Properties

Label 1764.4.k.bb.361.4
Level $1764$
Weight $4$
Character 1764.361
Analytic conductor $104.079$
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: \( k \) \(=\) \( 4 \)
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} + 27 x^{6} + 10 x^{5} + 446 x^{4} + 62 x^{3} + 3061 x^{2} + 2142 x + 14161\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{8}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 588)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.4
Root \(1.44795 - 2.50793i\) of defining polynomial
Character \(\chi\) \(=\) 1764.361
Dual form 1764.4.k.bb.1549.4

$q$-expansion

\(f(q)\) \(=\) \(q+(8.32734 + 14.4234i) q^{5} +O(q^{10})\) \(q+(8.32734 + 14.4234i) q^{5} +(-35.8800 + 62.1460i) q^{11} +65.3878 q^{13} +(-45.2158 + 78.3160i) q^{17} +(81.9092 + 141.871i) q^{19} +(39.6144 + 68.6142i) q^{23} +(-76.1893 + 131.964i) q^{25} +43.2995 q^{29} +(67.8179 - 117.464i) q^{31} +(-135.370 - 234.468i) q^{37} +152.241 q^{41} -177.641 q^{43} +(22.8166 + 39.5194i) q^{47} +(-79.2188 + 137.211i) q^{53} -1195.14 q^{55} +(195.884 - 339.282i) q^{59} +(275.647 + 477.435i) q^{61} +(544.506 + 943.113i) q^{65} +(-229.315 + 397.185i) q^{67} +486.786 q^{71} +(287.446 - 497.871i) q^{73} +(334.160 + 578.782i) q^{79} +76.2450 q^{83} -1506.11 q^{85} +(-683.401 - 1183.69i) q^{89} +(-1364.17 + 2362.82i) q^{95} -242.655 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 48q^{17} + 192q^{19} + 192q^{23} - 324q^{25} - 192q^{29} + 48q^{31} - 256q^{37} + 2016q^{41} - 224q^{43} - 864q^{47} - 648q^{53} - 4704q^{55} - 336q^{59} + 960q^{61} - 360q^{65} - 720q^{67} + 2688q^{71} + 672q^{73} + 1984q^{79} + 6240q^{83} + 1360q^{85} - 2160q^{89} - 3744q^{95} - 4032q^{97} + O(q^{100}) \)

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
\(5\) 8.32734 + 14.4234i 0.744820 + 1.29007i 0.950279 + 0.311401i \(0.100798\pi\)
−0.205459 + 0.978666i \(0.565869\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −35.8800 + 62.1460i −0.983475 + 1.70343i −0.334949 + 0.942236i \(0.608719\pi\)
−0.648526 + 0.761192i \(0.724614\pi\)
\(12\) 0 0
\(13\) 65.3878 1.39502 0.697512 0.716573i \(-0.254291\pi\)
0.697512 + 0.716573i \(0.254291\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −45.2158 + 78.3160i −0.645085 + 1.11732i 0.339197 + 0.940715i \(0.389845\pi\)
−0.984282 + 0.176604i \(0.943489\pi\)
\(18\) 0 0
\(19\) 81.9092 + 141.871i 0.989014 + 1.71302i 0.622526 + 0.782599i \(0.286107\pi\)
0.366488 + 0.930423i \(0.380560\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 39.6144 + 68.6142i 0.359138 + 0.622045i 0.987817 0.155619i \(-0.0497374\pi\)
−0.628679 + 0.777665i \(0.716404\pi\)
\(24\) 0 0
\(25\) −76.1893 + 131.964i −0.609514 + 1.05571i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 43.2995 0.277259 0.138630 0.990344i \(-0.455730\pi\)
0.138630 + 0.990344i \(0.455730\pi\)
\(30\) 0 0
\(31\) 67.8179 117.464i 0.392918 0.680554i −0.599915 0.800064i \(-0.704799\pi\)
0.992833 + 0.119510i \(0.0381323\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −135.370 234.468i −0.601479 1.04179i −0.992597 0.121452i \(-0.961245\pi\)
0.391118 0.920340i \(-0.372088\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 152.241 0.579902 0.289951 0.957041i \(-0.406361\pi\)
0.289951 + 0.957041i \(0.406361\pi\)
\(42\) 0 0
\(43\) −177.641 −0.630001 −0.315001 0.949091i \(-0.602005\pi\)
−0.315001 + 0.949091i \(0.602005\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 22.8166 + 39.5194i 0.0708114 + 0.122649i 0.899257 0.437420i \(-0.144108\pi\)
−0.828446 + 0.560069i \(0.810774\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −79.2188 + 137.211i −0.205312 + 0.355611i −0.950232 0.311543i \(-0.899154\pi\)
0.744920 + 0.667154i \(0.232488\pi\)
\(54\) 0 0
\(55\) −1195.14 −2.93005
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 195.884 339.282i 0.432237 0.748656i −0.564829 0.825208i \(-0.691058\pi\)
0.997066 + 0.0765519i \(0.0243911\pi\)
\(60\) 0 0
\(61\) 275.647 + 477.435i 0.578574 + 1.00212i 0.995643 + 0.0932450i \(0.0297240\pi\)
−0.417069 + 0.908875i \(0.636943\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 544.506 + 943.113i 1.03904 + 1.79967i
\(66\) 0 0
\(67\) −229.315 + 397.185i −0.418139 + 0.724237i −0.995752 0.0920729i \(-0.970651\pi\)
0.577614 + 0.816310i \(0.303984\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 486.786 0.813674 0.406837 0.913501i \(-0.366632\pi\)
0.406837 + 0.913501i \(0.366632\pi\)
\(72\) 0 0
\(73\) 287.446 497.871i 0.460863 0.798238i −0.538141 0.842855i \(-0.680873\pi\)
0.999004 + 0.0446164i \(0.0142066\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 334.160 + 578.782i 0.475897 + 0.824279i 0.999619 0.0276111i \(-0.00878999\pi\)
−0.523721 + 0.851890i \(0.675457\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 76.2450 0.100831 0.0504155 0.998728i \(-0.483945\pi\)
0.0504155 + 0.998728i \(0.483945\pi\)
\(84\) 0 0
\(85\) −1506.11 −1.92189
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −683.401 1183.69i −0.813937 1.40978i −0.910089 0.414413i \(-0.863987\pi\)
0.0961518 0.995367i \(-0.469347\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1364.17 + 2362.82i −1.47327 + 2.55179i
\(96\) 0 0
\(97\) −242.655 −0.253999 −0.127000 0.991903i \(-0.540535\pi\)
−0.127000 + 0.991903i \(0.540535\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −347.371 + 601.665i −0.342225 + 0.592751i −0.984846 0.173433i \(-0.944514\pi\)
0.642620 + 0.766185i \(0.277847\pi\)
\(102\) 0 0
\(103\) −87.1089 150.877i −0.0833310 0.144334i 0.821348 0.570428i \(-0.193223\pi\)
−0.904679 + 0.426094i \(0.859889\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 23.8025 + 41.2272i 0.0215054 + 0.0372485i 0.876578 0.481260i \(-0.159821\pi\)
−0.855072 + 0.518509i \(0.826487\pi\)
\(108\) 0 0
\(109\) 816.549 1414.30i 0.717534 1.24281i −0.244440 0.969664i \(-0.578604\pi\)
0.961974 0.273141i \(-0.0880626\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1127.29 0.938464 0.469232 0.883075i \(-0.344531\pi\)
0.469232 + 0.883075i \(0.344531\pi\)
\(114\) 0 0
\(115\) −659.766 + 1142.75i −0.534987 + 0.926624i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1909.25 3306.91i −1.43445 2.48453i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −455.982 −0.326274
\(126\) 0 0
\(127\) 2398.66 1.67596 0.837979 0.545702i \(-0.183737\pi\)
0.837979 + 0.545702i \(0.183737\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −774.482 1341.44i −0.516541 0.894675i −0.999816 0.0192062i \(-0.993886\pi\)
0.483275 0.875469i \(-0.339447\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 761.634 1319.19i 0.474969 0.822671i −0.524620 0.851337i \(-0.675793\pi\)
0.999589 + 0.0286657i \(0.00912583\pi\)
\(138\) 0 0
\(139\) −551.445 −0.336496 −0.168248 0.985745i \(-0.553811\pi\)
−0.168248 + 0.985745i \(0.553811\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2346.11 + 4063.59i −1.37197 + 2.37632i
\(144\) 0 0
\(145\) 360.570 + 624.526i 0.206508 + 0.357683i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1200.12 2078.67i −0.659851 1.14290i −0.980654 0.195749i \(-0.937286\pi\)
0.320803 0.947146i \(-0.396047\pi\)
\(150\) 0 0
\(151\) −310.780 + 538.287i −0.167490 + 0.290100i −0.937537 0.347887i \(-0.886899\pi\)
0.770047 + 0.637987i \(0.220233\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2258.97 1.17061
\(156\) 0 0
\(157\) −1345.79 + 2330.97i −0.684112 + 1.18492i 0.289603 + 0.957147i \(0.406477\pi\)
−0.973715 + 0.227770i \(0.926857\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −528.653 915.654i −0.254032 0.439997i 0.710600 0.703596i \(-0.248424\pi\)
−0.964632 + 0.263599i \(0.915090\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −243.840 −0.112987 −0.0564937 0.998403i \(-0.517992\pi\)
−0.0564937 + 0.998403i \(0.517992\pi\)
\(168\) 0 0
\(169\) 2078.56 0.946090
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 843.632 + 1461.21i 0.370753 + 0.642162i 0.989682 0.143285i \(-0.0457665\pi\)
−0.618929 + 0.785447i \(0.712433\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −179.570 + 311.024i −0.0749814 + 0.129872i −0.901078 0.433657i \(-0.857223\pi\)
0.826097 + 0.563528i \(0.190556\pi\)
\(180\) 0 0
\(181\) −2982.58 −1.22483 −0.612413 0.790538i \(-0.709801\pi\)
−0.612413 + 0.790538i \(0.709801\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2254.55 3904.99i 0.895988 1.55190i
\(186\) 0 0
\(187\) −3244.68 5619.96i −1.26885 2.19771i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1745.39 + 3023.10i 0.661213 + 1.14525i 0.980297 + 0.197528i \(0.0632914\pi\)
−0.319084 + 0.947726i \(0.603375\pi\)
\(192\) 0 0
\(193\) −802.114 + 1389.30i −0.299158 + 0.518156i −0.975943 0.218024i \(-0.930039\pi\)
0.676786 + 0.736180i \(0.263372\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 455.935 0.164893 0.0824467 0.996595i \(-0.473727\pi\)
0.0824467 + 0.996595i \(0.473727\pi\)
\(198\) 0 0
\(199\) 1042.08 1804.94i 0.371212 0.642958i −0.618541 0.785753i \(-0.712276\pi\)
0.989752 + 0.142795i \(0.0456091\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 1267.76 + 2195.83i 0.431923 + 0.748113i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −11755.6 −3.89068
\(210\) 0 0
\(211\) 2568.06 0.837880 0.418940 0.908014i \(-0.362402\pi\)
0.418940 + 0.908014i \(0.362402\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1479.28 2562.19i −0.469238 0.812743i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2956.56 + 5120.91i −0.899908 + 1.55869i
\(222\) 0 0
\(223\) 4659.55 1.39922 0.699612 0.714523i \(-0.253356\pi\)
0.699612 + 0.714523i \(0.253356\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2414.39 + 4181.84i −0.705940 + 1.22272i 0.260411 + 0.965498i \(0.416142\pi\)
−0.966351 + 0.257227i \(0.917191\pi\)
\(228\) 0 0
\(229\) −1611.29 2790.84i −0.464966 0.805345i 0.534234 0.845337i \(-0.320600\pi\)
−0.999200 + 0.0399917i \(0.987267\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2851.21 4938.44i −0.801670 1.38853i −0.918517 0.395383i \(-0.870612\pi\)
0.116847 0.993150i \(-0.462721\pi\)
\(234\) 0 0
\(235\) −380.003 + 658.184i −0.105484 + 0.182703i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1994.47 −0.539796 −0.269898 0.962889i \(-0.586990\pi\)
−0.269898 + 0.962889i \(0.586990\pi\)
\(240\) 0 0
\(241\) −710.443 + 1230.52i −0.189891 + 0.328900i −0.945214 0.326452i \(-0.894147\pi\)
0.755323 + 0.655353i \(0.227480\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 5355.86 + 9276.62i 1.37970 + 2.38971i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −3855.44 −0.969534 −0.484767 0.874643i \(-0.661096\pi\)
−0.484767 + 0.874643i \(0.661096\pi\)
\(252\) 0 0
\(253\) −5685.46 −1.41281
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2026.64 3510.24i −0.491900 0.851996i 0.508056 0.861324i \(-0.330364\pi\)
−0.999956 + 0.00932793i \(0.997031\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −250.393 + 433.694i −0.0587069 + 0.101683i −0.893885 0.448296i \(-0.852031\pi\)
0.835178 + 0.549979i \(0.185364\pi\)
\(264\) 0 0
\(265\) −2638.73 −0.611683
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1640.87 + 2842.07i −0.371917 + 0.644179i −0.989860 0.142043i \(-0.954633\pi\)
0.617943 + 0.786223i \(0.287966\pi\)
\(270\) 0 0
\(271\) 1662.18 + 2878.98i 0.372584 + 0.645334i 0.989962 0.141332i \(-0.0451386\pi\)
−0.617378 + 0.786666i \(0.711805\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −5467.34 9469.71i −1.19888 2.07653i
\(276\) 0 0
\(277\) −4309.87 + 7464.92i −0.934856 + 1.61922i −0.159966 + 0.987122i \(0.551139\pi\)
−0.774890 + 0.632096i \(0.782195\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 7710.80 1.63697 0.818484 0.574529i \(-0.194815\pi\)
0.818484 + 0.574529i \(0.194815\pi\)
\(282\) 0 0
\(283\) −69.6354 + 120.612i −0.0146268 + 0.0253344i −0.873246 0.487279i \(-0.837989\pi\)
0.858619 + 0.512614i \(0.171323\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1632.43 2827.46i −0.332268 0.575506i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −4437.09 −0.884702 −0.442351 0.896842i \(-0.645855\pi\)
−0.442351 + 0.896842i \(0.645855\pi\)
\(294\) 0 0
\(295\) 6524.79 1.28776
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2590.30 + 4486.53i 0.501006 + 0.867768i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −4590.82 + 7951.53i −0.861867 + 1.49280i
\(306\) 0 0
\(307\) 299.480 0.0556749 0.0278375 0.999612i \(-0.491138\pi\)
0.0278375 + 0.999612i \(0.491138\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −456.972 + 791.498i −0.0833199 + 0.144314i −0.904674 0.426104i \(-0.859886\pi\)
0.821354 + 0.570419i \(0.193219\pi\)
\(312\) 0 0
\(313\) 1943.69 + 3366.57i 0.351002 + 0.607954i 0.986425 0.164210i \(-0.0525075\pi\)
−0.635423 + 0.772164i \(0.719174\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1439.65 + 2493.55i 0.255075 + 0.441803i 0.964916 0.262559i \(-0.0845664\pi\)
−0.709841 + 0.704362i \(0.751233\pi\)
\(318\) 0 0
\(319\) −1553.59 + 2690.89i −0.272678 + 0.472291i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −14814.4 −2.55199
\(324\) 0 0
\(325\) −4981.85 + 8628.81i −0.850287 + 1.47274i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1405.39 + 2434.21i 0.233376 + 0.404219i 0.958799 0.284084i \(-0.0916894\pi\)
−0.725424 + 0.688303i \(0.758356\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −7638.34 −1.24575
\(336\) 0 0
\(337\) 2960.90 0.478607 0.239303 0.970945i \(-0.423081\pi\)
0.239303 + 0.970945i \(0.423081\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 4866.61 + 8429.22i 0.772850 + 1.33862i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2142.26 + 3710.50i −0.331419 + 0.574034i −0.982790 0.184725i \(-0.940861\pi\)
0.651371 + 0.758759i \(0.274194\pi\)
\(348\) 0 0
\(349\) −3295.36 −0.505434 −0.252717 0.967540i \(-0.581324\pi\)
−0.252717 + 0.967540i \(0.581324\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 5314.83 9205.56i 0.801360 1.38800i −0.117362 0.993089i \(-0.537444\pi\)
0.918721 0.394906i \(-0.129223\pi\)
\(354\) 0 0
\(355\) 4053.63 + 7021.09i 0.606040 + 1.04969i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1592.51 + 2758.30i 0.234121 + 0.405509i 0.959017 0.283349i \(-0.0914456\pi\)
−0.724896 + 0.688858i \(0.758112\pi\)
\(360\) 0 0
\(361\) −9988.74 + 17301.0i −1.45630 + 2.52238i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 9574.65 1.37304
\(366\) 0 0
\(367\) 3801.52 6584.42i 0.540702 0.936523i −0.458162 0.888869i \(-0.651492\pi\)
0.998864 0.0476544i \(-0.0151746\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −755.978 1309.39i −0.104941 0.181763i 0.808773 0.588121i \(-0.200132\pi\)
−0.913714 + 0.406358i \(0.866799\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2831.26 0.386783
\(378\) 0 0
\(379\) 8848.36 1.19923 0.599617 0.800287i \(-0.295320\pi\)
0.599617 + 0.800287i \(0.295320\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −2067.05 3580.24i −0.275774 0.477654i 0.694556 0.719438i \(-0.255601\pi\)
−0.970330 + 0.241784i \(0.922267\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3815.30 6608.30i 0.497284 0.861322i −0.502711 0.864455i \(-0.667664\pi\)
0.999995 + 0.00313295i \(0.000997251\pi\)
\(390\) 0 0
\(391\) −7164.79 −0.926698
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −5565.32 + 9639.43i −0.708916 + 1.22788i
\(396\) 0 0
\(397\) −5809.25 10061.9i −0.734403 1.27202i −0.954985 0.296654i \(-0.904129\pi\)
0.220582 0.975368i \(-0.429204\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1523.51 + 2638.79i 0.189726 + 0.328616i 0.945159 0.326611i \(-0.105907\pi\)
−0.755433 + 0.655226i \(0.772573\pi\)
\(402\) 0 0
\(403\) 4434.46 7680.71i 0.548130 0.949388i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 19428.3 2.36616
\(408\) 0 0
\(409\) 5791.42 10031.0i 0.700164 1.21272i −0.268244 0.963351i \(-0.586443\pi\)
0.968408 0.249369i \(-0.0802233\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 634.918 + 1099.71i 0.0751010 + 0.130079i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −1318.83 −0.153768 −0.0768841 0.997040i \(-0.524497\pi\)
−0.0768841 + 0.997040i \(0.524497\pi\)
\(420\) 0 0
\(421\) −9733.56 −1.12680 −0.563402 0.826183i \(-0.690508\pi\)
−0.563402 + 0.826183i \(0.690508\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −6889.92 11933.7i −0.786377 1.36204i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −7214.47 + 12495.8i −0.806284 + 1.39653i 0.109136 + 0.994027i \(0.465192\pi\)
−0.915421 + 0.402499i \(0.868142\pi\)
\(432\) 0 0
\(433\) 16620.3 1.84463 0.922313 0.386444i \(-0.126297\pi\)
0.922313 + 0.386444i \(0.126297\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6489.57 + 11240.3i −0.710385 + 1.23042i
\(438\) 0 0
\(439\) −257.783 446.493i −0.0280257 0.0485420i 0.851672 0.524075i \(-0.175589\pi\)
−0.879698 + 0.475533i \(0.842255\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4488.05 + 7773.52i 0.481340 + 0.833705i 0.999771 0.0214146i \(-0.00681700\pi\)
−0.518431 + 0.855120i \(0.673484\pi\)
\(444\) 0 0
\(445\) 11381.8 19713.9i 1.21247 2.10007i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −2106.83 −0.221442 −0.110721 0.993852i \(-0.535316\pi\)
−0.110721 + 0.993852i \(0.535316\pi\)
\(450\) 0 0
\(451\) −5462.39 + 9461.14i −0.570319 + 0.987822i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −7863.08 13619.3i −0.804857 1.39405i −0.916388 0.400292i \(-0.868909\pi\)
0.111531 0.993761i \(-0.464425\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −7921.16 −0.800272 −0.400136 0.916456i \(-0.631037\pi\)
−0.400136 + 0.916456i \(0.631037\pi\)
\(462\) 0 0
\(463\) −5821.20 −0.584306 −0.292153 0.956372i \(-0.594372\pi\)
−0.292153 + 0.956372i \(0.594372\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −5893.94 10208.6i −0.584024 1.01156i −0.994996 0.0999112i \(-0.968144\pi\)
0.410973 0.911648i \(-0.365189\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 6373.77 11039.7i 0.619590 1.07316i
\(474\) 0 0
\(475\) −24962.4 −2.41127
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 2657.21 4602.43i 0.253468 0.439020i −0.711010 0.703182i \(-0.751762\pi\)
0.964478 + 0.264162i \(0.0850954\pi\)
\(480\) 0 0
\(481\) −8851.56 15331.3i −0.839078 1.45332i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2020.67 3499.91i −0.189184 0.327676i
\(486\) 0 0
\(487\) 9331.34 16162.4i 0.868261 1.50387i 0.00448933 0.999990i \(-0.498571\pi\)
0.863772 0.503883i \(-0.168096\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −13519.1 −1.24259 −0.621293 0.783578i \(-0.713392\pi\)
−0.621293 + 0.783578i \(0.713392\pi\)
\(492\) 0 0
\(493\) −1957.82 + 3391.05i −0.178856 + 0.309787i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −6916.17 11979.2i −0.620462 1.07467i −0.989400 0.145217i \(-0.953612\pi\)
0.368938 0.929454i \(-0.379721\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 13469.0 1.19394 0.596970 0.802264i \(-0.296371\pi\)
0.596970 + 0.802264i \(0.296371\pi\)
\(504\) 0 0
\(505\) −11570.7 −1.01958
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −4387.62 7599.59i −0.382079 0.661779i 0.609281 0.792955i \(-0.291458\pi\)
−0.991359 + 0.131175i \(0.958125\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1450.77 2512.81i 0.124133 0.215005i
\(516\) 0 0
\(517\) −3274.63 −0.278565
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 830.382 1438.26i 0.0698267 0.120943i −0.828998 0.559251i \(-0.811089\pi\)
0.898825 + 0.438308i \(0.144422\pi\)
\(522\) 0 0
\(523\) 3162.41 + 5477.45i 0.264402 + 0.457958i 0.967407 0.253227i \(-0.0814920\pi\)
−0.703005 + 0.711185i \(0.748159\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6132.88 + 10622.5i 0.506931 + 0.878030i
\(528\) 0 0
\(529\) 2944.90 5100.71i 0.242040 0.419225i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 9954.68 0.808977
\(534\) 0 0
\(535\) −396.424 + 686.626i −0.0320353 + 0.0554868i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −2233.94 3869.30i −0.177532 0.307494i 0.763503 0.645804i \(-0.223478\pi\)
−0.941034 + 0.338311i \(0.890145\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 27198.7 2.13774
\(546\) 0 0
\(547\) −10939.0 −0.855063 −0.427531 0.904000i \(-0.640617\pi\)
−0.427531 + 0.904000i \(0.640617\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 3546.63 + 6142.94i 0.274213 + 0.474951i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −7162.57 + 12405.9i −0.544862 + 0.943728i 0.453754 + 0.891127i \(0.350085\pi\)
−0.998616 + 0.0526011i \(0.983249\pi\)
\(558\) 0 0
\(559\) −11615.6 −0.878867
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −10201.4 + 17669.4i −0.763656 + 1.32269i 0.177299 + 0.984157i \(0.443264\pi\)
−0.940954 + 0.338533i \(0.890069\pi\)
\(564\) 0 0
\(565\) 9387.32 + 16259.3i 0.698987 + 1.21068i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −3625.21 6279.05i −0.267095 0.462622i 0.701016 0.713146i \(-0.252730\pi\)
−0.968110 + 0.250524i \(0.919397\pi\)
\(570\) 0 0
\(571\) −12447.4 + 21559.5i −0.912272 + 1.58010i −0.101424 + 0.994843i \(0.532340\pi\)
−0.810847 + 0.585258i \(0.800993\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −12072.8 −0.875599
\(576\) 0 0
\(577\) 681.723 1180.78i 0.0491863 0.0851932i −0.840384 0.541991i \(-0.817671\pi\)
0.889570 + 0.456798i \(0.151004\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −5684.74 9846.26i −0.403839 0.699469i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 23854.8 1.67733 0.838666 0.544647i \(-0.183336\pi\)
0.838666 + 0.544647i \(0.183336\pi\)
\(588\) 0 0
\(589\) 22219.6 1.55440
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1718.60 2976.71i −0.119013 0.206136i 0.800364 0.599514i \(-0.204640\pi\)
−0.919377 + 0.393378i \(0.871306\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 8255.87 14299.6i 0.563148 0.975401i −0.434071 0.900879i \(-0.642923\pi\)
0.997219 0.0745226i \(-0.0237433\pi\)
\(600\) 0 0
\(601\) 11148.3 0.756656 0.378328 0.925672i \(-0.376499\pi\)
0.378328 + 0.925672i \(0.376499\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 31797.9 55075.6i 2.13681 3.70106i
\(606\) 0 0
\(607\) −195.791 339.121i −0.0130921 0.0226763i 0.859405 0.511295i \(-0.170834\pi\)
−0.872497 + 0.488619i \(0.837501\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1491.92 + 2584.09i 0.0987836 + 0.171098i
\(612\) 0 0
\(613\) 5870.25 10167.6i 0.386782 0.669925i −0.605233 0.796048i \(-0.706920\pi\)
0.992015 + 0.126123i \(0.0402534\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8818.95 0.575426 0.287713 0.957717i \(-0.407105\pi\)
0.287713 + 0.957717i \(0.407105\pi\)
\(618\) 0 0
\(619\) −7962.35 + 13791.2i −0.517018 + 0.895501i 0.482787 + 0.875738i \(0.339625\pi\)
−0.999805 + 0.0197631i \(0.993709\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 5726.55 + 9918.67i 0.366499 + 0.634795i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 24483.5 1.55202
\(630\) 0 0
\(631\) 29206.4 1.84261 0.921307 0.388836i \(-0.127123\pi\)
0.921307 + 0.388836i \(0.127123\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 19974.5 + 34596.8i 1.24829 + 2.16210i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 2806.51 4861.01i 0.172933 0.299530i −0.766511 0.642232i \(-0.778009\pi\)
0.939444 + 0.342702i \(0.111342\pi\)
\(642\) 0 0
\(643\) 18995.0 1.16499 0.582496 0.812834i \(-0.302076\pi\)
0.582496 + 0.812834i \(0.302076\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −11332.5 + 19628.5i −0.688604 + 1.19270i 0.283686 + 0.958917i \(0.408443\pi\)
−0.972290 + 0.233779i \(0.924891\pi\)
\(648\) 0 0
\(649\) 14056.7 + 24346.9i 0.850188 + 1.47257i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 13263.9 + 22973.7i 0.794879 + 1.37677i 0.922916 + 0.385001i \(0.125799\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(654\) 0 0
\(655\) 12898.8 22341.3i 0.769460 1.33274i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 17114.8 1.01168 0.505839 0.862628i \(-0.331183\pi\)
0.505839 + 0.862628i \(0.331183\pi\)
\(660\) 0 0
\(661\) −595.908 + 1032.14i −0.0350653 + 0.0607348i −0.883025 0.469325i \(-0.844497\pi\)
0.847960 + 0.530060i \(0.177831\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1715.29 + 2970.96i 0.0995744 + 0.172468i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −39560.9 −2.27605
\(672\) 0 0
\(673\) −9415.20 −0.539271 −0.269636 0.962962i \(-0.586903\pi\)
−0.269636 + 0.962962i \(0.586903\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −3958.56 6856.43i −0.224727 0.389238i 0.731511 0.681830i \(-0.238816\pi\)
−0.956237 + 0.292592i \(0.905482\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 14617.2 25317.7i 0.818904 1.41838i −0.0875859 0.996157i \(-0.527915\pi\)
0.906490 0.422227i \(-0.138751\pi\)
\(684\) 0 0
\(685\) 25369.6 1.41507
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −5179.94 + 8971.93i −0.286415 + 0.496086i
\(690\) 0 0
\(691\) −611.228 1058.68i −0.0336501 0.0582836i 0.848710 0.528859i \(-0.177380\pi\)
−0.882360 + 0.470575i \(0.844047\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4592.07 7953.70i −0.250629 0.434102i
\(696\) 0 0
\(697\) −6883.68 + 11922.9i −0.374086 + 0.647936i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −13054.6 −0.703372 −0.351686 0.936118i \(-0.614392\pi\)
−0.351686 + 0.936118i \(0.614392\pi\)
\(702\) 0 0
\(703\) 22176.1 38410.2i 1.18974 2.06069i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 9987.87 + 17299.5i 0.529058 + 0.916356i 0.999426 + 0.0338854i \(0.0107881\pi\)
−0.470367 + 0.882471i \(0.655879\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 10746.3 0.564447
\(714\) 0 0
\(715\) −78147.5 −4.08749
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −3256.13 5639.78i −0.168892 0.292529i 0.769139 0.639082i \(-0.220685\pi\)
−0.938030 + 0.346553i \(0.887352\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3298.96 + 5713.97i −0.168994 + 0.292705i
\(726\) 0 0
\(727\) −11437.1 −0.583467 −0.291733 0.956500i \(-0.594232\pi\)
−0.291733 + 0.956500i \(0.594232\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8032.19 13912.2i 0.406404 0.703913i
\(732\) 0 0
\(733\) −6138.02 10631.4i −0.309295 0.535714i 0.668914 0.743340i \(-0.266760\pi\)
−0.978208 + 0.207626i \(0.933426\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −16455.6 28502.0i −0.822458 1.42454i
\(738\) 0 0
\(739\) −6046.91 + 10473.5i −0.301000 + 0.521347i −0.976363 0.216138i \(-0.930654\pi\)
0.675363 + 0.737486i \(0.263987\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −34331.6 −1.69516 −0.847580 0.530668i \(-0.821941\pi\)
−0.847580 + 0.530668i \(0.821941\pi\)
\(744\) 0 0
\(745\) 19987.6 34619.6i 0.982940 1.70250i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 7278.96 + 12607.5i 0.353679 + 0.612590i 0.986891 0.161388i \(-0.0515971\pi\)
−0.633212 + 0.773979i \(0.718264\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −10351.9 −0.498998
\(756\) 0 0
\(757\) 27115.5 1.30189 0.650944 0.759126i \(-0.274373\pi\)
0.650944 + 0.759126i \(0.274373\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 5481.98 + 9495.06i 0.261132 + 0.452294i 0.966543 0.256504i \(-0.0825708\pi\)
−0.705411 + 0.708799i \(0.749238\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 12808.4 22184.9i 0.602981 1.04439i
\(768\) 0 0
\(769\) 7835.53 0.367434 0.183717 0.982979i \(-0.441187\pi\)
0.183717 + 0.982979i \(0.441187\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 14383.9 24913.6i 0.669278 1.15922i −0.308828 0.951118i \(-0.599937\pi\)
0.978106 0.208106i \(-0.0667298\pi\)
\(774\) 0 0
\(775\) 10334.0 + 17899.0i 0.478978 + 0.829614i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 12469.9 + 21598.5i 0.573531 + 0.993386i
\(780\) 0 0
\(781\) −17465.9 + 30251.8i −0.800228 + 1.38603i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −44827.3 −2.03816
\(786\) 0 0
\(787\) −1347.91 + 2334.65i −0.0610520 + 0.105745i −0.894936 0.446195i \(-0.852779\pi\)
0.833884 + 0.551940i \(0.186112\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 18024.0 + 31218.4i 0.807125 + 1.39798i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 12592.1 0.559643 0.279822 0.960052i \(-0.409725\pi\)
0.279822 + 0.960052i \(0.409725\pi\)
\(798\) 0 0
\(799\) −4126.67 −0.182717
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 20627.1 + 35727.2i 0.906495 + 1.57009i
\(804\) 0 0
\(805\)