Properties

Label 684.3.y.e.217.1
Level $684$
Weight $3$
Character 684.217
Analytic conductor $18.638$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 684.y (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.6376500822\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 228)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 217.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 684.217
Dual form 684.3.y.e.145.1

$q$-expansion

\(f(q)\) \(=\) \(q+(3.00000 - 5.19615i) q^{5} -5.00000 q^{7} +O(q^{10})\) \(q+(3.00000 - 5.19615i) q^{5} -5.00000 q^{7} +(-16.5000 + 9.52628i) q^{13} +(3.00000 - 5.19615i) q^{17} +(-13.0000 - 13.8564i) q^{19} +(-12.0000 - 20.7846i) q^{23} +(-5.50000 - 9.52628i) q^{25} +(-27.0000 + 15.5885i) q^{29} +29.4449i q^{31} +(-15.0000 + 25.9808i) q^{35} +60.6218i q^{37} +(36.0000 + 20.7846i) q^{41} +(12.5000 - 21.6506i) q^{43} +(-21.0000 - 36.3731i) q^{47} -24.0000 q^{49} +(-54.0000 + 31.1769i) q^{53} +(-63.0000 - 36.3731i) q^{59} +(-21.5000 - 37.2391i) q^{61} +114.315i q^{65} +(49.5000 - 28.5788i) q^{67} +(-54.0000 - 31.1769i) q^{71} +(-5.50000 + 9.52628i) q^{73} +(1.50000 + 0.866025i) q^{79} -126.000 q^{83} +(-18.0000 - 31.1769i) q^{85} +(9.00000 - 5.19615i) q^{89} +(82.5000 - 47.6314i) q^{91} +(-111.000 + 25.9808i) q^{95} +(-114.000 - 65.8179i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{5} - 10 q^{7} + O(q^{10}) \) \( 2 q + 6 q^{5} - 10 q^{7} - 33 q^{13} + 6 q^{17} - 26 q^{19} - 24 q^{23} - 11 q^{25} - 54 q^{29} - 30 q^{35} + 72 q^{41} + 25 q^{43} - 42 q^{47} - 48 q^{49} - 108 q^{53} - 126 q^{59} - 43 q^{61} + 99 q^{67} - 108 q^{71} - 11 q^{73} + 3 q^{79} - 252 q^{83} - 36 q^{85} + 18 q^{89} + 165 q^{91} - 222 q^{95} - 228 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(1\)

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
\(5\) 3.00000 5.19615i 0.600000 1.03923i −0.392820 0.919615i \(-0.628501\pi\)
0.992820 0.119615i \(-0.0381661\pi\)
\(6\) 0 0
\(7\) −5.00000 −0.714286 −0.357143 0.934050i \(-0.616249\pi\)
−0.357143 + 0.934050i \(0.616249\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −16.5000 + 9.52628i −1.26923 + 0.732791i −0.974842 0.222895i \(-0.928449\pi\)
−0.294388 + 0.955686i \(0.595116\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000 5.19615i 0.176471 0.305656i −0.764199 0.644981i \(-0.776865\pi\)
0.940669 + 0.339325i \(0.110199\pi\)
\(18\) 0 0
\(19\) −13.0000 13.8564i −0.684211 0.729285i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −12.0000 20.7846i −0.521739 0.903679i −0.999680 0.0252868i \(-0.991950\pi\)
0.477941 0.878392i \(-0.341383\pi\)
\(24\) 0 0
\(25\) −5.50000 9.52628i −0.220000 0.381051i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −27.0000 + 15.5885i −0.931034 + 0.537533i −0.887139 0.461503i \(-0.847310\pi\)
−0.0438959 + 0.999036i \(0.513977\pi\)
\(30\) 0 0
\(31\) 29.4449i 0.949834i 0.880031 + 0.474917i \(0.157522\pi\)
−0.880031 + 0.474917i \(0.842478\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −15.0000 + 25.9808i −0.428571 + 0.742307i
\(36\) 0 0
\(37\) 60.6218i 1.63843i 0.573489 + 0.819213i \(0.305590\pi\)
−0.573489 + 0.819213i \(0.694410\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 36.0000 + 20.7846i 0.878049 + 0.506942i 0.870015 0.493026i \(-0.164109\pi\)
0.00803422 + 0.999968i \(0.497443\pi\)
\(42\) 0 0
\(43\) 12.5000 21.6506i 0.290698 0.503503i −0.683277 0.730159i \(-0.739446\pi\)
0.973975 + 0.226656i \(0.0727793\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −21.0000 36.3731i −0.446809 0.773895i 0.551368 0.834262i \(-0.314106\pi\)
−0.998176 + 0.0603673i \(0.980773\pi\)
\(48\) 0 0
\(49\) −24.0000 −0.489796
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −54.0000 + 31.1769i −1.01887 + 0.588244i −0.913776 0.406219i \(-0.866847\pi\)
−0.105092 + 0.994462i \(0.533514\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −63.0000 36.3731i −1.06780 0.616493i −0.140218 0.990121i \(-0.544780\pi\)
−0.927579 + 0.373628i \(0.878114\pi\)
\(60\) 0 0
\(61\) −21.5000 37.2391i −0.352459 0.610477i 0.634221 0.773152i \(-0.281321\pi\)
−0.986680 + 0.162675i \(0.947988\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 114.315i 1.75870i
\(66\) 0 0
\(67\) 49.5000 28.5788i 0.738806 0.426550i −0.0828290 0.996564i \(-0.526396\pi\)
0.821635 + 0.570014i \(0.193062\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −54.0000 31.1769i −0.760563 0.439111i 0.0689346 0.997621i \(-0.478040\pi\)
−0.829498 + 0.558510i \(0.811373\pi\)
\(72\) 0 0
\(73\) −5.50000 + 9.52628i −0.0753425 + 0.130497i −0.901235 0.433330i \(-0.857338\pi\)
0.825893 + 0.563827i \(0.190672\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.50000 + 0.866025i 0.0189873 + 0.0109623i 0.509464 0.860492i \(-0.329844\pi\)
−0.490476 + 0.871455i \(0.663177\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −126.000 −1.51807 −0.759036 0.651048i \(-0.774329\pi\)
−0.759036 + 0.651048i \(0.774329\pi\)
\(84\) 0 0
\(85\) −18.0000 31.1769i −0.211765 0.366787i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.00000 5.19615i 0.101124 0.0583837i −0.448585 0.893740i \(-0.648072\pi\)
0.549709 + 0.835356i \(0.314739\pi\)
\(90\) 0 0
\(91\) 82.5000 47.6314i 0.906593 0.523422i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −111.000 + 25.9808i −1.16842 + 0.273482i
\(96\) 0 0
\(97\) −114.000 65.8179i −1.17526 0.678535i −0.220345 0.975422i \(-0.570718\pi\)
−0.954913 + 0.296887i \(0.904052\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 78.0000 + 135.100i 0.772277 + 1.33762i 0.936312 + 0.351169i \(0.114216\pi\)
−0.164035 + 0.986455i \(0.552451\pi\)
\(102\) 0 0
\(103\) 36.3731i 0.353137i −0.984288 0.176568i \(-0.943500\pi\)
0.984288 0.176568i \(-0.0564996\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 90.0000 + 51.9615i 0.825688 + 0.476711i 0.852374 0.522933i \(-0.175162\pi\)
−0.0266859 + 0.999644i \(0.508495\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 155.885i 1.37951i −0.724043 0.689755i \(-0.757718\pi\)
0.724043 0.689755i \(-0.242282\pi\)
\(114\) 0 0
\(115\) −144.000 −1.25217
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −15.0000 + 25.9808i −0.126050 + 0.218326i
\(120\) 0 0
\(121\) −121.000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 84.0000 0.672000
\(126\) 0 0
\(127\) −36.0000 + 20.7846i −0.283465 + 0.163658i −0.634991 0.772520i \(-0.718996\pi\)
0.351526 + 0.936178i \(0.385663\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 24.0000 41.5692i 0.183206 0.317322i −0.759764 0.650198i \(-0.774686\pi\)
0.942971 + 0.332876i \(0.108019\pi\)
\(132\) 0 0
\(133\) 65.0000 + 69.2820i 0.488722 + 0.520918i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −66.0000 114.315i −0.481752 0.834419i 0.518029 0.855363i \(-0.326666\pi\)
−0.999781 + 0.0209445i \(0.993333\pi\)
\(138\) 0 0
\(139\) −0.500000 0.866025i −0.00359712 0.00623040i 0.864221 0.503112i \(-0.167812\pi\)
−0.867818 + 0.496882i \(0.834478\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 187.061i 1.29008i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 90.0000 155.885i 0.604027 1.04621i −0.388178 0.921585i \(-0.626895\pi\)
0.992204 0.124621i \(-0.0397714\pi\)
\(150\) 0 0
\(151\) 235.559i 1.55999i 0.625784 + 0.779996i \(0.284779\pi\)
−0.625784 + 0.779996i \(0.715221\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 153.000 + 88.3346i 0.987097 + 0.569901i
\(156\) 0 0
\(157\) 38.5000 66.6840i 0.245223 0.424739i −0.716971 0.697103i \(-0.754472\pi\)
0.962194 + 0.272364i \(0.0878055\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 60.0000 + 103.923i 0.372671 + 0.645485i
\(162\) 0 0
\(163\) 145.000 0.889571 0.444785 0.895637i \(-0.353280\pi\)
0.444785 + 0.895637i \(0.353280\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 261.000 150.688i 1.56287 0.902326i 0.565910 0.824467i \(-0.308525\pi\)
0.996964 0.0778587i \(-0.0248083\pi\)
\(168\) 0 0
\(169\) 97.0000 168.009i 0.573964 0.994136i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 9.00000 + 5.19615i 0.0520231 + 0.0300356i 0.525786 0.850617i \(-0.323771\pi\)
−0.473763 + 0.880652i \(0.657105\pi\)
\(174\) 0 0
\(175\) 27.5000 + 47.6314i 0.157143 + 0.272179i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −66.0000 + 38.1051i −0.364641 + 0.210526i −0.671115 0.741354i \(-0.734184\pi\)
0.306474 + 0.951879i \(0.400851\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 315.000 + 181.865i 1.70270 + 0.983056i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 18.0000 0.0942408 0.0471204 0.998889i \(-0.484996\pi\)
0.0471204 + 0.998889i \(0.484996\pi\)
\(192\) 0 0
\(193\) 49.5000 + 28.5788i 0.256477 + 0.148077i 0.622726 0.782440i \(-0.286025\pi\)
−0.366250 + 0.930517i \(0.619358\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 90.0000 0.456853 0.228426 0.973561i \(-0.426642\pi\)
0.228426 + 0.973561i \(0.426642\pi\)
\(198\) 0 0
\(199\) 111.500 + 193.124i 0.560302 + 0.970471i 0.997470 + 0.0710910i \(0.0226481\pi\)
−0.437168 + 0.899380i \(0.644019\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 135.000 77.9423i 0.665025 0.383952i
\(204\) 0 0
\(205\) 216.000 124.708i 1.05366 0.608330i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −292.500 168.875i −1.38626 0.800355i −0.393365 0.919382i \(-0.628689\pi\)
−0.992891 + 0.119027i \(0.962022\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −75.0000 129.904i −0.348837 0.604204i
\(216\) 0 0
\(217\) 147.224i 0.678453i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 114.315i 0.517264i
\(222\) 0 0
\(223\) −370.500 213.908i −1.66143 0.959230i −0.972031 0.234853i \(-0.924539\pi\)
−0.689404 0.724377i \(-0.742128\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 176.669i 0.778278i −0.921179 0.389139i \(-0.872773\pi\)
0.921179 0.389139i \(-0.127227\pi\)
\(228\) 0 0
\(229\) 391.000 1.70742 0.853712 0.520746i \(-0.174346\pi\)
0.853712 + 0.520746i \(0.174346\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 72.0000 124.708i 0.309013 0.535226i −0.669134 0.743142i \(-0.733335\pi\)
0.978147 + 0.207916i \(0.0666680\pi\)
\(234\) 0 0
\(235\) −252.000 −1.07234
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −450.000 −1.88285 −0.941423 0.337229i \(-0.890510\pi\)
−0.941423 + 0.337229i \(0.890510\pi\)
\(240\) 0 0
\(241\) 67.5000 38.9711i 0.280083 0.161706i −0.353378 0.935481i \(-0.614967\pi\)
0.633461 + 0.773775i \(0.281634\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −72.0000 + 124.708i −0.293878 + 0.509011i
\(246\) 0 0
\(247\) 346.500 + 104.789i 1.40283 + 0.424247i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −21.0000 36.3731i −0.0836653 0.144913i 0.821156 0.570703i \(-0.193329\pi\)
−0.904822 + 0.425791i \(0.859996\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −414.000 + 239.023i −1.61089 + 0.930051i −0.621732 + 0.783230i \(0.713571\pi\)
−0.989163 + 0.146820i \(0.953096\pi\)
\(258\) 0 0
\(259\) 303.109i 1.17030i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 78.0000 135.100i 0.296578 0.513688i −0.678773 0.734348i \(-0.737488\pi\)
0.975351 + 0.220660i \(0.0708212\pi\)
\(264\) 0 0
\(265\) 374.123i 1.41178i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 459.000 + 265.004i 1.70632 + 0.985144i 0.939025 + 0.343850i \(0.111731\pi\)
0.767295 + 0.641294i \(0.221602\pi\)
\(270\) 0 0
\(271\) 257.000 445.137i 0.948339 1.64257i 0.199417 0.979915i \(-0.436095\pi\)
0.748923 0.662657i \(-0.230571\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 238.000 0.859206 0.429603 0.903018i \(-0.358654\pi\)
0.429603 + 0.903018i \(0.358654\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 225.000 129.904i 0.800712 0.462291i −0.0430082 0.999075i \(-0.513694\pi\)
0.843720 + 0.536784i \(0.180361\pi\)
\(282\) 0 0
\(283\) 113.000 195.722i 0.399293 0.691596i −0.594346 0.804210i \(-0.702589\pi\)
0.993639 + 0.112613i \(0.0359222\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −180.000 103.923i −0.627178 0.362101i
\(288\) 0 0
\(289\) 126.500 + 219.104i 0.437716 + 0.758147i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 31.1769i 0.106406i −0.998584 0.0532029i \(-0.983057\pi\)
0.998584 0.0532029i \(-0.0169430\pi\)
\(294\) 0 0
\(295\) −378.000 + 218.238i −1.28136 + 0.739791i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 396.000 + 228.631i 1.32441 + 0.764651i
\(300\) 0 0
\(301\) −62.5000 + 108.253i −0.207641 + 0.359645i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −258.000 −0.845902
\(306\) 0 0
\(307\) 54.0000 + 31.1769i 0.175896 + 0.101553i 0.585363 0.810771i \(-0.300952\pi\)
−0.409467 + 0.912325i \(0.634285\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 534.000 1.71704 0.858521 0.512779i \(-0.171384\pi\)
0.858521 + 0.512779i \(0.171384\pi\)
\(312\) 0 0
\(313\) −269.000 465.922i −0.859425 1.48857i −0.872478 0.488653i \(-0.837488\pi\)
0.0130534 0.999915i \(-0.495845\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −81.0000 + 46.7654i −0.255521 + 0.147525i −0.622289 0.782787i \(-0.713797\pi\)
0.366769 + 0.930312i \(0.380464\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −111.000 + 25.9808i −0.343653 + 0.0804358i
\(324\) 0 0
\(325\) 181.500 + 104.789i 0.558462 + 0.322428i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 105.000 + 181.865i 0.319149 + 0.552782i
\(330\) 0 0
\(331\) 278.860i 0.842478i −0.906950 0.421239i \(-0.861595\pi\)
0.906950 0.421239i \(-0.138405\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 342.946i 1.02372i
\(336\) 0 0
\(337\) 418.500 + 241.621i 1.24184 + 0.716977i 0.969469 0.245216i \(-0.0788588\pi\)
0.272371 + 0.962192i \(0.412192\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 365.000 1.06414
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −267.000 + 462.458i −0.769452 + 1.33273i 0.168408 + 0.985717i \(0.446137\pi\)
−0.937860 + 0.347013i \(0.887196\pi\)
\(348\) 0 0
\(349\) −187.000 −0.535817 −0.267908 0.963444i \(-0.586332\pi\)
−0.267908 + 0.963444i \(0.586332\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −438.000 −1.24079 −0.620397 0.784288i \(-0.713028\pi\)
−0.620397 + 0.784288i \(0.713028\pi\)
\(354\) 0 0
\(355\) −324.000 + 187.061i −0.912676 + 0.526934i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −168.000 + 290.985i −0.467967 + 0.810542i −0.999330 0.0366021i \(-0.988347\pi\)
0.531363 + 0.847144i \(0.321680\pi\)
\(360\) 0 0
\(361\) −23.0000 + 360.267i −0.0637119 + 0.997968i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 33.0000 + 57.1577i 0.0904110 + 0.156596i
\(366\) 0 0
\(367\) −50.5000 87.4686i −0.137602 0.238334i 0.788986 0.614411i \(-0.210606\pi\)
−0.926588 + 0.376077i \(0.877273\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 270.000 155.885i 0.727763 0.420174i
\(372\) 0 0
\(373\) 547.328i 1.46737i 0.679491 + 0.733684i \(0.262201\pi\)
−0.679491 + 0.733684i \(0.737799\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 297.000 514.419i 0.787798 1.36451i
\(378\) 0 0
\(379\) 46.7654i 0.123391i 0.998095 + 0.0616957i \(0.0196508\pi\)
−0.998095 + 0.0616957i \(0.980349\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −342.000 197.454i −0.892950 0.515545i −0.0180440 0.999837i \(-0.505744\pi\)
−0.874906 + 0.484292i \(0.839077\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −228.000 394.908i −0.586118 1.01519i −0.994735 0.102481i \(-0.967322\pi\)
0.408617 0.912706i \(-0.366011\pi\)
\(390\) 0 0
\(391\) −144.000 −0.368286
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 9.00000 5.19615i 0.0227848 0.0131548i
\(396\) 0 0
\(397\) −116.500 + 201.784i −0.293451 + 0.508272i −0.974623 0.223851i \(-0.928137\pi\)
0.681172 + 0.732123i \(0.261470\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −63.0000 36.3731i −0.157107 0.0907059i 0.419385 0.907808i \(-0.362246\pi\)
−0.576492 + 0.817102i \(0.695579\pi\)
\(402\) 0 0
\(403\) −280.500 485.840i −0.696030 1.20556i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −210.000 + 121.244i −0.513447 + 0.296439i −0.734250 0.678880i \(-0.762466\pi\)
0.220802 + 0.975319i \(0.429132\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 315.000 + 181.865i 0.762712 + 0.440352i
\(414\) 0 0
\(415\) −378.000 + 654.715i −0.910843 + 1.57763i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −84.0000 −0.200477 −0.100239 0.994963i \(-0.531961\pi\)
−0.100239 + 0.994963i \(0.531961\pi\)
\(420\) 0 0
\(421\) −678.000 391.443i −1.61045 0.929794i −0.989265 0.146131i \(-0.953318\pi\)
−0.621186 0.783663i \(-0.713349\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −66.0000 −0.155294
\(426\) 0 0
\(427\) 107.500 + 186.195i 0.251756 + 0.436055i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 333.000 192.258i 0.772622 0.446073i −0.0611873 0.998126i \(-0.519489\pi\)
0.833809 + 0.552053i \(0.186155\pi\)
\(432\) 0 0
\(433\) 253.500 146.358i 0.585450 0.338010i −0.177846 0.984058i \(-0.556913\pi\)
0.763296 + 0.646048i \(0.223580\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −132.000 + 436.477i −0.302059 + 0.998803i
\(438\) 0 0
\(439\) −97.5000 56.2917i −0.222096 0.128227i 0.384825 0.922990i \(-0.374262\pi\)
−0.606920 + 0.794763i \(0.707595\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 93.0000 + 161.081i 0.209932 + 0.363613i 0.951693 0.307051i \(-0.0993423\pi\)
−0.741761 + 0.670665i \(0.766009\pi\)
\(444\) 0 0
\(445\) 62.3538i 0.140121i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 135.100i 0.300891i −0.988618 0.150445i \(-0.951929\pi\)
0.988618 0.150445i \(-0.0480708\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 571.577i 1.25621i
\(456\) 0 0
\(457\) −565.000 −1.23632 −0.618162 0.786051i \(-0.712122\pi\)
−0.618162 + 0.786051i \(0.712122\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −69.0000 + 119.512i −0.149675 + 0.259244i −0.931107 0.364746i \(-0.881156\pi\)
0.781433 + 0.623990i \(0.214489\pi\)
\(462\) 0 0
\(463\) −139.000 −0.300216 −0.150108 0.988670i \(-0.547962\pi\)
−0.150108 + 0.988670i \(0.547962\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −888.000 −1.90150 −0.950749 0.309960i \(-0.899684\pi\)
−0.950749 + 0.309960i \(0.899684\pi\)
\(468\) 0 0
\(469\) −247.500 + 142.894i −0.527719 + 0.304678i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −60.5000 + 200.052i −0.127368 + 0.421162i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −348.000 602.754i −0.726514 1.25836i −0.958348 0.285603i \(-0.907806\pi\)
0.231834 0.972755i \(-0.425527\pi\)
\(480\) 0 0
\(481\) −577.500 1000.26i −1.20062 2.07954i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −684.000 + 394.908i −1.41031 + 0.814242i
\(486\) 0 0
\(487\) 214.774i 0.441015i 0.975385 + 0.220507i \(0.0707713\pi\)
−0.975385 + 0.220507i \(0.929229\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −468.000 + 810.600i −0.953157 + 1.65092i −0.214626 + 0.976696i \(0.568853\pi\)
−0.738531 + 0.674220i \(0.764480\pi\)
\(492\) 0 0
\(493\) 187.061i 0.379435i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 270.000 + 155.885i 0.543260 + 0.313651i
\(498\) 0 0
\(499\) 263.500 456.395i 0.528056 0.914620i −0.471409 0.881915i \(-0.656254\pi\)
0.999465 0.0327053i \(-0.0104123\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −195.000 337.750i −0.387674 0.671471i 0.604462 0.796634i \(-0.293388\pi\)
−0.992136 + 0.125163i \(0.960055\pi\)
\(504\) 0 0
\(505\) 936.000 1.85347
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −513.000 + 296.181i −1.00786 + 0.581887i −0.910564 0.413368i \(-0.864352\pi\)
−0.0972946 + 0.995256i \(0.531019\pi\)
\(510\) 0 0
\(511\) 27.5000 47.6314i 0.0538160 0.0932121i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −189.000 109.119i −0.366990 0.211882i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 103.923i 0.199468i −0.995014 0.0997342i \(-0.968201\pi\)
0.995014 0.0997342i \(-0.0317993\pi\)
\(522\) 0 0
\(523\) 118.500 68.4160i 0.226577 0.130815i −0.382415 0.923991i \(-0.624907\pi\)
0.608992 + 0.793176i \(0.291574\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 153.000 + 88.3346i 0.290323 + 0.167618i
\(528\) 0 0
\(529\) −23.5000 + 40.7032i −0.0444234 + 0.0769437i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −792.000 −1.48593
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 99.5000 + 172.339i 0.183919 + 0.318556i 0.943212 0.332192i \(-0.107788\pi\)
−0.759293 + 0.650749i \(0.774455\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 540.000 311.769i 0.990826 0.572053i
\(546\) 0 0
\(547\) 532.500 307.439i 0.973492 0.562046i 0.0731928 0.997318i \(-0.476681\pi\)
0.900299 + 0.435272i \(0.143348\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 567.000 + 171.473i 1.02904 + 0.311203i
\(552\) 0 0
\(553\) −7.50000 4.33013i −0.0135624 0.00783025i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 360.000 + 623.538i 0.646320 + 1.11946i 0.983995 + 0.178196i \(0.0570260\pi\)
−0.337675 + 0.941263i \(0.609641\pi\)
\(558\) 0 0
\(559\) 476.314i 0.852082i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 374.123i 0.664517i 0.943188 + 0.332258i \(0.107811\pi\)
−0.943188 + 0.332258i \(0.892189\pi\)
\(564\) 0 0
\(565\) −810.000 467.654i −1.43363 0.827706i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 290.985i 0.511396i −0.966757 0.255698i \(-0.917695\pi\)
0.966757 0.255698i \(-0.0823053\pi\)
\(570\) 0 0
\(571\) 95.0000 0.166375 0.0831874 0.996534i \(-0.473490\pi\)
0.0831874 + 0.996534i \(0.473490\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −132.000 + 228.631i −0.229565 + 0.397619i
\(576\) 0 0
\(577\) 874.000 1.51473 0.757366 0.652991i \(-0.226486\pi\)
0.757366 + 0.652991i \(0.226486\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 630.000 1.08434
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −327.000 + 566.381i −0.557070 + 0.964873i 0.440669 + 0.897669i \(0.354741\pi\)
−0.997739 + 0.0672038i \(0.978592\pi\)
\(588\) 0 0
\(589\) 408.000 382.783i 0.692699 0.649887i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −471.000 815.796i −0.794266 1.37571i −0.923304 0.384070i \(-0.874522\pi\)
0.129037 0.991640i \(-0.458811\pi\)
\(594\) 0 0
\(595\) 90.0000 + 155.885i 0.151261 + 0.261991i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −324.000 + 187.061i −0.540902 + 0.312290i −0.745444 0.666568i \(-0.767763\pi\)
0.204543 + 0.978858i \(0.434429\pi\)
\(600\) 0 0
\(601\) 646.055i 1.07497i 0.843274 + 0.537483i \(0.180625\pi\)
−0.843274 + 0.537483i \(0.819375\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −363.000 + 628.734i −0.600000 + 1.03923i
\(606\) 0 0
\(607\) 358.535i 0.590666i −0.955394 0.295333i \(-0.904569\pi\)
0.955394 0.295333i \(-0.0954307\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 693.000 + 400.104i 1.13421 + 0.654834i
\(612\) 0 0
\(613\) −383.000 + 663.375i −0.624796 + 1.08218i 0.363784 + 0.931483i \(0.381485\pi\)
−0.988580 + 0.150695i \(0.951849\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −270.000 467.654i −0.437601 0.757948i 0.559903 0.828558i \(-0.310839\pi\)
−0.997504 + 0.0706107i \(0.977505\pi\)
\(618\) 0 0
\(619\) 97.0000 0.156704 0.0783522 0.996926i \(-0.475034\pi\)
0.0783522 + 0.996926i \(0.475034\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −45.0000 + 25.9808i −0.0722311 + 0.0417027i
\(624\) 0 0
\(625\) 389.500 674.634i 0.623200 1.07941i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 315.000 + 181.865i 0.500795 + 0.289134i
\(630\) 0 0
\(631\) 224.500 + 388.845i 0.355784 + 0.616237i 0.987252 0.159166i \(-0.0508804\pi\)
−0.631467 + 0.775402i \(0.717547\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 249.415i 0.392780i
\(636\) 0 0
\(637\) 396.000 228.631i 0.621664 0.358918i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 513.000 + 296.181i 0.800312 + 0.462060i 0.843580 0.537003i \(-0.180444\pi\)
−0.0432682 + 0.999063i \(0.513777\pi\)
\(642\) 0 0
\(643\) −461.500 + 799.341i −0.717729 + 1.24314i 0.244168 + 0.969733i \(0.421485\pi\)
−0.961897 + 0.273411i \(0.911848\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −594.000 −0.918083 −0.459042 0.888415i \(-0.651807\pi\)
−0.459042 + 0.888415i \(0.651807\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −936.000 −1.43338 −0.716692 0.697390i \(-0.754345\pi\)
−0.716692 + 0.697390i \(0.754345\pi\)
\(654\) 0 0
\(655\) −144.000 249.415i −0.219847 0.380787i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 252.000 145.492i 0.382398 0.220777i −0.296463 0.955044i \(-0.595807\pi\)
0.678861 + 0.734267i \(0.262474\pi\)
\(660\) 0 0
\(661\) −672.000 + 387.979i −1.01664 + 0.586958i −0.913129 0.407670i \(-0.866341\pi\)
−0.103512 + 0.994628i \(0.533008\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 555.000 129.904i 0.834586 0.195344i
\(666\) 0 0
\(667\) 648.000 + 374.123i 0.971514 + 0.560904i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 961.288i 1.42836i −0.699961 0.714181i \(-0.746799\pi\)
0.699961 0.714181i \(-0.253201\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 259.808i 0.383763i −0.981418 0.191882i \(-0.938541\pi\)
0.981418 0.191882i \(-0.0614589\pi\)
\(678\) 0 0
\(679\) 570.000 + 329.090i 0.839470 + 0.484668i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1132.76i 1.65851i −0.558872 0.829254i \(-0.688766\pi\)
0.558872 0.829254i \(-0.311234\pi\)
\(684\) 0 0
\(685\) −792.000 −1.15620
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 594.000 1028.84i 0.862119 1.49323i
\(690\) 0 0
\(691\) −58.0000 −0.0839363 −0.0419682 0.999119i \(-0.513363\pi\)
−0.0419682 + 0.999119i \(0.513363\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.00000 −0.00863309
\(696\) 0 0
\(697\) 216.000 124.708i 0.309900 0.178921i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −171.000 + 296.181i −0.243937 + 0.422512i −0.961832 0.273640i \(-0.911772\pi\)
0.717895 + 0.696151i \(0.245106\pi\)
\(702\) 0 0
\(703\) 840.000 788.083i 1.19488 1.12103i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −390.000 675.500i −0.551627 0.955445i
\(708\) 0 0
\(709\) −272.500 471.984i −0.384344 0.665704i 0.607334 0.794447i \(-0.292239\pi\)
−0.991678 + 0.128743i \(0.958906\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 612.000 353.338i 0.858345 0.495566i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −534.000 + 924.915i −0.742698 + 1.28639i 0.208564 + 0.978009i \(0.433121\pi\)
−0.951262 + 0.308382i \(0.900212\pi\)
\(720\) 0 0
\(721\) 181.865i 0.252240i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 297.000 + 171.473i 0.409655 + 0.236515i
\(726\) 0 0
\(727\) 57.5000 99.5929i 0.0790922 0.136992i −0.823766 0.566929i \(-0.808131\pi\)
0.902858 + 0.429938i \(0.141465\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −75.0000 129.904i −0.102599 0.177707i
\(732\) 0 0
\(733\) −298.000 −0.406548 −0.203274 0.979122i \(-0.565158\pi\)
−0.203274 + 0.979122i \(0.565158\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −666.500 + 1154.41i −0.901894 + 1.56213i −0.0768619 + 0.997042i \(0.524490\pi\)
−0.825033 + 0.565085i \(0.808843\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 738.000 + 426.084i 0.993271 + 0.573465i 0.906250 0.422742i \(-0.138932\pi\)
0.0870202 + 0.996207i \(0.472266\pi\)
\(744\) 0 0
\(745\) −540.000 935.307i −0.724832 1.25545i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −343.500 + 198.320i −0.457390 + 0.264074i −0.710946 0.703246i \(-0.751733\pi\)
0.253556 + 0.967321i \(0.418400\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1224.00 + 706.677i 1.62119 + 0.935996i
\(756\) 0 0
\(757\) −117.500 + 203.516i −0.155218 + 0.268845i −0.933138 0.359517i \(-0.882941\pi\)
0.777920 + 0.628363i \(0.216275\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1158.00 1.52168 0.760841 0.648938i \(-0.224787\pi\)
0.760841 + 0.648938i \(0.224787\pi\)
\(762\) 0 0
\(763\) −450.000 259.808i −0.589777 0.340508i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1386.00 1.80704
\(768\) 0 0
\(769\) −492.500 853.035i −0.640442 1.10928i −0.985334 0.170636i \(-0.945418\pi\)
0.344892 0.938642i \(-0.387916\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 873.000 504.027i 1.12937 0.652040i 0.185591 0.982627i \(-0.440580\pi\)
0.943775 + 0.330587i \(0.107247\pi\)
\(774\) 0 0
\(775\) 280.500 161.947i 0.361935 0.208964i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −180.000 769.031i −0.231065 0.987202i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −231.000 400.104i −0.294268 0.509686i
\(786\) 0 0
\(787\) 1404.69i 1.78487i 0.451175 + 0.892435i \(0.351005\pi\)
−0.451175 + 0.892435i \(0.648995\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 779.423i 0.985364i
\(792\) 0 0
\(793\) 709.500 + 409.630i 0.894704 + 0.516557i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1267.86i 1.59079i −0.606090 0.795396i \(-0.707263\pi\)
0.606090 0.795396i \(-0.292737\pi\)
\(798\) 0 0
\(799\) −252.000 −0.315394
\(800\) 0 0
\(801\) 0 0