Properties

Label 2800.2.a.bo.1.1
Level $2800$
Weight $2$
Character 2800.1
Self dual yes
Analytic conductor $22.358$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.3581125660\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1400)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 2800.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.56155 q^{3} +1.00000 q^{7} -0.561553 q^{9} +O(q^{10})\) \(q-1.56155 q^{3} +1.00000 q^{7} -0.561553 q^{9} +6.12311 q^{11} -2.00000 q^{13} -1.56155 q^{17} -3.56155 q^{19} -1.56155 q^{21} +1.43845 q^{23} +5.56155 q^{27} +3.43845 q^{29} +9.12311 q^{31} -9.56155 q^{33} -8.80776 q^{37} +3.12311 q^{39} -2.43845 q^{41} +6.56155 q^{43} -8.24621 q^{47} +1.00000 q^{49} +2.43845 q^{51} +1.12311 q^{53} +5.56155 q^{57} -11.3693 q^{59} +11.1231 q^{61} -0.561553 q^{63} +7.87689 q^{67} -2.24621 q^{69} -1.68466 q^{71} -6.43845 q^{73} +6.12311 q^{77} +5.68466 q^{79} -7.00000 q^{81} -1.31534 q^{83} -5.36932 q^{87} +9.80776 q^{89} -2.00000 q^{91} -14.2462 q^{93} -6.00000 q^{97} -3.43845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + 2q^{7} + 3q^{9} + O(q^{10}) \) \( 2q + q^{3} + 2q^{7} + 3q^{9} + 4q^{11} - 4q^{13} + q^{17} - 3q^{19} + q^{21} + 7q^{23} + 7q^{27} + 11q^{29} + 10q^{31} - 15q^{33} + 3q^{37} - 2q^{39} - 9q^{41} + 9q^{43} + 2q^{49} + 9q^{51} - 6q^{53} + 7q^{57} + 2q^{59} + 14q^{61} + 3q^{63} + 24q^{67} + 12q^{69} + 9q^{71} - 17q^{73} + 4q^{77} - q^{79} - 14q^{81} - 15q^{83} + 14q^{87} - q^{89} - 4q^{91} - 12q^{93} - 12q^{97} - 11q^{99} + O(q^{100}) \)

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
\(3\) −1.56155 −0.901563 −0.450781 0.892634i \(-0.648855\pi\)
−0.450781 + 0.892634i \(0.648855\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) 6.12311 1.84619 0.923093 0.384577i \(-0.125653\pi\)
0.923093 + 0.384577i \(0.125653\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.56155 −0.378732 −0.189366 0.981907i \(-0.560643\pi\)
−0.189366 + 0.981907i \(0.560643\pi\)
\(18\) 0 0
\(19\) −3.56155 −0.817076 −0.408538 0.912741i \(-0.633961\pi\)
−0.408538 + 0.912741i \(0.633961\pi\)
\(20\) 0 0
\(21\) −1.56155 −0.340759
\(22\) 0 0
\(23\) 1.43845 0.299937 0.149968 0.988691i \(-0.452083\pi\)
0.149968 + 0.988691i \(0.452083\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.56155 1.07032
\(28\) 0 0
\(29\) 3.43845 0.638504 0.319252 0.947670i \(-0.396568\pi\)
0.319252 + 0.947670i \(0.396568\pi\)
\(30\) 0 0
\(31\) 9.12311 1.63856 0.819279 0.573395i \(-0.194374\pi\)
0.819279 + 0.573395i \(0.194374\pi\)
\(32\) 0 0
\(33\) −9.56155 −1.66445
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.80776 −1.44799 −0.723994 0.689807i \(-0.757696\pi\)
−0.723994 + 0.689807i \(0.757696\pi\)
\(38\) 0 0
\(39\) 3.12311 0.500097
\(40\) 0 0
\(41\) −2.43845 −0.380821 −0.190411 0.981705i \(-0.560982\pi\)
−0.190411 + 0.981705i \(0.560982\pi\)
\(42\) 0 0
\(43\) 6.56155 1.00063 0.500314 0.865844i \(-0.333218\pi\)
0.500314 + 0.865844i \(0.333218\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.24621 −1.20283 −0.601417 0.798935i \(-0.705397\pi\)
−0.601417 + 0.798935i \(0.705397\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.43845 0.341451
\(52\) 0 0
\(53\) 1.12311 0.154270 0.0771352 0.997021i \(-0.475423\pi\)
0.0771352 + 0.997021i \(0.475423\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.56155 0.736646
\(58\) 0 0
\(59\) −11.3693 −1.48016 −0.740079 0.672519i \(-0.765212\pi\)
−0.740079 + 0.672519i \(0.765212\pi\)
\(60\) 0 0
\(61\) 11.1231 1.42417 0.712084 0.702094i \(-0.247752\pi\)
0.712084 + 0.702094i \(0.247752\pi\)
\(62\) 0 0
\(63\) −0.561553 −0.0707490
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.87689 0.962316 0.481158 0.876634i \(-0.340216\pi\)
0.481158 + 0.876634i \(0.340216\pi\)
\(68\) 0 0
\(69\) −2.24621 −0.270412
\(70\) 0 0
\(71\) −1.68466 −0.199932 −0.0999661 0.994991i \(-0.531873\pi\)
−0.0999661 + 0.994991i \(0.531873\pi\)
\(72\) 0 0
\(73\) −6.43845 −0.753563 −0.376782 0.926302i \(-0.622969\pi\)
−0.376782 + 0.926302i \(0.622969\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.12311 0.697793
\(78\) 0 0
\(79\) 5.68466 0.639574 0.319787 0.947489i \(-0.396389\pi\)
0.319787 + 0.947489i \(0.396389\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) −1.31534 −0.144377 −0.0721887 0.997391i \(-0.522998\pi\)
−0.0721887 + 0.997391i \(0.522998\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −5.36932 −0.575651
\(88\) 0 0
\(89\) 9.80776 1.03962 0.519810 0.854282i \(-0.326003\pi\)
0.519810 + 0.854282i \(0.326003\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) −14.2462 −1.47726
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) −3.43845 −0.345577
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −1.12311 −0.110663 −0.0553314 0.998468i \(-0.517622\pi\)
−0.0553314 + 0.998468i \(0.517622\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.80776 0.754805 0.377403 0.926049i \(-0.376817\pi\)
0.377403 + 0.926049i \(0.376817\pi\)
\(108\) 0 0
\(109\) 19.9309 1.90903 0.954516 0.298161i \(-0.0963733\pi\)
0.954516 + 0.298161i \(0.0963733\pi\)
\(110\) 0 0
\(111\) 13.7538 1.30545
\(112\) 0 0
\(113\) −17.2462 −1.62239 −0.811194 0.584778i \(-0.801182\pi\)
−0.811194 + 0.584778i \(0.801182\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.12311 0.103831
\(118\) 0 0
\(119\) −1.56155 −0.143147
\(120\) 0 0
\(121\) 26.4924 2.40840
\(122\) 0 0
\(123\) 3.80776 0.343335
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 10.8078 0.959034 0.479517 0.877533i \(-0.340812\pi\)
0.479517 + 0.877533i \(0.340812\pi\)
\(128\) 0 0
\(129\) −10.2462 −0.902129
\(130\) 0 0
\(131\) 18.2462 1.59418 0.797089 0.603861i \(-0.206372\pi\)
0.797089 + 0.603861i \(0.206372\pi\)
\(132\) 0 0
\(133\) −3.56155 −0.308826
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 14.6847 1.25460 0.627298 0.778780i \(-0.284161\pi\)
0.627298 + 0.778780i \(0.284161\pi\)
\(138\) 0 0
\(139\) 7.31534 0.620479 0.310240 0.950658i \(-0.399591\pi\)
0.310240 + 0.950658i \(0.399591\pi\)
\(140\) 0 0
\(141\) 12.8769 1.08443
\(142\) 0 0
\(143\) −12.2462 −1.02408
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −1.56155 −0.128795
\(148\) 0 0
\(149\) 13.4384 1.10092 0.550460 0.834861i \(-0.314452\pi\)
0.550460 + 0.834861i \(0.314452\pi\)
\(150\) 0 0
\(151\) 1.43845 0.117059 0.0585296 0.998286i \(-0.481359\pi\)
0.0585296 + 0.998286i \(0.481359\pi\)
\(152\) 0 0
\(153\) 0.876894 0.0708927
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 8.49242 0.677769 0.338885 0.940828i \(-0.389950\pi\)
0.338885 + 0.940828i \(0.389950\pi\)
\(158\) 0 0
\(159\) −1.75379 −0.139084
\(160\) 0 0
\(161\) 1.43845 0.113366
\(162\) 0 0
\(163\) 14.9309 1.16948 0.584738 0.811222i \(-0.301197\pi\)
0.584738 + 0.811222i \(0.301197\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 21.6155 1.67266 0.836330 0.548227i \(-0.184697\pi\)
0.836330 + 0.548227i \(0.184697\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 0 0
\(173\) 18.4924 1.40595 0.702976 0.711213i \(-0.251854\pi\)
0.702976 + 0.711213i \(0.251854\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 17.7538 1.33446
\(178\) 0 0
\(179\) 20.6847 1.54604 0.773022 0.634379i \(-0.218744\pi\)
0.773022 + 0.634379i \(0.218744\pi\)
\(180\) 0 0
\(181\) −11.1231 −0.826774 −0.413387 0.910555i \(-0.635654\pi\)
−0.413387 + 0.910555i \(0.635654\pi\)
\(182\) 0 0
\(183\) −17.3693 −1.28398
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −9.56155 −0.699210
\(188\) 0 0
\(189\) 5.56155 0.404543
\(190\) 0 0
\(191\) −25.3693 −1.83566 −0.917830 0.396974i \(-0.870060\pi\)
−0.917830 + 0.396974i \(0.870060\pi\)
\(192\) 0 0
\(193\) 20.6155 1.48394 0.741969 0.670434i \(-0.233892\pi\)
0.741969 + 0.670434i \(0.233892\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −0.561553 −0.0400090 −0.0200045 0.999800i \(-0.506368\pi\)
−0.0200045 + 0.999800i \(0.506368\pi\)
\(198\) 0 0
\(199\) 4.87689 0.345714 0.172857 0.984947i \(-0.444700\pi\)
0.172857 + 0.984947i \(0.444700\pi\)
\(200\) 0 0
\(201\) −12.3002 −0.867588
\(202\) 0 0
\(203\) 3.43845 0.241332
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −0.807764 −0.0561435
\(208\) 0 0
\(209\) −21.8078 −1.50847
\(210\) 0 0
\(211\) −6.43845 −0.443241 −0.221620 0.975133i \(-0.571135\pi\)
−0.221620 + 0.975133i \(0.571135\pi\)
\(212\) 0 0
\(213\) 2.63068 0.180251
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 9.12311 0.619317
\(218\) 0 0
\(219\) 10.0540 0.679385
\(220\) 0 0
\(221\) 3.12311 0.210083
\(222\) 0 0
\(223\) −25.3693 −1.69886 −0.849428 0.527705i \(-0.823053\pi\)
−0.849428 + 0.527705i \(0.823053\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 11.3693 0.754608 0.377304 0.926089i \(-0.376851\pi\)
0.377304 + 0.926089i \(0.376851\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) −9.56155 −0.629104
\(232\) 0 0
\(233\) 0.561553 0.0367885 0.0183943 0.999831i \(-0.494145\pi\)
0.0183943 + 0.999831i \(0.494145\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −8.87689 −0.576616
\(238\) 0 0
\(239\) 4.00000 0.258738 0.129369 0.991596i \(-0.458705\pi\)
0.129369 + 0.991596i \(0.458705\pi\)
\(240\) 0 0
\(241\) −12.0540 −0.776465 −0.388232 0.921562i \(-0.626914\pi\)
−0.388232 + 0.921562i \(0.626914\pi\)
\(242\) 0 0
\(243\) −5.75379 −0.369106
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 7.12311 0.453232
\(248\) 0 0
\(249\) 2.05398 0.130165
\(250\) 0 0
\(251\) 24.6847 1.55808 0.779041 0.626973i \(-0.215706\pi\)
0.779041 + 0.626973i \(0.215706\pi\)
\(252\) 0 0
\(253\) 8.80776 0.553739
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −12.2462 −0.763898 −0.381949 0.924183i \(-0.624747\pi\)
−0.381949 + 0.924183i \(0.624747\pi\)
\(258\) 0 0
\(259\) −8.80776 −0.547288
\(260\) 0 0
\(261\) −1.93087 −0.119518
\(262\) 0 0
\(263\) 1.68466 0.103880 0.0519402 0.998650i \(-0.483459\pi\)
0.0519402 + 0.998650i \(0.483459\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −15.3153 −0.937284
\(268\) 0 0
\(269\) 28.2462 1.72220 0.861101 0.508434i \(-0.169775\pi\)
0.861101 + 0.508434i \(0.169775\pi\)
\(270\) 0 0
\(271\) 0.246211 0.0149563 0.00747813 0.999972i \(-0.497620\pi\)
0.00747813 + 0.999972i \(0.497620\pi\)
\(272\) 0 0
\(273\) 3.12311 0.189019
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −22.4924 −1.35144 −0.675719 0.737159i \(-0.736167\pi\)
−0.675719 + 0.737159i \(0.736167\pi\)
\(278\) 0 0
\(279\) −5.12311 −0.306712
\(280\) 0 0
\(281\) 7.43845 0.443741 0.221870 0.975076i \(-0.428784\pi\)
0.221870 + 0.975076i \(0.428784\pi\)
\(282\) 0 0
\(283\) −1.31534 −0.0781889 −0.0390945 0.999236i \(-0.512447\pi\)
−0.0390945 + 0.999236i \(0.512447\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.43845 −0.143937
\(288\) 0 0
\(289\) −14.5616 −0.856562
\(290\) 0 0
\(291\) 9.36932 0.549239
\(292\) 0 0
\(293\) 30.0000 1.75262 0.876309 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 34.0540 1.97601
\(298\) 0 0
\(299\) −2.87689 −0.166375
\(300\) 0 0
\(301\) 6.56155 0.378202
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 16.6847 0.952244 0.476122 0.879379i \(-0.342042\pi\)
0.476122 + 0.879379i \(0.342042\pi\)
\(308\) 0 0
\(309\) 1.75379 0.0997696
\(310\) 0 0
\(311\) −11.1231 −0.630733 −0.315367 0.948970i \(-0.602128\pi\)
−0.315367 + 0.948970i \(0.602128\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.31534 0.467036 0.233518 0.972352i \(-0.424976\pi\)
0.233518 + 0.972352i \(0.424976\pi\)
\(318\) 0 0
\(319\) 21.0540 1.17880
\(320\) 0 0
\(321\) −12.1922 −0.680504
\(322\) 0 0
\(323\) 5.56155 0.309453
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −31.1231 −1.72111
\(328\) 0 0
\(329\) −8.24621 −0.454628
\(330\) 0 0
\(331\) −27.0000 −1.48405 −0.742027 0.670370i \(-0.766135\pi\)
−0.742027 + 0.670370i \(0.766135\pi\)
\(332\) 0 0
\(333\) 4.94602 0.271040
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −6.68466 −0.364137 −0.182068 0.983286i \(-0.558279\pi\)
−0.182068 + 0.983286i \(0.558279\pi\)
\(338\) 0 0
\(339\) 26.9309 1.46268
\(340\) 0 0
\(341\) 55.8617 3.02508
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −18.6155 −0.999334 −0.499667 0.866218i \(-0.666544\pi\)
−0.499667 + 0.866218i \(0.666544\pi\)
\(348\) 0 0
\(349\) −21.1231 −1.13069 −0.565347 0.824853i \(-0.691258\pi\)
−0.565347 + 0.824853i \(0.691258\pi\)
\(350\) 0 0
\(351\) −11.1231 −0.593707
\(352\) 0 0
\(353\) 28.2462 1.50339 0.751697 0.659509i \(-0.229236\pi\)
0.751697 + 0.659509i \(0.229236\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 2.43845 0.129056
\(358\) 0 0
\(359\) 2.31534 0.122199 0.0610995 0.998132i \(-0.480539\pi\)
0.0610995 + 0.998132i \(0.480539\pi\)
\(360\) 0 0
\(361\) −6.31534 −0.332386
\(362\) 0 0
\(363\) −41.3693 −2.17133
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 34.2462 1.78764 0.893819 0.448428i \(-0.148016\pi\)
0.893819 + 0.448428i \(0.148016\pi\)
\(368\) 0 0
\(369\) 1.36932 0.0712838
\(370\) 0 0
\(371\) 1.12311 0.0583087
\(372\) 0 0
\(373\) 23.0540 1.19369 0.596845 0.802357i \(-0.296421\pi\)
0.596845 + 0.802357i \(0.296421\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −6.87689 −0.354178
\(378\) 0 0
\(379\) −32.8617 −1.68799 −0.843997 0.536348i \(-0.819804\pi\)
−0.843997 + 0.536348i \(0.819804\pi\)
\(380\) 0 0
\(381\) −16.8769 −0.864629
\(382\) 0 0
\(383\) −3.75379 −0.191810 −0.0959048 0.995391i \(-0.530574\pi\)
−0.0959048 + 0.995391i \(0.530574\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3.68466 −0.187302
\(388\) 0 0
\(389\) 6.56155 0.332684 0.166342 0.986068i \(-0.446804\pi\)
0.166342 + 0.986068i \(0.446804\pi\)
\(390\) 0 0
\(391\) −2.24621 −0.113596
\(392\) 0 0
\(393\) −28.4924 −1.43725
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −5.50758 −0.276417 −0.138209 0.990403i \(-0.544134\pi\)
−0.138209 + 0.990403i \(0.544134\pi\)
\(398\) 0 0
\(399\) 5.56155 0.278426
\(400\) 0 0
\(401\) 1.49242 0.0745280 0.0372640 0.999305i \(-0.488136\pi\)
0.0372640 + 0.999305i \(0.488136\pi\)
\(402\) 0 0
\(403\) −18.2462 −0.908909
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −53.9309 −2.67325
\(408\) 0 0
\(409\) −25.8078 −1.27611 −0.638056 0.769990i \(-0.720261\pi\)
−0.638056 + 0.769990i \(0.720261\pi\)
\(410\) 0 0
\(411\) −22.9309 −1.13110
\(412\) 0 0
\(413\) −11.3693 −0.559448
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −11.4233 −0.559401
\(418\) 0 0
\(419\) −1.80776 −0.0883151 −0.0441575 0.999025i \(-0.514060\pi\)
−0.0441575 + 0.999025i \(0.514060\pi\)
\(420\) 0 0
\(421\) 33.9309 1.65369 0.826845 0.562430i \(-0.190134\pi\)
0.826845 + 0.562430i \(0.190134\pi\)
\(422\) 0 0
\(423\) 4.63068 0.225152
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 11.1231 0.538285
\(428\) 0 0
\(429\) 19.1231 0.923272
\(430\) 0 0
\(431\) 30.2462 1.45691 0.728454 0.685094i \(-0.240239\pi\)
0.728454 + 0.685094i \(0.240239\pi\)
\(432\) 0 0
\(433\) −34.5464 −1.66019 −0.830097 0.557619i \(-0.811715\pi\)
−0.830097 + 0.557619i \(0.811715\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −5.12311 −0.245071
\(438\) 0 0
\(439\) −1.12311 −0.0536029 −0.0268015 0.999641i \(-0.508532\pi\)
−0.0268015 + 0.999641i \(0.508532\pi\)
\(440\) 0 0
\(441\) −0.561553 −0.0267406
\(442\) 0 0
\(443\) 18.4384 0.876037 0.438019 0.898966i \(-0.355680\pi\)
0.438019 + 0.898966i \(0.355680\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −20.9848 −0.992549
\(448\) 0 0
\(449\) −14.3693 −0.678130 −0.339065 0.940763i \(-0.610111\pi\)
−0.339065 + 0.940763i \(0.610111\pi\)
\(450\) 0 0
\(451\) −14.9309 −0.703067
\(452\) 0 0
\(453\) −2.24621 −0.105536
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 37.1080 1.73584 0.867918 0.496707i \(-0.165458\pi\)
0.867918 + 0.496707i \(0.165458\pi\)
\(458\) 0 0
\(459\) −8.68466 −0.405365
\(460\) 0 0
\(461\) −26.4924 −1.23388 −0.616938 0.787012i \(-0.711627\pi\)
−0.616938 + 0.787012i \(0.711627\pi\)
\(462\) 0 0
\(463\) 12.4924 0.580572 0.290286 0.956940i \(-0.406250\pi\)
0.290286 + 0.956940i \(0.406250\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −33.1231 −1.53275 −0.766377 0.642391i \(-0.777943\pi\)
−0.766377 + 0.642391i \(0.777943\pi\)
\(468\) 0 0
\(469\) 7.87689 0.363721
\(470\) 0 0
\(471\) −13.2614 −0.611052
\(472\) 0 0
\(473\) 40.1771 1.84734
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −0.630683 −0.0288770
\(478\) 0 0
\(479\) −8.73863 −0.399278 −0.199639 0.979869i \(-0.563977\pi\)
−0.199639 + 0.979869i \(0.563977\pi\)
\(480\) 0 0
\(481\) 17.6155 0.803199
\(482\) 0 0
\(483\) −2.24621 −0.102206
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 31.4384 1.42461 0.712306 0.701869i \(-0.247651\pi\)
0.712306 + 0.701869i \(0.247651\pi\)
\(488\) 0 0
\(489\) −23.3153 −1.05436
\(490\) 0 0
\(491\) −8.31534 −0.375266 −0.187633 0.982239i \(-0.560082\pi\)
−0.187633 + 0.982239i \(0.560082\pi\)
\(492\) 0 0
\(493\) −5.36932 −0.241822
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.68466 −0.0755673
\(498\) 0 0
\(499\) −16.4924 −0.738302 −0.369151 0.929369i \(-0.620352\pi\)
−0.369151 + 0.929369i \(0.620352\pi\)
\(500\) 0 0
\(501\) −33.7538 −1.50801
\(502\) 0 0
\(503\) 36.2462 1.61614 0.808069 0.589087i \(-0.200513\pi\)
0.808069 + 0.589087i \(0.200513\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 14.0540 0.624159
\(508\) 0 0
\(509\) 26.0000 1.15243 0.576215 0.817298i \(-0.304529\pi\)
0.576215 + 0.817298i \(0.304529\pi\)
\(510\) 0 0
\(511\) −6.43845 −0.284820
\(512\) 0 0
\(513\) −19.8078 −0.874534
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −50.4924 −2.22065
\(518\) 0 0
\(519\) −28.8769 −1.26755
\(520\) 0 0
\(521\) −31.5616 −1.38274 −0.691368 0.722502i \(-0.742992\pi\)
−0.691368 + 0.722502i \(0.742992\pi\)
\(522\) 0 0
\(523\) 40.6847 1.77902 0.889508 0.456920i \(-0.151047\pi\)
0.889508 + 0.456920i \(0.151047\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −14.2462 −0.620575
\(528\) 0 0
\(529\) −20.9309 −0.910038
\(530\) 0 0
\(531\) 6.38447 0.277062
\(532\) 0 0
\(533\) 4.87689 0.211242
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −32.3002 −1.39386
\(538\) 0 0
\(539\) 6.12311 0.263741
\(540\) 0 0
\(541\) −20.4233 −0.878066 −0.439033 0.898471i \(-0.644679\pi\)
−0.439033 + 0.898471i \(0.644679\pi\)
\(542\) 0 0
\(543\) 17.3693 0.745389
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −15.7386 −0.672935 −0.336468 0.941695i \(-0.609232\pi\)
−0.336468 + 0.941695i \(0.609232\pi\)
\(548\) 0 0
\(549\) −6.24621 −0.266582
\(550\) 0 0
\(551\) −12.2462 −0.521706
\(552\) 0 0
\(553\) 5.68466 0.241736
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −5.43845 −0.230434 −0.115217 0.993340i \(-0.536756\pi\)
−0.115217 + 0.993340i \(0.536756\pi\)
\(558\) 0 0
\(559\) −13.1231 −0.555048
\(560\) 0 0
\(561\) 14.9309 0.630382
\(562\) 0 0
\(563\) 1.12311 0.0473333 0.0236666 0.999720i \(-0.492466\pi\)
0.0236666 + 0.999720i \(0.492466\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −7.00000 −0.293972
\(568\) 0 0
\(569\) 45.9848 1.92778 0.963892 0.266292i \(-0.0857984\pi\)
0.963892 + 0.266292i \(0.0857984\pi\)
\(570\) 0 0
\(571\) −39.6847 −1.66075 −0.830376 0.557204i \(-0.811874\pi\)
−0.830376 + 0.557204i \(0.811874\pi\)
\(572\) 0 0
\(573\) 39.6155 1.65496
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −24.5464 −1.02188 −0.510940 0.859616i \(-0.670703\pi\)
−0.510940 + 0.859616i \(0.670703\pi\)
\(578\) 0 0
\(579\) −32.1922 −1.33786
\(580\) 0 0
\(581\) −1.31534 −0.0545696
\(582\) 0 0
\(583\) 6.87689 0.284812
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −37.1771 −1.53446 −0.767231 0.641371i \(-0.778366\pi\)
−0.767231 + 0.641371i \(0.778366\pi\)
\(588\) 0 0
\(589\) −32.4924 −1.33883
\(590\) 0 0
\(591\) 0.876894 0.0360706
\(592\) 0 0
\(593\) −24.6847 −1.01368 −0.506839 0.862041i \(-0.669186\pi\)
−0.506839 + 0.862041i \(0.669186\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −7.61553 −0.311683
\(598\) 0 0
\(599\) 29.3002 1.19717 0.598587 0.801058i \(-0.295729\pi\)
0.598587 + 0.801058i \(0.295729\pi\)
\(600\) 0 0
\(601\) −42.0540 −1.71542 −0.857709 0.514136i \(-0.828113\pi\)
−0.857709 + 0.514136i \(0.828113\pi\)
\(602\) 0 0
\(603\) −4.42329 −0.180130
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −12.6307 −0.512664 −0.256332 0.966589i \(-0.582514\pi\)
−0.256332 + 0.966589i \(0.582514\pi\)
\(608\) 0 0
\(609\) −5.36932 −0.217576
\(610\) 0 0
\(611\) 16.4924 0.667212
\(612\) 0 0
\(613\) −5.68466 −0.229601 −0.114801 0.993389i \(-0.536623\pi\)
−0.114801 + 0.993389i \(0.536623\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.80776 0.113036 0.0565182 0.998402i \(-0.482000\pi\)
0.0565182 + 0.998402i \(0.482000\pi\)
\(618\) 0 0
\(619\) −4.63068 −0.186123 −0.0930614 0.995660i \(-0.529665\pi\)
−0.0930614 + 0.995660i \(0.529665\pi\)
\(620\) 0 0
\(621\) 8.00000 0.321029
\(622\) 0 0
\(623\) 9.80776 0.392940
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 34.0540 1.35998
\(628\) 0 0
\(629\) 13.7538 0.548399
\(630\) 0 0
\(631\) 3.05398 0.121577 0.0607884 0.998151i \(-0.480639\pi\)
0.0607884 + 0.998151i \(0.480639\pi\)
\(632\) 0 0
\(633\) 10.0540 0.399610
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) 0.946025 0.0374242
\(640\) 0 0
\(641\) −20.5616 −0.812133 −0.406066 0.913844i \(-0.633100\pi\)
−0.406066 + 0.913844i \(0.633100\pi\)
\(642\) 0 0
\(643\) −46.7386 −1.84319 −0.921596 0.388151i \(-0.873114\pi\)
−0.921596 + 0.388151i \(0.873114\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −0.630683 −0.0247947 −0.0123974 0.999923i \(-0.503946\pi\)
−0.0123974 + 0.999923i \(0.503946\pi\)
\(648\) 0 0
\(649\) −69.6155 −2.73265
\(650\) 0 0
\(651\) −14.2462 −0.558353
\(652\) 0 0
\(653\) 23.8617 0.933782 0.466891 0.884315i \(-0.345374\pi\)
0.466891 + 0.884315i \(0.345374\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 3.61553 0.141055
\(658\) 0 0
\(659\) −14.4384 −0.562442 −0.281221 0.959643i \(-0.590739\pi\)
−0.281221 + 0.959643i \(0.590739\pi\)
\(660\) 0 0
\(661\) 10.4924 0.408108 0.204054 0.978960i \(-0.434588\pi\)
0.204054 + 0.978960i \(0.434588\pi\)
\(662\) 0 0
\(663\) −4.87689 −0.189403
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.94602 0.191511
\(668\) 0 0
\(669\) 39.6155 1.53162
\(670\) 0 0
\(671\) 68.1080 2.62928
\(672\) 0 0
\(673\) −13.5076 −0.520679 −0.260339 0.965517i \(-0.583834\pi\)
−0.260339 + 0.965517i \(0.583834\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −39.3693 −1.51309 −0.756543 0.653944i \(-0.773113\pi\)
−0.756543 + 0.653944i \(0.773113\pi\)
\(678\) 0 0
\(679\) −6.00000 −0.230259
\(680\) 0 0
\(681\) −17.7538 −0.680327
\(682\) 0 0
\(683\) 25.8769 0.990152 0.495076 0.868850i \(-0.335140\pi\)
0.495076 + 0.868850i \(0.335140\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 21.8617 0.834077
\(688\) 0 0
\(689\) −2.24621 −0.0855738
\(690\) 0 0
\(691\) 21.5616 0.820240 0.410120 0.912032i \(-0.365487\pi\)
0.410120 + 0.912032i \(0.365487\pi\)
\(692\) 0 0
\(693\) −3.43845 −0.130616
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 3.80776 0.144229
\(698\) 0 0
\(699\) −0.876894 −0.0331672
\(700\) 0 0
\(701\) −30.1080 −1.13716 −0.568581 0.822627i \(-0.692507\pi\)
−0.568581 + 0.822627i \(0.692507\pi\)
\(702\) 0 0
\(703\) 31.3693 1.18312
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 12.6307 0.474355 0.237178 0.971466i \(-0.423778\pi\)
0.237178 + 0.971466i \(0.423778\pi\)
\(710\) 0 0
\(711\) −3.19224 −0.119718
\(712\) 0 0
\(713\) 13.1231 0.491464
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −6.24621 −0.233269
\(718\) 0 0
\(719\) −21.7538 −0.811279 −0.405640 0.914033i \(-0.632951\pi\)
−0.405640 + 0.914033i \(0.632951\pi\)
\(720\) 0 0
\(721\) −1.12311 −0.0418266
\(722\) 0 0
\(723\) 18.8229 0.700032
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 39.6155 1.46926 0.734629 0.678469i \(-0.237356\pi\)
0.734629 + 0.678469i \(0.237356\pi\)
\(728\) 0 0
\(729\) 29.9848 1.11055
\(730\) 0 0
\(731\) −10.2462 −0.378970
\(732\) 0 0
\(733\) −5.75379 −0.212521 −0.106261 0.994338i \(-0.533888\pi\)
−0.106261 + 0.994338i \(0.533888\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 48.2311 1.77661
\(738\) 0 0
\(739\) 28.1771 1.03651 0.518255 0.855226i \(-0.326582\pi\)
0.518255 + 0.855226i \(0.326582\pi\)
\(740\) 0 0
\(741\) −11.1231 −0.408617
\(742\) 0 0
\(743\) 30.7386 1.12769 0.563846 0.825880i \(-0.309321\pi\)
0.563846 + 0.825880i \(0.309321\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0.738634 0.0270252
\(748\) 0 0
\(749\) 7.80776 0.285289
\(750\) 0 0
\(751\) −2.63068 −0.0959950 −0.0479975 0.998847i \(-0.515284\pi\)
−0.0479975 + 0.998847i \(0.515284\pi\)
\(752\) 0 0
\(753\) −38.5464 −1.40471
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −25.6847 −0.933525 −0.466762 0.884383i \(-0.654580\pi\)
−0.466762 + 0.884383i \(0.654580\pi\)
\(758\) 0 0
\(759\) −13.7538 −0.499231
\(760\) 0 0
\(761\) −27.4233 −0.994094 −0.497047 0.867724i \(-0.665582\pi\)
−0.497047 + 0.867724i \(0.665582\pi\)
\(762\) 0 0
\(763\) 19.9309 0.721546
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 22.7386 0.821044
\(768\) 0 0
\(769\) 20.3002 0.732043 0.366022 0.930606i \(-0.380720\pi\)
0.366022 + 0.930606i \(0.380720\pi\)
\(770\) 0 0
\(771\) 19.1231 0.688702
\(772\) 0 0
\(773\) −29.6155 −1.06520 −0.532598 0.846368i \(-0.678784\pi\)
−0.532598 + 0.846368i \(0.678784\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 13.7538 0.493414
\(778\) 0 0
\(779\) 8.68466 0.311160
\(780\) 0 0
\(781\) −10.3153 −0.369112
\(782\) 0 0
\(783\) 19.1231 0.683404
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 9.75379 0.347685 0.173843 0.984773i \(-0.444382\pi\)
0.173843 + 0.984773i \(0.444382\pi\)
\(788\) 0 0
\(789\) −2.63068 −0.0936548
\(790\) 0 0
\(791\) −17.2462 −0.613205
\(792\) 0 0
\(793\) −22.2462 −0.789986
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −0.384472 −0.0136187 −0.00680935 0.999977i \(-0.502167\pi\)
−0.00680935 + 0.999977i \(0.502167\pi\)
\(798\) 0 0
\(799\) 12.8769 0.455552
\(800\) 0 0
\(801\) −5.50758 −0.194601
\(802\) 0 0
\(803\) −39.4233 −1.39122
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −44.1080 −1.55267
\(808\) 0 0
\(809\) −12.4233 −0.436780 −0.218390 0.975862i \(-0.570080\pi\)
−0.218390 + 0.975862i \(0.570080\pi\)
\(810\) 0 0
\(811\) −2.38447 −0.0837301 −0.0418651 0.999123i \(-0.513330\pi\)
−0.0418651 + 0.999123i \(0.513330\pi\)
\(812\) 0 0
\(813\) −0.384472 −0.0134840
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −23.3693 −0.817589
\(818\) 0 0
\(819\) 1.12311 0.0392445
\(820\) 0 0
\(821\) −11.8617 −0.413978 −0.206989 0.978343i \(-0.566366\pi\)
−0.206989 + 0.978343i \(0.566366\pi\)
\(822\) 0 0
\(823\) 16.5616 0.577299 0.288650 0.957435i \(-0.406794\pi\)
0.288650 + 0.957435i \(0.406794\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 7.24621 0.251975 0.125988 0.992032i \(-0.459790\pi\)
0.125988 + 0.992032i \(0.459790\pi\)
\(828\) 0 0
\(829\) 14.2462 0.494791 0.247396 0.968915i \(-0.420425\pi\)
0.247396 + 0.968915i \(0.420425\pi\)
\(830\) 0 0
\(831\) 35.1231 1.21841
\(832\) 0 0
\(833\) −1.56155 −0.0541046
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 50.7386 1.75378
\(838\) 0 0
\(839\) −7.12311 −0.245917 −0.122958 0.992412i \(-0.539238\pi\)
−0.122958 + 0.992412i \(0.539238\pi\)
\(840\) 0 0
\(841\) −17.1771 −0.592313
\(842\) 0 0
\(843\) −11.6155 −0.400060
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 26.4924 0.910290
\(848\) 0 0
\(849\) 2.05398 0.0704923
\(850\) 0 0
\(851\) −12.6695 −0.434305
\(852\) 0 0
\(853\) −19.1231 −0.654763 −0.327381 0.944892i \(-0.606166\pi\)
−0.327381 + 0.944892i \(0.606166\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −42.0540 −1.43654 −0.718268 0.695766i \(-0.755065\pi\)
−0.718268 + 0.695766i \(0.755065\pi\)
\(858\) 0 0
\(859\) 11.1771 0.381357 0.190679 0.981653i \(-0.438931\pi\)
0.190679 + 0.981653i \(0.438931\pi\)
\(860\) 0 0
\(861\) 3.80776 0.129768
\(862\) 0 0
\(863\) −3.19224 −0.108665 −0.0543325 0.998523i \(-0.517303\pi\)
−0.0543325 + 0.998523i \(0.517303\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 22.7386 0.772244
\(868\) 0 0
\(869\) 34.8078 1.18077
\(870\) 0 0
\(871\) −15.7538 −0.533797
\(872\) 0 0
\(873\) 3.36932 0.114034
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 17.6155 0.594834 0.297417 0.954748i \(-0.403875\pi\)
0.297417 + 0.954748i \(0.403875\pi\)
\(878\) 0 0
\(879\) −46.8466 −1.58010
\(880\) 0 0
\(881\) 8.73863 0.294412 0.147206 0.989106i \(-0.452972\pi\)
0.147206 + 0.989106i \(0.452972\pi\)
\(882\) 0 0
\(883\) 31.4924 1.05980 0.529902 0.848059i \(-0.322229\pi\)
0.529902 + 0.848059i \(0.322229\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −8.38447 −0.281523 −0.140762 0.990044i \(-0.544955\pi\)
−0.140762 + 0.990044i \(0.544955\pi\)
\(888\) 0 0
\(889\) 10.8078 0.362481
\(890\) 0 0
\(891\) −42.8617 −1.43592
\(892\) 0 0
\(893\) 29.3693 0.982807
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 4.49242 0.149998
\(898\) 0 0
\(899\) 31.3693 1.04623
\(900\) 0 0
\(901\) −1.75379 −0.0584272
\(902\) 0 0
\(903\) −10.2462 −0.340973
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 4.00000 0.132818 0.0664089 0.997792i \(-0.478846\pi\)
0.0664089 + 0.997792i \(0.478846\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −22.4233 −0.742917 −0.371458 0.928450i \(-0.621142\pi\)
−0.371458 + 0.928450i \(0.621142\pi\)
\(912\) 0 0
\(913\) −8.05398 −0.266548
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 18.2462 0.602543
\(918\) 0 0
\(919\) −38.8078 −1.28015 −0.640075 0.768312i \(-0.721097\pi\)
−0.640075 + 0.768312i \(0.721097\pi\)
\(920\) 0 0
\(921\) −26.0540 −0.858508
\(922\) 0 0
\(923\) 3.36932 0.110902
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0.630683 0.0207144
\(928\) 0 0
\(929\) −34.3542 −1.12712 −0.563562 0.826074i \(-0.690569\pi\)
−0.563562 + 0.826074i \(0.690569\pi\)
\(930\) 0 0
\(931\) −3.56155 −0.116725
\(932\) 0 0
\(933\) 17.3693 0.568646
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 32.3002 1.05520 0.527601 0.849493i \(-0.323092\pi\)
0.527601 + 0.849493i \(0.323092\pi\)
\(938\) 0 0
\(939\) 34.3542 1.12111
\(940\) 0 0
\(941\) 23.3693 0.761818 0.380909 0.924613i \(-0.375611\pi\)
0.380909 + 0.924613i \(0.375611\pi\)
\(942\) 0 0
\(943\) −3.50758 −0.114222
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −16.4924 −0.535932 −0.267966 0.963428i \(-0.586351\pi\)
−0.267966 + 0.963428i \(0.586351\pi\)
\(948\) 0 0
\(949\) 12.8769 0.418002
\(950\) 0 0
\(951\) −12.9848 −0.421062
\(952\) 0 0
\(953\) 36.1231 1.17014 0.585071 0.810982i \(-0.301067\pi\)
0.585071 + 0.810982i \(0.301067\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −32.8769 −1.06276
\(958\) 0 0
\(959\) 14.6847 0.474192
\(960\) 0 0
\(961\) 52.2311 1.68487
\(962\) 0 0
\(963\) −4.38447 −0.141288
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −41.3693 −1.33035 −0.665174 0.746689i \(-0.731643\pi\)
−0.665174 + 0.746689i \(0.731643\pi\)
\(968\) 0 0
\(969\) −8.68466 −0.278991
\(970\) 0 0
\(971\) 14.6847 0.471253 0.235627 0.971844i \(-0.424286\pi\)
0.235627 + 0.971844i \(0.424286\pi\)
\(972\) 0 0
\(973\) 7.31534 0.234519
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 16.7538 0.536001 0.268001 0.963419i \(-0.413637\pi\)
0.268001 + 0.963419i \(0.413637\pi\)
\(978\) 0 0
\(979\) 60.0540 1.91933
\(980\) 0 0
\(981\) −11.1922 −0.357341
\(982\) 0 0
\(983\) 28.3542 0.904357 0.452179 0.891927i \(-0.350647\pi\)
0.452179 + 0.891927i \(0.350647\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 12.8769 0.409876
\(988\) 0 0
\(989\) 9.43845 0.300125
\(990\) 0 0
\(991\) −48.6695 −1.54604 −0.773019 0.634383i \(-0.781254\pi\)
−0.773019 + 0.634383i \(0.781254\pi\)
\(992\) 0 0
\(993\) 42.1619 1.33797
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 4.24621 0.134479 0.0672394 0.997737i \(-0.478581\pi\)
0.0672394 + 0.997737i \(0.478581\pi\)
\(998\) 0 0
\(999\) −48.9848 −1.54981
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.a.bo.1.1 2
4.3 odd 2 1400.2.a.o.1.2 2
5.2 odd 4 2800.2.g.v.449.3 4
5.3 odd 4 2800.2.g.v.449.2 4
5.4 even 2 2800.2.a.bj.1.2 2
20.3 even 4 1400.2.g.j.449.3 4
20.7 even 4 1400.2.g.j.449.2 4
20.19 odd 2 1400.2.a.q.1.1 yes 2
28.27 even 2 9800.2.a.bx.1.1 2
140.139 even 2 9800.2.a.bt.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1400.2.a.o.1.2 2 4.3 odd 2
1400.2.a.q.1.1 yes 2 20.19 odd 2
1400.2.g.j.449.2 4 20.7 even 4
1400.2.g.j.449.3 4 20.3 even 4
2800.2.a.bj.1.2 2 5.4 even 2
2800.2.a.bo.1.1 2 1.1 even 1 trivial
2800.2.g.v.449.2 4 5.3 odd 4
2800.2.g.v.449.3 4 5.2 odd 4
9800.2.a.bt.1.2 2 140.139 even 2
9800.2.a.bx.1.1 2 28.27 even 2