Properties

Label 5292.2.x.a.4409.8
Level $5292$
Weight $2$
Character 5292.4409
Analytic conductor $42.257$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5292.x (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(42.2568327497\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} - 156 x^{7} + 558 x^{6} - 837 x^{5} + 1782 x^{4} - 4131 x^{3} + 3645 x^{2} - 4374 x + 6561\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 4409.8
Root \(-0.544978 + 1.64408i\) of defining polynomial
Character \(\chi\) \(=\) 5292.4409
Dual form 5292.2.x.a.881.8

$q$-expansion

\(f(q)\) \(=\) \(q+(1.95741 + 3.39033i) q^{5} +O(q^{10})\) \(q+(1.95741 + 3.39033i) q^{5} +(-3.19958 - 1.84728i) q^{11} +(0.480242 - 0.277268i) q^{13} +5.83832 q^{17} +5.33973i q^{19} +(-1.96965 + 1.13718i) q^{23} +(-5.16291 + 8.94242i) q^{25} +(-3.53638 - 2.04173i) q^{29} +(-7.00132 + 4.04222i) q^{31} -7.79699 q^{37} +(-3.59234 - 6.22212i) q^{41} +(-0.754009 + 1.30598i) q^{43} +(-1.41416 + 2.44940i) q^{47} -0.0479960i q^{53} -14.4635i q^{55} +(-4.45656 - 7.71900i) q^{59} +(6.03343 + 3.48340i) q^{61} +(1.88006 + 1.08545i) q^{65} +(-0.587402 - 1.01741i) q^{67} +6.71061i q^{71} +4.07253i q^{73} +(1.97374 - 3.41861i) q^{79} +(-3.84674 + 6.66275i) q^{83} +(11.4280 + 19.7938i) q^{85} -5.42600 q^{89} +(-18.1035 + 10.4520i) q^{95} +(13.9874 + 8.07563i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + O(q^{10}) \) \( 16 q - 6 q^{11} - 3 q^{13} + 18 q^{17} + 21 q^{23} - 8 q^{25} - 6 q^{29} + 6 q^{31} - 2 q^{37} + 6 q^{41} - 2 q^{43} - 18 q^{47} - 15 q^{59} + 3 q^{61} - 39 q^{65} - 7 q^{67} - q^{79} + 6 q^{85} + 42 q^{89} + 6 q^{95} - 3 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(2647\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(1\)

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
\(5\) 1.95741 + 3.39033i 0.875381 + 1.51620i 0.856357 + 0.516385i \(0.172722\pi\)
0.0190238 + 0.999819i \(0.493944\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.19958 1.84728i −0.964710 0.556976i −0.0670908 0.997747i \(-0.521372\pi\)
−0.897620 + 0.440771i \(0.854705\pi\)
\(12\) 0 0
\(13\) 0.480242 0.277268i 0.133195 0.0769002i −0.431922 0.901911i \(-0.642164\pi\)
0.565117 + 0.825011i \(0.308831\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.83832 1.41600 0.708000 0.706213i \(-0.249598\pi\)
0.708000 + 0.706213i \(0.249598\pi\)
\(18\) 0 0
\(19\) 5.33973i 1.22502i 0.790464 + 0.612509i \(0.209840\pi\)
−0.790464 + 0.612509i \(0.790160\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.96965 + 1.13718i −0.410700 + 0.237118i −0.691090 0.722768i \(-0.742869\pi\)
0.280390 + 0.959886i \(0.409536\pi\)
\(24\) 0 0
\(25\) −5.16291 + 8.94242i −1.03258 + 1.78848i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.53638 2.04173i −0.656690 0.379140i 0.134325 0.990937i \(-0.457113\pi\)
−0.791014 + 0.611797i \(0.790447\pi\)
\(30\) 0 0
\(31\) −7.00132 + 4.04222i −1.25748 + 0.726004i −0.972583 0.232556i \(-0.925291\pi\)
−0.284892 + 0.958560i \(0.591958\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.79699 −1.28182 −0.640909 0.767617i \(-0.721442\pi\)
−0.640909 + 0.767617i \(0.721442\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.59234 6.22212i −0.561030 0.971732i −0.997407 0.0719684i \(-0.977072\pi\)
0.436377 0.899764i \(-0.356261\pi\)
\(42\) 0 0
\(43\) −0.754009 + 1.30598i −0.114985 + 0.199160i −0.917774 0.397103i \(-0.870015\pi\)
0.802789 + 0.596264i \(0.203349\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.41416 + 2.44940i −0.206277 + 0.357282i −0.950539 0.310606i \(-0.899468\pi\)
0.744262 + 0.667888i \(0.232801\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.0479960i 0.00659276i −0.999995 0.00329638i \(-0.998951\pi\)
0.999995 0.00329638i \(-0.00104927\pi\)
\(54\) 0 0
\(55\) 14.4635i 1.95026i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.45656 7.71900i −0.580195 1.00493i −0.995456 0.0952251i \(-0.969643\pi\)
0.415261 0.909703i \(-0.363690\pi\)
\(60\) 0 0
\(61\) 6.03343 + 3.48340i 0.772501 + 0.446004i 0.833766 0.552118i \(-0.186180\pi\)
−0.0612648 + 0.998122i \(0.519513\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.88006 + 1.08545i 0.233193 + 0.134634i
\(66\) 0 0
\(67\) −0.587402 1.01741i −0.0717626 0.124296i 0.827911 0.560859i \(-0.189529\pi\)
−0.899674 + 0.436563i \(0.856196\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.71061i 0.796403i 0.917298 + 0.398202i \(0.130366\pi\)
−0.917298 + 0.398202i \(0.869634\pi\)
\(72\) 0 0
\(73\) 4.07253i 0.476654i 0.971185 + 0.238327i \(0.0765990\pi\)
−0.971185 + 0.238327i \(0.923401\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.97374 3.41861i 0.222063 0.384624i −0.733371 0.679828i \(-0.762054\pi\)
0.955434 + 0.295204i \(0.0953877\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.84674 + 6.66275i −0.422235 + 0.731332i −0.996158 0.0875774i \(-0.972087\pi\)
0.573923 + 0.818909i \(0.305421\pi\)
\(84\) 0 0
\(85\) 11.4280 + 19.7938i 1.23954 + 2.14694i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5.42600 −0.575155 −0.287577 0.957757i \(-0.592850\pi\)
−0.287577 + 0.957757i \(0.592850\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −18.1035 + 10.4520i −1.85738 + 1.07236i
\(96\) 0 0
\(97\) 13.9874 + 8.07563i 1.42021 + 0.819956i 0.996316 0.0857571i \(-0.0273309\pi\)
0.423890 + 0.905714i \(0.360664\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −0.811750 + 1.40599i −0.0807722 + 0.139901i −0.903582 0.428416i \(-0.859072\pi\)
0.822810 + 0.568317i \(0.192405\pi\)
\(102\) 0 0
\(103\) 0.342653 0.197831i 0.0337626 0.0194929i −0.483024 0.875607i \(-0.660461\pi\)
0.516786 + 0.856114i \(0.327128\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.66700i 0.547850i 0.961751 + 0.273925i \(0.0883219\pi\)
−0.961751 + 0.273925i \(0.911678\pi\)
\(108\) 0 0
\(109\) 13.5133 1.29434 0.647171 0.762345i \(-0.275952\pi\)
0.647171 + 0.762345i \(0.275952\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.13651 + 0.656162i −0.106913 + 0.0617265i −0.552503 0.833511i \(-0.686327\pi\)
0.445590 + 0.895237i \(0.352994\pi\)
\(114\) 0 0
\(115\) −7.71082 4.45184i −0.719038 0.415137i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.32489 + 2.29477i 0.120444 + 0.208615i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −20.8496 −1.86485
\(126\) 0 0
\(127\) −17.3935 −1.54342 −0.771710 0.635975i \(-0.780598\pi\)
−0.771710 + 0.635975i \(0.780598\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −5.45361 9.44593i −0.476484 0.825295i 0.523153 0.852239i \(-0.324756\pi\)
−0.999637 + 0.0269442i \(0.991422\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 7.62547 + 4.40257i 0.651488 + 0.376137i 0.789026 0.614360i \(-0.210586\pi\)
−0.137538 + 0.990496i \(0.543919\pi\)
\(138\) 0 0
\(139\) −14.2352 + 8.21869i −1.20741 + 0.697100i −0.962193 0.272367i \(-0.912193\pi\)
−0.245220 + 0.969468i \(0.578860\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2.04876 −0.171326
\(144\) 0 0
\(145\) 15.9860i 1.32757i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 12.5814 7.26390i 1.03071 0.595082i 0.113523 0.993535i \(-0.463786\pi\)
0.917188 + 0.398454i \(0.130453\pi\)
\(150\) 0 0
\(151\) −2.80307 + 4.85505i −0.228110 + 0.395099i −0.957248 0.289268i \(-0.906588\pi\)
0.729138 + 0.684367i \(0.239921\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −27.4089 15.8246i −2.20154 1.27106i
\(156\) 0 0
\(157\) −15.4411 + 8.91493i −1.23233 + 0.711489i −0.967516 0.252809i \(-0.918645\pi\)
−0.264819 + 0.964298i \(0.585312\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1.15399 0.0903874 0.0451937 0.998978i \(-0.485610\pi\)
0.0451937 + 0.998978i \(0.485610\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.95550 + 15.5114i 0.692997 + 1.20031i 0.970851 + 0.239683i \(0.0770435\pi\)
−0.277854 + 0.960623i \(0.589623\pi\)
\(168\) 0 0
\(169\) −6.34625 + 10.9920i −0.488173 + 0.845540i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3.74814 6.49197i 0.284966 0.493576i −0.687635 0.726057i \(-0.741351\pi\)
0.972601 + 0.232481i \(0.0746843\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0.720974i 0.0538881i 0.999637 + 0.0269441i \(0.00857760\pi\)
−0.999637 + 0.0269441i \(0.991422\pi\)
\(180\) 0 0
\(181\) 5.07121i 0.376940i 0.982079 + 0.188470i \(0.0603529\pi\)
−0.982079 + 0.188470i \(0.939647\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −15.2619 26.4344i −1.12208 1.94350i
\(186\) 0 0
\(187\) −18.6802 10.7850i −1.36603 0.788678i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −11.0005 6.35111i −0.795965 0.459551i 0.0460934 0.998937i \(-0.485323\pi\)
−0.842058 + 0.539387i \(0.818656\pi\)
\(192\) 0 0
\(193\) 11.4076 + 19.7586i 0.821140 + 1.42226i 0.904834 + 0.425765i \(0.139995\pi\)
−0.0836931 + 0.996492i \(0.526672\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0.0311360i 0.00221835i 0.999999 + 0.00110918i \(0.000353062\pi\)
−0.999999 + 0.00110918i \(0.999647\pi\)
\(198\) 0 0
\(199\) 22.9952i 1.63008i −0.579402 0.815042i \(-0.696714\pi\)
0.579402 0.815042i \(-0.303286\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 14.0634 24.3585i 0.982229 1.70127i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 9.86397 17.0849i 0.682305 1.18179i
\(210\) 0 0
\(211\) 8.55841 + 14.8236i 0.589185 + 1.02050i 0.994339 + 0.106250i \(0.0338845\pi\)
−0.405154 + 0.914248i \(0.632782\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −5.90362 −0.402623
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.80380 1.61878i 0.188604 0.108891i
\(222\) 0 0
\(223\) 1.25230 + 0.723016i 0.0838602 + 0.0484167i 0.541344 0.840801i \(-0.317916\pi\)
−0.457484 + 0.889218i \(0.651249\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.23596 3.87280i 0.148406 0.257047i −0.782232 0.622987i \(-0.785919\pi\)
0.930639 + 0.365940i \(0.119252\pi\)
\(228\) 0 0
\(229\) −2.24072 + 1.29368i −0.148071 + 0.0854888i −0.572205 0.820111i \(-0.693912\pi\)
0.424134 + 0.905599i \(0.360579\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 17.4078i 1.14042i −0.821498 0.570211i \(-0.806862\pi\)
0.821498 0.570211i \(-0.193138\pi\)
\(234\) 0 0
\(235\) −11.0724 −0.722284
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −4.23642 + 2.44590i −0.274031 + 0.158212i −0.630718 0.776012i \(-0.717240\pi\)
0.356687 + 0.934224i \(0.383906\pi\)
\(240\) 0 0
\(241\) −7.04282 4.06618i −0.453668 0.261925i 0.255710 0.966754i \(-0.417691\pi\)
−0.709378 + 0.704828i \(0.751024\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.48053 + 2.56436i 0.0942041 + 0.163166i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 25.9341 1.63694 0.818472 0.574546i \(-0.194821\pi\)
0.818472 + 0.574546i \(0.194821\pi\)
\(252\) 0 0
\(253\) 8.40274 0.528276
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −15.4115 26.6935i −0.961344 1.66510i −0.719131 0.694874i \(-0.755460\pi\)
−0.242213 0.970223i \(-0.577873\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −15.6625 9.04276i −0.965792 0.557600i −0.0678413 0.997696i \(-0.521611\pi\)
−0.897951 + 0.440096i \(0.854944\pi\)
\(264\) 0 0
\(265\) 0.162723 0.0939479i 0.00999597 0.00577117i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −21.6406 −1.31945 −0.659725 0.751507i \(-0.729327\pi\)
−0.659725 + 0.751507i \(0.729327\pi\)
\(270\) 0 0
\(271\) 14.2551i 0.865937i −0.901409 0.432968i \(-0.857466\pi\)
0.901409 0.432968i \(-0.142534\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 33.0383 19.0747i 1.99229 1.15025i
\(276\) 0 0
\(277\) −4.40164 + 7.62386i −0.264469 + 0.458073i −0.967424 0.253160i \(-0.918530\pi\)
0.702956 + 0.711234i \(0.251863\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 16.6889 + 9.63537i 0.995579 + 0.574798i 0.906937 0.421266i \(-0.138414\pi\)
0.0886417 + 0.996064i \(0.471747\pi\)
\(282\) 0 0
\(283\) −8.32822 + 4.80830i −0.495061 + 0.285824i −0.726672 0.686985i \(-0.758934\pi\)
0.231611 + 0.972809i \(0.425601\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 17.0859 1.00506
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.22598 2.12346i −0.0716225 0.124054i 0.827990 0.560743i \(-0.189484\pi\)
−0.899613 + 0.436689i \(0.856151\pi\)
\(294\) 0 0
\(295\) 17.4467 30.2185i 1.01578 1.75939i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −0.630605 + 1.09224i −0.0364688 + 0.0631658i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 27.2738i 1.56169i
\(306\) 0 0
\(307\) 10.6839i 0.609760i 0.952391 + 0.304880i \(0.0986163\pi\)
−0.952391 + 0.304880i \(0.901384\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −10.3833 17.9843i −0.588780 1.01980i −0.994393 0.105752i \(-0.966275\pi\)
0.405612 0.914045i \(-0.367058\pi\)
\(312\) 0 0
\(313\) 3.40449 + 1.96558i 0.192433 + 0.111101i 0.593121 0.805113i \(-0.297896\pi\)
−0.400688 + 0.916215i \(0.631229\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.98369 + 1.14528i 0.111415 + 0.0643256i 0.554672 0.832069i \(-0.312844\pi\)
−0.443257 + 0.896395i \(0.646177\pi\)
\(318\) 0 0
\(319\) 7.54330 + 13.0654i 0.422344 + 0.731521i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 31.1750i 1.73462i
\(324\) 0 0
\(325\) 5.72603i 0.317623i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −3.46788 + 6.00655i −0.190612 + 0.330150i −0.945453 0.325758i \(-0.894381\pi\)
0.754841 + 0.655908i \(0.227714\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 2.29957 3.98298i 0.125639 0.217613i
\(336\) 0 0
\(337\) −9.59771 16.6237i −0.522821 0.905552i −0.999647 0.0265545i \(-0.991546\pi\)
0.476827 0.878997i \(-0.341787\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 29.8684 1.61747
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −7.35287 + 4.24518i −0.394723 + 0.227893i −0.684204 0.729290i \(-0.739850\pi\)
0.289482 + 0.957184i \(0.406517\pi\)
\(348\) 0 0
\(349\) −16.5478 9.55386i −0.885782 0.511407i −0.0132216 0.999913i \(-0.504209\pi\)
−0.872560 + 0.488506i \(0.837542\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −6.82951 + 11.8291i −0.363498 + 0.629597i −0.988534 0.150999i \(-0.951751\pi\)
0.625036 + 0.780596i \(0.285084\pi\)
\(354\) 0 0
\(355\) −22.7512 + 13.1354i −1.20751 + 0.697156i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 17.1945i 0.907490i 0.891132 + 0.453745i \(0.149912\pi\)
−0.891132 + 0.453745i \(0.850088\pi\)
\(360\) 0 0
\(361\) −9.51270 −0.500668
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −13.8073 + 7.97162i −0.722705 + 0.417254i
\(366\) 0 0
\(367\) −14.6001 8.42936i −0.762118 0.440009i 0.0679376 0.997690i \(-0.478358\pi\)
−0.830056 + 0.557680i \(0.811691\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0.704288 + 1.21986i 0.0364667 + 0.0631621i 0.883683 0.468086i \(-0.155056\pi\)
−0.847216 + 0.531248i \(0.821723\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.26442 −0.116624
\(378\) 0 0
\(379\) −0.598572 −0.0307466 −0.0153733 0.999882i \(-0.504894\pi\)
−0.0153733 + 0.999882i \(0.504894\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 4.26039 + 7.37921i 0.217696 + 0.377060i 0.954103 0.299478i \(-0.0968127\pi\)
−0.736407 + 0.676538i \(0.763479\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −29.9624 17.2988i −1.51915 0.877084i −0.999746 0.0225587i \(-0.992819\pi\)
−0.519409 0.854526i \(-0.673848\pi\)
\(390\) 0 0
\(391\) −11.4994 + 6.63920i −0.581551 + 0.335759i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 15.4537 0.777558
\(396\) 0 0
\(397\) 32.2821i 1.62019i 0.586296 + 0.810097i \(0.300586\pi\)
−0.586296 + 0.810097i \(0.699414\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 11.3473 6.55139i 0.566659 0.327161i −0.189155 0.981947i \(-0.560575\pi\)
0.755814 + 0.654787i \(0.227242\pi\)
\(402\) 0 0
\(403\) −2.24155 + 3.88248i −0.111660 + 0.193400i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 24.9471 + 14.4032i 1.23658 + 0.713941i
\(408\) 0 0
\(409\) 32.3493 18.6769i 1.59957 0.923513i 0.608002 0.793936i \(-0.291971\pi\)
0.991569 0.129577i \(-0.0413620\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −30.1186 −1.47846
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −14.1954 24.5871i −0.693490 1.20116i −0.970687 0.240346i \(-0.922739\pi\)
0.277198 0.960813i \(-0.410594\pi\)
\(420\) 0 0
\(421\) −17.3359 + 30.0267i −0.844901 + 1.46341i 0.0408054 + 0.999167i \(0.487008\pi\)
−0.885707 + 0.464245i \(0.846326\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −30.1427 + 52.2087i −1.46214 + 2.53249i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 15.2240i 0.733315i 0.930356 + 0.366657i \(0.119498\pi\)
−0.930356 + 0.366657i \(0.880502\pi\)
\(432\) 0 0
\(433\) 3.97041i 0.190806i −0.995439 0.0954028i \(-0.969586\pi\)
0.995439 0.0954028i \(-0.0304139\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.07222 10.5174i −0.290474 0.503115i
\(438\) 0 0
\(439\) 8.21910 + 4.74530i 0.392276 + 0.226481i 0.683146 0.730282i \(-0.260611\pi\)
−0.290870 + 0.956763i \(0.593945\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −28.3955 16.3942i −1.34911 0.778910i −0.360989 0.932570i \(-0.617561\pi\)
−0.988124 + 0.153660i \(0.950894\pi\)
\(444\) 0 0
\(445\) −10.6209 18.3960i −0.503479 0.872052i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0.658896i 0.0310952i −0.999879 0.0155476i \(-0.995051\pi\)
0.999879 0.0155476i \(-0.00494916\pi\)
\(450\) 0 0
\(451\) 26.5443i 1.24992i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −7.94514 + 13.7614i −0.371658 + 0.643730i −0.989821 0.142320i \(-0.954544\pi\)
0.618163 + 0.786050i \(0.287877\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9.81626 17.0023i 0.457189 0.791874i −0.541622 0.840622i \(-0.682190\pi\)
0.998811 + 0.0487477i \(0.0155230\pi\)
\(462\) 0 0
\(463\) 0.600159 + 1.03951i 0.0278918 + 0.0483099i 0.879634 0.475651i \(-0.157787\pi\)
−0.851743 + 0.523960i \(0.824454\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 38.5618 1.78443 0.892213 0.451614i \(-0.149152\pi\)
0.892213 + 0.451614i \(0.149152\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 4.82503 2.78573i 0.221855 0.128088i
\(474\) 0 0
\(475\) −47.7501 27.5685i −2.19093 1.26493i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3.61289 6.25771i 0.165077 0.285922i −0.771606 0.636101i \(-0.780546\pi\)
0.936683 + 0.350179i \(0.113879\pi\)
\(480\) 0 0
\(481\) −3.74444 + 2.16185i −0.170732 + 0.0985720i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 63.2293i 2.87110i
\(486\) 0 0
\(487\) −9.71539 −0.440247 −0.220123 0.975472i \(-0.570646\pi\)
−0.220123 + 0.975472i \(0.570646\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −17.2480 + 9.95814i −0.778392 + 0.449405i −0.835860 0.548943i \(-0.815031\pi\)
0.0574682 + 0.998347i \(0.481697\pi\)
\(492\) 0 0
\(493\) −20.6465 11.9203i −0.929872 0.536862i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −17.1920 29.7774i −0.769619 1.33302i −0.937770 0.347258i \(-0.887113\pi\)
0.168150 0.985761i \(-0.446221\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −1.22542 −0.0546388 −0.0273194 0.999627i \(-0.508697\pi\)
−0.0273194 + 0.999627i \(0.508697\pi\)
\(504\) 0 0
\(505\) −6.35571 −0.282826
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −5.05078 8.74820i −0.223872 0.387757i 0.732109 0.681188i \(-0.238536\pi\)
−0.955980 + 0.293431i \(0.905203\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.34143 + 0.774473i 0.0591103 + 0.0341274i
\(516\) 0 0
\(517\) 9.04947 5.22471i 0.397995 0.229783i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 21.0781 0.923446 0.461723 0.887024i \(-0.347231\pi\)
0.461723 + 0.887024i \(0.347231\pi\)
\(522\) 0 0
\(523\) 19.7145i 0.862057i 0.902338 + 0.431028i \(0.141849\pi\)
−0.902338 + 0.431028i \(0.858151\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −40.8760 + 23.5997i −1.78058 + 1.02802i
\(528\) 0 0
\(529\) −8.91366 + 15.4389i −0.387550 + 0.671257i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −3.45039 1.99208i −0.149453 0.0862866i
\(534\) 0 0
\(535\) −19.2130 + 11.0926i −0.830652 + 0.479577i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 8.44950 0.363272 0.181636 0.983366i \(-0.441861\pi\)
0.181636 + 0.983366i \(0.441861\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 26.4511 + 45.8147i 1.13304 + 1.96249i
\(546\) 0 0
\(547\) −4.02889 + 6.97824i −0.172263 + 0.298368i −0.939211 0.343342i \(-0.888441\pi\)
0.766948 + 0.641709i \(0.221774\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 10.9023 18.8833i 0.464453 0.804456i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 21.0495i 0.891895i −0.895059 0.445947i \(-0.852867\pi\)
0.895059 0.445947i \(-0.147133\pi\)
\(558\) 0 0
\(559\) 0.836249i 0.0353696i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 20.6410 + 35.7513i 0.869916 + 1.50674i 0.862082 + 0.506769i \(0.169160\pi\)
0.00783378 + 0.999969i \(0.497506\pi\)
\(564\) 0 0
\(565\) −4.44922 2.56876i −0.187180 0.108068i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 31.2691 + 18.0532i 1.31087 + 0.756829i 0.982240 0.187630i \(-0.0600807\pi\)
0.328627 + 0.944460i \(0.393414\pi\)
\(570\) 0 0
\(571\) 9.62111 + 16.6642i 0.402631 + 0.697377i 0.994043 0.108993i \(-0.0347625\pi\)
−0.591412 + 0.806370i \(0.701429\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 23.4846i 0.979375i
\(576\) 0 0
\(577\) 29.8031i 1.24072i 0.784318 + 0.620359i \(0.213013\pi\)
−0.784318 + 0.620359i \(0.786987\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −0.0886621 + 0.153567i −0.00367201 + 0.00636010i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −4.72218 + 8.17905i −0.194905 + 0.337586i −0.946869 0.321618i \(-0.895773\pi\)
0.751964 + 0.659204i \(0.229107\pi\)
\(588\) 0 0
\(589\) −21.5843 37.3852i −0.889367 1.54043i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 24.8352 1.01986 0.509929 0.860216i \(-0.329672\pi\)
0.509929 + 0.860216i \(0.329672\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −10.3052 + 5.94974i −0.421061 + 0.243100i −0.695531 0.718496i \(-0.744831\pi\)
0.274470 + 0.961596i \(0.411498\pi\)
\(600\) 0 0
\(601\) 22.1276 + 12.7754i 0.902604 + 0.521118i 0.878044 0.478580i \(-0.158848\pi\)
0.0245596 + 0.999698i \(0.492182\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −5.18669 + 8.98361i −0.210869 + 0.365236i
\(606\) 0 0
\(607\) 19.5544 11.2897i 0.793687 0.458235i −0.0475718 0.998868i \(-0.515148\pi\)
0.841259 + 0.540632i \(0.181815\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.56841i 0.0634510i
\(612\) 0 0
\(613\) 22.8588 0.923257 0.461628 0.887073i \(-0.347265\pi\)
0.461628 + 0.887073i \(0.347265\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.78792 + 1.03226i −0.0719791 + 0.0415572i −0.535558 0.844499i \(-0.679899\pi\)
0.463578 + 0.886056i \(0.346565\pi\)
\(618\) 0 0
\(619\) 28.2233 + 16.2947i 1.13439 + 0.654940i 0.945035 0.326969i \(-0.106027\pi\)
0.189354 + 0.981909i \(0.439361\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −14.9967 25.9751i −0.599870 1.03901i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −45.5213 −1.81505
\(630\) 0 0
\(631\) 38.4706 1.53149 0.765744 0.643145i \(-0.222371\pi\)
0.765744 + 0.643145i \(0.222371\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −34.0461 58.9696i −1.35108 2.34014i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 41.3645 + 23.8818i 1.63380 + 0.943274i 0.982907 + 0.184104i \(0.0589384\pi\)
0.650892 + 0.759170i \(0.274395\pi\)
\(642\) 0 0
\(643\) 29.2346 16.8786i 1.15290 0.665626i 0.203306 0.979115i \(-0.434831\pi\)
0.949592 + 0.313489i \(0.101498\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.07202 0.0421453 0.0210727 0.999778i \(-0.493292\pi\)
0.0210727 + 0.999778i \(0.493292\pi\)
\(648\) 0 0
\(649\) 32.9301i 1.29262i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 28.8503 16.6567i 1.12900 0.651828i 0.185317 0.982679i \(-0.440669\pi\)
0.943683 + 0.330851i \(0.107336\pi\)
\(654\) 0 0
\(655\) 21.3499 36.9791i 0.834210 1.44489i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 8.41890 + 4.86065i 0.327954 + 0.189344i 0.654932 0.755688i \(-0.272697\pi\)
−0.326979 + 0.945032i \(0.606031\pi\)
\(660\) 0 0
\(661\) −14.7856 + 8.53647i −0.575093 + 0.332030i −0.759181 0.650880i \(-0.774400\pi\)
0.184088 + 0.982910i \(0.441067\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 9.28724 0.359603
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −12.8696 22.2909i −0.496827 0.860529i
\(672\) 0 0
\(673\) −18.3359 + 31.7588i −0.706798 + 1.22421i 0.259240 + 0.965813i \(0.416528\pi\)
−0.966039 + 0.258398i \(0.916805\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −20.1769 + 34.9474i −0.775461 + 1.34314i 0.159073 + 0.987267i \(0.449149\pi\)
−0.934535 + 0.355872i \(0.884184\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 9.51083i 0.363922i 0.983306 + 0.181961i \(0.0582444\pi\)
−0.983306 + 0.181961i \(0.941756\pi\)
\(684\) 0 0
\(685\) 34.4705i 1.31705i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −0.0133077 0.0230497i −0.000506985 0.000878123i
\(690\) 0 0
\(691\) −6.67519 3.85392i −0.253936 0.146610i 0.367629 0.929972i \(-0.380170\pi\)
−0.621565 + 0.783362i \(0.713503\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −55.7282 32.1747i −2.11389 1.22046i
\(696\) 0 0
\(697\) −20.9732 36.3267i −0.794418 1.37597i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 15.6388i 0.590671i −0.955394 0.295336i \(-0.904569\pi\)
0.955394 0.295336i \(-0.0954314\pi\)
\(702\) 0 0
\(703\) 41.6338i 1.57025i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −6.72025 + 11.6398i −0.252384 + 0.437142i −0.964182 0.265242i \(-0.914548\pi\)
0.711797 + 0.702385i \(0.247881\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 9.19343 15.9235i 0.344297 0.596339i
\(714\) 0 0
\(715\) −4.01027 6.94599i −0.149976 0.259765i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 40.0619 1.49405 0.747027 0.664793i \(-0.231480\pi\)
0.747027 + 0.664793i \(0.231480\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 36.5161 21.0826i 1.35617 0.782986i
\(726\) 0 0
\(727\) 43.2091 + 24.9468i 1.60254 + 0.925225i 0.990978 + 0.134027i \(0.0427910\pi\)
0.611560 + 0.791198i \(0.290542\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −4.40214 + 7.62473i −0.162819 + 0.282011i
\(732\) 0 0
\(733\) 9.91430 5.72402i 0.366193 0.211422i −0.305601 0.952160i \(-0.598857\pi\)
0.671794 + 0.740738i \(0.265524\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.34038i 0.159880i
\(738\) 0 0
\(739\) −8.92607 −0.328351 −0.164175 0.986431i \(-0.552496\pi\)
−0.164175 + 0.986431i \(0.552496\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 45.8621 26.4785i 1.68252 0.971403i 0.722540 0.691329i \(-0.242974\pi\)
0.959979 0.280074i \(-0.0903589\pi\)
\(744\) 0 0
\(745\) 49.2541 + 28.4369i 1.80453 + 1.04185i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 13.2326 + 22.9195i 0.482865 + 0.836346i 0.999806 0.0196744i \(-0.00626295\pi\)
−0.516942 + 0.856021i \(0.672930\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −21.9470 −0.798733
\(756\) 0 0
\(757\) 8.46749 0.307756 0.153878 0.988090i \(-0.450824\pi\)
0.153878 + 0.988090i \(0.450824\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 26.9968 + 46.7599i 0.978635 + 1.69505i 0.667377 + 0.744720i \(0.267417\pi\)
0.311258 + 0.950325i \(0.399250\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −4.28046 2.47132i −0.154558 0.0892343i
\(768\) 0 0
\(769\) 30.1912 17.4309i 1.08872 0.628575i 0.155487 0.987838i \(-0.450305\pi\)
0.933236 + 0.359263i \(0.116972\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −2.12749 −0.0765207 −0.0382603 0.999268i \(-0.512182\pi\)
−0.0382603 + 0.999268i \(0.512182\pi\)
\(774\) 0 0
\(775\) 83.4784i 2.99863i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 33.2244 19.1821i 1.19039 0.687272i
\(780\) 0 0
\(781\) 12.3964 21.4712i 0.443577 0.768298i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −60.4492 34.9004i −2.15752 1.24565i
\(786\) 0 0
\(787\) 24.5457 14.1715i 0.874959 0.505158i 0.00596615 0.999982i \(-0.498101\pi\)
0.868993 + 0.494824i \(0.164768\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 3.86334 0.137191
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 18.9123 + 32.7570i 0.669907 + 1.16031i 0.977930 + 0.208935i \(0.0669996\pi\)
−0.308022 + 0.951379i \(0.599667\pi\)
\(798\) 0 0
\(799\) −8.25634 + 14.3004i −0.292088 + 0.505912i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 7.52311 13.0304i 0.265485 0.459833i
\(804\) 0 0
\(805\) 0 0
\(806\) 0