Properties

Label 1764.3.z.l.901.1
Level $1764$
Weight $3$
Character 1764.901
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 901.1
Root \(-0.662827 + 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 1764.901
Dual form 1764.3.z.l.325.1

$q$-expansion

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

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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 \(-0.514140\pi\)
−0.887374 + 0.461049i \(0.847473\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.29579 7.44053i −0.390527 0.676412i 0.601992 0.798502i \(-0.294374\pi\)
−0.992519 + 0.122090i \(0.961040\pi\)
\(12\) 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 \(-0.718254\pi\)
0.353707 + 0.935356i \(0.384921\pi\)
\(18\) 0 0
\(19\) −6.27088 3.62049i −0.330046 0.190552i 0.325815 0.945433i \(-0.394361\pi\)
−0.655862 + 0.754881i \(0.727695\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 14.0278 24.2969i 0.609905 1.05639i −0.381350 0.924431i \(-0.624541\pi\)
0.991256 0.131956i \(-0.0421258\pi\)
\(24\) 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 \(-0.440014\pi\)
0.944362 + 0.328907i \(0.106680\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −33.3185 + 57.7093i −0.900500 + 1.55971i −0.0736541 + 0.997284i \(0.523466\pi\)
−0.826846 + 0.562428i \(0.809867\pi\)
\(38\) 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 0.995100 + 0.0988687i \(0.0315224\pi\)
0.583173 + 0.812348i \(0.301811\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −22.2977 38.6207i −0.420711 0.728693i 0.575298 0.817944i \(-0.304886\pi\)
−0.996009 + 0.0892508i \(0.971553\pi\)
\(54\) 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 \(-0.848162\pi\)
−0.0465695 + 0.998915i \(0.514829\pi\)
\(60\) 0 0
\(61\) 27.7201 + 16.0042i 0.454427 + 0.262364i 0.709698 0.704506i \(-0.248831\pi\)
−0.255271 + 0.966870i \(0.582165\pi\)
\(62\) 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 0.987327 0.158700i \(-0.0507304\pi\)
−0.631102 + 0.775700i \(0.717397\pi\)
\(68\) 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 0.693825 0.720144i \(-0.255924\pi\)
0.970575 0.240798i \(-0.0774091\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 61.1537 105.921i 0.774098 1.34078i −0.161202 0.986921i \(-0.551537\pi\)
0.935300 0.353856i \(-0.115130\pi\)
\(80\) 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 \(-0.398385\pi\)
−0.665352 + 0.746530i \(0.731719\pi\)
\(90\) 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 \(-0.465239\pi\)
0.915360 + 0.402636i \(0.131906\pi\)
\(102\) 0 0
\(103\) −7.73523 4.46594i −0.0750994 0.0433586i 0.461980 0.886890i \(-0.347139\pi\)
−0.537079 + 0.843532i \(0.680472\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 86.9279 150.564i 0.812410 1.40714i −0.0987623 0.995111i \(-0.531488\pi\)
0.911173 0.412025i \(-0.135178\pi\)
\(108\) 0 0
\(109\) 80.1573 + 138.837i 0.735388 + 1.27373i 0.954553 + 0.298042i \(0.0963334\pi\)
−0.219165 + 0.975688i \(0.570333\pi\)
\(110\) 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 0.988420 + 0.151745i \(0.0484893\pi\)
0.625625 + 0.780124i \(0.284844\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 122.706 + 212.533i 0.895663 + 1.55133i 0.832981 + 0.553301i \(0.186632\pi\)
0.0626820 + 0.998034i \(0.480035\pi\)
\(138\) 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 −0.547940 0.836518i \(-0.684588\pi\)
0.998416 + 0.0562707i \(0.0179210\pi\)
\(150\) 0 0
\(151\) −99.5047 172.347i −0.658971 1.14137i −0.980882 0.194602i \(-0.937659\pi\)
0.321911 0.946770i \(-0.395675\pi\)
\(152\) 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 \(-0.745548\pi\)
0.272304 + 0.962211i \(0.412214\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 115.190 199.514i 0.706684 1.22401i −0.259396 0.965771i \(-0.583523\pi\)
0.966080 0.258242i \(-0.0831432\pi\)
\(164\) 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 \(-0.227947\pi\)
−0.191330 + 0.981526i \(0.561280\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 52.7470 + 91.3605i 0.294676 + 0.510394i 0.974909 0.222602i \(-0.0714550\pi\)
−0.680234 + 0.732995i \(0.738122\pi\)
\(180\) 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 0.117073 + 0.993123i \(0.462649\pi\)
−0.918606 + 0.395174i \(0.870684\pi\)
\(192\) 0 0
\(193\) 0.920499 + 1.59435i 0.00476943 + 0.00826089i 0.868400 0.495864i \(-0.165149\pi\)
−0.863631 + 0.504125i \(0.831815\pi\)
\(194\) 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 \(-0.637315\pi\)
0.577620 + 0.816305i \(0.303981\pi\)
\(200\) 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 \(-0.741310\pi\)
0.285091 + 0.958500i \(0.407976\pi\)
\(228\) 0 0
\(229\) 6.05426 + 3.49543i 0.0264378 + 0.0152639i 0.513161 0.858293i \(-0.328474\pi\)
−0.486723 + 0.873556i \(0.661808\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 168.023 291.024i 0.721128 1.24903i −0.239419 0.970916i \(-0.576957\pi\)
0.960548 0.278115i \(-0.0897096\pi\)
\(234\) 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 \(-0.624453\pi\)
0.610124 + 0.792306i \(0.291120\pi\)
\(242\) 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 \(-0.506183\pi\)
−0.875574 + 0.483084i \(0.839517\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 78.1866 + 135.423i 0.297288 + 0.514917i 0.975514 0.219935i \(-0.0705846\pi\)
−0.678227 + 0.734853i \(0.737251\pi\)
\(264\) 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 \(-0.454326\pi\)
0.928624 + 0.371022i \(0.120992\pi\)
\(270\) 0 0
\(271\) −181.076 104.544i −0.668178 0.385773i 0.127208 0.991876i \(-0.459398\pi\)
−0.795386 + 0.606103i \(0.792732\pi\)
\(272\) 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 0.999595 0.0284646i \(-0.00906180\pi\)
−0.524448 + 0.851442i \(0.675728\pi\)
\(278\) 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 \(-0.728578\pi\)
0.323191 + 0.946334i \(0.395244\pi\)
\(284\) 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 \(-0.836139\pi\)
−0.00881288 + 0.999961i \(0.502805\pi\)
\(312\) 0 0
\(313\) −44.6909 25.8023i −0.142782 0.0824355i 0.426907 0.904296i \(-0.359603\pi\)
−0.569689 + 0.821860i \(0.692936\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2.49696 + 4.32486i −0.00787683 + 0.0136431i −0.869937 0.493163i \(-0.835841\pi\)
0.862060 + 0.506806i \(0.169174\pi\)
\(318\) 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 0.285831 + 0.958280i \(0.407731\pi\)
−0.972810 + 0.231603i \(0.925603\pi\)
\(332\) 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 0.951277 0.308337i \(-0.0997723\pi\)
−0.742666 + 0.669662i \(0.766439\pi\)
\(348\) 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 \(-0.768771\pi\)
0.201440 + 0.979501i \(0.435438\pi\)
\(354\) 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 −0.818825 0.574043i \(-0.805374\pi\)
0.906549 + 0.422101i \(0.138707\pi\)
\(360\) 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 \(-0.398940\pi\)
0.978834 + 0.204654i \(0.0656071\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 16.2208 28.0953i 0.0434875 0.0753226i −0.843462 0.537188i \(-0.819486\pi\)
0.886950 + 0.461866i \(0.152820\pi\)
\(374\) 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 \(-0.344258\pi\)
−0.529422 + 0.848359i \(0.677591\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −180.064 311.880i −0.462890 0.801749i 0.536214 0.844082i \(-0.319854\pi\)
−0.999104 + 0.0423337i \(0.986521\pi\)
\(390\) 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 \(-0.0778722\pi\)
0.275352 + 0.961343i \(0.411206\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 34.2007 59.2373i 0.0852885 0.147724i −0.820226 0.572040i \(-0.806152\pi\)
0.905514 + 0.424316i \(0.139485\pi\)
\(402\) 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 \(-0.660687\pi\)
0.516179 + 0.856481i \(0.327354\pi\)
\(410\) 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 −0.974912 0.222590i \(-0.928549\pi\)
0.294688 0.955594i \(-0.404784\pi\)
\(432\) 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 \(-0.414522\pi\)
−0.702326 + 0.711855i \(0.747855\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −113.208 + 196.081i −0.255548 + 0.442622i −0.965044 0.262087i \(-0.915589\pi\)
0.709496 + 0.704709i \(0.248922\pi\)
\(444\) 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 −0.999720 0.0236483i \(-0.992472\pi\)
0.520340 + 0.853959i \(0.325805\pi\)
\(458\) 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 \(-0.525183\pi\)
−0.902832 + 0.429993i \(0.858516\pi\)
\(468\) 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 \(-0.764476\pi\)
0.214639 + 0.976694i \(0.431143\pi\)
\(480\) 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 0.547100 0.837067i \(-0.315732\pi\)
−0.998472 + 0.0552685i \(0.982399\pi\)
\(488\) 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 0.162103 + 0.986774i \(0.448172\pi\)
−0.935623 + 0.353002i \(0.885161\pi\)
\(500\) 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 \(-0.536063\pi\)
−0.917000 + 0.398886i \(0.869397\pi\)
\(510\) 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 \(-0.786571\pi\)
0.146382 + 0.989228i \(0.453237\pi\)
\(522\) 0 0
\(523\) 388.061 + 224.047i 0.741991 + 0.428389i 0.822793 0.568341i \(-0.192415\pi\)
−0.0808018 + 0.996730i \(0.525748\pi\)
\(524\) 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 −0.713019 0.701145i \(-0.752672\pi\)
0.963719 + 0.266920i \(0.0860058\pi\)
\(542\) 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 0.938078 0.346425i \(-0.112605\pi\)
−0.769052 + 0.639186i \(0.779271\pi\)
\(558\) 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 \(-0.608632\pi\)
0.648733 + 0.761016i \(0.275299\pi\)
\(564\) 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 0.414622 + 0.909994i \(0.363914\pi\)
−0.995389 + 0.0959241i \(0.969419\pi\)
\(570\) 0 0
\(571\) 266.989 + 462.438i 0.467581 + 0.809874i 0.999314 0.0370381i \(-0.0117923\pi\)
−0.531733 + 0.846912i \(0.678459\pi\)
\(572\) 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 \(-0.651923\pi\)
0.539563 + 0.841945i \(0.318589\pi\)
\(578\) 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 \(-0.300410\pi\)
−0.407912 + 0.913021i \(0.633743\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −322.568 558.704i −0.538511 0.932728i −0.998985 0.0450546i \(-0.985654\pi\)
0.460474 0.887673i \(-0.347680\pi\)
\(600\) 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 \(-0.265366\pi\)
−0.305130 + 0.952311i \(0.598700\pi\)
\(608\) 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 −0.960019 0.279936i \(-0.909687\pi\)
0.237578 0.971369i \(-0.423647\pi\)
\(614\) 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 −0.534658 + 0.845069i \(0.679560\pi\)
−0.999180 + 0.0404930i \(0.987107\pi\)
\(620\) 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 0.955771 + 0.294111i \(0.0950234\pi\)
−0.223178 + 0.974778i \(0.571643\pi\)
\(642\) 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 \(-0.600969\pi\)
0.666865 + 0.745178i \(0.267636\pi\)
\(648\) 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 −0.00478648 + 0.999989i \(0.501524\pi\)
−0.863622 + 0.504139i \(0.831810\pi\)
\(654\) 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 \(-0.723397\pi\)
0.338549 + 0.940949i \(0.390064\pi\)
\(662\) 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 −0.973744 0.227647i \(-0.926897\pi\)
−0.684020 0.729463i \(-0.739770\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 8.13060 + 14.0826i 0.0119042 + 0.0206188i 0.871916 0.489655i \(-0.162877\pi\)
−0.860012 + 0.510274i \(0.829544\pi\)
\(684\) 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 \(-0.349125\pi\)
−0.542333 + 0.840164i \(0.682459\pi\)
\(692\) 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 −0.340817 0.940130i \(-0.610704\pi\)
0.984585 0.174909i \(-0.0559631\pi\)
\(710\) 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 \(-0.457070\pi\)
−0.790930 + 0.611906i \(0.790403\pi\)
\(720\) 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 \(-0.551123\pi\)
−0.934839 + 0.355071i \(0.884457\pi\)
\(734\) 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 0.652290 0.757969i \(-0.273808\pi\)
−0.982566 + 0.185915i \(0.940475\pi\)
\(740\) 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 −0.660591 0.750746i \(-0.729694\pi\)
0.980460 + 0.196716i \(0.0630278\pi\)
\(752\) 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 \(-0.0468136\pi\)
0.367695 + 0.929946i \(0.380147\pi\)
\(762\) 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 \(-0.387258\pi\)
0.985685 + 0.168600i \(0.0539246\pi\)
\(774\) 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 \(-0.901065\pi\)
−0.211183 + 0.977447i \(0.567731\pi\)
\(788\) 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