Properties

Label 2352.3.m.f.1471.3
Level $2352$
Weight $3$
Character 2352.1471
Analytic conductor $64.087$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2352.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(64.0873581775\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-7})\)
Defining polynomial: \(x^{4} - x^{3} - x^{2} - 2 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1471.3
Root \(-0.895644 + 1.09445i\) of defining polynomial
Character \(\chi\) \(=\) 2352.1471
Dual form 2352.3.m.f.1471.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.73205i q^{3} -5.58258 q^{5} -3.00000 q^{9} +O(q^{10})\) \(q+1.73205i q^{3} -5.58258 q^{5} -3.00000 q^{9} -4.37780i q^{11} -17.1652 q^{13} -9.66930i q^{15} -0.747727 q^{17} +3.65480i q^{19} -9.86001i q^{23} +6.16515 q^{25} -5.19615i q^{27} -2.00000 q^{29} +17.8926i q^{31} +7.58258 q^{33} -7.49545 q^{37} -29.7309i q^{39} -76.5735 q^{41} -70.0448i q^{43} +16.7477 q^{45} +40.1232i q^{47} -1.29510i q^{51} +49.8258 q^{53} +24.4394i q^{55} -6.33030 q^{57} -89.3834i q^{59} +120.991 q^{61} +95.8258 q^{65} +23.3748i q^{67} +17.0780 q^{69} +66.0484i q^{71} +40.5045 q^{73} +10.6784i q^{75} +95.5488i q^{79} +9.00000 q^{81} -115.650i q^{83} +4.17424 q^{85} -3.46410i q^{87} -34.9038 q^{89} -30.9909 q^{93} -20.4032i q^{95} -52.5045 q^{97} +13.1334i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{5} - 12q^{9} + O(q^{10}) \) \( 4q - 4q^{5} - 12q^{9} - 32q^{13} + 52q^{17} - 12q^{25} - 8q^{29} + 12q^{33} + 80q^{37} - 68q^{41} + 12q^{45} + 16q^{53} + 48q^{57} + 264q^{61} + 200q^{65} - 60q^{69} + 272q^{73} + 36q^{81} + 200q^{85} + 172q^{89} + 96q^{93} - 320q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(3\) 1.73205i 0.577350i
\(4\) 0 0
\(5\) −5.58258 −1.11652 −0.558258 0.829668i \(-0.688530\pi\)
−0.558258 + 0.829668i \(0.688530\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) − 4.37780i − 0.397982i −0.980001 0.198991i \(-0.936234\pi\)
0.980001 0.198991i \(-0.0637665\pi\)
\(12\) 0 0
\(13\) −17.1652 −1.32040 −0.660198 0.751091i \(-0.729528\pi\)
−0.660198 + 0.751091i \(0.729528\pi\)
\(14\) 0 0
\(15\) − 9.66930i − 0.644620i
\(16\) 0 0
\(17\) −0.747727 −0.0439839 −0.0219920 0.999758i \(-0.507001\pi\)
−0.0219920 + 0.999758i \(0.507001\pi\)
\(18\) 0 0
\(19\) 3.65480i 0.192358i 0.995364 + 0.0961790i \(0.0306621\pi\)
−0.995364 + 0.0961790i \(0.969338\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 9.86001i − 0.428696i −0.976757 0.214348i \(-0.931237\pi\)
0.976757 0.214348i \(-0.0687626\pi\)
\(24\) 0 0
\(25\) 6.16515 0.246606
\(26\) 0 0
\(27\) − 5.19615i − 0.192450i
\(28\) 0 0
\(29\) −2.00000 −0.0689655 −0.0344828 0.999405i \(-0.510978\pi\)
−0.0344828 + 0.999405i \(0.510978\pi\)
\(30\) 0 0
\(31\) 17.8926i 0.577181i 0.957453 + 0.288590i \(0.0931866\pi\)
−0.957453 + 0.288590i \(0.906813\pi\)
\(32\) 0 0
\(33\) 7.58258 0.229775
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.49545 −0.202580 −0.101290 0.994857i \(-0.532297\pi\)
−0.101290 + 0.994857i \(0.532297\pi\)
\(38\) 0 0
\(39\) − 29.7309i − 0.762331i
\(40\) 0 0
\(41\) −76.5735 −1.86765 −0.933823 0.357735i \(-0.883549\pi\)
−0.933823 + 0.357735i \(0.883549\pi\)
\(42\) 0 0
\(43\) − 70.0448i − 1.62895i −0.580199 0.814475i \(-0.697025\pi\)
0.580199 0.814475i \(-0.302975\pi\)
\(44\) 0 0
\(45\) 16.7477 0.372172
\(46\) 0 0
\(47\) 40.1232i 0.853686i 0.904326 + 0.426843i \(0.140374\pi\)
−0.904326 + 0.426843i \(0.859626\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) − 1.29510i − 0.0253941i
\(52\) 0 0
\(53\) 49.8258 0.940109 0.470054 0.882637i \(-0.344234\pi\)
0.470054 + 0.882637i \(0.344234\pi\)
\(54\) 0 0
\(55\) 24.4394i 0.444353i
\(56\) 0 0
\(57\) −6.33030 −0.111058
\(58\) 0 0
\(59\) − 89.3834i − 1.51497i −0.652850 0.757487i \(-0.726427\pi\)
0.652850 0.757487i \(-0.273573\pi\)
\(60\) 0 0
\(61\) 120.991 1.98346 0.991729 0.128351i \(-0.0409685\pi\)
0.991729 + 0.128351i \(0.0409685\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 95.8258 1.47424
\(66\) 0 0
\(67\) 23.3748i 0.348878i 0.984668 + 0.174439i \(0.0558112\pi\)
−0.984668 + 0.174439i \(0.944189\pi\)
\(68\) 0 0
\(69\) 17.0780 0.247508
\(70\) 0 0
\(71\) 66.0484i 0.930260i 0.885242 + 0.465130i \(0.153992\pi\)
−0.885242 + 0.465130i \(0.846008\pi\)
\(72\) 0 0
\(73\) 40.5045 0.554857 0.277428 0.960746i \(-0.410518\pi\)
0.277428 + 0.960746i \(0.410518\pi\)
\(74\) 0 0
\(75\) 10.6784i 0.142378i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 95.5488i 1.20948i 0.796423 + 0.604740i \(0.206723\pi\)
−0.796423 + 0.604740i \(0.793277\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) − 115.650i − 1.39338i −0.717374 0.696688i \(-0.754656\pi\)
0.717374 0.696688i \(-0.245344\pi\)
\(84\) 0 0
\(85\) 4.17424 0.0491087
\(86\) 0 0
\(87\) − 3.46410i − 0.0398173i
\(88\) 0 0
\(89\) −34.9038 −0.392177 −0.196089 0.980586i \(-0.562824\pi\)
−0.196089 + 0.980586i \(0.562824\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −30.9909 −0.333236
\(94\) 0 0
\(95\) − 20.4032i − 0.214771i
\(96\) 0 0
\(97\) −52.5045 −0.541284 −0.270642 0.962680i \(-0.587236\pi\)
−0.270642 + 0.962680i \(0.587236\pi\)
\(98\) 0 0
\(99\) 13.1334i 0.132661i
\(100\) 0 0
\(101\) 113.583 1.12458 0.562290 0.826940i \(-0.309920\pi\)
0.562290 + 0.826940i \(0.309920\pi\)
\(102\) 0 0
\(103\) − 50.0230i − 0.485660i −0.970069 0.242830i \(-0.921924\pi\)
0.970069 0.242830i \(-0.0780758\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 160.533i 1.50031i 0.661265 + 0.750153i \(0.270020\pi\)
−0.661265 + 0.750153i \(0.729980\pi\)
\(108\) 0 0
\(109\) −205.303 −1.88351 −0.941757 0.336294i \(-0.890826\pi\)
−0.941757 + 0.336294i \(0.890826\pi\)
\(110\) 0 0
\(111\) − 12.9825i − 0.116960i
\(112\) 0 0
\(113\) 179.321 1.58691 0.793457 0.608627i \(-0.208279\pi\)
0.793457 + 0.608627i \(0.208279\pi\)
\(114\) 0 0
\(115\) 55.0442i 0.478645i
\(116\) 0 0
\(117\) 51.4955 0.440132
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 101.835 0.841610
\(122\) 0 0
\(123\) − 132.629i − 1.07829i
\(124\) 0 0
\(125\) 105.147 0.841176
\(126\) 0 0
\(127\) 130.571i 1.02812i 0.857754 + 0.514060i \(0.171859\pi\)
−0.857754 + 0.514060i \(0.828141\pi\)
\(128\) 0 0
\(129\) 121.321 0.940475
\(130\) 0 0
\(131\) − 71.1890i − 0.543428i −0.962378 0.271714i \(-0.912410\pi\)
0.962378 0.271714i \(-0.0875904\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 29.0079i 0.214873i
\(136\) 0 0
\(137\) −99.8439 −0.728788 −0.364394 0.931245i \(-0.618724\pi\)
−0.364394 + 0.931245i \(0.618724\pi\)
\(138\) 0 0
\(139\) 162.400i 1.16834i 0.811630 + 0.584172i \(0.198581\pi\)
−0.811630 + 0.584172i \(0.801419\pi\)
\(140\) 0 0
\(141\) −69.4955 −0.492876
\(142\) 0 0
\(143\) 75.1456i 0.525494i
\(144\) 0 0
\(145\) 11.1652 0.0770010
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 293.165 1.96755 0.983776 0.179403i \(-0.0574166\pi\)
0.983776 + 0.179403i \(0.0574166\pi\)
\(150\) 0 0
\(151\) 120.449i 0.797677i 0.917021 + 0.398839i \(0.130587\pi\)
−0.917021 + 0.398839i \(0.869413\pi\)
\(152\) 0 0
\(153\) 2.24318 0.0146613
\(154\) 0 0
\(155\) − 99.8868i − 0.644431i
\(156\) 0 0
\(157\) 163.982 1.04447 0.522235 0.852802i \(-0.325098\pi\)
0.522235 + 0.852802i \(0.325098\pi\)
\(158\) 0 0
\(159\) 86.3007i 0.542772i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 226.803i 1.39143i 0.718317 + 0.695716i \(0.244913\pi\)
−0.718317 + 0.695716i \(0.755087\pi\)
\(164\) 0 0
\(165\) −42.3303 −0.256547
\(166\) 0 0
\(167\) − 226.803i − 1.35810i −0.734090 0.679052i \(-0.762391\pi\)
0.734090 0.679052i \(-0.237609\pi\)
\(168\) 0 0
\(169\) 125.642 0.743446
\(170\) 0 0
\(171\) − 10.9644i − 0.0641193i
\(172\) 0 0
\(173\) 108.922 0.629607 0.314803 0.949157i \(-0.398061\pi\)
0.314803 + 0.949157i \(0.398061\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 154.817 0.874670
\(178\) 0 0
\(179\) − 91.2506i − 0.509780i −0.966970 0.254890i \(-0.917961\pi\)
0.966970 0.254890i \(-0.0820393\pi\)
\(180\) 0 0
\(181\) −135.495 −0.748594 −0.374297 0.927309i \(-0.622116\pi\)
−0.374297 + 0.927309i \(0.622116\pi\)
\(182\) 0 0
\(183\) 209.562i 1.14515i
\(184\) 0 0
\(185\) 41.8439 0.226183
\(186\) 0 0
\(187\) 3.27340i 0.0175048i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 106.934i − 0.559866i −0.960020 0.279933i \(-0.909688\pi\)
0.960020 0.279933i \(-0.0903123\pi\)
\(192\) 0 0
\(193\) −116.505 −0.603650 −0.301825 0.953363i \(-0.597596\pi\)
−0.301825 + 0.953363i \(0.597596\pi\)
\(194\) 0 0
\(195\) 165.975i 0.851154i
\(196\) 0 0
\(197\) 38.8348 0.197131 0.0985656 0.995131i \(-0.468575\pi\)
0.0985656 + 0.995131i \(0.468575\pi\)
\(198\) 0 0
\(199\) 353.116i 1.77445i 0.461334 + 0.887227i \(0.347371\pi\)
−0.461334 + 0.887227i \(0.652629\pi\)
\(200\) 0 0
\(201\) −40.4864 −0.201425
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 427.477 2.08525
\(206\) 0 0
\(207\) 29.5800i 0.142899i
\(208\) 0 0
\(209\) 16.0000 0.0765550
\(210\) 0 0
\(211\) − 54.0592i − 0.256205i −0.991761 0.128102i \(-0.959111\pi\)
0.991761 0.128102i \(-0.0408886\pi\)
\(212\) 0 0
\(213\) −114.399 −0.537086
\(214\) 0 0
\(215\) 391.031i 1.81875i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 70.1559i 0.320347i
\(220\) 0 0
\(221\) 12.8348 0.0580762
\(222\) 0 0
\(223\) 93.3400i 0.418565i 0.977855 + 0.209283i \(0.0671129\pi\)
−0.977855 + 0.209283i \(0.932887\pi\)
\(224\) 0 0
\(225\) −18.4955 −0.0822020
\(226\) 0 0
\(227\) 170.233i 0.749927i 0.927040 + 0.374964i \(0.122345\pi\)
−0.927040 + 0.374964i \(0.877655\pi\)
\(228\) 0 0
\(229\) 227.844 0.994952 0.497476 0.867478i \(-0.334260\pi\)
0.497476 + 0.867478i \(0.334260\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 378.486 1.62440 0.812202 0.583376i \(-0.198268\pi\)
0.812202 + 0.583376i \(0.198268\pi\)
\(234\) 0 0
\(235\) − 223.991i − 0.953153i
\(236\) 0 0
\(237\) −165.495 −0.698293
\(238\) 0 0
\(239\) − 205.535i − 0.859977i −0.902834 0.429989i \(-0.858518\pi\)
0.902834 0.429989i \(-0.141482\pi\)
\(240\) 0 0
\(241\) −40.4682 −0.167918 −0.0839589 0.996469i \(-0.526756\pi\)
−0.0839589 + 0.996469i \(0.526756\pi\)
\(242\) 0 0
\(243\) 15.5885i 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 62.7352i − 0.253989i
\(248\) 0 0
\(249\) 200.312 0.804466
\(250\) 0 0
\(251\) − 332.856i − 1.32612i −0.748567 0.663059i \(-0.769258\pi\)
0.748567 0.663059i \(-0.230742\pi\)
\(252\) 0 0
\(253\) −43.1652 −0.170613
\(254\) 0 0
\(255\) 7.23000i 0.0283529i
\(256\) 0 0
\(257\) 205.372 0.799113 0.399556 0.916709i \(-0.369164\pi\)
0.399556 + 0.916709i \(0.369164\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 6.00000 0.0229885
\(262\) 0 0
\(263\) 45.4064i 0.172648i 0.996267 + 0.0863240i \(0.0275120\pi\)
−0.996267 + 0.0863240i \(0.972488\pi\)
\(264\) 0 0
\(265\) −278.156 −1.04965
\(266\) 0 0
\(267\) − 60.4551i − 0.226424i
\(268\) 0 0
\(269\) 202.069 0.751186 0.375593 0.926785i \(-0.377439\pi\)
0.375593 + 0.926785i \(0.377439\pi\)
\(270\) 0 0
\(271\) 279.798i 1.03246i 0.856449 + 0.516232i \(0.172666\pi\)
−0.856449 + 0.516232i \(0.827334\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 26.9898i − 0.0981448i
\(276\) 0 0
\(277\) 135.459 0.489022 0.244511 0.969647i \(-0.421373\pi\)
0.244511 + 0.969647i \(0.421373\pi\)
\(278\) 0 0
\(279\) − 53.6778i − 0.192394i
\(280\) 0 0
\(281\) 20.8530 0.0742101 0.0371050 0.999311i \(-0.488186\pi\)
0.0371050 + 0.999311i \(0.488186\pi\)
\(282\) 0 0
\(283\) 113.743i 0.401920i 0.979599 + 0.200960i \(0.0644061\pi\)
−0.979599 + 0.200960i \(0.935594\pi\)
\(284\) 0 0
\(285\) 35.3394 0.123998
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −288.441 −0.998065
\(290\) 0 0
\(291\) − 90.9405i − 0.312510i
\(292\) 0 0
\(293\) 121.931 0.416147 0.208073 0.978113i \(-0.433281\pi\)
0.208073 + 0.978113i \(0.433281\pi\)
\(294\) 0 0
\(295\) 498.990i 1.69149i
\(296\) 0 0
\(297\) −22.7477 −0.0765917
\(298\) 0 0
\(299\) 169.248i 0.566048i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 196.731i 0.649277i
\(304\) 0 0
\(305\) −675.441 −2.21456
\(306\) 0 0
\(307\) − 337.131i − 1.09815i −0.835775 0.549073i \(-0.814981\pi\)
0.835775 0.549073i \(-0.185019\pi\)
\(308\) 0 0
\(309\) 86.6424 0.280396
\(310\) 0 0
\(311\) 246.762i 0.793447i 0.917938 + 0.396724i \(0.129853\pi\)
−0.917938 + 0.396724i \(0.870147\pi\)
\(312\) 0 0
\(313\) −274.624 −0.877394 −0.438697 0.898635i \(-0.644560\pi\)
−0.438697 + 0.898635i \(0.644560\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −129.441 −0.408331 −0.204165 0.978936i \(-0.565448\pi\)
−0.204165 + 0.978936i \(0.565448\pi\)
\(318\) 0 0
\(319\) 8.75560i 0.0274470i
\(320\) 0 0
\(321\) −278.051 −0.866202
\(322\) 0 0
\(323\) − 2.73279i − 0.00846066i
\(324\) 0 0
\(325\) −105.826 −0.325618
\(326\) 0 0
\(327\) − 355.595i − 1.08745i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 219.573i 0.663363i 0.943391 + 0.331682i \(0.107616\pi\)
−0.943391 + 0.331682i \(0.892384\pi\)
\(332\) 0 0
\(333\) 22.4864 0.0675266
\(334\) 0 0
\(335\) − 130.492i − 0.389527i
\(336\) 0 0
\(337\) −132.955 −0.394524 −0.197262 0.980351i \(-0.563205\pi\)
−0.197262 + 0.980351i \(0.563205\pi\)
\(338\) 0 0
\(339\) 310.593i 0.916205i
\(340\) 0 0
\(341\) 78.3303 0.229708
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −95.3394 −0.276346
\(346\) 0 0
\(347\) 453.282i 1.30629i 0.757234 + 0.653144i \(0.226550\pi\)
−0.757234 + 0.653144i \(0.773450\pi\)
\(348\) 0 0
\(349\) −525.615 −1.50606 −0.753030 0.657986i \(-0.771409\pi\)
−0.753030 + 0.657986i \(0.771409\pi\)
\(350\) 0 0
\(351\) 89.1927i 0.254110i
\(352\) 0 0
\(353\) 389.858 1.10441 0.552207 0.833707i \(-0.313786\pi\)
0.552207 + 0.833707i \(0.313786\pi\)
\(354\) 0 0
\(355\) − 368.720i − 1.03865i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 692.718i − 1.92958i −0.263030 0.964788i \(-0.584722\pi\)
0.263030 0.964788i \(-0.415278\pi\)
\(360\) 0 0
\(361\) 347.642 0.962998
\(362\) 0 0
\(363\) 176.383i 0.485904i
\(364\) 0 0
\(365\) −226.120 −0.619506
\(366\) 0 0
\(367\) − 556.863i − 1.51734i −0.651476 0.758669i \(-0.725850\pi\)
0.651476 0.758669i \(-0.274150\pi\)
\(368\) 0 0
\(369\) 229.720 0.622549
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 549.267 1.47256 0.736282 0.676674i \(-0.236580\pi\)
0.736282 + 0.676674i \(0.236580\pi\)
\(374\) 0 0
\(375\) 182.120i 0.485653i
\(376\) 0 0
\(377\) 34.3303 0.0910618
\(378\) 0 0
\(379\) − 232.667i − 0.613897i −0.951726 0.306948i \(-0.900692\pi\)
0.951726 0.306948i \(-0.0993079\pi\)
\(380\) 0 0
\(381\) −226.156 −0.593585
\(382\) 0 0
\(383\) 394.526i 1.03009i 0.857162 + 0.515047i \(0.172226\pi\)
−0.857162 + 0.515047i \(0.827774\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 210.135i 0.542983i
\(388\) 0 0
\(389\) 3.98182 0.0102360 0.00511802 0.999987i \(-0.498371\pi\)
0.00511802 + 0.999987i \(0.498371\pi\)
\(390\) 0 0
\(391\) 7.37259i 0.0188557i
\(392\) 0 0
\(393\) 123.303 0.313748
\(394\) 0 0
\(395\) − 533.409i − 1.35040i
\(396\) 0 0
\(397\) 476.606 1.20052 0.600260 0.799805i \(-0.295064\pi\)
0.600260 + 0.799805i \(0.295064\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −157.441 −0.392621 −0.196310 0.980542i \(-0.562896\pi\)
−0.196310 + 0.980542i \(0.562896\pi\)
\(402\) 0 0
\(403\) − 307.129i − 0.762108i
\(404\) 0 0
\(405\) −50.2432 −0.124057
\(406\) 0 0
\(407\) 32.8136i 0.0806231i
\(408\) 0 0
\(409\) 112.156 0.274220 0.137110 0.990556i \(-0.456219\pi\)
0.137110 + 0.990556i \(0.456219\pi\)
\(410\) 0 0
\(411\) − 172.935i − 0.420766i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 645.626i 1.55573i
\(416\) 0 0
\(417\) −281.285 −0.674544
\(418\) 0 0
\(419\) − 699.065i − 1.66841i −0.551452 0.834207i \(-0.685926\pi\)
0.551452 0.834207i \(-0.314074\pi\)
\(420\) 0 0
\(421\) 198.762 0.472119 0.236060 0.971739i \(-0.424144\pi\)
0.236060 + 0.971739i \(0.424144\pi\)
\(422\) 0 0
\(423\) − 120.370i − 0.284562i
\(424\) 0 0
\(425\) −4.60985 −0.0108467
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −130.156 −0.303394
\(430\) 0 0
\(431\) 513.871i 1.19228i 0.802882 + 0.596138i \(0.203299\pi\)
−0.802882 + 0.596138i \(0.796701\pi\)
\(432\) 0 0
\(433\) 654.900 1.51247 0.756236 0.654299i \(-0.227036\pi\)
0.756236 + 0.654299i \(0.227036\pi\)
\(434\) 0 0
\(435\) 19.3386i 0.0444566i
\(436\) 0 0
\(437\) 36.0364 0.0824631
\(438\) 0 0
\(439\) − 143.141i − 0.326061i −0.986621 0.163031i \(-0.947873\pi\)
0.986621 0.163031i \(-0.0521269\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 46.8690i 0.105799i 0.998600 + 0.0528996i \(0.0168463\pi\)
−0.998600 + 0.0528996i \(0.983154\pi\)
\(444\) 0 0
\(445\) 194.853 0.437872
\(446\) 0 0
\(447\) 507.777i 1.13597i
\(448\) 0 0
\(449\) −292.955 −0.652460 −0.326230 0.945290i \(-0.605778\pi\)
−0.326230 + 0.945290i \(0.605778\pi\)
\(450\) 0 0
\(451\) 335.224i 0.743289i
\(452\) 0 0
\(453\) −208.624 −0.460539
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −424.018 −0.927830 −0.463915 0.885880i \(-0.653556\pi\)
−0.463915 + 0.885880i \(0.653556\pi\)
\(458\) 0 0
\(459\) 3.88530i 0.00846471i
\(460\) 0 0
\(461\) 344.886 0.748125 0.374062 0.927404i \(-0.377965\pi\)
0.374062 + 0.927404i \(0.377965\pi\)
\(462\) 0 0
\(463\) 319.381i 0.689807i 0.938638 + 0.344903i \(0.112088\pi\)
−0.938638 + 0.344903i \(0.887912\pi\)
\(464\) 0 0
\(465\) 173.009 0.372063
\(466\) 0 0
\(467\) − 306.065i − 0.655385i −0.944784 0.327692i \(-0.893729\pi\)
0.944784 0.327692i \(-0.106271\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 284.025i 0.603025i
\(472\) 0 0
\(473\) −306.642 −0.648293
\(474\) 0 0
\(475\) 22.5324i 0.0474366i
\(476\) 0 0
\(477\) −149.477 −0.313370
\(478\) 0 0
\(479\) − 355.769i − 0.742734i −0.928486 0.371367i \(-0.878889\pi\)
0.928486 0.371367i \(-0.121111\pi\)
\(480\) 0 0
\(481\) 128.661 0.267486
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 293.111 0.604352
\(486\) 0 0
\(487\) − 765.313i − 1.57148i −0.618554 0.785742i \(-0.712281\pi\)
0.618554 0.785742i \(-0.287719\pi\)
\(488\) 0 0
\(489\) −392.835 −0.803343
\(490\) 0 0
\(491\) − 975.169i − 1.98609i −0.117749 0.993043i \(-0.537568\pi\)
0.117749 0.993043i \(-0.462432\pi\)
\(492\) 0 0
\(493\) 1.49545 0.00303338
\(494\) 0 0
\(495\) − 73.3182i − 0.148118i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) − 472.358i − 0.946610i −0.880899 0.473305i \(-0.843061\pi\)
0.880899 0.473305i \(-0.156939\pi\)
\(500\) 0 0
\(501\) 392.835 0.784101
\(502\) 0 0
\(503\) 513.165i 1.02021i 0.860113 + 0.510104i \(0.170393\pi\)
−0.860113 + 0.510104i \(0.829607\pi\)
\(504\) 0 0
\(505\) −634.083 −1.25561
\(506\) 0 0
\(507\) 217.619i 0.429229i
\(508\) 0 0
\(509\) −361.546 −0.710307 −0.355153 0.934808i \(-0.615571\pi\)
−0.355153 + 0.934808i \(0.615571\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 18.9909 0.0370193
\(514\) 0 0
\(515\) 279.257i 0.542247i
\(516\) 0 0
\(517\) 175.652 0.339751
\(518\) 0 0
\(519\) 188.658i 0.363504i
\(520\) 0 0
\(521\) −71.4265 −0.137095 −0.0685475 0.997648i \(-0.521836\pi\)
−0.0685475 + 0.997648i \(0.521836\pi\)
\(522\) 0 0
\(523\) 587.246i 1.12284i 0.827531 + 0.561420i \(0.189745\pi\)
−0.827531 + 0.561420i \(0.810255\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 13.3788i − 0.0253867i
\(528\) 0 0
\(529\) 431.780 0.816220
\(530\) 0 0
\(531\) 268.150i 0.504991i
\(532\) 0 0
\(533\) 1314.40 2.46603
\(534\) 0 0
\(535\) − 896.186i − 1.67511i
\(536\) 0 0
\(537\) 158.051 0.294322
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −436.505 −0.806848 −0.403424 0.915013i \(-0.632180\pi\)
−0.403424 + 0.915013i \(0.632180\pi\)
\(542\) 0 0
\(543\) − 234.685i − 0.432201i
\(544\) 0 0
\(545\) 1146.12 2.10297
\(546\) 0 0
\(547\) 25.2188i 0.0461039i 0.999734 + 0.0230519i \(0.00733831\pi\)
−0.999734 + 0.0230519i \(0.992662\pi\)
\(548\) 0 0
\(549\) −362.973 −0.661153
\(550\) 0 0
\(551\) − 7.30960i − 0.0132661i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 72.4758i 0.130587i
\(556\) 0 0
\(557\) 58.1742 0.104442 0.0522210 0.998636i \(-0.483370\pi\)
0.0522210 + 0.998636i \(0.483370\pi\)
\(558\) 0 0
\(559\) 1202.33i 2.15086i
\(560\) 0 0
\(561\) −5.66970 −0.0101064
\(562\) 0 0
\(563\) 565.396i 1.00426i 0.864793 + 0.502128i \(0.167449\pi\)
−0.864793 + 0.502128i \(0.832551\pi\)
\(564\) 0 0
\(565\) −1001.07 −1.77181
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 573.056 1.00713 0.503564 0.863958i \(-0.332022\pi\)
0.503564 + 0.863958i \(0.332022\pi\)
\(570\) 0 0
\(571\) − 685.750i − 1.20096i −0.799639 0.600481i \(-0.794976\pi\)
0.799639 0.600481i \(-0.205024\pi\)
\(572\) 0 0
\(573\) 185.216 0.323239
\(574\) 0 0
\(575\) − 60.7884i − 0.105719i
\(576\) 0 0
\(577\) 278.900 0.483362 0.241681 0.970356i \(-0.422301\pi\)
0.241681 + 0.970356i \(0.422301\pi\)
\(578\) 0 0
\(579\) − 201.792i − 0.348518i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) − 218.127i − 0.374146i
\(584\) 0 0
\(585\) −287.477 −0.491414
\(586\) 0 0
\(587\) − 1169.74i − 1.99274i −0.0851245 0.996370i \(-0.527129\pi\)
0.0851245 0.996370i \(-0.472871\pi\)
\(588\) 0 0
\(589\) −65.3939 −0.111025
\(590\) 0 0
\(591\) 67.2639i 0.113814i
\(592\) 0 0
\(593\) −969.023 −1.63410 −0.817052 0.576564i \(-0.804393\pi\)
−0.817052 + 0.576564i \(0.804393\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −611.615 −1.02448
\(598\) 0 0
\(599\) − 656.488i − 1.09597i −0.836487 0.547987i \(-0.815394\pi\)
0.836487 0.547987i \(-0.184606\pi\)
\(600\) 0 0
\(601\) −312.955 −0.520723 −0.260362 0.965511i \(-0.583842\pi\)
−0.260362 + 0.965511i \(0.583842\pi\)
\(602\) 0 0
\(603\) − 70.1244i − 0.116293i
\(604\) 0 0
\(605\) −568.501 −0.939671
\(606\) 0 0
\(607\) − 113.584i − 0.187124i −0.995613 0.0935618i \(-0.970175\pi\)
0.995613 0.0935618i \(-0.0298253\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 688.721i − 1.12720i
\(612\) 0 0
\(613\) 10.0000 0.0163132 0.00815661 0.999967i \(-0.497404\pi\)
0.00815661 + 0.999967i \(0.497404\pi\)
\(614\) 0 0
\(615\) 740.412i 1.20392i
\(616\) 0 0
\(617\) 984.323 1.59534 0.797668 0.603096i \(-0.206067\pi\)
0.797668 + 0.603096i \(0.206067\pi\)
\(618\) 0 0
\(619\) 266.864i 0.431120i 0.976491 + 0.215560i \(0.0691578\pi\)
−0.976491 + 0.215560i \(0.930842\pi\)
\(620\) 0 0
\(621\) −51.2341 −0.0825026
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −741.120 −1.18579
\(626\) 0 0
\(627\) 27.7128i 0.0441991i
\(628\) 0 0
\(629\) 5.60455 0.00891026
\(630\) 0 0
\(631\) 609.443i 0.965837i 0.875665 + 0.482918i \(0.160423\pi\)
−0.875665 + 0.482918i \(0.839577\pi\)
\(632\) 0 0
\(633\) 93.6333 0.147920
\(634\) 0 0
\(635\) − 728.924i − 1.14791i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) − 198.145i − 0.310087i
\(640\) 0 0
\(641\) 528.120 0.823900 0.411950 0.911207i \(-0.364848\pi\)
0.411950 + 0.911207i \(0.364848\pi\)
\(642\) 0 0
\(643\) 812.666i 1.26387i 0.775023 + 0.631933i \(0.217738\pi\)
−0.775023 + 0.631933i \(0.782262\pi\)
\(644\) 0 0
\(645\) −677.285 −1.05005
\(646\) 0 0
\(647\) − 1153.37i − 1.78265i −0.453369 0.891323i \(-0.649778\pi\)
0.453369 0.891323i \(-0.350222\pi\)
\(648\) 0 0
\(649\) −391.303 −0.602932
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −994.000 −1.52221 −0.761103 0.648632i \(-0.775342\pi\)
−0.761103 + 0.648632i \(0.775342\pi\)
\(654\) 0 0
\(655\) 397.418i 0.606745i
\(656\) 0 0
\(657\) −121.514 −0.184952
\(658\) 0 0
\(659\) − 1223.14i − 1.85605i −0.372516 0.928026i \(-0.621505\pi\)
0.372516 0.928026i \(-0.378495\pi\)
\(660\) 0 0
\(661\) −24.0545 −0.0363911 −0.0181956 0.999834i \(-0.505792\pi\)
−0.0181956 + 0.999834i \(0.505792\pi\)
\(662\) 0 0
\(663\) 22.2306i 0.0335303i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 19.7200i 0.0295652i
\(668\) 0 0
\(669\) −161.670 −0.241659
\(670\) 0 0
\(671\) − 529.674i − 0.789380i
\(672\) 0 0
\(673\) 653.789 0.971455 0.485728 0.874110i \(-0.338555\pi\)
0.485728 + 0.874110i \(0.338555\pi\)
\(674\) 0 0
\(675\) − 32.0351i − 0.0474594i
\(676\) 0 0
\(677\) −366.069 −0.540722 −0.270361 0.962759i \(-0.587143\pi\)
−0.270361 + 0.962759i \(0.587143\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −294.853 −0.432971
\(682\) 0 0
\(683\) 971.912i 1.42300i 0.702684 + 0.711502i \(0.251985\pi\)
−0.702684 + 0.711502i \(0.748015\pi\)
\(684\) 0 0
\(685\) 557.386 0.813703
\(686\) 0 0
\(687\) 394.637i 0.574436i
\(688\) 0 0
\(689\) −855.267 −1.24132
\(690\) 0 0
\(691\) − 1028.20i − 1.48799i −0.668184 0.743996i \(-0.732928\pi\)
0.668184 0.743996i \(-0.267072\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 906.610i − 1.30447i
\(696\) 0 0
\(697\) 57.2561 0.0821464
\(698\) 0 0
\(699\) 655.558i 0.937851i
\(700\) 0 0
\(701\) −276.918 −0.395033 −0.197517 0.980300i \(-0.563288\pi\)
−0.197517 + 0.980300i \(0.563288\pi\)
\(702\) 0 0
\(703\) − 27.3944i − 0.0389679i
\(704\) 0 0
\(705\) 387.964 0.550303
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 51.2485 0.0722828 0.0361414 0.999347i \(-0.488493\pi\)
0.0361414 + 0.999347i \(0.488493\pi\)
\(710\) 0 0
\(711\) − 286.647i − 0.403160i
\(712\) 0 0
\(713\) 176.421 0.247435
\(714\) 0 0
\(715\) − 419.506i − 0.586722i
\(716\) 0 0
\(717\) 355.996 0.496508
\(718\) 0 0
\(719\) 898.093i 1.24909i 0.780990 + 0.624543i \(0.214715\pi\)
−0.780990 + 0.624543i \(0.785285\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) − 70.0929i − 0.0969474i
\(724\) 0 0
\(725\) −12.3303 −0.0170073
\(726\) 0 0
\(727\) 107.100i 0.147318i 0.997283 + 0.0736590i \(0.0234677\pi\)
−0.997283 + 0.0736590i \(0.976532\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) 0 0
\(731\) 52.3744i 0.0716476i
\(732\) 0 0
\(733\) 541.368 0.738565 0.369283 0.929317i \(-0.379603\pi\)
0.369283 + 0.929317i \(0.379603\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 102.330 0.138847
\(738\) 0 0
\(739\) 241.423i 0.326688i 0.986569 + 0.163344i \(0.0522281\pi\)
−0.986569 + 0.163344i \(0.947772\pi\)
\(740\) 0 0
\(741\) 108.661 0.146640
\(742\) 0 0
\(743\) − 593.753i − 0.799129i −0.916705 0.399564i \(-0.869161\pi\)
0.916705 0.399564i \(-0.130839\pi\)
\(744\) 0 0
\(745\) −1636.62 −2.19680
\(746\) 0 0
\(747\) 346.951i 0.464459i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) − 918.781i − 1.22341i −0.791086 0.611705i \(-0.790484\pi\)
0.791086 0.611705i \(-0.209516\pi\)
\(752\) 0 0
\(753\) 576.523 0.765634
\(754\) 0 0
\(755\) − 672.417i − 0.890619i
\(756\) 0 0
\(757\) −1407.80 −1.85971 −0.929855 0.367927i \(-0.880068\pi\)
−0.929855 + 0.367927i \(0.880068\pi\)
\(758\) 0 0
\(759\) − 74.7642i − 0.0985036i
\(760\) 0 0
\(761\) 295.042 0.387703 0.193851 0.981031i \(-0.437902\pi\)
0.193851 + 0.981031i \(0.437902\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −12.5227 −0.0163696
\(766\) 0 0
\(767\) 1534.28i 2.00037i
\(768\) 0 0
\(769\) −116.991 −0.152134 −0.0760669 0.997103i \(-0.524236\pi\)
−0.0760669 + 0.997103i \(0.524236\pi\)
\(770\) 0 0
\(771\) 355.715i 0.461368i
\(772\) 0 0
\(773\) −105.408 −0.136363 −0.0681813 0.997673i \(-0.521720\pi\)
−0.0681813 + 0.997673i \(0.521720\pi\)
\(774\) 0 0
\(775\) 110.311i 0.142336i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 279.861i − 0.359257i
\(780\) 0 0
\(781\) 289.147 0.370227
\(782\) 0 0
\(783\) 10.3923i 0.0132724i
\(784\) 0 0
\(785\) −915.441 −1.16617
\(786\) 0 0
\(787\) 542.148i 0.688879i 0.938809 + 0.344439i \(0.111931\pi\)
−0.938809 + 0.344439i \(0.888069\pi\)
\(788\) 0 0
\(789\) −78.6462 −0.0996783
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −2076.83 −2.61895
\(794\) 0 0
\(795\) − 481.780i − 0.606013i
\(796\) 0 0
\(797\) 296.886 0.372504 0.186252 0.982502i \(-0.440366\pi\)
0.186252 + 0.982502i \(0.440366\pi\)
\(798\) 0 0
\(799\) − 30.0012i − 0.0375485i
\(800\) 0 0
\(801\) 104.711 0.130726
\(802\) 0 0
\(803\) − 177.321i − 0.220823i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 349.994i 0.433697i
\(808\) 0 0
\(809\) 1066.83 1.31870 0.659349 0.751837i \(-0.270832\pi\)
0.659349 + 0.751837i \(0.270832\pi\)
\(810\) 0 0
\(811\) − 9.82020i − 0.0121088i −0.999982 0.00605438i \(-0.998073\pi\)
0.999982 0.00605438i \(-0.00192718\pi\)
\(812\) 0 0
\(813\) −484.624 −0.596094
\(814\) 0 0
\(815\) − 1266.15i − 1.55355i
\(816\) 0 0
\(817\) 256.000 0.313341
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −928.395 −1.13081 −0.565405 0.824813i \(-0.691280\pi\)
−0.565405 + 0.824813i \(0.691280\pi\)
\(822\) 0 0
\(823\) 396.291i 0.481520i 0.970585 + 0.240760i \(0.0773966\pi\)
−0.970585 + 0.240760i \(0.922603\pi\)
\(824\) 0 0
\(825\) 46.7477 0.0566639
\(826\) 0 0
\(827\) 1094.28i 1.32320i 0.749858 + 0.661599i \(0.230122\pi\)
−0.749858 + 0.661599i \(0.769878\pi\)
\(828\) 0 0
\(829\) 107.074 0.129161 0.0645804 0.997913i \(-0.479429\pi\)
0.0645804 + 0.997913i \(0.479429\pi\)
\(830\) 0 0
\(831\) 234.622i 0.282337i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 1266.15i 1.51634i
\(836\) 0 0
\(837\) 92.9727 0.111079
\(838\) 0 0
\(839\) − 1110.28i − 1.32333i −0.749798 0.661667i \(-0.769849\pi\)
0.749798 0.661667i \(-0.230151\pi\)
\(840\) 0 0
\(841\) −837.000 −0.995244
\(842\) 0 0
\(843\) 36.1185i 0.0428452i
\(844\) 0 0
\(845\) −701.408 −0.830069
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −197.009 −0.232048
\(850\) 0 0
\(851\) 73.9052i 0.0868451i
\(852\) 0 0
\(853\) −657.506 −0.770816 −0.385408 0.922746i \(-0.625939\pi\)
−0.385408 + 0.922746i \(0.625939\pi\)
\(854\) 0 0
\(855\) 61.2096i 0.0715902i
\(856\) 0 0
\(857\) −664.958 −0.775914 −0.387957 0.921677i \(-0.626819\pi\)
−0.387957 + 0.921677i \(0.626819\pi\)
\(858\) 0 0
\(859\) − 1071.96i − 1.24792i −0.781456 0.623961i \(-0.785523\pi\)
0.781456 0.623961i \(-0.214477\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 612.073i − 0.709239i −0.935011 0.354619i \(-0.884610\pi\)
0.935011 0.354619i \(-0.115390\pi\)
\(864\) 0 0
\(865\) −608.065 −0.702965
\(866\) 0 0
\(867\) − 499.594i − 0.576233i
\(868\) 0 0
\(869\) 418.294 0.481351
\(870\) 0 0
\(871\) − 401.232i − 0.460657i
\(872\) 0 0
\(873\) 157.514 0.180428
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1385.19 −1.57947 −0.789734 0.613449i \(-0.789782\pi\)
−0.789734 + 0.613449i \(0.789782\pi\)
\(878\) 0 0
\(879\) 211.191i 0.240263i
\(880\) 0 0
\(881\) 623.325 0.707520 0.353760 0.935336i \(-0.384903\pi\)
0.353760 + 0.935336i \(0.384903\pi\)
\(882\) 0 0
\(883\) − 636.791i − 0.721168i −0.932727 0.360584i \(-0.882577\pi\)
0.932727 0.360584i \(-0.117423\pi\)
\(884\) 0 0
\(885\) −864.276 −0.976583
\(886\) 0 0
\(887\) − 282.866i − 0.318902i −0.987206 0.159451i \(-0.949028\pi\)
0.987206 0.159451i \(-0.0509723\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) − 39.4002i − 0.0442202i
\(892\) 0 0
\(893\) −146.642 −0.164213
\(894\) 0 0
\(895\) 509.414i 0.569177i
\(896\) 0 0
\(897\) −293.147 −0.326808
\(898\) 0 0
\(899\) − 35.7852i − 0.0398056i
\(900\) 0 0
\(901\) −37.2561 −0.0413497
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 756.414 0.835816
\(906\) 0 0
\(907\) 1419.65i 1.56521i 0.622517 + 0.782607i \(0.286110\pi\)
−0.622517 + 0.782607i \(0.713890\pi\)
\(908\) 0 0
\(909\) −340.748 −0.374860
\(910\) 0 0
\(911\) 1170.46i 1.28481i 0.766365 + 0.642405i \(0.222063\pi\)
−0.766365 + 0.642405i \(0.777937\pi\)
\(912\) 0 0
\(913\) −506.294 −0.554539
\(914\) 0 0
\(915\) − 1169.90i − 1.27858i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) − 944.239i − 1.02746i −0.857951 0.513732i \(-0.828263\pi\)
0.857951 0.513732i \(-0.171737\pi\)
\(920\) 0 0
\(921\) 583.927 0.634014
\(922\) 0 0
\(923\) − 1133.73i − 1.22831i
\(924\) 0 0
\(925\) −46.2106 −0.0499574
\(926\) 0 0
\(927\) 150.069i 0.161887i
\(928\) 0 0
\(929\) −1253.20 −1.34897 −0.674487 0.738286i \(-0.735635\pi\)
−0.674487 + 0.738286i \(0.735635\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −427.405 −0.458097
\(934\) 0 0
\(935\) − 18.2740i − 0.0195444i
\(936\) 0 0
\(937\) −470.936 −0.502600 −0.251300 0.967909i \(-0.580858\pi\)
−0.251300 + 0.967909i \(0.580858\pi\)
\(938\) 0 0
\(939\) − 475.663i − 0.506564i
\(940\) 0 0
\(941\) 816.951 0.868173 0.434086 0.900871i \(-0.357071\pi\)
0.434086 + 0.900871i \(0.357071\pi\)
\(942\) 0 0
\(943\) 755.015i 0.800652i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 994.889i − 1.05057i −0.850927 0.525284i \(-0.823959\pi\)
0.850927 0.525284i \(-0.176041\pi\)
\(948\) 0 0
\(949\) −695.267 −0.732631
\(950\) 0 0
\(951\) − 224.198i − 0.235750i
\(952\) 0 0
\(953\) −1293.85 −1.35766 −0.678832 0.734293i \(-0.737514\pi\)
−0.678832 + 0.734293i \(0.737514\pi\)
\(954\) 0 0
\(955\) 596.970i 0.625099i
\(956\) 0 0
\(957\) −15.1652 −0.0158466
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 640.855 0.666862
\(962\) 0 0
\(963\) − 481.598i − 0.500102i
\(964\) 0 0
\(965\) 650.395 0.673985
\(966\) 0 0
\(967\) 1071.60i 1.10817i 0.832460 + 0.554085i \(0.186932\pi\)
−0.832460 + 0.554085i \(0.813068\pi\)
\(968\) 0 0
\(969\) 4.73334 0.00488477
\(970\) 0 0
\(971\) 1433.12i 1.47593i 0.674842 + 0.737963i \(0.264212\pi\)
−0.674842 + 0.737963i \(0.735788\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) − 183.296i − 0.187995i
\(976\) 0 0
\(977\) −126.559 −0.129538 −0.0647692 0.997900i \(-0.520631\pi\)
−0.0647692 + 0.997900i \(0.520631\pi\)
\(978\) 0 0
\(979\) 152.802i 0.156080i
\(980\) 0 0
\(981\) 615.909 0.627838
\(982\) 0 0
\(983\) 18.1480i 0.0184619i 0.999957 + 0.00923094i \(0.00293834\pi\)
−0.999957 + 0.00923094i \(0.997062\pi\)
\(984\) 0 0
\(985\) −216.798 −0.220100
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −690.642 −0.698324
\(990\) 0 0
\(991\) 643.338i 0.649181i 0.945855 + 0.324590i \(0.105226\pi\)
−0.945855 + 0.324590i \(0.894774\pi\)
\(992\) 0 0
\(993\) −380.312 −0.382993
\(994\) 0 0
\(995\) − 1971.30i − 1.98120i
\(996\) 0 0
\(997\) −631.358 −0.633257 −0.316629 0.948550i \(-0.602551\pi\)
−0.316629 + 0.948550i \(0.602551\pi\)
\(998\) 0 0
\(999\) 38.9475i 0.0389865i
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 2352.3.m.f.1471.3 4
4.3 odd 2 inner 2352.3.m.f.1471.1 4
7.6 odd 2 336.3.m.c.127.2 4
21.20 even 2 1008.3.m.b.127.1 4
28.27 even 2 336.3.m.c.127.4 yes 4
56.13 odd 2 1344.3.m.a.127.3 4
56.27 even 2 1344.3.m.a.127.1 4
84.83 odd 2 1008.3.m.b.127.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.3.m.c.127.2 4 7.6 odd 2
336.3.m.c.127.4 yes 4 28.27 even 2
1008.3.m.b.127.1 4 21.20 even 2
1008.3.m.b.127.2 4 84.83 odd 2
1344.3.m.a.127.1 4 56.27 even 2
1344.3.m.a.127.3 4 56.13 odd 2
2352.3.m.f.1471.1 4 4.3 odd 2 inner
2352.3.m.f.1471.3 4 1.1 even 1 trivial