409215\pi\)
\(860\) 0 0
\(861\) −8.00000 −0.272639
\(862\) 0 0
\(863\) − 42.2462i − 1.43808i −0.694970 0.719039i \(-0.744582\pi\)
0.694970 0.719039i \(-0.255418\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 9.75379i 0.331256i
\(868\) 0 0
\(869\) 16.8078 0.570164
\(870\) 0 0
\(871\) 28.4924 0.965429
\(872\) 0 0
\(873\) 52.7386i 1.78493i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 23.7538i − 0.802108i −0.916054 0.401054i \(-0.868644\pi\)
0.916054 0.401054i \(-0.131356\pi\)
\(878\) 0 0
\(879\) −24.8078 −0.836745
\(880\) 0 0
\(881\) 45.8617 1.54512 0.772561 0.634941i \(-0.218976\pi\)
0.772561 + 0.634941i \(0.218976\pi\)
\(882\) 0 0
\(883\) 24.4924i 0.824236i 0.911131 + 0.412118i \(0.135211\pi\)
−0.911131 + 0.412118i \(0.864789\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 12.4924i 0.419454i 0.977760 + 0.209727i \(0.0672576\pi\)
−0.977760 + 0.209727i \(0.932742\pi\)
\(888\) 0 0
\(889\) 10.2462 0.343647
\(890\) 0 0
\(891\) 17.9309 0.600707
\(892\) 0 0
\(893\) − 4.13826i − 0.138482i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 59.8617i 1.99873i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −14.2462 −0.474610
\(902\) 0 0
\(903\) 23.3693i 0.777682i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 50.1080i − 1.66381i −0.554920 0.831904i \(-0.687251\pi\)
0.554920 0.831904i \(-0.312749\pi\)
\(908\) 0 0
\(909\) −0.876894 −0.0290848
\(910\) 0 0
\(911\) −4.49242 −0.148841 −0.0744203 0.997227i \(-0.523711\pi\)
−0.0744203 + 0.997227i \(0.523711\pi\)
\(912\) 0 0
\(913\) − 10.2462i − 0.339100i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 9.12311i 0.301271i
\(918\) 0 0
\(919\) −13.3002 −0.438733 −0.219366 0.975643i \(-0.570399\pi\)
−0.219366 + 0.975643i \(0.570399\pi\)
\(920\) 0 0
\(921\) 81.1619 2.67438
\(922\) 0 0
\(923\) 36.4924i 1.20116i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 5.12311i − 0.168265i
\(928\) 0 0
\(929\) 52.1080 1.70961 0.854803 0.518952i \(-0.173678\pi\)
0.854803 + 0.518952i \(0.173678\pi\)
\(930\) 0 0
\(931\) −1.12311 −0.0368083
\(932\) 0 0
\(933\) − 24.6307i − 0.806372i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 22.6695i 0.740580i 0.928916 + 0.370290i \(0.120742\pi\)
−0.928916 + 0.370290i \(0.879258\pi\)
\(938\) 0 0
\(939\) −80.1771 −2.61648
\(940\) 0 0
\(941\) −13.8617 −0.451880 −0.225940 0.974141i \(-0.572545\pi\)
−0.225940 + 0.974141i \(0.572545\pi\)
\(942\) 0 0
\(943\) 16.0000i 0.521032i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 4.00000i − 0.129983i −0.997886 0.0649913i \(-0.979298\pi\)
0.997886 0.0649913i \(-0.0207020\pi\)
\(948\) 0 0
\(949\) −19.3693 −0.628755
\(950\) 0 0
\(951\) −57.6155 −1.86831
\(952\) 0 0
\(953\) 24.8769i 0.805842i 0.915235 + 0.402921i \(0.132005\pi\)
−0.915235 + 0.402921i \(0.867995\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 37.3002i 1.20574i
\(958\) 0 0
\(959\) 8.87689 0.286650
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) − 40.4924i − 1.30485i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 26.8769i 0.864303i 0.901801 + 0.432151i \(0.142245\pi\)
−0.901801 + 0.432151i \(0.857755\pi\)
\(968\) 0 0
\(969\) −13.1231 −0.421575
\(970\) 0 0
\(971\) 49.4773 1.58780 0.793901 0.608048i \(-0.208047\pi\)
0.793901 + 0.608048i \(0.208047\pi\)
\(972\) 0 0
\(973\) − 6.87689i − 0.220463i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 49.2311i − 1.57504i −0.616288 0.787521i \(-0.711364\pi\)
0.616288 0.787521i \(-0.288636\pi\)
\(978\) 0 0
\(979\) 18.2462 0.583151
\(980\) 0 0
\(981\) 62.9848 2.01095
\(982\) 0 0
\(983\) − 10.4233i − 0.332451i −0.986088 0.166226i \(-0.946842\pi\)
0.986088 0.166226i \(-0.0531580\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 9.43845i − 0.300429i
\(988\) 0 0
\(989\) 46.7386 1.48620
\(990\) 0 0
\(991\) 20.4924 0.650963 0.325482 0.945548i \(-0.394474\pi\)
0.325482 + 0.945548i \(0.394474\pi\)
\(992\) 0 0
\(993\) 30.7386i 0.975461i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 9.68466i 0.306716i 0.988171 + 0.153358i \(0.0490088\pi\)
−0.988171 + 0.153358i \(0.950991\pi\)
\(998\) 0 0
\(999\) −8.63068 −0.273063
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.t.449.1 4
4.3 odd 2 175.2.b.b.99.3 4
5.2 odd 4 2800.2.a.bi.1.1 2
5.3 odd 4 560.2.a.i.1.2 2
5.4 even 2 inner 2800.2.g.t.449.4 4
12.11 even 2 1575.2.d.e.1324.2 4
15.8 even 4 5040.2.a.bt.1.2 2
20.3 even 4 35.2.a.b.1.2 2
20.7 even 4 175.2.a.f.1.1 2
20.19 odd 2 175.2.b.b.99.2 4
28.27 even 2 1225.2.b.f.99.3 4
35.13 even 4 3920.2.a.bs.1.1 2
40.3 even 4 2240.2.a.bh.1.2 2
40.13 odd 4 2240.2.a.bd.1.1 2
60.23 odd 4 315.2.a.e.1.1 2
60.47 odd 4 1575.2.a.p.1.2 2
60.59 even 2 1575.2.d.e.1324.3 4
140.3 odd 12 245.2.e.h.226.1 4
140.23 even 12 245.2.e.i.116.1 4
140.27 odd 4 1225.2.a.s.1.1 2
140.83 odd 4 245.2.a.d.1.2 2
140.103 odd 12 245.2.e.h.116.1 4
140.123 even 12 245.2.e.i.226.1 4
140.139 even 2 1225.2.b.f.99.2 4
220.43 odd 4 4235.2.a.m.1.1 2
260.103 even 4 5915.2.a.l.1.1 2
420.83 even 4 2205.2.a.x.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.2 2 20.3 even 4
175.2.a.f.1.1 2 20.7 even 4
175.2.b.b.99.2 4 20.19 odd 2
175.2.b.b.99.3 4 4.3 odd 2
245.2.a.d.1.2 2 140.83 odd 4
245.2.e.h.116.1 4 140.103 odd 12
245.2.e.h.226.1 4 140.3 odd 12
245.2.e.i.116.1 4 140.23 even 12
245.2.e.i.226.1 4 140.123 even 12
315.2.a.e.1.1 2 60.23 odd 4
560.2.a.i.1.2 2 5.3 odd 4
1225.2.a.s.1.1 2 140.27 odd 4
1225.2.b.f.99.2 4 140.139 even 2
1225.2.b.f.99.3 4 28.27 even 2
1575.2.a.p.1.2 2 60.47 odd 4
1575.2.d.e.1324.2 4 12.11 even 2
1575.2.d.e.1324.3 4 60.59 even 2
2205.2.a.x.1.1 2 420.83 even 4
2240.2.a.bd.1.1 2 40.13 odd 4
2240.2.a.bh.1.2 2 40.3 even 4
2800.2.a.bi.1.1 2 5.2 odd 4
2800.2.g.t.449.1 4 1.1 even 1 trivial
2800.2.g.t.449.4 4 5.4 even 2 inner
3920.2.a.bs.1.1 2 35.13 even 4
4235.2.a.m.1.1 2 220.43 odd 4
5040.2.a.bt.1.2 2 15.8 even 4
5915.2.a.l.1.1 2 260.103 even 4