Properties

Label 1323.2.c.d.1322.11
Level $1323$
Weight $2$
Character 1323.1322
Analytic conductor $10.564$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.5642081874\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 9 x^{10} + 59 x^{8} - 180 x^{6} + 403 x^{4} - 198 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 189)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1322.11
Root \(0.617942 + 0.356769i\) of defining polynomial
Character \(\chi\) \(=\) 1323.1322
Dual form 1323.2.c.d.1322.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.49086i q^{2} -4.20440 q^{4} -1.23588 q^{5} -5.49086i q^{8} +O(q^{10})\) \(q+2.49086i q^{2} -4.20440 q^{4} -1.23588 q^{5} -5.49086i q^{8} -3.07842i q^{10} +5.49086i q^{11} +2.96793i q^{13} +5.26819 q^{16} -4.31430 q^{17} -5.55019i q^{19} +5.19615 q^{20} -13.6770 q^{22} +1.63148i q^{23} -3.47259 q^{25} -7.39272 q^{26} -0.509136i q^{29} -8.13244i q^{31} +2.14061i q^{32} -10.7463i q^{34} -3.06379 q^{37} +13.8248 q^{38} +6.78607i q^{40} -0.354034 q^{41} -0.0637877 q^{43} -23.0858i q^{44} -4.06379 q^{46} +9.86449 q^{47} -8.64975i q^{50} -12.4784i q^{52} -4.12234i q^{53} -6.78607i q^{55} +1.26819 q^{58} +3.07842 q^{59} -6.89655i q^{61} +20.2568 q^{62} +5.20440 q^{64} -3.66802i q^{65} -12.3320 q^{67} +18.1391 q^{68} +4.63148i q^{71} +7.03869i q^{73} -7.63148i q^{74} +23.3352i q^{76} -0.331977 q^{79} -6.51087 q^{80} -0.881850i q^{82} -7.39272 q^{83} +5.33198 q^{85} -0.158887i q^{86} +30.1496 q^{88} -3.43245 q^{89} -6.85939i q^{92} +24.5711i q^{94} +6.85939i q^{95} -4.45644i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 16q^{4} + O(q^{10}) \) \( 12q - 16q^{4} + 8q^{16} - 40q^{22} + 48q^{25} - 16q^{37} + 20q^{43} - 28q^{46} - 40q^{58} + 28q^{64} - 72q^{67} + 72q^{79} - 12q^{85} + 148q^{88} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.49086i 1.76131i 0.473761 + 0.880653i \(0.342896\pi\)
−0.473761 + 0.880653i \(0.657104\pi\)
\(3\) 0 0
\(4\) −4.20440 −2.10220
\(5\) −1.23588 −0.552704 −0.276352 0.961056i \(-0.589126\pi\)
−0.276352 + 0.961056i \(0.589126\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) − 5.49086i − 1.94131i
\(9\) 0 0
\(10\) − 3.07842i − 0.973481i
\(11\) 5.49086i 1.65556i 0.561055 + 0.827779i \(0.310396\pi\)
−0.561055 + 0.827779i \(0.689604\pi\)
\(12\) 0 0
\(13\) 2.96793i 0.823157i 0.911374 + 0.411578i \(0.135022\pi\)
−0.911374 + 0.411578i \(0.864978\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 5.26819 1.31705
\(17\) −4.31430 −1.04637 −0.523186 0.852219i \(-0.675257\pi\)
−0.523186 + 0.852219i \(0.675257\pi\)
\(18\) 0 0
\(19\) − 5.55019i − 1.27330i −0.771153 0.636650i \(-0.780320\pi\)
0.771153 0.636650i \(-0.219680\pi\)
\(20\) 5.19615 1.16190
\(21\) 0 0
\(22\) −13.6770 −2.91594
\(23\) 1.63148i 0.340187i 0.985428 + 0.170093i \(0.0544069\pi\)
−0.985428 + 0.170093i \(0.945593\pi\)
\(24\) 0 0
\(25\) −3.47259 −0.694518
\(26\) −7.39272 −1.44983
\(27\) 0 0
\(28\) 0 0
\(29\) − 0.509136i − 0.0945443i −0.998882 0.0472721i \(-0.984947\pi\)
0.998882 0.0472721i \(-0.0150528\pi\)
\(30\) 0 0
\(31\) − 8.13244i − 1.46063i −0.683111 0.730314i \(-0.739374\pi\)
0.683111 0.730314i \(-0.260626\pi\)
\(32\) 2.14061i 0.378411i
\(33\) 0 0
\(34\) − 10.7463i − 1.84298i
\(35\) 0 0
\(36\) 0 0
\(37\) −3.06379 −0.503684 −0.251842 0.967768i \(-0.581036\pi\)
−0.251842 + 0.967768i \(0.581036\pi\)
\(38\) 13.8248 2.24267
\(39\) 0 0
\(40\) 6.78607i 1.07297i
\(41\) −0.354034 −0.0552908 −0.0276454 0.999618i \(-0.508801\pi\)
−0.0276454 + 0.999618i \(0.508801\pi\)
\(42\) 0 0
\(43\) −0.0637877 −0.00972754 −0.00486377 0.999988i \(-0.501548\pi\)
−0.00486377 + 0.999988i \(0.501548\pi\)
\(44\) − 23.0858i − 3.48031i
\(45\) 0 0
\(46\) −4.06379 −0.599173
\(47\) 9.86449 1.43888 0.719442 0.694553i \(-0.244398\pi\)
0.719442 + 0.694553i \(0.244398\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) − 8.64975i − 1.22326i
\(51\) 0 0
\(52\) − 12.4784i − 1.73044i
\(53\) − 4.12234i − 0.566247i −0.959083 0.283124i \(-0.908629\pi\)
0.959083 0.283124i \(-0.0913706\pi\)
\(54\) 0 0
\(55\) − 6.78607i − 0.915034i
\(56\) 0 0
\(57\) 0 0
\(58\) 1.26819 0.166521
\(59\) 3.07842 0.400776 0.200388 0.979717i \(-0.435780\pi\)
0.200388 + 0.979717i \(0.435780\pi\)
\(60\) 0 0
\(61\) − 6.89655i − 0.883013i −0.897258 0.441507i \(-0.854444\pi\)
0.897258 0.441507i \(-0.145556\pi\)
\(62\) 20.2568 2.57262
\(63\) 0 0
\(64\) 5.20440 0.650550
\(65\) − 3.66802i − 0.454962i
\(66\) 0 0
\(67\) −12.3320 −1.50659 −0.753295 0.657682i \(-0.771537\pi\)
−0.753295 + 0.657682i \(0.771537\pi\)
\(68\) 18.1391 2.19968
\(69\) 0 0
\(70\) 0 0
\(71\) 4.63148i 0.549655i 0.961494 + 0.274828i \(0.0886208\pi\)
−0.961494 + 0.274828i \(0.911379\pi\)
\(72\) 0 0
\(73\) 7.03869i 0.823816i 0.911225 + 0.411908i \(0.135137\pi\)
−0.911225 + 0.411908i \(0.864863\pi\)
\(74\) − 7.63148i − 0.887141i
\(75\) 0 0
\(76\) 23.3352i 2.67673i
\(77\) 0 0
\(78\) 0 0
\(79\) −0.331977 −0.0373503 −0.0186752 0.999826i \(-0.505945\pi\)
−0.0186752 + 0.999826i \(0.505945\pi\)
\(80\) −6.51087 −0.727937
\(81\) 0 0
\(82\) − 0.881850i − 0.0973841i
\(83\) −7.39272 −0.811457 −0.405728 0.913994i \(-0.632982\pi\)
−0.405728 + 0.913994i \(0.632982\pi\)
\(84\) 0 0
\(85\) 5.33198 0.578334
\(86\) − 0.158887i − 0.0171332i
\(87\) 0 0
\(88\) 30.1496 3.21396
\(89\) −3.43245 −0.363839 −0.181920 0.983313i \(-0.558231\pi\)
−0.181920 + 0.983313i \(0.558231\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 6.85939i − 0.715140i
\(93\) 0 0
\(94\) 24.5711i 2.53432i
\(95\) 6.85939i 0.703758i
\(96\) 0 0
\(97\) − 4.45644i − 0.452482i −0.974071 0.226241i \(-0.927356\pi\)
0.974071 0.226241i \(-0.0726438\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 14.6002 1.46002
\(101\) −9.15642 −0.911098 −0.455549 0.890211i \(-0.650557\pi\)
−0.455549 + 0.890211i \(0.650557\pi\)
\(102\) 0 0
\(103\) − 4.20382i − 0.414215i −0.978318 0.207107i \(-0.933595\pi\)
0.978318 0.207107i \(-0.0664049\pi\)
\(104\) 16.2965 1.59801
\(105\) 0 0
\(106\) 10.2682 0.997335
\(107\) − 10.3137i − 0.997063i −0.866871 0.498532i \(-0.833873\pi\)
0.866871 0.498532i \(-0.166127\pi\)
\(108\) 0 0
\(109\) 0.127575 0.0122195 0.00610976 0.999981i \(-0.498055\pi\)
0.00610976 + 0.999981i \(0.498055\pi\)
\(110\) 16.9032 1.61165
\(111\) 0 0
\(112\) 0 0
\(113\) 2.29950i 0.216319i 0.994134 + 0.108159i \(0.0344957\pi\)
−0.994134 + 0.108159i \(0.965504\pi\)
\(114\) 0 0
\(115\) − 2.01632i − 0.188023i
\(116\) 2.14061i 0.198751i
\(117\) 0 0
\(118\) 7.66792i 0.705889i
\(119\) 0 0
\(120\) 0 0
\(121\) −19.1496 −1.74087
\(122\) 17.1784 1.55526
\(123\) 0 0
\(124\) 34.1920i 3.07054i
\(125\) 10.4711 0.936567
\(126\) 0 0
\(127\) −0.386795 −0.0343225 −0.0171613 0.999853i \(-0.505463\pi\)
−0.0171613 + 0.999853i \(0.505463\pi\)
\(128\) 17.2447i 1.52423i
\(129\) 0 0
\(130\) 9.13654 0.801328
\(131\) −20.8634 −1.82285 −0.911424 0.411469i \(-0.865016\pi\)
−0.911424 + 0.411469i \(0.865016\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) − 30.7173i − 2.65357i
\(135\) 0 0
\(136\) 23.6892i 2.03134i
\(137\) − 0.700500i − 0.0598477i −0.999552 0.0299239i \(-0.990474\pi\)
0.999552 0.0299239i \(-0.00952648\pi\)
\(138\) 0 0
\(139\) − 11.5965i − 0.983606i −0.870707 0.491803i \(-0.836338\pi\)
0.870707 0.491803i \(-0.163662\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −11.5364 −0.968111
\(143\) −16.2965 −1.36278
\(144\) 0 0
\(145\) 0.629233i 0.0522550i
\(146\) −17.5324 −1.45099
\(147\) 0 0
\(148\) 12.8814 1.05884
\(149\) 0.350250i 0.0286936i 0.999897 + 0.0143468i \(0.00456688\pi\)
−0.999897 + 0.0143468i \(0.995433\pi\)
\(150\) 0 0
\(151\) 4.65498 0.378817 0.189409 0.981898i \(-0.439343\pi\)
0.189409 + 0.981898i \(0.439343\pi\)
\(152\) −30.4753 −2.47187
\(153\) 0 0
\(154\) 0 0
\(155\) 10.0507i 0.807296i
\(156\) 0 0
\(157\) 1.23588i 0.0986343i 0.998783 + 0.0493171i \(0.0157045\pi\)
−0.998783 + 0.0493171i \(0.984295\pi\)
\(158\) − 0.826910i − 0.0657854i
\(159\) 0 0
\(160\) − 2.64555i − 0.209149i
\(161\) 0 0
\(162\) 0 0
\(163\) −19.6132 −1.53622 −0.768112 0.640315i \(-0.778804\pi\)
−0.768112 + 0.640315i \(0.778804\pi\)
\(164\) 1.48850 0.116232
\(165\) 0 0
\(166\) − 18.4143i − 1.42922i
\(167\) −23.6892 −1.83313 −0.916564 0.399887i \(-0.869049\pi\)
−0.916564 + 0.399887i \(0.869049\pi\)
\(168\) 0 0
\(169\) 4.19136 0.322413
\(170\) 13.2812i 1.01862i
\(171\) 0 0
\(172\) 0.268189 0.0204492
\(173\) −11.9822 −0.910992 −0.455496 0.890238i \(-0.650538\pi\)
−0.455496 + 0.890238i \(0.650538\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 28.9269i 2.18045i
\(177\) 0 0
\(178\) − 8.54977i − 0.640832i
\(179\) 22.3853i 1.67316i 0.547848 + 0.836578i \(0.315447\pi\)
−0.547848 + 0.836578i \(0.684553\pi\)
\(180\) 0 0
\(181\) 8.90380i 0.661815i 0.943663 + 0.330907i \(0.107355\pi\)
−0.943663 + 0.330907i \(0.892645\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 8.95822 0.660409
\(185\) 3.78649 0.278388
\(186\) 0 0
\(187\) − 23.6892i − 1.73233i
\(188\) −41.4743 −3.02482
\(189\) 0 0
\(190\) −17.0858 −1.23953
\(191\) 11.4909i 0.831450i 0.909490 + 0.415725i \(0.136472\pi\)
−0.909490 + 0.415725i \(0.863528\pi\)
\(192\) 0 0
\(193\) 11.0220 0.793381 0.396691 0.917952i \(-0.370159\pi\)
0.396691 + 0.917952i \(0.370159\pi\)
\(194\) 11.1004 0.796960
\(195\) 0 0
\(196\) 0 0
\(197\) 19.8228i 1.41232i 0.708053 + 0.706159i \(0.249574\pi\)
−0.708053 + 0.706159i \(0.750426\pi\)
\(198\) 0 0
\(199\) 8.66025i 0.613909i 0.951724 + 0.306955i \(0.0993100\pi\)
−0.951724 + 0.306955i \(0.900690\pi\)
\(200\) 19.0675i 1.34828i
\(201\) 0 0
\(202\) − 22.8074i − 1.60472i
\(203\) 0 0
\(204\) 0 0
\(205\) 0.437545 0.0305595
\(206\) 10.4711 0.729559
\(207\) 0 0
\(208\) 15.6356i 1.08414i
\(209\) 30.4753 2.10802
\(210\) 0 0
\(211\) 22.1626 1.52574 0.762869 0.646553i \(-0.223790\pi\)
0.762869 + 0.646553i \(0.223790\pi\)
\(212\) 17.3320i 1.19037i
\(213\) 0 0
\(214\) 25.6900 1.75613
\(215\) 0.0788343 0.00537645
\(216\) 0 0
\(217\) 0 0
\(218\) 0.317773i 0.0215223i
\(219\) 0 0
\(220\) 28.5314i 1.92358i
\(221\) − 12.8046i − 0.861328i
\(222\) 0 0
\(223\) 18.7457i 1.25531i 0.778493 + 0.627653i \(0.215984\pi\)
−0.778493 + 0.627653i \(0.784016\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −5.72774 −0.381004
\(227\) 14.1788 0.941079 0.470540 0.882379i \(-0.344059\pi\)
0.470540 + 0.882379i \(0.344059\pi\)
\(228\) 0 0
\(229\) − 4.07075i − 0.269003i −0.990913 0.134501i \(-0.957057\pi\)
0.990913 0.134501i \(-0.0429433\pi\)
\(230\) 5.02237 0.331165
\(231\) 0 0
\(232\) −2.79560 −0.183540
\(233\) 6.96345i 0.456191i 0.973639 + 0.228096i \(0.0732499\pi\)
−0.973639 + 0.228096i \(0.926750\pi\)
\(234\) 0 0
\(235\) −12.1914 −0.795277
\(236\) −12.9429 −0.842511
\(237\) 0 0
\(238\) 0 0
\(239\) − 26.5401i − 1.71674i −0.513033 0.858369i \(-0.671478\pi\)
0.513033 0.858369i \(-0.328522\pi\)
\(240\) 0 0
\(241\) − 13.6038i − 0.876297i −0.898903 0.438149i \(-0.855634\pi\)
0.898903 0.438149i \(-0.144366\pi\)
\(242\) − 47.6990i − 3.06621i
\(243\) 0 0
\(244\) 28.9959i 1.85627i
\(245\) 0 0
\(246\) 0 0
\(247\) 16.4726 1.04813
\(248\) −44.6541 −2.83554
\(249\) 0 0
\(250\) 26.0822i 1.64958i
\(251\) 2.55060 0.160993 0.0804963 0.996755i \(-0.474349\pi\)
0.0804963 + 0.996755i \(0.474349\pi\)
\(252\) 0 0
\(253\) −8.95822 −0.563198
\(254\) − 0.963454i − 0.0604525i
\(255\) 0 0
\(256\) −32.5453 −2.03408
\(257\) 18.0602 1.12657 0.563283 0.826264i \(-0.309538\pi\)
0.563283 + 0.826264i \(0.309538\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 15.4218i 0.956422i
\(261\) 0 0
\(262\) − 51.9680i − 3.21059i
\(263\) 8.06902i 0.497557i 0.968560 + 0.248779i \(0.0800292\pi\)
−0.968560 + 0.248779i \(0.919971\pi\)
\(264\) 0 0
\(265\) 5.09474i 0.312967i
\(266\) 0 0
\(267\) 0 0
\(268\) 51.8486 3.16716
\(269\) −2.09515 −0.127744 −0.0638718 0.997958i \(-0.520345\pi\)
−0.0638718 + 0.997958i \(0.520345\pi\)
\(270\) 0 0
\(271\) 24.6027i 1.49451i 0.664537 + 0.747255i \(0.268629\pi\)
−0.664537 + 0.747255i \(0.731371\pi\)
\(272\) −22.7286 −1.37812
\(273\) 0 0
\(274\) 1.74485 0.105410
\(275\) − 19.0675i − 1.14981i
\(276\) 0 0
\(277\) −21.6990 −1.30377 −0.651883 0.758319i \(-0.726021\pi\)
−0.651883 + 0.758319i \(0.726021\pi\)
\(278\) 28.8854 1.73243
\(279\) 0 0
\(280\) 0 0
\(281\) 22.6315i 1.35008i 0.737781 + 0.675040i \(0.235874\pi\)
−0.737781 + 0.675040i \(0.764126\pi\)
\(282\) 0 0
\(283\) 9.54210i 0.567219i 0.958940 + 0.283610i \(0.0915320\pi\)
−0.958940 + 0.283610i \(0.908468\pi\)
\(284\) − 19.4726i − 1.15549i
\(285\) 0 0
\(286\) − 40.5924i − 2.40028i
\(287\) 0 0
\(288\) 0 0
\(289\) 1.61320 0.0948944
\(290\) −1.56733 −0.0920371
\(291\) 0 0
\(292\) − 29.5935i − 1.73183i
\(293\) −26.3348 −1.53850 −0.769248 0.638951i \(-0.779369\pi\)
−0.769248 + 0.638951i \(0.779369\pi\)
\(294\) 0 0
\(295\) −3.80457 −0.221511
\(296\) 16.8228i 0.977808i
\(297\) 0 0
\(298\) −0.872425 −0.0505382
\(299\) −4.84212 −0.280027
\(300\) 0 0
\(301\) 0 0
\(302\) 11.5949i 0.667213i
\(303\) 0 0
\(304\) − 29.2394i − 1.67700i
\(305\) 8.52334i 0.488045i
\(306\) 0 0
\(307\) − 6.81772i − 0.389108i −0.980892 0.194554i \(-0.937674\pi\)
0.980892 0.194554i \(-0.0623259\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −25.0350 −1.42190
\(311\) −26.9414 −1.52771 −0.763855 0.645388i \(-0.776696\pi\)
−0.763855 + 0.645388i \(0.776696\pi\)
\(312\) 0 0
\(313\) 23.1614i 1.30916i 0.755992 + 0.654581i \(0.227155\pi\)
−0.755992 + 0.654581i \(0.772845\pi\)
\(314\) −3.07842 −0.173725
\(315\) 0 0
\(316\) 1.39576 0.0785179
\(317\) − 25.0675i − 1.40793i −0.710234 0.703966i \(-0.751411\pi\)
0.710234 0.703966i \(-0.248589\pi\)
\(318\) 0 0
\(319\) 2.79560 0.156523
\(320\) −6.43204 −0.359562
\(321\) 0 0
\(322\) 0 0
\(323\) 23.9452i 1.33235i
\(324\) 0 0
\(325\) − 10.3064i − 0.571697i
\(326\) − 48.8538i − 2.70576i
\(327\) 0 0
\(328\) 1.94395i 0.107337i
\(329\) 0 0
\(330\) 0 0
\(331\) −8.39576 −0.461473 −0.230736 0.973016i \(-0.574114\pi\)
−0.230736 + 0.973016i \(0.574114\pi\)
\(332\) 31.0820 1.70584
\(333\) 0 0
\(334\) − 59.0067i − 3.22870i
\(335\) 15.2409 0.832699
\(336\) 0 0
\(337\) −28.2902 −1.54107 −0.770533 0.637401i \(-0.780010\pi\)
−0.770533 + 0.637401i \(0.780010\pi\)
\(338\) 10.4401i 0.567867i
\(339\) 0 0
\(340\) −22.4178 −1.21577
\(341\) 44.6541 2.41816
\(342\) 0 0
\(343\) 0 0
\(344\) 0.350250i 0.0188842i
\(345\) 0 0
\(346\) − 29.8461i − 1.60454i
\(347\) 9.96345i 0.534866i 0.963576 + 0.267433i \(0.0861754\pi\)
−0.963576 + 0.267433i \(0.913825\pi\)
\(348\) 0 0
\(349\) − 22.4624i − 1.20239i −0.799104 0.601193i \(-0.794692\pi\)
0.799104 0.601193i \(-0.205308\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −11.7538 −0.626481
\(353\) −32.0652 −1.70666 −0.853330 0.521371i \(-0.825421\pi\)
−0.853330 + 0.521371i \(0.825421\pi\)
\(354\) 0 0
\(355\) − 5.72397i − 0.303797i
\(356\) 14.4314 0.764863
\(357\) 0 0
\(358\) −55.7587 −2.94694
\(359\) 11.8736i 0.626664i 0.949644 + 0.313332i \(0.101445\pi\)
−0.949644 + 0.313332i \(0.898555\pi\)
\(360\) 0 0
\(361\) −11.8046 −0.621293
\(362\) −22.1782 −1.16566
\(363\) 0 0
\(364\) 0 0
\(365\) − 8.69900i − 0.455326i
\(366\) 0 0
\(367\) 13.5721i 0.708460i 0.935158 + 0.354230i \(0.115257\pi\)
−0.935158 + 0.354230i \(0.884743\pi\)
\(368\) 8.59493i 0.448042i
\(369\) 0 0
\(370\) 9.43162i 0.490327i
\(371\) 0 0
\(372\) 0 0
\(373\) 30.1716 1.56223 0.781113 0.624390i \(-0.214652\pi\)
0.781113 + 0.624390i \(0.214652\pi\)
\(374\) 59.0067 3.05116
\(375\) 0 0
\(376\) − 54.1646i − 2.79332i
\(377\) 1.51108 0.0778248
\(378\) 0 0
\(379\) −12.6770 −0.651173 −0.325587 0.945512i \(-0.605562\pi\)
−0.325587 + 0.945512i \(0.605562\pi\)
\(380\) − 28.8396i − 1.47944i
\(381\) 0 0
\(382\) −28.6222 −1.46444
\(383\) −31.9864 −1.63443 −0.817214 0.576334i \(-0.804483\pi\)
−0.817214 + 0.576334i \(0.804483\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 27.4543i 1.39739i
\(387\) 0 0
\(388\) 18.7366i 0.951209i
\(389\) 7.57666i 0.384152i 0.981380 + 0.192076i \(0.0615220\pi\)
−0.981380 + 0.192076i \(0.938478\pi\)
\(390\) 0 0
\(391\) − 7.03869i − 0.355962i
\(392\) 0 0
\(393\) 0 0
\(394\) −49.3760 −2.48753
\(395\) 0.410285 0.0206437
\(396\) 0 0
\(397\) − 0.951618i − 0.0477603i −0.999715 0.0238802i \(-0.992398\pi\)
0.999715 0.0238802i \(-0.00760202\pi\)
\(398\) −21.5715 −1.08128
\(399\) 0 0
\(400\) −18.2943 −0.914713
\(401\) − 21.0325i − 1.05031i −0.851006 0.525156i \(-0.824007\pi\)
0.851006 0.525156i \(-0.175993\pi\)
\(402\) 0 0
\(403\) 24.1365 1.20233
\(404\) 38.4973 1.91531
\(405\) 0 0
\(406\) 0 0
\(407\) − 16.8228i − 0.833877i
\(408\) 0 0
\(409\) − 20.1463i − 0.996171i −0.867128 0.498085i \(-0.834037\pi\)
0.867128 0.498085i \(-0.165963\pi\)
\(410\) 1.08986i 0.0538246i
\(411\) 0 0
\(412\) 17.6745i 0.870762i
\(413\) 0 0
\(414\) 0 0
\(415\) 9.13654 0.448495
\(416\) −6.35320 −0.311491
\(417\) 0 0
\(418\) 75.9099i 3.71287i
\(419\) −9.58929 −0.468467 −0.234234 0.972180i \(-0.575258\pi\)
−0.234234 + 0.972180i \(0.575258\pi\)
\(420\) 0 0
\(421\) 8.61320 0.419782 0.209891 0.977725i \(-0.432689\pi\)
0.209891 + 0.977725i \(0.432689\pi\)
\(422\) 55.2041i 2.68729i
\(423\) 0 0
\(424\) −22.6352 −1.09926
\(425\) 14.9818 0.726724
\(426\) 0 0
\(427\) 0 0
\(428\) 43.3630i 2.09603i
\(429\) 0 0
\(430\) 0.196365i 0.00946958i
\(431\) 11.5457i 0.556136i 0.960561 + 0.278068i \(0.0896940\pi\)
−0.960561 + 0.278068i \(0.910306\pi\)
\(432\) 0 0
\(433\) − 22.8707i − 1.09910i −0.835462 0.549548i \(-0.814800\pi\)
0.835462 0.549548i \(-0.185200\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −0.536379 −0.0256879
\(437\) 9.05500 0.433160
\(438\) 0 0
\(439\) 0.252617i 0.0120567i 0.999982 + 0.00602837i \(0.00191890\pi\)
−0.999982 + 0.00602837i \(0.998081\pi\)
\(440\) −37.2614 −1.77637
\(441\) 0 0
\(442\) 31.8944 1.51706
\(443\) 21.7721i 1.03442i 0.855858 + 0.517212i \(0.173030\pi\)
−0.855858 + 0.517212i \(0.826970\pi\)
\(444\) 0 0
\(445\) 4.24211 0.201095
\(446\) −46.6930 −2.21098
\(447\) 0 0
\(448\) 0 0
\(449\) − 10.7904i − 0.509229i −0.967043 0.254614i \(-0.918051\pi\)
0.967043 0.254614i \(-0.0819485\pi\)
\(450\) 0 0
\(451\) − 1.94395i − 0.0915371i
\(452\) − 9.66802i − 0.454746i
\(453\) 0 0
\(454\) 35.3174i 1.65753i
\(455\) 0 0
\(456\) 0 0
\(457\) 35.0220 1.63826 0.819130 0.573608i \(-0.194457\pi\)
0.819130 + 0.573608i \(0.194457\pi\)
\(458\) 10.1397 0.473797
\(459\) 0 0
\(460\) 8.47741i 0.395261i
\(461\) −32.9471 −1.53450 −0.767249 0.641349i \(-0.778375\pi\)
−0.767249 + 0.641349i \(0.778375\pi\)
\(462\) 0 0
\(463\) −5.22641 −0.242892 −0.121446 0.992598i \(-0.538753\pi\)
−0.121446 + 0.992598i \(0.538753\pi\)
\(464\) − 2.68223i − 0.124519i
\(465\) 0 0
\(466\) −17.3450 −0.803492
\(467\) −21.6665 −1.00260 −0.501302 0.865272i \(-0.667145\pi\)
−0.501302 + 0.865272i \(0.667145\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) − 30.3670i − 1.40073i
\(471\) 0 0
\(472\) − 16.9032i − 0.778032i
\(473\) − 0.350250i − 0.0161045i
\(474\) 0 0
\(475\) 19.2735i 0.884330i
\(476\) 0 0
\(477\) 0 0
\(478\) 66.1078 3.02370
\(479\) −22.4308 −1.02489 −0.512444 0.858721i \(-0.671260\pi\)
−0.512444 + 0.858721i \(0.671260\pi\)
\(480\) 0 0
\(481\) − 9.09312i − 0.414611i
\(482\) 33.8852 1.54343
\(483\) 0 0
\(484\) 80.5125 3.65966
\(485\) 5.50764i 0.250089i
\(486\) 0 0
\(487\) 22.1626 1.00428 0.502142 0.864785i \(-0.332545\pi\)
0.502142 + 0.864785i \(0.332545\pi\)
\(488\) −37.8680 −1.71421
\(489\) 0 0
\(490\) 0 0
\(491\) − 15.4543i − 0.697444i −0.937226 0.348722i \(-0.886616\pi\)
0.937226 0.348722i \(-0.113384\pi\)
\(492\) 0 0
\(493\) 2.19657i 0.0989285i
\(494\) 41.0310i 1.84607i
\(495\) 0 0
\(496\) − 42.8432i − 1.92372i
\(497\) 0 0
\(498\) 0 0
\(499\) −1.34502 −0.0602112 −0.0301056 0.999547i \(-0.509584\pi\)
−0.0301056 + 0.999547i \(0.509584\pi\)
\(500\) −44.0249 −1.96885
\(501\) 0 0
\(502\) 6.35320i 0.283557i
\(503\) 18.1391 0.808781 0.404390 0.914586i \(-0.367484\pi\)
0.404390 + 0.914586i \(0.367484\pi\)
\(504\) 0 0
\(505\) 11.3163 0.503568
\(506\) − 22.3137i − 0.991965i
\(507\) 0 0
\(508\) 1.62624 0.0721529
\(509\) −28.1838 −1.24922 −0.624612 0.780935i \(-0.714743\pi\)
−0.624612 + 0.780935i \(0.714743\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 46.5767i − 2.05842i
\(513\) 0 0
\(514\) 44.9856i 1.98423i
\(515\) 5.19543i 0.228938i
\(516\) 0 0
\(517\) 54.1646i 2.38215i
\(518\) 0 0
\(519\) 0 0
\(520\) −20.1406 −0.883224
\(521\) 34.1830 1.49758 0.748792 0.662806i \(-0.230634\pi\)
0.748792 + 0.662806i \(0.230634\pi\)
\(522\) 0 0
\(523\) 36.8531i 1.61147i 0.592273 + 0.805737i \(0.298231\pi\)
−0.592273 + 0.805737i \(0.701769\pi\)
\(524\) 87.7183 3.83199
\(525\) 0 0
\(526\) −20.0988 −0.876351
\(527\) 35.0858i 1.52836i
\(528\) 0 0
\(529\) 20.3383 0.884273
\(530\) −12.6903 −0.551231
\(531\) 0 0
\(532\) 0 0
\(533\) − 1.05075i − 0.0455130i
\(534\) 0 0
\(535\) 12.7465i 0.551081i
\(536\) 67.7132i 2.92476i
\(537\) 0 0
\(538\) − 5.21874i − 0.224996i
\(539\) 0 0
\(540\) 0 0
\(541\) −38.4178 −1.65171 −0.825855 0.563883i \(-0.809307\pi\)
−0.825855 + 0.563883i \(0.809307\pi\)
\(542\) −61.2821 −2.63229
\(543\) 0 0
\(544\) − 9.23526i − 0.395958i
\(545\) −0.157669 −0.00675378
\(546\) 0 0
\(547\) 11.9959 0.512909 0.256454 0.966556i \(-0.417446\pi\)
0.256454 + 0.966556i \(0.417446\pi\)
\(548\) 2.94518i 0.125812i
\(549\) 0 0
\(550\) 47.4946 2.02518
\(551\) −2.82580 −0.120383
\(552\) 0 0
\(553\) 0 0
\(554\) − 54.0492i − 2.29633i
\(555\) 0 0
\(556\) 48.7565i 2.06774i
\(557\) − 8.70457i − 0.368824i −0.982849 0.184412i \(-0.940962\pi\)
0.982849 0.184412i \(-0.0590381\pi\)
\(558\) 0 0
\(559\) − 0.189318i − 0.00800729i
\(560\) 0 0
\(561\) 0 0
\(562\) −56.3719 −2.37791
\(563\) 23.4366 0.987736 0.493868 0.869537i \(-0.335583\pi\)
0.493868 + 0.869537i \(0.335583\pi\)
\(564\) 0 0
\(565\) − 2.84192i − 0.119560i
\(566\) −23.7681 −0.999047
\(567\) 0 0
\(568\) 25.4308 1.06705
\(569\) − 34.3137i − 1.43851i −0.694749 0.719253i \(-0.744484\pi\)
0.694749 0.719253i \(-0.255516\pi\)
\(570\) 0 0
\(571\) 21.4726 0.898600 0.449300 0.893381i \(-0.351673\pi\)
0.449300 + 0.893381i \(0.351673\pi\)
\(572\) 68.5171 2.86485
\(573\) 0 0
\(574\) 0 0
\(575\) − 5.66545i − 0.236266i
\(576\) 0 0
\(577\) 28.6193i 1.19144i 0.803194 + 0.595718i \(0.203132\pi\)
−0.803194 + 0.595718i \(0.796868\pi\)
\(578\) 4.01827i 0.167138i
\(579\) 0 0
\(580\) − 2.64555i − 0.109850i
\(581\) 0 0
\(582\) 0 0
\(583\) 22.6352 0.937455
\(584\) 38.6485 1.59929
\(585\) 0 0
\(586\) − 65.5964i − 2.70976i
\(587\) −18.5157 −0.764224 −0.382112 0.924116i \(-0.624803\pi\)
−0.382112 + 0.924116i \(0.624803\pi\)
\(588\) 0 0
\(589\) −45.1365 −1.85982
\(590\) − 9.47666i − 0.390148i
\(591\) 0 0
\(592\) −16.1406 −0.663375
\(593\) 18.7457 0.769794 0.384897 0.922960i \(-0.374237\pi\)
0.384897 + 0.922960i \(0.374237\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) − 1.47259i − 0.0603197i
\(597\) 0 0
\(598\) − 12.0611i − 0.493213i
\(599\) − 25.8721i − 1.05710i −0.848901 0.528552i \(-0.822735\pi\)
0.848901 0.528552i \(-0.177265\pi\)
\(600\) 0 0
\(601\) 14.5735i 0.594467i 0.954805 + 0.297234i \(0.0960640\pi\)
−0.954805 + 0.297234i \(0.903936\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −19.5714 −0.796350
\(605\) 23.6667 0.962187
\(606\) 0 0
\(607\) 22.2796i 0.904300i 0.891942 + 0.452150i \(0.149343\pi\)
−0.891942 + 0.452150i \(0.850657\pi\)
\(608\) 11.8808 0.481830
\(609\) 0 0
\(610\) −21.2305 −0.859597
\(611\) 29.2772i 1.18443i
\(612\) 0 0
\(613\) 25.2223 1.01872 0.509360 0.860553i \(-0.329882\pi\)
0.509360 + 0.860553i \(0.329882\pi\)
\(614\) 16.9820 0.685338
\(615\) 0 0
\(616\) 0 0
\(617\) − 2.03655i − 0.0819882i −0.999159 0.0409941i \(-0.986948\pi\)
0.999159 0.0409941i \(-0.0130525\pi\)
\(618\) 0 0
\(619\) − 38.0011i − 1.52739i −0.645575 0.763697i \(-0.723382\pi\)
0.645575 0.763697i \(-0.276618\pi\)
\(620\) − 42.2574i − 1.69710i
\(621\) 0 0
\(622\) − 67.1075i − 2.69076i
\(623\) 0 0
\(624\) 0 0
\(625\) 4.42184 0.176874
\(626\) −57.6920 −2.30583
\(627\) 0 0
\(628\) − 5.19615i − 0.207349i
\(629\) 13.2181 0.527040
\(630\) 0 0
\(631\) 2.80457 0.111648 0.0558240 0.998441i \(-0.482221\pi\)
0.0558240 + 0.998441i \(0.482221\pi\)
\(632\) 1.82284i 0.0725087i
\(633\) 0 0
\(634\) 62.4398 2.47980
\(635\) 0.478034 0.0189702
\(636\) 0 0
\(637\) 0 0
\(638\) 6.96345i 0.275686i
\(639\) 0 0
\(640\) − 21.3124i − 0.842448i
\(641\) 1.87766i 0.0741631i 0.999312 + 0.0370815i \(0.0118061\pi\)
−0.999312 + 0.0370815i \(0.988194\pi\)
\(642\) 0 0
\(643\) − 14.5735i − 0.574724i −0.957822 0.287362i \(-0.907222\pi\)
0.957822 0.287362i \(-0.0927783\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −59.6442 −2.34667
\(647\) −36.4584 −1.43333 −0.716663 0.697419i \(-0.754332\pi\)
−0.716663 + 0.697419i \(0.754332\pi\)
\(648\) 0 0
\(649\) 16.9032i 0.663508i
\(650\) 25.6719 1.00693
\(651\) 0 0
\(652\) 82.4618 3.22945
\(653\) − 15.0716i − 0.589797i −0.955529 0.294898i \(-0.904714\pi\)
0.955529 0.294898i \(-0.0952858\pi\)
\(654\) 0 0
\(655\) 25.7848 1.00750
\(656\) −1.86512 −0.0728206
\(657\) 0 0
\(658\) 0 0
\(659\) 31.4670i 1.22578i 0.790168 + 0.612891i \(0.209993\pi\)
−0.790168 + 0.612891i \(0.790007\pi\)
\(660\) 0 0
\(661\) 13.7524i 0.534906i 0.963571 + 0.267453i \(0.0861820\pi\)
−0.963571 + 0.267453i \(0.913818\pi\)
\(662\) − 20.9127i − 0.812795i
\(663\) 0 0
\(664\) 40.5924i 1.57529i
\(665\) 0 0
\(666\) 0 0
\(667\) 0.830645 0.0321627
\(668\) 99.5991 3.85360
\(669\) 0 0
\(670\) 37.9630i 1.46664i
\(671\) 37.8680 1.46188
\(672\) 0 0
\(673\) 11.7448 0.452731 0.226365 0.974042i \(-0.427316\pi\)
0.226365 + 0.974042i \(0.427316\pi\)
\(674\) − 70.4670i − 2.71429i
\(675\) 0 0
\(676\) −17.6222 −0.677776
\(677\) 21.7453 0.835740 0.417870 0.908507i \(-0.362777\pi\)
0.417870 + 0.908507i \(0.362777\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) − 29.2772i − 1.12273i
\(681\) 0 0
\(682\) 111.227i 4.25911i
\(683\) − 12.8919i − 0.493293i −0.969105 0.246647i \(-0.920671\pi\)
0.969105 0.246647i \(-0.0793287\pi\)
\(684\) 0 0
\(685\) 0.865736i 0.0330781i
\(686\) 0 0
\(687\) 0 0
\(688\) −0.336046 −0.0128116
\(689\) 12.2348 0.466110
\(690\) 0 0
\(691\) − 10.3381i − 0.393279i −0.980476 0.196639i \(-0.936997\pi\)
0.980476 0.196639i \(-0.0630028\pi\)
\(692\) 50.3781 1.91509
\(693\) 0 0
\(694\) −24.8176 −0.942063
\(695\) 14.3320i 0.543643i
\(696\) 0 0
\(697\) 1.52741 0.0578547
\(698\) 55.9508 2.11777
\(699\) 0 0
\(700\) 0 0
\(701\) 41.2056i 1.55631i 0.628071 + 0.778156i \(0.283845\pi\)
−0.628071 + 0.778156i \(0.716155\pi\)
\(702\) 0 0
\(703\) 17.0046i 0.641340i
\(704\) 28.5767i 1.07702i
\(705\) 0 0
\(706\) − 79.8701i − 3.00595i
\(707\) 0 0
\(708\) 0 0
\(709\) 28.6002 1.07410 0.537051 0.843550i \(-0.319538\pi\)
0.537051 + 0.843550i \(0.319538\pi\)
\(710\) 14.2576 0.535079
\(711\) 0 0
\(712\) 18.8471i 0.706326i
\(713\) 13.2679 0.496886
\(714\) 0 0
\(715\) 20.1406 0.753216
\(716\) − 94.1168i − 3.51731i
\(717\) 0 0
\(718\) −29.5755 −1.10375
\(719\) 1.76370 0.0657749 0.0328875 0.999459i \(-0.489530\pi\)
0.0328875 + 0.999459i \(0.489530\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 29.4036i − 1.09429i
\(723\) 0 0
\(724\) − 37.4352i − 1.39127i
\(725\) 1.76802i 0.0656627i
\(726\) 0 0
\(727\) 20.0358i 0.743088i 0.928415 + 0.371544i \(0.121171\pi\)
−0.928415 + 0.371544i \(0.878829\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 21.6680 0.801970
\(731\) 0.275200 0.0101786
\(732\) 0 0
\(733\) 15.4237i 0.569689i 0.958574 + 0.284844i \(0.0919419\pi\)
−0.958574 + 0.284844i \(0.908058\pi\)
\(734\) −33.8064 −1.24782
\(735\) 0 0
\(736\) −3.49236 −0.128730
\(737\) − 67.7132i − 2.49425i
\(738\) 0 0
\(739\) −32.4569 −1.19395 −0.596973 0.802261i \(-0.703630\pi\)
−0.596973 + 0.802261i \(0.703630\pi\)
\(740\) −15.9199 −0.585227
\(741\) 0 0
\(742\) 0 0
\(743\) 7.52184i 0.275950i 0.990436 + 0.137975i \(0.0440593\pi\)
−0.990436 + 0.137975i \(0.955941\pi\)
\(744\) 0 0
\(745\) − 0.432868i − 0.0158591i
\(746\) 75.1533i 2.75156i
\(747\) 0 0
\(748\) 99.5991i 3.64170i
\(749\) 0 0
\(750\) 0 0
\(751\) 25.8463 0.943147 0.471573 0.881827i \(-0.343686\pi\)
0.471573 + 0.881827i \(0.343686\pi\)
\(752\) 51.9680 1.89508
\(753\) 0 0
\(754\) 3.76390i 0.137073i
\(755\) −5.75302 −0.209374
\(756\) 0 0
\(757\) 33.5103 1.21795 0.608976 0.793188i \(-0.291580\pi\)
0.608976 + 0.793188i \(0.291580\pi\)
\(758\) − 31.5767i − 1.14692i
\(759\) 0 0
\(760\) 37.6640 1.36622
\(761\) 8.73002 0.316463 0.158232 0.987402i \(-0.449421\pi\)
0.158232 + 0.987402i \(0.449421\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) − 48.3122i − 1.74787i
\(765\) 0 0
\(766\) − 79.6738i − 2.87873i
\(767\) 9.13654i 0.329902i
\(768\) 0 0
\(769\) − 5.04495i − 0.181926i −0.995854 0.0909628i \(-0.971006\pi\)
0.995854 0.0909628i \(-0.0289944\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −46.3409 −1.66785
\(773\) −5.80280 −0.208712 −0.104356 0.994540i \(-0.533278\pi\)
−0.104356 + 0.994540i \(0.533278\pi\)
\(774\) 0 0
\(775\) 28.2406i 1.01443i
\(776\) −24.4697 −0.878410
\(777\) 0 0
\(778\) −18.8724 −0.676609
\(779\) 1.96495i 0.0704018i
\(780\) 0 0
\(781\) −25.4308 −0.909986
\(782\) 17.5324 0.626958
\(783\) 0 0
\(784\) 0 0
\(785\) − 1.52741i − 0.0545156i
\(786\) 0 0
\(787\) 24.7042i 0.880608i 0.897849 + 0.440304i \(0.145129\pi\)
−0.897849 + 0.440304i \(0.854871\pi\)
\(788\) − 83.3432i − 2.96898i
\(789\) 0 0
\(790\) 1.02196i 0.0363599i
\(791\) 0 0
\(792\) 0 0
\(793\) 20.4685 0.726859
\(794\) 2.37035 0.0841206
\(795\) 0 0
\(796\) − 36.4112i − 1.29056i
\(797\) −4.39314 −0.155613 −0.0778064 0.996968i \(-0.524792\pi\)
−0.0778064 + 0.996968i \(0.524792\pi\)
\(798\) 0 0
\(799\) −42.5584 −1.50561
\(800\) − 7.43348i − 0.262813i
\(801\) 0 0
\(802\) 52.3890 1.84992
\(803\) −38.6485 −1.36387
\(804\) 0 0
\(805\) 0 0
\(806\) 60.1208i 2.11767i
\(807\) 0 0
\(808\) 50.2767i 1.76873i
\(809\) − 50.7315i − 1.78362i −0.452406 0.891812i \(-0.649434\pi\)
0.452406 0.891812i \(-0.350566\pi\)
\(810\) 0 0
\(811\) 31.8126i 1.11709i 0.829474 + 0.558546i \(0.188641\pi\)
−0.829474 + 0.558546i \(0.811359\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 41.9034 1.46871
\(815\) 24.2396 0.849078
\(816\) 0 0
\(817\) 0.354034i 0.0123861i
\(818\) 50.1817 1.75456
\(819\) 0 0
\(820\) −1.83961 −0.0642421
\(821\) 25.9437i 0.905441i 0.891653 + 0.452720i \(0.149546\pi\)
−0.891653 + 0.452720i \(0.850454\pi\)
\(822\) 0 0
\(823\) −0.140614 −0.00490149 −0.00245074 0.999997i \(-0.500780\pi\)
−0.00245074 + 0.999997i \(0.500780\pi\)
\(824\) −23.0826 −0.804120
\(825\) 0 0
\(826\) 0 0
\(827\) − 16.1772i − 0.562535i −0.959629 0.281267i \(-0.909245\pi\)
0.959629 0.281267i \(-0.0907548\pi\)
\(828\) 0 0
\(829\) 33.3237i 1.15738i 0.815548 + 0.578690i \(0.196436\pi\)
−0.815548 + 0.578690i \(0.803564\pi\)
\(830\) 22.7579i 0.789938i
\(831\) 0 0
\(832\) 15.4463i 0.535505i
\(833\) 0 0
\(834\) 0 0
\(835\) 29.2772 1.01318
\(836\) −128.130 −4.43148
\(837\) 0 0
\(838\) − 23.8856i − 0.825115i
\(839\) 21.4701 0.741230 0.370615 0.928787i \(-0.379147\pi\)
0.370615 + 0.928787i \(0.379147\pi\)
\(840\) 0 0
\(841\) 28.7408 0.991061
\(842\) 21.4543i 0.739365i
\(843\) 0 0
\(844\) −93.1806 −3.20741
\(845\) −5.18004 −0.178199
\(846\) 0 0
\(847\) 0 0
\(848\) − 21.7173i − 0.745774i
\(849\) 0 0
\(850\) 37.3176i 1.27998i
\(851\) − 4.99850i − 0.171346i
\(852\) 0 0
\(853\) − 25.1842i − 0.862291i −0.902282 0.431146i \(-0.858110\pi\)
0.902282 0.431146i \(-0.141890\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −56.6311 −1.93561
\(857\) 29.0366 0.991871 0.495936 0.868359i \(-0.334825\pi\)
0.495936 + 0.868359i \(0.334825\pi\)
\(858\) 0 0
\(859\) − 23.8314i − 0.813116i −0.913625 0.406558i \(-0.866729\pi\)
0.913625 0.406558i \(-0.133271\pi\)
\(860\) −0.331451 −0.0113024
\(861\) 0 0
\(862\) −28.7587 −0.979526
\(863\) 31.8777i 1.08513i 0.840014 + 0.542564i \(0.182546\pi\)
−0.840014 + 0.542564i \(0.817454\pi\)
\(864\) 0 0
\(865\) 14.8086 0.503509
\(866\) 56.9678 1.93584
\(867\) 0 0
\(868\) 0 0
\(869\) − 1.82284i − 0.0618356i
\(870\) 0 0
\(871\) − 36.6005i − 1.24016i
\(872\) − 0.700500i − 0.0237219i
\(873\) 0 0
\(874\) 22.5548i 0.762927i
\(875\) 0 0
\(876\) 0 0
\(877\) −2.28123 −0.0770316 −0.0385158 0.999258i \(-0.512263\pi\)
−0.0385158 + 0.999258i \(0.512263\pi\)
\(878\) −0.629233 −0.0212356
\(879\) 0 0
\(880\) − 35.7503i − 1.20514i
\(881\) −11.8808 −0.400275 −0.200137 0.979768i \(-0.564139\pi\)
−0.200137 + 0.979768i \(0.564139\pi\)
\(882\) 0 0
\(883\) −49.7120 −1.67294 −0.836472 0.548010i \(-0.815385\pi\)
−0.836472 + 0.548010i \(0.815385\pi\)
\(884\) 53.8355i 1.81069i
\(885\) 0 0
\(886\) −54.2313 −1.82194
\(887\) −12.3588 −0.414969 −0.207485 0.978238i \(-0.566528\pi\)
−0.207485 + 0.978238i \(0.566528\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 10.5665i 0.354191i
\(891\) 0 0
\(892\) − 78.8145i − 2.63890i
\(893\) − 54.7497i − 1.83213i
\(894\) 0 0
\(895\) − 27.6656i − 0.924760i
\(896\) 0 0
\(897\) 0 0
\(898\) 26.8773 0.896908
\(899\) −4.14052 −0.138094
\(900\) 0 0
\(901\) 17.7850i 0.592505i
\(902\) 4.84212 0.161225
\(903\) 0 0
\(904\) 12.6262 0.419943
\(905\) − 11.0041i − 0.365788i
\(906\) 0 0
\(907\) −35.7628 −1.18748 −0.593742 0.804656i \(-0.702350\pi\)
−0.593742 + 0.804656i \(0.702350\pi\)
\(908\) −59.6133 −1.97834
\(909\) 0 0
\(910\) 0 0
\(911\) − 47.9228i − 1.58775i −0.608078 0.793877i \(-0.708059\pi\)
0.608078 0.793877i \(-0.291941\pi\)
\(912\) 0 0
\(913\) − 40.5924i − 1.34341i
\(914\) 87.2350i 2.88548i
\(915\) 0 0
\(916\) 17.1151i 0.565498i
\(917\) 0 0
\(918\) 0 0
\(919\) −34.4816 −1.13744 −0.568721 0.822531i \(-0.692562\pi\)
−0.568721 + 0.822531i \(0.692562\pi\)
\(920\) −11.0713 −0.365011
\(921\) 0 0
\(922\) − 82.0667i − 2.70272i
\(923\) −13.7459 −0.452453
\(924\) 0 0
\(925\) 10.6393 0.349817
\(926\) − 13.0183i − 0.427807i
\(927\) 0 0
\(928\) 1.08986 0.0357766
\(929\) 58.3549 1.91456 0.957280 0.289161i \(-0.0933763\pi\)
0.957280 + 0.289161i \(0.0933763\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 29.2772i − 0.959005i
\(933\) 0 0
\(934\) − 53.9682i − 1.76589i
\(935\) 29.2772i 0.957465i
\(936\) 0 0
\(937\) − 29.8596i − 0.975471i −0.872992 0.487735i \(-0.837823\pi\)
0.872992 0.487735i \(-0.162177\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 51.2574 1.67183
\(941\) −16.2177 −0.528682 −0.264341 0.964429i \(-0.585154\pi\)
−0.264341 + 0.964429i \(0.585154\pi\)
\(942\) 0 0
\(943\) − 0.577598i − 0.0188092i
\(944\) 16.2177 0.527841
\(945\) 0 0
\(946\) 0.872425 0.0283650
\(947\) 28.1939i 0.916180i 0.888906 + 0.458090i \(0.151466\pi\)
−0.888906 + 0.458090i \(0.848534\pi\)
\(948\) 0 0
\(949\) −20.8904 −0.678130
\(950\) −48.0077 −1.55758
\(951\) 0 0
\(952\) 0 0
\(953\) − 56.3488i − 1.82532i −0.408725 0.912658i \(-0.634027\pi\)
0.408725 0.912658i \(-0.365973\pi\)
\(954\) 0 0
\(955\) − 14.2014i − 0.459546i
\(956\) 111.585i 3.60893i
\(957\) 0 0
\(958\) − 55.8720i − 1.80514i
\(959\) 0 0
\(960\) 0 0
\(961\) −35.1365 −1.13344
\(962\) 22.6497 0.730257
\(963\) 0 0
\(964\) 57.1958i 1.84215i
\(965\) −13.6219 −0.438505
\(966\) 0 0
\(967\) 34.0310 1.09436 0.547181 0.837014i \(-0.315701\pi\)
0.547181 + 0.837014i \(0.315701\pi\)
\(968\) 105.148i 3.37958i
\(969\) 0 0
\(970\) −13.7188 −0.440483
\(971\) −32.6945 −1.04922 −0.524608 0.851344i \(-0.675788\pi\)
−0.524608 + 0.851344i \(0.675788\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 55.2041i 1.76885i
\(975\) 0 0
\(976\) − 36.3324i − 1.16297i
\(977\) 24.6625i 0.789022i 0.918891 + 0.394511i \(0.129086\pi\)
−0.918891 + 0.394511i \(0.870914\pi\)
\(978\) 0 0
\(979\) − 18.8471i − 0.602357i
\(980\) 0 0
\(981\) 0 0
\(982\) 38.4946 1.22841
\(983\) 17.7915 0.567461 0.283730 0.958904i \(-0.408428\pi\)
0.283730 + 0.958904i \(0.408428\pi\)
\(984\) 0 0
\(985\) − 24.4987i − 0.780594i
\(986\) −5.47135 −0.174243
\(987\) 0 0
\(988\) −69.2574 −2.20337
\(989\) − 0.104068i − 0.00330918i
\(990\) 0 0
\(991\) −31.9034 −1.01344 −0.506722 0.862109i \(-0.669143\pi\)
−0.506722 + 0.862109i \(0.669143\pi\)
\(992\) 17.4084 0.552718
\(993\) 0 0
\(994\) 0 0
\(995\) − 10.7031i − 0.339310i
\(996\) 0 0
\(997\) 38.9553i 1.23373i 0.787070 + 0.616864i \(0.211597\pi\)
−0.787070 + 0.616864i \(0.788403\pi\)
\(998\) − 3.35025i − 0.106050i
\(999\) 0 0
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 1323.2.c.d.1322.11 12
3.2 odd 2 inner 1323.2.c.d.1322.2 12
7.2 even 3 189.2.p.d.80.6 yes 12
7.3 odd 6 189.2.p.d.26.1 12
7.6 odd 2 inner 1323.2.c.d.1322.12 12
21.2 odd 6 189.2.p.d.80.1 yes 12
21.17 even 6 189.2.p.d.26.6 yes 12
21.20 even 2 inner 1323.2.c.d.1322.1 12
63.2 odd 6 567.2.s.f.458.6 12
63.16 even 3 567.2.s.f.458.1 12
63.23 odd 6 567.2.i.f.269.1 12
63.31 odd 6 567.2.s.f.26.6 12
63.38 even 6 567.2.i.f.215.1 12
63.52 odd 6 567.2.i.f.215.6 12
63.58 even 3 567.2.i.f.269.6 12
63.59 even 6 567.2.s.f.26.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.p.d.26.1 12 7.3 odd 6
189.2.p.d.26.6 yes 12 21.17 even 6
189.2.p.d.80.1 yes 12 21.2 odd 6
189.2.p.d.80.6 yes 12 7.2 even 3
567.2.i.f.215.1 12 63.38 even 6
567.2.i.f.215.6 12 63.52 odd 6
567.2.i.f.269.1 12 63.23 odd 6
567.2.i.f.269.6 12 63.58 even 3
567.2.s.f.26.1 12 63.59 even 6
567.2.s.f.26.6 12 63.31 odd 6
567.2.s.f.458.1 12 63.16 even 3
567.2.s.f.458.6 12 63.2 odd 6
1323.2.c.d.1322.1 12 21.20 even 2 inner
1323.2.c.d.1322.2 12 3.2 odd 2 inner
1323.2.c.d.1322.11 12 1.1 even 1 trivial
1323.2.c.d.1322.12 12 7.6 odd 2 inner