Properties

Label 3380.2.a.r.1.7
Level $3380$
Weight $2$
Character 3380.1
Self dual yes
Analytic conductor $26.989$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9894358832\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - x^{8} - 19 x^{7} + 16 x^{6} + 106 x^{5} - 87 x^{4} - 153 x^{3} + 149 x^{2} - 26 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-1.65879\) of defining polynomial
Character \(\chi\) \(=\) 3380.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.65879 q^{3} -1.00000 q^{5} +4.25414 q^{7} -0.248414 q^{9} +O(q^{10})\) \(q+1.65879 q^{3} -1.00000 q^{5} +4.25414 q^{7} -0.248414 q^{9} +1.49959 q^{11} -1.65879 q^{15} +5.70395 q^{17} -0.636838 q^{19} +7.05673 q^{21} +3.01186 q^{23} +1.00000 q^{25} -5.38844 q^{27} +6.58768 q^{29} +1.96883 q^{31} +2.48750 q^{33} -4.25414 q^{35} -7.56859 q^{37} -10.0713 q^{41} +5.06615 q^{43} +0.248414 q^{45} -3.08005 q^{47} +11.0977 q^{49} +9.46166 q^{51} +2.26608 q^{53} -1.49959 q^{55} -1.05638 q^{57} +0.969910 q^{59} -11.0500 q^{61} -1.05679 q^{63} +13.1252 q^{67} +4.99604 q^{69} -7.08679 q^{71} +12.5266 q^{73} +1.65879 q^{75} +6.37945 q^{77} +13.7459 q^{79} -8.19305 q^{81} +5.57487 q^{83} -5.70395 q^{85} +10.9276 q^{87} -9.73995 q^{89} +3.26587 q^{93} +0.636838 q^{95} -10.0261 q^{97} -0.372519 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - q^{3} - 9q^{5} - q^{7} + 12q^{9} + O(q^{10}) \) \( 9q - q^{3} - 9q^{5} - q^{7} + 12q^{9} + 7q^{11} + q^{15} + 13q^{17} + 4q^{19} + 3q^{21} + 12q^{23} + 9q^{25} - 4q^{27} + 16q^{29} - 13q^{31} - 34q^{33} + q^{35} - q^{37} + 6q^{41} + q^{43} - 12q^{45} + 2q^{47} + 20q^{49} + 11q^{51} + 30q^{53} - 7q^{55} - 38q^{57} - 15q^{59} + 21q^{61} + 17q^{63} + 7q^{67} + 15q^{69} + 7q^{71} + 28q^{73} - q^{75} + 46q^{77} + 31q^{79} + 41q^{81} - 45q^{83} - 13q^{85} + 28q^{87} + 41q^{89} + 11q^{93} - 4q^{95} - 8q^{97} + 81q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.65879 0.957703 0.478852 0.877896i \(-0.341053\pi\)
0.478852 + 0.877896i \(0.341053\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.25414 1.60791 0.803957 0.594687i \(-0.202724\pi\)
0.803957 + 0.594687i \(0.202724\pi\)
\(8\) 0 0
\(9\) −0.248414 −0.0828048
\(10\) 0 0
\(11\) 1.49959 0.452142 0.226071 0.974111i \(-0.427412\pi\)
0.226071 + 0.974111i \(0.427412\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) −1.65879 −0.428298
\(16\) 0 0
\(17\) 5.70395 1.38341 0.691706 0.722180i \(-0.256860\pi\)
0.691706 + 0.722180i \(0.256860\pi\)
\(18\) 0 0
\(19\) −0.636838 −0.146101 −0.0730504 0.997328i \(-0.523273\pi\)
−0.0730504 + 0.997328i \(0.523273\pi\)
\(20\) 0 0
\(21\) 7.05673 1.53990
\(22\) 0 0
\(23\) 3.01186 0.628016 0.314008 0.949420i \(-0.398328\pi\)
0.314008 + 0.949420i \(0.398328\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.38844 −1.03701
\(28\) 0 0
\(29\) 6.58768 1.22330 0.611651 0.791128i \(-0.290506\pi\)
0.611651 + 0.791128i \(0.290506\pi\)
\(30\) 0 0
\(31\) 1.96883 0.353612 0.176806 0.984246i \(-0.443424\pi\)
0.176806 + 0.984246i \(0.443424\pi\)
\(32\) 0 0
\(33\) 2.48750 0.433018
\(34\) 0 0
\(35\) −4.25414 −0.719081
\(36\) 0 0
\(37\) −7.56859 −1.24427 −0.622134 0.782911i \(-0.713734\pi\)
−0.622134 + 0.782911i \(0.713734\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.0713 −1.57288 −0.786440 0.617667i \(-0.788078\pi\)
−0.786440 + 0.617667i \(0.788078\pi\)
\(42\) 0 0
\(43\) 5.06615 0.772581 0.386290 0.922377i \(-0.373756\pi\)
0.386290 + 0.922377i \(0.373756\pi\)
\(44\) 0 0
\(45\) 0.248414 0.0370314
\(46\) 0 0
\(47\) −3.08005 −0.449272 −0.224636 0.974443i \(-0.572119\pi\)
−0.224636 + 0.974443i \(0.572119\pi\)
\(48\) 0 0
\(49\) 11.0977 1.58539
\(50\) 0 0
\(51\) 9.46166 1.32490
\(52\) 0 0
\(53\) 2.26608 0.311270 0.155635 0.987815i \(-0.450258\pi\)
0.155635 + 0.987815i \(0.450258\pi\)
\(54\) 0 0
\(55\) −1.49959 −0.202204
\(56\) 0 0
\(57\) −1.05638 −0.139921
\(58\) 0 0
\(59\) 0.969910 0.126271 0.0631357 0.998005i \(-0.479890\pi\)
0.0631357 + 0.998005i \(0.479890\pi\)
\(60\) 0 0
\(61\) −11.0500 −1.41481 −0.707404 0.706809i \(-0.750134\pi\)
−0.707404 + 0.706809i \(0.750134\pi\)
\(62\) 0 0
\(63\) −1.05679 −0.133143
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 13.1252 1.60350 0.801750 0.597659i \(-0.203902\pi\)
0.801750 + 0.597659i \(0.203902\pi\)
\(68\) 0 0
\(69\) 4.99604 0.601453
\(70\) 0 0
\(71\) −7.08679 −0.841047 −0.420524 0.907282i \(-0.638154\pi\)
−0.420524 + 0.907282i \(0.638154\pi\)
\(72\) 0 0
\(73\) 12.5266 1.46613 0.733064 0.680159i \(-0.238089\pi\)
0.733064 + 0.680159i \(0.238089\pi\)
\(74\) 0 0
\(75\) 1.65879 0.191541
\(76\) 0 0
\(77\) 6.37945 0.727006
\(78\) 0 0
\(79\) 13.7459 1.54653 0.773265 0.634083i \(-0.218622\pi\)
0.773265 + 0.634083i \(0.218622\pi\)
\(80\) 0 0
\(81\) −8.19305 −0.910339
\(82\) 0 0
\(83\) 5.57487 0.611922 0.305961 0.952044i \(-0.401022\pi\)
0.305961 + 0.952044i \(0.401022\pi\)
\(84\) 0 0
\(85\) −5.70395 −0.618680
\(86\) 0 0
\(87\) 10.9276 1.17156
\(88\) 0 0
\(89\) −9.73995 −1.03243 −0.516217 0.856458i \(-0.672660\pi\)
−0.516217 + 0.856458i \(0.672660\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 3.26587 0.338655
\(94\) 0 0
\(95\) 0.636838 0.0653382
\(96\) 0 0
\(97\) −10.0261 −1.01800 −0.508999 0.860767i \(-0.669984\pi\)
−0.508999 + 0.860767i \(0.669984\pi\)
\(98\) 0 0
\(99\) −0.372519 −0.0374395
\(100\) 0 0
\(101\) 13.8952 1.38263 0.691314 0.722555i \(-0.257032\pi\)
0.691314 + 0.722555i \(0.257032\pi\)
\(102\) 0 0
\(103\) 16.6183 1.63745 0.818724 0.574187i \(-0.194682\pi\)
0.818724 + 0.574187i \(0.194682\pi\)
\(104\) 0 0
\(105\) −7.05673 −0.688666
\(106\) 0 0
\(107\) 18.2026 1.75971 0.879855 0.475243i \(-0.157640\pi\)
0.879855 + 0.475243i \(0.157640\pi\)
\(108\) 0 0
\(109\) −16.3445 −1.56552 −0.782758 0.622326i \(-0.786188\pi\)
−0.782758 + 0.622326i \(0.786188\pi\)
\(110\) 0 0
\(111\) −12.5547 −1.19164
\(112\) 0 0
\(113\) 11.6082 1.09200 0.546002 0.837784i \(-0.316149\pi\)
0.546002 + 0.837784i \(0.316149\pi\)
\(114\) 0 0
\(115\) −3.01186 −0.280857
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 24.2654 2.22441
\(120\) 0 0
\(121\) −8.75124 −0.795567
\(122\) 0 0
\(123\) −16.7062 −1.50635
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 2.51211 0.222914 0.111457 0.993769i \(-0.464448\pi\)
0.111457 + 0.993769i \(0.464448\pi\)
\(128\) 0 0
\(129\) 8.40368 0.739903
\(130\) 0 0
\(131\) −19.9291 −1.74122 −0.870608 0.491978i \(-0.836274\pi\)
−0.870608 + 0.491978i \(0.836274\pi\)
\(132\) 0 0
\(133\) −2.70920 −0.234918
\(134\) 0 0
\(135\) 5.38844 0.463763
\(136\) 0 0
\(137\) −11.3552 −0.970141 −0.485070 0.874475i \(-0.661206\pi\)
−0.485070 + 0.874475i \(0.661206\pi\)
\(138\) 0 0
\(139\) −17.4904 −1.48352 −0.741759 0.670666i \(-0.766008\pi\)
−0.741759 + 0.670666i \(0.766008\pi\)
\(140\) 0 0
\(141\) −5.10916 −0.430269
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −6.58768 −0.547077
\(146\) 0 0
\(147\) 18.4088 1.51833
\(148\) 0 0
\(149\) 2.27148 0.186087 0.0930434 0.995662i \(-0.470340\pi\)
0.0930434 + 0.995662i \(0.470340\pi\)
\(150\) 0 0
\(151\) 16.4094 1.33538 0.667689 0.744440i \(-0.267284\pi\)
0.667689 + 0.744440i \(0.267284\pi\)
\(152\) 0 0
\(153\) −1.41694 −0.114553
\(154\) 0 0
\(155\) −1.96883 −0.158140
\(156\) 0 0
\(157\) −1.21588 −0.0970377 −0.0485188 0.998822i \(-0.515450\pi\)
−0.0485188 + 0.998822i \(0.515450\pi\)
\(158\) 0 0
\(159\) 3.75895 0.298104
\(160\) 0 0
\(161\) 12.8129 1.00980
\(162\) 0 0
\(163\) 11.4749 0.898784 0.449392 0.893335i \(-0.351641\pi\)
0.449392 + 0.893335i \(0.351641\pi\)
\(164\) 0 0
\(165\) −2.48750 −0.193652
\(166\) 0 0
\(167\) −6.54231 −0.506259 −0.253129 0.967432i \(-0.581460\pi\)
−0.253129 + 0.967432i \(0.581460\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 0.158200 0.0120978
\(172\) 0 0
\(173\) −16.6643 −1.26696 −0.633481 0.773758i \(-0.718375\pi\)
−0.633481 + 0.773758i \(0.718375\pi\)
\(174\) 0 0
\(175\) 4.25414 0.321583
\(176\) 0 0
\(177\) 1.60888 0.120931
\(178\) 0 0
\(179\) −0.958705 −0.0716569 −0.0358285 0.999358i \(-0.511407\pi\)
−0.0358285 + 0.999358i \(0.511407\pi\)
\(180\) 0 0
\(181\) 5.31264 0.394885 0.197443 0.980314i \(-0.436736\pi\)
0.197443 + 0.980314i \(0.436736\pi\)
\(182\) 0 0
\(183\) −18.3297 −1.35497
\(184\) 0 0
\(185\) 7.56859 0.556454
\(186\) 0 0
\(187\) 8.55357 0.625499
\(188\) 0 0
\(189\) −22.9232 −1.66742
\(190\) 0 0
\(191\) 4.70838 0.340686 0.170343 0.985385i \(-0.445512\pi\)
0.170343 + 0.985385i \(0.445512\pi\)
\(192\) 0 0
\(193\) 16.3532 1.17713 0.588563 0.808451i \(-0.299694\pi\)
0.588563 + 0.808451i \(0.299694\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −13.4002 −0.954728 −0.477364 0.878706i \(-0.658408\pi\)
−0.477364 + 0.878706i \(0.658408\pi\)
\(198\) 0 0
\(199\) 3.34363 0.237023 0.118512 0.992953i \(-0.462188\pi\)
0.118512 + 0.992953i \(0.462188\pi\)
\(200\) 0 0
\(201\) 21.7720 1.53568
\(202\) 0 0
\(203\) 28.0249 1.96697
\(204\) 0 0
\(205\) 10.0713 0.703413
\(206\) 0 0
\(207\) −0.748189 −0.0520028
\(208\) 0 0
\(209\) −0.954994 −0.0660583
\(210\) 0 0
\(211\) 5.87702 0.404591 0.202295 0.979325i \(-0.435160\pi\)
0.202295 + 0.979325i \(0.435160\pi\)
\(212\) 0 0
\(213\) −11.7555 −0.805474
\(214\) 0 0
\(215\) −5.06615 −0.345509
\(216\) 0 0
\(217\) 8.37567 0.568577
\(218\) 0 0
\(219\) 20.7790 1.40412
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 17.1066 1.14554 0.572772 0.819715i \(-0.305868\pi\)
0.572772 + 0.819715i \(0.305868\pi\)
\(224\) 0 0
\(225\) −0.248414 −0.0165610
\(226\) 0 0
\(227\) −12.9169 −0.857323 −0.428661 0.903465i \(-0.641015\pi\)
−0.428661 + 0.903465i \(0.641015\pi\)
\(228\) 0 0
\(229\) 22.2246 1.46864 0.734322 0.678801i \(-0.237500\pi\)
0.734322 + 0.678801i \(0.237500\pi\)
\(230\) 0 0
\(231\) 10.5822 0.696256
\(232\) 0 0
\(233\) −5.15849 −0.337944 −0.168972 0.985621i \(-0.554045\pi\)
−0.168972 + 0.985621i \(0.554045\pi\)
\(234\) 0 0
\(235\) 3.08005 0.200920
\(236\) 0 0
\(237\) 22.8015 1.48112
\(238\) 0 0
\(239\) −30.6724 −1.98403 −0.992016 0.126109i \(-0.959751\pi\)
−0.992016 + 0.126109i \(0.959751\pi\)
\(240\) 0 0
\(241\) 17.1733 1.10623 0.553115 0.833105i \(-0.313439\pi\)
0.553115 + 0.833105i \(0.313439\pi\)
\(242\) 0 0
\(243\) 2.57477 0.165171
\(244\) 0 0
\(245\) −11.0977 −0.709008
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 9.24755 0.586039
\(250\) 0 0
\(251\) −4.82309 −0.304431 −0.152215 0.988347i \(-0.548641\pi\)
−0.152215 + 0.988347i \(0.548641\pi\)
\(252\) 0 0
\(253\) 4.51654 0.283953
\(254\) 0 0
\(255\) −9.46166 −0.592512
\(256\) 0 0
\(257\) −22.9376 −1.43081 −0.715403 0.698712i \(-0.753757\pi\)
−0.715403 + 0.698712i \(0.753757\pi\)
\(258\) 0 0
\(259\) −32.1978 −2.00068
\(260\) 0 0
\(261\) −1.63648 −0.101295
\(262\) 0 0
\(263\) 12.2220 0.753639 0.376820 0.926287i \(-0.377018\pi\)
0.376820 + 0.926287i \(0.377018\pi\)
\(264\) 0 0
\(265\) −2.26608 −0.139204
\(266\) 0 0
\(267\) −16.1565 −0.988764
\(268\) 0 0
\(269\) −29.3278 −1.78815 −0.894073 0.447922i \(-0.852164\pi\)
−0.894073 + 0.447922i \(0.852164\pi\)
\(270\) 0 0
\(271\) 12.9094 0.784190 0.392095 0.919925i \(-0.371750\pi\)
0.392095 + 0.919925i \(0.371750\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.49959 0.0904285
\(276\) 0 0
\(277\) 3.40741 0.204731 0.102366 0.994747i \(-0.467359\pi\)
0.102366 + 0.994747i \(0.467359\pi\)
\(278\) 0 0
\(279\) −0.489085 −0.0292807
\(280\) 0 0
\(281\) 20.7553 1.23816 0.619080 0.785328i \(-0.287506\pi\)
0.619080 + 0.785328i \(0.287506\pi\)
\(282\) 0 0
\(283\) −1.01218 −0.0601679 −0.0300839 0.999547i \(-0.509577\pi\)
−0.0300839 + 0.999547i \(0.509577\pi\)
\(284\) 0 0
\(285\) 1.05638 0.0625746
\(286\) 0 0
\(287\) −42.8449 −2.52906
\(288\) 0 0
\(289\) 15.5350 0.913826
\(290\) 0 0
\(291\) −16.6312 −0.974940
\(292\) 0 0
\(293\) −6.59637 −0.385364 −0.192682 0.981261i \(-0.561719\pi\)
−0.192682 + 0.981261i \(0.561719\pi\)
\(294\) 0 0
\(295\) −0.969910 −0.0564703
\(296\) 0 0
\(297\) −8.08043 −0.468874
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 21.5521 1.24224
\(302\) 0 0
\(303\) 23.0493 1.32415
\(304\) 0 0
\(305\) 11.0500 0.632722
\(306\) 0 0
\(307\) −11.7321 −0.669587 −0.334793 0.942292i \(-0.608666\pi\)
−0.334793 + 0.942292i \(0.608666\pi\)
\(308\) 0 0
\(309\) 27.5663 1.56819
\(310\) 0 0
\(311\) −11.1618 −0.632925 −0.316463 0.948605i \(-0.602495\pi\)
−0.316463 + 0.948605i \(0.602495\pi\)
\(312\) 0 0
\(313\) 14.5600 0.822980 0.411490 0.911414i \(-0.365008\pi\)
0.411490 + 0.911414i \(0.365008\pi\)
\(314\) 0 0
\(315\) 1.05679 0.0595434
\(316\) 0 0
\(317\) 8.31408 0.466965 0.233483 0.972361i \(-0.424988\pi\)
0.233483 + 0.972361i \(0.424988\pi\)
\(318\) 0 0
\(319\) 9.87880 0.553107
\(320\) 0 0
\(321\) 30.1943 1.68528
\(322\) 0 0
\(323\) −3.63250 −0.202117
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −27.1120 −1.49930
\(328\) 0 0
\(329\) −13.1030 −0.722391
\(330\) 0 0
\(331\) −35.8927 −1.97284 −0.986421 0.164235i \(-0.947485\pi\)
−0.986421 + 0.164235i \(0.947485\pi\)
\(332\) 0 0
\(333\) 1.88015 0.103031
\(334\) 0 0
\(335\) −13.1252 −0.717107
\(336\) 0 0
\(337\) 19.7590 1.07634 0.538171 0.842836i \(-0.319115\pi\)
0.538171 + 0.842836i \(0.319115\pi\)
\(338\) 0 0
\(339\) 19.2555 1.04582
\(340\) 0 0
\(341\) 2.95242 0.159883
\(342\) 0 0
\(343\) 17.4323 0.941256
\(344\) 0 0
\(345\) −4.99604 −0.268978
\(346\) 0 0
\(347\) 10.8791 0.584020 0.292010 0.956415i \(-0.405676\pi\)
0.292010 + 0.956415i \(0.405676\pi\)
\(348\) 0 0
\(349\) −4.96959 −0.266016 −0.133008 0.991115i \(-0.542464\pi\)
−0.133008 + 0.991115i \(0.542464\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −15.6620 −0.833606 −0.416803 0.908997i \(-0.636849\pi\)
−0.416803 + 0.908997i \(0.636849\pi\)
\(354\) 0 0
\(355\) 7.08679 0.376128
\(356\) 0 0
\(357\) 40.2512 2.13032
\(358\) 0 0
\(359\) −23.1319 −1.22086 −0.610428 0.792072i \(-0.709002\pi\)
−0.610428 + 0.792072i \(0.709002\pi\)
\(360\) 0 0
\(361\) −18.5944 −0.978655
\(362\) 0 0
\(363\) −14.5165 −0.761917
\(364\) 0 0
\(365\) −12.5266 −0.655673
\(366\) 0 0
\(367\) −18.9594 −0.989671 −0.494836 0.868987i \(-0.664772\pi\)
−0.494836 + 0.868987i \(0.664772\pi\)
\(368\) 0 0
\(369\) 2.50187 0.130242
\(370\) 0 0
\(371\) 9.64022 0.500495
\(372\) 0 0
\(373\) 3.11921 0.161507 0.0807534 0.996734i \(-0.474267\pi\)
0.0807534 + 0.996734i \(0.474267\pi\)
\(374\) 0 0
\(375\) −1.65879 −0.0856596
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −4.68573 −0.240690 −0.120345 0.992732i \(-0.538400\pi\)
−0.120345 + 0.992732i \(0.538400\pi\)
\(380\) 0 0
\(381\) 4.16707 0.213485
\(382\) 0 0
\(383\) −18.9467 −0.968130 −0.484065 0.875032i \(-0.660840\pi\)
−0.484065 + 0.875032i \(0.660840\pi\)
\(384\) 0 0
\(385\) −6.37945 −0.325127
\(386\) 0 0
\(387\) −1.25850 −0.0639734
\(388\) 0 0
\(389\) 15.7332 0.797702 0.398851 0.917016i \(-0.369409\pi\)
0.398851 + 0.917016i \(0.369409\pi\)
\(390\) 0 0
\(391\) 17.1795 0.868805
\(392\) 0 0
\(393\) −33.0582 −1.66757
\(394\) 0 0
\(395\) −13.7459 −0.691630
\(396\) 0 0
\(397\) −13.0486 −0.654890 −0.327445 0.944870i \(-0.606188\pi\)
−0.327445 + 0.944870i \(0.606188\pi\)
\(398\) 0 0
\(399\) −4.49400 −0.224981
\(400\) 0 0
\(401\) 2.09694 0.104716 0.0523582 0.998628i \(-0.483326\pi\)
0.0523582 + 0.998628i \(0.483326\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 8.19305 0.407116
\(406\) 0 0
\(407\) −11.3497 −0.562586
\(408\) 0 0
\(409\) −10.0168 −0.495298 −0.247649 0.968850i \(-0.579658\pi\)
−0.247649 + 0.968850i \(0.579658\pi\)
\(410\) 0 0
\(411\) −18.8359 −0.929107
\(412\) 0 0
\(413\) 4.12613 0.203034
\(414\) 0 0
\(415\) −5.57487 −0.273660
\(416\) 0 0
\(417\) −29.0129 −1.42077
\(418\) 0 0
\(419\) −38.7042 −1.89083 −0.945413 0.325876i \(-0.894341\pi\)
−0.945413 + 0.325876i \(0.894341\pi\)
\(420\) 0 0
\(421\) 4.88277 0.237972 0.118986 0.992896i \(-0.462036\pi\)
0.118986 + 0.992896i \(0.462036\pi\)
\(422\) 0 0
\(423\) 0.765129 0.0372019
\(424\) 0 0
\(425\) 5.70395 0.276682
\(426\) 0 0
\(427\) −47.0083 −2.27489
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 8.48596 0.408754 0.204377 0.978892i \(-0.434483\pi\)
0.204377 + 0.978892i \(0.434483\pi\)
\(432\) 0 0
\(433\) −34.6384 −1.66461 −0.832307 0.554315i \(-0.812980\pi\)
−0.832307 + 0.554315i \(0.812980\pi\)
\(434\) 0 0
\(435\) −10.9276 −0.523938
\(436\) 0 0
\(437\) −1.91807 −0.0917537
\(438\) 0 0
\(439\) 36.2573 1.73047 0.865234 0.501368i \(-0.167170\pi\)
0.865234 + 0.501368i \(0.167170\pi\)
\(440\) 0 0
\(441\) −2.75683 −0.131278
\(442\) 0 0
\(443\) 21.1718 1.00590 0.502950 0.864315i \(-0.332248\pi\)
0.502950 + 0.864315i \(0.332248\pi\)
\(444\) 0 0
\(445\) 9.73995 0.461718
\(446\) 0 0
\(447\) 3.76791 0.178216
\(448\) 0 0
\(449\) −4.18848 −0.197667 −0.0988333 0.995104i \(-0.531511\pi\)
−0.0988333 + 0.995104i \(0.531511\pi\)
\(450\) 0 0
\(451\) −15.1028 −0.711165
\(452\) 0 0
\(453\) 27.2198 1.27890
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −35.5088 −1.66103 −0.830516 0.556994i \(-0.811954\pi\)
−0.830516 + 0.556994i \(0.811954\pi\)
\(458\) 0 0
\(459\) −30.7354 −1.43460
\(460\) 0 0
\(461\) −4.40907 −0.205351 −0.102676 0.994715i \(-0.532740\pi\)
−0.102676 + 0.994715i \(0.532740\pi\)
\(462\) 0 0
\(463\) 17.5773 0.816885 0.408442 0.912784i \(-0.366072\pi\)
0.408442 + 0.912784i \(0.366072\pi\)
\(464\) 0 0
\(465\) −3.26587 −0.151451
\(466\) 0 0
\(467\) −4.21327 −0.194967 −0.0974835 0.995237i \(-0.531079\pi\)
−0.0974835 + 0.995237i \(0.531079\pi\)
\(468\) 0 0
\(469\) 55.8366 2.57829
\(470\) 0 0
\(471\) −2.01689 −0.0929333
\(472\) 0 0
\(473\) 7.59713 0.349316
\(474\) 0 0
\(475\) −0.636838 −0.0292202
\(476\) 0 0
\(477\) −0.562926 −0.0257746
\(478\) 0 0
\(479\) 20.7016 0.945880 0.472940 0.881095i \(-0.343193\pi\)
0.472940 + 0.881095i \(0.343193\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 21.2539 0.967085
\(484\) 0 0
\(485\) 10.0261 0.455263
\(486\) 0 0
\(487\) 15.4795 0.701442 0.350721 0.936480i \(-0.385937\pi\)
0.350721 + 0.936480i \(0.385937\pi\)
\(488\) 0 0
\(489\) 19.0345 0.860768
\(490\) 0 0
\(491\) −34.3346 −1.54950 −0.774750 0.632268i \(-0.782124\pi\)
−0.774750 + 0.632268i \(0.782124\pi\)
\(492\) 0 0
\(493\) 37.5758 1.69233
\(494\) 0 0
\(495\) 0.372519 0.0167435
\(496\) 0 0
\(497\) −30.1482 −1.35233
\(498\) 0 0
\(499\) 21.1450 0.946579 0.473290 0.880907i \(-0.343066\pi\)
0.473290 + 0.880907i \(0.343066\pi\)
\(500\) 0 0
\(501\) −10.8523 −0.484846
\(502\) 0 0
\(503\) 3.46540 0.154515 0.0772573 0.997011i \(-0.475384\pi\)
0.0772573 + 0.997011i \(0.475384\pi\)
\(504\) 0 0
\(505\) −13.8952 −0.618330
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 17.5573 0.778213 0.389107 0.921193i \(-0.372784\pi\)
0.389107 + 0.921193i \(0.372784\pi\)
\(510\) 0 0
\(511\) 53.2900 2.35741
\(512\) 0 0
\(513\) 3.43157 0.151507
\(514\) 0 0
\(515\) −16.6183 −0.732289
\(516\) 0 0
\(517\) −4.61880 −0.203135
\(518\) 0 0
\(519\) −27.6426 −1.21337
\(520\) 0 0
\(521\) 39.7429 1.74117 0.870585 0.492018i \(-0.163741\pi\)
0.870585 + 0.492018i \(0.163741\pi\)
\(522\) 0 0
\(523\) −27.7468 −1.21328 −0.606641 0.794976i \(-0.707483\pi\)
−0.606641 + 0.794976i \(0.707483\pi\)
\(524\) 0 0
\(525\) 7.05673 0.307981
\(526\) 0 0
\(527\) 11.2301 0.489190
\(528\) 0 0
\(529\) −13.9287 −0.605596
\(530\) 0 0
\(531\) −0.240939 −0.0104559
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −18.2026 −0.786966
\(536\) 0 0
\(537\) −1.59029 −0.0686261
\(538\) 0 0
\(539\) 16.6420 0.716822
\(540\) 0 0
\(541\) −43.1354 −1.85453 −0.927267 0.374400i \(-0.877849\pi\)
−0.927267 + 0.374400i \(0.877849\pi\)
\(542\) 0 0
\(543\) 8.81255 0.378183
\(544\) 0 0
\(545\) 16.3445 0.700120
\(546\) 0 0
\(547\) −22.4698 −0.960740 −0.480370 0.877066i \(-0.659498\pi\)
−0.480370 + 0.877066i \(0.659498\pi\)
\(548\) 0 0
\(549\) 2.74498 0.117153
\(550\) 0 0
\(551\) −4.19529 −0.178725
\(552\) 0 0
\(553\) 58.4769 2.48669
\(554\) 0 0
\(555\) 12.5547 0.532917
\(556\) 0 0
\(557\) −24.2107 −1.02584 −0.512920 0.858436i \(-0.671436\pi\)
−0.512920 + 0.858436i \(0.671436\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 14.1886 0.599042
\(562\) 0 0
\(563\) −8.87480 −0.374028 −0.187014 0.982357i \(-0.559881\pi\)
−0.187014 + 0.982357i \(0.559881\pi\)
\(564\) 0 0
\(565\) −11.6082 −0.488359
\(566\) 0 0
\(567\) −34.8544 −1.46375
\(568\) 0 0
\(569\) −27.1897 −1.13985 −0.569927 0.821695i \(-0.693028\pi\)
−0.569927 + 0.821695i \(0.693028\pi\)
\(570\) 0 0
\(571\) 13.5393 0.566604 0.283302 0.959031i \(-0.408570\pi\)
0.283302 + 0.959031i \(0.408570\pi\)
\(572\) 0 0
\(573\) 7.81021 0.326276
\(574\) 0 0
\(575\) 3.01186 0.125603
\(576\) 0 0
\(577\) 40.3084 1.67806 0.839031 0.544084i \(-0.183123\pi\)
0.839031 + 0.544084i \(0.183123\pi\)
\(578\) 0 0
\(579\) 27.1265 1.12734
\(580\) 0 0
\(581\) 23.7163 0.983918
\(582\) 0 0
\(583\) 3.39818 0.140738
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −17.9302 −0.740060 −0.370030 0.929020i \(-0.620653\pi\)
−0.370030 + 0.929020i \(0.620653\pi\)
\(588\) 0 0
\(589\) −1.25382 −0.0516629
\(590\) 0 0
\(591\) −22.2282 −0.914346
\(592\) 0 0
\(593\) −44.4851 −1.82678 −0.913392 0.407081i \(-0.866547\pi\)
−0.913392 + 0.407081i \(0.866547\pi\)
\(594\) 0 0
\(595\) −24.2654 −0.994785
\(596\) 0 0
\(597\) 5.54637 0.226998
\(598\) 0 0
\(599\) 0.145172 0.00593155 0.00296577 0.999996i \(-0.499056\pi\)
0.00296577 + 0.999996i \(0.499056\pi\)
\(600\) 0 0
\(601\) 19.1209 0.779958 0.389979 0.920824i \(-0.372482\pi\)
0.389979 + 0.920824i \(0.372482\pi\)
\(602\) 0 0
\(603\) −3.26049 −0.132778
\(604\) 0 0
\(605\) 8.75124 0.355789
\(606\) 0 0
\(607\) −33.6276 −1.36490 −0.682451 0.730931i \(-0.739086\pi\)
−0.682451 + 0.730931i \(0.739086\pi\)
\(608\) 0 0
\(609\) 46.4875 1.88377
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −31.2683 −1.26292 −0.631458 0.775410i \(-0.717543\pi\)
−0.631458 + 0.775410i \(0.717543\pi\)
\(614\) 0 0
\(615\) 16.7062 0.673661
\(616\) 0 0
\(617\) 8.22626 0.331177 0.165588 0.986195i \(-0.447048\pi\)
0.165588 + 0.986195i \(0.447048\pi\)
\(618\) 0 0
\(619\) −17.6835 −0.710759 −0.355379 0.934722i \(-0.615648\pi\)
−0.355379 + 0.934722i \(0.615648\pi\)
\(620\) 0 0
\(621\) −16.2292 −0.651256
\(622\) 0 0
\(623\) −41.4351 −1.66006
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −1.58414 −0.0632643
\(628\) 0 0
\(629\) −43.1708 −1.72133
\(630\) 0 0
\(631\) 32.8689 1.30849 0.654245 0.756283i \(-0.272987\pi\)
0.654245 + 0.756283i \(0.272987\pi\)
\(632\) 0 0
\(633\) 9.74874 0.387478
\(634\) 0 0
\(635\) −2.51211 −0.0996901
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 1.76046 0.0696428
\(640\) 0 0
\(641\) −14.9749 −0.591471 −0.295736 0.955270i \(-0.595565\pi\)
−0.295736 + 0.955270i \(0.595565\pi\)
\(642\) 0 0
\(643\) 22.0696 0.870341 0.435170 0.900348i \(-0.356688\pi\)
0.435170 + 0.900348i \(0.356688\pi\)
\(644\) 0 0
\(645\) −8.40368 −0.330895
\(646\) 0 0
\(647\) 30.2132 1.18780 0.593902 0.804538i \(-0.297587\pi\)
0.593902 + 0.804538i \(0.297587\pi\)
\(648\) 0 0
\(649\) 1.45446 0.0570927
\(650\) 0 0
\(651\) 13.8935 0.544528
\(652\) 0 0
\(653\) 30.5523 1.19561 0.597803 0.801643i \(-0.296041\pi\)
0.597803 + 0.801643i \(0.296041\pi\)
\(654\) 0 0
\(655\) 19.9291 0.778695
\(656\) 0 0
\(657\) −3.11179 −0.121402
\(658\) 0 0
\(659\) −33.9179 −1.32125 −0.660626 0.750715i \(-0.729709\pi\)
−0.660626 + 0.750715i \(0.729709\pi\)
\(660\) 0 0
\(661\) −17.9952 −0.699931 −0.349965 0.936763i \(-0.613807\pi\)
−0.349965 + 0.936763i \(0.613807\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.70920 0.105058
\(666\) 0 0
\(667\) 19.8412 0.768254
\(668\) 0 0
\(669\) 28.3763 1.09709
\(670\) 0 0
\(671\) −16.5704 −0.639695
\(672\) 0 0
\(673\) −29.6026 −1.14110 −0.570548 0.821265i \(-0.693269\pi\)
−0.570548 + 0.821265i \(0.693269\pi\)
\(674\) 0 0
\(675\) −5.38844 −0.207401
\(676\) 0 0
\(677\) 3.06371 0.117748 0.0588740 0.998265i \(-0.481249\pi\)
0.0588740 + 0.998265i \(0.481249\pi\)
\(678\) 0 0
\(679\) −42.6525 −1.63685
\(680\) 0 0
\(681\) −21.4264 −0.821060
\(682\) 0 0
\(683\) −24.2517 −0.927964 −0.463982 0.885845i \(-0.653580\pi\)
−0.463982 + 0.885845i \(0.653580\pi\)
\(684\) 0 0
\(685\) 11.3552 0.433860
\(686\) 0 0
\(687\) 36.8660 1.40653
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −26.4961 −1.00796 −0.503979 0.863716i \(-0.668131\pi\)
−0.503979 + 0.863716i \(0.668131\pi\)
\(692\) 0 0
\(693\) −1.58475 −0.0601996
\(694\) 0 0
\(695\) 17.4904 0.663449
\(696\) 0 0
\(697\) −57.4464 −2.17594
\(698\) 0 0
\(699\) −8.55685 −0.323650
\(700\) 0 0
\(701\) −31.4444 −1.18764 −0.593819 0.804599i \(-0.702380\pi\)
−0.593819 + 0.804599i \(0.702380\pi\)
\(702\) 0 0
\(703\) 4.81997 0.181788
\(704\) 0 0
\(705\) 5.10916 0.192422
\(706\) 0 0
\(707\) 59.1123 2.22315
\(708\) 0 0
\(709\) 34.6756 1.30227 0.651134 0.758963i \(-0.274294\pi\)
0.651134 + 0.758963i \(0.274294\pi\)
\(710\) 0 0
\(711\) −3.41467 −0.128060
\(712\) 0 0
\(713\) 5.92983 0.222074
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −50.8791 −1.90011
\(718\) 0 0
\(719\) −25.4213 −0.948053 −0.474027 0.880511i \(-0.657200\pi\)
−0.474027 + 0.880511i \(0.657200\pi\)
\(720\) 0 0
\(721\) 70.6966 2.63288
\(722\) 0 0
\(723\) 28.4869 1.05944
\(724\) 0 0
\(725\) 6.58768 0.244660
\(726\) 0 0
\(727\) −5.13306 −0.190375 −0.0951873 0.995459i \(-0.530345\pi\)
−0.0951873 + 0.995459i \(0.530345\pi\)
\(728\) 0 0
\(729\) 28.8501 1.06852
\(730\) 0 0
\(731\) 28.8971 1.06880
\(732\) 0 0
\(733\) −22.2813 −0.822979 −0.411490 0.911414i \(-0.634991\pi\)
−0.411490 + 0.911414i \(0.634991\pi\)
\(734\) 0 0
\(735\) −18.4088 −0.679019
\(736\) 0 0
\(737\) 19.6824 0.725011
\(738\) 0 0
\(739\) −1.27189 −0.0467871 −0.0233935 0.999726i \(-0.507447\pi\)
−0.0233935 + 0.999726i \(0.507447\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 23.3848 0.857906 0.428953 0.903327i \(-0.358883\pi\)
0.428953 + 0.903327i \(0.358883\pi\)
\(744\) 0 0
\(745\) −2.27148 −0.0832205
\(746\) 0 0
\(747\) −1.38488 −0.0506701
\(748\) 0 0
\(749\) 77.4364 2.82946
\(750\) 0 0
\(751\) −2.31524 −0.0844843 −0.0422422 0.999107i \(-0.513450\pi\)
−0.0422422 + 0.999107i \(0.513450\pi\)
\(752\) 0 0
\(753\) −8.00050 −0.291554
\(754\) 0 0
\(755\) −16.4094 −0.597199
\(756\) 0 0
\(757\) 41.4080 1.50500 0.752500 0.658592i \(-0.228848\pi\)
0.752500 + 0.658592i \(0.228848\pi\)
\(758\) 0 0
\(759\) 7.49200 0.271942
\(760\) 0 0
\(761\) 23.5369 0.853212 0.426606 0.904438i \(-0.359709\pi\)
0.426606 + 0.904438i \(0.359709\pi\)
\(762\) 0 0
\(763\) −69.5317 −2.51722
\(764\) 0 0
\(765\) 1.41694 0.0512297
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −8.84837 −0.319080 −0.159540 0.987191i \(-0.551001\pi\)
−0.159540 + 0.987191i \(0.551001\pi\)
\(770\) 0 0
\(771\) −38.0486 −1.37029
\(772\) 0 0
\(773\) 6.15591 0.221413 0.110706 0.993853i \(-0.464689\pi\)
0.110706 + 0.993853i \(0.464689\pi\)
\(774\) 0 0
\(775\) 1.96883 0.0707223
\(776\) 0 0
\(777\) −53.4095 −1.91605
\(778\) 0 0
\(779\) 6.41382 0.229799
\(780\) 0 0
\(781\) −10.6273 −0.380273
\(782\) 0 0
\(783\) −35.4973 −1.26857
\(784\) 0 0
\(785\) 1.21588 0.0433966
\(786\) 0 0
\(787\) 7.12007 0.253803 0.126901 0.991915i \(-0.459497\pi\)
0.126901 + 0.991915i \(0.459497\pi\)
\(788\) 0 0
\(789\) 20.2737 0.721763
\(790\) 0 0
\(791\) 49.3828 1.75585
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −3.75895 −0.133316
\(796\) 0 0
\(797\) 4.93234 0.174712 0.0873562 0.996177i \(-0.472158\pi\)
0.0873562 + 0.996177i \(0.472158\pi\)
\(798\) 0 0
\(799\) −17.5685 −0.621528
\(800\) 0 0
\(801\) 2.41954 0.0854904
\(802\) 0 0
\(803\) 18.7847 0.662899
\(804\) 0 0
\(805\) −12.8129 −0.451595
\(806\) 0 0
\(807\) −48.6486 −1.71251
\(808\) 0 0
\(809\) −47.6258 −1.67443 −0.837217 0.546871i \(-0.815819\pi\)
−0.837217 + 0.546871i \(0.815819\pi\)
\(810\) 0 0
\(811\) −15.2644 −0.536005 −0.268002 0.963418i \(-0.586363\pi\)
−0.268002 + 0.963418i \(0.586363\pi\)
\(812\) 0 0
\(813\) 21.4140 0.751021
\(814\) 0 0
\(815\) −11.4749 −0.401948
\(816\) 0 0
\(817\) −3.22632 −0.112875
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 24.3914 0.851267 0.425634 0.904896i \(-0.360051\pi\)
0.425634 + 0.904896i \(0.360051\pi\)
\(822\) 0 0
\(823\) 7.63554 0.266158 0.133079 0.991105i \(-0.457514\pi\)
0.133079 + 0.991105i \(0.457514\pi\)
\(824\) 0 0
\(825\) 2.48750 0.0866036
\(826\) 0 0
\(827\) −32.8801 −1.14335 −0.571677 0.820479i \(-0.693707\pi\)
−0.571677 + 0.820479i \(0.693707\pi\)
\(828\) 0 0
\(829\) 2.73117 0.0948576 0.0474288 0.998875i \(-0.484897\pi\)
0.0474288 + 0.998875i \(0.484897\pi\)
\(830\) 0 0
\(831\) 5.65217 0.196072
\(832\) 0 0
\(833\) 63.3009 2.19325
\(834\) 0 0
\(835\) 6.54231 0.226406
\(836\) 0 0
\(837\) −10.6089 −0.366697
\(838\) 0 0
\(839\) 32.0624 1.10692 0.553459 0.832877i \(-0.313308\pi\)
0.553459 + 0.832877i \(0.313308\pi\)
\(840\) 0 0
\(841\) 14.3976 0.496468
\(842\) 0 0
\(843\) 34.4287 1.18579
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −37.2290 −1.27920
\(848\) 0 0
\(849\) −1.67900 −0.0576230
\(850\) 0 0
\(851\) −22.7955 −0.781420
\(852\) 0 0
\(853\) −39.7908 −1.36241 −0.681206 0.732092i \(-0.738544\pi\)
−0.681206 + 0.732092i \(0.738544\pi\)
\(854\) 0 0
\(855\) −0.158200 −0.00541032
\(856\) 0 0
\(857\) −42.3226 −1.44571 −0.722856 0.690999i \(-0.757171\pi\)
−0.722856 + 0.690999i \(0.757171\pi\)
\(858\) 0 0
\(859\) 18.1451 0.619104 0.309552 0.950883i \(-0.399821\pi\)
0.309552 + 0.950883i \(0.399821\pi\)
\(860\) 0 0
\(861\) −71.0707 −2.42208
\(862\) 0 0
\(863\) −19.3054 −0.657165 −0.328582 0.944475i \(-0.606571\pi\)
−0.328582 + 0.944475i \(0.606571\pi\)
\(864\) 0 0
\(865\) 16.6643 0.566603
\(866\) 0 0
\(867\) 25.7694 0.875174
\(868\) 0 0
\(869\) 20.6131 0.699252
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 2.49063 0.0842951
\(874\) 0 0
\(875\) −4.25414 −0.143816
\(876\) 0 0
\(877\) −14.1636 −0.478270 −0.239135 0.970986i \(-0.576864\pi\)
−0.239135 + 0.970986i \(0.576864\pi\)
\(878\) 0 0
\(879\) −10.9420 −0.369064
\(880\) 0 0
\(881\) 17.0142 0.573222 0.286611 0.958047i \(-0.407471\pi\)
0.286611 + 0.958047i \(0.407471\pi\)
\(882\) 0 0
\(883\) 0.539474 0.0181548 0.00907738 0.999959i \(-0.497111\pi\)
0.00907738 + 0.999959i \(0.497111\pi\)
\(884\) 0 0
\(885\) −1.60888 −0.0540818
\(886\) 0 0
\(887\) 7.96905 0.267574 0.133787 0.991010i \(-0.457286\pi\)
0.133787 + 0.991010i \(0.457286\pi\)
\(888\) 0 0
\(889\) 10.6869 0.358427
\(890\) 0 0
\(891\) −12.2862 −0.411603
\(892\) 0 0
\(893\) 1.96150 0.0656390
\(894\) 0 0
\(895\) 0.958705 0.0320460
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 12.9700 0.432574
\(900\) 0 0
\(901\) 12.9256 0.430614
\(902\) 0 0
\(903\) 35.7505 1.18970
\(904\) 0 0
\(905\) −5.31264 −0.176598
\(906\) 0 0
\(907\) −43.8197 −1.45501 −0.727505 0.686103i \(-0.759320\pi\)
−0.727505 + 0.686103i \(0.759320\pi\)
\(908\) 0 0
\(909\) −3.45178 −0.114488
\(910\) 0 0
\(911\) −8.95127 −0.296569 −0.148284 0.988945i \(-0.547375\pi\)
−0.148284 + 0.988945i \(0.547375\pi\)
\(912\) 0 0
\(913\) 8.36000 0.276676
\(914\) 0 0
\(915\) 18.3297 0.605960
\(916\) 0 0
\(917\) −84.7813 −2.79973
\(918\) 0 0
\(919\) 41.1943 1.35887 0.679437 0.733734i \(-0.262224\pi\)
0.679437 + 0.733734i \(0.262224\pi\)
\(920\) 0 0
\(921\) −19.4611 −0.641265
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −7.56859 −0.248854
\(926\) 0 0
\(927\) −4.12822 −0.135589
\(928\) 0 0
\(929\) −7.32126 −0.240203 −0.120101 0.992762i \(-0.538322\pi\)
−0.120101 + 0.992762i \(0.538322\pi\)
\(930\) 0 0
\(931\) −7.06746 −0.231627
\(932\) 0 0
\(933\) −18.5150 −0.606154
\(934\) 0 0
\(935\) −8.55357 −0.279732
\(936\) 0 0
\(937\) −6.88696 −0.224987 −0.112494 0.993652i \(-0.535884\pi\)
−0.112494 + 0.993652i \(0.535884\pi\)
\(938\) 0 0
\(939\) 24.1520 0.788171
\(940\) 0 0
\(941\) 51.9727 1.69426 0.847131 0.531384i \(-0.178328\pi\)
0.847131 + 0.531384i \(0.178328\pi\)
\(942\) 0 0
\(943\) −30.3335 −0.987794
\(944\) 0 0
\(945\) 22.9232 0.745691
\(946\) 0 0
\(947\) −44.0866 −1.43262 −0.716310 0.697782i \(-0.754170\pi\)
−0.716310 + 0.697782i \(0.754170\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 13.7913 0.447214
\(952\) 0 0
\(953\) −16.9713 −0.549755 −0.274877 0.961479i \(-0.588637\pi\)
−0.274877 + 0.961479i \(0.588637\pi\)
\(954\) 0 0
\(955\) −4.70838 −0.152359
\(956\) 0 0
\(957\) 16.3869 0.529712
\(958\) 0 0
\(959\) −48.3067 −1.55990
\(960\) 0 0
\(961\) −27.1237 −0.874959
\(962\) 0 0
\(963\) −4.52178 −0.145712
\(964\) 0 0
\(965\) −16.3532 −0.526427
\(966\) 0 0
\(967\) −8.23698 −0.264883 −0.132442 0.991191i \(-0.542282\pi\)
−0.132442 + 0.991191i \(0.542282\pi\)
\(968\) 0 0
\(969\) −6.02555 −0.193568
\(970\) 0 0
\(971\) −26.1164 −0.838114 −0.419057 0.907960i \(-0.637639\pi\)
−0.419057 + 0.907960i \(0.637639\pi\)
\(972\) 0 0
\(973\) −74.4067 −2.38537
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 7.16938 0.229369 0.114684 0.993402i \(-0.463414\pi\)
0.114684 + 0.993402i \(0.463414\pi\)
\(978\) 0 0
\(979\) −14.6059 −0.466807
\(980\) 0 0
\(981\) 4.06020 0.129632
\(982\) 0 0
\(983\) −28.4268 −0.906675 −0.453338 0.891339i \(-0.649767\pi\)
−0.453338 + 0.891339i \(0.649767\pi\)
\(984\) 0 0
\(985\) 13.4002 0.426967
\(986\) 0 0
\(987\) −21.7351 −0.691836
\(988\) 0 0
\(989\) 15.2585 0.485193
\(990\) 0 0
\(991\) −40.0219 −1.27134 −0.635669 0.771962i \(-0.719276\pi\)
−0.635669 + 0.771962i \(0.719276\pi\)
\(992\) 0 0
\(993\) −59.5385 −1.88940
\(994\) 0 0
\(995\) −3.34363 −0.106000
\(996\) 0 0
\(997\) 53.5360 1.69550 0.847751 0.530395i \(-0.177956\pi\)
0.847751 + 0.530395i \(0.177956\pi\)
\(998\) 0 0
\(999\) 40.7829 1.29031
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 3380.2.a.r.1.7 9
13.5 odd 4 3380.2.f.j.3041.14 18
13.8 odd 4 3380.2.f.j.3041.13 18
13.12 even 2 3380.2.a.s.1.7 yes 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3380.2.a.r.1.7 9 1.1 even 1 trivial
3380.2.a.s.1.7 yes 9 13.12 even 2
3380.2.f.j.3041.13 18 13.8 odd 4
3380.2.f.j.3041.14 18 13.5 odd 4