Properties

Label 1764.4.k.j.361.1
Level $1764$
Weight $4$
Character 1764.361
Analytic conductor $104.079$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1764.k (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(104.079369250\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 588)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1764.361
Dual form 1764.4.k.j.1549.1

$q$-expansion

\(f(q)\) \(=\) \(q+(2.00000 + 3.46410i) q^{5} +O(q^{10})\) \(q+(2.00000 + 3.46410i) q^{5} +(-10.0000 + 17.3205i) q^{11} -4.00000 q^{13} +(12.0000 - 20.7846i) q^{17} +(-22.0000 - 38.1051i) q^{19} +(36.0000 + 62.3538i) q^{23} +(54.5000 - 94.3968i) q^{25} +38.0000 q^{29} +(-92.0000 + 159.349i) q^{31} +(15.0000 + 25.9808i) q^{37} +216.000 q^{41} -164.000 q^{43} +(260.000 + 450.333i) q^{47} +(-73.0000 + 126.440i) q^{53} -80.0000 q^{55} +(230.000 - 398.372i) q^{59} +(-314.000 - 543.864i) q^{61} +(-8.00000 - 13.8564i) q^{65} +(-278.000 + 481.510i) q^{67} -592.000 q^{71} +(-512.000 + 886.810i) q^{73} +(52.0000 + 90.0666i) q^{79} +324.000 q^{83} +96.0000 q^{85} +(448.000 + 775.959i) q^{89} +(88.0000 - 152.420i) q^{95} -920.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} + O(q^{10}) \) \( 2q + 4q^{5} - 20q^{11} - 8q^{13} + 24q^{17} - 44q^{19} + 72q^{23} + 109q^{25} + 76q^{29} - 184q^{31} + 30q^{37} + 432q^{41} - 328q^{43} + 520q^{47} - 146q^{53} - 160q^{55} + 460q^{59} - 628q^{61} - 16q^{65} - 556q^{67} - 1184q^{71} - 1024q^{73} + 104q^{79} + 648q^{83} + 192q^{85} + 896q^{89} + 176q^{95} - 1840q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 2.00000 + 3.46410i 0.178885 + 0.309839i 0.941499 0.337016i \(-0.109418\pi\)
−0.762614 + 0.646854i \(0.776084\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −10.0000 + 17.3205i −0.274101 + 0.474757i −0.969908 0.243472i \(-0.921714\pi\)
0.695807 + 0.718229i \(0.255047\pi\)
\(12\) 0 0
\(13\) −4.00000 −0.0853385 −0.0426692 0.999089i \(-0.513586\pi\)
−0.0426692 + 0.999089i \(0.513586\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 12.0000 20.7846i 0.171202 0.296530i −0.767639 0.640883i \(-0.778568\pi\)
0.938840 + 0.344353i \(0.111902\pi\)
\(18\) 0 0
\(19\) −22.0000 38.1051i −0.265639 0.460101i 0.702092 0.712087i \(-0.252250\pi\)
−0.967731 + 0.251986i \(0.918916\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 36.0000 + 62.3538i 0.326370 + 0.565290i 0.981789 0.189976i \(-0.0608410\pi\)
−0.655418 + 0.755266i \(0.727508\pi\)
\(24\) 0 0
\(25\) 54.5000 94.3968i 0.436000 0.755174i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 38.0000 0.243325 0.121662 0.992572i \(-0.461177\pi\)
0.121662 + 0.992572i \(0.461177\pi\)
\(30\) 0 0
\(31\) −92.0000 + 159.349i −0.533022 + 0.923222i 0.466234 + 0.884661i \(0.345610\pi\)
−0.999256 + 0.0385601i \(0.987723\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 15.0000 + 25.9808i 0.0666482 + 0.115438i 0.897424 0.441169i \(-0.145436\pi\)
−0.830776 + 0.556607i \(0.812103\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 216.000 0.822769 0.411385 0.911462i \(-0.365045\pi\)
0.411385 + 0.911462i \(0.365045\pi\)
\(42\) 0 0
\(43\) −164.000 −0.581622 −0.290811 0.956780i \(-0.593925\pi\)
−0.290811 + 0.956780i \(0.593925\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 260.000 + 450.333i 0.806913 + 1.39761i 0.914992 + 0.403472i \(0.132197\pi\)
−0.108079 + 0.994142i \(0.534470\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −73.0000 + 126.440i −0.189195 + 0.327695i −0.944982 0.327122i \(-0.893921\pi\)
0.755787 + 0.654817i \(0.227254\pi\)
\(54\) 0 0
\(55\) −80.0000 −0.196131
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 230.000 398.372i 0.507516 0.879044i −0.492446 0.870343i \(-0.663897\pi\)
0.999962 0.00870069i \(-0.00276955\pi\)
\(60\) 0 0
\(61\) −314.000 543.864i −0.659075 1.14155i −0.980855 0.194737i \(-0.937615\pi\)
0.321780 0.946814i \(-0.395719\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −8.00000 13.8564i −0.0152658 0.0264412i
\(66\) 0 0
\(67\) −278.000 + 481.510i −0.506912 + 0.877997i 0.493056 + 0.869998i \(0.335880\pi\)
−0.999968 + 0.00799979i \(0.997454\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −592.000 −0.989542 −0.494771 0.869023i \(-0.664748\pi\)
−0.494771 + 0.869023i \(0.664748\pi\)
\(72\) 0 0
\(73\) −512.000 + 886.810i −0.820891 + 1.42183i 0.0841280 + 0.996455i \(0.473190\pi\)
−0.905019 + 0.425371i \(0.860144\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 52.0000 + 90.0666i 0.0740564 + 0.128269i 0.900676 0.434492i \(-0.143072\pi\)
−0.826619 + 0.562762i \(0.809739\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 324.000 0.428477 0.214239 0.976781i \(-0.431273\pi\)
0.214239 + 0.976781i \(0.431273\pi\)
\(84\) 0 0
\(85\) 96.0000 0.122502
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 448.000 + 775.959i 0.533572 + 0.924174i 0.999231 + 0.0392095i \(0.0124840\pi\)
−0.465659 + 0.884964i \(0.654183\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 88.0000 152.420i 0.0950380 0.164611i
\(96\) 0 0
\(97\) −920.000 −0.963009 −0.481504 0.876444i \(-0.659909\pi\)
−0.481504 + 0.876444i \(0.659909\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 554.000 959.556i 0.545793 0.945341i −0.452764 0.891630i \(-0.649562\pi\)
0.998557 0.0537102i \(-0.0171047\pi\)
\(102\) 0 0
\(103\) −724.000 1254.00i −0.692600 1.19962i −0.970983 0.239149i \(-0.923132\pi\)
0.278383 0.960470i \(-0.410202\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 658.000 + 1139.69i 0.594498 + 1.02970i 0.993618 + 0.112802i \(0.0359824\pi\)
−0.399120 + 0.916899i \(0.630684\pi\)
\(108\) 0 0
\(109\) 43.0000 74.4782i 0.0377858 0.0654469i −0.846514 0.532366i \(-0.821303\pi\)
0.884300 + 0.466919i \(0.154636\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1778.00 −1.48018 −0.740089 0.672509i \(-0.765217\pi\)
−0.740089 + 0.672509i \(0.765217\pi\)
\(114\) 0 0
\(115\) −144.000 + 249.415i −0.116766 + 0.202244i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 465.500 + 806.270i 0.349737 + 0.605762i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 936.000 0.669747
\(126\) 0 0
\(127\) −928.000 −0.648399 −0.324200 0.945989i \(-0.605095\pi\)
−0.324200 + 0.945989i \(0.605095\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 702.000 + 1215.90i 0.468199 + 0.810944i 0.999339 0.0363397i \(-0.0115698\pi\)
−0.531141 + 0.847284i \(0.678236\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −685.000 + 1186.45i −0.427179 + 0.739895i −0.996621 0.0821359i \(-0.973826\pi\)
0.569442 + 0.822031i \(0.307159\pi\)
\(138\) 0 0
\(139\) 516.000 0.314867 0.157434 0.987530i \(-0.449678\pi\)
0.157434 + 0.987530i \(0.449678\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 40.0000 69.2820i 0.0233914 0.0405151i
\(144\) 0 0
\(145\) 76.0000 + 131.636i 0.0435273 + 0.0753915i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 695.000 + 1203.78i 0.382125 + 0.661860i 0.991366 0.131125i \(-0.0418591\pi\)
−0.609241 + 0.792985i \(0.708526\pi\)
\(150\) 0 0
\(151\) −68.0000 + 117.779i −0.0366474 + 0.0634752i −0.883767 0.467927i \(-0.845001\pi\)
0.847120 + 0.531402i \(0.178335\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −736.000 −0.381400
\(156\) 0 0
\(157\) 74.0000 128.172i 0.0376168 0.0651543i −0.846604 0.532223i \(-0.821357\pi\)
0.884221 + 0.467069i \(0.154690\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 606.000 + 1049.62i 0.291200 + 0.504373i 0.974094 0.226145i \(-0.0726122\pi\)
−0.682894 + 0.730518i \(0.739279\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1976.00 −0.915614 −0.457807 0.889052i \(-0.651365\pi\)
−0.457807 + 0.889052i \(0.651365\pi\)
\(168\) 0 0
\(169\) −2181.00 −0.992717
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1346.00 2331.34i −0.591529 1.02456i −0.994027 0.109137i \(-0.965191\pi\)
0.402498 0.915421i \(-0.368142\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1290.00 + 2234.35i −0.538654 + 0.932977i 0.460322 + 0.887752i \(0.347734\pi\)
−0.998977 + 0.0452249i \(0.985600\pi\)
\(180\) 0 0
\(181\) −2036.00 −0.836103 −0.418052 0.908423i \(-0.637287\pi\)
−0.418052 + 0.908423i \(0.637287\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −60.0000 + 103.923i −0.0238448 + 0.0413004i
\(186\) 0 0
\(187\) 240.000 + 415.692i 0.0938531 + 0.162558i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1980.00 + 3429.46i 0.750093 + 1.29920i 0.947777 + 0.318933i \(0.103325\pi\)
−0.197684 + 0.980266i \(0.563342\pi\)
\(192\) 0 0
\(193\) −1.00000 + 1.73205i −0.000372962 + 0.000645988i −0.866212 0.499677i \(-0.833452\pi\)
0.865839 + 0.500323i \(0.166785\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3774.00 −1.36491 −0.682453 0.730930i \(-0.739087\pi\)
−0.682453 + 0.730930i \(0.739087\pi\)
\(198\) 0 0
\(199\) 1780.00 3083.05i 0.634075 1.09825i −0.352636 0.935761i \(-0.614715\pi\)
0.986710 0.162489i \(-0.0519521\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 432.000 + 748.246i 0.147181 + 0.254926i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 880.000 0.291248
\(210\) 0 0
\(211\) −2692.00 −0.878317 −0.439159 0.898410i \(-0.644723\pi\)
−0.439159 + 0.898410i \(0.644723\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −328.000 568.113i −0.104044 0.180209i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −48.0000 + 83.1384i −0.0146101 + 0.0253054i
\(222\) 0 0
\(223\) 4528.00 1.35972 0.679859 0.733342i \(-0.262041\pi\)
0.679859 + 0.733342i \(0.262041\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1826.00 + 3162.72i −0.533903 + 0.924746i 0.465313 + 0.885146i \(0.345942\pi\)
−0.999216 + 0.0396002i \(0.987392\pi\)
\(228\) 0 0
\(229\) 2402.00 + 4160.39i 0.693138 + 1.20055i 0.970804 + 0.239873i \(0.0771057\pi\)
−0.277666 + 0.960678i \(0.589561\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1379.00 + 2388.50i 0.387731 + 0.671570i 0.992144 0.125102i \(-0.0399257\pi\)
−0.604413 + 0.796671i \(0.706592\pi\)
\(234\) 0 0
\(235\) −1040.00 + 1801.33i −0.288690 + 0.500026i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6528.00 −1.76678 −0.883392 0.468635i \(-0.844746\pi\)
−0.883392 + 0.468635i \(0.844746\pi\)
\(240\) 0 0
\(241\) 28.0000 48.4974i 0.00748398 0.0129626i −0.862259 0.506467i \(-0.830951\pi\)
0.869743 + 0.493505i \(0.164284\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 88.0000 + 152.420i 0.0226693 + 0.0392643i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −4900.00 −1.23221 −0.616106 0.787663i \(-0.711291\pi\)
−0.616106 + 0.787663i \(0.711291\pi\)
\(252\) 0 0
\(253\) −1440.00 −0.357834
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3392.00 5875.12i −0.823296 1.42599i −0.903214 0.429190i \(-0.858799\pi\)
0.0799181 0.996801i \(-0.474534\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −2272.00 + 3935.22i −0.532690 + 0.922646i 0.466581 + 0.884478i \(0.345486\pi\)
−0.999271 + 0.0381681i \(0.987848\pi\)
\(264\) 0 0
\(265\) −584.000 −0.135377
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2026.00 + 3509.13i −0.459210 + 0.795374i −0.998919 0.0464767i \(-0.985201\pi\)
0.539710 + 0.841851i \(0.318534\pi\)
\(270\) 0 0
\(271\) 1376.00 + 2383.30i 0.308436 + 0.534226i 0.978020 0.208510i \(-0.0668612\pi\)
−0.669585 + 0.742736i \(0.733528\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1090.00 + 1887.94i 0.239016 + 0.413988i
\(276\) 0 0
\(277\) −2183.00 + 3781.07i −0.473515 + 0.820153i −0.999540 0.0303164i \(-0.990348\pi\)
0.526025 + 0.850469i \(0.323682\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −7734.00 −1.64189 −0.820946 0.571006i \(-0.806553\pi\)
−0.820946 + 0.571006i \(0.806553\pi\)
\(282\) 0 0
\(283\) 2026.00 3509.13i 0.425559 0.737090i −0.570913 0.821010i \(-0.693411\pi\)
0.996472 + 0.0839204i \(0.0267442\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2168.50 + 3755.95i 0.441380 + 0.764493i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 3420.00 0.681906 0.340953 0.940080i \(-0.389250\pi\)
0.340953 + 0.940080i \(0.389250\pi\)
\(294\) 0 0
\(295\) 1840.00 0.363149
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −144.000 249.415i −0.0278520 0.0482410i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1256.00 2175.46i 0.235798 0.408414i
\(306\) 0 0
\(307\) 7324.00 1.36157 0.680786 0.732482i \(-0.261638\pi\)
0.680786 + 0.732482i \(0.261638\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −2096.00 + 3630.38i −0.382165 + 0.661929i −0.991371 0.131083i \(-0.958155\pi\)
0.609207 + 0.793012i \(0.291488\pi\)
\(312\) 0 0
\(313\) 3420.00 + 5923.61i 0.617603 + 1.06972i 0.989922 + 0.141615i \(0.0452296\pi\)
−0.372318 + 0.928105i \(0.621437\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3315.00 + 5741.75i 0.587347 + 1.01731i 0.994578 + 0.103990i \(0.0331609\pi\)
−0.407232 + 0.913325i \(0.633506\pi\)
\(318\) 0 0
\(319\) −380.000 + 658.179i −0.0666957 + 0.115520i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −1056.00 −0.181911
\(324\) 0 0
\(325\) −218.000 + 377.587i −0.0372076 + 0.0644454i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 3434.00 + 5947.86i 0.570241 + 0.987686i 0.996541 + 0.0831042i \(0.0264834\pi\)
−0.426300 + 0.904582i \(0.640183\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2224.00 −0.362717
\(336\) 0 0
\(337\) −7378.00 −1.19260 −0.596299 0.802763i \(-0.703363\pi\)
−0.596299 + 0.802763i \(0.703363\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1840.00 3186.97i −0.292204 0.506112i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1338.00 + 2317.48i −0.206996 + 0.358528i −0.950767 0.309907i \(-0.899702\pi\)
0.743771 + 0.668435i \(0.233035\pi\)
\(348\) 0 0
\(349\) −5124.00 −0.785907 −0.392953 0.919558i \(-0.628547\pi\)
−0.392953 + 0.919558i \(0.628547\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2280.00 3949.08i 0.343774 0.595434i −0.641356 0.767243i \(-0.721628\pi\)
0.985130 + 0.171809i \(0.0549613\pi\)
\(354\) 0 0
\(355\) −1184.00 2050.75i −0.177015 0.306598i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1828.00 + 3166.19i 0.268741 + 0.465474i 0.968537 0.248869i \(-0.0800590\pi\)
−0.699796 + 0.714343i \(0.746726\pi\)
\(360\) 0 0
\(361\) 2461.50 4263.44i 0.358872 0.621584i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −4096.00 −0.587382
\(366\) 0 0
\(367\) 808.000 1399.50i 0.114924 0.199055i −0.802825 0.596215i \(-0.796671\pi\)
0.917750 + 0.397160i \(0.130004\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1367.00 2367.71i −0.189760 0.328674i 0.755410 0.655252i \(-0.227438\pi\)
−0.945170 + 0.326578i \(0.894104\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −152.000 −0.0207650
\(378\) 0 0
\(379\) −1380.00 −0.187034 −0.0935169 0.995618i \(-0.529811\pi\)
−0.0935169 + 0.995618i \(0.529811\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3444.00 + 5965.18i 0.459478 + 0.795840i 0.998933 0.0461746i \(-0.0147031\pi\)
−0.539455 + 0.842014i \(0.681370\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1023.00 + 1771.89i −0.133337 + 0.230947i −0.924961 0.380062i \(-0.875903\pi\)
0.791624 + 0.611009i \(0.209236\pi\)
\(390\) 0 0
\(391\) 1728.00 0.223501
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −208.000 + 360.267i −0.0264952 + 0.0458911i
\(396\) 0 0
\(397\) −1558.00 2698.54i −0.196962 0.341148i 0.750580 0.660779i \(-0.229774\pi\)
−0.947542 + 0.319632i \(0.896441\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1479.00 + 2561.70i 0.184184 + 0.319016i 0.943301 0.331938i \(-0.107702\pi\)
−0.759117 + 0.650954i \(0.774369\pi\)
\(402\) 0 0
\(403\) 368.000 637.395i 0.0454873 0.0787863i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −600.000 −0.0730735
\(408\) 0 0
\(409\) −3972.00 + 6879.71i −0.480202 + 0.831735i −0.999742 0.0227114i \(-0.992770\pi\)
0.519540 + 0.854446i \(0.326103\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 648.000 + 1122.37i 0.0766484 + 0.132759i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −4084.00 −0.476173 −0.238086 0.971244i \(-0.576520\pi\)
−0.238086 + 0.971244i \(0.576520\pi\)
\(420\) 0 0
\(421\) −6306.00 −0.730013 −0.365007 0.931005i \(-0.618933\pi\)
−0.365007 + 0.931005i \(0.618933\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1308.00 2265.52i −0.149288 0.258574i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −5912.00 + 10239.9i −0.660722 + 1.14440i 0.319705 + 0.947517i \(0.396416\pi\)
−0.980426 + 0.196886i \(0.936917\pi\)
\(432\) 0 0
\(433\) −4504.00 −0.499881 −0.249940 0.968261i \(-0.580411\pi\)
−0.249940 + 0.968261i \(0.580411\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1584.00 2743.57i 0.173394 0.300326i
\(438\) 0 0
\(439\) −6528.00 11306.8i −0.709714 1.22926i −0.964963 0.262385i \(-0.915491\pi\)
0.255249 0.966875i \(-0.417842\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 66.0000 + 114.315i 0.00707845 + 0.0122602i 0.869543 0.493857i \(-0.164414\pi\)
−0.862464 + 0.506118i \(0.831080\pi\)
\(444\) 0 0
\(445\) −1792.00 + 3103.84i −0.190897 + 0.330642i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −4866.00 −0.511449 −0.255725 0.966750i \(-0.582314\pi\)
−0.255725 + 0.966750i \(0.582314\pi\)
\(450\) 0 0
\(451\) −2160.00 + 3741.23i −0.225522 + 0.390616i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −5053.00 8752.05i −0.517220 0.895851i −0.999800 0.0199990i \(-0.993634\pi\)
0.482580 0.875852i \(-0.339700\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 18036.0 1.82217 0.911085 0.412219i \(-0.135246\pi\)
0.911085 + 0.412219i \(0.135246\pi\)
\(462\) 0 0
\(463\) 5288.00 0.530787 0.265393 0.964140i \(-0.414498\pi\)
0.265393 + 0.964140i \(0.414498\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 7582.00 + 13132.4i 0.751291 + 1.30128i 0.947197 + 0.320652i \(0.103902\pi\)
−0.195906 + 0.980623i \(0.562765\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1640.00 2840.56i 0.159423 0.276129i
\(474\) 0 0
\(475\) −4796.00 −0.463275
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3948.00 6838.14i 0.376594 0.652281i −0.613970 0.789329i \(-0.710428\pi\)
0.990564 + 0.137049i \(0.0437617\pi\)
\(480\) 0 0
\(481\) −60.0000 103.923i −0.00568766 0.00985132i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1840.00 3186.97i −0.172268 0.298377i
\(486\) 0 0
\(487\) −1460.00 + 2528.79i −0.135850 + 0.235299i −0.925922 0.377715i \(-0.876710\pi\)
0.790072 + 0.613014i \(0.210043\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 7932.00 0.729055 0.364528 0.931193i \(-0.381230\pi\)
0.364528 + 0.931193i \(0.381230\pi\)
\(492\) 0 0
\(493\) 456.000 789.815i 0.0416576 0.0721531i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 1002.00 + 1735.51i 0.0898911 + 0.155696i 0.907465 0.420128i \(-0.138015\pi\)
−0.817574 + 0.575824i \(0.804681\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −4496.00 −0.398542 −0.199271 0.979944i \(-0.563857\pi\)
−0.199271 + 0.979944i \(0.563857\pi\)
\(504\) 0 0
\(505\) 4432.00 0.390537
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 6310.00 + 10929.2i 0.549481 + 0.951729i 0.998310 + 0.0581114i \(0.0185079\pi\)
−0.448829 + 0.893618i \(0.648159\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2896.00 5016.02i 0.247792 0.429189i
\(516\) 0 0
\(517\) −10400.0 −0.884703
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 9004.00 15595.4i 0.757145 1.31141i −0.187156 0.982330i \(-0.559927\pi\)
0.944301 0.329083i \(-0.106740\pi\)
\(522\) 0 0
\(523\) −6646.00 11511.2i −0.555658 0.962428i −0.997852 0.0655088i \(-0.979133\pi\)
0.442194 0.896920i \(-0.354200\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2208.00 + 3824.37i 0.182509 + 0.316114i
\(528\) 0 0
\(529\) 3491.50 6047.46i 0.286965 0.497038i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −864.000 −0.0702139
\(534\) 0 0
\(535\) −2632.00 + 4558.76i −0.212694 + 0.368397i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 4285.00 + 7421.84i 0.340530 + 0.589815i 0.984531 0.175210i \(-0.0560603\pi\)
−0.644002 + 0.765024i \(0.722727\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 344.000 0.0270373
\(546\) 0 0
\(547\) −1916.00 −0.149766 −0.0748832 0.997192i \(-0.523858\pi\)
−0.0748832 + 0.997192i \(0.523858\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −836.000 1447.99i −0.0646367 0.111954i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9963.00 17256.4i 0.757892 1.31271i −0.186032 0.982544i \(-0.559563\pi\)
0.943924 0.330164i \(-0.107104\pi\)
\(558\) 0 0
\(559\) 656.000 0.0496348
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2122.00 3675.41i 0.158848 0.275133i −0.775605 0.631218i \(-0.782555\pi\)
0.934454 + 0.356085i \(0.115889\pi\)
\(564\) 0 0
\(565\) −3556.00 6159.17i −0.264782 0.458617i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −11397.0 19740.2i −0.839696 1.45440i −0.890149 0.455670i \(-0.849400\pi\)
0.0504527 0.998726i \(-0.483934\pi\)
\(570\) 0 0
\(571\) −7014.00 + 12148.6i −0.514057 + 0.890374i 0.485810 + 0.874065i \(0.338525\pi\)
−0.999867 + 0.0163089i \(0.994809\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 7848.00 0.569190
\(576\) 0 0
\(577\) −4184.00 + 7246.90i −0.301876 + 0.522864i −0.976561 0.215242i \(-0.930946\pi\)
0.674685 + 0.738106i \(0.264279\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1460.00 2528.79i −0.103717 0.179643i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −52.0000 −0.00365634 −0.00182817 0.999998i \(-0.500582\pi\)
−0.00182817 + 0.999998i \(0.500582\pi\)
\(588\) 0 0
\(589\) 8096.00 0.566366
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 2904.00 + 5029.88i 0.201101 + 0.348317i 0.948883 0.315627i \(-0.102215\pi\)
−0.747782 + 0.663944i \(0.768881\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −5232.00 + 9062.09i −0.356884 + 0.618142i −0.987439 0.158003i \(-0.949494\pi\)
0.630554 + 0.776145i \(0.282828\pi\)
\(600\) 0 0
\(601\) 1184.00 0.0803600 0.0401800 0.999192i \(-0.487207\pi\)
0.0401800 + 0.999192i \(0.487207\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1862.00 + 3225.08i −0.125126 + 0.216724i
\(606\) 0 0
\(607\) −6576.00 11390.0i −0.439723 0.761622i 0.557945 0.829878i \(-0.311590\pi\)
−0.997668 + 0.0682559i \(0.978257\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1040.00 1801.33i −0.0688607 0.119270i
\(612\) 0 0
\(613\) 9167.00 15877.7i 0.603999 1.04616i −0.388209 0.921571i \(-0.626906\pi\)
0.992209 0.124586i \(-0.0397604\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8122.00 0.529950 0.264975 0.964255i \(-0.414636\pi\)
0.264975 + 0.964255i \(0.414636\pi\)
\(618\) 0 0
\(619\) −2990.00 + 5178.83i −0.194149 + 0.336276i −0.946621 0.322348i \(-0.895528\pi\)
0.752472 + 0.658624i \(0.228861\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −4940.50 8557.20i −0.316192 0.547661i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 720.000 0.0456411
\(630\) 0 0
\(631\) 12528.0 0.790383 0.395192 0.918599i \(-0.370678\pi\)
0.395192 + 0.918599i \(0.370678\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1856.00 3214.69i −0.115989 0.200899i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 10399.0 18011.6i 0.640773 1.10985i −0.344487 0.938791i \(-0.611947\pi\)
0.985260 0.171061i \(-0.0547196\pi\)
\(642\) 0 0
\(643\) −1932.00 −0.118492 −0.0592462 0.998243i \(-0.518870\pi\)
−0.0592462 + 0.998243i \(0.518870\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −4212.00 + 7295.40i −0.255936 + 0.443295i −0.965149 0.261699i \(-0.915717\pi\)
0.709213 + 0.704994i \(0.249050\pi\)
\(648\) 0 0
\(649\) 4600.00 + 7967.43i 0.278222 + 0.481894i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −8875.00 15372.0i −0.531862 0.921211i −0.999308 0.0371899i \(-0.988159\pi\)
0.467447 0.884021i \(-0.345174\pi\)
\(654\) 0 0
\(655\) −2808.00 + 4863.60i −0.167508 + 0.290132i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 27580.0 1.63029 0.815147 0.579254i \(-0.196656\pi\)
0.815147 + 0.579254i \(0.196656\pi\)
\(660\) 0 0
\(661\) 4646.00 8047.11i 0.273386 0.473519i −0.696340 0.717712i \(-0.745190\pi\)
0.969727 + 0.244193i \(0.0785229\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1368.00 + 2369.45i 0.0794141 + 0.137549i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 12560.0 0.722613
\(672\) 0 0
\(673\) 11486.0 0.657879 0.328940 0.944351i \(-0.393309\pi\)
0.328940 + 0.944351i \(0.393309\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3558.00 + 6162.64i 0.201987 + 0.349851i 0.949168 0.314768i \(-0.101927\pi\)
−0.747182 + 0.664620i \(0.768594\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −3806.00 + 6592.19i −0.213225 + 0.369316i −0.952722 0.303843i \(-0.901730\pi\)
0.739497 + 0.673160i \(0.235063\pi\)
\(684\) 0 0
\(685\) −5480.00 −0.305664
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 292.000 505.759i 0.0161456 0.0279650i
\(690\) 0 0
\(691\) 10786.0 + 18681.9i 0.593804 + 1.02850i 0.993714 + 0.111945i \(0.0357080\pi\)
−0.399910 + 0.916554i \(0.630959\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1032.00 + 1787.48i 0.0563252 + 0.0975581i
\(696\) 0 0
\(697\) 2592.00 4489.48i 0.140859 0.243976i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1702.00 0.0917028 0.0458514 0.998948i \(-0.485400\pi\)
0.0458514 + 0.998948i \(0.485400\pi\)
\(702\) 0 0
\(703\) 660.000 1143.15i 0.0354088 0.0613298i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −3185.00 5516.58i −0.168710 0.292214i 0.769257 0.638940i \(-0.220627\pi\)
−0.937966 + 0.346726i \(0.887293\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −13248.0 −0.695851
\(714\) 0 0
\(715\) 320.000 0.0167375
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −4404.00 7627.95i −0.228430 0.395653i 0.728913 0.684607i \(-0.240026\pi\)
−0.957343 + 0.288954i \(0.906693\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2071.00 3587.08i 0.106090 0.183753i
\(726\) 0 0
\(727\) 17768.0 0.906436 0.453218 0.891400i \(-0.350276\pi\)
0.453218 + 0.891400i \(0.350276\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1968.00 + 3408.68i −0.0995747 + 0.172468i
\(732\) 0 0
\(733\) 2782.00 + 4818.57i 0.140185 + 0.242807i 0.927566 0.373659i \(-0.121897\pi\)
−0.787381 + 0.616466i \(0.788564\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5560.00 9630.20i −0.277890 0.481320i
\(738\) 0 0
\(739\) 8782.00 15210.9i 0.437146 0.757160i −0.560322 0.828275i \(-0.689323\pi\)
0.997468 + 0.0711154i \(0.0226559\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 38280.0 1.89012 0.945059 0.326901i \(-0.106004\pi\)
0.945059 + 0.326901i \(0.106004\pi\)
\(744\) 0 0
\(745\) −2780.00 + 4815.10i −0.136713 + 0.236794i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −18096.0 31343.2i −0.879271 1.52294i −0.852142 0.523310i \(-0.824697\pi\)
−0.0271284 0.999632i \(-0.508636\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −544.000 −0.0262228
\(756\) 0 0
\(757\) −14.0000 −0.000672178 −0.000336089 1.00000i \(-0.500107\pi\)
−0.000336089 1.00000i \(0.500107\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −13252.0 22953.1i −0.631254 1.09336i −0.987296 0.158894i \(-0.949207\pi\)
0.356041 0.934470i \(-0.384126\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −920.000 + 1593.49i −0.0433107 + 0.0750163i
\(768\) 0 0
\(769\) 40184.0 1.88436 0.942180 0.335109i \(-0.108773\pi\)
0.942180 + 0.335109i \(0.108773\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −17670.0 + 30605.3i −0.822181 + 1.42406i 0.0818742 + 0.996643i \(0.473909\pi\)
−0.904055 + 0.427416i \(0.859424\pi\)
\(774\) 0 0
\(775\) 10028.0 + 17369.0i 0.464795 + 0.805049i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4752.00 8230.71i −0.218560 0.378557i
\(780\) 0 0
\(781\) 5920.00 10253.7i 0.271235 0.469792i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 592.000 0.0269164
\(786\) 0 0
\(787\) 7426.00 12862.2i 0.336351 0.582577i −0.647392 0.762157i \(-0.724140\pi\)
0.983743 + 0.179580i \(0.0574738\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1256.00 + 2175.46i 0.0562445 + 0.0974183i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −19788.0 −0.879457 −0.439728 0.898131i \(-0.644925\pi\)
−0.439728 + 0.898131i \(0.644925\pi\)
\(798\) 0 0
\(799\) 12480.0 0.552579
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −10240.0 17736.2i −0.450015 0.779448i