Properties

Label 405.3.h.b.269.1
Level $405$
Weight $3$
Character 405.269
Analytic conductor $11.035$
Analytic rank $0$
Dimension $2$
CM discriminant -15
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 405.h (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.0354507066\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 269.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 405.269
Dual form 405.3.h.b.134.1

$q$-expansion

\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{2} +(1.50000 + 2.59808i) q^{4} +(2.50000 + 4.33013i) q^{5} +7.00000 q^{8} +O(q^{10})\) \(q+(0.500000 - 0.866025i) q^{2} +(1.50000 + 2.59808i) q^{4} +(2.50000 + 4.33013i) q^{5} +7.00000 q^{8} +5.00000 q^{10} +(-2.50000 + 4.33013i) q^{16} +14.0000 q^{17} -22.0000 q^{19} +(-7.50000 + 12.9904i) q^{20} +(17.0000 + 29.4449i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(-1.00000 - 1.73205i) q^{31} +(16.5000 + 28.5788i) q^{32} +(7.00000 - 12.1244i) q^{34} +(-11.0000 + 19.0526i) q^{38} +(17.5000 + 30.3109i) q^{40} +34.0000 q^{46} +(-7.00000 + 12.1244i) q^{47} +(-24.5000 - 42.4352i) q^{49} +(12.5000 + 21.6506i) q^{50} +86.0000 q^{53} +(59.0000 - 102.191i) q^{61} -2.00000 q^{62} +13.0000 q^{64} +(21.0000 + 36.3731i) q^{68} +(-33.0000 - 57.1577i) q^{76} +(-49.0000 + 84.8705i) q^{79} -25.0000 q^{80} +(77.0000 - 133.368i) q^{83} +(35.0000 + 60.6218i) q^{85} +(-51.0000 + 88.3346i) q^{92} +(7.00000 + 12.1244i) q^{94} +(-55.0000 - 95.2628i) q^{95} -49.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 3 q^{4} + 5 q^{5} + 14 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 3 q^{4} + 5 q^{5} + 14 q^{8} + 10 q^{10} - 5 q^{16} + 28 q^{17} - 44 q^{19} - 15 q^{20} + 34 q^{23} - 25 q^{25} - 2 q^{31} + 33 q^{32} + 14 q^{34} - 22 q^{38} + 35 q^{40} + 68 q^{46} - 14 q^{47} - 49 q^{49} + 25 q^{50} + 172 q^{53} + 118 q^{61} - 4 q^{62} + 26 q^{64} + 42 q^{68} - 66 q^{76} - 98 q^{79} - 50 q^{80} + 154 q^{83} + 70 q^{85} - 102 q^{92} + 14 q^{94} - 110 q^{95} - 98 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\) \(82\) \(326\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 0.866025i 0.250000 0.433013i −0.713525 0.700629i \(-0.752903\pi\)
0.963525 + 0.267617i \(0.0862360\pi\)
\(3\) 0 0
\(4\) 1.50000 + 2.59808i 0.375000 + 0.649519i
\(5\) 2.50000 + 4.33013i 0.500000 + 0.866025i
\(6\) 0 0
\(7\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(8\) 7.00000 0.875000
\(9\) 0 0
\(10\) 5.00000 0.500000
\(11\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) 0 0
\(13\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −2.50000 + 4.33013i −0.156250 + 0.270633i
\(17\) 14.0000 0.823529 0.411765 0.911290i \(-0.364913\pi\)
0.411765 + 0.911290i \(0.364913\pi\)
\(18\) 0 0
\(19\) −22.0000 −1.15789 −0.578947 0.815365i \(-0.696536\pi\)
−0.578947 + 0.815365i \(0.696536\pi\)
\(20\) −7.50000 + 12.9904i −0.375000 + 0.649519i
\(21\) 0 0
\(22\) 0 0
\(23\) 17.0000 + 29.4449i 0.739130 + 1.28021i 0.952887 + 0.303325i \(0.0980966\pi\)
−0.213757 + 0.976887i \(0.568570\pi\)
\(24\) 0 0
\(25\) −12.5000 + 21.6506i −0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) −1.00000 1.73205i −0.0322581 0.0558726i 0.849446 0.527676i \(-0.176937\pi\)
−0.881704 + 0.471803i \(0.843603\pi\)
\(32\) 16.5000 + 28.5788i 0.515625 + 0.893089i
\(33\) 0 0
\(34\) 7.00000 12.1244i 0.205882 0.356599i
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) −11.0000 + 19.0526i −0.289474 + 0.501383i
\(39\) 0 0
\(40\) 17.5000 + 30.3109i 0.437500 + 0.757772i
\(41\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(42\) 0 0
\(43\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 34.0000 0.739130
\(47\) −7.00000 + 12.1244i −0.148936 + 0.257965i −0.930835 0.365441i \(-0.880918\pi\)
0.781898 + 0.623406i \(0.214252\pi\)
\(48\) 0 0
\(49\) −24.5000 42.4352i −0.500000 0.866025i
\(50\) 12.5000 + 21.6506i 0.250000 + 0.433013i
\(51\) 0 0
\(52\) 0 0
\(53\) 86.0000 1.62264 0.811321 0.584601i \(-0.198749\pi\)
0.811321 + 0.584601i \(0.198749\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) 59.0000 102.191i 0.967213 1.67526i 0.263665 0.964614i \(-0.415069\pi\)
0.703548 0.710648i \(-0.251598\pi\)
\(62\) −2.00000 −0.0322581
\(63\) 0 0
\(64\) 13.0000 0.203125
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) 21.0000 + 36.3731i 0.308824 + 0.534898i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −33.0000 57.1577i −0.434211 0.752075i
\(77\) 0 0
\(78\) 0 0
\(79\) −49.0000 + 84.8705i −0.620253 + 1.07431i 0.369185 + 0.929356i \(0.379637\pi\)
−0.989438 + 0.144954i \(0.953697\pi\)
\(80\) −25.0000 −0.312500
\(81\) 0 0
\(82\) 0 0
\(83\) 77.0000 133.368i 0.927711 1.60684i 0.140568 0.990071i \(-0.455107\pi\)
0.787142 0.616771i \(-0.211560\pi\)
\(84\) 0 0
\(85\) 35.0000 + 60.6218i 0.411765 + 0.713197i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −51.0000 + 88.3346i −0.554348 + 0.960159i
\(93\) 0 0
\(94\) 7.00000 + 12.1244i 0.0744681 + 0.128983i
\(95\) −55.0000 95.2628i −0.578947 1.00277i
\(96\) 0 0
\(97\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(98\) −49.0000 −0.500000
\(99\) 0 0
\(100\) −75.0000 −0.750000
\(101\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(102\) 0 0
\(103\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 43.0000 74.4782i 0.405660 0.702624i
\(107\) −106.000 −0.990654 −0.495327 0.868707i \(-0.664952\pi\)
−0.495327 + 0.868707i \(0.664952\pi\)
\(108\) 0 0
\(109\) −22.0000 −0.201835 −0.100917 0.994895i \(-0.532178\pi\)
−0.100917 + 0.994895i \(0.532178\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −103.000 178.401i −0.911504 1.57877i −0.811940 0.583741i \(-0.801588\pi\)
−0.0995644 0.995031i \(-0.531745\pi\)
\(114\) 0 0
\(115\) −85.0000 + 147.224i −0.739130 + 1.28021i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −60.5000 104.789i −0.500000 0.866025i
\(122\) −59.0000 102.191i −0.483607 0.837631i
\(123\) 0 0
\(124\) 3.00000 5.19615i 0.0241935 0.0419045i
\(125\) −125.000 −1.00000
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −59.5000 + 103.057i −0.464844 + 0.805133i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 98.0000 0.720588
\(137\) 113.000 195.722i 0.824818 1.42863i −0.0772412 0.997012i \(-0.524611\pi\)
0.902059 0.431613i \(-0.142056\pi\)
\(138\) 0 0
\(139\) 131.000 + 226.899i 0.942446 + 1.63236i 0.760786 + 0.649003i \(0.224814\pi\)
0.181660 + 0.983361i \(0.441853\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(150\) 0 0
\(151\) 119.000 206.114i 0.788079 1.36499i −0.139063 0.990284i \(-0.544409\pi\)
0.927142 0.374710i \(-0.122258\pi\)
\(152\) −154.000 −1.01316
\(153\) 0 0
\(154\) 0 0
\(155\) 5.00000 8.66025i 0.0322581 0.0558726i
\(156\) 0 0
\(157\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(158\) 49.0000 + 84.8705i 0.310127 + 0.537155i
\(159\) 0 0
\(160\) −82.5000 + 142.894i −0.515625 + 0.893089i
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −77.0000 133.368i −0.463855 0.803421i
\(167\) −127.000 219.970i −0.760479 1.31719i −0.942604 0.333913i \(-0.891631\pi\)
0.182125 0.983275i \(-0.441702\pi\)
\(168\) 0 0
\(169\) −84.5000 + 146.358i −0.500000 + 0.866025i
\(170\) 70.0000 0.411765
\(171\) 0 0
\(172\) 0 0
\(173\) 77.0000 133.368i 0.445087 0.770913i −0.552972 0.833200i \(-0.686506\pi\)
0.998058 + 0.0622873i \(0.0198395\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 122.000 0.674033 0.337017 0.941499i \(-0.390582\pi\)
0.337017 + 0.941499i \(0.390582\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 119.000 + 206.114i 0.646739 + 1.12019i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −42.0000 −0.223404
\(189\) 0 0
\(190\) −110.000 −0.578947
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) 0 0
\(193\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 73.5000 127.306i 0.375000 0.649519i
\(197\) 374.000 1.89848 0.949239 0.314557i \(-0.101856\pi\)
0.949239 + 0.314557i \(0.101856\pi\)
\(198\) 0 0
\(199\) −142.000 −0.713568 −0.356784 0.934187i \(-0.616127\pi\)
−0.356784 + 0.934187i \(0.616127\pi\)
\(200\) −87.5000 + 151.554i −0.437500 + 0.757772i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −181.000 313.501i −0.857820 1.48579i −0.874004 0.485919i \(-0.838485\pi\)
0.0161841 0.999869i \(-0.494848\pi\)
\(212\) 129.000 + 223.435i 0.608491 + 1.05394i
\(213\) 0 0
\(214\) −53.0000 + 91.7987i −0.247664 + 0.428966i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −11.0000 + 19.0526i −0.0504587 + 0.0873971i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −206.000 −0.911504
\(227\) −67.0000 + 116.047i −0.295154 + 0.511222i −0.975021 0.222114i \(-0.928704\pi\)
0.679867 + 0.733336i \(0.262038\pi\)
\(228\) 0 0
\(229\) −109.000 188.794i −0.475983 0.824426i 0.523639 0.851940i \(-0.324574\pi\)
−0.999621 + 0.0275144i \(0.991241\pi\)
\(230\) 85.0000 + 147.224i 0.369565 + 0.640106i
\(231\) 0 0
\(232\) 0 0
\(233\) −34.0000 −0.145923 −0.0729614 0.997335i \(-0.523245\pi\)
−0.0729614 + 0.997335i \(0.523245\pi\)
\(234\) 0 0
\(235\) −70.0000 −0.297872
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(240\) 0 0
\(241\) 239.000 413.960i 0.991701 1.71768i 0.384511 0.923120i \(-0.374370\pi\)
0.607190 0.794557i \(-0.292297\pi\)
\(242\) −121.000 −0.500000
\(243\) 0 0
\(244\) 354.000 1.45082
\(245\) 122.500 212.176i 0.500000 0.866025i
\(246\) 0 0
\(247\) 0 0
\(248\) −7.00000 12.1244i −0.0282258 0.0488885i
\(249\) 0 0
\(250\) −62.5000 + 108.253i −0.250000 + 0.433013i
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 85.5000 + 148.090i 0.333984 + 0.578478i
\(257\) 233.000 + 403.568i 0.906615 + 1.57030i 0.818735 + 0.574172i \(0.194676\pi\)
0.0878799 + 0.996131i \(0.471991\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −223.000 + 386.247i −0.847909 + 1.46862i 0.0351622 + 0.999382i \(0.488805\pi\)
−0.883071 + 0.469239i \(0.844528\pi\)
\(264\) 0 0
\(265\) 215.000 + 372.391i 0.811321 + 1.40525i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 482.000 1.77860 0.889299 0.457326i \(-0.151193\pi\)
0.889299 + 0.457326i \(0.151193\pi\)
\(272\) −35.0000 + 60.6218i −0.128676 + 0.222874i
\(273\) 0 0
\(274\) −113.000 195.722i −0.412409 0.714313i
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(278\) 262.000 0.942446
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(282\) 0 0
\(283\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −93.0000 −0.321799
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 197.000 + 341.214i 0.672355 + 1.16455i 0.977235 + 0.212162i \(0.0680505\pi\)
−0.304880 + 0.952391i \(0.598616\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) −119.000 206.114i −0.394040 0.682497i
\(303\) 0 0
\(304\) 55.0000 95.2628i 0.180921 0.313364i
\(305\) 590.000 1.93443
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −5.00000 8.66025i −0.0161290 0.0279363i
\(311\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(312\) 0 0
\(313\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −294.000 −0.930380
\(317\) −67.0000 + 116.047i −0.211356 + 0.366080i −0.952139 0.305664i \(-0.901121\pi\)
0.740783 + 0.671745i \(0.234455\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 32.5000 + 56.2917i 0.101562 + 0.175911i
\(321\) 0 0
\(322\) 0 0
\(323\) −308.000 −0.953560
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −61.0000 + 105.655i −0.184290 + 0.319200i −0.943337 0.331836i \(-0.892332\pi\)
0.759047 + 0.651036i \(0.225665\pi\)
\(332\) 462.000 1.39157
\(333\) 0 0
\(334\) −254.000 −0.760479
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(338\) 84.5000 + 146.358i 0.250000 + 0.433013i
\(339\) 0 0
\(340\) −105.000 + 181.865i −0.308824 + 0.534898i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −77.0000 133.368i −0.222543 0.385456i
\(347\) 293.000 + 507.491i 0.844380 + 1.46251i 0.886158 + 0.463383i \(0.153365\pi\)
−0.0417778 + 0.999127i \(0.513302\pi\)
\(348\) 0 0
\(349\) −229.000 + 396.640i −0.656160 + 1.13650i 0.325441 + 0.945562i \(0.394487\pi\)
−0.981602 + 0.190941i \(0.938846\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 137.000 237.291i 0.388102 0.672212i −0.604092 0.796914i \(-0.706464\pi\)
0.992194 + 0.124702i \(0.0397975\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 123.000 0.340720
\(362\) 61.0000 105.655i 0.168508 0.291865i
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(368\) −170.000 −0.461957
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −49.0000 + 84.8705i −0.130319 + 0.225719i
\(377\) 0 0
\(378\) 0 0
\(379\) −742.000 −1.95778 −0.978892 0.204379i \(-0.934482\pi\)
−0.978892 + 0.204379i \(0.934482\pi\)
\(380\) 165.000 285.788i 0.434211 0.752075i
\(381\) 0 0
\(382\) 0 0
\(383\) −343.000 594.093i −0.895561 1.55116i −0.833108 0.553110i \(-0.813441\pi\)
−0.0624530 0.998048i \(-0.519892\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(390\) 0 0
\(391\) 238.000 + 412.228i 0.608696 + 1.05429i
\(392\) −171.500 297.047i −0.437500 0.757772i
\(393\) 0 0
\(394\) 187.000 323.894i 0.474619 0.822065i
\(395\) −490.000 −1.24051
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) −71.0000 + 122.976i −0.178392 + 0.308984i
\(399\) 0 0
\(400\) −62.5000 108.253i −0.156250 0.270633i
\(401\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 71.0000 + 122.976i 0.173594 + 0.300674i 0.939674 0.342072i \(-0.111129\pi\)
−0.766080 + 0.642746i \(0.777795\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 770.000 1.85542
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(420\) 0 0
\(421\) −301.000 + 521.347i −0.714964 + 1.23835i 0.248009 + 0.968758i \(0.420224\pi\)
−0.962973 + 0.269597i \(0.913110\pi\)
\(422\) −362.000 −0.857820
\(423\) 0 0
\(424\) 602.000 1.41981
\(425\) −175.000 + 303.109i −0.411765 + 0.713197i
\(426\) 0 0
\(427\) 0 0
\(428\) −159.000 275.396i −0.371495 0.643449i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −33.0000 57.1577i −0.0756881 0.131096i
\(437\) −374.000 647.787i −0.855835 1.48235i
\(438\) 0 0
\(439\) 311.000 538.668i 0.708428 1.22703i −0.257012 0.966408i \(-0.582738\pi\)
0.965440 0.260625i \(-0.0839288\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −283.000 + 490.170i −0.638826 + 1.10648i 0.346864 + 0.937915i \(0.387246\pi\)
−0.985691 + 0.168564i \(0.946087\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 309.000 535.204i 0.683628 1.18408i
\(453\) 0 0
\(454\) 67.0000 + 116.047i 0.147577 + 0.255611i
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(458\) −218.000 −0.475983
\(459\) 0 0
\(460\) −510.000 −1.10870
\(461\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(462\) 0 0
\(463\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −17.0000 + 29.4449i −0.0364807 + 0.0631864i
\(467\) −346.000 −0.740899 −0.370450 0.928853i \(-0.620796\pi\)
−0.370450 + 0.928853i \(0.620796\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −35.0000 + 60.6218i −0.0744681 + 0.128983i
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 275.000 476.314i 0.578947 1.00277i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −239.000 413.960i −0.495851 0.858838i
\(483\) 0 0
\(484\) 181.500 314.367i 0.375000 0.649519i
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 413.000 715.337i 0.846311 1.46585i
\(489\) 0 0
\(490\) −122.500 212.176i −0.250000 0.433013i
\(491\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 10.0000 0.0201613
\(497\) 0 0
\(498\) 0 0
\(499\) −469.000 812.332i −0.939880 1.62792i −0.765692 0.643207i \(-0.777603\pi\)
−0.174187 0.984713i \(-0.555730\pi\)
\(500\) −187.500 324.760i −0.375000 0.649519i
\(501\) 0 0
\(502\) 0 0
\(503\) −994.000 −1.97614 −0.988072 0.153995i \(-0.950786\pi\)
−0.988072 + 0.153995i \(0.950786\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −305.000 −0.595703
\(513\) 0 0
\(514\) 466.000 0.906615
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 223.000 + 386.247i 0.423954 + 0.734311i
\(527\) −14.0000 24.2487i −0.0265655 0.0460127i
\(528\) 0 0
\(529\) −313.500 + 542.998i −0.592628 + 1.02646i
\(530\) 430.000 0.811321
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −265.000 458.993i −0.495327 0.857932i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −1078.00 −1.99261 −0.996303 0.0859072i \(-0.972621\pi\)
−0.996303 + 0.0859072i \(0.972621\pi\)
\(542\) 241.000 417.424i 0.444649 0.770155i
\(543\) 0 0
\(544\) 231.000 + 400.104i 0.424632 + 0.735485i
\(545\) −55.0000 95.2628i −0.100917 0.174794i
\(546\) 0 0
\(547\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(548\) 678.000 1.23723
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −393.000 + 680.696i −0.706835 + 1.22427i
\(557\) 614.000 1.10233 0.551167 0.834395i \(-0.314183\pi\)
0.551167 + 0.834395i \(0.314183\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 77.0000 + 133.368i 0.136767 + 0.236888i 0.926271 0.376858i \(-0.122995\pi\)
−0.789504 + 0.613746i \(0.789662\pi\)
\(564\) 0 0
\(565\) 515.000 892.006i 0.911504 1.57877i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(570\) 0 0
\(571\) 179.000 + 310.037i 0.313485 + 0.542972i 0.979114 0.203310i \(-0.0651701\pi\)
−0.665629 + 0.746283i \(0.731837\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −850.000 −1.47826
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −46.5000 + 80.5404i −0.0804498 + 0.139343i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 394.000 0.672355
\(587\) −427.000 + 739.586i −0.727428 + 1.25994i 0.230539 + 0.973063i \(0.425951\pi\)
−0.957967 + 0.286879i \(0.907382\pi\)
\(588\) 0 0
\(589\) 22.0000 + 38.1051i 0.0373514 + 0.0646946i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1166.00 1.96627 0.983137 0.182873i \(-0.0585396\pi\)
0.983137 + 0.182873i \(0.0585396\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(600\) 0 0
\(601\) −121.000 + 209.578i −0.201331 + 0.348716i −0.948958 0.315404i \(-0.897860\pi\)
0.747626 + 0.664119i \(0.231193\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 714.000 1.18212
\(605\) 302.500 523.945i 0.500000 0.866025i
\(606\) 0 0
\(607\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(608\) −363.000 628.734i −0.597039 1.03410i
\(609\) 0 0
\(610\) 295.000 510.955i 0.483607 0.837631i
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 593.000 + 1027.11i 0.961102 + 1.66468i 0.719741 + 0.694242i \(0.244260\pi\)
0.241361 + 0.970435i \(0.422406\pi\)
\(618\) 0 0
\(619\) −349.000 + 604.486i −0.563813 + 0.976552i 0.433346 + 0.901227i \(0.357333\pi\)
−0.997159 + 0.0753247i \(0.976001\pi\)
\(620\) 30.0000 0.0483871
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −312.500 541.266i −0.500000 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −238.000 −0.377179 −0.188590 0.982056i \(-0.560392\pi\)
−0.188590 + 0.982056i \(0.560392\pi\)
\(632\) −343.000 + 594.093i −0.542722 + 0.940021i
\(633\) 0 0
\(634\) 67.0000 + 116.047i 0.105678 + 0.183040i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −595.000 −0.929688
\(641\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(642\) 0 0
\(643\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −154.000 + 266.736i −0.238390 + 0.412904i
\(647\) −706.000 −1.09119 −0.545595 0.838049i \(-0.683696\pi\)
−0.545595 + 0.838049i \(0.683696\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 557.000 + 964.752i 0.852986 + 1.47742i 0.878501 + 0.477741i \(0.158544\pi\)
−0.0255145 + 0.999674i \(0.508122\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(660\) 0 0
\(661\) 419.000 + 725.729i 0.633888 + 1.09793i 0.986750 + 0.162251i \(0.0518753\pi\)
−0.352862 + 0.935676i \(0.614791\pi\)
\(662\) 61.0000 + 105.655i 0.0921450 + 0.159600i
\(663\) 0 0
\(664\) 539.000 933.575i 0.811747 1.40599i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 381.000 659.911i 0.570359 0.987891i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −507.000 −0.750000
\(677\) −187.000 + 323.894i −0.276219 + 0.478425i −0.970442 0.241335i \(-0.922415\pi\)
0.694223 + 0.719760i \(0.255748\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 245.000 + 424.352i 0.360294 + 0.624048i
\(681\) 0 0
\(682\) 0 0
\(683\) 86.0000 0.125915 0.0629575 0.998016i \(-0.479947\pi\)
0.0629575 + 0.998016i \(0.479947\pi\)
\(684\) 0 0
\(685\) 1130.00 1.64964
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −661.000 + 1144.89i −0.956585 + 1.65685i −0.225885 + 0.974154i \(0.572527\pi\)
−0.730699 + 0.682699i \(0.760806\pi\)
\(692\) 462.000 0.667630
\(693\) 0 0
\(694\) 586.000 0.844380
\(695\) −655.000 + 1134.49i −0.942446 + 1.63236i
\(696\) 0 0
\(697\) 0 0
\(698\) 229.000 + 396.640i 0.328080 + 0.568252i
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −137.000 237.291i −0.194051 0.336106i
\(707\) 0 0
\(708\) 0 0
\(709\) 371.000 642.591i 0.523272 0.906334i −0.476361 0.879250i \(-0.658044\pi\)
0.999633 0.0270842i \(-0.00862223\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 34.0000 58.8897i 0.0476858 0.0825943i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 61.5000 106.521i 0.0851801 0.147536i
\(723\) 0 0
\(724\) 183.000 + 316.965i 0.252762 + 0.437797i
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −561.000 + 971.681i −0.762228 + 1.32022i
\(737\) 0 0
\(738\) 0 0
\(739\) −1462.00 −1.97835 −0.989175 0.146744i \(-0.953121\pi\)
−0.989175 + 0.146744i \(0.953121\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 257.000 + 445.137i 0.345895 + 0.599108i 0.985516 0.169583i \(-0.0542420\pi\)
−0.639621 + 0.768690i \(0.720909\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 719.000 + 1245.34i 0.957390 + 1.65825i 0.728801 + 0.684725i \(0.240078\pi\)
0.228589 + 0.973523i \(0.426589\pi\)
\(752\) −35.0000 60.6218i −0.0465426 0.0806141i
\(753\) 0 0
\(754\) 0 0
\(755\) 1190.00 1.57616
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) −371.000 + 642.591i −0.489446 + 0.847745i
\(759\) 0 0
\(760\) −385.000 666.840i −0.506579 0.877420i
\(761\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) −686.000 −0.895561
\(767\) 0 0
\(768\) 0 0
\(769\) −289.000 500.563i −0.375813 0.650927i 0.614636 0.788811i \(-0.289303\pi\)
−0.990448 + 0.137884i \(0.955970\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1526.00 1.97413 0.987063 0.160330i \(-0.0512560\pi\)
0.987063 + 0.160330i \(0.0512560\pi\)
\(774\) 0 0
\(775\) 50.0000 0.0645161
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 476.000 0.608696
\(783\) 0 0
\(784\) 245.000 0.312500
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(788\) 561.000 + 971.681i 0.711929 + 1.23310i
\(789\) 0 0
\(790\) −245.000 + 424.352i −0.310127 + 0.537155i
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −213.000 368.927i −0.267588 0.463476i
\(797\) 413.000 + 715.337i 0.518193 + 0.897537i 0.999777 + 0.0211367i \(0.00672852\pi\)
−0.481583 + 0.876400i \(0.659938\pi\)
\(798\) 0 0
\(799\) −98.0000 + 169.741i −0.122653 + 0.212442i
\(800\) −825.000 −1.03125
\(801\) 0 0
\(802\)