Properties

Label 684.3.y.a.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: \(x^{2} - x + 1\)
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.a.217.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.00000 - 1.73205i) q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+(-1.00000 - 1.73205i) q^{5} -1.00000 q^{7} -16.0000 q^{11} +(13.5000 + 7.79423i) q^{13} +(11.0000 + 19.0526i) q^{17} +19.0000 q^{19} +(20.0000 - 34.6410i) q^{23} +(10.5000 - 18.1865i) q^{25} +(-15.0000 - 8.66025i) q^{29} -50.2295i q^{31} +(1.00000 + 1.73205i) q^{35} +15.5885i q^{37} +(12.0000 - 6.92820i) q^{41} +(24.5000 + 42.4352i) q^{43} +(23.0000 - 39.8372i) q^{47} -48.0000 q^{49} +(42.0000 + 24.2487i) q^{53} +(16.0000 + 27.7128i) q^{55} +(57.0000 - 32.9090i) q^{59} +(48.5000 - 84.0045i) q^{61} -31.1769i q^{65} +(-22.5000 - 12.9904i) q^{67} +(42.0000 - 24.2487i) q^{71} +(-17.5000 - 30.3109i) q^{73} +16.0000 q^{77} +(-76.5000 + 44.1673i) q^{79} +146.000 q^{83} +(22.0000 - 38.1051i) q^{85} +(33.0000 + 19.0526i) q^{89} +(-13.5000 - 7.79423i) q^{91} +(-19.0000 - 32.9090i) q^{95} +(54.0000 - 31.1769i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{7} + O(q^{10}) \) \( 2 q - 2 q^{5} - 2 q^{7} - 32 q^{11} + 27 q^{13} + 22 q^{17} + 38 q^{19} + 40 q^{23} + 21 q^{25} - 30 q^{29} + 2 q^{35} + 24 q^{41} + 49 q^{43} + 46 q^{47} - 96 q^{49} + 84 q^{53} + 32 q^{55} + 114 q^{59} + 97 q^{61} - 45 q^{67} + 84 q^{71} - 35 q^{73} + 32 q^{77} - 153 q^{79} + 292 q^{83} + 44 q^{85} + 66 q^{89} - 27 q^{91} - 38 q^{95} + 108 q^{97} + O(q^{100}) \)

Character values

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

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

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
\(5\) −1.00000 1.73205i −0.200000 0.346410i 0.748528 0.663103i \(-0.230761\pi\)
−0.948528 + 0.316693i \(0.897428\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.142857 −0.0714286 0.997446i \(-0.522756\pi\)
−0.0714286 + 0.997446i \(0.522756\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −16.0000 −1.45455 −0.727273 0.686349i \(-0.759212\pi\)
−0.727273 + 0.686349i \(0.759212\pi\)
\(12\) 0 0
\(13\) 13.5000 + 7.79423i 1.03846 + 0.599556i 0.919397 0.393330i \(-0.128677\pi\)
0.119064 + 0.992887i \(0.462011\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 11.0000 + 19.0526i 0.647059 + 1.12074i 0.983822 + 0.179149i \(0.0573345\pi\)
−0.336763 + 0.941589i \(0.609332\pi\)
\(18\) 0 0
\(19\) 19.0000 1.00000
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 20.0000 34.6410i 0.869565 1.50613i 0.00712357 0.999975i \(-0.497732\pi\)
0.862442 0.506157i \(-0.168934\pi\)
\(24\) 0 0
\(25\) 10.5000 18.1865i 0.420000 0.727461i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −15.0000 8.66025i −0.517241 0.298629i 0.218564 0.975823i \(-0.429863\pi\)
−0.735805 + 0.677193i \(0.763196\pi\)
\(30\) 0 0
\(31\) 50.2295i 1.62031i −0.586219 0.810153i \(-0.699384\pi\)
0.586219 0.810153i \(-0.300616\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.00000 + 1.73205i 0.0285714 + 0.0494872i
\(36\) 0 0
\(37\) 15.5885i 0.421310i 0.977561 + 0.210655i \(0.0675596\pi\)
−0.977561 + 0.210655i \(0.932440\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 12.0000 6.92820i 0.292683 0.168981i −0.346468 0.938062i \(-0.612619\pi\)
0.639151 + 0.769081i \(0.279286\pi\)
\(42\) 0 0
\(43\) 24.5000 + 42.4352i 0.569767 + 0.986866i 0.996589 + 0.0825301i \(0.0263001\pi\)
−0.426821 + 0.904336i \(0.640367\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 23.0000 39.8372i 0.489362 0.847599i −0.510563 0.859840i \(-0.670563\pi\)
0.999925 + 0.0122408i \(0.00389646\pi\)
\(48\) 0 0
\(49\) −48.0000 −0.979592
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 42.0000 + 24.2487i 0.792453 + 0.457523i 0.840825 0.541307i \(-0.182070\pi\)
−0.0483725 + 0.998829i \(0.515403\pi\)
\(54\) 0 0
\(55\) 16.0000 + 27.7128i 0.290909 + 0.503869i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 57.0000 32.9090i 0.966102 0.557779i 0.0680561 0.997681i \(-0.478320\pi\)
0.898046 + 0.439902i \(0.144987\pi\)
\(60\) 0 0
\(61\) 48.5000 84.0045i 0.795082 1.37712i −0.127705 0.991812i \(-0.540761\pi\)
0.922787 0.385310i \(-0.125906\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 31.1769i 0.479645i
\(66\) 0 0
\(67\) −22.5000 12.9904i −0.335821 0.193886i 0.322602 0.946535i \(-0.395443\pi\)
−0.658422 + 0.752649i \(0.728776\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 42.0000 24.2487i 0.591549 0.341531i −0.174161 0.984717i \(-0.555721\pi\)
0.765710 + 0.643186i \(0.222388\pi\)
\(72\) 0 0
\(73\) −17.5000 30.3109i −0.239726 0.415218i 0.720910 0.693029i \(-0.243724\pi\)
−0.960636 + 0.277811i \(0.910391\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 16.0000 0.207792
\(78\) 0 0
\(79\) −76.5000 + 44.1673i −0.968354 + 0.559080i −0.898734 0.438494i \(-0.855512\pi\)
−0.0696203 + 0.997574i \(0.522179\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 146.000 1.75904 0.879518 0.475865i \(-0.157865\pi\)
0.879518 + 0.475865i \(0.157865\pi\)
\(84\) 0 0
\(85\) 22.0000 38.1051i 0.258824 0.448296i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 33.0000 + 19.0526i 0.370787 + 0.214074i 0.673802 0.738912i \(-0.264660\pi\)
−0.303015 + 0.952986i \(0.597993\pi\)
\(90\) 0 0
\(91\) −13.5000 7.79423i −0.148352 0.0856509i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −19.0000 32.9090i −0.200000 0.346410i
\(96\) 0 0
\(97\) 54.0000 31.1769i 0.556701 0.321411i −0.195119 0.980780i \(-0.562509\pi\)
0.751820 + 0.659368i \(0.229176\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 14.0000 24.2487i 0.138614 0.240086i −0.788358 0.615216i \(-0.789069\pi\)
0.926972 + 0.375130i \(0.122402\pi\)
\(102\) 0 0
\(103\) 126.440i 1.22757i 0.789473 + 0.613785i \(0.210354\pi\)
−0.789473 + 0.613785i \(0.789646\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 83.1384i 0.776995i −0.921450 0.388497i \(-0.872994\pi\)
0.921450 0.388497i \(-0.127006\pi\)
\(108\) 0 0
\(109\) −66.0000 + 38.1051i −0.605505 + 0.349588i −0.771204 0.636588i \(-0.780345\pi\)
0.165699 + 0.986176i \(0.447012\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 183.597i 1.62476i 0.583131 + 0.812378i \(0.301827\pi\)
−0.583131 + 0.812378i \(0.698173\pi\)
\(114\) 0 0
\(115\) −80.0000 −0.695652
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −11.0000 19.0526i −0.0924370 0.160106i
\(120\) 0 0
\(121\) 135.000 1.11570
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −92.0000 −0.736000
\(126\) 0 0
\(127\) 24.0000 + 13.8564i 0.188976 + 0.109106i 0.591503 0.806303i \(-0.298535\pi\)
−0.402527 + 0.915408i \(0.631868\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −64.0000 110.851i −0.488550 0.846193i 0.511364 0.859364i \(-0.329141\pi\)
−0.999913 + 0.0131717i \(0.995807\pi\)
\(132\) 0 0
\(133\) −19.0000 −0.142857
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −58.0000 + 100.459i −0.423358 + 0.733277i −0.996265 0.0863428i \(-0.972482\pi\)
0.572908 + 0.819620i \(0.305815\pi\)
\(138\) 0 0
\(139\) 87.5000 151.554i 0.629496 1.09032i −0.358156 0.933662i \(-0.616595\pi\)
0.987653 0.156658i \(-0.0500721\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −216.000 124.708i −1.51049 0.872082i
\(144\) 0 0
\(145\) 34.6410i 0.238904i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 74.0000 + 128.172i 0.496644 + 0.860213i 0.999993 0.00387051i \(-0.00123202\pi\)
−0.503348 + 0.864084i \(0.667899\pi\)
\(150\) 0 0
\(151\) 193.990i 1.28470i −0.766411 0.642350i \(-0.777960\pi\)
0.766411 0.642350i \(-0.222040\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −87.0000 + 50.2295i −0.561290 + 0.324061i
\(156\) 0 0
\(157\) 96.5000 + 167.143i 0.614650 + 1.06460i 0.990446 + 0.137902i \(0.0440359\pi\)
−0.375796 + 0.926702i \(0.622631\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −20.0000 + 34.6410i −0.124224 + 0.215162i
\(162\) 0 0
\(163\) 233.000 1.42945 0.714724 0.699407i \(-0.246552\pi\)
0.714724 + 0.699407i \(0.246552\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −63.0000 36.3731i −0.377246 0.217803i 0.299374 0.954136i \(-0.403222\pi\)
−0.676619 + 0.736333i \(0.736556\pi\)
\(168\) 0 0
\(169\) 37.0000 + 64.0859i 0.218935 + 0.379206i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −99.0000 + 57.1577i −0.572254 + 0.330391i −0.758049 0.652197i \(-0.773847\pi\)
0.185795 + 0.982589i \(0.440514\pi\)
\(174\) 0 0
\(175\) −10.5000 + 18.1865i −0.0600000 + 0.103923i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 83.1384i 0.464461i 0.972661 + 0.232230i \(0.0746023\pi\)
−0.972661 + 0.232230i \(0.925398\pi\)
\(180\) 0 0
\(181\) 42.0000 + 24.2487i 0.232044 + 0.133971i 0.611515 0.791233i \(-0.290561\pi\)
−0.379471 + 0.925204i \(0.623894\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 27.0000 15.5885i 0.145946 0.0842619i
\(186\) 0 0
\(187\) −176.000 304.841i −0.941176 1.63017i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −70.0000 −0.366492 −0.183246 0.983067i \(-0.558661\pi\)
−0.183246 + 0.983067i \(0.558661\pi\)
\(192\) 0 0
\(193\) −262.500 + 151.554i −1.36010 + 0.785256i −0.989637 0.143590i \(-0.954135\pi\)
−0.370466 + 0.928846i \(0.620802\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.00000 0.0101523 0.00507614 0.999987i \(-0.498384\pi\)
0.00507614 + 0.999987i \(0.498384\pi\)
\(198\) 0 0
\(199\) −98.5000 + 170.607i −0.494975 + 0.857322i −0.999983 0.00579283i \(-0.998156\pi\)
0.505008 + 0.863114i \(0.331489\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 15.0000 + 8.66025i 0.0738916 + 0.0426613i
\(204\) 0 0
\(205\) −24.0000 13.8564i −0.117073 0.0675922i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −304.000 −1.45455
\(210\) 0 0
\(211\) −316.500 + 182.731i −1.50000 + 0.866025i −0.500000 + 0.866025i \(0.666667\pi\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 49.0000 84.8705i 0.227907 0.394746i
\(216\) 0 0
\(217\) 50.2295i 0.231472i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 342.946i 1.55179i
\(222\) 0 0
\(223\) 211.500 122.110i 0.948430 0.547577i 0.0558375 0.998440i \(-0.482217\pi\)
0.892593 + 0.450863i \(0.148884\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 148.956i 0.656195i 0.944644 + 0.328098i \(0.106408\pi\)
−0.944644 + 0.328098i \(0.893592\pi\)
\(228\) 0 0
\(229\) 83.0000 0.362445 0.181223 0.983442i \(-0.441995\pi\)
0.181223 + 0.983442i \(0.441995\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −40.0000 69.2820i −0.171674 0.297348i 0.767331 0.641251i \(-0.221584\pi\)
−0.939005 + 0.343903i \(0.888251\pi\)
\(234\) 0 0
\(235\) −92.0000 −0.391489
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −58.0000 −0.242678 −0.121339 0.992611i \(-0.538719\pi\)
−0.121339 + 0.992611i \(0.538719\pi\)
\(240\) 0 0
\(241\) −244.500 141.162i −1.01452 0.585735i −0.102010 0.994783i \(-0.532527\pi\)
−0.912513 + 0.409048i \(0.865861\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 48.0000 + 83.1384i 0.195918 + 0.339341i
\(246\) 0 0
\(247\) 256.500 + 148.090i 1.03846 + 0.599556i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −133.000 + 230.363i −0.529880 + 0.917780i 0.469512 + 0.882926i \(0.344430\pi\)
−0.999392 + 0.0348538i \(0.988903\pi\)
\(252\) 0 0
\(253\) −320.000 + 554.256i −1.26482 + 2.19074i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 162.000 + 93.5307i 0.630350 + 0.363933i 0.780888 0.624671i \(-0.214767\pi\)
−0.150537 + 0.988604i \(0.548100\pi\)
\(258\) 0 0
\(259\) 15.5885i 0.0601871i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −82.0000 142.028i −0.311787 0.540031i 0.666962 0.745092i \(-0.267594\pi\)
−0.978749 + 0.205060i \(0.934261\pi\)
\(264\) 0 0
\(265\) 96.9948i 0.366018i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 447.000 258.076i 1.66171 0.959389i 0.689811 0.723989i \(-0.257693\pi\)
0.971899 0.235400i \(-0.0756399\pi\)
\(270\) 0 0
\(271\) −7.00000 12.1244i −0.0258303 0.0447393i 0.852821 0.522203i \(-0.174890\pi\)
−0.878652 + 0.477464i \(0.841556\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −168.000 + 290.985i −0.610909 + 1.05813i
\(276\) 0 0
\(277\) 206.000 0.743682 0.371841 0.928296i \(-0.378727\pi\)
0.371841 + 0.928296i \(0.378727\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −87.0000 50.2295i −0.309609 0.178753i 0.337143 0.941454i \(-0.390539\pi\)
−0.646751 + 0.762701i \(0.723873\pi\)
\(282\) 0 0
\(283\) −271.000 469.386i −0.957597 1.65861i −0.728310 0.685248i \(-0.759694\pi\)
−0.229288 0.973359i \(-0.573640\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −12.0000 + 6.92820i −0.0418118 + 0.0241401i
\(288\) 0 0
\(289\) −97.5000 + 168.875i −0.337370 + 0.584342i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 481.510i 1.64338i −0.569935 0.821690i \(-0.693032\pi\)
0.569935 0.821690i \(-0.306968\pi\)
\(294\) 0 0
\(295\) −114.000 65.8179i −0.386441 0.223112i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 540.000 311.769i 1.80602 1.04271i
\(300\) 0 0
\(301\) −24.5000 42.4352i −0.0813953 0.140981i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −194.000 −0.636066
\(306\) 0 0
\(307\) −138.000 + 79.6743i −0.449511 + 0.259526i −0.707624 0.706589i \(-0.750233\pi\)
0.258112 + 0.966115i \(0.416900\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −322.000 −1.03537 −0.517685 0.855571i \(-0.673206\pi\)
−0.517685 + 0.855571i \(0.673206\pi\)
\(312\) 0 0
\(313\) −13.0000 + 22.5167i −0.0415335 + 0.0719382i −0.886045 0.463599i \(-0.846558\pi\)
0.844511 + 0.535538i \(0.179891\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 291.000 + 168.009i 0.917981 + 0.529997i 0.882990 0.469391i \(-0.155527\pi\)
0.0349907 + 0.999388i \(0.488860\pi\)
\(318\) 0 0
\(319\) 240.000 + 138.564i 0.752351 + 0.434370i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 209.000 + 361.999i 0.647059 + 1.12074i
\(324\) 0 0
\(325\) 283.500 163.679i 0.872308 0.503627i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −23.0000 + 39.8372i −0.0699088 + 0.121086i
\(330\) 0 0
\(331\) 219.970i 0.664563i −0.943180 0.332282i \(-0.892182\pi\)
0.943180 0.332282i \(-0.107818\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 51.9615i 0.155109i
\(336\) 0 0
\(337\) −505.500 + 291.851i −1.50000 + 0.866025i −0.500000 + 0.866025i \(0.666667\pi\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 803.672i 2.35681i
\(342\) 0 0
\(343\) 97.0000 0.282799
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −235.000 407.032i −0.677233 1.17300i −0.975811 0.218617i \(-0.929845\pi\)
0.298577 0.954385i \(-0.403488\pi\)
\(348\) 0 0
\(349\) −295.000 −0.845272 −0.422636 0.906299i \(-0.638895\pi\)
−0.422636 + 0.906299i \(0.638895\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −262.000 −0.742210 −0.371105 0.928591i \(-0.621021\pi\)
−0.371105 + 0.928591i \(0.621021\pi\)
\(354\) 0 0
\(355\) −84.0000 48.4974i −0.236620 0.136612i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −232.000 401.836i −0.646240 1.11932i −0.984014 0.178093i \(-0.943007\pi\)
0.337774 0.941227i \(-0.390326\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −35.0000 + 60.6218i −0.0958904 + 0.166087i
\(366\) 0 0
\(367\) −116.500 + 201.784i −0.317439 + 0.549820i −0.979953 0.199229i \(-0.936156\pi\)
0.662514 + 0.749049i \(0.269489\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −42.0000 24.2487i −0.113208 0.0653604i
\(372\) 0 0
\(373\) 533.472i 1.43022i −0.699013 0.715109i \(-0.746377\pi\)
0.699013 0.715109i \(-0.253623\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −135.000 233.827i −0.358090 0.620230i
\(378\) 0 0
\(379\) 161.081i 0.425015i 0.977159 + 0.212508i \(0.0681630\pi\)
−0.977159 + 0.212508i \(0.931837\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 498.000 287.520i 1.30026 0.750706i 0.319812 0.947481i \(-0.396380\pi\)
0.980449 + 0.196775i \(0.0630468\pi\)
\(384\) 0 0
\(385\) −16.0000 27.7128i −0.0415584 0.0719813i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −76.0000 + 131.636i −0.195373 + 0.338396i −0.947023 0.321167i \(-0.895925\pi\)
0.751650 + 0.659562i \(0.229258\pi\)
\(390\) 0 0
\(391\) 880.000 2.25064
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 153.000 + 88.3346i 0.387342 + 0.223632i
\(396\) 0 0
\(397\) 249.500 + 432.147i 0.628463 + 1.08853i 0.987860 + 0.155346i \(0.0496492\pi\)
−0.359397 + 0.933185i \(0.617017\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −87.0000 + 50.2295i −0.216958 + 0.125261i −0.604541 0.796574i \(-0.706643\pi\)
0.387583 + 0.921835i \(0.373310\pi\)
\(402\) 0 0
\(403\) 391.500 678.098i 0.971464 1.68263i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 249.415i 0.612814i
\(408\) 0 0
\(409\) 342.000 + 197.454i 0.836186 + 0.482772i 0.855966 0.517032i \(-0.172963\pi\)
−0.0197801 + 0.999804i \(0.506297\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −57.0000 + 32.9090i −0.138015 + 0.0796827i
\(414\) 0 0
\(415\) −146.000 252.879i −0.351807 0.609348i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −628.000 −1.49881 −0.749403 0.662114i \(-0.769660\pi\)
−0.749403 + 0.662114i \(0.769660\pi\)
\(420\) 0 0
\(421\) 558.000 322.161i 1.32542 0.765229i 0.340829 0.940125i \(-0.389292\pi\)
0.984587 + 0.174896i \(0.0559590\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 462.000 1.08706
\(426\) 0 0
\(427\) −48.5000 + 84.0045i −0.113583 + 0.196732i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 393.000 + 226.899i 0.911833 + 0.526447i 0.881020 0.473078i \(-0.156857\pi\)
0.0308125 + 0.999525i \(0.490191\pi\)
\(432\) 0 0
\(433\) 433.500 + 250.281i 1.00115 + 0.578017i 0.908590 0.417690i \(-0.137160\pi\)
0.0925650 + 0.995707i \(0.470493\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 380.000 658.179i 0.869565 1.50613i
\(438\) 0 0
\(439\) 292.500 168.875i 0.666287 0.384681i −0.128381 0.991725i \(-0.540978\pi\)
0.794668 + 0.607044i \(0.207645\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 365.000 632.199i 0.823928 1.42708i −0.0788082 0.996890i \(-0.525111\pi\)
0.902736 0.430195i \(-0.141555\pi\)
\(444\) 0 0
\(445\) 76.2102i 0.171259i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 446.869i 0.995254i −0.867391 0.497627i \(-0.834205\pi\)
0.867391 0.497627i \(-0.165795\pi\)
\(450\) 0 0
\(451\) −192.000 + 110.851i −0.425721 + 0.245790i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 31.1769i 0.0685207i
\(456\) 0 0
\(457\) −397.000 −0.868709 −0.434354 0.900742i \(-0.643023\pi\)
−0.434354 + 0.900742i \(0.643023\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 287.000 + 497.099i 0.622560 + 1.07830i 0.989007 + 0.147866i \(0.0472405\pi\)
−0.366448 + 0.930439i \(0.619426\pi\)
\(462\) 0 0
\(463\) −367.000 −0.792657 −0.396328 0.918109i \(-0.629716\pi\)
−0.396328 + 0.918109i \(0.629716\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −424.000 −0.907923 −0.453961 0.891021i \(-0.649990\pi\)
−0.453961 + 0.891021i \(0.649990\pi\)
\(468\) 0 0
\(469\) 22.5000 + 12.9904i 0.0479744 + 0.0276980i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −392.000 678.964i −0.828753 1.43544i
\(474\) 0 0
\(475\) 199.500 345.544i 0.420000 0.727461i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −364.000 + 630.466i −0.759916 + 1.31621i 0.182976 + 0.983117i \(0.441427\pi\)
−0.942893 + 0.333097i \(0.891906\pi\)
\(480\) 0 0
\(481\) −121.500 + 210.444i −0.252599 + 0.437514i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −108.000 62.3538i −0.222680 0.128565i
\(486\) 0 0
\(487\) 270.200i 0.554825i 0.960751 + 0.277413i \(0.0894769\pi\)
−0.960751 + 0.277413i \(0.910523\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 380.000 + 658.179i 0.773931 + 1.34049i 0.935393 + 0.353609i \(0.115046\pi\)
−0.161463 + 0.986879i \(0.551621\pi\)
\(492\) 0 0
\(493\) 381.051i 0.772923i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −42.0000 + 24.2487i −0.0845070 + 0.0487902i
\(498\) 0 0
\(499\) 231.500 + 400.970i 0.463928 + 0.803547i 0.999152 0.0411632i \(-0.0131064\pi\)
−0.535225 + 0.844710i \(0.679773\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 65.0000 112.583i 0.129225 0.223824i −0.794152 0.607720i \(-0.792084\pi\)
0.923376 + 0.383896i \(0.125418\pi\)
\(504\) 0 0
\(505\) −56.0000 −0.110891
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 459.000 + 265.004i 0.901768 + 0.520636i 0.877773 0.479076i \(-0.159028\pi\)
0.0239947 + 0.999712i \(0.492362\pi\)
\(510\) 0 0
\(511\) 17.5000 + 30.3109i 0.0342466 + 0.0593168i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 219.000 126.440i 0.425243 0.245514i
\(516\) 0 0
\(517\) −368.000 + 637.395i −0.711799 + 1.23287i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 602.754i 1.15692i 0.815712 + 0.578458i \(0.196345\pi\)
−0.815712 + 0.578458i \(0.803655\pi\)
\(522\) 0 0
\(523\) −277.500 160.215i −0.530593 0.306338i 0.210665 0.977558i \(-0.432437\pi\)
−0.741258 + 0.671220i \(0.765770\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 957.000 552.524i 1.81594 1.04843i
\(528\) 0 0
\(529\) −535.500 927.513i −1.01229 1.75333i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 216.000 0.405253
\(534\) 0 0
\(535\) −144.000 + 83.1384i −0.269159 + 0.155399i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 768.000 1.42486
\(540\) 0 0
\(541\) −182.500 + 316.099i −0.337338 + 0.584287i −0.983931 0.178548i \(-0.942860\pi\)
0.646593 + 0.762835i \(0.276193\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 132.000 + 76.2102i 0.242202 + 0.139835i
\(546\) 0 0
\(547\) −199.500 115.181i −0.364717 0.210569i 0.306431 0.951893i \(-0.400865\pi\)
−0.671148 + 0.741324i \(0.734198\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −285.000 164.545i −0.517241 0.298629i
\(552\) 0 0
\(553\) 76.5000 44.1673i 0.138336 0.0798685i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 320.000 554.256i 0.574506 0.995074i −0.421589 0.906787i \(-0.638527\pi\)
0.996095 0.0882870i \(-0.0281392\pi\)
\(558\) 0 0
\(559\) 763.834i 1.36643i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 96.9948i 0.172282i −0.996283 0.0861411i \(-0.972546\pi\)
0.996283 0.0861411i \(-0.0274536\pi\)
\(564\) 0 0
\(565\) 318.000 183.597i 0.562832 0.324951i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 152.420i 0.267874i −0.990990 0.133937i \(-0.957238\pi\)
0.990990 0.133937i \(-0.0427620\pi\)
\(570\) 0 0
\(571\) 335.000 0.586690 0.293345 0.956007i \(-0.405232\pi\)
0.293345 + 0.956007i \(0.405232\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −420.000 727.461i −0.730435 1.26515i
\(576\) 0 0
\(577\) 746.000 1.29289 0.646447 0.762959i \(-0.276254\pi\)
0.646447 + 0.762959i \(0.276254\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −146.000 −0.251291
\(582\) 0 0
\(583\) −672.000 387.979i −1.15266 0.665488i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 89.0000 + 154.153i 0.151618 + 0.262611i 0.931823 0.362914i \(-0.118218\pi\)
−0.780204 + 0.625525i \(0.784885\pi\)
\(588\) 0 0
\(589\) 954.360i 1.62031i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 137.000 237.291i 0.231029 0.400153i −0.727082 0.686550i \(-0.759124\pi\)
0.958111 + 0.286397i \(0.0924575\pi\)
\(594\) 0 0
\(595\) −22.0000 + 38.1051i −0.0369748 + 0.0640422i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 252.000 + 145.492i 0.420701 + 0.242892i 0.695377 0.718645i \(-0.255237\pi\)
−0.274676 + 0.961537i \(0.588571\pi\)
\(600\) 0 0
\(601\) 1020.18i 1.69747i −0.528820 0.848734i \(-0.677366\pi\)
0.528820 0.848734i \(-0.322634\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −135.000 233.827i −0.223140 0.386491i
\(606\) 0 0
\(607\) 157.617i 0.259665i 0.991536 + 0.129832i \(0.0414440\pi\)
−0.991536 + 0.129832i \(0.958556\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 621.000 358.535i 1.01637 0.586800i
\(612\) 0 0
\(613\) 401.000 + 694.552i 0.654160 + 1.13304i 0.982104 + 0.188341i \(0.0603110\pi\)
−0.327944 + 0.944697i \(0.606356\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −382.000 + 661.643i −0.619125 + 1.07236i 0.370521 + 0.928824i \(0.379179\pi\)
−0.989646 + 0.143531i \(0.954154\pi\)
\(618\) 0 0
\(619\) −127.000 −0.205170 −0.102585 0.994724i \(-0.532711\pi\)
−0.102585 + 0.994724i \(0.532711\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −33.0000 19.0526i −0.0529695 0.0305820i
\(624\) 0 0
\(625\) −170.500 295.315i −0.272800 0.472503i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −297.000 + 171.473i −0.472178 + 0.272612i
\(630\) 0 0
\(631\) −161.500 + 279.726i −0.255943 + 0.443306i −0.965151 0.261693i \(-0.915719\pi\)
0.709208 + 0.704999i \(0.249053\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 55.4256i 0.0872845i
\(636\) 0 0
\(637\) −648.000 374.123i −1.01727 0.587320i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 873.000 504.027i 1.36193 0.786313i 0.372053 0.928211i \(-0.378654\pi\)
0.989881 + 0.141898i \(0.0453206\pi\)
\(642\) 0 0
\(643\) 222.500 + 385.381i 0.346034 + 0.599349i 0.985541 0.169437i \(-0.0541948\pi\)
−0.639507 + 0.768785i \(0.720861\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −106.000 −0.163833 −0.0819165 0.996639i \(-0.526104\pi\)
−0.0819165 + 0.996639i \(0.526104\pi\)
\(648\) 0 0
\(649\) −912.000 + 526.543i −1.40524 + 0.811315i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −328.000 −0.502297 −0.251149 0.967949i \(-0.580808\pi\)
−0.251149 + 0.967949i \(0.580808\pi\)
\(654\) 0 0
\(655\) −128.000 + 221.703i −0.195420 + 0.338477i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −348.000 200.918i −0.528073 0.304883i 0.212158 0.977235i \(-0.431951\pi\)
−0.740231 + 0.672352i \(0.765284\pi\)
\(660\) 0 0
\(661\) 828.000 + 478.046i 1.25265 + 0.723216i 0.971635 0.236488i \(-0.0759963\pi\)
0.281013 + 0.959704i \(0.409330\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 19.0000 + 32.9090i 0.0285714 + 0.0494872i
\(666\) 0 0
\(667\) −600.000 + 346.410i −0.899550 + 0.519356i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −776.000 + 1344.07i −1.15648 + 2.00309i
\(672\) 0 0
\(673\) 521.347i 0.774662i −0.921941 0.387331i \(-0.873397\pi\)
0.921941 0.387331i \(-0.126603\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 439.941i 0.649839i 0.945742 + 0.324919i \(0.105337\pi\)
−0.945742 + 0.324919i \(0.894663\pi\)
\(678\) 0 0
\(679\) −54.0000 + 31.1769i −0.0795287 + 0.0459159i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 717.069i 1.04988i 0.851139 + 0.524941i \(0.175913\pi\)
−0.851139 + 0.524941i \(0.824087\pi\)
\(684\) 0 0
\(685\) 232.000 0.338686
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 378.000 + 654.715i 0.548621 + 0.950240i
\(690\) 0 0
\(691\) 422.000 0.610709 0.305355 0.952239i \(-0.401225\pi\)
0.305355 + 0.952239i \(0.401225\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −350.000 −0.503597
\(696\) 0 0
\(697\) 264.000 + 152.420i 0.378766 + 0.218681i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 185.000 + 320.429i 0.263909 + 0.457103i 0.967277 0.253722i \(-0.0816549\pi\)
−0.703368 + 0.710825i \(0.748322\pi\)
\(702\) 0 0
\(703\) 296.181i 0.421310i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −14.0000 + 24.2487i −0.0198020 + 0.0342980i
\(708\) 0 0
\(709\) 45.5000 78.8083i 0.0641749 0.111154i −0.832153 0.554546i \(-0.812892\pi\)
0.896328 + 0.443392i \(0.146225\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1740.00 1004.59i −2.44039 1.40896i
\(714\) 0 0
\(715\) 498.831i 0.697665i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −502.000 869.490i −0.698192 1.20930i −0.969093 0.246697i \(-0.920655\pi\)
0.270901 0.962607i \(-0.412678\pi\)
\(720\) 0 0
\(721\) 126.440i 0.175367i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −315.000 + 181.865i −0.434483 + 0.250849i
\(726\) 0 0
\(727\) 147.500 + 255.477i 0.202889 + 0.351413i 0.949458 0.313894i \(-0.101634\pi\)
−0.746569 + 0.665308i \(0.768300\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −539.000 + 933.575i −0.737346 + 1.27712i
\(732\) 0 0
\(733\) −250.000 −0.341064 −0.170532 0.985352i \(-0.554549\pi\)
−0.170532 + 0.985352i \(0.554549\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 360.000 + 207.846i 0.488467 + 0.282016i
\(738\) 0 0
\(739\) 285.500 + 494.501i 0.386333 + 0.669148i 0.991953 0.126606i \(-0.0404083\pi\)
−0.605620 + 0.795754i \(0.707075\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −6.00000 + 3.46410i −0.00807537 + 0.00466232i −0.504032 0.863685i \(-0.668151\pi\)
0.495957 + 0.868347i \(0.334817\pi\)
\(744\) 0 0
\(745\) 148.000 256.344i 0.198658 0.344085i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 83.1384i 0.110999i
\(750\) 0 0
\(751\) −841.500 485.840i −1.12051 0.646924i −0.178976 0.983853i \(-0.557278\pi\)
−0.941530 + 0.336929i \(0.890612\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −336.000 + 193.990i −0.445033 + 0.256940i
\(756\) 0 0
\(757\) −119.500 206.980i −0.157860 0.273421i 0.776237 0.630442i \(-0.217126\pi\)
−0.934097 + 0.357020i \(0.883793\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1402.00 −1.84231 −0.921156 0.389193i \(-0.872754\pi\)
−0.921156 + 0.389193i \(0.872754\pi\)
\(762\) 0 0
\(763\) 66.0000 38.1051i 0.0865007 0.0499412i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1026.00 1.33768
\(768\) 0 0
\(769\) 51.5000 89.2006i 0.0669701 0.115996i −0.830596 0.556875i \(-0.812000\pi\)
0.897566 + 0.440880i \(0.145333\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 69.0000 + 39.8372i 0.0892626 + 0.0515358i 0.543967 0.839107i \(-0.316922\pi\)
−0.454704 + 0.890642i \(0.650255\pi\)
\(774\) 0 0
\(775\) −913.500 527.409i −1.17871 0.680528i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 228.000 131.636i 0.292683 0.168981i
\(780\) 0 0
\(781\) −672.000 + 387.979i −0.860435 + 0.496773i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 193.000 334.286i 0.245860 0.425842i
\(786\) 0 0
\(787\) 188.794i 0.239890i 0.992781 + 0.119945i \(0.0382718\pi\)
−0.992781 + 0.119945i \(0.961728\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 183.597i 0.232108i
\(792\) 0 0
\(793\) 1309.50 756.040i 1.65132 0.953392i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 630.466i 0.791050i 0.918455 + 0.395525i \(0.129437\pi\)
−0.918455 + 0.395525i \(0.870563\pi\)
\(798\) 0 0
\(799\) 1012.00 1.26658
\(800\) 0