Properties

Label 960.3.j.e.319.1
Level $960$
Weight $3$
Character 960.319
Analytic conductor $26.158$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 960.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.1581053786\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.389136420864.4
Defining polynomial: \(x^{8} + 5 x^{6} + 24 x^{4} + 80 x^{2} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 319.1
Root \(1.52274 - 1.29664i\) of defining polynomial
Character \(\chi\) \(=\) 960.319
Dual form 960.3.j.e.319.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205 q^{3} +(-4.27492 - 2.59328i) q^{5} +0.837253 q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} +(-4.27492 - 2.59328i) q^{5} +0.837253 q^{7} +3.00000 q^{9} -15.7955i q^{11} +5.18655i q^{13} +(7.40437 + 4.49169i) q^{15} -27.3586i q^{17} +17.9667i q^{19} -1.45017 q^{21} -19.1101 q^{23} +(11.5498 + 22.1721i) q^{25} -5.19615 q^{27} +45.6495 q^{29} -13.6243i q^{31} +27.3586i q^{33} +(-3.57919 - 2.17123i) q^{35} -15.5597i q^{37} -8.98337i q^{39} +13.2990 q^{41} -27.9430 q^{43} +(-12.8248 - 7.77983i) q^{45} -55.6558 q^{47} -48.2990 q^{49} +47.3865i q^{51} -15.5597i q^{53} +(-40.9621 + 67.5245i) q^{55} -31.1193i q^{57} +87.6625i q^{59} -38.0000 q^{61} +2.51176 q^{63} +(13.4502 - 22.1721i) q^{65} -92.2015 q^{67} +33.0997 q^{69} +130.707i q^{71} -54.7173i q^{73} +(-20.0049 - 38.4032i) q^{75} -13.2249i q^{77} +13.6243i q^{79} +9.00000 q^{81} -59.0048 q^{83} +(-70.9485 + 116.956i) q^{85} -79.0673 q^{87} +39.8007 q^{89} +4.34246i q^{91} +23.5980i q^{93} +(46.5927 - 76.8064i) q^{95} +168.821i q^{97} -47.3865i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{5} + 24q^{9} + O(q^{10}) \) \( 8q - 4q^{5} + 24q^{9} - 72q^{21} + 32q^{25} + 184q^{29} - 256q^{41} - 12q^{45} - 24q^{49} - 304q^{61} + 168q^{65} + 144q^{69} + 72q^{81} - 24q^{85} + 560q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.73205 −0.577350
\(4\) 0 0
\(5\) −4.27492 2.59328i −0.854983 0.518655i
\(6\) 0 0
\(7\) 0.837253 0.119608 0.0598038 0.998210i \(-0.480952\pi\)
0.0598038 + 0.998210i \(0.480952\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) 15.7955i 1.43596i −0.696066 0.717978i \(-0.745068\pi\)
0.696066 0.717978i \(-0.254932\pi\)
\(12\) 0 0
\(13\) 5.18655i 0.398966i 0.979901 + 0.199483i \(0.0639262\pi\)
−0.979901 + 0.199483i \(0.936074\pi\)
\(14\) 0 0
\(15\) 7.40437 + 4.49169i 0.493625 + 0.299446i
\(16\) 0 0
\(17\) 27.3586i 1.60933i −0.593728 0.804666i \(-0.702344\pi\)
0.593728 0.804666i \(-0.297656\pi\)
\(18\) 0 0
\(19\) 17.9667i 0.945618i 0.881165 + 0.472809i \(0.156760\pi\)
−0.881165 + 0.472809i \(0.843240\pi\)
\(20\) 0 0
\(21\) −1.45017 −0.0690555
\(22\) 0 0
\(23\) −19.1101 −0.830874 −0.415437 0.909622i \(-0.636371\pi\)
−0.415437 + 0.909622i \(0.636371\pi\)
\(24\) 0 0
\(25\) 11.5498 + 22.1721i 0.461993 + 0.886883i
\(26\) 0 0
\(27\) −5.19615 −0.192450
\(28\) 0 0
\(29\) 45.6495 1.57412 0.787060 0.616876i \(-0.211602\pi\)
0.787060 + 0.616876i \(0.211602\pi\)
\(30\) 0 0
\(31\) 13.6243i 0.439493i −0.975557 0.219747i \(-0.929477\pi\)
0.975557 0.219747i \(-0.0705230\pi\)
\(32\) 0 0
\(33\) 27.3586i 0.829050i
\(34\) 0 0
\(35\) −3.57919 2.17123i −0.102263 0.0620351i
\(36\) 0 0
\(37\) 15.5597i 0.420531i −0.977644 0.210266i \(-0.932567\pi\)
0.977644 0.210266i \(-0.0674329\pi\)
\(38\) 0 0
\(39\) 8.98337i 0.230343i
\(40\) 0 0
\(41\) 13.2990 0.324366 0.162183 0.986761i \(-0.448147\pi\)
0.162183 + 0.986761i \(0.448147\pi\)
\(42\) 0 0
\(43\) −27.9430 −0.649837 −0.324918 0.945742i \(-0.605337\pi\)
−0.324918 + 0.945742i \(0.605337\pi\)
\(44\) 0 0
\(45\) −12.8248 7.77983i −0.284994 0.172885i
\(46\) 0 0
\(47\) −55.6558 −1.18417 −0.592083 0.805877i \(-0.701694\pi\)
−0.592083 + 0.805877i \(0.701694\pi\)
\(48\) 0 0
\(49\) −48.2990 −0.985694
\(50\) 0 0
\(51\) 47.3865i 0.929148i
\(52\) 0 0
\(53\) 15.5597i 0.293578i −0.989168 0.146789i \(-0.953106\pi\)
0.989168 0.146789i \(-0.0468939\pi\)
\(54\) 0 0
\(55\) −40.9621 + 67.5245i −0.744766 + 1.22772i
\(56\) 0 0
\(57\) 31.1193i 0.545953i
\(58\) 0 0
\(59\) 87.6625i 1.48581i 0.669400 + 0.742903i \(0.266551\pi\)
−0.669400 + 0.742903i \(0.733449\pi\)
\(60\) 0 0
\(61\) −38.0000 −0.622951 −0.311475 0.950254i \(-0.600823\pi\)
−0.311475 + 0.950254i \(0.600823\pi\)
\(62\) 0 0
\(63\) 2.51176 0.0398692
\(64\) 0 0
\(65\) 13.4502 22.1721i 0.206926 0.341109i
\(66\) 0 0
\(67\) −92.2015 −1.37614 −0.688071 0.725643i \(-0.741542\pi\)
−0.688071 + 0.725643i \(0.741542\pi\)
\(68\) 0 0
\(69\) 33.0997 0.479705
\(70\) 0 0
\(71\) 130.707i 1.84094i 0.390816 + 0.920469i \(0.372193\pi\)
−0.390816 + 0.920469i \(0.627807\pi\)
\(72\) 0 0
\(73\) 54.7173i 0.749552i −0.927115 0.374776i \(-0.877720\pi\)
0.927115 0.374776i \(-0.122280\pi\)
\(74\) 0 0
\(75\) −20.0049 38.4032i −0.266732 0.512042i
\(76\) 0 0
\(77\) 13.2249i 0.171751i
\(78\) 0 0
\(79\) 13.6243i 0.172459i 0.996275 + 0.0862297i \(0.0274819\pi\)
−0.996275 + 0.0862297i \(0.972518\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) −59.0048 −0.710901 −0.355451 0.934695i \(-0.615673\pi\)
−0.355451 + 0.934695i \(0.615673\pi\)
\(84\) 0 0
\(85\) −70.9485 + 116.956i −0.834688 + 1.37595i
\(86\) 0 0
\(87\) −79.0673 −0.908819
\(88\) 0 0
\(89\) 39.8007 0.447198 0.223599 0.974681i \(-0.428219\pi\)
0.223599 + 0.974681i \(0.428219\pi\)
\(90\) 0 0
\(91\) 4.34246i 0.0477193i
\(92\) 0 0
\(93\) 23.5980i 0.253741i
\(94\) 0 0
\(95\) 46.5927 76.8064i 0.490450 0.808488i
\(96\) 0 0
\(97\) 168.821i 1.74043i 0.492675 + 0.870214i \(0.336019\pi\)
−0.492675 + 0.870214i \(0.663981\pi\)
\(98\) 0 0
\(99\) 47.3865i 0.478652i
\(100\) 0 0
\(101\) −44.5498 −0.441087 −0.220544 0.975377i \(-0.570783\pi\)
−0.220544 + 0.975377i \(0.570783\pi\)
\(102\) 0 0
\(103\) −126.466 −1.22782 −0.613911 0.789375i \(-0.710405\pi\)
−0.613911 + 0.789375i \(0.710405\pi\)
\(104\) 0 0
\(105\) 6.19934 + 3.76068i 0.0590413 + 0.0358160i
\(106\) 0 0
\(107\) −104.383 −0.975546 −0.487773 0.872971i \(-0.662191\pi\)
−0.487773 + 0.872971i \(0.662191\pi\)
\(108\) 0 0
\(109\) 0.501656 0.00460235 0.00230117 0.999997i \(-0.499268\pi\)
0.00230117 + 0.999997i \(0.499268\pi\)
\(110\) 0 0
\(111\) 26.9501i 0.242794i
\(112\) 0 0
\(113\) 16.9855i 0.150314i 0.997172 + 0.0751572i \(0.0239459\pi\)
−0.997172 + 0.0751572i \(0.976054\pi\)
\(114\) 0 0
\(115\) 81.6941 + 49.5578i 0.710384 + 0.430937i
\(116\) 0 0
\(117\) 15.5597i 0.132989i
\(118\) 0 0
\(119\) 22.9061i 0.192488i
\(120\) 0 0
\(121\) −128.498 −1.06197
\(122\) 0 0
\(123\) −23.0346 −0.187273
\(124\) 0 0
\(125\) 8.12376 124.736i 0.0649901 0.997886i
\(126\) 0 0
\(127\) 8.45598 0.0665825 0.0332913 0.999446i \(-0.489401\pi\)
0.0332913 + 0.999446i \(0.489401\pi\)
\(128\) 0 0
\(129\) 48.3987 0.375184
\(130\) 0 0
\(131\) 51.7290i 0.394878i 0.980315 + 0.197439i \(0.0632624\pi\)
−0.980315 + 0.197439i \(0.936738\pi\)
\(132\) 0 0
\(133\) 15.0427i 0.113103i
\(134\) 0 0
\(135\) 22.2131 + 13.4751i 0.164542 + 0.0998153i
\(136\) 0 0
\(137\) 53.8083i 0.392762i 0.980528 + 0.196381i \(0.0629189\pi\)
−0.980528 + 0.196381i \(0.937081\pi\)
\(138\) 0 0
\(139\) 13.6243i 0.0980165i 0.998798 + 0.0490082i \(0.0156061\pi\)
−0.998798 + 0.0490082i \(0.984394\pi\)
\(140\) 0 0
\(141\) 96.3987 0.683679
\(142\) 0 0
\(143\) 81.9243 0.572897
\(144\) 0 0
\(145\) −195.148 118.382i −1.34585 0.816426i
\(146\) 0 0
\(147\) 83.6563 0.569091
\(148\) 0 0
\(149\) 33.6495 0.225836 0.112918 0.993604i \(-0.463980\pi\)
0.112918 + 0.993604i \(0.463980\pi\)
\(150\) 0 0
\(151\) 139.988i 0.927076i −0.886077 0.463538i \(-0.846580\pi\)
0.886077 0.463538i \(-0.153420\pi\)
\(152\) 0 0
\(153\) 82.0759i 0.536444i
\(154\) 0 0
\(155\) −35.3315 + 58.2427i −0.227945 + 0.375759i
\(156\) 0 0
\(157\) 21.2631i 0.135434i 0.997705 + 0.0677170i \(0.0215715\pi\)
−0.997705 + 0.0677170i \(0.978428\pi\)
\(158\) 0 0
\(159\) 26.9501i 0.169498i
\(160\) 0 0
\(161\) −16.0000 −0.0993789
\(162\) 0 0
\(163\) 210.211 1.28964 0.644819 0.764335i \(-0.276933\pi\)
0.644819 + 0.764335i \(0.276933\pi\)
\(164\) 0 0
\(165\) 70.9485 116.956i 0.429991 0.708824i
\(166\) 0 0
\(167\) −238.384 −1.42745 −0.713725 0.700426i \(-0.752994\pi\)
−0.713725 + 0.700426i \(0.752994\pi\)
\(168\) 0 0
\(169\) 142.100 0.840826
\(170\) 0 0
\(171\) 53.9002i 0.315206i
\(172\) 0 0
\(173\) 2.33481i 0.0134960i −0.999977 0.00674800i \(-0.997852\pi\)
0.999977 0.00674800i \(-0.00214797\pi\)
\(174\) 0 0
\(175\) 9.67014 + 18.5637i 0.0552579 + 0.106078i
\(176\) 0 0
\(177\) 151.836i 0.857830i
\(178\) 0 0
\(179\) 227.054i 1.26846i −0.773145 0.634229i \(-0.781318\pi\)
0.773145 0.634229i \(-0.218682\pi\)
\(180\) 0 0
\(181\) −114.096 −0.630367 −0.315183 0.949031i \(-0.602066\pi\)
−0.315183 + 0.949031i \(0.602066\pi\)
\(182\) 0 0
\(183\) 65.8179 0.359661
\(184\) 0 0
\(185\) −40.3505 + 66.5163i −0.218111 + 0.359547i
\(186\) 0 0
\(187\) −432.144 −2.31093
\(188\) 0 0
\(189\) −4.35050 −0.0230185
\(190\) 0 0
\(191\) 139.392i 0.729798i −0.931047 0.364899i \(-0.881103\pi\)
0.931047 0.364899i \(-0.118897\pi\)
\(192\) 0 0
\(193\) 182.046i 0.943245i 0.881801 + 0.471623i \(0.156331\pi\)
−0.881801 + 0.471623i \(0.843669\pi\)
\(194\) 0 0
\(195\) −23.2964 + 38.4032i −0.119469 + 0.196939i
\(196\) 0 0
\(197\) 258.027i 1.30978i 0.755724 + 0.654890i \(0.227285\pi\)
−0.755724 + 0.654890i \(0.772715\pi\)
\(198\) 0 0
\(199\) 256.474i 1.28881i 0.764683 + 0.644407i \(0.222896\pi\)
−0.764683 + 0.644407i \(0.777104\pi\)
\(200\) 0 0
\(201\) 159.698 0.794516
\(202\) 0 0
\(203\) 38.2202 0.188277
\(204\) 0 0
\(205\) −56.8522 34.4880i −0.277328 0.168234i
\(206\) 0 0
\(207\) −57.3303 −0.276958
\(208\) 0 0
\(209\) 283.794 1.35787
\(210\) 0 0
\(211\) 211.855i 1.00405i −0.864852 0.502027i \(-0.832588\pi\)
0.864852 0.502027i \(-0.167412\pi\)
\(212\) 0 0
\(213\) 226.390i 1.06287i
\(214\) 0 0
\(215\) 119.454 + 72.4639i 0.555600 + 0.337041i
\(216\) 0 0
\(217\) 11.4070i 0.0525667i
\(218\) 0 0
\(219\) 94.7731i 0.432754i
\(220\) 0 0
\(221\) 141.897 0.642068
\(222\) 0 0
\(223\) 349.843 1.56880 0.784401 0.620255i \(-0.212971\pi\)
0.784401 + 0.620255i \(0.212971\pi\)
\(224\) 0 0
\(225\) 34.6495 + 66.5163i 0.153998 + 0.295628i
\(226\) 0 0
\(227\) 185.554 0.817418 0.408709 0.912665i \(-0.365979\pi\)
0.408709 + 0.912665i \(0.365979\pi\)
\(228\) 0 0
\(229\) −263.897 −1.15239 −0.576194 0.817313i \(-0.695463\pi\)
−0.576194 + 0.817313i \(0.695463\pi\)
\(230\) 0 0
\(231\) 22.9061i 0.0991607i
\(232\) 0 0
\(233\) 58.4780i 0.250978i −0.992095 0.125489i \(-0.959950\pi\)
0.992095 0.125489i \(-0.0400500\pi\)
\(234\) 0 0
\(235\) 237.924 + 144.331i 1.01244 + 0.614174i
\(236\) 0 0
\(237\) 23.5980i 0.0995694i
\(238\) 0 0
\(239\) 113.337i 0.474212i 0.971484 + 0.237106i \(0.0761989\pi\)
−0.971484 + 0.237106i \(0.923801\pi\)
\(240\) 0 0
\(241\) −77.7940 −0.322797 −0.161398 0.986889i \(-0.551600\pi\)
−0.161398 + 0.986889i \(0.551600\pi\)
\(242\) 0 0
\(243\) −15.5885 −0.0641500
\(244\) 0 0
\(245\) 206.474 + 125.253i 0.842752 + 0.511235i
\(246\) 0 0
\(247\) −93.1855 −0.377269
\(248\) 0 0
\(249\) 102.199 0.410439
\(250\) 0 0
\(251\) 106.226i 0.423212i 0.977355 + 0.211606i \(0.0678693\pi\)
−0.977355 + 0.211606i \(0.932131\pi\)
\(252\) 0 0
\(253\) 301.854i 1.19310i
\(254\) 0 0
\(255\) 122.886 202.574i 0.481908 0.794406i
\(256\) 0 0
\(257\) 381.078i 1.48279i −0.671067 0.741397i \(-0.734164\pi\)
0.671067 0.741397i \(-0.265836\pi\)
\(258\) 0 0
\(259\) 13.0274i 0.0502988i
\(260\) 0 0
\(261\) 136.949 0.524707
\(262\) 0 0
\(263\) −11.4914 −0.0436934 −0.0218467 0.999761i \(-0.506955\pi\)
−0.0218467 + 0.999761i \(0.506955\pi\)
\(264\) 0 0
\(265\) −40.3505 + 66.5163i −0.152266 + 0.251005i
\(266\) 0 0
\(267\) −68.9368 −0.258190
\(268\) 0 0
\(269\) 77.9518 0.289784 0.144892 0.989447i \(-0.453717\pi\)
0.144892 + 0.989447i \(0.453717\pi\)
\(270\) 0 0
\(271\) 86.6851i 0.319871i 0.987127 + 0.159936i \(0.0511287\pi\)
−0.987127 + 0.159936i \(0.948871\pi\)
\(272\) 0 0
\(273\) 7.52136i 0.0275508i
\(274\) 0 0
\(275\) 350.220 182.436i 1.27353 0.663402i
\(276\) 0 0
\(277\) 287.328i 1.03729i −0.854991 0.518643i \(-0.826437\pi\)
0.854991 0.518643i \(-0.173563\pi\)
\(278\) 0 0
\(279\) 40.8729i 0.146498i
\(280\) 0 0
\(281\) −224.598 −0.799281 −0.399641 0.916672i \(-0.630865\pi\)
−0.399641 + 0.916672i \(0.630865\pi\)
\(282\) 0 0
\(283\) 84.1224 0.297252 0.148626 0.988893i \(-0.452515\pi\)
0.148626 + 0.988893i \(0.452515\pi\)
\(284\) 0 0
\(285\) −80.7010 + 133.033i −0.283161 + 0.466781i
\(286\) 0 0
\(287\) 11.1346 0.0387967
\(288\) 0 0
\(289\) −459.495 −1.58995
\(290\) 0 0
\(291\) 292.407i 1.00484i
\(292\) 0 0
\(293\) 246.620i 0.841706i −0.907129 0.420853i \(-0.861731\pi\)
0.907129 0.420853i \(-0.138269\pi\)
\(294\) 0 0
\(295\) 227.333 374.750i 0.770621 1.27034i
\(296\) 0 0
\(297\) 82.0759i 0.276350i
\(298\) 0 0
\(299\) 99.1156i 0.331490i
\(300\) 0 0
\(301\) −23.3954 −0.0777255
\(302\) 0 0
\(303\) 77.1626 0.254662
\(304\) 0 0
\(305\) 162.447 + 98.5445i 0.532613 + 0.323097i
\(306\) 0 0
\(307\) 115.811 0.377236 0.188618 0.982051i \(-0.439599\pi\)
0.188618 + 0.982051i \(0.439599\pi\)
\(308\) 0 0
\(309\) 219.045 0.708883
\(310\) 0 0
\(311\) 203.767i 0.655201i −0.944816 0.327600i \(-0.893760\pi\)
0.944816 0.327600i \(-0.106240\pi\)
\(312\) 0 0
\(313\) 99.0614i 0.316490i 0.987400 + 0.158245i \(0.0505836\pi\)
−0.987400 + 0.158245i \(0.949416\pi\)
\(314\) 0 0
\(315\) −10.7376 6.51369i −0.0340875 0.0206784i
\(316\) 0 0
\(317\) 471.192i 1.48641i −0.669063 0.743206i \(-0.733304\pi\)
0.669063 0.743206i \(-0.266696\pi\)
\(318\) 0 0
\(319\) 721.057i 2.26037i
\(320\) 0 0
\(321\) 180.797 0.563232
\(322\) 0 0
\(323\) 491.546 1.52181
\(324\) 0 0
\(325\) −114.997 + 59.9038i −0.353836 + 0.184319i
\(326\) 0 0
\(327\) −0.868893 −0.00265717
\(328\) 0 0
\(329\) −46.5980 −0.141635
\(330\) 0 0
\(331\) 270.695i 0.817810i 0.912577 + 0.408905i \(0.134089\pi\)
−0.912577 + 0.408905i \(0.865911\pi\)
\(332\) 0 0
\(333\) 46.6790i 0.140177i
\(334\) 0 0
\(335\) 394.154 + 239.104i 1.17658 + 0.713743i
\(336\) 0 0
\(337\) 377.317i 1.11964i 0.828615 + 0.559818i \(0.189129\pi\)
−0.828615 + 0.559818i \(0.810871\pi\)
\(338\) 0 0
\(339\) 29.4198i 0.0867841i
\(340\) 0 0
\(341\) −215.203 −0.631093
\(342\) 0 0
\(343\) −81.4639 −0.237504
\(344\) 0 0
\(345\) −141.498 85.8366i −0.410140 0.248802i
\(346\) 0 0
\(347\) −462.222 −1.33205 −0.666025 0.745929i \(-0.732006\pi\)
−0.666025 + 0.745929i \(0.732006\pi\)
\(348\) 0 0
\(349\) −200.598 −0.574779 −0.287390 0.957814i \(-0.592787\pi\)
−0.287390 + 0.957814i \(0.592787\pi\)
\(350\) 0 0
\(351\) 26.9501i 0.0767810i
\(352\) 0 0
\(353\) 250.897i 0.710757i −0.934722 0.355379i \(-0.884352\pi\)
0.934722 0.355379i \(-0.115648\pi\)
\(354\) 0 0
\(355\) 338.958 558.760i 0.954812 1.57397i
\(356\) 0 0
\(357\) 39.6746i 0.111133i
\(358\) 0 0
\(359\) 215.601i 0.600560i −0.953851 0.300280i \(-0.902920\pi\)
0.953851 0.300280i \(-0.0970801\pi\)
\(360\) 0 0
\(361\) 38.1960 0.105806
\(362\) 0 0
\(363\) 222.566 0.613129
\(364\) 0 0
\(365\) −141.897 + 233.912i −0.388759 + 0.640854i
\(366\) 0 0
\(367\) −67.0637 −0.182735 −0.0913675 0.995817i \(-0.529124\pi\)
−0.0913675 + 0.995817i \(0.529124\pi\)
\(368\) 0 0
\(369\) 39.8970 0.108122
\(370\) 0 0
\(371\) 13.0274i 0.0351142i
\(372\) 0 0
\(373\) 567.402i 1.52119i 0.649230 + 0.760593i \(0.275091\pi\)
−0.649230 + 0.760593i \(0.724909\pi\)
\(374\) 0 0
\(375\) −14.0708 + 216.049i −0.0375220 + 0.576130i
\(376\) 0 0
\(377\) 236.764i 0.628020i
\(378\) 0 0
\(379\) 240.298i 0.634031i −0.948420 0.317016i \(-0.897319\pi\)
0.948420 0.317016i \(-0.102681\pi\)
\(380\) 0 0
\(381\) −14.6462 −0.0384414
\(382\) 0 0
\(383\) 670.068 1.74952 0.874762 0.484553i \(-0.161018\pi\)
0.874762 + 0.484553i \(0.161018\pi\)
\(384\) 0 0
\(385\) −34.2957 + 56.5351i −0.0890797 + 0.146845i
\(386\) 0 0
\(387\) −83.8290 −0.216612
\(388\) 0 0
\(389\) 474.640 1.22015 0.610077 0.792342i \(-0.291139\pi\)
0.610077 + 0.792342i \(0.291139\pi\)
\(390\) 0 0
\(391\) 522.826i 1.33715i
\(392\) 0 0
\(393\) 89.5973i 0.227983i
\(394\) 0 0
\(395\) 35.3315 58.2427i 0.0894469 0.147450i
\(396\) 0 0
\(397\) 499.460i 1.25809i 0.777371 + 0.629043i \(0.216553\pi\)
−0.777371 + 0.629043i \(0.783447\pi\)
\(398\) 0 0
\(399\) 26.0548i 0.0653001i
\(400\) 0 0
\(401\) 344.694 0.859587 0.429793 0.902927i \(-0.358586\pi\)
0.429793 + 0.902927i \(0.358586\pi\)
\(402\) 0 0
\(403\) 70.6631 0.175343
\(404\) 0 0
\(405\) −38.4743 23.3395i −0.0949982 0.0576284i
\(406\) 0 0
\(407\) −245.773 −0.603864
\(408\) 0 0
\(409\) −501.890 −1.22712 −0.613558 0.789650i \(-0.710262\pi\)
−0.613558 + 0.789650i \(0.710262\pi\)
\(410\) 0 0
\(411\) 93.1988i 0.226761i
\(412\) 0 0
\(413\) 73.3957i 0.177714i
\(414\) 0 0
\(415\) 252.241 + 153.016i 0.607809 + 0.368713i
\(416\) 0 0
\(417\) 23.5980i 0.0565898i
\(418\) 0 0
\(419\) 218.369i 0.521167i 0.965451 + 0.260584i \(0.0839150\pi\)
−0.965451 + 0.260584i \(0.916085\pi\)
\(420\) 0 0
\(421\) −281.698 −0.669116 −0.334558 0.942375i \(-0.608587\pi\)
−0.334558 + 0.942375i \(0.608587\pi\)
\(422\) 0 0
\(423\) −166.967 −0.394722
\(424\) 0 0
\(425\) 606.598 315.988i 1.42729 0.743501i
\(426\) 0 0
\(427\) −31.8156 −0.0745097
\(428\) 0 0
\(429\) −141.897 −0.330762
\(430\) 0 0
\(431\) 441.081i 1.02339i −0.859167 0.511694i \(-0.829018\pi\)
0.859167 0.511694i \(-0.170982\pi\)
\(432\) 0 0
\(433\) 123.443i 0.285089i 0.989788 + 0.142544i \(0.0455283\pi\)
−0.989788 + 0.142544i \(0.954472\pi\)
\(434\) 0 0
\(435\) 338.006 + 205.043i 0.777025 + 0.471364i
\(436\) 0 0
\(437\) 343.346i 0.785690i
\(438\) 0 0
\(439\) 330.728i 0.753368i 0.926342 + 0.376684i \(0.122936\pi\)
−0.926342 + 0.376684i \(0.877064\pi\)
\(440\) 0 0
\(441\) −144.897 −0.328565
\(442\) 0 0
\(443\) −154.952 −0.349780 −0.174890 0.984588i \(-0.555957\pi\)
−0.174890 + 0.984588i \(0.555957\pi\)
\(444\) 0 0
\(445\) −170.145 103.214i −0.382347 0.231942i
\(446\) 0 0
\(447\) −58.2826 −0.130386
\(448\) 0 0
\(449\) −95.8970 −0.213579 −0.106790 0.994282i \(-0.534057\pi\)
−0.106790 + 0.994282i \(0.534057\pi\)
\(450\) 0 0
\(451\) 210.065i 0.465775i
\(452\) 0 0
\(453\) 242.467i 0.535247i
\(454\) 0 0
\(455\) 11.2612 18.5637i 0.0247499 0.0407992i
\(456\) 0 0
\(457\) 485.718i 1.06284i −0.847108 0.531420i \(-0.821659\pi\)
0.847108 0.531420i \(-0.178341\pi\)
\(458\) 0 0
\(459\) 142.160i 0.309716i
\(460\) 0 0
\(461\) −353.650 −0.767136 −0.383568 0.923513i \(-0.625305\pi\)
−0.383568 + 0.923513i \(0.625305\pi\)
\(462\) 0 0
\(463\) 421.720 0.910842 0.455421 0.890276i \(-0.349489\pi\)
0.455421 + 0.890276i \(0.349489\pi\)
\(464\) 0 0
\(465\) 61.1960 100.879i 0.131604 0.216945i
\(466\) 0 0
\(467\) −640.974 −1.37254 −0.686268 0.727349i \(-0.740752\pi\)
−0.686268 + 0.727349i \(0.740752\pi\)
\(468\) 0 0
\(469\) −77.1960 −0.164597
\(470\) 0 0
\(471\) 36.8289i 0.0781929i
\(472\) 0 0
\(473\) 441.374i 0.933137i
\(474\) 0 0
\(475\) −398.360 + 207.513i −0.838653 + 0.436869i
\(476\) 0 0
\(477\) 46.6790i 0.0978595i
\(478\) 0 0
\(479\) 221.137i 0.461664i −0.972994 0.230832i \(-0.925855\pi\)
0.972994 0.230832i \(-0.0741448\pi\)
\(480\) 0 0
\(481\) 80.7010 0.167778
\(482\) 0 0
\(483\) 27.7128 0.0573764
\(484\) 0 0
\(485\) 437.801 721.698i 0.902682 1.48804i
\(486\) 0 0
\(487\) −889.949 −1.82741 −0.913705 0.406377i \(-0.866792\pi\)
−0.913705 + 0.406377i \(0.866792\pi\)
\(488\) 0 0
\(489\) −364.096 −0.744573
\(490\) 0 0
\(491\) 552.843i 1.12595i −0.826473 0.562977i \(-0.809656\pi\)
0.826473 0.562977i \(-0.190344\pi\)
\(492\) 0 0
\(493\) 1248.91i 2.53328i
\(494\) 0 0
\(495\) −122.886 + 202.574i −0.248255 + 0.409240i
\(496\) 0 0
\(497\) 109.435i 0.220190i
\(498\) 0 0
\(499\) 533.302i 1.06874i 0.845250 + 0.534371i \(0.179451\pi\)
−0.845250 + 0.534371i \(0.820549\pi\)
\(500\) 0 0
\(501\) 412.894 0.824139
\(502\) 0 0
\(503\) −574.914 −1.14297 −0.571485 0.820612i \(-0.693633\pi\)
−0.571485 + 0.820612i \(0.693633\pi\)
\(504\) 0 0
\(505\) 190.447 + 115.530i 0.377122 + 0.228772i
\(506\) 0 0
\(507\) −246.124 −0.485451
\(508\) 0 0
\(509\) −207.547 −0.407753 −0.203877 0.978997i \(-0.565354\pi\)
−0.203877 + 0.978997i \(0.565354\pi\)
\(510\) 0 0
\(511\) 45.8122i 0.0896521i
\(512\) 0 0
\(513\) 93.3580i 0.181984i
\(514\) 0 0
\(515\) 540.630 + 327.960i 1.04977 + 0.636816i
\(516\) 0 0
\(517\) 879.112i 1.70041i
\(518\) 0 0
\(519\) 4.04401i 0.00779192i
\(520\) 0 0
\(521\) −712.900 −1.36833 −0.684165 0.729327i \(-0.739833\pi\)
−0.684165 + 0.729327i \(0.739833\pi\)
\(522\) 0 0
\(523\) 139.548 0.266822 0.133411 0.991061i \(-0.457407\pi\)
0.133411 + 0.991061i \(0.457407\pi\)
\(524\) 0 0
\(525\) −16.7492 32.1532i −0.0319032 0.0612442i
\(526\) 0 0
\(527\) −372.742 −0.707290
\(528\) 0 0
\(529\) −163.804 −0.309648
\(530\) 0 0
\(531\) 262.988i 0.495268i
\(532\) 0 0
\(533\) 68.9760i 0.129411i
\(534\) 0 0
\(535\) 446.230 + 270.695i 0.834075 + 0.505972i
\(536\) 0 0
\(537\) 393.269i 0.732345i
\(538\) 0 0
\(539\) 762.908i 1.41541i
\(540\) 0 0
\(541\) 946.688 1.74988 0.874942 0.484227i \(-0.160899\pi\)
0.874942 + 0.484227i \(0.160899\pi\)
\(542\) 0 0
\(543\) 197.621 0.363942
\(544\) 0 0
\(545\) −2.14454 1.30093i −0.00393493 0.00238703i
\(546\) 0 0
\(547\) 50.3388 0.0920271 0.0460136 0.998941i \(-0.485348\pi\)
0.0460136 + 0.998941i \(0.485348\pi\)
\(548\) 0 0
\(549\) −114.000 −0.207650
\(550\) 0 0
\(551\) 820.173i 1.48852i
\(552\) 0 0
\(553\) 11.4070i 0.0206275i
\(554\) 0 0
\(555\) 69.8891 115.210i 0.125926 0.207585i
\(556\) 0 0
\(557\) 790.157i 1.41859i 0.704910 + 0.709297i \(0.250987\pi\)
−0.704910 + 0.709297i \(0.749013\pi\)
\(558\) 0 0
\(559\) 144.928i 0.259263i
\(560\) 0 0
\(561\) 748.495 1.33422
\(562\) 0 0
\(563\) −354.133 −0.629010 −0.314505 0.949256i \(-0.601838\pi\)
−0.314505 + 0.949256i \(0.601838\pi\)
\(564\) 0 0
\(565\) 44.0482 72.6117i 0.0779614 0.128516i
\(566\) 0 0
\(567\) 7.53528 0.0132897
\(568\) 0 0
\(569\) 55.4983 0.0975366 0.0487683 0.998810i \(-0.484470\pi\)
0.0487683 + 0.998810i \(0.484470\pi\)
\(570\) 0 0
\(571\) 791.134i 1.38552i 0.721167 + 0.692762i \(0.243606\pi\)
−0.721167 + 0.692762i \(0.756394\pi\)
\(572\) 0 0
\(573\) 241.433i 0.421349i
\(574\) 0 0
\(575\) −220.719 423.711i −0.383858 0.736888i
\(576\) 0 0
\(577\) 201.759i 0.349668i −0.984598 0.174834i \(-0.944061\pi\)
0.984598 0.174834i \(-0.0559389\pi\)
\(578\) 0 0
\(579\) 315.313i 0.544583i
\(580\) 0 0
\(581\) −49.4020 −0.0850292
\(582\) 0 0
\(583\) −245.773 −0.421566
\(584\) 0 0
\(585\) 40.3505 66.5163i 0.0689752 0.113703i
\(586\) 0 0
\(587\) −444.556 −0.757335 −0.378668 0.925533i \(-0.623618\pi\)
−0.378668 + 0.925533i \(0.623618\pi\)
\(588\) 0 0
\(589\) 244.784 0.415593
\(590\) 0 0
\(591\) 446.915i 0.756202i
\(592\) 0 0
\(593\) 563.908i 0.950942i −0.879731 0.475471i \(-0.842278\pi\)
0.879731 0.475471i \(-0.157722\pi\)
\(594\) 0 0
\(595\) −59.4019 + 97.9217i −0.0998351 + 0.164574i
\(596\) 0 0
\(597\) 444.226i 0.744097i
\(598\) 0 0
\(599\) 845.034i 1.41074i 0.708839 + 0.705371i \(0.249219\pi\)
−0.708839 + 0.705371i \(0.750781\pi\)
\(600\) 0 0
\(601\) 672.296 1.11863 0.559314 0.828956i \(-0.311065\pi\)
0.559314 + 0.828956i \(0.311065\pi\)
\(602\) 0 0
\(603\) −276.604 −0.458714
\(604\) 0 0
\(605\) 549.320 + 333.232i 0.907967 + 0.550796i
\(606\) 0 0
\(607\) −882.664 −1.45414 −0.727071 0.686562i \(-0.759119\pi\)
−0.727071 + 0.686562i \(0.759119\pi\)
\(608\) 0 0
\(609\) −66.1993 −0.108702
\(610\) 0 0
\(611\) 288.662i 0.472442i
\(612\) 0 0
\(613\) 469.374i 0.765701i −0.923810 0.382850i \(-0.874943\pi\)
0.923810 0.382850i \(-0.125057\pi\)
\(614\) 0 0
\(615\) 98.4708 + 59.7350i 0.160115 + 0.0971300i
\(616\) 0 0
\(617\) 218.994i 0.354934i −0.984127 0.177467i \(-0.943210\pi\)
0.984127 0.177467i \(-0.0567902\pi\)
\(618\) 0 0
\(619\) 879.610i 1.42102i −0.703689 0.710509i \(-0.748465\pi\)
0.703689 0.710509i \(-0.251535\pi\)
\(620\) 0 0
\(621\) 99.2990 0.159902
\(622\) 0 0
\(623\) 33.3232 0.0534884
\(624\) 0 0
\(625\) −358.203 + 512.168i −0.573124 + 0.819468i
\(626\) 0 0
\(627\) −491.546 −0.783964
\(628\) 0 0
\(629\) −425.691 −0.676774
\(630\) 0 0
\(631\) 635.566i 1.00724i −0.863926 0.503618i \(-0.832002\pi\)
0.863926 0.503618i \(-0.167998\pi\)
\(632\) 0 0
\(633\) 366.944i 0.579691i
\(634\) 0 0
\(635\) −36.1486 21.9287i −0.0569270 0.0345334i
\(636\) 0 0
\(637\) 250.505i 0.393258i
\(638\) 0 0
\(639\) 392.120i 0.613646i
\(640\) 0 0
\(641\) −296.309 −0.462260 −0.231130 0.972923i \(-0.574242\pi\)
−0.231130 + 0.972923i \(0.574242\pi\)
\(642\) 0 0
\(643\) −591.032 −0.919179 −0.459590 0.888131i \(-0.652003\pi\)
−0.459590 + 0.888131i \(0.652003\pi\)
\(644\) 0 0
\(645\) −206.900 125.511i −0.320776 0.194591i
\(646\) 0 0
\(647\) −166.507 −0.257352 −0.128676 0.991687i \(-0.541073\pi\)
−0.128676 + 0.991687i \(0.541073\pi\)
\(648\) 0 0
\(649\) 1384.67 2.13355
\(650\) 0 0
\(651\) 19.7575i 0.0303494i
\(652\) 0 0
\(653\) 621.335i 0.951509i 0.879578 + 0.475754i \(0.157825\pi\)
−0.879578 + 0.475754i \(0.842175\pi\)
\(654\) 0 0
\(655\) 134.148 221.137i 0.204806 0.337614i
\(656\) 0 0
\(657\) 164.152i 0.249851i
\(658\) 0 0
\(659\) 702.113i 1.06542i −0.846297 0.532711i \(-0.821173\pi\)
0.846297 0.532711i \(-0.178827\pi\)
\(660\) 0 0
\(661\) −358.193 −0.541895 −0.270948 0.962594i \(-0.587337\pi\)
−0.270948 + 0.962594i \(0.587337\pi\)
\(662\) 0 0
\(663\) −245.773 −0.370698
\(664\) 0 0
\(665\) 39.0099 64.3064i 0.0586616 0.0967013i
\(666\) 0 0
\(667\) −872.367 −1.30790
\(668\) 0 0
\(669\) −605.945 −0.905748
\(670\) 0 0
\(671\) 600.230i 0.894530i
\(672\) 0 0
\(673\) 714.176i 1.06118i −0.847628 0.530592i \(-0.821970\pi\)
0.847628 0.530592i \(-0.178030\pi\)
\(674\) 0 0
\(675\) −60.0147 115.210i −0.0889107 0.170681i
\(676\) 0 0
\(677\) 509.833i 0.753077i −0.926401 0.376538i \(-0.877114\pi\)
0.926401 0.376538i \(-0.122886\pi\)
\(678\) 0 0
\(679\) 141.346i 0.208168i
\(680\) 0 0
\(681\) −321.389 −0.471936
\(682\) 0 0
\(683\) 1263.93 1.85055 0.925275 0.379298i \(-0.123834\pi\)
0.925275 + 0.379298i \(0.123834\pi\)
\(684\) 0 0
\(685\) 139.540 230.026i 0.203708 0.335805i
\(686\) 0 0
\(687\) 457.083 0.665332
\(688\) 0 0
\(689\) 80.7010 0.117128
\(690\) 0 0
\(691\) 512.351i 0.741463i 0.928740 + 0.370731i \(0.120893\pi\)
−0.928740 + 0.370731i \(0.879107\pi\)
\(692\) 0 0
\(693\) 39.6746i 0.0572504i
\(694\) 0 0
\(695\) 35.3315 58.2427i 0.0508368 0.0838024i
\(696\) 0 0
\(697\) 363.843i 0.522012i
\(698\) 0 0
\(699\) 101.287i 0.144902i
\(700\) 0 0
\(701\) −1092.03 −1.55781 −0.778907 0.627139i \(-0.784226\pi\)
−0.778907 + 0.627139i \(0.784226\pi\)
\(702\) 0 0
\(703\) 279.556 0.397662
\(704\) 0 0
\(705\) −412.096 249.988i −0.584534 0.354593i
\(706\) 0 0
\(707\) −37.2995 −0.0527574
\(708\) 0 0
\(709\) −416.887 −0.587993 −0.293997 0.955806i \(-0.594985\pi\)
−0.293997 + 0.955806i \(0.594985\pi\)
\(710\) 0 0
\(711\) 40.8729i 0.0574864i
\(712\) 0 0
\(713\) 260.362i 0.365163i
\(714\) 0 0
\(715\) −350.220 212.452i −0.489818 0.297136i
\(716\) 0 0
\(717\) 196.305i 0.273787i
\(718\) 0 0
\(719\) 395.268i 0.549747i 0.961480 + 0.274874i \(0.0886361\pi\)
−0.961480 + 0.274874i \(0.911364\pi\)
\(720\) 0 0
\(721\) −105.884 −0.146857
\(722\) 0 0
\(723\) 134.743 0.186367
\(724\) 0 0
\(725\) 527.244 + 1012.14i 0.727233 + 1.39606i
\(726\) 0 0
\(727\) −597.583 −0.821985 −0.410993 0.911639i \(-0.634818\pi\)
−0.410993 + 0.911639i \(0.634818\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) 764.482i 1.04580i
\(732\) 0 0
\(733\) 23.8650i 0.0325580i −0.999867 0.0162790i \(-0.994818\pi\)
0.999867 0.0162790i \(-0.00518200\pi\)
\(734\) 0 0
\(735\) −357.624 216.944i −0.486563 0.295162i
\(736\) 0 0
\(737\) 1456.37i 1.97608i
\(738\) 0 0
\(739\) 125.767i 0.170186i 0.996373 + 0.0850928i \(0.0271187\pi\)
−0.996373 + 0.0850928i \(0.972881\pi\)
\(740\) 0 0
\(741\) 161.402 0.217816
\(742\) 0 0
\(743\) −148.841 −0.200325 −0.100162 0.994971i \(-0.531936\pi\)
−0.100162 + 0.994971i \(0.531936\pi\)
\(744\) 0 0
\(745\) −143.849 87.2625i −0.193086 0.117131i
\(746\) 0 0
\(747\) −177.014 −0.236967
\(748\) 0 0
\(749\) −87.3954 −0.116683
\(750\) 0 0
\(751\) 463.390i 0.617030i 0.951219 + 0.308515i \(0.0998321\pi\)
−0.951219 + 0.308515i \(0.900168\pi\)
\(752\) 0 0
\(753\) 183.989i 0.244341i
\(754\) 0 0
\(755\) −363.029 + 598.439i −0.480833 + 0.792634i
\(756\) 0 0
\(757\) 719.363i 0.950281i −0.879910 0.475141i \(-0.842397\pi\)
0.879910 0.475141i \(-0.157603\pi\)
\(758\) 0 0
\(759\) 522.826i 0.688836i
\(760\) 0 0
\(761\) −1107.49 −1.45530 −0.727651 0.685947i \(-0.759388\pi\)
−0.727651 + 0.685947i \(0.759388\pi\)
\(762\) 0 0
\(763\) 0.420013 0.000550476
\(764\) 0 0
\(765\) −212.846 + 350.868i −0.278229 + 0.458651i
\(766\) 0 0
\(767\) −454.666 −0.592785
\(768\) 0 0
\(769\) −231.691 −0.301289 −0.150644 0.988588i \(-0.548135\pi\)
−0.150644 + 0.988588i \(0.548135\pi\)
\(770\) 0 0
\(771\) 660.047i 0.856092i
\(772\) 0 0
\(773\) 519.956i 0.672647i 0.941746 + 0.336324i \(0.109184\pi\)
−0.941746 + 0.336324i \(0.890816\pi\)
\(774\) 0 0
\(775\) 302.079 157.358i 0.389779 0.203043i
\(776\) 0 0
\(777\) 22.5641i 0.0290400i
\(778\) 0 0
\(779\) 238.940i 0.306726i
\(780\) 0 0
\(781\) 2064.58 2.64351
\(782\) 0 0
\(783\) −237.202 −0.302940
\(784\) 0 0
\(785\) 55.1412 90.8982i 0.0702436 0.115794i
\(786\) 0 0
\(787\) 46.0288 0.0584864 0.0292432 0.999572i \(-0.490690\pi\)
0.0292432 + 0.999572i \(0.490690\pi\)
\(788\) 0 0
\(789\) 19.9036 0.0252264
\(790\) 0 0
\(791\) 14.2212i 0.0179788i
\(792\) 0 0
\(793\) 197.089i 0.248536i
\(794\) 0 0
\(795\) 69.8891 115.210i 0.0879108 0.144918i
\(796\) 0 0
\(797\) 15.3098i 0.0192093i 0.999954 + 0.00960463i \(0.00305730\pi\)
−0.999954 + 0.00960463i \(0.996943\pi\)
\(798\) 0 0
\(799\) 1522.67i 1.90572i
\(800\) 0 0
\(801\) 119.402 0.14906