Properties

Label 1764.4.k.g.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.g.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 0.762614 0.646854i \(-0.223916\pi\)
−0.941499 + 0.337016i \(0.890582\pi\)
\(6\) 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 \(-0.486414\pi\)
0.0426692 + 0.999089i \(0.486414\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −12.0000 + 20.7846i −0.171202 + 0.296530i −0.938840 0.344353i \(-0.888098\pi\)
0.767639 + 0.640883i \(0.221432\pi\)
\(18\) 0 0
\(19\) 22.0000 + 38.1051i 0.265639 + 0.460101i 0.967731 0.251986i \(-0.0810837\pi\)
−0.702092 + 0.712087i \(0.747750\pi\)
\(20\) 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 \(-0.654390\pi\)
0.999256 0.0385601i \(-0.0122771\pi\)
\(32\) 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 \(-0.634955\pi\)
−0.411385 + 0.911462i \(0.634955\pi\)
\(42\) 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 \(-0.867803\pi\)
0.108079 0.994142i \(-0.465530\pi\)
\(48\) 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 \(0.336103\pi\)
−0.999962 + 0.00870069i \(0.997230\pi\)
\(60\) 0 0
\(61\) 314.000 + 543.864i 0.659075 + 1.14155i 0.980855 + 0.194737i \(0.0623854\pi\)
−0.321780 + 0.946814i \(0.604281\pi\)
\(62\) 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 \(-0.526810\pi\)
0.905019 0.425371i \(-0.139856\pi\)
\(74\) 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 \(-0.568727\pi\)
−0.214239 + 0.976781i \(0.568727\pi\)
\(84\) 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 \(-0.987516\pi\)
0.465659 0.884964i \(-0.345817\pi\)
\(90\) 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 \(-0.340091\pi\)
0.481504 + 0.876444i \(0.340091\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −554.000 + 959.556i −0.545793 + 0.945341i 0.452764 + 0.891630i \(0.350438\pi\)
−0.998557 + 0.0537102i \(0.982895\pi\)
\(102\) 0 0
\(103\) 724.000 + 1254.00i 0.692600 + 1.19962i 0.970983 + 0.239149i \(0.0768684\pi\)
−0.278383 + 0.960470i \(0.589798\pi\)
\(104\) 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 0.531141 0.847284i \(-0.321764\pi\)
−0.999339 + 0.0363397i \(0.988430\pi\)
\(132\) 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 \(-0.550322\pi\)
−0.157434 + 0.987530i \(0.550322\pi\)
\(140\) 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 −0.884221 0.467069i \(-0.845310\pi\)
0.846604 + 0.532223i \(0.178643\pi\)
\(158\) 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 \(-0.348635\pi\)
0.457807 + 0.889052i \(0.348635\pi\)
\(168\) 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 \(0.0348087\pi\)
−0.402498 + 0.915421i \(0.631858\pi\)
\(174\) 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 \(-0.362713\pi\)
0.418052 + 0.908423i \(0.362713\pi\)
\(182\) 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 \(0.385285\pi\)
−0.986710 + 0.162489i \(0.948048\pi\)
\(200\) 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 \(-0.737959\pi\)
−0.679859 + 0.733342i \(0.737959\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1826.00 3162.72i 0.533903 0.924746i −0.465313 0.885146i \(-0.654058\pi\)
0.999216 0.0396002i \(-0.0126084\pi\)
\(228\) 0 0
\(229\) −2402.00 4160.39i −0.693138 1.20055i −0.970804 0.239873i \(-0.922894\pi\)
0.277666 0.960678i \(-0.410439\pi\)
\(230\) 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 −0.869743 0.493505i \(-0.835716\pi\)
0.862259 + 0.506467i \(0.169049\pi\)
\(242\) 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 \(-0.288709\pi\)
0.616106 + 0.787663i \(0.288709\pi\)
\(252\) 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 \(0.141201\pi\)
−0.0799181 + 0.996801i \(0.525466\pi\)
\(258\) 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 −0.539710 0.841851i \(-0.681466\pi\)
0.998919 + 0.0464767i \(0.0147993\pi\)
\(270\) 0 0
\(271\) −1376.00 2383.30i −0.308436 0.534226i 0.669585 0.742736i \(-0.266472\pi\)
−0.978020 + 0.208510i \(0.933139\pi\)
\(272\) 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 −0.996472 0.0839204i \(-0.973256\pi\)
0.570913 + 0.821010i \(0.306589\pi\)
\(284\) 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 \(-0.610750\pi\)
−0.340953 + 0.940080i \(0.610750\pi\)
\(294\) 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 \(-0.738362\pi\)
−0.680786 + 0.732482i \(0.738362\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 2096.00 3630.38i 0.382165 0.661929i −0.609207 0.793012i \(-0.708512\pi\)
0.991371 + 0.131083i \(0.0418453\pi\)
\(312\) 0 0
\(313\) −3420.00 5923.61i −0.617603 1.06972i −0.989922 0.141615i \(-0.954770\pi\)
0.372318 0.928105i \(-0.378563\pi\)
\(314\) 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 \(-0.371453\pi\)
0.392953 + 0.919558i \(0.371453\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −2280.00 + 3949.08i −0.343774 + 0.595434i −0.985130 0.171809i \(-0.945039\pi\)
0.641356 + 0.767243i \(0.278372\pi\)
\(354\) 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 −0.917750 0.397160i \(-0.869996\pi\)
0.802825 + 0.596215i \(0.203329\pi\)
\(368\) 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 0.539455 0.842014i \(-0.318630\pi\)
−0.998933 + 0.0461746i \(0.985297\pi\)
\(384\) 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 0.947542 0.319632i \(-0.103559\pi\)
−0.750580 + 0.660779i \(0.770226\pi\)
\(398\) 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 −0.519540 0.854446i \(-0.673897\pi\)
0.999742 + 0.0227114i \(0.00722990\pi\)
\(410\) 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 \(-0.423480\pi\)
0.238086 + 0.971244i \(0.423480\pi\)
\(420\) 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 \(-0.419589\pi\)
0.249940 + 0.968261i \(0.419589\pi\)
\(434\) 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 \(0.0845092\pi\)
−0.255249 + 0.966875i \(0.582158\pi\)
\(440\) 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 \(-0.864754\pi\)
−0.911085 + 0.412219i \(0.864754\pi\)
\(462\) 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 \(-0.896098\pi\)
0.195906 0.980623i \(-0.437235\pi\)
\(468\) 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 −0.990564 0.137049i \(-0.956238\pi\)
0.613970 + 0.789329i \(0.289572\pi\)
\(480\) 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 \(-0.436143\pi\)
0.199271 + 0.979944i \(0.436143\pi\)
\(504\) 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 \(-0.981492\pi\)
0.448829 0.893618i \(-0.351841\pi\)
\(510\) 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 \(0.440073\pi\)
−0.944301 + 0.329083i \(0.893260\pi\)
\(522\) 0 0
\(523\) 6646.00 + 11511.2i 0.555658 + 0.962428i 0.997852 + 0.0655088i \(0.0208670\pi\)
−0.442194 + 0.896920i \(0.645800\pi\)
\(524\) 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 −0.934454 0.356085i \(-0.884111\pi\)
0.775605 + 0.631218i \(0.217445\pi\)
\(564\) 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 −0.674685 0.738106i \(-0.735721\pi\)
0.976561 + 0.215242i \(0.0690540\pi\)
\(578\) 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 \(-0.499418\pi\)
0.00182817 + 0.999998i \(0.499418\pi\)
\(588\) 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 0.747782 0.663944i \(-0.231119\pi\)
−0.948883 + 0.315627i \(0.897785\pi\)
\(594\) 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 \(-0.512793\pi\)
−0.0401800 + 0.999192i \(0.512793\pi\)
\(602\) 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 0.997668 0.0682559i \(-0.0217434\pi\)
−0.557945 + 0.829878i \(0.688410\pi\)
\(608\) 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 −0.752472 0.658624i \(-0.771139\pi\)
0.946621 + 0.322348i \(0.104472\pi\)
\(620\) 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 \(-0.481130\pi\)
0.0592462 + 0.998243i \(0.481130\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4212.00 7295.40i 0.255936 0.443295i −0.709213 0.704994i \(-0.750950\pi\)
0.965149 + 0.261699i \(0.0842829\pi\)
\(648\) 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 −0.969727 0.244193i \(-0.921477\pi\)
0.696340 + 0.717712i \(0.254810\pi\)
\(662\) 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 0.747182 0.664620i \(-0.231406\pi\)
−0.949168 + 0.314768i \(0.898073\pi\)
\(678\) 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 \(-0.964292\pi\)
0.399910 0.916554i \(-0.369041\pi\)
\(692\) 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 0.957343 0.288954i \(-0.0933073\pi\)
−0.728913 + 0.684607i \(0.759974\pi\)
\(720\) 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 \(-0.649724\pi\)
−0.453218 + 0.891400i \(0.649724\pi\)
\(728\) 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 0.787381 0.616466i \(-0.211436\pi\)
−0.927566 + 0.373659i \(0.878103\pi\)
\(734\) 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 \(0.0507930\pi\)
−0.356041 + 0.934470i \(0.615874\pi\)
\(762\) 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 \(-0.891227\pi\)
−0.942180 + 0.335109i \(0.891227\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 17670.0 30605.3i 0.822181 1.42406i −0.0818742 0.996643i \(-0.526091\pi\)
0.904055 0.427416i \(-0.140576\pi\)
\(774\) 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 −0.983743 0.179580i \(-0.942526\pi\)
0.647392 + 0.762157i \(0.275860\pi\)
\(788\) 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 \(-0.355075\pi\)
0.439728 + 0.898131i \(0.355075\pi\)
\(798\) 0 0
\(799\) 12480.0 0.552579
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 10240.0 + 17736.2i