Properties

Label 1400.2.g.j.449.3
Level $1400$
Weight $2$
Character 1400.449
Analytic conductor $11.179$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.1790562830\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.3
Root \(1.56155i\) of defining polynomial
Character \(\chi\) \(=\) 1400.449
Dual form 1400.2.g.j.449.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.56155i q^{3} +1.00000i q^{7} +0.561553 q^{9} +O(q^{10})\) \(q+1.56155i q^{3} +1.00000i q^{7} +0.561553 q^{9} -6.12311 q^{11} -2.00000i q^{13} +1.56155i q^{17} -3.56155 q^{19} -1.56155 q^{21} -1.43845i q^{23} +5.56155i q^{27} -3.43845 q^{29} -9.12311 q^{31} -9.56155i q^{33} +8.80776i q^{37} +3.12311 q^{39} -2.43845 q^{41} -6.56155i q^{43} -8.24621i q^{47} -1.00000 q^{49} -2.43845 q^{51} +1.12311i q^{53} -5.56155i q^{57} -11.3693 q^{59} +11.1231 q^{61} +0.561553i q^{63} +7.87689i q^{67} +2.24621 q^{69} +1.68466 q^{71} -6.43845i q^{73} -6.12311i q^{77} +5.68466 q^{79} -7.00000 q^{81} +1.31534i q^{83} -5.36932i q^{87} -9.80776 q^{89} +2.00000 q^{91} -14.2462i q^{93} +6.00000i q^{97} -3.43845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 6q^{9} + O(q^{10}) \) \( 4q - 6q^{9} - 8q^{11} - 6q^{19} + 2q^{21} - 22q^{29} - 20q^{31} - 4q^{39} - 18q^{41} - 4q^{49} - 18q^{51} + 4q^{59} + 28q^{61} - 24q^{69} - 18q^{71} - 2q^{79} - 28q^{81} + 2q^{89} + 8q^{91} - 22q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 1.56155i 0.901563i 0.892634 + 0.450781i \(0.148855\pi\)
−0.892634 + 0.450781i \(0.851145\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0.561553 0.187184
\(10\) 0 0
\(11\) −6.12311 −1.84619 −0.923093 0.384577i \(-0.874347\pi\)
−0.923093 + 0.384577i \(0.874347\pi\)
\(12\) 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
\(17\) 1.56155i 0.378732i 0.981907 + 0.189366i \(0.0606433\pi\)
−0.981907 + 0.189366i \(0.939357\pi\)
\(18\) 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
\(23\) − 1.43845i − 0.299937i −0.988691 0.149968i \(-0.952083\pi\)
0.988691 0.149968i \(-0.0479172\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.56155i 1.07032i
\(28\) 0 0
\(29\) −3.43845 −0.638504 −0.319252 0.947670i \(-0.603432\pi\)
−0.319252 + 0.947670i \(0.603432\pi\)
\(30\) 0 0
\(31\) −9.12311 −1.63856 −0.819279 0.573395i \(-0.805626\pi\)
−0.819279 + 0.573395i \(0.805626\pi\)
\(32\) 0 0
\(33\) − 9.56155i − 1.66445i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.80776i 1.44799i 0.689807 + 0.723994i \(0.257696\pi\)
−0.689807 + 0.723994i \(0.742304\pi\)
\(38\) 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
\(43\) − 6.56155i − 1.00063i −0.865844 0.500314i \(-0.833218\pi\)
0.865844 0.500314i \(-0.166782\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 8.24621i − 1.20283i −0.798935 0.601417i \(-0.794603\pi\)
0.798935 0.601417i \(-0.205397\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −2.43845 −0.341451
\(52\) 0 0
\(53\) 1.12311i 0.154270i 0.997021 + 0.0771352i \(0.0245773\pi\)
−0.997021 + 0.0771352i \(0.975423\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 5.56155i − 0.736646i
\(58\) 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
\(63\) 0.561553i 0.0707490i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.87689i 0.962316i 0.876634 + 0.481158i \(0.159784\pi\)
−0.876634 + 0.481158i \(0.840216\pi\)
\(68\) 0 0
\(69\) 2.24621 0.270412
\(70\) 0 0
\(71\) 1.68466 0.199932 0.0999661 0.994991i \(-0.468127\pi\)
0.0999661 + 0.994991i \(0.468127\pi\)
\(72\) 0 0
\(73\) − 6.43845i − 0.753563i −0.926302 0.376782i \(-0.877031\pi\)
0.926302 0.376782i \(-0.122969\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 6.12311i − 0.697793i
\(78\) 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
\(83\) 1.31534i 0.144377i 0.997391 + 0.0721887i \(0.0229984\pi\)
−0.997391 + 0.0721887i \(0.977002\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 5.36932i − 0.575651i
\(88\) 0 0
\(89\) −9.80776 −1.03962 −0.519810 0.854282i \(-0.673997\pi\)
−0.519810 + 0.854282i \(0.673997\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) − 14.2462i − 1.47726i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.00000i 0.609208i 0.952479 + 0.304604i \(0.0985241\pi\)
−0.952479 + 0.304604i \(0.901476\pi\)
\(98\) 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
\(103\) 1.12311i 0.110663i 0.998468 + 0.0553314i \(0.0176215\pi\)
−0.998468 + 0.0553314i \(0.982378\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.80776i 0.754805i 0.926049 + 0.377403i \(0.123183\pi\)
−0.926049 + 0.377403i \(0.876817\pi\)
\(108\) 0 0
\(109\) −19.9309 −1.90903 −0.954516 0.298161i \(-0.903627\pi\)
−0.954516 + 0.298161i \(0.903627\pi\)
\(110\) 0 0
\(111\) −13.7538 −1.30545
\(112\) 0 0
\(113\) − 17.2462i − 1.62239i −0.584778 0.811194i \(-0.698818\pi\)
0.584778 0.811194i \(-0.301182\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 1.12311i − 0.103831i
\(118\) 0 0
\(119\) −1.56155 −0.143147
\(120\) 0 0
\(121\) 26.4924 2.40840
\(122\) 0 0
\(123\) − 3.80776i − 0.343335i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 10.8078i 0.959034i 0.877533 + 0.479517i \(0.159188\pi\)
−0.877533 + 0.479517i \(0.840812\pi\)
\(128\) 0 0
\(129\) 10.2462 0.902129
\(130\) 0 0
\(131\) −18.2462 −1.59418 −0.797089 0.603861i \(-0.793628\pi\)
−0.797089 + 0.603861i \(0.793628\pi\)
\(132\) 0 0
\(133\) − 3.56155i − 0.308826i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 14.6847i − 1.25460i −0.778780 0.627298i \(-0.784161\pi\)
0.778780 0.627298i \(-0.215839\pi\)
\(138\) 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
\(143\) 12.2462i 1.02408i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 1.56155i − 0.128795i
\(148\) 0 0
\(149\) −13.4384 −1.10092 −0.550460 0.834861i \(-0.685548\pi\)
−0.550460 + 0.834861i \(0.685548\pi\)
\(150\) 0 0
\(151\) −1.43845 −0.117059 −0.0585296 0.998286i \(-0.518641\pi\)
−0.0585296 + 0.998286i \(0.518641\pi\)
\(152\) 0 0
\(153\) 0.876894i 0.0708927i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 8.49242i − 0.677769i −0.940828 0.338885i \(-0.889950\pi\)
0.940828 0.338885i \(-0.110050\pi\)
\(158\) 0 0
\(159\) −1.75379 −0.139084
\(160\) 0 0
\(161\) 1.43845 0.113366
\(162\) 0 0
\(163\) − 14.9309i − 1.16948i −0.811222 0.584738i \(-0.801197\pi\)
0.811222 0.584738i \(-0.198803\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 21.6155i 1.67266i 0.548227 + 0.836330i \(0.315303\pi\)
−0.548227 + 0.836330i \(0.684697\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) 18.4924i 1.40595i 0.711213 + 0.702976i \(0.248146\pi\)
−0.711213 + 0.702976i \(0.751854\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 17.7538i − 1.33446i
\(178\) 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
\(183\) 17.3693i 1.28398i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 9.56155i − 0.699210i
\(188\) 0 0
\(189\) −5.56155 −0.404543
\(190\) 0 0
\(191\) 25.3693 1.83566 0.917830 0.396974i \(-0.129940\pi\)
0.917830 + 0.396974i \(0.129940\pi\)
\(192\) 0 0
\(193\) 20.6155i 1.48394i 0.670434 + 0.741969i \(0.266108\pi\)
−0.670434 + 0.741969i \(0.733892\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0.561553i 0.0400090i 0.999800 + 0.0200045i \(0.00636805\pi\)
−0.999800 + 0.0200045i \(0.993632\pi\)
\(198\) 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
\(203\) − 3.43845i − 0.241332i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 0.807764i − 0.0561435i
\(208\) 0 0
\(209\) 21.8078 1.50847
\(210\) 0 0
\(211\) 6.43845 0.443241 0.221620 0.975133i \(-0.428865\pi\)
0.221620 + 0.975133i \(0.428865\pi\)
\(212\) 0 0
\(213\) 2.63068i 0.180251i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 9.12311i − 0.619317i
\(218\) 0 0
\(219\) 10.0540 0.679385
\(220\) 0 0
\(221\) 3.12311 0.210083
\(222\) 0 0
\(223\) 25.3693i 1.69886i 0.527705 + 0.849428i \(0.323053\pi\)
−0.527705 + 0.849428i \(0.676947\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 11.3693i 0.754608i 0.926089 + 0.377304i \(0.123149\pi\)
−0.926089 + 0.377304i \(0.876851\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 9.56155 0.629104
\(232\) 0 0
\(233\) 0.561553i 0.0367885i 0.999831 + 0.0183943i \(0.00585541\pi\)
−0.999831 + 0.0183943i \(0.994145\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 8.87689i 0.576616i
\(238\) 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
\(243\) 5.75379i 0.369106i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 7.12311i 0.453232i
\(248\) 0 0
\(249\) −2.05398 −0.130165
\(250\) 0 0
\(251\) −24.6847 −1.55808 −0.779041 0.626973i \(-0.784294\pi\)
−0.779041 + 0.626973i \(0.784294\pi\)
\(252\) 0 0
\(253\) 8.80776i 0.553739i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 12.2462i 0.763898i 0.924183 + 0.381949i \(0.124747\pi\)
−0.924183 + 0.381949i \(0.875253\pi\)
\(258\) 0 0
\(259\) −8.80776 −0.547288
\(260\) 0 0
\(261\) −1.93087 −0.119518
\(262\) 0 0
\(263\) − 1.68466i − 0.103880i −0.998650 0.0519402i \(-0.983459\pi\)
0.998650 0.0519402i \(-0.0165405\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 15.3153i − 0.937284i
\(268\) 0 0
\(269\) −28.2462 −1.72220 −0.861101 0.508434i \(-0.830225\pi\)
−0.861101 + 0.508434i \(0.830225\pi\)
\(270\) 0 0
\(271\) −0.246211 −0.0149563 −0.00747813 0.999972i \(-0.502380\pi\)
−0.00747813 + 0.999972i \(0.502380\pi\)
\(272\) 0 0
\(273\) 3.12311i 0.189019i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 22.4924i 1.35144i 0.737159 + 0.675719i \(0.236167\pi\)
−0.737159 + 0.675719i \(0.763833\pi\)
\(278\) 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
\(283\) 1.31534i 0.0781889i 0.999236 + 0.0390945i \(0.0124473\pi\)
−0.999236 + 0.0390945i \(0.987553\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 2.43845i − 0.143937i
\(288\) 0 0
\(289\) 14.5616 0.856562
\(290\) 0 0
\(291\) −9.36932 −0.549239
\(292\) 0 0
\(293\) 30.0000i 1.75262i 0.481749 + 0.876309i \(0.340002\pi\)
−0.481749 + 0.876309i \(0.659998\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 34.0540i − 1.97601i
\(298\) 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
\(307\) 16.6847i 0.952244i 0.879379 + 0.476122i \(0.157958\pi\)
−0.879379 + 0.476122i \(0.842042\pi\)
\(308\) 0 0
\(309\) −1.75379 −0.0997696
\(310\) 0 0
\(311\) 11.1231 0.630733 0.315367 0.948970i \(-0.397872\pi\)
0.315367 + 0.948970i \(0.397872\pi\)
\(312\) 0 0
\(313\) − 22.0000i − 1.24351i −0.783210 0.621757i \(-0.786419\pi\)
0.783210 0.621757i \(-0.213581\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 8.31534i − 0.467036i −0.972352 0.233518i \(-0.924976\pi\)
0.972352 0.233518i \(-0.0750238\pi\)
\(318\) 0 0
\(319\) 21.0540 1.17880
\(320\) 0 0
\(321\) −12.1922 −0.680504
\(322\) 0 0
\(323\) − 5.56155i − 0.309453i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 31.1231i − 1.72111i
\(328\) 0 0
\(329\) 8.24621 0.454628
\(330\) 0 0
\(331\) 27.0000 1.48405 0.742027 0.670370i \(-0.233865\pi\)
0.742027 + 0.670370i \(0.233865\pi\)
\(332\) 0 0
\(333\) 4.94602i 0.271040i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 6.68466i 0.364137i 0.983286 + 0.182068i \(0.0582792\pi\)
−0.983286 + 0.182068i \(0.941721\pi\)
\(338\) 0 0
\(339\) 26.9309 1.46268
\(340\) 0 0
\(341\) 55.8617 3.02508
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 18.6155i − 0.999334i −0.866218 0.499667i \(-0.833456\pi\)
0.866218 0.499667i \(-0.166544\pi\)
\(348\) 0 0
\(349\) 21.1231 1.13069 0.565347 0.824853i \(-0.308742\pi\)
0.565347 + 0.824853i \(0.308742\pi\)
\(350\) 0 0
\(351\) 11.1231 0.593707
\(352\) 0 0
\(353\) 28.2462i 1.50339i 0.659509 + 0.751697i \(0.270764\pi\)
−0.659509 + 0.751697i \(0.729236\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 2.43845i − 0.129056i
\(358\) 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
\(363\) 41.3693i 2.17133i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 34.2462i 1.78764i 0.448428 + 0.893819i \(0.351984\pi\)
−0.448428 + 0.893819i \(0.648016\pi\)
\(368\) 0 0
\(369\) −1.36932 −0.0712838
\(370\) 0 0
\(371\) −1.12311 −0.0583087
\(372\) 0 0
\(373\) 23.0540i 1.19369i 0.802357 + 0.596845i \(0.203579\pi\)
−0.802357 + 0.596845i \(0.796421\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.87689i 0.354178i
\(378\) 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
\(383\) 3.75379i 0.191810i 0.995391 + 0.0959048i \(0.0305744\pi\)
−0.995391 + 0.0959048i \(0.969426\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 3.68466i − 0.187302i
\(388\) 0 0
\(389\) −6.56155 −0.332684 −0.166342 0.986068i \(-0.553196\pi\)
−0.166342 + 0.986068i \(0.553196\pi\)
\(390\) 0 0
\(391\) 2.24621 0.113596
\(392\) 0 0
\(393\) − 28.4924i − 1.43725i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 5.50758i 0.276417i 0.990403 + 0.138209i \(0.0441345\pi\)
−0.990403 + 0.138209i \(0.955866\pi\)
\(398\) 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
\(403\) 18.2462i 0.908909i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 53.9309i − 2.67325i
\(408\) 0 0
\(409\) 25.8078 1.27611 0.638056 0.769990i \(-0.279739\pi\)
0.638056 + 0.769990i \(0.279739\pi\)
\(410\) 0 0
\(411\) 22.9309 1.13110
\(412\) 0 0
\(413\) − 11.3693i − 0.559448i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 11.4233i 0.559401i
\(418\) 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
\(423\) − 4.63068i − 0.225152i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 11.1231i 0.538285i
\(428\) 0 0
\(429\) −19.1231 −0.923272
\(430\) 0 0
\(431\) −30.2462 −1.45691 −0.728454 0.685094i \(-0.759761\pi\)
−0.728454 + 0.685094i \(0.759761\pi\)
\(432\) 0 0
\(433\) − 34.5464i − 1.66019i −0.557619 0.830097i \(-0.688285\pi\)
0.557619 0.830097i \(-0.311715\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 5.12311i 0.245071i
\(438\) 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
\(443\) − 18.4384i − 0.876037i −0.898966 0.438019i \(-0.855680\pi\)
0.898966 0.438019i \(-0.144320\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 20.9848i − 0.992549i
\(448\) 0 0
\(449\) 14.3693 0.678130 0.339065 0.940763i \(-0.389889\pi\)
0.339065 + 0.940763i \(0.389889\pi\)
\(450\) 0 0
\(451\) 14.9309 0.703067
\(452\) 0 0
\(453\) − 2.24621i − 0.105536i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 37.1080i − 1.73584i −0.496707 0.867918i \(-0.665458\pi\)
0.496707 0.867918i \(-0.334542\pi\)
\(458\) 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
\(463\) − 12.4924i − 0.580572i −0.956940 0.290286i \(-0.906250\pi\)
0.956940 0.290286i \(-0.0937505\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 33.1231i − 1.53275i −0.642391 0.766377i \(-0.722057\pi\)
0.642391 0.766377i \(-0.277943\pi\)
\(468\) 0 0
\(469\) −7.87689 −0.363721
\(470\) 0 0
\(471\) 13.2614 0.611052
\(472\) 0 0
\(473\) 40.1771i 1.84734i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0.630683i 0.0288770i
\(478\) 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
\(483\) 2.24621i 0.102206i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 31.4384i 1.42461i 0.701869 + 0.712306i \(0.252349\pi\)
−0.701869 + 0.712306i \(0.747651\pi\)
\(488\) 0 0
\(489\) 23.3153 1.05436
\(490\) 0 0
\(491\) 8.31534 0.375266 0.187633 0.982239i \(-0.439918\pi\)
0.187633 + 0.982239i \(0.439918\pi\)
\(492\) 0 0
\(493\) − 5.36932i − 0.241822i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.68466i 0.0755673i
\(498\) 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
\(503\) − 36.2462i − 1.61614i −0.589087 0.808069i \(-0.700513\pi\)
0.589087 0.808069i \(-0.299487\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 14.0540i 0.624159i
\(508\) 0 0
\(509\) −26.0000 −1.15243 −0.576215 0.817298i \(-0.695471\pi\)
−0.576215 + 0.817298i \(0.695471\pi\)
\(510\) 0 0
\(511\) 6.43845 0.284820
\(512\) 0 0
\(513\) − 19.8078i − 0.874534i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 50.4924i 2.22065i
\(518\) 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
\(523\) − 40.6847i − 1.77902i −0.456920 0.889508i \(-0.651047\pi\)
0.456920 0.889508i \(-0.348953\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 14.2462i − 0.620575i
\(528\) 0 0
\(529\) 20.9309 0.910038
\(530\) 0 0
\(531\) −6.38447 −0.277062
\(532\) 0 0
\(533\) 4.87689i 0.211242i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 32.3002i 1.39386i
\(538\) 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
\(543\) − 17.3693i − 0.745389i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 15.7386i − 0.672935i −0.941695 0.336468i \(-0.890768\pi\)
0.941695 0.336468i \(-0.109232\pi\)
\(548\) 0 0
\(549\) 6.24621 0.266582
\(550\) 0 0
\(551\) 12.2462 0.521706
\(552\) 0 0
\(553\) 5.68466i 0.241736i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 5.43845i 0.230434i 0.993340 + 0.115217i \(0.0367564\pi\)
−0.993340 + 0.115217i \(0.963244\pi\)
\(558\) 0 0
\(559\) −13.1231 −0.555048
\(560\) 0 0
\(561\) 14.9309 0.630382
\(562\) 0 0
\(563\) − 1.12311i − 0.0473333i −0.999720 0.0236666i \(-0.992466\pi\)
0.999720 0.0236666i \(-0.00753403\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 7.00000i − 0.293972i
\(568\) 0 0
\(569\) −45.9848 −1.92778 −0.963892 0.266292i \(-0.914202\pi\)
−0.963892 + 0.266292i \(0.914202\pi\)
\(570\) 0 0
\(571\) 39.6847 1.66075 0.830376 0.557204i \(-0.188126\pi\)
0.830376 + 0.557204i \(0.188126\pi\)
\(572\) 0 0
\(573\) 39.6155i 1.65496i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 24.5464i 1.02188i 0.859616 + 0.510940i \(0.170703\pi\)
−0.859616 + 0.510940i \(0.829297\pi\)
\(578\) 0 0
\(579\) −32.1922 −1.33786
\(580\) 0 0
\(581\) −1.31534 −0.0545696
\(582\) 0 0
\(583\) − 6.87689i − 0.284812i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 37.1771i − 1.53446i −0.641371 0.767231i \(-0.721634\pi\)
0.641371 0.767231i \(-0.278366\pi\)
\(588\) 0 0
\(589\) 32.4924 1.33883
\(590\) 0 0
\(591\) −0.876894 −0.0360706
\(592\) 0 0
\(593\) − 24.6847i − 1.01368i −0.862041 0.506839i \(-0.830814\pi\)
0.862041 0.506839i \(-0.169186\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 7.61553i 0.311683i
\(598\) 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
\(603\) 4.42329i 0.180130i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 12.6307i − 0.512664i −0.966589 0.256332i \(-0.917486\pi\)
0.966589 0.256332i \(-0.0825140\pi\)
\(608\) 0 0
\(609\) 5.36932 0.217576
\(610\) 0 0
\(611\) −16.4924 −0.667212
\(612\) 0 0
\(613\) − 5.68466i − 0.229601i −0.993389 0.114801i \(-0.963377\pi\)
0.993389 0.114801i \(-0.0366229\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 2.80776i − 0.113036i −0.998402 0.0565182i \(-0.982000\pi\)
0.998402 0.0565182i \(-0.0179999\pi\)
\(618\) 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
\(623\) − 9.80776i − 0.392940i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 34.0540i 1.35998i
\(628\) 0 0
\(629\) −13.7538 −0.548399
\(630\) 0 0
\(631\) −3.05398 −0.121577 −0.0607884 0.998151i \(-0.519361\pi\)
−0.0607884 + 0.998151i \(0.519361\pi\)
\(632\) 0 0
\(633\) 10.0540i 0.399610i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2.00000i 0.0792429i
\(638\) 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
\(643\) 46.7386i 1.84319i 0.388151 + 0.921596i \(0.373114\pi\)
−0.388151 + 0.921596i \(0.626886\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 0.630683i − 0.0247947i −0.999923 0.0123974i \(-0.996054\pi\)
0.999923 0.0123974i \(-0.00394630\pi\)
\(648\) 0 0
\(649\) 69.6155 2.73265
\(650\) 0 0
\(651\) 14.2462 0.558353
\(652\) 0 0
\(653\) 23.8617i 0.933782i 0.884315 + 0.466891i \(0.154626\pi\)
−0.884315 + 0.466891i \(0.845374\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 3.61553i − 0.141055i
\(658\) 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
\(663\) 4.87689i 0.189403i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.94602i 0.191511i
\(668\) 0 0
\(669\) −39.6155 −1.53162
\(670\) 0 0
\(671\) −68.1080 −2.62928
\(672\) 0 0
\(673\) − 13.5076i − 0.520679i −0.965517 0.260339i \(-0.916166\pi\)
0.965517 0.260339i \(-0.0838345\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 39.3693i 1.51309i 0.653944 + 0.756543i \(0.273113\pi\)
−0.653944 + 0.756543i \(0.726887\pi\)
\(678\) 0 0
\(679\) −6.00000 −0.230259
\(680\) 0 0
\(681\) −17.7538 −0.680327
\(682\) 0 0
\(683\) − 25.8769i − 0.990152i −0.868850 0.495076i \(-0.835140\pi\)
0.868850 0.495076i \(-0.164860\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 21.8617i 0.834077i
\(688\) 0 0
\(689\) 2.24621 0.0855738
\(690\) 0 0
\(691\) −21.5616 −0.820240 −0.410120 0.912032i \(-0.634513\pi\)
−0.410120 + 0.912032i \(0.634513\pi\)
\(692\) 0 0
\(693\) − 3.43845i − 0.130616i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 3.80776i − 0.144229i
\(698\) 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
\(703\) − 31.3693i − 1.18312i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −12.6307 −0.474355 −0.237178 0.971466i \(-0.576222\pi\)
−0.237178 + 0.971466i \(0.576222\pi\)
\(710\) 0 0
\(711\) 3.19224 0.119718
\(712\) 0 0
\(713\) 13.1231i 0.491464i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 6.24621i 0.233269i
\(718\) 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
\(723\) − 18.8229i − 0.700032i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 39.6155i 1.46926i 0.678469 + 0.734629i \(0.262644\pi\)
−0.678469 + 0.734629i \(0.737356\pi\)
\(728\) 0 0
\(729\) −29.9848 −1.11055
\(730\) 0 0
\(731\) 10.2462 0.378970
\(732\) 0 0
\(733\) − 5.75379i − 0.212521i −0.994338 0.106261i \(-0.966112\pi\)
0.994338 0.106261i \(-0.0338878\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 48.2311i − 1.77661i
\(738\) 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
\(743\) − 30.7386i − 1.12769i −0.825880 0.563846i \(-0.809321\pi\)
0.825880 0.563846i \(-0.190679\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0.738634i 0.0270252i
\(748\) 0 0
\(749\) −7.80776 −0.285289
\(750\) 0 0
\(751\) 2.63068 0.0959950 0.0479975 0.998847i \(-0.484716\pi\)
0.0479975 + 0.998847i \(0.484716\pi\)
\(752\) 0 0
\(753\) − 38.5464i − 1.40471i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 25.6847i 0.933525i 0.884383 + 0.466762i \(0.154580\pi\)
−0.884383 + 0.466762i \(0.845420\pi\)
\(758\) 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
\(763\) − 19.9309i − 0.721546i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 22.7386i 0.821044i
\(768\) 0 0
\(769\) −20.3002 −0.732043 −0.366022 0.930606i \(-0.619280\pi\)
−0.366022 + 0.930606i \(0.619280\pi\)
\(770\) 0 0
\(771\) −19.1231 −0.688702
\(772\) 0 0
\(773\) − 29.6155i − 1.06520i −0.846368 0.532598i \(-0.821216\pi\)
0.846368 0.532598i \(-0.178784\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 13.7538i − 0.493414i
\(778\) 0 0
\(779\) 8.68466 0.311160
\(780\) 0 0
\(781\) −10.3153 −0.369112
\(782\) 0 0
\(783\) − 19.1231i − 0.683404i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 9.75379i 0.347685i 0.984773 + 0.173843i \(0.0556184\pi\)
−0.984773 + 0.173843i \(0.944382\pi\)
\(788\) 0 0
\(789\) 2.63068 0.0936548
\(790\) 0 0
\(791\) 17.2462 0.613205
\(792\) 0 0
\(793\) − 22.2462i − 0.789986i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0.384472i 0.0136187i 0.999977 + 0.00680935i \(0.00216750\pi\)
−0.999977 + 0.00680935i \(0.997833\pi\)
\(798\) 0 0
\(799\) 12.8769 0.455552
\(800\) 0 0
\(801\) −5.50758 −0.194601
\(802\) 0 0
\(803\) 39.4233i 1.39122i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 44.1080i − 1.55267i
\(808\) 0 0
\(809\) 12.4233 0.436780 0.218390 0.975862i \(-0.429920\pi\)
0.218390 + 0.975862i \(0.429920\pi\)
\(810\) 0 0
\(811\) 2.38447 0.0837301 0.0418651 0.999123i \(-0.486670\pi\)
0.0418651 + 0.999123i \(0.486670\pi\)
\(812\) 0 0
\(813\) − 0.384472i − 0.0134840i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 23.3693i 0.817589i
\(818\) 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
\(823\) − 16.5616i − 0.577299i −0.957435 0.288650i \(-0.906794\pi\)
0.957435 0.288650i \(-0.0932063\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 7.24621i 0.251975i 0.992032 + 0.125988i \(0.0402100\pi\)
−0.992032 + 0.125988i \(0.959790\pi\)
\(828\) 0 0
\(829\) −14.2462 −0.494791 −0.247396 0.968915i \(-0.579575\pi\)
−0.247396 + 0.968915i \(0.579575\pi\)
\(830\) 0 0
\(831\) −35.1231 −1.21841
\(832\) 0 0
\(833\) − 1.56155i − 0.0541046i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 50.7386i − 1.75378i
\(838\) 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
\(843\) 11.6155i 0.400060i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 26.4924i 0.910290i
\(848\) 0 0
\(849\) −2.05398 −0.0704923
\(850\) 0 0
\(851\) 12.6695 0.434305
\(852\) 0 0
\(853\) − 19.1231i − 0.654763i −0.944892 0.327381i \(-0.893834\pi\)
0.944892 0.327381i \(-0.106166\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 42.0540i 1.43654i 0.695766 + 0.718268i \(0.255065\pi\)
−0.695766 + 0.718268i \(0.744935\pi\)
\(858\) 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
\(863\) 3.19224i 0.108665i 0.998523 + 0.0543325i \(0.0173031\pi\)
−0.998523 + 0.0543325i \(0.982697\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 22.7386i 0.772244i
\(868\) 0 0
\(869\) −34.8078 −1.18077
\(870\) 0 0
\(871\) 15.7538 0.533797
\(872\) 0 0
\(873\) 3.36932i 0.114034i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 17.6155i − 0.594834i −0.954748 0.297417i \(-0.903875\pi\)
0.954748 0.297417i \(-0.0961252\pi\)
\(878\) 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
\(883\) − 31.4924i − 1.05980i −0.848059 0.529902i \(-0.822229\pi\)
0.848059 0.529902i \(-0.177771\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 8.38447i − 0.281523i −0.990044 0.140762i \(-0.955045\pi\)
0.990044 0.140762i \(-0.0449551\pi\)
\(888\) 0 0
\(889\) −10.8078 −0.362481
\(890\) 0 0
\(891\) 42.8617 1.43592
\(892\) 0 0
\(893\) 29.3693i 0.982807i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 4.49242i − 0.149998i
\(898\) 0 0
\(899\) 31.3693 1.04623
\(900\) 0 0
\(901\) −1.75379 −0.0584272
\(902\) 0 0
\(903\) 10.2462i 0.340973i
\(904\) 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
\(911\) 22.4233 0.742917 0.371458 0.928450i \(-0.378858\pi\)
0.371458 + 0.928450i \(0.378858\pi\)
\(912\) 0 0
\(913\) − 8.05398i − 0.266548i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 18.2462i − 0.602543i
\(918\) 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
\(923\) − 3.36932i − 0.110902i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0.630683i 0.0207144i
\(928\) 0 0
\(929\) 34.3542 1.12712 0.563562 0.826074i \(-0.309431\pi\)
0.563562 + 0.826074i \(0.309431\pi\)
\(930\) 0 0
\(931\) 3.56155 0.116725
\(932\) 0 0
\(933\) 17.3693i 0.568646i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 32.3002i − 1.05520i −0.849493 0.527601i \(-0.823092\pi\)
0.849493 0.527601i \(-0.176908\pi\)
\(938\) 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
\(943\) 3.50758i 0.114222i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 16.4924i − 0.535932i −0.963428 0.267966i \(-0.913649\pi\)
0.963428 0.267966i \(-0.0863514\pi\)
\(948\) 0 0
\(949\) −12.8769 −0.418002
\(950\) 0 0
\(951\) 12.9848 0.421062
\(952\) 0 0
\(953\) 36.1231i 1.17014i 0.810982 + 0.585071i \(0.198933\pi\)
−0.810982 + 0.585071i \(0.801067\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 32.8769i 1.06276i
\(958\) 0 0
\(959\) 14.6847 0.474192
\(960\) 0 0
\(961\) 52.2311 1.68487
\(962\) 0 0
\(963\) 4.38447i 0.141288i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 41.3693i − 1.33035i −0.746689 0.665174i \(-0.768357\pi\)
0.746689 0.665174i \(-0.231643\pi\)
\(968\) 0 0
\(969\) 8.68466 0.278991
\(970\) 0 0
\(971\) −14.6847 −0.471253 −0.235627 0.971844i \(-0.575714\pi\)
−0.235627 + 0.971844i \(0.575714\pi\)
\(972\) 0 0
\(973\) 7.31534i 0.234519i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 16.7538i − 0.536001i −0.963419 0.268001i \(-0.913637\pi\)
0.963419 0.268001i \(-0.0863629\pi\)
\(978\) 0 0
\(979\) 60.0540 1.91933
\(980\) 0 0
\(981\) −11.1922 −0.357341
\(982\) 0 0
\(983\) − 28.3542i − 0.904357i −0.891927 0.452179i \(-0.850647\pi\)
0.891927 0.452179i \(-0.149353\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 12.8769i 0.409876i
\(988\) 0 0
\(989\) −9.43845 −0.300125
\(990\) 0 0
\(991\) 48.6695 1.54604 0.773019 0.634383i \(-0.218746\pi\)
0.773019 + 0.634383i \(0.218746\pi\)
\(992\) 0 0
\(993\) 42.1619i 1.33797i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 4.24621i − 0.134479i −0.997737 0.0672394i \(-0.978581\pi\)
0.997737 0.0672394i \(-0.0214191\pi\)
\(998\) 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 1400.2.g.j.449.3 4
4.3 odd 2 2800.2.g.v.449.2 4
5.2 odd 4 1400.2.a.o.1.2 2
5.3 odd 4 1400.2.a.q.1.1 yes 2
5.4 even 2 inner 1400.2.g.j.449.2 4
20.3 even 4 2800.2.a.bj.1.2 2
20.7 even 4 2800.2.a.bo.1.1 2
20.19 odd 2 2800.2.g.v.449.3 4
35.13 even 4 9800.2.a.bt.1.2 2
35.27 even 4 9800.2.a.bx.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1400.2.a.o.1.2 2 5.2 odd 4
1400.2.a.q.1.1 yes 2 5.3 odd 4
1400.2.g.j.449.2 4 5.4 even 2 inner
1400.2.g.j.449.3 4 1.1 even 1 trivial
2800.2.a.bj.1.2 2 20.3 even 4
2800.2.a.bo.1.1 2 20.7 even 4
2800.2.g.v.449.2 4 4.3 odd 2
2800.2.g.v.449.3 4 20.19 odd 2
9800.2.a.bt.1.2 2 35.13 even 4
9800.2.a.bx.1.1 2 35.27 even 4