Properties

Label 1764.3.z.m.325.4
Level $1764$
Weight $3$
Character 1764.325
Analytic conductor $48.066$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1764.z (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.0655186332\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.339738624.1
Defining polynomial: \(x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 7^{2} \)
Twist minimal: no (minimal twist has level 588)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 325.4
Root \(0.662827 + 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 1764.325
Dual form 1764.3.z.m.901.4

$q$-expansion

\(f(q)\) \(=\) \(q+(4.65891 - 2.68982i) q^{5} +O(q^{10})\) \(q+(4.65891 - 2.68982i) q^{5} +(-4.29579 + 7.44053i) q^{11} +21.0158i q^{13} +(4.75120 + 2.74311i) q^{17} +(6.27088 - 3.62049i) q^{19} +(14.0278 + 24.2969i) q^{23} +(1.97027 - 3.41261i) q^{25} -40.3447 q^{29} +(-35.0828 - 20.2550i) q^{31} +(-33.3185 - 57.7093i) q^{37} +33.6357i q^{41} +0.932907 q^{43} +(-74.1789 + 42.8272i) q^{47} +(-22.2977 + 38.6207i) q^{53} +46.2197i q^{55} +(55.1615 + 31.8475i) q^{59} +(-27.7201 + 16.0042i) q^{61} +(56.5288 + 97.9108i) q^{65} +(23.8671 - 41.3390i) q^{67} -14.9676 q^{71} +(-121.501 - 70.1488i) q^{73} +(61.1537 + 105.921i) q^{79} +33.1852i q^{83} +29.5139 q^{85} +(31.2848 - 18.0623i) q^{89} +(19.4769 - 33.7351i) q^{95} -16.2175i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 48q^{17} + 96q^{19} - 8q^{23} - 36q^{25} - 80q^{29} - 48q^{31} - 64q^{37} - 112q^{43} - 264q^{47} - 72q^{53} + 168q^{59} + 144q^{61} + 120q^{65} + 32q^{67} - 224q^{71} - 336q^{73} + 216q^{79} - 96q^{85} + 96q^{89} - 136q^{95} + O(q^{100}) \)

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{6}\right)\)

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
\(5\) 4.65891 2.68982i 0.931781 0.537964i 0.0444067 0.999014i \(-0.485860\pi\)
0.887374 + 0.461049i \(0.152527\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.29579 + 7.44053i −0.390527 + 0.676412i −0.992519 0.122090i \(-0.961040\pi\)
0.601992 + 0.798502i \(0.294374\pi\)
\(12\) 0 0
\(13\) 21.0158i 1.61660i 0.588769 + 0.808301i \(0.299613\pi\)
−0.588769 + 0.808301i \(0.700387\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.75120 + 2.74311i 0.279483 + 0.161359i 0.633189 0.773997i \(-0.281746\pi\)
−0.353707 + 0.935356i \(0.615079\pi\)
\(18\) 0 0
\(19\) 6.27088 3.62049i 0.330046 0.190552i −0.325815 0.945433i \(-0.605639\pi\)
0.655862 + 0.754881i \(0.272305\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 14.0278 + 24.2969i 0.609905 + 1.05639i 0.991256 + 0.131956i \(0.0421258\pi\)
−0.381350 + 0.924431i \(0.624541\pi\)
\(24\) 0 0
\(25\) 1.97027 3.41261i 0.0788107 0.136504i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −40.3447 −1.39120 −0.695599 0.718431i \(-0.744861\pi\)
−0.695599 + 0.718431i \(0.744861\pi\)
\(30\) 0 0
\(31\) −35.0828 20.2550i −1.13170 0.653388i −0.187340 0.982295i \(-0.559986\pi\)
−0.944362 + 0.328907i \(0.893320\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −33.3185 57.7093i −0.900500 1.55971i −0.826846 0.562428i \(-0.809867\pi\)
−0.0736541 0.997284i \(-0.523466\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 33.6357i 0.820383i 0.911999 + 0.410191i \(0.134538\pi\)
−0.911999 + 0.410191i \(0.865462\pi\)
\(42\) 0 0
\(43\) 0.932907 0.0216955 0.0108478 0.999941i \(-0.496547\pi\)
0.0108478 + 0.999941i \(0.496547\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −74.1789 + 42.8272i −1.57827 + 0.911217i −0.583173 + 0.812348i \(0.698189\pi\)
−0.995100 + 0.0988687i \(0.968478\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −22.2977 + 38.6207i −0.420711 + 0.728693i −0.996009 0.0892508i \(-0.971553\pi\)
0.575298 + 0.817944i \(0.304886\pi\)
\(54\) 0 0
\(55\) 46.2197i 0.840358i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 55.1615 + 31.8475i 0.934940 + 0.539788i 0.888371 0.459127i \(-0.151838\pi\)
0.0465695 + 0.998915i \(0.485171\pi\)
\(60\) 0 0
\(61\) −27.7201 + 16.0042i −0.454427 + 0.262364i −0.709698 0.704506i \(-0.751169\pi\)
0.255271 + 0.966870i \(0.417835\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 56.5288 + 97.9108i 0.869674 + 1.50632i
\(66\) 0 0
\(67\) 23.8671 41.3390i 0.356225 0.617000i −0.631102 0.775700i \(-0.717397\pi\)
0.987327 + 0.158700i \(0.0507304\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −14.9676 −0.210811 −0.105405 0.994429i \(-0.533614\pi\)
−0.105405 + 0.994429i \(0.533614\pi\)
\(72\) 0 0
\(73\) −121.501 70.1488i −1.66440 0.960942i −0.970575 0.240798i \(-0.922591\pi\)
−0.693825 0.720144i \(-0.744076\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 61.1537 + 105.921i 0.774098 + 1.34078i 0.935300 + 0.353856i \(0.115130\pi\)
−0.161202 + 0.986921i \(0.551537\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 33.1852i 0.399822i 0.979814 + 0.199911i \(0.0640652\pi\)
−0.979814 + 0.199911i \(0.935935\pi\)
\(84\) 0 0
\(85\) 29.5139 0.347222
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 31.2848 18.0623i 0.351515 0.202947i −0.313838 0.949477i \(-0.601615\pi\)
0.665352 + 0.746530i \(0.268281\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 19.4769 33.7351i 0.205021 0.355106i
\(96\) 0 0
\(97\) 16.2175i 0.167191i −0.996500 0.0835956i \(-0.973360\pi\)
0.996500 0.0835956i \(-0.0266404\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −103.459 59.7322i −1.02435 0.591407i −0.108987 0.994043i \(-0.534761\pi\)
−0.915360 + 0.402636i \(0.868094\pi\)
\(102\) 0 0
\(103\) 7.73523 4.46594i 0.0750994 0.0433586i −0.461980 0.886890i \(-0.652861\pi\)
0.537079 + 0.843532i \(0.319528\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 86.9279 + 150.564i 0.812410 + 1.40714i 0.911173 + 0.412025i \(0.135178\pi\)
−0.0987623 + 0.995111i \(0.531488\pi\)
\(108\) 0 0
\(109\) 80.1573 138.837i 0.735388 1.27373i −0.219165 0.975688i \(-0.570333\pi\)
0.954553 0.298042i \(-0.0963334\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 81.4420 0.720725 0.360363 0.932812i \(-0.382653\pi\)
0.360363 + 0.932812i \(0.382653\pi\)
\(114\) 0 0
\(115\) 130.709 + 75.4646i 1.13660 + 0.656214i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 23.5923 + 40.8631i 0.194978 + 0.337711i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 113.292i 0.906339i
\(126\) 0 0
\(127\) 117.172 0.922613 0.461307 0.887241i \(-0.347381\pi\)
0.461307 + 0.887241i \(0.347381\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −211.440 + 122.075i −1.61404 + 0.931869i −0.625625 + 0.780124i \(0.715156\pi\)
−0.988420 + 0.151745i \(0.951511\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 122.706 212.533i 0.895663 1.55133i 0.0626820 0.998034i \(-0.480035\pi\)
0.832981 0.553301i \(-0.186632\pi\)
\(138\) 0 0
\(139\) 17.1371i 0.123288i 0.998098 + 0.0616441i \(0.0196344\pi\)
−0.998098 + 0.0616441i \(0.980366\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −156.369 90.2797i −1.09349 0.631326i
\(144\) 0 0
\(145\) −187.962 + 108.520i −1.29629 + 0.748414i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 67.1209 + 116.257i 0.450476 + 0.780247i 0.998416 0.0562707i \(-0.0179210\pi\)
−0.547940 + 0.836518i \(0.684588\pi\)
\(150\) 0 0
\(151\) −99.5047 + 172.347i −0.658971 + 1.14137i 0.321911 + 0.946770i \(0.395675\pi\)
−0.980882 + 0.194602i \(0.937659\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −217.930 −1.40600
\(156\) 0 0
\(157\) 66.7004 + 38.5095i 0.424843 + 0.245283i 0.697147 0.716928i \(-0.254452\pi\)
−0.272304 + 0.962211i \(0.587786\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 115.190 + 199.514i 0.706684 + 1.22401i 0.966080 + 0.258242i \(0.0831432\pi\)
−0.259396 + 0.965771i \(0.583523\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 229.231i 1.37264i 0.727298 + 0.686321i \(0.240776\pi\)
−0.727298 + 0.686321i \(0.759224\pi\)
\(168\) 0 0
\(169\) −272.665 −1.61340
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −97.4044 + 56.2365i −0.563031 + 0.325066i −0.754361 0.656459i \(-0.772053\pi\)
0.191330 + 0.981526i \(0.438720\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 52.7470 91.3605i 0.294676 0.510394i −0.680234 0.732995i \(-0.738122\pi\)
0.974909 + 0.222602i \(0.0714550\pi\)
\(180\) 0 0
\(181\) 15.2683i 0.0843553i −0.999110 0.0421776i \(-0.986570\pi\)
0.999110 0.0421776i \(-0.0134295\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −310.456 179.242i −1.67814 0.968874i
\(186\) 0 0
\(187\) −40.8204 + 23.5677i −0.218291 + 0.126030i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −153.093 265.165i −0.801534 1.38830i −0.918606 0.395174i \(-0.870684\pi\)
0.117073 0.993123i \(-0.462649\pi\)
\(192\) 0 0
\(193\) 0.920499 1.59435i 0.00476943 0.00826089i −0.863631 0.504125i \(-0.831815\pi\)
0.868400 + 0.495864i \(0.165149\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 255.334 1.29611 0.648056 0.761593i \(-0.275582\pi\)
0.648056 + 0.761593i \(0.275582\pi\)
\(198\) 0 0
\(199\) −31.7384 18.3242i −0.159489 0.0920812i 0.418131 0.908387i \(-0.362685\pi\)
−0.577620 + 0.816305i \(0.696019\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 90.4740 + 156.706i 0.441337 + 0.764417i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 62.2116i 0.297663i
\(210\) 0 0
\(211\) −126.571 −0.599862 −0.299931 0.953961i \(-0.596964\pi\)
−0.299931 + 0.953961i \(0.596964\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4.34633 2.50935i 0.0202155 0.0116714i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −57.6487 + 99.8505i −0.260854 + 0.451812i
\(222\) 0 0
\(223\) 212.193i 0.951536i 0.879571 + 0.475768i \(0.157830\pi\)
−0.879571 + 0.475768i \(0.842170\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 91.3560 + 52.7444i 0.402449 + 0.232354i 0.687540 0.726146i \(-0.258690\pi\)
−0.285091 + 0.958500i \(0.592024\pi\)
\(228\) 0 0
\(229\) −6.05426 + 3.49543i −0.0264378 + 0.0152639i −0.513161 0.858293i \(-0.671526\pi\)
0.486723 + 0.873556i \(0.338192\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 168.023 + 291.024i 0.721128 + 1.24903i 0.960548 + 0.278115i \(0.0897096\pi\)
−0.239419 + 0.970916i \(0.576957\pi\)
\(234\) 0 0
\(235\) −230.395 + 399.056i −0.980404 + 1.69811i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 232.382 0.972309 0.486155 0.873873i \(-0.338399\pi\)
0.486155 + 0.873873i \(0.338399\pi\)
\(240\) 0 0
\(241\) −55.1958 31.8673i −0.229028 0.132229i 0.381095 0.924536i \(-0.375547\pi\)
−0.610124 + 0.792306i \(0.708880\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 76.0876 + 131.788i 0.308047 + 0.533553i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 415.450i 1.65518i 0.561334 + 0.827589i \(0.310288\pi\)
−0.561334 + 0.827589i \(0.689712\pi\)
\(252\) 0 0
\(253\) −241.043 −0.952737
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 230.014 132.799i 0.894998 0.516727i 0.0194240 0.999811i \(-0.493817\pi\)
0.875574 + 0.483084i \(0.160483\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 78.1866 135.423i 0.297288 0.514917i −0.678227 0.734853i \(-0.737251\pi\)
0.975514 + 0.219935i \(0.0705846\pi\)
\(264\) 0 0
\(265\) 239.907i 0.905310i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −288.266 166.431i −1.07162 0.618701i −0.142998 0.989723i \(-0.545674\pi\)
−0.928624 + 0.371022i \(0.879008\pi\)
\(270\) 0 0
\(271\) 181.076 104.544i 0.668178 0.385773i −0.127208 0.991876i \(-0.540602\pi\)
0.795386 + 0.606103i \(0.207268\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 16.9277 + 29.3197i 0.0615554 + 0.106617i
\(276\) 0 0
\(277\) 131.616 227.965i 0.475146 0.822978i −0.524448 0.851442i \(-0.675728\pi\)
0.999595 + 0.0284646i \(0.00906180\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −391.519 −1.39331 −0.696653 0.717408i \(-0.745328\pi\)
−0.696653 + 0.717408i \(0.745328\pi\)
\(282\) 0 0
\(283\) 94.7376 + 54.6968i 0.334762 + 0.193275i 0.657953 0.753059i \(-0.271422\pi\)
−0.323191 + 0.946334i \(0.604756\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −129.451 224.215i −0.447926 0.775831i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 35.3685i 0.120712i 0.998177 + 0.0603558i \(0.0192235\pi\)
−0.998177 + 0.0603558i \(0.980776\pi\)
\(294\) 0 0
\(295\) 342.656 1.16155
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −510.619 + 294.806i −1.70776 + 0.985974i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −86.0968 + 149.124i −0.282284 + 0.488931i
\(306\) 0 0
\(307\) 125.621i 0.409189i −0.978847 0.204594i \(-0.934412\pi\)
0.978847 0.204594i \(-0.0655875\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 273.435 + 157.868i 0.879211 + 0.507613i 0.870398 0.492348i \(-0.163861\pi\)
0.00881288 + 0.999961i \(0.497195\pi\)
\(312\) 0 0
\(313\) 44.6909 25.8023i 0.142782 0.0824355i −0.426907 0.904296i \(-0.640397\pi\)
0.569689 + 0.821860i \(0.307064\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2.49696 4.32486i −0.00787683 0.0136431i 0.862060 0.506806i \(-0.169174\pi\)
−0.869937 + 0.493163i \(0.835841\pi\)
\(318\) 0 0
\(319\) 173.313 300.186i 0.543300 0.941023i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 39.7256 0.122990
\(324\) 0 0
\(325\) 71.7187 + 41.4068i 0.220673 + 0.127406i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −227.390 393.851i −0.686980 1.18988i −0.972810 0.231603i \(-0.925603\pi\)
0.285831 0.958280i \(-0.407731\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 256.792i 0.766545i
\(336\) 0 0
\(337\) −183.824 −0.545471 −0.272736 0.962089i \(-0.587928\pi\)
−0.272736 + 0.962089i \(0.587928\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 301.417 174.023i 0.883920 0.510331i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 72.3882 125.380i 0.208611 0.361326i −0.742666 0.669662i \(-0.766439\pi\)
0.951277 + 0.308337i \(0.0997723\pi\)
\(348\) 0 0
\(349\) 187.069i 0.536015i −0.963417 0.268007i \(-0.913635\pi\)
0.963417 0.268007i \(-0.0863651\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 192.778 + 111.300i 0.546113 + 0.315299i 0.747553 0.664202i \(-0.231229\pi\)
−0.201440 + 0.979501i \(0.564562\pi\)
\(354\) 0 0
\(355\) −69.7324 + 40.2600i −0.196429 + 0.113409i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 31.4928 + 54.5472i 0.0877237 + 0.151942i 0.906549 0.422101i \(-0.138707\pi\)
−0.818825 + 0.574043i \(0.805374\pi\)
\(360\) 0 0
\(361\) −154.284 + 267.228i −0.427380 + 0.740243i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −754.750 −2.06781
\(366\) 0 0
\(367\) −473.803 273.550i −1.29102 0.745368i −0.312181 0.950023i \(-0.601060\pi\)
−0.978834 + 0.204654i \(0.934393\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 16.2208 + 28.0953i 0.0434875 + 0.0753226i 0.886950 0.461866i \(-0.152820\pi\)
−0.843462 + 0.537188i \(0.819486\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 847.878i 2.24901i
\(378\) 0 0
\(379\) 508.859 1.34263 0.671317 0.741170i \(-0.265729\pi\)
0.671317 + 0.741170i \(0.265729\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 22.7625 13.1419i 0.0594321 0.0343131i −0.469990 0.882672i \(-0.655742\pi\)
0.529422 + 0.848359i \(0.322409\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −180.064 + 311.880i −0.462890 + 0.801749i −0.999104 0.0423337i \(-0.986521\pi\)
0.536214 + 0.844082i \(0.319854\pi\)
\(390\) 0 0
\(391\) 153.919i 0.393656i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 569.819 + 328.985i 1.44258 + 0.832874i
\(396\) 0 0
\(397\) −494.494 + 285.496i −1.24558 + 0.719134i −0.970224 0.242210i \(-0.922128\pi\)
−0.275352 + 0.961343i \(0.588794\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 34.2007 + 59.2373i 0.0852885 + 0.147724i 0.905514 0.424316i \(-0.139485\pi\)
−0.820226 + 0.572040i \(0.806152\pi\)
\(402\) 0 0
\(403\) 425.677 737.293i 1.05627 1.82951i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 572.518 1.40668
\(408\) 0 0
\(409\) −13.3064 7.68246i −0.0325340 0.0187835i 0.483645 0.875264i \(-0.339313\pi\)
−0.516179 + 0.856481i \(0.672646\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 89.2622 + 154.607i 0.215090 + 0.372546i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 366.079i 0.873696i −0.899535 0.436848i \(-0.856095\pi\)
0.899535 0.436848i \(-0.143905\pi\)
\(420\) 0 0
\(421\) 607.135 1.44213 0.721063 0.692870i \(-0.243654\pi\)
0.721063 + 0.692870i \(0.243654\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 18.7223 10.8093i 0.0440525 0.0254337i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −293.177 + 507.797i −0.680224 + 1.17818i 0.294688 + 0.955594i \(0.404784\pi\)
−0.974912 + 0.222590i \(0.928549\pi\)
\(432\) 0 0
\(433\) 518.769i 1.19808i −0.800719 0.599040i \(-0.795549\pi\)
0.800719 0.599040i \(-0.204451\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 175.933 + 101.575i 0.402594 + 0.232438i
\(438\) 0 0
\(439\) 191.845 110.762i 0.437005 0.252305i −0.265321 0.964160i \(-0.585478\pi\)
0.702326 + 0.711855i \(0.252145\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −113.208 196.081i −0.255548 0.442622i 0.709496 0.704709i \(-0.248922\pi\)
−0.965044 + 0.262087i \(0.915589\pi\)
\(444\) 0 0
\(445\) 97.1686 168.301i 0.218356 0.378205i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 378.422 0.842810 0.421405 0.906873i \(-0.361537\pi\)
0.421405 + 0.906873i \(0.361537\pi\)
\(450\) 0 0
\(451\) −250.268 144.492i −0.554917 0.320382i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −219.077 379.452i −0.479380 0.830311i 0.520340 0.853959i \(-0.325805\pi\)
−0.999720 + 0.0236483i \(0.992472\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 46.3981i 0.100647i 0.998733 + 0.0503234i \(0.0160252\pi\)
−0.998733 + 0.0503234i \(0.983975\pi\)
\(462\) 0 0
\(463\) 367.455 0.793639 0.396820 0.917897i \(-0.370114\pi\)
0.396820 + 0.917897i \(0.370114\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 458.530 264.733i 0.981863 0.566879i 0.0790311 0.996872i \(-0.474817\pi\)
0.902832 + 0.429993i \(0.141484\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −4.00758 + 6.94133i −0.00847268 + 0.0146751i
\(474\) 0 0
\(475\) 28.5334i 0.0600702i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 250.940 + 144.880i 0.523883 + 0.302464i 0.738522 0.674229i \(-0.235524\pi\)
−0.214639 + 0.976694i \(0.568857\pi\)
\(480\) 0 0
\(481\) 1212.81 700.216i 2.52143 1.45575i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −43.6223 75.5560i −0.0899428 0.155786i
\(486\) 0 0
\(487\) −219.818 + 380.736i −0.451372 + 0.781799i −0.998472 0.0552685i \(-0.982399\pi\)
0.547100 + 0.837067i \(0.315732\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 320.561 0.652874 0.326437 0.945219i \(-0.394152\pi\)
0.326437 + 0.945219i \(0.394152\pi\)
\(492\) 0 0
\(493\) −191.686 110.670i −0.388815 0.224483i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −385.986 668.548i −0.773520 1.33978i −0.935623 0.353002i \(-0.885161\pi\)
0.162103 0.986774i \(-0.448172\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 101.632i 0.202052i 0.994884 + 0.101026i \(0.0322126\pi\)
−0.994884 + 0.101026i \(0.967787\pi\)
\(504\) 0 0
\(505\) −642.675 −1.27262
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 524.298 302.704i 1.03006 0.594703i 0.113055 0.993589i \(-0.463937\pi\)
0.917000 + 0.398886i \(0.130603\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 24.0252 41.6128i 0.0466508 0.0808015i
\(516\) 0 0
\(517\) 735.907i 1.42342i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 331.942 + 191.647i 0.637124 + 0.367844i 0.783506 0.621384i \(-0.213429\pi\)
−0.146382 + 0.989228i \(0.546763\pi\)
\(522\) 0 0
\(523\) −388.061 + 224.047i −0.741991 + 0.428389i −0.822793 0.568341i \(-0.807585\pi\)
0.0808018 + 0.996730i \(0.474252\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −111.124 192.472i −0.210861 0.365221i
\(528\) 0 0
\(529\) −129.059 + 223.538i −0.243969 + 0.422566i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −706.882 −1.32623
\(534\) 0 0
\(535\) 809.978 + 467.641i 1.51398 + 0.874095i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 135.629 + 234.916i 0.250700 + 0.434226i 0.963719 0.266920i \(-0.0860058\pi\)
−0.713019 + 0.701145i \(0.752672\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 862.435i 1.58245i
\(546\) 0 0
\(547\) −590.544 −1.07961 −0.539803 0.841791i \(-0.681501\pi\)
−0.539803 + 0.841791i \(0.681501\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −252.997 + 146.068i −0.459159 + 0.265096i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 94.1473 163.068i 0.169026 0.292761i −0.769052 0.639186i \(-0.779271\pi\)
0.938078 + 0.346425i \(0.112605\pi\)
\(558\) 0 0
\(559\) 19.6058i 0.0350730i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −176.805 102.078i −0.314041 0.181312i 0.334692 0.942327i \(-0.391368\pi\)
−0.648733 + 0.761016i \(0.724701\pi\)
\(564\) 0 0
\(565\) 379.430 219.064i 0.671558 0.387724i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −330.456 572.367i −0.580767 1.00592i −0.995389 0.0959241i \(-0.969419\pi\)
0.414622 0.909994i \(-0.363914\pi\)
\(570\) 0 0
\(571\) 266.989 462.438i 0.467581 0.809874i −0.531733 0.846912i \(-0.678459\pi\)
0.999314 + 0.0370381i \(0.0117923\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 110.554 0.192268
\(576\) 0 0
\(577\) −46.2750 26.7169i −0.0801993 0.0463031i 0.459364 0.888248i \(-0.348077\pi\)
−0.539563 + 0.841945i \(0.681411\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −191.573 331.813i −0.328598 0.569148i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 413.063i 0.703684i 0.936059 + 0.351842i \(0.114445\pi\)
−0.936059 + 0.351842i \(0.885555\pi\)
\(588\) 0 0
\(589\) −293.333 −0.498018
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −106.047 + 61.2261i −0.178831 + 0.103248i −0.586743 0.809773i \(-0.699590\pi\)
0.407912 + 0.913021i \(0.366257\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −322.568 + 558.704i −0.538511 + 0.932728i 0.460474 + 0.887673i \(0.347680\pi\)
−0.998985 + 0.0450546i \(0.985654\pi\)
\(600\) 0 0
\(601\) 683.488i 1.13725i 0.822596 + 0.568626i \(0.192525\pi\)
−0.822596 + 0.568626i \(0.807475\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 219.829 + 126.918i 0.363353 + 0.209782i
\(606\) 0 0
\(607\) −222.788 + 128.627i −0.367031 + 0.211905i −0.672161 0.740405i \(-0.734634\pi\)
0.305130 + 0.952311i \(0.401300\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −900.049 1558.93i −1.47307 2.55144i
\(612\) 0 0
\(613\) −442.856 + 767.050i −0.722441 + 1.25130i 0.237578 + 0.971369i \(0.423647\pi\)
−0.960019 + 0.279936i \(0.909687\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −46.0724 −0.0746717 −0.0373358 0.999303i \(-0.511887\pi\)
−0.0373358 + 0.999303i \(0.511887\pi\)
\(618\) 0 0
\(619\) 949.446 + 548.163i 1.53384 + 0.885562i 0.999180 + 0.0404930i \(0.0128928\pi\)
0.534658 + 0.845069i \(0.320440\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 353.993 + 613.134i 0.566388 + 0.981014i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 365.585i 0.581217i
\(630\) 0 0
\(631\) 606.319 0.960886 0.480443 0.877026i \(-0.340476\pi\)
0.480443 + 0.877026i \(0.340476\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 545.893 315.171i 0.859674 0.496333i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 469.592 813.357i 0.732593 1.26889i −0.223178 0.974778i \(-0.571643\pi\)
0.955771 0.294111i \(-0.0950234\pi\)
\(642\) 0 0
\(643\) 992.960i 1.54426i 0.635464 + 0.772131i \(0.280809\pi\)
−0.635464 + 0.772131i \(0.719191\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −229.656 132.592i −0.354955 0.204933i 0.311910 0.950112i \(-0.399031\pi\)
−0.666865 + 0.745178i \(0.732364\pi\)
\(648\) 0 0
\(649\) −473.925 + 273.620i −0.730238 + 0.421603i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −567.071 982.196i −0.868409 1.50413i −0.863622 0.504139i \(-0.831810\pi\)
−0.00478648 0.999989i \(-0.501524\pi\)
\(654\) 0 0
\(655\) −656.719 + 1137.47i −1.00262 + 1.73660i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 924.147 1.40235 0.701174 0.712990i \(-0.252660\pi\)
0.701174 + 0.712990i \(0.252660\pi\)
\(660\) 0 0
\(661\) 202.968 + 117.184i 0.307062 + 0.177282i 0.645611 0.763666i \(-0.276603\pi\)
−0.338549 + 0.940949i \(0.609936\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −565.948 980.251i −0.848498 1.46964i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 275.003i 0.409840i
\(672\) 0 0
\(673\) −465.127 −0.691125 −0.345563 0.938396i \(-0.612312\pi\)
−0.345563 + 0.938396i \(0.612312\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1122.31 647.964i 1.65776 0.957110i 0.684020 0.729463i \(-0.260230\pi\)
0.973744 0.227647i \(-0.0731032\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 8.13060 14.0826i 0.0119042 0.0206188i −0.860012 0.510274i \(-0.829544\pi\)
0.871916 + 0.489655i \(0.162877\pi\)
\(684\) 0 0
\(685\) 1320.23i 1.92734i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −811.647 468.604i −1.17801 0.680123i
\(690\) 0 0
\(691\) 59.3542 34.2682i 0.0858962 0.0495922i −0.456437 0.889756i \(-0.650875\pi\)
0.542333 + 0.840164i \(0.317541\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 46.0956 + 79.8400i 0.0663246 + 0.114878i
\(696\) 0 0
\(697\) −92.2664 + 159.810i −0.132376 + 0.229283i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −923.360 −1.31720 −0.658602 0.752491i \(-0.728852\pi\)
−0.658602 + 0.752491i \(0.728852\pi\)
\(702\) 0 0
\(703\) −417.872 241.259i −0.594413 0.343185i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 456.431 + 790.562i 0.643768 + 1.11504i 0.984585 + 0.174909i \(0.0559631\pi\)
−0.340817 + 0.940130i \(0.610704\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1136.54i 1.59402i
\(714\) 0 0
\(715\) −971.345 −1.35852
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 472.001 272.510i 0.656469 0.379013i −0.134461 0.990919i \(-0.542930\pi\)
0.790930 + 0.611906i \(0.209597\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −79.4899 + 137.681i −0.109641 + 0.189904i
\(726\) 0 0
\(727\) 750.292i 1.03204i −0.856577 0.516019i \(-0.827413\pi\)
0.856577 0.516019i \(-0.172587\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 4.43243 + 2.55907i 0.00606352 + 0.00350078i
\(732\) 0 0
\(733\) 802.458 463.299i 1.09476 0.632059i 0.159919 0.987130i \(-0.448877\pi\)
0.934839 + 0.355071i \(0.115543\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 205.056 + 355.167i 0.278231 + 0.481910i
\(738\) 0 0
\(739\) −244.073 + 422.748i −0.330275 + 0.572054i −0.982566 0.185915i \(-0.940475\pi\)
0.652290 + 0.757969i \(0.273808\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1091.72 1.46935 0.734674 0.678421i \(-0.237335\pi\)
0.734674 + 0.678421i \(0.237335\pi\)
\(744\) 0 0
\(745\) 625.420 + 361.086i 0.839490 + 0.484680i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 240.222 + 416.076i 0.319869 + 0.554029i 0.980460 0.196716i \(-0.0630278\pi\)
−0.660591 + 0.750746i \(0.729694\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1070.60i 1.41801i
\(756\) 0 0
\(757\) 941.400 1.24359 0.621796 0.783179i \(-0.286403\pi\)
0.621796 + 0.783179i \(0.286403\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1032.60 + 596.172i −1.35690 + 0.783407i −0.989205 0.146540i \(-0.953186\pi\)
−0.367695 + 0.929946i \(0.619853\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −669.301 + 1159.26i −0.872622 + 1.51143i
\(768\) 0 0
\(769\) 908.294i 1.18114i 0.806988 + 0.590568i \(0.201096\pi\)
−0.806988 + 0.590568i \(0.798904\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1030.03 594.690i −1.33251 0.769328i −0.346830 0.937928i \(-0.612742\pi\)
−0.985685 + 0.168600i \(0.946075\pi\)
\(774\) 0 0
\(775\) −138.245 + 79.8158i −0.178381 + 0.102988i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 121.778 + 210.925i 0.156326 + 0.270764i
\(780\) 0 0
\(781\) 64.2976 111.367i 0.0823272 0.142595i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 414.334 0.527814
\(786\) 0 0
\(787\) 915.492 + 528.559i 1.16327 + 0.671613i 0.952085 0.305834i \(-0.0989352\pi\)
0.211183 + 0.977447i \(0.432269\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −336.341 582.560i −0.424138 0.734628i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1037.94i 1.30231i 0.758945 + 0.651154i \(0.225715\pi\)
−0.758945 + 0.651154i \(0.774285\pi\)
\(798\) 0 0
\(799\) −469.919 −0.588133
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1043.89 602.689i 1.29999 0.750547i
\(804\) 0 0