Properties

Label 684.3.y.e.145.1
Level $684$
Weight $3$
Character 684.145
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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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 145.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 684.145
Dual form 684.3.y.e.217.1

$q$-expansion

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

Character values

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

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

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 0.992820 + 0.119615i \(0.0381661\pi\)
−0.392820 + 0.919615i \(0.628501\pi\)
\(6\) 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 −0.294388 0.955686i \(-0.595116\pi\)
−0.974842 + 0.222895i \(0.928449\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000 + 5.19615i 0.176471 + 0.305656i 0.940669 0.339325i \(-0.110199\pi\)
−0.764199 + 0.644981i \(0.776865\pi\)
\(18\) 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 0.477941 + 0.878392i \(0.341383\pi\)
−0.999680 + 0.0252868i \(0.991950\pi\)
\(24\) 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 −0.0438959 0.999036i \(-0.513977\pi\)
−0.887139 + 0.461503i \(0.847310\pi\)
\(30\) 0 0
\(31\) 29.4449i 0.949834i −0.880031 0.474917i \(-0.842478\pi\)
0.880031 0.474917i \(-0.157522\pi\)
\(32\) 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 \(-0.694410\pi\)
0.573489 0.819213i \(-0.305590\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 36.0000 20.7846i 0.878049 0.506942i 0.00803422 0.999968i \(-0.497443\pi\)
0.870015 + 0.493026i \(0.164109\pi\)
\(42\) 0 0
\(43\) 12.5000 + 21.6506i 0.290698 + 0.503503i 0.973975 0.226656i \(-0.0727793\pi\)
−0.683277 + 0.730159i \(0.739446\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −21.0000 + 36.3731i −0.446809 + 0.773895i −0.998176 0.0603673i \(-0.980773\pi\)
0.551368 + 0.834262i \(0.314106\pi\)
\(48\) 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 −0.105092 0.994462i \(-0.533514\pi\)
−0.913776 + 0.406219i \(0.866847\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −63.0000 + 36.3731i −1.06780 + 0.616493i −0.927579 0.373628i \(-0.878114\pi\)
−0.140218 + 0.990121i \(0.544780\pi\)
\(60\) 0 0
\(61\) −21.5000 + 37.2391i −0.352459 + 0.610477i −0.986680 0.162675i \(-0.947988\pi\)
0.634221 + 0.773152i \(0.281321\pi\)
\(62\) 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 0.821635 0.570014i \(-0.193062\pi\)
−0.0828290 + 0.996564i \(0.526396\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −54.0000 + 31.1769i −0.760563 + 0.439111i −0.829498 0.558510i \(-0.811373\pi\)
0.0689346 + 0.997621i \(0.478040\pi\)
\(72\) 0 0
\(73\) −5.50000 9.52628i −0.0753425 0.130497i 0.825893 0.563827i \(-0.190672\pi\)
−0.901235 + 0.433330i \(0.857338\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.50000 0.866025i 0.0189873 0.0109623i −0.490476 0.871455i \(-0.663177\pi\)
0.509464 + 0.860492i \(0.329844\pi\)
\(80\) 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 0.549709 0.835356i \(-0.314739\pi\)
−0.448585 + 0.893740i \(0.648072\pi\)
\(90\) 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 −0.954913 0.296887i \(-0.904052\pi\)
−0.220345 + 0.975422i \(0.570718\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 78.0000 135.100i 0.772277 1.33762i −0.164035 0.986455i \(-0.552451\pi\)
0.936312 0.351169i \(-0.114216\pi\)
\(102\) 0 0
\(103\) 36.3731i 0.353137i 0.984288 + 0.176568i \(0.0564996\pi\)
−0.984288 + 0.176568i \(0.943500\pi\)
\(104\) 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 −0.0266859 0.999644i \(-0.508495\pi\)
0.852374 + 0.522933i \(0.175162\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 155.885i 1.37951i 0.724043 + 0.689755i \(0.242282\pi\)
−0.724043 + 0.689755i \(0.757718\pi\)
\(114\) 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 0.351526 0.936178i \(-0.385663\pi\)
−0.634991 + 0.772520i \(0.718996\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 24.0000 + 41.5692i 0.183206 + 0.317322i 0.942971 0.332876i \(-0.108019\pi\)
−0.759764 + 0.650198i \(0.774686\pi\)
\(132\) 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 −0.999781 0.0209445i \(-0.993333\pi\)
0.518029 + 0.855363i \(0.326666\pi\)
\(138\) 0 0
\(139\) −0.500000 + 0.866025i −0.00359712 + 0.00623040i −0.867818 0.496882i \(-0.834478\pi\)
0.864221 + 0.503112i \(0.167812\pi\)
\(140\) 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 0.992204 + 0.124621i \(0.0397714\pi\)
−0.388178 + 0.921585i \(0.626895\pi\)
\(150\) 0 0
\(151\) 235.559i 1.55999i −0.625784 0.779996i \(-0.715221\pi\)
0.625784 0.779996i \(-0.284779\pi\)
\(152\) 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 0.962194 0.272364i \(-0.0878055\pi\)
−0.716971 + 0.697103i \(0.754472\pi\)
\(158\) 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 0.996964 + 0.0778587i \(0.0248083\pi\)
0.565910 + 0.824467i \(0.308525\pi\)
\(168\) 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 −0.473763 0.880652i \(-0.657105\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(174\) 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 0.306474 0.951879i \(-0.400851\pi\)
−0.671115 + 0.741354i \(0.734184\pi\)
\(182\) 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 −0.366250 0.930517i \(-0.619358\pi\)
0.622726 + 0.782440i \(0.286025\pi\)
\(194\) 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 −0.437168 0.899380i \(-0.644019\pi\)
0.997470 0.0710910i \(-0.0226481\pi\)
\(200\) 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 −0.992891 0.119027i \(-0.962022\pi\)
−0.393365 + 0.919382i \(0.628689\pi\)
\(212\) 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 −0.689404 + 0.724377i \(0.742128\pi\)
−0.972031 + 0.234853i \(0.924539\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 176.669i 0.778278i 0.921179 + 0.389139i \(0.127227\pi\)
−0.921179 + 0.389139i \(0.872773\pi\)
\(228\) 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 0.978147 0.207916i \(-0.0666680\pi\)
−0.669134 + 0.743142i \(0.733335\pi\)
\(234\) 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 0.633461 0.773775i \(-0.281634\pi\)
−0.353378 + 0.935481i \(0.614967\pi\)
\(242\) 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 −0.904822 0.425791i \(-0.859996\pi\)
0.821156 + 0.570703i \(0.193329\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −414.000 239.023i −1.61089 0.930051i −0.989163 0.146820i \(-0.953096\pi\)
−0.621732 0.783230i \(-0.713571\pi\)
\(258\) 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 0.975351 0.220660i \(-0.0708212\pi\)
−0.678773 + 0.734348i \(0.737488\pi\)
\(264\) 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 0.767295 0.641294i \(-0.221602\pi\)
0.939025 0.343850i \(-0.111731\pi\)
\(270\) 0 0
\(271\) 257.000 + 445.137i 0.948339 + 1.64257i 0.748923 + 0.662657i \(0.230571\pi\)
0.199417 + 0.979915i \(0.436095\pi\)
\(272\) 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 0.843720 0.536784i \(-0.180361\pi\)
−0.0430082 + 0.999075i \(0.513694\pi\)
\(282\) 0 0
\(283\) 113.000 + 195.722i 0.399293 + 0.691596i 0.993639 0.112613i \(-0.0359222\pi\)
−0.594346 + 0.804210i \(0.702589\pi\)
\(284\) 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 \(0.0169430\pi\)
−0.998584 + 0.0532029i \(0.983057\pi\)
\(294\) 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 −0.409467 0.912325i \(-0.634285\pi\)
0.585363 + 0.810771i \(0.300952\pi\)
\(308\) 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 0.0130534 + 0.999915i \(0.495845\pi\)
−0.872478 + 0.488653i \(0.837488\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −81.0000 46.7654i −0.255521 0.147525i 0.366769 0.930312i \(-0.380464\pi\)
−0.622289 + 0.782787i \(0.713797\pi\)
\(318\) 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 \(0.138405\pi\)
−0.906950 + 0.421239i \(0.861595\pi\)
\(332\) 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 0.272371 0.962192i \(-0.412192\pi\)
0.969469 + 0.245216i \(0.0788588\pi\)
\(338\) 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 −0.937860 0.347013i \(-0.887196\pi\)
0.168408 0.985717i \(-0.446137\pi\)
\(348\) 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 0.531363 0.847144i \(-0.321680\pi\)
−0.999330 + 0.0366021i \(0.988347\pi\)
\(360\) 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 −0.926588 0.376077i \(-0.877273\pi\)
0.788986 + 0.614411i \(0.210606\pi\)
\(368\) 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 \(-0.737799\pi\)
0.679491 0.733684i \(-0.262201\pi\)
\(374\) 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 \(-0.980349\pi\)
0.998095 0.0616957i \(-0.0196508\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −342.000 + 197.454i −0.892950 + 0.515545i −0.874906 0.484292i \(-0.839077\pi\)
−0.0180440 + 0.999837i \(0.505744\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −228.000 + 394.908i −0.586118 + 1.01519i 0.408617 + 0.912706i \(0.366011\pi\)
−0.994735 + 0.102481i \(0.967322\pi\)
\(390\) 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 0.681172 0.732123i \(-0.261470\pi\)
−0.974623 + 0.223851i \(0.928137\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −63.0000 + 36.3731i −0.157107 + 0.0907059i −0.576492 0.817102i \(-0.695579\pi\)
0.419385 + 0.907808i \(0.362246\pi\)
\(402\) 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 0.220802 0.975319i \(-0.429132\pi\)
−0.734250 + 0.678880i \(0.762466\pi\)
\(410\) 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 −0.621186 + 0.783663i \(0.713349\pi\)
−0.989265 + 0.146131i \(0.953318\pi\)
\(422\) 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 0.833809 0.552053i \(-0.186155\pi\)
−0.0611873 + 0.998126i \(0.519489\pi\)
\(432\) 0 0
\(433\) 253.500 + 146.358i 0.585450 + 0.338010i 0.763296 0.646048i \(-0.223580\pi\)
−0.177846 + 0.984058i \(0.556913\pi\)
\(434\) 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 −0.606920 0.794763i \(-0.707595\pi\)
0.384825 + 0.922990i \(0.374262\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 93.0000 161.081i 0.209932 0.363613i −0.741761 0.670665i \(-0.766009\pi\)
0.951693 + 0.307051i \(0.0993423\pi\)
\(444\) 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 \(0.0480708\pi\)
−0.988618 + 0.150445i \(0.951929\pi\)
\(450\) 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 0.781433 0.623990i \(-0.214489\pi\)
−0.931107 + 0.364746i \(0.881156\pi\)
\(462\) 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 0.231834 + 0.972755i \(0.425527\pi\)
−0.958348 + 0.285603i \(0.907806\pi\)
\(480\) 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 \(-0.929229\pi\)
0.975385 0.220507i \(-0.0707713\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −468.000 810.600i −0.953157 1.65092i −0.738531 0.674220i \(-0.764480\pi\)
−0.214626 0.976696i \(-0.568853\pi\)
\(492\) 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 0.999465 + 0.0327053i \(0.0104123\pi\)
−0.471409 + 0.881915i \(0.656254\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −195.000 + 337.750i −0.387674 + 0.671471i −0.992136 0.125163i \(-0.960055\pi\)
0.604462 + 0.796634i \(0.293388\pi\)
\(504\) 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 −0.0972946 0.995256i \(-0.531019\pi\)
−0.910564 + 0.413368i \(0.864352\pi\)
\(510\) 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 \(0.0317993\pi\)
−0.995014 + 0.0997342i \(0.968201\pi\)
\(522\) 0 0
\(523\) 118.500 + 68.4160i 0.226577 + 0.130815i 0.608992 0.793176i \(-0.291574\pi\)
−0.382415 + 0.923991i \(0.624907\pi\)
\(524\) 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 −0.759293 0.650749i \(-0.774455\pi\)
0.943212 + 0.332192i \(0.107788\pi\)
\(542\) 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 0.900299 0.435272i \(-0.143348\pi\)
0.0731928 + 0.997318i \(0.476681\pi\)
\(548\) 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 −0.337675 0.941263i \(-0.609641\pi\)
0.983995 0.178196i \(-0.0570260\pi\)
\(558\) 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 \(-0.892189\pi\)
0.943188 0.332258i \(-0.107811\pi\)
\(564\) 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 \(0.0823053\pi\)
−0.966757 + 0.255698i \(0.917695\pi\)
\(570\) 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 −0.997739 0.0672038i \(-0.978592\pi\)
0.440669 0.897669i \(-0.354741\pi\)
\(588\) 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 0.129037 + 0.991640i \(0.458811\pi\)
−0.923304 + 0.384070i \(0.874522\pi\)
\(594\) 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 0.204543 0.978858i \(-0.434429\pi\)
−0.745444 + 0.666568i \(0.767763\pi\)
\(600\) 0 0
\(601\) 646.055i 1.07497i −0.843274 0.537483i \(-0.819375\pi\)
0.843274 0.537483i \(-0.180625\pi\)
\(602\) 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 \(0.0954307\pi\)
−0.955394 + 0.295333i \(0.904569\pi\)
\(608\) 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 −0.988580 0.150695i \(-0.951849\pi\)
0.363784 0.931483i \(-0.381485\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −270.000 + 467.654i −0.437601 + 0.757948i −0.997504 0.0706107i \(-0.977505\pi\)
0.559903 + 0.828558i \(0.310839\pi\)
\(618\) 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 −0.631467 0.775402i \(-0.717547\pi\)
0.987252 + 0.159166i \(0.0508804\pi\)
\(632\) 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 −0.0432682 0.999063i \(-0.513777\pi\)
0.843580 + 0.537003i \(0.180444\pi\)
\(642\) 0 0
\(643\) −461.500 799.341i −0.717729 1.24314i −0.961897 0.273411i \(-0.911848\pi\)
0.244168 0.969733i \(-0.421485\pi\)
\(644\) 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 0.678861 0.734267i \(-0.262474\pi\)
−0.296463 + 0.955044i \(0.595807\pi\)
\(660\) 0 0
\(661\) −672.000 387.979i −1.01664 0.586958i −0.103512 0.994628i \(-0.533008\pi\)
−0.913129 + 0.407670i \(0.866341\pi\)
\(662\) 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 \(0.253201\pi\)
−0.699961 + 0.714181i \(0.746799\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 259.808i 0.383763i 0.981418 + 0.191882i \(0.0614589\pi\)
−0.981418 + 0.191882i \(0.938541\pi\)
\(678\) 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 \(0.311234\pi\)
−0.558872 + 0.829254i \(0.688766\pi\)
\(684\) 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 0.717895 0.696151i \(-0.245106\pi\)
−0.961832 + 0.273640i \(0.911772\pi\)
\(702\) 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 −0.991678 0.128743i \(-0.958906\pi\)
0.607334 + 0.794447i \(0.292239\pi\)
\(710\) 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 −0.951262 0.308382i \(-0.900212\pi\)
0.208564 0.978009i \(-0.433121\pi\)
\(720\) 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 0.902858 0.429938i \(-0.141465\pi\)
−0.823766 + 0.566929i \(0.808131\pi\)
\(728\) 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 −0.825033 0.565085i \(-0.808843\pi\)
−0.0768619 0.997042i \(-0.524490\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 738.000 426.084i 0.993271 0.573465i 0.0870202 0.996207i \(-0.472266\pi\)
0.906250 + 0.422742i \(0.138932\pi\)
\(744\) 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 0.253556 0.967321i \(-0.418400\pi\)
−0.710946 + 0.703246i \(0.751733\pi\)
\(752\) 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 0.777920 0.628363i \(-0.216275\pi\)
−0.933138 + 0.359517i \(0.882941\pi\)
\(758\) 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 0.344892 + 0.938642i \(0.387916\pi\)
−0.985334 + 0.170636i \(0.945418\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 873.000 + 504.027i 1.12937 + 0.652040i 0.943775 0.330587i \(-0.107247\pi\)
0.185591 + 0.982627i \(0.440580\pi\)
\(774\) 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 \(-0.648995\pi\)
0.451175 0.892435i \(-0.351005\pi\)
\(788\) 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 \(0.292737\pi\)
−0.606090 + 0.795396i \(0.707263\pi\)
\(798\) 0 0
\(799\) −252.000 −0.315394
\(800\) 0