Properties

Label 912.3.o.c.721.3
Level $912$
Weight $3$
Character 912.721
Analytic conductor $24.850$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 912.o (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.8502001097\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.219615408.1
Defining polynomial: \(x^{6} - 3 x^{5} + 22 x^{4} - 39 x^{3} + 112 x^{2} - 93 x + 39\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: no (minimal twist has level 228)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 721.3
Root \(0.500000 - 0.460304i\) of defining polynomial
Character \(\chi\) \(=\) 912.721
Dual form 912.3.o.c.721.6

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205i q^{3} +5.83539 q^{5} -5.24085 q^{7} -3.00000 q^{9} +O(q^{10})\) \(q-1.73205i q^{3} +5.83539 q^{5} -5.24085 q^{7} -3.00000 q^{9} -1.24085 q^{11} -5.89843i q^{13} -10.1072i q^{15} -1.57007 q^{17} +(-10.6708 - 15.7205i) q^{19} +9.07742i q^{21} -27.5579 q^{23} +9.05177 q^{25} +5.19615i q^{27} +15.9159i q^{29} -53.2554i q^{31} +2.14922i q^{33} -30.5824 q^{35} -10.0175i q^{37} -10.2164 q^{39} -69.8102i q^{41} +52.9116 q^{43} -17.5062 q^{45} -12.2135 q^{47} -21.5335 q^{49} +2.71944i q^{51} -40.4288i q^{53} -7.24085 q^{55} +(-27.2287 + 18.4823i) q^{57} +75.8194i q^{59} -28.0275 q^{61} +15.7226 q^{63} -34.4197i q^{65} -47.3570i q^{67} +47.7318i q^{69} +56.3447i q^{71} -74.9363 q^{73} -15.6781i q^{75} +6.50312 q^{77} +38.2585i q^{79} +9.00000 q^{81} +42.6584 q^{83} -9.16198 q^{85} +27.5672 q^{87} -24.5128i q^{89} +30.9128i q^{91} -92.2411 q^{93} +(-62.2682 - 91.7353i) q^{95} -3.19999i q^{97} +3.72255 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{5} + 2 q^{7} - 18 q^{9} + O(q^{10}) \) \( 6 q - 2 q^{5} + 2 q^{7} - 18 q^{9} + 26 q^{11} - 50 q^{17} + 10 q^{19} - 28 q^{23} + 28 q^{25} - 2 q^{35} - 72 q^{39} + 210 q^{43} + 6 q^{45} - 22 q^{47} - 36 q^{49} - 10 q^{55} + 48 q^{57} + 214 q^{61} - 6 q^{63} + 102 q^{73} + 266 q^{77} + 54 q^{81} + 404 q^{83} + 370 q^{85} + 144 q^{87} - 120 q^{93} - 358 q^{95} - 78 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 1.73205i 0.577350i
\(4\) 0 0
\(5\) 5.83539 1.16708 0.583539 0.812085i \(-0.301668\pi\)
0.583539 + 0.812085i \(0.301668\pi\)
\(6\) 0 0
\(7\) −5.24085 −0.748693 −0.374347 0.927289i \(-0.622133\pi\)
−0.374347 + 0.927289i \(0.622133\pi\)
\(8\) 0 0
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) −1.24085 −0.112805 −0.0564023 0.998408i \(-0.517963\pi\)
−0.0564023 + 0.998408i \(0.517963\pi\)
\(12\) 0 0
\(13\) 5.89843i 0.453726i −0.973927 0.226863i \(-0.927153\pi\)
0.973927 0.226863i \(-0.0728469\pi\)
\(14\) 0 0
\(15\) 10.1072i 0.673813i
\(16\) 0 0
\(17\) −1.57007 −0.0923572 −0.0461786 0.998933i \(-0.514704\pi\)
−0.0461786 + 0.998933i \(0.514704\pi\)
\(18\) 0 0
\(19\) −10.6708 15.7205i −0.561620 0.827395i
\(20\) 0 0
\(21\) 9.07742i 0.432258i
\(22\) 0 0
\(23\) −27.5579 −1.19817 −0.599086 0.800685i \(-0.704469\pi\)
−0.599086 + 0.800685i \(0.704469\pi\)
\(24\) 0 0
\(25\) 9.05177 0.362071
\(26\) 0 0
\(27\) 5.19615i 0.192450i
\(28\) 0 0
\(29\) 15.9159i 0.548826i 0.961612 + 0.274413i \(0.0884835\pi\)
−0.961612 + 0.274413i \(0.911517\pi\)
\(30\) 0 0
\(31\) 53.2554i 1.71792i −0.512046 0.858958i \(-0.671112\pi\)
0.512046 0.858958i \(-0.328888\pi\)
\(32\) 0 0
\(33\) 2.14922i 0.0651278i
\(34\) 0 0
\(35\) −30.5824 −0.873783
\(36\) 0 0
\(37\) 10.0175i 0.270744i −0.990795 0.135372i \(-0.956777\pi\)
0.990795 0.135372i \(-0.0432229\pi\)
\(38\) 0 0
\(39\) −10.2164 −0.261959
\(40\) 0 0
\(41\) 69.8102i 1.70269i −0.524607 0.851344i \(-0.675788\pi\)
0.524607 0.851344i \(-0.324212\pi\)
\(42\) 0 0
\(43\) 52.9116 1.23050 0.615252 0.788331i \(-0.289054\pi\)
0.615252 + 0.788331i \(0.289054\pi\)
\(44\) 0 0
\(45\) −17.5062 −0.389026
\(46\) 0 0
\(47\) −12.2135 −0.259863 −0.129931 0.991523i \(-0.541476\pi\)
−0.129931 + 0.991523i \(0.541476\pi\)
\(48\) 0 0
\(49\) −21.5335 −0.439459
\(50\) 0 0
\(51\) 2.71944i 0.0533224i
\(52\) 0 0
\(53\) 40.4288i 0.762807i −0.924409 0.381403i \(-0.875441\pi\)
0.924409 0.381403i \(-0.124559\pi\)
\(54\) 0 0
\(55\) −7.24085 −0.131652
\(56\) 0 0
\(57\) −27.2287 + 18.4823i −0.477697 + 0.324251i
\(58\) 0 0
\(59\) 75.8194i 1.28507i 0.766255 + 0.642537i \(0.222118\pi\)
−0.766255 + 0.642537i \(0.777882\pi\)
\(60\) 0 0
\(61\) −28.0275 −0.459468 −0.229734 0.973254i \(-0.573786\pi\)
−0.229734 + 0.973254i \(0.573786\pi\)
\(62\) 0 0
\(63\) 15.7226 0.249564
\(64\) 0 0
\(65\) 34.4197i 0.529533i
\(66\) 0 0
\(67\) 47.3570i 0.706820i −0.935468 0.353410i \(-0.885022\pi\)
0.935468 0.353410i \(-0.114978\pi\)
\(68\) 0 0
\(69\) 47.7318i 0.691765i
\(70\) 0 0
\(71\) 56.3447i 0.793588i 0.917908 + 0.396794i \(0.129877\pi\)
−0.917908 + 0.396794i \(0.870123\pi\)
\(72\) 0 0
\(73\) −74.9363 −1.02652 −0.513262 0.858232i \(-0.671563\pi\)
−0.513262 + 0.858232i \(0.671563\pi\)
\(74\) 0 0
\(75\) 15.6781i 0.209042i
\(76\) 0 0
\(77\) 6.50312 0.0844561
\(78\) 0 0
\(79\) 38.2585i 0.484285i 0.970241 + 0.242143i \(0.0778502\pi\)
−0.970241 + 0.242143i \(0.922150\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 42.6584 0.513957 0.256979 0.966417i \(-0.417273\pi\)
0.256979 + 0.966417i \(0.417273\pi\)
\(84\) 0 0
\(85\) −9.16198 −0.107788
\(86\) 0 0
\(87\) 27.5672 0.316865
\(88\) 0 0
\(89\) 24.5128i 0.275425i −0.990472 0.137712i \(-0.956025\pi\)
0.990472 0.137712i \(-0.0439750\pi\)
\(90\) 0 0
\(91\) 30.9128i 0.339701i
\(92\) 0 0
\(93\) −92.2411 −0.991839
\(94\) 0 0
\(95\) −62.2682 91.7353i −0.655454 0.965635i
\(96\) 0 0
\(97\) 3.19999i 0.0329896i −0.999864 0.0164948i \(-0.994749\pi\)
0.999864 0.0164948i \(-0.00525070\pi\)
\(98\) 0 0
\(99\) 3.72255 0.0376016
\(100\) 0 0
\(101\) −67.7650 −0.670941 −0.335471 0.942051i \(-0.608895\pi\)
−0.335471 + 0.942051i \(0.608895\pi\)
\(102\) 0 0
\(103\) 50.0554i 0.485975i 0.970029 + 0.242987i \(0.0781274\pi\)
−0.970029 + 0.242987i \(0.921873\pi\)
\(104\) 0 0
\(105\) 52.9703i 0.504479i
\(106\) 0 0
\(107\) 128.605i 1.20192i −0.799279 0.600960i \(-0.794785\pi\)
0.799279 0.600960i \(-0.205215\pi\)
\(108\) 0 0
\(109\) 40.9303i 0.375508i −0.982216 0.187754i \(-0.939879\pi\)
0.982216 0.187754i \(-0.0601207\pi\)
\(110\) 0 0
\(111\) −17.3508 −0.156314
\(112\) 0 0
\(113\) 161.546i 1.42961i −0.699326 0.714803i \(-0.746516\pi\)
0.699326 0.714803i \(-0.253484\pi\)
\(114\) 0 0
\(115\) −160.811 −1.39836
\(116\) 0 0
\(117\) 17.6953i 0.151242i
\(118\) 0 0
\(119\) 8.22851 0.0691472
\(120\) 0 0
\(121\) −119.460 −0.987275
\(122\) 0 0
\(123\) −120.915 −0.983048
\(124\) 0 0
\(125\) −93.0641 −0.744513
\(126\) 0 0
\(127\) 26.4617i 0.208360i −0.994558 0.104180i \(-0.966778\pi\)
0.994558 0.104180i \(-0.0332218\pi\)
\(128\) 0 0
\(129\) 91.6456i 0.710431i
\(130\) 0 0
\(131\) −23.7282 −0.181132 −0.0905658 0.995890i \(-0.528868\pi\)
−0.0905658 + 0.995890i \(0.528868\pi\)
\(132\) 0 0
\(133\) 55.9240 + 82.3889i 0.420481 + 0.619465i
\(134\) 0 0
\(135\) 30.3216i 0.224604i
\(136\) 0 0
\(137\) 127.778 0.932683 0.466342 0.884605i \(-0.345572\pi\)
0.466342 + 0.884605i \(0.345572\pi\)
\(138\) 0 0
\(139\) −104.967 −0.755156 −0.377578 0.925978i \(-0.623243\pi\)
−0.377578 + 0.925978i \(0.623243\pi\)
\(140\) 0 0
\(141\) 21.1545i 0.150032i
\(142\) 0 0
\(143\) 7.31908i 0.0511824i
\(144\) 0 0
\(145\) 92.8758i 0.640522i
\(146\) 0 0
\(147\) 37.2971i 0.253722i
\(148\) 0 0
\(149\) 144.372 0.968940 0.484470 0.874808i \(-0.339012\pi\)
0.484470 + 0.874808i \(0.339012\pi\)
\(150\) 0 0
\(151\) 34.0021i 0.225180i −0.993642 0.112590i \(-0.964085\pi\)
0.993642 0.112590i \(-0.0359146\pi\)
\(152\) 0 0
\(153\) 4.71022 0.0307857
\(154\) 0 0
\(155\) 310.766i 2.00494i
\(156\) 0 0
\(157\) −167.385 −1.06615 −0.533073 0.846069i \(-0.678963\pi\)
−0.533073 + 0.846069i \(0.678963\pi\)
\(158\) 0 0
\(159\) −70.0247 −0.440407
\(160\) 0 0
\(161\) 144.427 0.897063
\(162\) 0 0
\(163\) −212.281 −1.30234 −0.651169 0.758933i \(-0.725721\pi\)
−0.651169 + 0.758933i \(0.725721\pi\)
\(164\) 0 0
\(165\) 12.5415i 0.0760092i
\(166\) 0 0
\(167\) 52.3097i 0.313231i 0.987660 + 0.156616i \(0.0500584\pi\)
−0.987660 + 0.156616i \(0.949942\pi\)
\(168\) 0 0
\(169\) 134.208 0.794133
\(170\) 0 0
\(171\) 32.0123 + 47.1615i 0.187207 + 0.275798i
\(172\) 0 0
\(173\) 14.9128i 0.0862014i 0.999071 + 0.0431007i \(0.0137236\pi\)
−0.999071 + 0.0431007i \(0.986276\pi\)
\(174\) 0 0
\(175\) −47.4390 −0.271080
\(176\) 0 0
\(177\) 131.323 0.741938
\(178\) 0 0
\(179\) 183.640i 1.02592i −0.858412 0.512961i \(-0.828548\pi\)
0.858412 0.512961i \(-0.171452\pi\)
\(180\) 0 0
\(181\) 87.8964i 0.485616i 0.970074 + 0.242808i \(0.0780684\pi\)
−0.970074 + 0.242808i \(0.921932\pi\)
\(182\) 0 0
\(183\) 48.5451i 0.265274i
\(184\) 0 0
\(185\) 58.4561i 0.315979i
\(186\) 0 0
\(187\) 1.94823 0.0104183
\(188\) 0 0
\(189\) 27.2323i 0.144086i
\(190\) 0 0
\(191\) 269.940 1.41330 0.706649 0.707565i \(-0.250206\pi\)
0.706649 + 0.707565i \(0.250206\pi\)
\(192\) 0 0
\(193\) 361.323i 1.87214i 0.351814 + 0.936070i \(0.385565\pi\)
−0.351814 + 0.936070i \(0.614435\pi\)
\(194\) 0 0
\(195\) −59.6166 −0.305726
\(196\) 0 0
\(197\) −24.6863 −0.125311 −0.0626556 0.998035i \(-0.519957\pi\)
−0.0626556 + 0.998035i \(0.519957\pi\)
\(198\) 0 0
\(199\) 339.851 1.70779 0.853897 0.520441i \(-0.174233\pi\)
0.853897 + 0.520441i \(0.174233\pi\)
\(200\) 0 0
\(201\) −82.0247 −0.408083
\(202\) 0 0
\(203\) 83.4131i 0.410902i
\(204\) 0 0
\(205\) 407.370i 1.98717i
\(206\) 0 0
\(207\) 82.6738 0.399390
\(208\) 0 0
\(209\) 13.2409 + 19.5068i 0.0633534 + 0.0933341i
\(210\) 0 0
\(211\) 376.489i 1.78431i 0.451730 + 0.892155i \(0.350807\pi\)
−0.451730 + 0.892155i \(0.649193\pi\)
\(212\) 0 0
\(213\) 97.5919 0.458178
\(214\) 0 0
\(215\) 308.760 1.43609
\(216\) 0 0
\(217\) 279.104i 1.28619i
\(218\) 0 0
\(219\) 129.793i 0.592664i
\(220\) 0 0
\(221\) 9.26096i 0.0419048i
\(222\) 0 0
\(223\) 370.168i 1.65995i 0.557804 + 0.829973i \(0.311644\pi\)
−0.557804 + 0.829973i \(0.688356\pi\)
\(224\) 0 0
\(225\) −27.1553 −0.120690
\(226\) 0 0
\(227\) 341.595i 1.50482i −0.658693 0.752411i \(-0.728891\pi\)
0.658693 0.752411i \(-0.271109\pi\)
\(228\) 0 0
\(229\) 369.046 1.61155 0.805777 0.592219i \(-0.201748\pi\)
0.805777 + 0.592219i \(0.201748\pi\)
\(230\) 0 0
\(231\) 11.2637i 0.0487607i
\(232\) 0 0
\(233\) 75.5098 0.324076 0.162038 0.986784i \(-0.448193\pi\)
0.162038 + 0.986784i \(0.448193\pi\)
\(234\) 0 0
\(235\) −71.2708 −0.303280
\(236\) 0 0
\(237\) 66.2657 0.279602
\(238\) 0 0
\(239\) 399.287 1.67066 0.835329 0.549750i \(-0.185277\pi\)
0.835329 + 0.549750i \(0.185277\pi\)
\(240\) 0 0
\(241\) 228.357i 0.947541i −0.880648 0.473771i \(-0.842893\pi\)
0.880648 0.473771i \(-0.157107\pi\)
\(242\) 0 0
\(243\) 15.5885i 0.0641500i
\(244\) 0 0
\(245\) −125.656 −0.512883
\(246\) 0 0
\(247\) −92.7264 + 62.9409i −0.375410 + 0.254821i
\(248\) 0 0
\(249\) 73.8866i 0.296733i
\(250\) 0 0
\(251\) 15.0827 0.0600904 0.0300452 0.999549i \(-0.490435\pi\)
0.0300452 + 0.999549i \(0.490435\pi\)
\(252\) 0 0
\(253\) 34.1953 0.135159
\(254\) 0 0
\(255\) 15.8690i 0.0622314i
\(256\) 0 0
\(257\) 223.762i 0.870669i 0.900269 + 0.435335i \(0.143370\pi\)
−0.900269 + 0.435335i \(0.856630\pi\)
\(258\) 0 0
\(259\) 52.5003i 0.202704i
\(260\) 0 0
\(261\) 47.7478i 0.182942i
\(262\) 0 0
\(263\) 50.2140 0.190928 0.0954638 0.995433i \(-0.469567\pi\)
0.0954638 + 0.995433i \(0.469567\pi\)
\(264\) 0 0
\(265\) 235.918i 0.890255i
\(266\) 0 0
\(267\) −42.4574 −0.159017
\(268\) 0 0
\(269\) 329.132i 1.22354i 0.791035 + 0.611770i \(0.209542\pi\)
−0.791035 + 0.611770i \(0.790458\pi\)
\(270\) 0 0
\(271\) 283.879 1.04752 0.523762 0.851864i \(-0.324528\pi\)
0.523762 + 0.851864i \(0.324528\pi\)
\(272\) 0 0
\(273\) 53.5426 0.196127
\(274\) 0 0
\(275\) −11.2319 −0.0408433
\(276\) 0 0
\(277\) 167.315 0.604025 0.302012 0.953304i \(-0.402342\pi\)
0.302012 + 0.953304i \(0.402342\pi\)
\(278\) 0 0
\(279\) 159.766i 0.572639i
\(280\) 0 0
\(281\) 491.818i 1.75024i −0.483903 0.875122i \(-0.660781\pi\)
0.483903 0.875122i \(-0.339219\pi\)
\(282\) 0 0
\(283\) 41.3743 0.146199 0.0730995 0.997325i \(-0.476711\pi\)
0.0730995 + 0.997325i \(0.476711\pi\)
\(284\) 0 0
\(285\) −158.890 + 107.852i −0.557510 + 0.378427i
\(286\) 0 0
\(287\) 365.865i 1.27479i
\(288\) 0 0
\(289\) −286.535 −0.991470
\(290\) 0 0
\(291\) −5.54255 −0.0190466
\(292\) 0 0
\(293\) 416.306i 1.42084i 0.703778 + 0.710420i \(0.251495\pi\)
−0.703778 + 0.710420i \(0.748505\pi\)
\(294\) 0 0
\(295\) 442.436i 1.49978i
\(296\) 0 0
\(297\) 6.44765i 0.0217093i
\(298\) 0 0
\(299\) 162.549i 0.543641i
\(300\) 0 0
\(301\) −277.302 −0.921269
\(302\) 0 0
\(303\) 117.372i 0.387368i
\(304\) 0 0
\(305\) −163.551 −0.536234
\(306\) 0 0
\(307\) 536.008i 1.74595i −0.487762 0.872976i \(-0.662187\pi\)
0.487762 0.872976i \(-0.337813\pi\)
\(308\) 0 0
\(309\) 86.6985 0.280578
\(310\) 0 0
\(311\) 405.293 1.30319 0.651596 0.758566i \(-0.274100\pi\)
0.651596 + 0.758566i \(0.274100\pi\)
\(312\) 0 0
\(313\) −64.9274 −0.207436 −0.103718 0.994607i \(-0.533074\pi\)
−0.103718 + 0.994607i \(0.533074\pi\)
\(314\) 0 0
\(315\) 91.7472 0.291261
\(316\) 0 0
\(317\) 466.133i 1.47045i −0.677823 0.735225i \(-0.737076\pi\)
0.677823 0.735225i \(-0.262924\pi\)
\(318\) 0 0
\(319\) 19.7493i 0.0619101i
\(320\) 0 0
\(321\) −222.751 −0.693928
\(322\) 0 0
\(323\) 16.7539 + 24.6823i 0.0518696 + 0.0764159i
\(324\) 0 0
\(325\) 53.3913i 0.164281i
\(326\) 0 0
\(327\) −70.8934 −0.216799
\(328\) 0 0
\(329\) 64.0094 0.194557
\(330\) 0 0
\(331\) 423.761i 1.28024i −0.768273 0.640122i \(-0.778884\pi\)
0.768273 0.640122i \(-0.221116\pi\)
\(332\) 0 0
\(333\) 30.0525i 0.0902479i
\(334\) 0 0
\(335\) 276.346i 0.824915i
\(336\) 0 0
\(337\) 204.478i 0.606760i 0.952870 + 0.303380i \(0.0981151\pi\)
−0.952870 + 0.303380i \(0.901885\pi\)
\(338\) 0 0
\(339\) −279.805 −0.825384
\(340\) 0 0
\(341\) 66.0820i 0.193789i
\(342\) 0 0
\(343\) 369.655 1.07771
\(344\) 0 0
\(345\) 278.533i 0.807343i
\(346\) 0 0
\(347\) −384.045 −1.10676 −0.553380 0.832929i \(-0.686662\pi\)
−0.553380 + 0.832929i \(0.686662\pi\)
\(348\) 0 0
\(349\) 376.053 1.07751 0.538757 0.842461i \(-0.318894\pi\)
0.538757 + 0.842461i \(0.318894\pi\)
\(350\) 0 0
\(351\) 30.6492 0.0873195
\(352\) 0 0
\(353\) 695.360 1.96986 0.984929 0.172959i \(-0.0553329\pi\)
0.984929 + 0.172959i \(0.0553329\pi\)
\(354\) 0 0
\(355\) 328.793i 0.926179i
\(356\) 0 0
\(357\) 14.2522i 0.0399221i
\(358\) 0 0
\(359\) −456.970 −1.27290 −0.636448 0.771319i \(-0.719597\pi\)
−0.636448 + 0.771319i \(0.719597\pi\)
\(360\) 0 0
\(361\) −133.269 + 335.500i −0.369166 + 0.929363i
\(362\) 0 0
\(363\) 206.911i 0.570004i
\(364\) 0 0
\(365\) −437.283 −1.19803
\(366\) 0 0
\(367\) −337.233 −0.918891 −0.459446 0.888206i \(-0.651952\pi\)
−0.459446 + 0.888206i \(0.651952\pi\)
\(368\) 0 0
\(369\) 209.431i 0.567563i
\(370\) 0 0
\(371\) 211.881i 0.571108i
\(372\) 0 0
\(373\) 343.151i 0.919977i 0.887925 + 0.459988i \(0.152146\pi\)
−0.887925 + 0.459988i \(0.847854\pi\)
\(374\) 0 0
\(375\) 161.192i 0.429845i
\(376\) 0 0
\(377\) 93.8791 0.249016
\(378\) 0 0
\(379\) 397.158i 1.04791i −0.851746 0.523955i \(-0.824456\pi\)
0.851746 0.523955i \(-0.175544\pi\)
\(380\) 0 0
\(381\) −45.8330 −0.120297
\(382\) 0 0
\(383\) 474.318i 1.23843i −0.785222 0.619214i \(-0.787451\pi\)
0.785222 0.619214i \(-0.212549\pi\)
\(384\) 0 0
\(385\) 37.9482 0.0985668
\(386\) 0 0
\(387\) −158.735 −0.410168
\(388\) 0 0
\(389\) 539.232 1.38620 0.693101 0.720841i \(-0.256244\pi\)
0.693101 + 0.720841i \(0.256244\pi\)
\(390\) 0 0
\(391\) 43.2679 0.110660
\(392\) 0 0
\(393\) 41.0985i 0.104576i
\(394\) 0 0
\(395\) 223.254i 0.565199i
\(396\) 0 0
\(397\) 421.186 1.06092 0.530461 0.847709i \(-0.322019\pi\)
0.530461 + 0.847709i \(0.322019\pi\)
\(398\) 0 0
\(399\) 142.702 96.8632i 0.357648 0.242765i
\(400\) 0 0
\(401\) 433.330i 1.08062i −0.841465 0.540312i \(-0.818306\pi\)
0.841465 0.540312i \(-0.181694\pi\)
\(402\) 0 0
\(403\) −314.123 −0.779463
\(404\) 0 0
\(405\) 52.5185 0.129675
\(406\) 0 0
\(407\) 12.4302i 0.0305411i
\(408\) 0 0
\(409\) 569.984i 1.39361i 0.717263 + 0.696803i \(0.245395\pi\)
−0.717263 + 0.696803i \(0.754605\pi\)
\(410\) 0 0
\(411\) 221.317i 0.538485i
\(412\) 0 0
\(413\) 397.358i 0.962126i
\(414\) 0 0
\(415\) 248.929 0.599828
\(416\) 0 0
\(417\) 181.808i 0.435989i
\(418\) 0 0
\(419\) −156.568 −0.373672 −0.186836 0.982391i \(-0.559823\pi\)
−0.186836 + 0.982391i \(0.559823\pi\)
\(420\) 0 0
\(421\) 367.275i 0.872387i −0.899853 0.436193i \(-0.856326\pi\)
0.899853 0.436193i \(-0.143674\pi\)
\(422\) 0 0
\(423\) 36.6406 0.0866209
\(424\) 0 0
\(425\) −14.2119 −0.0334398
\(426\) 0 0
\(427\) 146.888 0.344000
\(428\) 0 0
\(429\) 12.6770 0.0295502
\(430\) 0 0
\(431\) 289.780i 0.672343i 0.941801 + 0.336171i \(0.109132\pi\)
−0.941801 + 0.336171i \(0.890868\pi\)
\(432\) 0 0
\(433\) 284.531i 0.657116i −0.944484 0.328558i \(-0.893437\pi\)
0.944484 0.328558i \(-0.106563\pi\)
\(434\) 0 0
\(435\) 160.866 0.369806
\(436\) 0 0
\(437\) 294.065 + 433.225i 0.672917 + 0.991361i
\(438\) 0 0
\(439\) 628.180i 1.43093i 0.698646 + 0.715467i \(0.253786\pi\)
−0.698646 + 0.715467i \(0.746214\pi\)
\(440\) 0 0
\(441\) 64.6004 0.146486
\(442\) 0 0
\(443\) 416.332 0.939801 0.469900 0.882720i \(-0.344290\pi\)
0.469900 + 0.882720i \(0.344290\pi\)
\(444\) 0 0
\(445\) 143.042i 0.321442i
\(446\) 0 0
\(447\) 250.060i 0.559418i
\(448\) 0 0
\(449\) 499.834i 1.11322i 0.830775 + 0.556608i \(0.187897\pi\)
−0.830775 + 0.556608i \(0.812103\pi\)
\(450\) 0 0
\(451\) 86.6241i 0.192071i
\(452\) 0 0
\(453\) −58.8934 −0.130008
\(454\) 0 0
\(455\) 180.388i 0.396458i
\(456\) 0 0
\(457\) 143.028 0.312973 0.156486 0.987680i \(-0.449983\pi\)
0.156486 + 0.987680i \(0.449983\pi\)
\(458\) 0 0
\(459\) 8.15833i 0.0177741i
\(460\) 0 0
\(461\) 562.915 1.22107 0.610537 0.791988i \(-0.290954\pi\)
0.610537 + 0.791988i \(0.290954\pi\)
\(462\) 0 0
\(463\) −253.173 −0.546810 −0.273405 0.961899i \(-0.588150\pi\)
−0.273405 + 0.961899i \(0.588150\pi\)
\(464\) 0 0
\(465\) −538.263 −1.15755
\(466\) 0 0
\(467\) −246.479 −0.527793 −0.263896 0.964551i \(-0.585008\pi\)
−0.263896 + 0.964551i \(0.585008\pi\)
\(468\) 0 0
\(469\) 248.191i 0.529192i
\(470\) 0 0
\(471\) 289.919i 0.615539i
\(472\) 0 0
\(473\) −65.6555 −0.138806
\(474\) 0 0
\(475\) −96.5895 142.299i −0.203346 0.299576i
\(476\) 0 0
\(477\) 121.286i 0.254269i
\(478\) 0 0
\(479\) 223.144 0.465853 0.232927 0.972494i \(-0.425170\pi\)
0.232927 + 0.972494i \(0.425170\pi\)
\(480\) 0 0
\(481\) −59.0876 −0.122843
\(482\) 0 0
\(483\) 250.155i 0.517919i
\(484\) 0 0
\(485\) 18.6732i 0.0385015i
\(486\) 0 0
\(487\) 674.619i 1.38526i −0.721295 0.692628i \(-0.756453\pi\)
0.721295 0.692628i \(-0.243547\pi\)
\(488\) 0 0
\(489\) 367.682i 0.751905i
\(490\) 0 0
\(491\) −31.6662 −0.0644933 −0.0322466 0.999480i \(-0.510266\pi\)
−0.0322466 + 0.999480i \(0.510266\pi\)
\(492\) 0 0
\(493\) 24.9892i 0.0506880i
\(494\) 0 0
\(495\) 21.7226 0.0438839
\(496\) 0 0
\(497\) 295.294i 0.594153i
\(498\) 0 0
\(499\) −151.589 −0.303786 −0.151893 0.988397i \(-0.548537\pi\)
−0.151893 + 0.988397i \(0.548537\pi\)
\(500\) 0 0
\(501\) 90.6030 0.180844
\(502\) 0 0
\(503\) −355.069 −0.705903 −0.352952 0.935642i \(-0.614822\pi\)
−0.352952 + 0.935642i \(0.614822\pi\)
\(504\) 0 0
\(505\) −395.435 −0.783040
\(506\) 0 0
\(507\) 232.456i 0.458493i
\(508\) 0 0
\(509\) 411.332i 0.808118i −0.914733 0.404059i \(-0.867599\pi\)
0.914733 0.404059i \(-0.132401\pi\)
\(510\) 0 0
\(511\) 392.730 0.768552
\(512\) 0 0
\(513\) 81.6862 55.4470i 0.159232 0.108084i
\(514\) 0 0
\(515\) 292.093i 0.567171i
\(516\) 0 0
\(517\) 15.1552 0.0293137
\(518\) 0 0
\(519\) 25.8298 0.0497684
\(520\) 0 0
\(521\) 538.069i 1.03276i −0.856359 0.516381i \(-0.827279\pi\)
0.856359 0.516381i \(-0.172721\pi\)
\(522\) 0 0
\(523\) 367.164i 0.702035i −0.936369 0.351017i \(-0.885836\pi\)
0.936369 0.351017i \(-0.114164\pi\)
\(524\) 0 0
\(525\) 82.1668i 0.156508i
\(526\) 0 0
\(527\) 83.6148i 0.158662i
\(528\) 0 0
\(529\) 230.440 0.435615
\(530\) 0 0
\(531\) 227.458i 0.428358i
\(532\) 0 0
\(533\) −411.771 −0.772554
\(534\) 0 0
\(535\) 750.463i 1.40273i
\(536\) 0 0
\(537\) −318.074 −0.592317
\(538\) 0 0
\(539\) 26.7198 0.0495730
\(540\) 0 0
\(541\) −873.057 −1.61378 −0.806892 0.590699i \(-0.798852\pi\)
−0.806892 + 0.590699i \(0.798852\pi\)
\(542\) 0 0
\(543\) 152.241 0.280370
\(544\) 0 0
\(545\) 238.844i 0.438247i
\(546\) 0 0
\(547\) 855.495i 1.56398i 0.623293 + 0.781988i \(0.285794\pi\)
−0.623293 + 0.781988i \(0.714206\pi\)
\(548\) 0 0
\(549\) 84.0826 0.153156
\(550\) 0 0
\(551\) 250.207 169.836i 0.454096 0.308232i
\(552\) 0 0
\(553\) 200.507i 0.362581i
\(554\) 0 0
\(555\) −101.249 −0.182431
\(556\) 0 0
\(557\) −140.061 −0.251455 −0.125728 0.992065i \(-0.540127\pi\)
−0.125728 + 0.992065i \(0.540127\pi\)
\(558\) 0 0
\(559\) 312.096i 0.558311i
\(560\) 0 0
\(561\) 3.37443i 0.00601502i
\(562\) 0 0
\(563\) 10.5527i 0.0187437i −0.999956 0.00937185i \(-0.997017\pi\)
0.999956 0.00937185i \(-0.00298320\pi\)
\(564\) 0 0
\(565\) 942.681i 1.66846i
\(566\) 0 0
\(567\) −47.1677 −0.0831881
\(568\) 0 0
\(569\) 971.756i 1.70783i −0.520412 0.853916i \(-0.674221\pi\)
0.520412 0.853916i \(-0.325779\pi\)
\(570\) 0 0
\(571\) 895.734 1.56871 0.784355 0.620312i \(-0.212994\pi\)
0.784355 + 0.620312i \(0.212994\pi\)
\(572\) 0 0
\(573\) 467.549i 0.815968i
\(574\) 0 0
\(575\) −249.448 −0.433823
\(576\) 0 0
\(577\) 209.918 0.363809 0.181905 0.983316i \(-0.441774\pi\)
0.181905 + 0.983316i \(0.441774\pi\)
\(578\) 0 0
\(579\) 625.830 1.08088
\(580\) 0 0
\(581\) −223.567 −0.384796
\(582\) 0 0
\(583\) 50.1661i 0.0860482i
\(584\) 0 0
\(585\) 103.259i 0.176511i
\(586\) 0 0
\(587\) −1032.25 −1.75852 −0.879258 0.476345i \(-0.841961\pi\)
−0.879258 + 0.476345i \(0.841961\pi\)
\(588\) 0 0
\(589\) −837.202 + 568.277i −1.42140 + 0.964816i
\(590\) 0 0
\(591\) 42.7579i 0.0723484i
\(592\) 0 0
\(593\) −1030.88 −1.73842 −0.869210 0.494442i \(-0.835372\pi\)
−0.869210 + 0.494442i \(0.835372\pi\)
\(594\) 0 0
\(595\) 48.0166 0.0807001
\(596\) 0 0
\(597\) 588.640i 0.985996i
\(598\) 0 0
\(599\) 83.7382i 0.139797i 0.997554 + 0.0698983i \(0.0222675\pi\)
−0.997554 + 0.0698983i \(0.977733\pi\)
\(600\) 0 0
\(601\) 293.467i 0.488298i 0.969738 + 0.244149i \(0.0785086\pi\)
−0.969738 + 0.244149i \(0.921491\pi\)
\(602\) 0 0
\(603\) 142.071i 0.235607i
\(604\) 0 0
\(605\) −697.097 −1.15223
\(606\) 0 0
\(607\) 418.633i 0.689676i 0.938662 + 0.344838i \(0.112066\pi\)
−0.938662 + 0.344838i \(0.887934\pi\)
\(608\) 0 0
\(609\) −144.476 −0.237234
\(610\) 0 0
\(611\) 72.0408i 0.117906i
\(612\) 0 0
\(613\) −377.709 −0.616165 −0.308083 0.951360i \(-0.599687\pi\)
−0.308083 + 0.951360i \(0.599687\pi\)
\(614\) 0 0
\(615\) −705.586 −1.14729
\(616\) 0 0
\(617\) 845.723 1.37070 0.685351 0.728213i \(-0.259649\pi\)
0.685351 + 0.728213i \(0.259649\pi\)
\(618\) 0 0
\(619\) −410.355 −0.662932 −0.331466 0.943467i \(-0.607543\pi\)
−0.331466 + 0.943467i \(0.607543\pi\)
\(620\) 0 0
\(621\) 143.195i 0.230588i
\(622\) 0 0
\(623\) 128.468i 0.206209i
\(624\) 0 0
\(625\) −769.360 −1.23098
\(626\) 0 0
\(627\) 33.7868 22.9338i 0.0538864 0.0365771i
\(628\) 0 0
\(629\) 15.7282i 0.0250051i
\(630\) 0 0
\(631\) −313.898 −0.497462 −0.248731 0.968573i \(-0.580013\pi\)
−0.248731 + 0.968573i \(0.580013\pi\)
\(632\) 0 0
\(633\) 652.099 1.03017
\(634\) 0 0
\(635\) 154.414i 0.243172i
\(636\) 0 0
\(637\) 127.014i 0.199394i
\(638\) 0 0
\(639\) 169.034i 0.264529i
\(640\) 0 0
\(641\) 425.842i 0.664339i −0.943220 0.332170i \(-0.892219\pi\)
0.943220 0.332170i \(-0.107781\pi\)
\(642\) 0 0
\(643\) −1038.06 −1.61441 −0.807204 0.590272i \(-0.799020\pi\)
−0.807204 + 0.590272i \(0.799020\pi\)
\(644\) 0 0
\(645\) 534.788i 0.829129i
\(646\) 0 0
\(647\) −174.684 −0.269991 −0.134995 0.990846i \(-0.543102\pi\)
−0.134995 + 0.990846i \(0.543102\pi\)
\(648\) 0 0
\(649\) 94.0806i 0.144962i
\(650\) 0 0
\(651\) 483.422 0.742583
\(652\) 0 0
\(653\) −1247.54 −1.91048 −0.955238 0.295839i \(-0.904401\pi\)
−0.955238 + 0.295839i \(0.904401\pi\)
\(654\) 0 0
\(655\) −138.464 −0.211395
\(656\) 0 0
\(657\) 224.809 0.342175
\(658\) 0 0
\(659\) 746.281i 1.13244i −0.824253 0.566222i \(-0.808405\pi\)
0.824253 0.566222i \(-0.191595\pi\)
\(660\) 0 0
\(661\) 73.1406i 0.110651i 0.998468 + 0.0553257i \(0.0176197\pi\)
−0.998468 + 0.0553257i \(0.982380\pi\)
\(662\) 0 0
\(663\) 16.0405 0.0241938
\(664\) 0 0
\(665\) 326.338 + 480.771i 0.490734 + 0.722964i
\(666\) 0 0
\(667\) 438.611i 0.657587i
\(668\) 0 0
\(669\) 641.150 0.958370
\(670\) 0 0
\(671\) 34.7780 0.0518301
\(672\) 0 0
\(673\) 939.481i 1.39596i −0.716117 0.697980i \(-0.754082\pi\)
0.716117 0.697980i \(-0.245918\pi\)
\(674\) 0 0
\(675\) 47.0344i 0.0696806i
\(676\) 0 0
\(677\) 666.060i 0.983841i 0.870640 + 0.491920i \(0.163705\pi\)
−0.870640 + 0.491920i \(0.836295\pi\)
\(678\) 0 0
\(679\) 16.7707i 0.0246991i
\(680\) 0 0
\(681\) −591.660 −0.868810
\(682\) 0 0
\(683\) 79.9509i 0.117058i −0.998286 0.0585292i \(-0.981359\pi\)
0.998286 0.0585292i \(-0.0186411\pi\)
\(684\) 0 0
\(685\) 745.632 1.08851
\(686\) 0 0
\(687\) 639.206i 0.930431i
\(688\) 0 0
\(689\) −238.466 −0.346105
\(690\) 0 0
\(691\) −208.333 −0.301495 −0.150748 0.988572i \(-0.548168\pi\)
−0.150748 + 0.988572i \(0.548168\pi\)
\(692\) 0 0
\(693\) −19.5094 −0.0281520
\(694\) 0 0
\(695\) −612.521 −0.881326
\(696\) 0 0
\(697\) 109.607i 0.157256i
\(698\) 0 0
\(699\) 130.787i 0.187106i
\(700\) 0 0
\(701\) 21.0082 0.0299688 0.0149844 0.999888i \(-0.495230\pi\)
0.0149844 + 0.999888i \(0.495230\pi\)
\(702\) 0 0
\(703\) −157.480 + 106.895i −0.224012 + 0.152055i
\(704\) 0 0
\(705\) 123.445i 0.175099i
\(706\) 0 0
\(707\) 355.147 0.502329
\(708\) 0 0
\(709\) 230.915 0.325691 0.162845 0.986652i \(-0.447933\pi\)
0.162845 + 0.986652i \(0.447933\pi\)
\(710\) 0 0
\(711\) 114.776i 0.161428i
\(712\) 0 0
\(713\) 1467.61i 2.05836i
\(714\) 0 0
\(715\) 42.7097i 0.0597338i
\(716\) 0 0
\(717\) 691.586i 0.964555i
\(718\) 0 0
\(719\) 846.640 1.17752 0.588762 0.808306i \(-0.299615\pi\)
0.588762 + 0.808306i \(0.299615\pi\)
\(720\) 0 0
\(721\) 262.333i 0.363846i
\(722\) 0 0
\(723\) −395.527 −0.547063
\(724\) 0 0
\(725\) 144.068i 0.198714i
\(726\) 0 0
\(727\) −1078.59 −1.48362 −0.741810 0.670610i \(-0.766032\pi\)
−0.741810 + 0.670610i \(0.766032\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) 0 0
\(731\) −83.0751 −0.113646
\(732\) 0 0
\(733\) −648.893 −0.885256 −0.442628 0.896705i \(-0.645954\pi\)
−0.442628 + 0.896705i \(0.645954\pi\)
\(734\) 0 0
\(735\) 217.643i 0.296113i
\(736\) 0 0
\(737\) 58.7630i 0.0797326i
\(738\) 0 0
\(739\) 1022.09 1.38307 0.691534 0.722344i \(-0.256935\pi\)
0.691534 + 0.722344i \(0.256935\pi\)
\(740\) 0 0
\(741\) 109.017 + 160.607i 0.147121 + 0.216743i
\(742\) 0 0
\(743\) 257.760i 0.346918i −0.984841 0.173459i \(-0.944506\pi\)
0.984841 0.173459i \(-0.0554944\pi\)
\(744\) 0 0
\(745\) 842.468 1.13083
\(746\) 0 0
\(747\) −127.975 −0.171319
\(748\) 0 0
\(749\) 674.002i 0.899869i
\(750\) 0 0
\(751\) 553.605i 0.737157i 0.929597 + 0.368578i \(0.120155\pi\)
−0.929597 + 0.368578i \(0.879845\pi\)
\(752\) 0 0
\(753\) 26.1240i 0.0346932i
\(754\) 0 0
\(755\) 198.416i 0.262802i
\(756\) 0 0
\(757\) 456.061 0.602458 0.301229 0.953552i \(-0.402603\pi\)
0.301229 + 0.953552i \(0.402603\pi\)
\(758\) 0 0
\(759\) 59.2280i 0.0780343i
\(760\) 0 0
\(761\) 536.682 0.705233 0.352616 0.935768i \(-0.385292\pi\)
0.352616 + 0.935768i \(0.385292\pi\)
\(762\) 0 0
\(763\) 214.510i 0.281140i
\(764\) 0 0
\(765\) 27.4859 0.0359293
\(766\) 0 0
\(767\) 447.215 0.583071
\(768\) 0 0
\(769\) 814.347 1.05897 0.529484 0.848320i \(-0.322385\pi\)
0.529484 + 0.848320i \(0.322385\pi\)
\(770\) 0 0
\(771\) 387.567 0.502681
\(772\) 0 0
\(773\) 1451.94i 1.87831i 0.343490 + 0.939156i \(0.388391\pi\)
−0.343490 + 0.939156i \(0.611609\pi\)
\(774\) 0 0
\(775\) 482.056i 0.622008i
\(776\) 0 0
\(777\) 90.9332 0.117031
\(778\) 0 0
\(779\) −1097.45 + 744.930i −1.40880 + 0.956264i
\(780\) 0 0
\(781\) 69.9154i 0.0895204i
\(782\) 0 0
\(783\) −82.7017 −0.105622
\(784\) 0 0
\(785\) −976.756 −1.24427
\(786\) 0 0
\(787\) 389.048i 0.494343i −0.968972 0.247172i \(-0.920499\pi\)
0.968972 0.247172i \(-0.0795012\pi\)
\(788\) 0 0
\(789\) 86.9731i 0.110232i
\(790\) 0 0
\(791\) 846.636i 1.07034i
\(792\) 0 0
\(793\) 165.318i 0.208472i
\(794\) 0 0
\(795\) −408.621 −0.513989
\(796\) 0 0
\(797\) 1243.63i 1.56039i 0.625539 + 0.780193i \(0.284879\pi\)
−0.625539 + 0.780193i \(0.715121\pi\)
\(798\) 0 0
\(799\) 19.1761 0.0240002
\(800\) 0 0
\(801\) 73.5385i 0.0918083i
\(802\) 0 0