Properties

Label 1764.3.c.h.197.7
Level $1764$
Weight $3$
Character 1764.197
Analytic conductor $48.066$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1764.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.0655186332\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.224054542336.12
Defining polynomial: \(x^{8} + 199 x^{4} + 14641\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 197.7
Root \(-2.67196 - 1.96485i\) of defining polynomial
Character \(\chi\) \(=\) 1764.197
Dual form 1764.3.c.h.197.2

$q$-expansion

\(f(q)\) \(=\) \(q+9.55744i q^{5} +O(q^{10})\) \(q+9.55744i q^{5} -9.27362i q^{11} +4.24264 q^{13} +15.5574i q^{17} +34.8919 q^{19} +17.7589i q^{23} -66.3446 q^{25} +26.2442i q^{29} +32.0635 q^{31} +55.3446 q^{37} -38.4426i q^{41} +29.3446 q^{43} +68.2298i q^{47} +1.08929i q^{53} +88.6320 q^{55} +68.2298i q^{59} -47.6198 q^{61} +40.5488i q^{65} -8.65537 q^{67} +95.7031i q^{71} -90.0463 q^{73} -148.689 q^{79} -88.8851i q^{83} -148.689 q^{85} -76.2467i q^{89} +333.477i q^{95} -12.7519 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 216q^{25} + 128q^{37} - 80q^{43} - 384q^{67} - 560q^{79} - 560q^{85} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 9.55744i 1.91149i 0.294200 + 0.955744i \(0.404947\pi\)
−0.294200 + 0.955744i \(0.595053\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 9.27362i − 0.843056i −0.906815 0.421528i \(-0.861494\pi\)
0.906815 0.421528i \(-0.138506\pi\)
\(12\) 0 0
\(13\) 4.24264 0.326357 0.163178 0.986597i \(-0.447825\pi\)
0.163178 + 0.986597i \(0.447825\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 15.5574i 0.915143i 0.889173 + 0.457572i \(0.151281\pi\)
−0.889173 + 0.457572i \(0.848719\pi\)
\(18\) 0 0
\(19\) 34.8919 1.83642 0.918209 0.396097i \(-0.129636\pi\)
0.918209 + 0.396097i \(0.129636\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 17.7589i 0.772126i 0.922472 + 0.386063i \(0.126165\pi\)
−0.922472 + 0.386063i \(0.873835\pi\)
\(24\) 0 0
\(25\) −66.3446 −2.65379
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 26.2442i 0.904972i 0.891772 + 0.452486i \(0.149463\pi\)
−0.891772 + 0.452486i \(0.850537\pi\)
\(30\) 0 0
\(31\) 32.0635 1.03431 0.517153 0.855893i \(-0.326992\pi\)
0.517153 + 0.855893i \(0.326992\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 55.3446 1.49580 0.747900 0.663811i \(-0.231062\pi\)
0.747900 + 0.663811i \(0.231062\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 38.4426i − 0.937623i −0.883298 0.468812i \(-0.844682\pi\)
0.883298 0.468812i \(-0.155318\pi\)
\(42\) 0 0
\(43\) 29.3446 0.682433 0.341217 0.939985i \(-0.389161\pi\)
0.341217 + 0.939985i \(0.389161\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 68.2298i 1.45170i 0.687855 + 0.725848i \(0.258553\pi\)
−0.687855 + 0.725848i \(0.741447\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.08929i 0.0205527i 0.999947 + 0.0102763i \(0.00327112\pi\)
−0.999947 + 0.0102763i \(0.996729\pi\)
\(54\) 0 0
\(55\) 88.6320 1.61149
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 68.2298i 1.15644i 0.815882 + 0.578218i \(0.196252\pi\)
−0.815882 + 0.578218i \(0.803748\pi\)
\(60\) 0 0
\(61\) −47.6198 −0.780653 −0.390327 0.920676i \(-0.627638\pi\)
−0.390327 + 0.920676i \(0.627638\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 40.5488i 0.623827i
\(66\) 0 0
\(67\) −8.65537 −0.129185 −0.0645923 0.997912i \(-0.520575\pi\)
−0.0645923 + 0.997912i \(0.520575\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 95.7031i 1.34793i 0.738763 + 0.673966i \(0.235410\pi\)
−0.738763 + 0.673966i \(0.764590\pi\)
\(72\) 0 0
\(73\) −90.0463 −1.23351 −0.616755 0.787155i \(-0.711553\pi\)
−0.616755 + 0.787155i \(0.711553\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −148.689 −1.88214 −0.941071 0.338208i \(-0.890179\pi\)
−0.941071 + 0.338208i \(0.890179\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 88.8851i − 1.07091i −0.844565 0.535453i \(-0.820141\pi\)
0.844565 0.535453i \(-0.179859\pi\)
\(84\) 0 0
\(85\) −148.689 −1.74929
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 76.2467i − 0.856705i −0.903612 0.428352i \(-0.859094\pi\)
0.903612 0.428352i \(-0.140906\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 333.477i 3.51029i
\(96\) 0 0
\(97\) −12.7519 −0.131463 −0.0657314 0.997837i \(-0.520938\pi\)
−0.0657314 + 0.997837i \(0.520938\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 5.36158i − 0.0530849i −0.999648 0.0265425i \(-0.991550\pi\)
0.999648 0.0265425i \(-0.00844972\pi\)
\(102\) 0 0
\(103\) −136.715 −1.32733 −0.663667 0.748029i \(-0.731001\pi\)
−0.663667 + 0.748029i \(0.731001\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 149.166i − 1.39408i −0.717034 0.697038i \(-0.754501\pi\)
0.717034 0.697038i \(-0.245499\pi\)
\(108\) 0 0
\(109\) 28.6893 0.263204 0.131602 0.991303i \(-0.457988\pi\)
0.131602 + 0.991303i \(0.457988\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 145.712i − 1.28949i −0.764399 0.644743i \(-0.776964\pi\)
0.764399 0.644743i \(-0.223036\pi\)
\(114\) 0 0
\(115\) −169.730 −1.47591
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 35.0000 0.289256
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 395.149i − 3.16119i
\(126\) 0 0
\(127\) −119.345 −0.939722 −0.469861 0.882741i \(-0.655696\pi\)
−0.469861 + 0.882741i \(0.655696\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 96.0000i 0.732824i 0.930453 + 0.366412i \(0.119414\pi\)
−0.930453 + 0.366412i \(0.880586\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 172.071i 1.25599i 0.778217 + 0.627995i \(0.216124\pi\)
−0.778217 + 0.627995i \(0.783876\pi\)
\(138\) 0 0
\(139\) 211.205 1.51946 0.759731 0.650238i \(-0.225331\pi\)
0.759731 + 0.650238i \(0.225331\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 39.3446i − 0.275137i
\(144\) 0 0
\(145\) −250.827 −1.72984
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 151.044i 1.01372i 0.862029 + 0.506859i \(0.169193\pi\)
−0.862029 + 0.506859i \(0.830807\pi\)
\(150\) 0 0
\(151\) 12.0339 0.0796947 0.0398473 0.999206i \(-0.487313\pi\)
0.0398473 + 0.999206i \(0.487313\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 306.445i 1.97706i
\(156\) 0 0
\(157\) −160.757 −1.02393 −0.511965 0.859006i \(-0.671082\pi\)
−0.511965 + 0.859006i \(0.671082\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −258.034 −1.58303 −0.791515 0.611150i \(-0.790707\pi\)
−0.791515 + 0.611150i \(0.790707\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 107.838i 0.645737i 0.946444 + 0.322868i \(0.104647\pi\)
−0.946444 + 0.322868i \(0.895353\pi\)
\(168\) 0 0
\(169\) −151.000 −0.893491
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 25.3277i 0.146403i 0.997317 + 0.0732014i \(0.0233216\pi\)
−0.997317 + 0.0732014i \(0.976678\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 56.0572i 0.313169i 0.987665 + 0.156584i \(0.0500483\pi\)
−0.987665 + 0.156584i \(0.949952\pi\)
\(180\) 0 0
\(181\) 118.307 0.653627 0.326814 0.945089i \(-0.394025\pi\)
0.326814 + 0.945089i \(0.394025\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 528.953i 2.85920i
\(186\) 0 0
\(187\) 144.274 0.771517
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 94.1264i 0.492809i 0.969167 + 0.246404i \(0.0792491\pi\)
−0.969167 + 0.246404i \(0.920751\pi\)
\(192\) 0 0
\(193\) −305.379 −1.58227 −0.791136 0.611640i \(-0.790510\pi\)
−0.791136 + 0.611640i \(0.790510\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 369.483i 1.87555i 0.347249 + 0.937773i \(0.387116\pi\)
−0.347249 + 0.937773i \(0.612884\pi\)
\(198\) 0 0
\(199\) 268.749 1.35050 0.675248 0.737591i \(-0.264037\pi\)
0.675248 + 0.737591i \(0.264037\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 367.412 1.79226
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 323.574i − 1.54820i
\(210\) 0 0
\(211\) 296.068 1.40316 0.701582 0.712588i \(-0.252477\pi\)
0.701582 + 0.712588i \(0.252477\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 280.460i 1.30446i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 66.0046i 0.298663i
\(222\) 0 0
\(223\) 298.934 1.34051 0.670256 0.742130i \(-0.266184\pi\)
0.670256 + 0.742130i \(0.266184\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 132.689i − 0.584534i −0.956337 0.292267i \(-0.905590\pi\)
0.956337 0.292267i \(-0.0944096\pi\)
\(228\) 0 0
\(229\) 235.295 1.02749 0.513744 0.857944i \(-0.328258\pi\)
0.513744 + 0.857944i \(0.328258\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 95.7031i − 0.410743i −0.978684 0.205371i \(-0.934160\pi\)
0.978684 0.205371i \(-0.0658403\pi\)
\(234\) 0 0
\(235\) −652.102 −2.77490
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 382.397i − 1.59999i −0.600009 0.799994i \(-0.704836\pi\)
0.600009 0.799994i \(-0.295164\pi\)
\(240\) 0 0
\(241\) 8.06980 0.0334846 0.0167423 0.999860i \(-0.494671\pi\)
0.0167423 + 0.999860i \(0.494671\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 148.034 0.599328
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 216.264i − 0.861608i −0.902445 0.430804i \(-0.858230\pi\)
0.902445 0.430804i \(-0.141770\pi\)
\(252\) 0 0
\(253\) 164.689 0.650946
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 288.740i 1.12350i 0.827306 + 0.561751i \(0.189872\pi\)
−0.827306 + 0.561751i \(0.810128\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 67.4668i − 0.256528i −0.991740 0.128264i \(-0.959060\pi\)
0.991740 0.128264i \(-0.0409404\pi\)
\(264\) 0 0
\(265\) −10.4108 −0.0392862
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 58.4087i 0.217133i 0.994089 + 0.108566i \(0.0346260\pi\)
−0.994089 + 0.108566i \(0.965374\pi\)
\(270\) 0 0
\(271\) 138.665 0.511678 0.255839 0.966719i \(-0.417648\pi\)
0.255839 + 0.966719i \(0.417648\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 615.255i 2.23729i
\(276\) 0 0
\(277\) −128.689 −0.464582 −0.232291 0.972646i \(-0.574622\pi\)
−0.232291 + 0.972646i \(0.574622\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 225.534i 0.802611i 0.915944 + 0.401306i \(0.131443\pi\)
−0.915944 + 0.401306i \(0.868557\pi\)
\(282\) 0 0
\(283\) 23.5303 0.0831459 0.0415729 0.999135i \(-0.486763\pi\)
0.0415729 + 0.999135i \(0.486763\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 46.9661 0.162512
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 301.855i 1.03022i 0.857124 + 0.515111i \(0.172249\pi\)
−0.857124 + 0.515111i \(0.827751\pi\)
\(294\) 0 0
\(295\) −652.102 −2.21051
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 75.3446i 0.251989i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 455.124i − 1.49221i
\(306\) 0 0
\(307\) −320.611 −1.04434 −0.522168 0.852843i \(-0.674877\pi\)
−0.522168 + 0.852843i \(0.674877\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 391.906i 1.26015i 0.776535 + 0.630074i \(0.216975\pi\)
−0.776535 + 0.630074i \(0.783025\pi\)
\(312\) 0 0
\(313\) 399.296 1.27570 0.637852 0.770159i \(-0.279823\pi\)
0.637852 + 0.770159i \(0.279823\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 573.129i − 1.80798i −0.427555 0.903990i \(-0.640625\pi\)
0.427555 0.903990i \(-0.359375\pi\)
\(318\) 0 0
\(319\) 243.379 0.762942
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 542.829i 1.68058i
\(324\) 0 0
\(325\) −281.476 −0.866081
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −422.689 −1.27701 −0.638503 0.769619i \(-0.720446\pi\)
−0.638503 + 0.769619i \(0.720446\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 82.7232i − 0.246935i
\(336\) 0 0
\(337\) 299.345 0.888263 0.444132 0.895962i \(-0.353512\pi\)
0.444132 + 0.895962i \(0.353512\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 297.345i − 0.871978i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 424.823i 1.22427i 0.790751 + 0.612137i \(0.209690\pi\)
−0.790751 + 0.612137i \(0.790310\pi\)
\(348\) 0 0
\(349\) 182.505 0.522938 0.261469 0.965212i \(-0.415793\pi\)
0.261469 + 0.965212i \(0.415793\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 211.166i − 0.598203i −0.954221 0.299102i \(-0.903313\pi\)
0.954221 0.299102i \(-0.0966869\pi\)
\(354\) 0 0
\(355\) −914.677 −2.57655
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 92.5498i 0.257799i 0.991658 + 0.128899i \(0.0411444\pi\)
−0.991658 + 0.128899i \(0.958856\pi\)
\(360\) 0 0
\(361\) 856.446 2.37243
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 860.612i − 2.35784i
\(366\) 0 0
\(367\) 303.616 0.827293 0.413646 0.910438i \(-0.364255\pi\)
0.413646 + 0.910438i \(0.364255\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −347.446 −0.931491 −0.465746 0.884919i \(-0.654214\pi\)
−0.465746 + 0.884919i \(0.654214\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 111.345i 0.295344i
\(378\) 0 0
\(379\) 233.379 0.615774 0.307887 0.951423i \(-0.400378\pi\)
0.307887 + 0.951423i \(0.400378\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 220.987i − 0.576989i −0.957482 0.288495i \(-0.906845\pi\)
0.957482 0.288495i \(-0.0931547\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 130.992i 0.336740i 0.985724 + 0.168370i \(0.0538503\pi\)
−0.985724 + 0.168370i \(0.946150\pi\)
\(390\) 0 0
\(391\) −276.283 −0.706606
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 1421.09i − 3.59769i
\(396\) 0 0
\(397\) −196.648 −0.495334 −0.247667 0.968845i \(-0.579664\pi\)
−0.247667 + 0.968845i \(0.579664\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 635.981i − 1.58599i −0.609230 0.792993i \(-0.708522\pi\)
0.609230 0.792993i \(-0.291478\pi\)
\(402\) 0 0
\(403\) 136.034 0.337553
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 513.245i − 1.26104i
\(408\) 0 0
\(409\) −70.2473 −0.171754 −0.0858768 0.996306i \(-0.527369\pi\)
−0.0858768 + 0.996306i \(0.527369\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 849.514 2.04702
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 436.196i 1.04104i 0.853849 + 0.520520i \(0.174262\pi\)
−0.853849 + 0.520520i \(0.825738\pi\)
\(420\) 0 0
\(421\) 7.37852 0.0175262 0.00876309 0.999962i \(-0.497211\pi\)
0.00876309 + 0.999962i \(0.497211\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 1032.15i − 2.42859i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 200.451i 0.465083i 0.972586 + 0.232542i \(0.0747041\pi\)
−0.972586 + 0.232542i \(0.925296\pi\)
\(432\) 0 0
\(433\) 454.106 1.04874 0.524372 0.851489i \(-0.324300\pi\)
0.524372 + 0.851489i \(0.324300\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 619.642i 1.41795i
\(438\) 0 0
\(439\) −262.213 −0.597296 −0.298648 0.954363i \(-0.596536\pi\)
−0.298648 + 0.954363i \(0.596536\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 232.671i 0.525218i 0.964902 + 0.262609i \(0.0845829\pi\)
−0.964902 + 0.262609i \(0.915417\pi\)
\(444\) 0 0
\(445\) 728.723 1.63758
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 92.0197i 0.204944i 0.994736 + 0.102472i \(0.0326752\pi\)
−0.994736 + 0.102472i \(0.967325\pi\)
\(450\) 0 0
\(451\) −356.502 −0.790469
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −694.825 −1.52040 −0.760202 0.649687i \(-0.774900\pi\)
−0.760202 + 0.649687i \(0.774900\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 432.476i 0.938127i 0.883164 + 0.469063i \(0.155408\pi\)
−0.883164 + 0.469063i \(0.844592\pi\)
\(462\) 0 0
\(463\) −310.169 −0.669912 −0.334956 0.942234i \(-0.608722\pi\)
−0.334956 + 0.942234i \(0.608722\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 415.804i − 0.890373i −0.895438 0.445186i \(-0.853137\pi\)
0.895438 0.445186i \(-0.146863\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 272.131i − 0.575330i
\(474\) 0 0
\(475\) −2314.89 −4.87346
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 286.358i − 0.597824i −0.954281 0.298912i \(-0.903376\pi\)
0.954281 0.298912i \(-0.0966238\pi\)
\(480\) 0 0
\(481\) 234.807 0.488165
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 121.875i − 0.251289i
\(486\) 0 0
\(487\) 485.412 0.996740 0.498370 0.866964i \(-0.333932\pi\)
0.498370 + 0.866964i \(0.333932\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 138.130i − 0.281323i −0.990058 0.140661i \(-0.955077\pi\)
0.990058 0.140661i \(-0.0449229\pi\)
\(492\) 0 0
\(493\) −408.292 −0.828179
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −145.345 −0.291272 −0.145636 0.989338i \(-0.546523\pi\)
−0.145636 + 0.989338i \(0.546523\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 619.115i 1.23084i 0.788198 + 0.615422i \(0.211015\pi\)
−0.788198 + 0.615422i \(0.788985\pi\)
\(504\) 0 0
\(505\) 51.2430 0.101471
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 231.923i 0.455644i 0.973703 + 0.227822i \(0.0731604\pi\)
−0.973703 + 0.227822i \(0.926840\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 1306.65i − 2.53718i
\(516\) 0 0
\(517\) 632.737 1.22386
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 52.8343i 0.101409i 0.998714 + 0.0507047i \(0.0161467\pi\)
−0.998714 + 0.0507047i \(0.983853\pi\)
\(522\) 0 0
\(523\) 56.6165 0.108253 0.0541266 0.998534i \(-0.482763\pi\)
0.0541266 + 0.998534i \(0.482763\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 498.826i 0.946539i
\(528\) 0 0
\(529\) 213.621 0.403821
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 163.098i − 0.306000i
\(534\) 0 0
\(535\) 1425.65 2.66476
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 91.2430 0.168656 0.0843281 0.996438i \(-0.473126\pi\)
0.0843281 + 0.996438i \(0.473126\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 274.196i 0.503112i
\(546\) 0 0
\(547\) 356.825 0.652331 0.326165 0.945313i \(-0.394243\pi\)
0.326165 + 0.945313i \(0.394243\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 915.710i 1.66191i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 938.598i − 1.68510i −0.538621 0.842548i \(-0.681055\pi\)
0.538621 0.842548i \(-0.318945\pi\)
\(558\) 0 0
\(559\) 124.499 0.222717
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 1013.15i − 1.79955i −0.436350 0.899777i \(-0.643729\pi\)
0.436350 0.899777i \(-0.356271\pi\)
\(564\) 0 0
\(565\) 1392.63 2.46484
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 967.279i − 1.69996i −0.526812 0.849982i \(-0.676613\pi\)
0.526812 0.849982i \(-0.323387\pi\)
\(570\) 0 0
\(571\) −631.514 −1.10598 −0.552990 0.833188i \(-0.686513\pi\)
−0.552990 + 0.833188i \(0.686513\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 1178.21i − 2.04906i
\(576\) 0 0
\(577\) 614.352 1.06473 0.532367 0.846513i \(-0.321303\pi\)
0.532367 + 0.846513i \(0.321303\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 10.1017 0.0173271
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 411.243i 0.700584i 0.936641 + 0.350292i \(0.113918\pi\)
−0.936641 + 0.350292i \(0.886082\pi\)
\(588\) 0 0
\(589\) 1118.76 1.89942
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 381.923i 0.644052i 0.946731 + 0.322026i \(0.104364\pi\)
−0.946731 + 0.322026i \(0.895636\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 245.285i − 0.409491i −0.978815 0.204745i \(-0.934363\pi\)
0.978815 0.204745i \(-0.0656366\pi\)
\(600\) 0 0
\(601\) −168.243 −0.279939 −0.139970 0.990156i \(-0.544700\pi\)
−0.139970 + 0.990156i \(0.544700\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 334.510i 0.552910i
\(606\) 0 0
\(607\) 558.367 0.919879 0.459940 0.887950i \(-0.347871\pi\)
0.459940 + 0.887950i \(0.347871\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 289.474i 0.473771i
\(612\) 0 0
\(613\) −260.169 −0.424420 −0.212210 0.977224i \(-0.568066\pi\)
−0.212210 + 0.977224i \(0.568066\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 351.007i 0.568893i 0.958692 + 0.284447i \(0.0918099\pi\)
−0.958692 + 0.284447i \(0.908190\pi\)
\(618\) 0 0
\(619\) −317.807 −0.513419 −0.256710 0.966489i \(-0.582638\pi\)
−0.256710 + 0.966489i \(0.582638\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 2117.99 3.38879
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 861.021i 1.36887i
\(630\) 0 0
\(631\) 297.966 0.472213 0.236106 0.971727i \(-0.424129\pi\)
0.236106 + 0.971727i \(0.424129\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 1140.63i − 1.79627i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 761.912i − 1.18863i −0.804232 0.594315i \(-0.797423\pi\)
0.804232 0.594315i \(-0.202577\pi\)
\(642\) 0 0
\(643\) −21.5808 −0.0335626 −0.0167813 0.999859i \(-0.505342\pi\)
−0.0167813 + 0.999859i \(0.505342\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 104.554i − 0.161598i −0.996730 0.0807988i \(-0.974253\pi\)
0.996730 0.0807988i \(-0.0257471\pi\)
\(648\) 0 0
\(649\) 632.737 0.974941
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 359.722i − 0.550875i −0.961319 0.275438i \(-0.911177\pi\)
0.961319 0.275438i \(-0.0888228\pi\)
\(654\) 0 0
\(655\) −917.514 −1.40078
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 983.046i 1.49172i 0.666100 + 0.745862i \(0.267962\pi\)
−0.666100 + 0.745862i \(0.732038\pi\)
\(660\) 0 0
\(661\) 511.626 0.774018 0.387009 0.922076i \(-0.373508\pi\)
0.387009 + 0.922076i \(0.373508\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −466.068 −0.698752
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 441.608i 0.658135i
\(672\) 0 0
\(673\) −49.8644 −0.0740928 −0.0370464 0.999314i \(-0.511795\pi\)
−0.0370464 + 0.999314i \(0.511795\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 564.051i − 0.833162i −0.909099 0.416581i \(-0.863228\pi\)
0.909099 0.416581i \(-0.136772\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 132.425i 0.193887i 0.995290 + 0.0969434i \(0.0309066\pi\)
−0.995290 + 0.0969434i \(0.969093\pi\)
\(684\) 0 0
\(685\) −1644.55 −2.40081
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 4.62148i 0.00670751i
\(690\) 0 0
\(691\) 1102.11 1.59495 0.797476 0.603351i \(-0.206168\pi\)
0.797476 + 0.603351i \(0.206168\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2018.58i 2.90443i
\(696\) 0 0
\(697\) 598.068 0.858060
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 302.130i − 0.430999i −0.976504 0.215500i \(-0.930862\pi\)
0.976504 0.215500i \(-0.0691380\pi\)
\(702\) 0 0
\(703\) 1931.08 2.74691
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −12.6215 −0.0178018 −0.00890090 0.999960i \(-0.502833\pi\)
−0.00890090 + 0.999960i \(0.502833\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 569.412i 0.798615i
\(714\) 0 0
\(715\) 376.034 0.525922
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 759.608i 1.05648i 0.849095 + 0.528239i \(0.177148\pi\)
−0.849095 + 0.528239i \(0.822852\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 1741.16i − 2.40160i
\(726\) 0 0
\(727\) 829.041 1.14036 0.570179 0.821520i \(-0.306874\pi\)
0.570179 + 0.821520i \(0.306874\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 456.527i 0.624524i
\(732\) 0 0
\(733\) −1354.52 −1.84791 −0.923957 0.382496i \(-0.875064\pi\)
−0.923957 + 0.382496i \(0.875064\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 80.2666i 0.108910i
\(738\) 0 0
\(739\) 34.0339 0.0460540 0.0230270 0.999735i \(-0.492670\pi\)
0.0230270 + 0.999735i \(0.492670\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 61.0163i − 0.0821215i −0.999157 0.0410607i \(-0.986926\pi\)
0.999157 0.0410607i \(-0.0130737\pi\)
\(744\) 0 0
\(745\) −1443.59 −1.93771
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −140.136 −0.186599 −0.0932993 0.995638i \(-0.529741\pi\)
−0.0932993 + 0.995638i \(0.529741\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 115.013i 0.152335i
\(756\) 0 0
\(757\) 518.000 0.684280 0.342140 0.939649i \(-0.388848\pi\)
0.342140 + 0.939649i \(0.388848\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 136.145i − 0.178903i −0.995991 0.0894514i \(-0.971489\pi\)
0.995991 0.0894514i \(-0.0285114\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 289.474i 0.377411i
\(768\) 0 0
\(769\) −237.244 −0.308510 −0.154255 0.988031i \(-0.549298\pi\)
−0.154255 + 0.988031i \(0.549298\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 863.301i 1.11682i 0.829565 + 0.558410i \(0.188588\pi\)
−0.829565 + 0.558410i \(0.811412\pi\)
\(774\) 0 0
\(775\) −2127.24 −2.74483
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 1341.33i − 1.72187i
\(780\) 0 0
\(781\) 887.514 1.13638
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 1536.42i − 1.95723i
\(786\) 0 0
\(787\) −764.842 −0.971845 −0.485922 0.874002i \(-0.661516\pi\)
−0.485922 + 0.874002i \(0.661516\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −202.034 −0.254772
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 1413.60i − 1.77365i −0.462106 0.886825i \(-0.652906\pi\)
0.462106 0.886825i \(-0.347094\pi\)
\(798\) 0 0
\(799\) −1061.48 −1.32851
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 835.055i 1.03992i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 721.278i 0.891568i 0.895141 + 0.445784i \(0.147075\pi\)
−0.895141 + 0.445784i \(0.852925\pi\)
\(810\) 0 0
\(811\) 928.795 1.14525 0.572623 0.819819i \(-0.305926\pi\)
0.572623 + 0.819819i \(0.305926\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 2466.14i − 3.02594i
\(816\) 0 0
\(817\) 1023.89 1.25323
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 1012.19i − 1.23287i −0.787406 0.616434i \(-0.788577\pi\)
0.787406 0.616434i \(-0.211423\pi\)
\(822\) 0 0
\(823\) 1331.58 1.61796 0.808980 0.587835i \(-0.200020\pi\)
0.808980 + 0.587835i \(0.200020\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 96.8217i − 0.117076i −0.998285 0.0585379i \(-0.981356\pi\)
0.998285 0.0585379i \(-0.0186438\pi\)
\(828\) 0 0
\(829\) −274.845 −0.331538 −0.165769 0.986165i \(-0.553011\pi\)
−0.165769 + 0.986165i \(0.553011\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −1030.66 −1.23432
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1196.53i 1.42614i 0.701091 + 0.713072i \(0.252697\pi\)
−0.701091 + 0.713072i \(0.747303\pi\)
\(840\) 0 0
\(841\) 152.243 0.181026
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 1443.17i − 1.70790i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 982.860i 1.15495i
\(852\) 0 0
\(853\) −761.454 −0.892678 −0.446339 0.894864i \(-0.647272\pi\)
−0.446339 + 0.894864i \(0.647272\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 204.416i − 0.238525i −0.992863 0.119263i \(-0.961947\pi\)
0.992863 0.119263i \(-0.0380531\pi\)
\(858\) 0 0
\(859\) −56.4487 −0.0657144 −0.0328572 0.999460i \(-0.510461\pi\)
−0.0328572 + 0.999460i \(0.510461\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 887.673i − 1.02859i −0.857613 0.514295i \(-0.828054\pi\)
0.857613 0.514295i \(-0.171946\pi\)
\(864\) 0 0
\(865\) −242.068 −0.279847
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1378.89i 1.58675i
\(870\) 0 0
\(871\) −36.7216 −0.0421603
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 77.1413 0.0879604 0.0439802 0.999032i \(-0.485996\pi\)
0.0439802 + 0.999032i \(0.485996\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 918.740i − 1.04284i −0.853301 0.521419i \(-0.825403\pi\)
0.853301 0.521419i \(-0.174597\pi\)
\(882\) 0 0
\(883\) 6.72316 0.00761399 0.00380700 0.999993i \(-0.498788\pi\)
0.00380700 + 0.999993i \(0.498788\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 157.744i − 0.177840i −0.996039 0.0889199i \(-0.971658\pi\)
0.996039 0.0889199i \(-0.0283415\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 2380.67i 2.66592i
\(894\) 0 0
\(895\) −535.763 −0.598618
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 841.480i 0.936018i
\(900\) 0 0
\(901\) −16.9466 −0.0188087
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1130.71i 1.24940i
\(906\) 0 0
\(907\) 612.169 0.674939 0.337469 0.941337i \(-0.390429\pi\)
0.337469 + 0.941337i \(0.390429\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 570.650i − 0.626399i −0.949687 0.313200i \(-0.898599\pi\)
0.949687 0.313200i \(-0.101401\pi\)
\(912\) 0 0
\(913\) −824.287 −0.902833
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −1266.93 −1.37859 −0.689296 0.724480i \(-0.742080\pi\)
−0.689296 + 0.724480i \(0.742080\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 406.034i 0.439907i
\(924\) 0 0
\(925\) −3671.82 −3.96953
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 1377.34i − 1.48260i −0.671174 0.741300i \(-0.734210\pi\)
0.671174 0.741300i \(-0.265790\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1378.89i 1.47475i
\(936\) 0 0
\(937\) 754.751 0.805497 0.402748 0.915311i \(-0.368055\pi\)
0.402748 + 0.915311i \(0.368055\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 760.126i − 0.807785i −0.914806 0.403893i \(-0.867657\pi\)
0.914806 0.403893i \(-0.132343\pi\)
\(942\) 0 0
\(943\) 682.698 0.723964
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 38.1119i 0.0402448i 0.999798 + 0.0201224i \(0.00640560\pi\)
−0.999798 + 0.0201224i \(0.993594\pi\)
\(948\) 0 0
\(949\) −382.034 −0.402565
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 1140.96i − 1.19723i −0.801039 0.598613i \(-0.795719\pi\)
0.801039 0.598613i \(-0.204281\pi\)
\(954\) 0 0
\(955\) −899.608 −0.941997
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 67.0678 0.0697896
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 2918.64i − 3.02449i
\(966\) 0 0
\(967\) 1212.79 1.25418 0.627089 0.778947i \(-0.284246\pi\)
0.627089 + 0.778947i \(0.284246\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 475.115i − 0.489305i −0.969611 0.244652i \(-0.921326\pi\)
0.969611 0.244652i \(-0.0786738\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 1335.82i − 1.36726i −0.729827 0.683632i \(-0.760399\pi\)
0.729827 0.683632i \(-0.239601\pi\)
\(978\) 0 0
\(979\) −707.083 −0.722250
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 108.750i − 0.110630i −0.998469 0.0553151i \(-0.982384\pi\)
0.998469 0.0553151i \(-0.0176163\pi\)
\(984\) 0 0
\(985\) −3531.31 −3.58508
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 521.128i 0.526925i
\(990\) 0 0
\(991\) 550.859 0.555861 0.277931 0.960601i \(-0.410351\pi\)
0.277931 + 0.960601i \(0.410351\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 2568.55i 2.58145i
\(996\) 0 0
\(997\) 71.4617 0.0716767 0.0358384 0.999358i \(-0.488590\pi\)
0.0358384 + 0.999358i \(0.488590\pi\)
\(998\) 0 0
\(999\) 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 1764.3.c.h.197.7 yes 8
3.2 odd 2 inner 1764.3.c.h.197.2 yes 8
7.2 even 3 1764.3.bk.g.557.8 16
7.3 odd 6 1764.3.bk.g.1745.7 16
7.4 even 3 1764.3.bk.g.1745.1 16
7.5 odd 6 1764.3.bk.g.557.2 16
7.6 odd 2 inner 1764.3.c.h.197.1 8
21.2 odd 6 1764.3.bk.g.557.1 16
21.5 even 6 1764.3.bk.g.557.7 16
21.11 odd 6 1764.3.bk.g.1745.8 16
21.17 even 6 1764.3.bk.g.1745.2 16
21.20 even 2 inner 1764.3.c.h.197.8 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1764.3.c.h.197.1 8 7.6 odd 2 inner
1764.3.c.h.197.2 yes 8 3.2 odd 2 inner
1764.3.c.h.197.7 yes 8 1.1 even 1 trivial
1764.3.c.h.197.8 yes 8 21.20 even 2 inner
1764.3.bk.g.557.1 16 21.2 odd 6
1764.3.bk.g.557.2 16 7.5 odd 6
1764.3.bk.g.557.7 16 21.5 even 6
1764.3.bk.g.557.8 16 7.2 even 3
1764.3.bk.g.1745.1 16 7.4 even 3
1764.3.bk.g.1745.2 16 21.17 even 6
1764.3.bk.g.1745.7 16 7.3 odd 6
1764.3.bk.g.1745.8 16 21.11 odd 6