Properties

Label 1323.2.c.d.1322.2
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 1322.2
Root \(-0.617942 - 0.356769i\) of defining polynomial
Character \(\chi\) \(=\) 1323.1322
Dual form 1323.2.c.d.1322.12

$q$-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 \(-0.657104\pi\)
0.473761 0.880653i \(-0.342896\pi\)
\(3\) 0 0
\(4\) −4.20440 −2.10220
\(5\) 1.23588 0.552704 0.276352 0.961056i \(-0.410874\pi\)
0.276352 + 0.961056i \(0.410874\pi\)
\(6\) 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 \(-0.689604\pi\)
0.561055 0.827779i \(-0.310396\pi\)
\(12\) 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 \(-0.324743\pi\)
0.523186 + 0.852219i \(0.324743\pi\)
\(18\) 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 \(-0.945593\pi\)
0.985428 0.170093i \(-0.0544069\pi\)
\(24\) 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 \(0.0150528\pi\)
−0.998882 + 0.0472721i \(0.984947\pi\)
\(30\) 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 \(-0.491199\pi\)
0.0276454 + 0.999618i \(0.491199\pi\)
\(42\) 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 \(-0.755602\pi\)
−0.719442 + 0.694553i \(0.755602\pi\)
\(48\) 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 \(0.0913706\pi\)
−0.959083 + 0.283124i \(0.908629\pi\)
\(54\) 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 \(-0.564220\pi\)
−0.200388 + 0.979717i \(0.564220\pi\)
\(60\) 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 \(-0.911379\pi\)
0.961494 0.274828i \(-0.0886208\pi\)
\(72\) 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 \(-0.367018\pi\)
0.405728 + 0.913994i \(0.367018\pi\)
\(84\) 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 \(-0.441769\pi\)
0.181920 + 0.983313i \(0.441769\pi\)
\(90\) 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 \(-0.349443\pi\)
0.455549 + 0.890211i \(0.349443\pi\)
\(102\) 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 \(0.166127\pi\)
−0.866871 + 0.498532i \(0.833873\pi\)
\(108\) 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 \(-0.965504\pi\)
0.994134 0.108159i \(-0.0344957\pi\)
\(114\) 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 \(-0.134984\pi\)
0.911424 + 0.411469i \(0.134984\pi\)
\(132\) 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 \(0.00952648\pi\)
−0.999552 + 0.0299239i \(0.990474\pi\)
\(138\) 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 \(-0.995433\pi\)
0.999897 0.0143468i \(-0.00456688\pi\)
\(150\) 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 \(-0.130951\pi\)
0.916564 + 0.399887i \(0.130951\pi\)
\(168\) 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 \(-0.349462\pi\)
0.455496 + 0.890238i \(0.349462\pi\)
\(174\) 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 \(-0.684553\pi\)
0.547848 0.836578i \(-0.315447\pi\)
\(180\) 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 \(-0.863528\pi\)
0.909490 0.415725i \(-0.136472\pi\)
\(192\) 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 \(-0.750426\pi\)
0.708053 0.706159i \(-0.249574\pi\)
\(198\) 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 \(-0.655941\pi\)
−0.470540 + 0.882379i \(0.655941\pi\)
\(228\) 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 \(-0.926750\pi\)
0.973639 0.228096i \(-0.0732499\pi\)
\(234\) 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 \(0.328522\pi\)
−0.513033 + 0.858369i \(0.671478\pi\)
\(240\) 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 \(-0.525651\pi\)
−0.0804963 + 0.996755i \(0.525651\pi\)
\(252\) 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 \(-0.690462\pi\)
−0.563283 + 0.826264i \(0.690462\pi\)
\(258\) 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 \(-0.919971\pi\)
0.968560 0.248779i \(-0.0800292\pi\)
\(264\) 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 \(-0.479655\pi\)
0.0638718 + 0.997958i \(0.479655\pi\)
\(270\) 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 \(-0.764126\pi\)
0.737781 0.675040i \(-0.235874\pi\)
\(282\) 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 \(-0.220631\pi\)
0.769248 + 0.638951i \(0.220631\pi\)
\(294\) 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 \(-0.223304\pi\)
0.763855 + 0.645388i \(0.223304\pi\)
\(312\) 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 \(0.248589\pi\)
−0.710234 + 0.703966i \(0.751411\pi\)
\(318\) 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 \(-0.913825\pi\)
0.963576 0.267433i \(-0.0861754\pi\)
\(348\) 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 \(-0.174579\pi\)
0.853330 + 0.521371i \(0.174579\pi\)
\(354\) 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 \(-0.898555\pi\)
0.949644 0.313332i \(-0.101445\pi\)
\(360\) 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 \(-0.195517\pi\)
0.817214 + 0.576334i \(0.195517\pi\)
\(384\) 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 \(-0.938478\pi\)
0.981380 0.192076i \(-0.0615220\pi\)
\(390\) 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 \(0.175993\pi\)
−0.851006 + 0.525156i \(0.824007\pi\)
\(402\) 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 \(-0.424742\pi\)
0.234234 + 0.972180i \(0.424742\pi\)
\(420\) 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 \(-0.910306\pi\)
0.960561 0.278068i \(-0.0896940\pi\)
\(432\) 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 \(-0.826970\pi\)
0.855858 0.517212i \(-0.173030\pi\)
\(444\) 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 \(0.0819485\pi\)
−0.967043 + 0.254614i \(0.918051\pi\)
\(450\) 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 \(-0.221625\pi\)
0.767249 + 0.641349i \(0.221625\pi\)
\(462\) 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 \(-0.332855\pi\)
0.501302 + 0.865272i \(0.332855\pi\)
\(468\) 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 \(-0.328740\pi\)
0.512444 + 0.858721i \(0.328740\pi\)
\(480\) 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 \(0.113384\pi\)
−0.937226 + 0.348722i \(0.886616\pi\)
\(492\) 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 \(-0.632516\pi\)
−0.404390 + 0.914586i \(0.632516\pi\)
\(504\) 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 \(-0.285257\pi\)
0.624612 + 0.780935i \(0.285257\pi\)
\(510\) 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 \(-0.769366\pi\)
−0.748792 + 0.662806i \(0.769366\pi\)
\(522\) 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 \(0.0590381\pi\)
−0.982849 + 0.184412i \(0.940962\pi\)
\(558\) 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 \(-0.664417\pi\)
−0.493868 + 0.869537i \(0.664417\pi\)
\(564\) 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 \(0.255516\pi\)
−0.694749 + 0.719253i \(0.744484\pi\)
\(570\) 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 \(-0.375197\pi\)
0.382112 + 0.924116i \(0.375197\pi\)
\(588\) 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 \(-0.625763\pi\)
−0.384897 + 0.922960i \(0.625763\pi\)
\(594\) 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 \(0.177265\pi\)
−0.848901 + 0.528552i \(0.822735\pi\)
\(600\) 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 \(0.0130525\pi\)
−0.999159 + 0.0409941i \(0.986948\pi\)
\(618\) 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 \(-0.988194\pi\)
0.999312 0.0370815i \(-0.0118061\pi\)
\(642\) 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 \(-0.245668\pi\)
0.716663 + 0.697419i \(0.245668\pi\)
\(648\) 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 \(0.0952858\pi\)
−0.955529 + 0.294898i \(0.904714\pi\)
\(654\) 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 \(-0.790007\pi\)
0.790168 0.612891i \(-0.209993\pi\)
\(660\) 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 \(-0.637223\pi\)
−0.417870 + 0.908507i \(0.637223\pi\)
\(678\) 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 \(0.0793287\pi\)
−0.969105 + 0.246647i \(0.920671\pi\)
\(684\) 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 \(-0.716155\pi\)
0.628071 0.778156i \(-0.283845\pi\)
\(702\) 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 \(-0.510470\pi\)
−0.0328875 + 0.999459i \(0.510470\pi\)
\(720\) 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 \(-0.955941\pi\)
0.990436 0.137975i \(-0.0440593\pi\)
\(744\) 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 \(-0.550579\pi\)
−0.158232 + 0.987402i \(0.550579\pi\)
\(762\) 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 \(-0.466722\pi\)
0.104356 + 0.994540i \(0.466722\pi\)
\(774\) 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 \(-0.475208\pi\)
0.0778064 + 0.996968i \(0.475208\pi\)
\(798\) 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 \(0.350566\pi\)
−0.452406 + 0.891812i \(0.649434\pi\)
\(810\) 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 \(-0.850454\pi\)
0.891653 0.452720i \(-0.149546\pi\)
\(822\) 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 \(0.0907548\pi\)
−0.959629 + 0.281267i \(0.909245\pi\)
\(828\) 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 \(-0.620853\pi\)
−0.370615 + 0.928787i \(0.620853\pi\)
\(840\) 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 \(-0.665175\pi\)
−0.495936 + 0.868359i \(0.665175\pi\)
\(858\) 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 \(-0.817454\pi\)
0.840014 0.542564i \(-0.182546\pi\)
\(864\) 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 \(-0.435861\pi\)
0.200137 + 0.979768i \(0.435861\pi\)
\(882\) 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 \(-0.433472\pi\)
0.207485 + 0.978238i \(0.433472\pi\)
\(888\) 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 \(0.291941\pi\)
−0.608078 + 0.793877i \(0.708059\pi\)
\(912\) 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 \(-0.906624\pi\)
−0.957280 + 0.289161i \(0.906624\pi\)
\(930\) 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 \(-0.414846\pi\)
0.264341 + 0.964429i \(0.414846\pi\)
\(942\) 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 \(-0.848534\pi\)
0.888906 0.458090i \(-0.151466\pi\)
\(948\) 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 \(0.365973\pi\)
−0.408725 + 0.912658i \(0.634027\pi\)
\(954\) 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 \(-0.324212\pi\)
0.524608 + 0.851344i \(0.324212\pi\)
\(972\) 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 \(-0.870914\pi\)
0.918891 0.394511i \(-0.129086\pi\)
\(978\) 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 \(-0.591572\pi\)
−0.283730 + 0.958904i \(0.591572\pi\)
\(984\) 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.1322.2 12
3.2 odd 2 inner 1323.2.c.d.1322.11 12
7.2 even 3 189.2.p.d.80.1 yes 12
7.3 odd 6 189.2.p.d.26.6 yes 12
7.6 odd 2 inner 1323.2.c.d.1322.1 12
21.2 odd 6 189.2.p.d.80.6 yes 12
21.17 even 6 189.2.p.d.26.1 12
21.20 even 2 inner 1323.2.c.d.1322.12 12
63.2 odd 6 567.2.s.f.458.1 12
63.16 even 3 567.2.s.f.458.6 12
63.23 odd 6 567.2.i.f.269.6 12
63.31 odd 6 567.2.s.f.26.1 12
63.38 even 6 567.2.i.f.215.6 12
63.52 odd 6 567.2.i.f.215.1 12
63.58 even 3 567.2.i.f.269.1 12
63.59 even 6 567.2.s.f.26.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.p.d.26.1 12 21.17 even 6
189.2.p.d.26.6 yes 12 7.3 odd 6
189.2.p.d.80.1 yes 12 7.2 even 3
189.2.p.d.80.6 yes 12 21.2 odd 6
567.2.i.f.215.1 12 63.52 odd 6
567.2.i.f.215.6 12 63.38 even 6
567.2.i.f.269.1 12 63.58 even 3
567.2.i.f.269.6 12 63.23 odd 6
567.2.s.f.26.1 12 63.31 odd 6
567.2.s.f.26.6 12 63.59 even 6
567.2.s.f.458.1 12 63.2 odd 6
567.2.s.f.458.6 12 63.16 even 3
1323.2.c.d.1322.1 12 7.6 odd 2 inner
1323.2.c.d.1322.2 12 1.1 even 1 trivial
1323.2.c.d.1322.11 12 3.2 odd 2 inner
1323.2.c.d.1322.12 12 21.20 even 2 inner