Properties

Label 1040.2.k.c.961.1
Level $1040$
Weight $2$
Character 1040.961
Analytic conductor $8.304$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.30444181021\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.9144576.1
Defining polynomial: \(x^{6} + 12 x^{4} + 36 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 961.1
Root \(-2.60168i\) of defining polynomial
Character \(\chi\) \(=\) 1040.961
Dual form 1040.2.k.c.961.2

$q$-expansion

\(f(q)\) \(=\) \(q-2.60168 q^{3} -1.00000i q^{5} -2.76873i q^{7} +3.76873 q^{9} +O(q^{10})\) \(q-2.60168 q^{3} -1.00000i q^{5} -2.76873i q^{7} +3.76873 q^{9} -2.16706i q^{11} +(-0.167055 - 3.60168i) q^{13} +2.60168i q^{15} +5.03630i q^{19} +7.20336i q^{21} +4.93579 q^{23} -1.00000 q^{25} -2.00000 q^{27} -4.43462 q^{29} -3.37041i q^{31} +5.63798i q^{33} -2.76873 q^{35} -3.97209i q^{37} +(0.434624 + 9.37041i) q^{39} +1.66589i q^{41} -9.80504 q^{43} -3.76873i q^{45} -11.6380i q^{47} -0.665890 q^{49} -12.7408 q^{53} -2.16706 q^{55} -13.1028i q^{57} +8.16706i q^{59} -10.3062 q^{61} -10.4346i q^{63} +(-3.60168 + 0.167055i) q^{65} -7.10284i q^{67} -12.8413 q^{69} +16.1112i q^{71} +15.9721i q^{73} +2.60168 q^{75} -6.00000 q^{77} -11.9442 q^{79} -6.10284 q^{81} -3.97209i q^{83} +11.5375 q^{87} -7.94419i q^{89} +(-9.97209 + 0.462531i) q^{91} +8.76873i q^{93} +5.03630 q^{95} -0.462531i q^{97} -8.16706i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{9} + O(q^{10}) \) \( 6 q + 6 q^{9} + 12 q^{23} - 6 q^{25} - 12 q^{27} - 12 q^{29} - 12 q^{39} - 12 q^{43} - 6 q^{49} - 12 q^{53} - 12 q^{55} - 12 q^{61} - 6 q^{65} - 36 q^{77} + 24 q^{79} - 18 q^{81} + 36 q^{87} - 12 q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) −2.60168 −1.50208 −0.751040 0.660257i \(-0.770448\pi\)
−0.751040 + 0.660257i \(0.770448\pi\)
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 2.76873i 1.04648i −0.852184 0.523242i \(-0.824723\pi\)
0.852184 0.523242i \(-0.175277\pi\)
\(8\) 0 0
\(9\) 3.76873 1.25624
\(10\) 0 0
\(11\) 2.16706i 0.653392i −0.945130 0.326696i \(-0.894065\pi\)
0.945130 0.326696i \(-0.105935\pi\)
\(12\) 0 0
\(13\) −0.167055 3.60168i −0.0463328 0.998926i
\(14\) 0 0
\(15\) 2.60168i 0.671751i
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 5.03630i 1.15541i 0.816247 + 0.577704i \(0.196051\pi\)
−0.816247 + 0.577704i \(0.803949\pi\)
\(20\) 0 0
\(21\) 7.20336i 1.57190i
\(22\) 0 0
\(23\) 4.93579 1.02918 0.514592 0.857435i \(-0.327944\pi\)
0.514592 + 0.857435i \(0.327944\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −2.00000 −0.384900
\(28\) 0 0
\(29\) −4.43462 −0.823489 −0.411744 0.911299i \(-0.635080\pi\)
−0.411744 + 0.911299i \(0.635080\pi\)
\(30\) 0 0
\(31\) 3.37041i 0.605344i −0.953095 0.302672i \(-0.902121\pi\)
0.953095 0.302672i \(-0.0978787\pi\)
\(32\) 0 0
\(33\) 5.63798i 0.981447i
\(34\) 0 0
\(35\) −2.76873 −0.468002
\(36\) 0 0
\(37\) 3.97209i 0.653008i −0.945196 0.326504i \(-0.894129\pi\)
0.945196 0.326504i \(-0.105871\pi\)
\(38\) 0 0
\(39\) 0.434624 + 9.37041i 0.0695955 + 1.50047i
\(40\) 0 0
\(41\) 1.66589i 0.260168i 0.991503 + 0.130084i \(0.0415247\pi\)
−0.991503 + 0.130084i \(0.958475\pi\)
\(42\) 0 0
\(43\) −9.80504 −1.49525 −0.747627 0.664119i \(-0.768807\pi\)
−0.747627 + 0.664119i \(0.768807\pi\)
\(44\) 0 0
\(45\) 3.76873i 0.561810i
\(46\) 0 0
\(47\) 11.6380i 1.69757i −0.528735 0.848787i \(-0.677333\pi\)
0.528735 0.848787i \(-0.322667\pi\)
\(48\) 0 0
\(49\) −0.665890 −0.0951271
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −12.7408 −1.75009 −0.875044 0.484044i \(-0.839167\pi\)
−0.875044 + 0.484044i \(0.839167\pi\)
\(54\) 0 0
\(55\) −2.16706 −0.292206
\(56\) 0 0
\(57\) 13.1028i 1.73551i
\(58\) 0 0
\(59\) 8.16706i 1.06326i 0.846977 + 0.531630i \(0.178420\pi\)
−0.846977 + 0.531630i \(0.821580\pi\)
\(60\) 0 0
\(61\) −10.3062 −1.31957 −0.659787 0.751453i \(-0.729354\pi\)
−0.659787 + 0.751453i \(0.729354\pi\)
\(62\) 0 0
\(63\) 10.4346i 1.31464i
\(64\) 0 0
\(65\) −3.60168 + 0.167055i −0.446733 + 0.0207206i
\(66\) 0 0
\(67\) 7.10284i 0.867751i −0.900973 0.433875i \(-0.857146\pi\)
0.900973 0.433875i \(-0.142854\pi\)
\(68\) 0 0
\(69\) −12.8413 −1.54592
\(70\) 0 0
\(71\) 16.1112i 1.91205i 0.293281 + 0.956026i \(0.405253\pi\)
−0.293281 + 0.956026i \(0.594747\pi\)
\(72\) 0 0
\(73\) 15.9721i 1.86939i 0.355448 + 0.934696i \(0.384328\pi\)
−0.355448 + 0.934696i \(0.615672\pi\)
\(74\) 0 0
\(75\) 2.60168 0.300416
\(76\) 0 0
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) −11.9442 −1.34383 −0.671913 0.740630i \(-0.734527\pi\)
−0.671913 + 0.740630i \(0.734527\pi\)
\(80\) 0 0
\(81\) −6.10284 −0.678094
\(82\) 0 0
\(83\) 3.97209i 0.435994i −0.975949 0.217997i \(-0.930048\pi\)
0.975949 0.217997i \(-0.0699523\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 11.5375 1.23695
\(88\) 0 0
\(89\) 7.94419i 0.842082i −0.907042 0.421041i \(-0.861665\pi\)
0.907042 0.421041i \(-0.138335\pi\)
\(90\) 0 0
\(91\) −9.97209 + 0.462531i −1.04536 + 0.0484865i
\(92\) 0 0
\(93\) 8.76873i 0.909275i
\(94\) 0 0
\(95\) 5.03630 0.516714
\(96\) 0 0
\(97\) 0.462531i 0.0469629i −0.999724 0.0234815i \(-0.992525\pi\)
0.999724 0.0234815i \(-0.00747507\pi\)
\(98\) 0 0
\(99\) 8.16706i 0.820820i
\(100\) 0 0
\(101\) 12.7408 1.26776 0.633880 0.773432i \(-0.281461\pi\)
0.633880 + 0.773432i \(0.281461\pi\)
\(102\) 0 0
\(103\) 4.13915 0.407842 0.203921 0.978987i \(-0.434631\pi\)
0.203921 + 0.978987i \(0.434631\pi\)
\(104\) 0 0
\(105\) 7.20336 0.702976
\(106\) 0 0
\(107\) −7.06421 −0.682923 −0.341462 0.939896i \(-0.610922\pi\)
−0.341462 + 0.939896i \(0.610922\pi\)
\(108\) 0 0
\(109\) 14.8692i 1.42422i −0.702070 0.712108i \(-0.747741\pi\)
0.702070 0.712108i \(-0.252259\pi\)
\(110\) 0 0
\(111\) 10.3341i 0.980870i
\(112\) 0 0
\(113\) 3.13075 0.294516 0.147258 0.989098i \(-0.452955\pi\)
0.147258 + 0.989098i \(0.452955\pi\)
\(114\) 0 0
\(115\) 4.93579i 0.460265i
\(116\) 0 0
\(117\) −0.629587 13.5738i −0.0582053 1.25490i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 6.30387 0.573079
\(122\) 0 0
\(123\) 4.33411i 0.390794i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) −16.1391 −1.43212 −0.716059 0.698040i \(-0.754056\pi\)
−0.716059 + 0.698040i \(0.754056\pi\)
\(128\) 0 0
\(129\) 25.5096 2.24599
\(130\) 0 0
\(131\) 8.86925 0.774910 0.387455 0.921889i \(-0.373354\pi\)
0.387455 + 0.921889i \(0.373354\pi\)
\(132\) 0 0
\(133\) 13.9442 1.20911
\(134\) 0 0
\(135\) 2.00000i 0.172133i
\(136\) 0 0
\(137\) 1.66589i 0.142327i −0.997465 0.0711633i \(-0.977329\pi\)
0.997465 0.0711633i \(-0.0226711\pi\)
\(138\) 0 0
\(139\) −19.2760 −1.63497 −0.817483 0.575953i \(-0.804631\pi\)
−0.817483 + 0.575953i \(0.804631\pi\)
\(140\) 0 0
\(141\) 30.2783i 2.54989i
\(142\) 0 0
\(143\) −7.80504 + 0.362018i −0.652690 + 0.0302734i
\(144\) 0 0
\(145\) 4.43462i 0.368275i
\(146\) 0 0
\(147\) 1.73243 0.142889
\(148\) 0 0
\(149\) 13.9442i 1.14235i 0.820828 + 0.571176i \(0.193513\pi\)
−0.820828 + 0.571176i \(0.806487\pi\)
\(150\) 0 0
\(151\) 6.96370i 0.566698i −0.959017 0.283349i \(-0.908555\pi\)
0.959017 0.283349i \(-0.0914454\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3.37041 −0.270718
\(156\) 0 0
\(157\) 12.3341 0.984369 0.492185 0.870491i \(-0.336199\pi\)
0.492185 + 0.870491i \(0.336199\pi\)
\(158\) 0 0
\(159\) 33.1475 2.62877
\(160\) 0 0
\(161\) 13.6659i 1.07702i
\(162\) 0 0
\(163\) 20.5072i 1.60625i 0.595810 + 0.803125i \(0.296831\pi\)
−0.595810 + 0.803125i \(0.703169\pi\)
\(164\) 0 0
\(165\) 5.63798 0.438916
\(166\) 0 0
\(167\) 19.3039i 1.49378i −0.664948 0.746889i \(-0.731547\pi\)
0.664948 0.746889i \(-0.268453\pi\)
\(168\) 0 0
\(169\) −12.9442 + 1.20336i −0.995707 + 0.0925660i
\(170\) 0 0
\(171\) 18.9805i 1.45147i
\(172\) 0 0
\(173\) 9.61007 0.730640 0.365320 0.930882i \(-0.380959\pi\)
0.365320 + 0.930882i \(0.380959\pi\)
\(174\) 0 0
\(175\) 2.76873i 0.209297i
\(176\) 0 0
\(177\) 21.2481i 1.59710i
\(178\) 0 0
\(179\) 9.87158 0.737836 0.368918 0.929462i \(-0.379728\pi\)
0.368918 + 0.929462i \(0.379728\pi\)
\(180\) 0 0
\(181\) −1.97209 −0.146584 −0.0732922 0.997311i \(-0.523351\pi\)
−0.0732922 + 0.997311i \(0.523351\pi\)
\(182\) 0 0
\(183\) 26.8134 1.98211
\(184\) 0 0
\(185\) −3.97209 −0.292034
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 5.53747i 0.402792i
\(190\) 0 0
\(191\) 5.73850 0.415223 0.207611 0.978211i \(-0.433431\pi\)
0.207611 + 0.978211i \(0.433431\pi\)
\(192\) 0 0
\(193\) 4.07261i 0.293153i −0.989199 0.146576i \(-0.953175\pi\)
0.989199 0.146576i \(-0.0468254\pi\)
\(194\) 0 0
\(195\) 9.37041 0.434624i 0.671029 0.0311241i
\(196\) 0 0
\(197\) 21.6101i 1.53965i 0.638253 + 0.769827i \(0.279658\pi\)
−0.638253 + 0.769827i \(0.720342\pi\)
\(198\) 0 0
\(199\) 11.6101 0.823016 0.411508 0.911406i \(-0.365002\pi\)
0.411508 + 0.911406i \(0.365002\pi\)
\(200\) 0 0
\(201\) 18.4793i 1.30343i
\(202\) 0 0
\(203\) 12.2783i 0.861767i
\(204\) 0 0
\(205\) 1.66589 0.116351
\(206\) 0 0
\(207\) 18.6017 1.29291
\(208\) 0 0
\(209\) 10.9139 0.754933
\(210\) 0 0
\(211\) −16.2783 −1.12064 −0.560322 0.828275i \(-0.689323\pi\)
−0.560322 + 0.828275i \(0.689323\pi\)
\(212\) 0 0
\(213\) 41.9163i 2.87206i
\(214\) 0 0
\(215\) 9.80504i 0.668698i
\(216\) 0 0
\(217\) −9.33178 −0.633482
\(218\) 0 0
\(219\) 41.5543i 2.80798i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 11.4370i 0.765875i 0.923774 + 0.382938i \(0.125088\pi\)
−0.923774 + 0.382938i \(0.874912\pi\)
\(224\) 0 0
\(225\) −3.76873 −0.251249
\(226\) 0 0
\(227\) 15.9721i 1.06011i −0.847965 0.530053i \(-0.822172\pi\)
0.847965 0.530053i \(-0.177828\pi\)
\(228\) 0 0
\(229\) 2.12842i 0.140650i 0.997524 + 0.0703250i \(0.0224036\pi\)
−0.997524 + 0.0703250i \(0.977596\pi\)
\(230\) 0 0
\(231\) 15.6101 1.02707
\(232\) 0 0
\(233\) −2.86925 −0.187971 −0.0939853 0.995574i \(-0.529961\pi\)
−0.0939853 + 0.995574i \(0.529961\pi\)
\(234\) 0 0
\(235\) −11.6380 −0.759178
\(236\) 0 0
\(237\) 31.0749 2.01853
\(238\) 0 0
\(239\) 3.16939i 0.205011i 0.994732 + 0.102505i \(0.0326859\pi\)
−0.994732 + 0.102505i \(0.967314\pi\)
\(240\) 0 0
\(241\) 4.79664i 0.308979i 0.987994 + 0.154489i \(0.0493733\pi\)
−0.987994 + 0.154489i \(0.950627\pi\)
\(242\) 0 0
\(243\) 21.8776 1.40345
\(244\) 0 0
\(245\) 0.665890i 0.0425421i
\(246\) 0 0
\(247\) 18.1391 0.841340i 1.15417 0.0535332i
\(248\) 0 0
\(249\) 10.3341i 0.654898i
\(250\) 0 0
\(251\) −1.00233 −0.0632666 −0.0316333 0.999500i \(-0.510071\pi\)
−0.0316333 + 0.999500i \(0.510071\pi\)
\(252\) 0 0
\(253\) 10.6961i 0.672460i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −19.4817 −1.21523 −0.607616 0.794231i \(-0.707874\pi\)
−0.607616 + 0.794231i \(0.707874\pi\)
\(258\) 0 0
\(259\) −10.9977 −0.683362
\(260\) 0 0
\(261\) −16.7129 −1.03450
\(262\) 0 0
\(263\) 4.19496 0.258672 0.129336 0.991601i \(-0.458715\pi\)
0.129336 + 0.991601i \(0.458715\pi\)
\(264\) 0 0
\(265\) 12.7408i 0.782663i
\(266\) 0 0
\(267\) 20.6682i 1.26487i
\(268\) 0 0
\(269\) 30.4793 1.85836 0.929179 0.369631i \(-0.120516\pi\)
0.929179 + 0.369631i \(0.120516\pi\)
\(270\) 0 0
\(271\) 4.83528i 0.293722i 0.989157 + 0.146861i \(0.0469170\pi\)
−0.989157 + 0.146861i \(0.953083\pi\)
\(272\) 0 0
\(273\) 25.9442 1.20336i 1.57021 0.0728306i
\(274\) 0 0
\(275\) 2.16706i 0.130678i
\(276\) 0 0
\(277\) 3.00233 0.180393 0.0901963 0.995924i \(-0.471251\pi\)
0.0901963 + 0.995924i \(0.471251\pi\)
\(278\) 0 0
\(279\) 12.7022i 0.760460i
\(280\) 0 0
\(281\) 21.6101i 1.28915i −0.764542 0.644574i \(-0.777035\pi\)
0.764542 0.644574i \(-0.222965\pi\)
\(282\) 0 0
\(283\) −2.47326 −0.147020 −0.0735100 0.997294i \(-0.523420\pi\)
−0.0735100 + 0.997294i \(0.523420\pi\)
\(284\) 0 0
\(285\) −13.1028 −0.776146
\(286\) 0 0
\(287\) 4.61241 0.272262
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 1.20336i 0.0705421i
\(292\) 0 0
\(293\) 23.9163i 1.39720i −0.715511 0.698602i \(-0.753806\pi\)
0.715511 0.698602i \(-0.246194\pi\)
\(294\) 0 0
\(295\) 8.16706 0.475504
\(296\) 0 0
\(297\) 4.33411i 0.251491i
\(298\) 0 0
\(299\) −0.824549 17.7771i −0.0476849 1.02808i
\(300\) 0 0
\(301\) 27.1475i 1.56476i
\(302\) 0 0
\(303\) −33.1475 −1.90428
\(304\) 0 0
\(305\) 10.3062i 0.590131i
\(306\) 0 0
\(307\) 13.5654i 0.774217i −0.922034 0.387108i \(-0.873474\pi\)
0.922034 0.387108i \(-0.126526\pi\)
\(308\) 0 0
\(309\) −10.7687 −0.612612
\(310\) 0 0
\(311\) 2.12842 0.120692 0.0603458 0.998178i \(-0.480780\pi\)
0.0603458 + 0.998178i \(0.480780\pi\)
\(312\) 0 0
\(313\) −22.6077 −1.27787 −0.638933 0.769263i \(-0.720624\pi\)
−0.638933 + 0.769263i \(0.720624\pi\)
\(314\) 0 0
\(315\) −10.4346 −0.587924
\(316\) 0 0
\(317\) 10.6357i 0.597358i −0.954354 0.298679i \(-0.903454\pi\)
0.954354 0.298679i \(-0.0965459\pi\)
\(318\) 0 0
\(319\) 9.61007i 0.538061i
\(320\) 0 0
\(321\) 18.3788 1.02581
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0.167055 + 3.60168i 0.00926655 + 0.199785i
\(326\) 0 0
\(327\) 38.6850i 2.13929i
\(328\) 0 0
\(329\) −32.2225 −1.77648
\(330\) 0 0
\(331\) 1.16472i 0.0640190i 0.999488 + 0.0320095i \(0.0101907\pi\)
−0.999488 + 0.0320095i \(0.989809\pi\)
\(332\) 0 0
\(333\) 14.9698i 0.820338i
\(334\) 0 0
\(335\) −7.10284 −0.388070
\(336\) 0 0
\(337\) −20.0000 −1.08947 −0.544735 0.838608i \(-0.683370\pi\)
−0.544735 + 0.838608i \(0.683370\pi\)
\(338\) 0 0
\(339\) −8.14521 −0.442387
\(340\) 0 0
\(341\) −7.30387 −0.395527
\(342\) 0 0
\(343\) 17.5375i 0.946934i
\(344\) 0 0
\(345\) 12.8413i 0.691355i
\(346\) 0 0
\(347\) −4.67429 −0.250929 −0.125464 0.992098i \(-0.540042\pi\)
−0.125464 + 0.992098i \(0.540042\pi\)
\(348\) 0 0
\(349\) 4.33411i 0.232000i −0.993249 0.116000i \(-0.962993\pi\)
0.993249 0.116000i \(-0.0370072\pi\)
\(350\) 0 0
\(351\) 0.334110 + 7.20336i 0.0178335 + 0.384487i
\(352\) 0 0
\(353\) 21.3085i 1.13414i 0.823670 + 0.567069i \(0.191923\pi\)
−0.823670 + 0.567069i \(0.808077\pi\)
\(354\) 0 0
\(355\) 16.1112 0.855096
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 12.7795i 0.674474i −0.941420 0.337237i \(-0.890508\pi\)
0.941420 0.337237i \(-0.109492\pi\)
\(360\) 0 0
\(361\) −6.36435 −0.334966
\(362\) 0 0
\(363\) −16.4007 −0.860811
\(364\) 0 0
\(365\) 15.9721 0.836018
\(366\) 0 0
\(367\) −12.1950 −0.636572 −0.318286 0.947995i \(-0.603107\pi\)
−0.318286 + 0.947995i \(0.603107\pi\)
\(368\) 0 0
\(369\) 6.27830i 0.326835i
\(370\) 0 0
\(371\) 35.2760i 1.83144i
\(372\) 0 0
\(373\) −28.2225 −1.46130 −0.730652 0.682750i \(-0.760784\pi\)
−0.730652 + 0.682750i \(0.760784\pi\)
\(374\) 0 0
\(375\) 2.60168i 0.134350i
\(376\) 0 0
\(377\) 0.740827 + 15.9721i 0.0381545 + 0.822605i
\(378\) 0 0
\(379\) 23.3146i 1.19759i −0.800902 0.598795i \(-0.795646\pi\)
0.800902 0.598795i \(-0.204354\pi\)
\(380\) 0 0
\(381\) 41.9889 2.15116
\(382\) 0 0
\(383\) 8.02791i 0.410207i −0.978740 0.205103i \(-0.934247\pi\)
0.978740 0.205103i \(-0.0657531\pi\)
\(384\) 0 0
\(385\) 6.00000i 0.305788i
\(386\) 0 0
\(387\) −36.9526 −1.87841
\(388\) 0 0
\(389\) 23.7385 1.20359 0.601795 0.798651i \(-0.294453\pi\)
0.601795 + 0.798651i \(0.294453\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −23.0749 −1.16398
\(394\) 0 0
\(395\) 11.9442i 0.600977i
\(396\) 0 0
\(397\) 3.04703i 0.152926i −0.997072 0.0764630i \(-0.975637\pi\)
0.997072 0.0764630i \(-0.0243627\pi\)
\(398\) 0 0
\(399\) −36.2783 −1.81619
\(400\) 0 0
\(401\) 31.2201i 1.55906i −0.626365 0.779530i \(-0.715458\pi\)
0.626365 0.779530i \(-0.284542\pi\)
\(402\) 0 0
\(403\) −12.1391 + 0.563045i −0.604694 + 0.0280473i
\(404\) 0 0
\(405\) 6.10284i 0.303253i
\(406\) 0 0
\(407\) −8.60774 −0.426670
\(408\) 0 0
\(409\) 33.4090i 1.65197i −0.563691 0.825986i \(-0.690619\pi\)
0.563691 0.825986i \(-0.309381\pi\)
\(410\) 0 0
\(411\) 4.33411i 0.213786i
\(412\) 0 0
\(413\) 22.6124 1.11268
\(414\) 0 0
\(415\) −3.97209 −0.194982
\(416\) 0 0
\(417\) 50.1499 2.45585
\(418\) 0 0
\(419\) 30.7408 1.50179 0.750894 0.660423i \(-0.229623\pi\)
0.750894 + 0.660423i \(0.229623\pi\)
\(420\) 0 0
\(421\) 19.2806i 0.939680i −0.882752 0.469840i \(-0.844312\pi\)
0.882752 0.469840i \(-0.155688\pi\)
\(422\) 0 0
\(423\) 43.8605i 2.13257i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 28.5351i 1.38091i
\(428\) 0 0
\(429\) 20.3062 0.941854i 0.980393 0.0454731i
\(430\) 0 0
\(431\) 2.44535i 0.117788i −0.998264 0.0588942i \(-0.981243\pi\)
0.998264 0.0588942i \(-0.0187575\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 11.5375i 0.553179i
\(436\) 0 0
\(437\) 24.8581i 1.18913i
\(438\) 0 0
\(439\) 11.6659 0.556783 0.278391 0.960468i \(-0.410199\pi\)
0.278391 + 0.960468i \(0.410199\pi\)
\(440\) 0 0
\(441\) −2.50956 −0.119503
\(442\) 0 0
\(443\) −20.8074 −0.988588 −0.494294 0.869295i \(-0.664573\pi\)
−0.494294 + 0.869295i \(0.664573\pi\)
\(444\) 0 0
\(445\) −7.94419 −0.376590
\(446\) 0 0
\(447\) 36.2783i 1.71590i
\(448\) 0 0
\(449\) 30.2783i 1.42892i 0.699676 + 0.714461i \(0.253328\pi\)
−0.699676 + 0.714461i \(0.746672\pi\)
\(450\) 0 0
\(451\) 3.61007 0.169992
\(452\) 0 0
\(453\) 18.1173i 0.851225i
\(454\) 0 0
\(455\) 0.462531 + 9.97209i 0.0216838 + 0.467499i
\(456\) 0 0
\(457\) 22.6124i 1.05776i −0.848695 0.528882i \(-0.822611\pi\)
0.848695 0.528882i \(-0.177389\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 18.9419i 0.882210i −0.897455 0.441105i \(-0.854587\pi\)
0.897455 0.441105i \(-0.145413\pi\)
\(462\) 0 0
\(463\) 11.6380i 0.540863i −0.962739 0.270431i \(-0.912834\pi\)
0.962739 0.270431i \(-0.0871663\pi\)
\(464\) 0 0
\(465\) 8.76873 0.406640
\(466\) 0 0
\(467\) 25.0642 1.15983 0.579917 0.814676i \(-0.303085\pi\)
0.579917 + 0.814676i \(0.303085\pi\)
\(468\) 0 0
\(469\) −19.6659 −0.908086
\(470\) 0 0
\(471\) −32.0894 −1.47860
\(472\) 0 0
\(473\) 21.2481i 0.976987i
\(474\) 0 0
\(475\) 5.03630i 0.231081i
\(476\) 0 0
\(477\) −48.0168 −2.19854
\(478\) 0 0
\(479\) 18.4407i 0.842577i 0.906927 + 0.421288i \(0.138422\pi\)
−0.906927 + 0.421288i \(0.861578\pi\)
\(480\) 0 0
\(481\) −14.3062 + 0.663559i −0.652307 + 0.0302557i
\(482\) 0 0
\(483\) 35.5543i 1.61777i
\(484\) 0 0
\(485\) −0.462531 −0.0210025
\(486\) 0 0
\(487\) 25.1196i 1.13828i 0.822241 + 0.569140i \(0.192724\pi\)
−0.822241 + 0.569140i \(0.807276\pi\)
\(488\) 0 0
\(489\) 53.3532i 2.41272i
\(490\) 0 0
\(491\) 5.73850 0.258975 0.129487 0.991581i \(-0.458667\pi\)
0.129487 + 0.991581i \(0.458667\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −8.16706 −0.367082
\(496\) 0 0
\(497\) 44.6077 2.00093
\(498\) 0 0
\(499\) 33.3704i 1.49386i −0.664900 0.746932i \(-0.731526\pi\)
0.664900 0.746932i \(-0.268474\pi\)
\(500\) 0 0
\(501\) 50.2225i 2.24377i
\(502\) 0 0
\(503\) 12.6789 0.565326 0.282663 0.959219i \(-0.408782\pi\)
0.282663 + 0.959219i \(0.408782\pi\)
\(504\) 0 0
\(505\) 12.7408i 0.566959i
\(506\) 0 0
\(507\) 33.6766 3.13075i 1.49563 0.139042i
\(508\) 0 0
\(509\) 17.6147i 0.780759i −0.920654 0.390380i \(-0.872344\pi\)
0.920654 0.390380i \(-0.127656\pi\)
\(510\) 0 0
\(511\) 44.2225 1.95629
\(512\) 0 0
\(513\) 10.0726i 0.444716i
\(514\) 0 0
\(515\) 4.13915i 0.182393i
\(516\) 0 0
\(517\) −25.2201 −1.10918
\(518\) 0 0
\(519\) −25.0023 −1.09748
\(520\) 0 0
\(521\) −16.4346 −0.720014 −0.360007 0.932950i \(-0.617226\pi\)
−0.360007 + 0.932950i \(0.617226\pi\)
\(522\) 0 0
\(523\) 20.4733 0.895233 0.447617 0.894226i \(-0.352273\pi\)
0.447617 + 0.894226i \(0.352273\pi\)
\(524\) 0 0
\(525\) 7.20336i 0.314380i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.36202 0.0592182
\(530\) 0 0
\(531\) 30.7795i 1.33571i
\(532\) 0 0
\(533\) 6.00000 0.278295i 0.259889 0.0120543i
\(534\) 0 0
\(535\) 7.06421i 0.305412i
\(536\) 0 0
\(537\) −25.6827 −1.10829
\(538\) 0 0
\(539\) 1.44302i 0.0621553i
\(540\) 0 0
\(541\) 41.2928i 1.77531i 0.460505 + 0.887657i \(0.347668\pi\)
−0.460505 + 0.887657i \(0.652332\pi\)
\(542\) 0 0
\(543\) 5.13075 0.220182
\(544\) 0 0
\(545\) −14.8692 −0.636929
\(546\) 0 0
\(547\) 10.4174 0.445418 0.222709 0.974885i \(-0.428510\pi\)
0.222709 + 0.974885i \(0.428510\pi\)
\(548\) 0 0
\(549\) −38.8413 −1.65771
\(550\) 0 0
\(551\) 22.3341i 0.951465i
\(552\) 0 0
\(553\) 33.0703i 1.40629i
\(554\) 0 0
\(555\) 10.3341 0.438659
\(556\) 0 0
\(557\) 20.5845i 0.872193i 0.899900 + 0.436097i \(0.143639\pi\)
−0.899900 + 0.436097i \(0.856361\pi\)
\(558\) 0 0
\(559\) 1.63798 + 35.3146i 0.0692793 + 1.49365i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 23.6766 0.997850 0.498925 0.866645i \(-0.333728\pi\)
0.498925 + 0.866645i \(0.333728\pi\)
\(564\) 0 0
\(565\) 3.13075i 0.131712i
\(566\) 0 0
\(567\) 16.8972i 0.709614i
\(568\) 0 0
\(569\) 5.43695 0.227929 0.113965 0.993485i \(-0.463645\pi\)
0.113965 + 0.993485i \(0.463645\pi\)
\(570\) 0 0
\(571\) −15.2760 −0.639279 −0.319640 0.947539i \(-0.603562\pi\)
−0.319640 + 0.947539i \(0.603562\pi\)
\(572\) 0 0
\(573\) −14.9297 −0.623698
\(574\) 0 0
\(575\) −4.93579 −0.205837
\(576\) 0 0
\(577\) 1.76640i 0.0735363i 0.999324 + 0.0367682i \(0.0117063\pi\)
−0.999324 + 0.0367682i \(0.988294\pi\)
\(578\) 0 0
\(579\) 10.5956i 0.440339i
\(580\) 0 0
\(581\) −10.9977 −0.456260
\(582\) 0 0
\(583\) 27.6101i 1.14349i
\(584\) 0 0
\(585\) −13.5738 + 0.629587i −0.561206 + 0.0260302i
\(586\) 0 0
\(587\) 22.9139i 0.945760i −0.881127 0.472880i \(-0.843214\pi\)
0.881127 0.472880i \(-0.156786\pi\)
\(588\) 0 0
\(589\) 16.9744 0.699419
\(590\) 0 0
\(591\) 56.2225i 2.31268i
\(592\) 0 0
\(593\) 6.72404i 0.276123i −0.990424 0.138062i \(-0.955913\pi\)
0.990424 0.138062i \(-0.0440872\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −30.2057 −1.23624
\(598\) 0 0
\(599\) 19.8669 0.811740 0.405870 0.913931i \(-0.366969\pi\)
0.405870 + 0.913931i \(0.366969\pi\)
\(600\) 0 0
\(601\) −33.2201 −1.35508 −0.677539 0.735487i \(-0.736954\pi\)
−0.677539 + 0.735487i \(0.736954\pi\)
\(602\) 0 0
\(603\) 26.7687i 1.09011i
\(604\) 0 0
\(605\) 6.30387i 0.256289i
\(606\) 0 0
\(607\) −14.4174 −0.585186 −0.292593 0.956237i \(-0.594518\pi\)
−0.292593 + 0.956237i \(0.594518\pi\)
\(608\) 0 0
\(609\) 31.9442i 1.29444i
\(610\) 0 0
\(611\) −41.9163 + 1.94419i −1.69575 + 0.0786533i
\(612\) 0 0
\(613\) 11.3364i 0.457875i −0.973441 0.228937i \(-0.926475\pi\)
0.973441 0.228937i \(-0.0735251\pi\)
\(614\) 0 0
\(615\) −4.33411 −0.174768
\(616\) 0 0
\(617\) 34.8907i 1.40465i −0.711858 0.702323i \(-0.752146\pi\)
0.711858 0.702323i \(-0.247854\pi\)
\(618\) 0 0
\(619\) 32.6464i 1.31217i −0.754688 0.656084i \(-0.772212\pi\)
0.754688 0.656084i \(-0.227788\pi\)
\(620\) 0 0
\(621\) −9.87158 −0.396133
\(622\) 0 0
\(623\) −21.9953 −0.881225
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −28.3946 −1.13397
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 47.3146i 1.88356i −0.336224 0.941782i \(-0.609150\pi\)
0.336224 0.941782i \(-0.390850\pi\)
\(632\) 0 0
\(633\) 42.3509 1.68330
\(634\) 0 0
\(635\) 16.1391i 0.640463i
\(636\) 0 0
\(637\) 0.111240 + 2.39832i 0.00440750 + 0.0950249i
\(638\) 0 0
\(639\) 60.7190i 2.40201i
\(640\) 0 0
\(641\) 43.4817 1.71742 0.858711 0.512460i \(-0.171266\pi\)
0.858711 + 0.512460i \(0.171266\pi\)
\(642\) 0 0
\(643\) 20.3062i 0.800798i 0.916341 + 0.400399i \(0.131129\pi\)
−0.916341 + 0.400399i \(0.868871\pi\)
\(644\) 0 0
\(645\) 25.5096i 1.00444i
\(646\) 0 0
\(647\) −30.8027 −1.21098 −0.605490 0.795853i \(-0.707023\pi\)
−0.605490 + 0.795853i \(0.707023\pi\)
\(648\) 0 0
\(649\) 17.6985 0.694725
\(650\) 0 0
\(651\) 24.2783 0.951541
\(652\) 0 0
\(653\) 3.61007 0.141273 0.0706366 0.997502i \(-0.477497\pi\)
0.0706366 + 0.997502i \(0.477497\pi\)
\(654\) 0 0
\(655\) 8.86925i 0.346550i
\(656\) 0 0
\(657\) 60.1946i 2.34841i
\(658\) 0 0
\(659\) 6.74083 0.262585 0.131293 0.991344i \(-0.458087\pi\)
0.131293 + 0.991344i \(0.458087\pi\)
\(660\) 0 0
\(661\) 26.6682i 1.03727i −0.854995 0.518637i \(-0.826440\pi\)
0.854995 0.518637i \(-0.173560\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 13.9442i 0.540732i
\(666\) 0 0
\(667\) −21.8884 −0.847521
\(668\) 0 0
\(669\) 29.7553i 1.15041i
\(670\) 0 0
\(671\) 22.3341i 0.862199i
\(672\) 0 0
\(673\) −8.61241 −0.331984 −0.165992 0.986127i \(-0.553083\pi\)
−0.165992 + 0.986127i \(0.553083\pi\)
\(674\) 0 0
\(675\) 2.00000 0.0769800
\(676\) 0 0
\(677\) 8.12842 0.312401 0.156200 0.987725i \(-0.450075\pi\)
0.156200 + 0.987725i \(0.450075\pi\)
\(678\) 0 0
\(679\) −1.28063 −0.0491459
\(680\) 0 0
\(681\) 41.5543i 1.59236i
\(682\) 0 0
\(683\) 8.02791i 0.307179i 0.988135 + 0.153590i \(0.0490834\pi\)
−0.988135 + 0.153590i \(0.950917\pi\)
\(684\) 0 0
\(685\) −1.66589 −0.0636504
\(686\) 0 0
\(687\) 5.53747i 0.211268i
\(688\) 0 0
\(689\) 2.12842 + 45.8884i 0.0810864 + 1.74821i
\(690\) 0 0
\(691\) 22.1112i 0.841151i −0.907257 0.420576i \(-0.861828\pi\)
0.907257 0.420576i \(-0.138172\pi\)
\(692\) 0 0
\(693\) −22.6124 −0.858974
\(694\) 0 0
\(695\) 19.2760i 0.731179i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 7.46486 0.282347
\(700\) 0 0
\(701\) −6.47932 −0.244721 −0.122360 0.992486i \(-0.539046\pi\)
−0.122360 + 0.992486i \(0.539046\pi\)
\(702\) 0 0
\(703\) 20.0047 0.754490
\(704\) 0 0
\(705\) 30.2783 1.14035
\(706\) 0 0
\(707\) 35.2760i 1.32669i
\(708\) 0 0
\(709\) 2.85246i 0.107126i 0.998564 + 0.0535631i \(0.0170578\pi\)
−0.998564 + 0.0535631i \(0.982942\pi\)
\(710\) 0 0
\(711\) −45.0145 −1.68817
\(712\) 0 0
\(713\) 16.6357i 0.623010i
\(714\) 0 0
\(715\) 0.362018 + 7.80504i 0.0135387 + 0.291892i
\(716\) 0 0
\(717\) 8.24573i 0.307942i
\(718\) 0 0
\(719\) −42.0941 −1.56984 −0.784922 0.619595i \(-0.787297\pi\)
−0.784922 + 0.619595i \(0.787297\pi\)
\(720\) 0 0
\(721\) 11.4602i 0.426800i
\(722\) 0 0
\(723\) 12.4793i 0.464111i
\(724\) 0 0
\(725\) 4.43462 0.164698
\(726\) 0 0
\(727\) 32.8027 1.21659 0.608293 0.793713i \(-0.291855\pi\)
0.608293 + 0.793713i \(0.291855\pi\)
\(728\) 0 0
\(729\) −38.6101 −1.43000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 2.38993i 0.0882739i 0.999025 + 0.0441370i \(0.0140538\pi\)
−0.999025 + 0.0441370i \(0.985946\pi\)
\(734\) 0 0
\(735\) 1.73243i 0.0639017i
\(736\) 0 0
\(737\) −15.3923 −0.566981
\(738\) 0 0
\(739\) 41.4988i 1.52656i 0.646068 + 0.763280i \(0.276412\pi\)
−0.646068 + 0.763280i \(0.723588\pi\)
\(740\) 0 0
\(741\) −47.1922 + 2.18890i −1.73365 + 0.0804112i
\(742\) 0 0
\(743\) 39.8046i 1.46029i −0.683292 0.730145i \(-0.739452\pi\)
0.683292 0.730145i \(-0.260548\pi\)
\(744\) 0 0
\(745\) 13.9442 0.510875
\(746\) 0 0
\(747\) 14.9698i 0.547715i
\(748\) 0 0
\(749\) 19.5589i 0.714667i
\(750\) 0 0
\(751\) −15.6659 −0.571656 −0.285828 0.958281i \(-0.592269\pi\)
−0.285828 + 0.958281i \(0.592269\pi\)
\(752\) 0 0
\(753\) 2.60774 0.0950315
\(754\) 0 0
\(755\) −6.96370 −0.253435
\(756\) 0 0
\(757\) 32.2736 1.17301 0.586503 0.809947i \(-0.300504\pi\)
0.586503 + 0.809947i \(0.300504\pi\)
\(758\) 0 0
\(759\) 27.8279i 1.01009i
\(760\) 0 0
\(761\) 13.2806i 0.481422i −0.970597 0.240711i \(-0.922619\pi\)
0.970597 0.240711i \(-0.0773807\pi\)
\(762\) 0 0
\(763\) −41.1690 −1.49042
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 29.4151 1.36435i 1.06212 0.0492638i
\(768\) 0 0
\(769\) 12.2010i 0.439980i −0.975502 0.219990i \(-0.929397\pi\)
0.975502 0.219990i \(-0.0706025\pi\)
\(770\) 0 0
\(771\) 50.6850 1.82538
\(772\) 0 0
\(773\) 27.9721i 1.00609i 0.864261 + 0.503043i \(0.167786\pi\)
−0.864261 + 0.503043i \(0.832214\pi\)
\(774\) 0 0
\(775\) 3.37041i 0.121069i
\(776\) 0 0
\(777\) 28.6124 1.02646
\(778\) 0 0
\(779\) −8.38993 −0.300600
\(780\) 0 0
\(781\) 34.9139 1.24932
\(782\) 0 0
\(783\) 8.86925 0.316961
\(784\) 0 0
\(785\) 12.3341i 0.440223i
\(786\) 0 0
\(787\) 10.2336i 0.364788i −0.983225 0.182394i \(-0.941615\pi\)
0.983225 0.182394i \(-0.0583847\pi\)
\(788\) 0 0
\(789\) −10.9139 −0.388547
\(790\) 0 0
\(791\) 8.66822i 0.308206i
\(792\) 0 0
\(793\) 1.72170 + 37.1196i 0.0611395 + 1.31816i
\(794\) 0 0
\(795\) 33.1475i 1.17562i
\(796\) 0 0
\(797\) −23.3532 −0.827214 −0.413607 0.910456i \(-0.635731\pi\)
−0.413607 + 0.910456i \(0.635731\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 29.9395i 1.05786i
\(802\) 0 0
\(803\) 34.6124 1.22145
\(804\) 0 0
\(805\) −13.6659 −0.481659