Properties

Label 1764.3.z.m.325.3
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 325.3
Root \(-1.60021 - 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 1764.325
Dual form 1764.3.z.m.901.3

$q$-expansion

\(f(q)\) \(=\) \(q+(0.804540 - 0.464502i) q^{5} +O(q^{10})\) \(q+(0.804540 - 0.464502i) q^{5} +(4.84490 - 8.39161i) q^{11} -15.9753i q^{13} +(-9.14154 - 5.27787i) q^{17} +(6.25313 - 3.61025i) q^{19} +(-5.65003 - 9.78614i) q^{23} +(-12.0685 + 20.9032i) q^{25} -46.3148 q^{29} +(-0.418333 - 0.241525i) q^{31} +(-1.24065 - 2.14887i) q^{37} +55.8520i q^{41} +60.6786 q^{43} +(-31.6850 + 18.2933i) q^{47} +(-14.2615 + 24.7016i) q^{53} -9.00185i q^{55} +(-81.4683 - 47.0358i) q^{59} +(95.4301 - 55.0966i) q^{61} +(-7.42055 - 12.8528i) q^{65} +(-41.0155 + 71.0409i) q^{67} -127.349 q^{71} +(-40.0577 - 23.1273i) q^{73} +(-9.35016 - 16.1949i) q^{79} -59.6357i q^{83} -9.80632 q^{85} +(61.5988 - 35.5641i) q^{89} +(3.35393 - 5.80917i) q^{95} -102.239i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 48q^{17} + 96q^{19} - 8q^{23} - 36q^{25} - 80q^{29} - 48q^{31} - 64q^{37} - 112q^{43} - 264q^{47} - 72q^{53} + 168q^{59} + 144q^{61} + 120q^{65} + 32q^{67} - 224q^{71} - 336q^{73} + 216q^{79} - 96q^{85} + 96q^{89} - 136q^{95} + O(q^{100}) \)

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
\(5\) 0.804540 0.464502i 0.160908 0.0929003i −0.417384 0.908730i \(-0.637053\pi\)
0.578292 + 0.815830i \(0.303720\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.84490 8.39161i 0.440445 0.762874i −0.557277 0.830327i \(-0.688154\pi\)
0.997722 + 0.0674529i \(0.0214872\pi\)
\(12\) 0 0
\(13\) 15.9753i 1.22887i −0.788968 0.614434i \(-0.789384\pi\)
0.788968 0.614434i \(-0.210616\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −9.14154 5.27787i −0.537738 0.310463i 0.206424 0.978463i \(-0.433817\pi\)
−0.744162 + 0.668000i \(0.767151\pi\)
\(18\) 0 0
\(19\) 6.25313 3.61025i 0.329112 0.190013i −0.326335 0.945254i \(-0.605814\pi\)
0.655447 + 0.755241i \(0.272480\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.65003 9.78614i −0.245654 0.425484i 0.716662 0.697421i \(-0.245669\pi\)
−0.962315 + 0.271937i \(0.912336\pi\)
\(24\) 0 0
\(25\) −12.0685 + 20.9032i −0.482739 + 0.836129i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −46.3148 −1.59706 −0.798532 0.601953i \(-0.794390\pi\)
−0.798532 + 0.601953i \(0.794390\pi\)
\(30\) 0 0
\(31\) −0.418333 0.241525i −0.0134946 0.00779111i 0.493238 0.869895i \(-0.335813\pi\)
−0.506732 + 0.862104i \(0.669147\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.24065 2.14887i −0.0335312 0.0580777i 0.848773 0.528758i \(-0.177342\pi\)
−0.882304 + 0.470680i \(0.844009\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 55.8520i 1.36224i 0.732170 + 0.681122i \(0.238508\pi\)
−0.732170 + 0.681122i \(0.761492\pi\)
\(42\) 0 0
\(43\) 60.6786 1.41113 0.705566 0.708645i \(-0.250693\pi\)
0.705566 + 0.708645i \(0.250693\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −31.6850 + 18.2933i −0.674149 + 0.389220i −0.797647 0.603125i \(-0.793922\pi\)
0.123498 + 0.992345i \(0.460589\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −14.2615 + 24.7016i −0.269084 + 0.466068i −0.968626 0.248524i \(-0.920054\pi\)
0.699541 + 0.714592i \(0.253388\pi\)
\(54\) 0 0
\(55\) 9.00185i 0.163670i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −81.4683 47.0358i −1.38082 0.797216i −0.388563 0.921422i \(-0.627028\pi\)
−0.992256 + 0.124206i \(0.960362\pi\)
\(60\) 0 0
\(61\) 95.4301 55.0966i 1.56443 0.903223i 0.567628 0.823285i \(-0.307861\pi\)
0.996800 0.0799382i \(-0.0254723\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −7.42055 12.8528i −0.114162 0.197735i
\(66\) 0 0
\(67\) −41.0155 + 71.0409i −0.612171 + 1.06031i 0.378702 + 0.925518i \(0.376370\pi\)
−0.990874 + 0.134793i \(0.956963\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −127.349 −1.79365 −0.896827 0.442382i \(-0.854133\pi\)
−0.896827 + 0.442382i \(0.854133\pi\)
\(72\) 0 0
\(73\) −40.0577 23.1273i −0.548735 0.316812i 0.199877 0.979821i \(-0.435946\pi\)
−0.748612 + 0.663009i \(0.769279\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −9.35016 16.1949i −0.118356 0.204999i 0.800760 0.598985i \(-0.204429\pi\)
−0.919116 + 0.393986i \(0.871096\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 59.6357i 0.718502i −0.933241 0.359251i \(-0.883032\pi\)
0.933241 0.359251i \(-0.116968\pi\)
\(84\) 0 0
\(85\) −9.80632 −0.115368
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 61.5988 35.5641i 0.692121 0.399596i −0.112285 0.993676i \(-0.535817\pi\)
0.804406 + 0.594080i \(0.202484\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.35393 5.80917i 0.0353045 0.0611492i
\(96\) 0 0
\(97\) 102.239i 1.05401i −0.849861 0.527007i \(-0.823314\pi\)
0.849861 0.527007i \(-0.176686\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −94.3357 54.4647i −0.934017 0.539255i −0.0459371 0.998944i \(-0.514627\pi\)
−0.888080 + 0.459690i \(0.847961\pi\)
\(102\) 0 0
\(103\) −0.647083 + 0.373594i −0.00628236 + 0.00362712i −0.503138 0.864206i \(-0.667821\pi\)
0.496856 + 0.867833i \(0.334488\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.64630 + 13.2438i 0.0714608 + 0.123774i 0.899542 0.436835i \(-0.143901\pi\)
−0.828081 + 0.560609i \(0.810567\pi\)
\(108\) 0 0
\(109\) 27.1116 46.9587i 0.248730 0.430814i −0.714443 0.699693i \(-0.753320\pi\)
0.963174 + 0.268879i \(0.0866533\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −65.2511 −0.577444 −0.288722 0.957413i \(-0.593230\pi\)
−0.288722 + 0.957413i \(0.593230\pi\)
\(114\) 0 0
\(115\) −9.09136 5.24890i −0.0790553 0.0456426i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 13.5539 + 23.4761i 0.112016 + 0.194017i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 45.6484i 0.365187i
\(126\) 0 0
\(127\) −235.761 −1.85639 −0.928193 0.372098i \(-0.878639\pi\)
−0.928193 + 0.372098i \(0.878639\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 196.180 113.265i 1.49756 0.864616i 0.497563 0.867428i \(-0.334228\pi\)
0.999996 + 0.00281117i \(0.000894824\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −126.872 + 219.748i −0.926070 + 1.60400i −0.136238 + 0.990676i \(0.543501\pi\)
−0.789832 + 0.613323i \(0.789832\pi\)
\(138\) 0 0
\(139\) 148.040i 1.06503i 0.846419 + 0.532517i \(0.178754\pi\)
−0.846419 + 0.532517i \(0.821246\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −134.058 77.3986i −0.937471 0.541249i
\(144\) 0 0
\(145\) −37.2622 + 21.5133i −0.256980 + 0.148368i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −122.732 212.579i −0.823708 1.42670i −0.902903 0.429845i \(-0.858568\pi\)
0.0791950 0.996859i \(-0.474765\pi\)
\(150\) 0 0
\(151\) −88.1270 + 152.640i −0.583623 + 1.01086i 0.411423 + 0.911445i \(0.365032\pi\)
−0.995046 + 0.0994194i \(0.968301\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.448754 −0.00289519
\(156\) 0 0
\(157\) −179.836 103.828i −1.14545 0.661326i −0.197675 0.980268i \(-0.563339\pi\)
−0.947774 + 0.318942i \(0.896673\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −47.5121 82.2933i −0.291485 0.504867i 0.682676 0.730721i \(-0.260816\pi\)
−0.974161 + 0.225854i \(0.927483\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 158.478i 0.948972i −0.880263 0.474486i \(-0.842634\pi\)
0.880263 0.474486i \(-0.157366\pi\)
\(168\) 0 0
\(169\) −86.2098 −0.510117
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 187.054 107.995i 1.08123 0.624251i 0.150005 0.988685i \(-0.452071\pi\)
0.931229 + 0.364434i \(0.118738\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −131.029 + 226.950i −0.732008 + 1.26787i 0.224016 + 0.974585i \(0.428083\pi\)
−0.956024 + 0.293289i \(0.905250\pi\)
\(180\) 0 0
\(181\) 83.8554i 0.463290i 0.972800 + 0.231645i \(0.0744107\pi\)
−0.972800 + 0.231645i \(0.925589\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.99631 1.15257i −0.0107909 0.00623011i
\(186\) 0 0
\(187\) −88.5797 + 51.1415i −0.473688 + 0.273484i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 116.846 + 202.384i 0.611760 + 1.05960i 0.990944 + 0.134278i \(0.0428716\pi\)
−0.379183 + 0.925322i \(0.623795\pi\)
\(192\) 0 0
\(193\) 111.819 193.677i 0.579375 1.00351i −0.416176 0.909284i \(-0.636630\pi\)
0.995551 0.0942227i \(-0.0300366\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 133.006 0.675158 0.337579 0.941297i \(-0.390392\pi\)
0.337579 + 0.941297i \(0.390392\pi\)
\(198\) 0 0
\(199\) −52.5277 30.3269i −0.263959 0.152397i 0.362180 0.932108i \(-0.382032\pi\)
−0.626139 + 0.779711i \(0.715366\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 25.9433 + 44.9352i 0.126553 + 0.219196i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 69.9651i 0.334761i
\(210\) 0 0
\(211\) −169.145 −0.801637 −0.400819 0.916157i \(-0.631274\pi\)
−0.400819 + 0.916157i \(0.631274\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 48.8184 28.1853i 0.227062 0.131095i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −84.3155 + 146.039i −0.381518 + 0.660809i
\(222\) 0 0
\(223\) 162.093i 0.726874i 0.931619 + 0.363437i \(0.118397\pi\)
−0.931619 + 0.363437i \(0.881603\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −306.755 177.105i −1.35134 0.780198i −0.362905 0.931826i \(-0.618215\pi\)
−0.988438 + 0.151628i \(0.951548\pi\)
\(228\) 0 0
\(229\) 113.844 65.7279i 0.497136 0.287021i −0.230394 0.973097i \(-0.574002\pi\)
0.727530 + 0.686076i \(0.240668\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −71.4744 123.797i −0.306757 0.531319i 0.670894 0.741553i \(-0.265911\pi\)
−0.977651 + 0.210234i \(0.932577\pi\)
\(234\) 0 0
\(235\) −16.9946 + 29.4354i −0.0723173 + 0.125257i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −47.5259 −0.198853 −0.0994266 0.995045i \(-0.531701\pi\)
−0.0994266 + 0.995045i \(0.531701\pi\)
\(240\) 0 0
\(241\) −205.380 118.576i −0.852198 0.492017i 0.00919389 0.999958i \(-0.497073\pi\)
−0.861392 + 0.507941i \(0.830407\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −57.6747 99.8955i −0.233501 0.404435i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 309.248i 1.23206i −0.787722 0.616031i \(-0.788739\pi\)
0.787722 0.616031i \(-0.211261\pi\)
\(252\) 0 0
\(253\) −109.495 −0.432788
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 155.317 89.6724i 0.604347 0.348920i −0.166403 0.986058i \(-0.553215\pi\)
0.770750 + 0.637138i \(0.219882\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 37.3848 64.7523i 0.142147 0.246206i −0.786158 0.618026i \(-0.787933\pi\)
0.928305 + 0.371820i \(0.121266\pi\)
\(264\) 0 0
\(265\) 26.4979i 0.0999921i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 49.2164 + 28.4151i 0.182961 + 0.105632i 0.588683 0.808364i \(-0.299647\pi\)
−0.405722 + 0.913996i \(0.632980\pi\)
\(270\) 0 0
\(271\) −11.6951 + 6.75218i −0.0431554 + 0.0249158i −0.521422 0.853299i \(-0.674598\pi\)
0.478267 + 0.878214i \(0.341265\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 116.941 + 202.548i 0.425240 + 0.736538i
\(276\) 0 0
\(277\) 124.726 216.032i 0.450274 0.779897i −0.548129 0.836394i \(-0.684660\pi\)
0.998403 + 0.0564965i \(0.0179930\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −200.268 −0.712696 −0.356348 0.934353i \(-0.615978\pi\)
−0.356348 + 0.934353i \(0.615978\pi\)
\(282\) 0 0
\(283\) 59.5549 + 34.3840i 0.210441 + 0.121498i 0.601516 0.798860i \(-0.294563\pi\)
−0.391075 + 0.920359i \(0.627897\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −88.7881 153.786i −0.307225 0.532130i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 253.164i 0.864040i 0.901864 + 0.432020i \(0.142199\pi\)
−0.901864 + 0.432020i \(0.857801\pi\)
\(294\) 0 0
\(295\) −87.3927 −0.296247
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −156.336 + 90.2609i −0.522864 + 0.301876i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 51.1849 88.6549i 0.167819 0.290672i
\(306\) 0 0
\(307\) 529.913i 1.72610i −0.505116 0.863051i \(-0.668550\pi\)
0.505116 0.863051i \(-0.331450\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −26.4186 15.2528i −0.0849473 0.0490444i 0.456925 0.889505i \(-0.348951\pi\)
−0.541872 + 0.840461i \(0.682284\pi\)
\(312\) 0 0
\(313\) −7.44956 + 4.30101i −0.0238005 + 0.0137412i −0.511853 0.859073i \(-0.671041\pi\)
0.488053 + 0.872814i \(0.337707\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −240.076 415.824i −0.757337 1.31175i −0.944204 0.329362i \(-0.893167\pi\)
0.186866 0.982385i \(-0.440167\pi\)
\(318\) 0 0
\(319\) −224.391 + 388.656i −0.703419 + 1.21836i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −76.2176 −0.235968
\(324\) 0 0
\(325\) 333.935 + 192.797i 1.02749 + 0.593223i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 44.5098 + 77.0932i 0.134471 + 0.232910i 0.925395 0.379004i \(-0.123733\pi\)
−0.790924 + 0.611914i \(0.790400\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 76.2070i 0.227484i
\(336\) 0 0
\(337\) 495.701 1.47092 0.735461 0.677567i \(-0.236966\pi\)
0.735461 + 0.677567i \(0.236966\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −4.05356 + 2.34032i −0.0118873 + 0.00686312i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −30.1767 + 52.2676i −0.0869645 + 0.150627i −0.906227 0.422792i \(-0.861050\pi\)
0.819262 + 0.573419i \(0.194383\pi\)
\(348\) 0 0
\(349\) 72.2171i 0.206926i 0.994633 + 0.103463i \(0.0329923\pi\)
−0.994633 + 0.103463i \(0.967008\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 332.000 + 191.680i 0.940510 + 0.543004i 0.890120 0.455726i \(-0.150620\pi\)
0.0503901 + 0.998730i \(0.483954\pi\)
\(354\) 0 0
\(355\) −102.458 + 59.1540i −0.288613 + 0.166631i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −40.0224 69.3209i −0.111483 0.193094i 0.804885 0.593430i \(-0.202227\pi\)
−0.916368 + 0.400336i \(0.868893\pi\)
\(360\) 0 0
\(361\) −154.432 + 267.485i −0.427790 + 0.740954i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −42.9707 −0.117728
\(366\) 0 0
\(367\) 70.5218 + 40.7158i 0.192157 + 0.110942i 0.592992 0.805208i \(-0.297946\pi\)
−0.400835 + 0.916150i \(0.631280\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 266.126 + 460.944i 0.713475 + 1.23577i 0.963545 + 0.267547i \(0.0862131\pi\)
−0.250070 + 0.968228i \(0.580454\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 739.893i 1.96258i
\(378\) 0 0
\(379\) 440.518 1.16232 0.581159 0.813790i \(-0.302600\pi\)
0.581159 + 0.813790i \(0.302600\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −605.603 + 349.645i −1.58121 + 0.912911i −0.586525 + 0.809931i \(0.699504\pi\)
−0.994683 + 0.102980i \(0.967162\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 312.231 540.800i 0.802650 1.39023i −0.115216 0.993340i \(-0.536756\pi\)
0.917866 0.396891i \(-0.129911\pi\)
\(390\) 0 0
\(391\) 119.281i 0.305065i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −15.0452 8.68632i −0.0380890 0.0219907i
\(396\) 0 0
\(397\) 547.801 316.273i 1.37985 0.796658i 0.387711 0.921781i \(-0.373266\pi\)
0.992141 + 0.125123i \(0.0399325\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 233.671 + 404.731i 0.582721 + 1.00930i 0.995155 + 0.0983156i \(0.0313455\pi\)
−0.412434 + 0.910988i \(0.635321\pi\)
\(402\) 0 0
\(403\) −3.85842 + 6.68298i −0.00957425 + 0.0165831i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −24.0434 −0.0590746
\(408\) 0 0
\(409\) −280.848 162.148i −0.686671 0.396450i 0.115693 0.993285i \(-0.463091\pi\)
−0.802364 + 0.596835i \(0.796425\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −27.7009 47.9793i −0.0667491 0.115613i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 28.8211i 0.0687854i −0.999408 0.0343927i \(-0.989050\pi\)
0.999408 0.0343927i \(-0.0109497\pi\)
\(420\) 0 0
\(421\) −0.326830 −0.000776319 −0.000388160 1.00000i \(-0.500124\pi\)
−0.000388160 1.00000i \(0.500124\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 220.649 127.392i 0.519174 0.299745i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 195.669 338.908i 0.453988 0.786329i −0.544642 0.838669i \(-0.683334\pi\)
0.998629 + 0.0523393i \(0.0166677\pi\)
\(432\) 0 0
\(433\) 470.579i 1.08679i −0.839478 0.543394i \(-0.817139\pi\)
0.839478 0.543394i \(-0.182861\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −70.6607 40.7960i −0.161695 0.0933547i
\(438\) 0 0
\(439\) −116.395 + 67.2006i −0.265136 + 0.153077i −0.626675 0.779280i \(-0.715585\pi\)
0.361539 + 0.932357i \(0.382251\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 164.528 + 284.971i 0.371395 + 0.643275i 0.989780 0.142600i \(-0.0455462\pi\)
−0.618385 + 0.785875i \(0.712213\pi\)
\(444\) 0 0
\(445\) 33.0391 57.2254i 0.0742452 0.128596i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −130.592 −0.290851 −0.145426 0.989369i \(-0.546455\pi\)
−0.145426 + 0.989369i \(0.546455\pi\)
\(450\) 0 0
\(451\) 468.688 + 270.597i 1.03922 + 0.599994i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −158.141 273.908i −0.346041 0.599361i 0.639501 0.768790i \(-0.279141\pi\)
−0.985542 + 0.169429i \(0.945808\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 31.4853i 0.0682979i −0.999417 0.0341490i \(-0.989128\pi\)
0.999417 0.0341490i \(-0.0108721\pi\)
\(462\) 0 0
\(463\) 667.424 1.44152 0.720761 0.693184i \(-0.243793\pi\)
0.720761 + 0.693184i \(0.243793\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −748.502 + 432.148i −1.60279 + 0.925370i −0.611861 + 0.790965i \(0.709579\pi\)
−0.990927 + 0.134405i \(0.957088\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 293.982 509.192i 0.621526 1.07651i
\(474\) 0 0
\(475\) 174.281i 0.366907i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 542.476 + 313.198i 1.13252 + 0.653859i 0.944567 0.328320i \(-0.106482\pi\)
0.187950 + 0.982179i \(0.439816\pi\)
\(480\) 0 0
\(481\) −34.3289 + 19.8198i −0.0713698 + 0.0412054i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −47.4904 82.2557i −0.0979183 0.169599i
\(486\) 0 0
\(487\) 242.924 420.756i 0.498817 0.863976i −0.501182 0.865342i \(-0.667101\pi\)
0.999999 + 0.00136550i \(0.000434653\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 6.00051 0.0122210 0.00611049 0.999981i \(-0.498055\pi\)
0.00611049 + 0.999981i \(0.498055\pi\)
\(492\) 0 0
\(493\) 423.389 + 244.444i 0.858801 + 0.495829i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 343.775 + 595.437i 0.688929 + 1.19326i 0.972185 + 0.234215i \(0.0752520\pi\)
−0.283256 + 0.959044i \(0.591415\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 435.270i 0.865348i −0.901550 0.432674i \(-0.857570\pi\)
0.901550 0.432674i \(-0.142430\pi\)
\(504\) 0 0
\(505\) −101.196 −0.200388
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −418.381 + 241.553i −0.821967 + 0.474563i −0.851094 0.525013i \(-0.824061\pi\)
0.0291272 + 0.999576i \(0.490727\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −0.347070 + 0.601142i −0.000673921 + 0.00116727i
\(516\) 0 0
\(517\) 354.517i 0.685720i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 272.425 + 157.285i 0.522888 + 0.301890i 0.738116 0.674674i \(-0.235716\pi\)
−0.215227 + 0.976564i \(0.569049\pi\)
\(522\) 0 0
\(523\) −135.591 + 78.2835i −0.259256 + 0.149682i −0.623995 0.781428i \(-0.714492\pi\)
0.364739 + 0.931110i \(0.381158\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.54947 + 4.41581i 0.00483771 + 0.00837915i
\(528\) 0 0
\(529\) 200.654 347.543i 0.379309 0.656982i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 892.251 1.67402
\(534\) 0 0
\(535\) 12.3035 + 7.10344i 0.0229972 + 0.0132775i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −386.033 668.628i −0.713554 1.23591i −0.963515 0.267656i \(-0.913751\pi\)
0.249960 0.968256i \(-0.419582\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 50.3736i 0.0924285i
\(546\) 0 0
\(547\) 48.0113 0.0877721 0.0438860 0.999037i \(-0.486026\pi\)
0.0438860 + 0.999037i \(0.486026\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −289.613 + 167.208i −0.525613 + 0.303463i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 75.7027 131.121i 0.135911 0.235406i −0.790034 0.613063i \(-0.789937\pi\)
0.925945 + 0.377658i \(0.123270\pi\)
\(558\) 0 0
\(559\) 969.359i 1.73409i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 732.918 + 423.150i 1.30181 + 0.751599i 0.980714 0.195448i \(-0.0626161\pi\)
0.321094 + 0.947047i \(0.395949\pi\)
\(564\) 0 0
\(565\) −52.4972 + 30.3093i −0.0929153 + 0.0536447i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −10.7887 18.6865i −0.0189608 0.0328410i 0.856389 0.516331i \(-0.172702\pi\)
−0.875350 + 0.483490i \(0.839369\pi\)
\(570\) 0 0
\(571\) −141.623 + 245.298i −0.248026 + 0.429593i −0.962978 0.269580i \(-0.913115\pi\)
0.714952 + 0.699173i \(0.246448\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 272.749 0.474346
\(576\) 0 0
\(577\) 406.431 + 234.653i 0.704387 + 0.406678i 0.808979 0.587837i \(-0.200021\pi\)
−0.104593 + 0.994515i \(0.533354\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 138.191 + 239.353i 0.237034 + 0.410555i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 554.500i 0.944634i 0.881429 + 0.472317i \(0.156582\pi\)
−0.881429 + 0.472317i \(0.843418\pi\)
\(588\) 0 0
\(589\) −3.48785 −0.00592165
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 707.094 408.241i 1.19240 0.688433i 0.233550 0.972345i \(-0.424966\pi\)
0.958850 + 0.283912i \(0.0916323\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −434.341 + 752.301i −0.725111 + 1.25593i 0.233818 + 0.972280i \(0.424878\pi\)
−0.958928 + 0.283648i \(0.908455\pi\)
\(600\) 0 0
\(601\) 705.861i 1.17448i 0.809413 + 0.587239i \(0.199785\pi\)
−0.809413 + 0.587239i \(0.800215\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 21.8093 + 12.5916i 0.0360485 + 0.0208126i
\(606\) 0 0
\(607\) −464.026 + 267.906i −0.764458 + 0.441360i −0.830894 0.556431i \(-0.812170\pi\)
0.0664361 + 0.997791i \(0.478837\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 292.241 + 506.177i 0.478300 + 0.828440i
\(612\) 0 0
\(613\) 142.247 246.379i 0.232050 0.401923i −0.726361 0.687313i \(-0.758790\pi\)
0.958411 + 0.285390i \(0.0921232\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 766.565 1.24241 0.621203 0.783650i \(-0.286644\pi\)
0.621203 + 0.783650i \(0.286644\pi\)
\(618\) 0 0
\(619\) −470.257 271.503i −0.759705 0.438616i 0.0694849 0.997583i \(-0.477864\pi\)
−0.829190 + 0.558967i \(0.811198\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −280.508 485.854i −0.448813 0.777367i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 26.1920i 0.0416408i
\(630\) 0 0
\(631\) −967.080 −1.53261 −0.766307 0.642474i \(-0.777908\pi\)
−0.766307 + 0.642474i \(0.777908\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −189.679 + 109.511i −0.298708 + 0.172459i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 489.512 847.860i 0.763669 1.32271i −0.177278 0.984161i \(-0.556729\pi\)
0.940947 0.338553i \(-0.109937\pi\)
\(642\) 0 0
\(643\) 991.244i 1.54159i 0.637081 + 0.770797i \(0.280142\pi\)
−0.637081 + 0.770797i \(0.719858\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1067.31 + 616.209i 1.64962 + 0.952409i 0.977222 + 0.212222i \(0.0680698\pi\)
0.672400 + 0.740188i \(0.265264\pi\)
\(648\) 0 0
\(649\) −789.412 + 455.767i −1.21635 + 0.702260i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −390.535 676.426i −0.598062 1.03587i −0.993107 0.117212i \(-0.962604\pi\)
0.395045 0.918662i \(-0.370729\pi\)
\(654\) 0 0
\(655\) 105.223 182.252i 0.160646 0.278247i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 549.328 0.833578 0.416789 0.909003i \(-0.363155\pi\)
0.416789 + 0.909003i \(0.363155\pi\)
\(660\) 0 0
\(661\) 358.214 + 206.815i 0.541927 + 0.312882i 0.745860 0.666103i \(-0.232039\pi\)
−0.203932 + 0.978985i \(0.565372\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 261.680 + 453.244i 0.392324 + 0.679526i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1067.75i 1.59128i
\(672\) 0 0
\(673\) 553.924 0.823067 0.411533 0.911395i \(-0.364993\pi\)
0.411533 + 0.911395i \(0.364993\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −95.5972 + 55.1930i −0.141207 + 0.0815259i −0.568939 0.822380i \(-0.692646\pi\)
0.427732 + 0.903906i \(0.359313\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −97.8156 + 169.422i −0.143215 + 0.248055i −0.928705 0.370818i \(-0.879077\pi\)
0.785491 + 0.618873i \(0.212411\pi\)
\(684\) 0 0
\(685\) 235.728i 0.344129i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 394.615 + 227.831i 0.572736 + 0.330669i
\(690\) 0 0
\(691\) −574.132 + 331.475i −0.830871 + 0.479703i −0.854151 0.520025i \(-0.825922\pi\)
0.0232799 + 0.999729i \(0.492589\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 68.7647 + 119.104i 0.0989421 + 0.171373i
\(696\) 0 0
\(697\) 294.780 510.573i 0.422926 0.732530i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1236.29 1.76361 0.881807 0.471610i \(-0.156327\pi\)
0.881807 + 0.471610i \(0.156327\pi\)
\(702\) 0 0
\(703\) −15.5159 8.95812i −0.0220710 0.0127427i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 450.250 + 779.856i 0.635050 + 1.09994i 0.986505 + 0.163734i \(0.0523537\pi\)
−0.351455 + 0.936205i \(0.614313\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 5.45848i 0.00765566i
\(714\) 0 0
\(715\) −143.807 −0.201129
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 103.145 59.5509i 0.143456 0.0828246i −0.426554 0.904462i \(-0.640273\pi\)
0.570010 + 0.821638i \(0.306939\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 558.950 968.129i 0.770965 1.33535i
\(726\) 0 0
\(727\) 815.672i 1.12197i −0.827826 0.560985i \(-0.810423\pi\)
0.827826 0.560985i \(-0.189577\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −554.696 320.254i −0.758819 0.438104i
\(732\) 0 0
\(733\) −193.766 + 111.871i −0.264346 + 0.152620i −0.626316 0.779570i \(-0.715438\pi\)
0.361969 + 0.932190i \(0.382104\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 397.432 + 688.372i 0.539256 + 0.934019i
\(738\) 0 0
\(739\) 636.897 1103.14i 0.861836 1.49274i −0.00831892 0.999965i \(-0.502648\pi\)
0.870155 0.492778i \(-0.164019\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 204.339 0.275019 0.137510 0.990500i \(-0.456090\pi\)
0.137510 + 0.990500i \(0.456090\pi\)
\(744\) 0 0
\(745\) −197.486 114.019i −0.265082 0.153045i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 193.280 + 334.770i 0.257363 + 0.445766i 0.965535 0.260275i \(-0.0838131\pi\)
−0.708172 + 0.706040i \(0.750480\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 163.741i 0.216875i
\(756\) 0 0
\(757\) −1254.03 −1.65658 −0.828290 0.560300i \(-0.810686\pi\)
−0.828290 + 0.560300i \(0.810686\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 702.009 405.305i 0.922483 0.532596i 0.0380564 0.999276i \(-0.487883\pi\)
0.884426 + 0.466680i \(0.154550\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −751.410 + 1301.48i −0.979674 + 1.69684i
\(768\) 0 0
\(769\) 1338.07i 1.74002i 0.493036 + 0.870009i \(0.335887\pi\)
−0.493036 + 0.870009i \(0.664113\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −446.977 258.062i −0.578236 0.333845i 0.182196 0.983262i \(-0.441680\pi\)
−0.760432 + 0.649417i \(0.775013\pi\)
\(774\) 0 0
\(775\) 10.0973 5.82967i 0.0130287 0.00752215i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 201.639 + 349.250i 0.258844 + 0.448331i
\(780\) 0 0
\(781\) −616.995 + 1068.67i −0.790006 + 1.36833i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −192.913 −0.245749
\(786\) 0 0
\(787\) −1302.69 752.107i −1.65526 0.955663i −0.974860 0.222819i \(-0.928474\pi\)
−0.680396 0.732844i \(-0.738192\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −880.184 1524.52i −1.10994 1.92248i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 147.647i 0.185254i −0.995701 0.0926268i \(-0.970474\pi\)
0.995701 0.0926268i \(-0.0295263\pi\)
\(798\) 0 0
\(799\) 386.200 0.483354
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −388.151 + 224.099i −0.483375 + 0.279077i
\(804\) 0