Properties

Label 1152.3.m.c.415.7
Level $1152$
Weight $3$
Character 1152.415
Analytic conductor $31.390$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1152.m (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(31.3897264543\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + 1408 x^{6} + 3584 x^{5} + 2560 x^{4} - 4096 x^{3} - 24576 x^{2} + 65536\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{28} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 415.7
Root \(0.125358 + 1.99607i\) of defining polynomial
Character \(\chi\) \(=\) 1152.415
Dual form 1152.3.m.c.991.7

$q$-expansion

\(f(q)\) \(=\) \(q+(3.32679 + 3.32679i) q^{5} +4.04088 q^{7} +O(q^{10})\) \(q+(3.32679 + 3.32679i) q^{5} +4.04088 q^{7} +(-6.82458 + 6.82458i) q^{11} +(-4.29091 + 4.29091i) q^{13} -30.1192 q^{17} +(-19.7548 - 19.7548i) q^{19} -28.2345 q^{23} -2.86488i q^{25} +(-21.3607 + 21.3607i) q^{29} -38.0396i q^{31} +(13.4432 + 13.4432i) q^{35} +(42.8916 + 42.8916i) q^{37} -48.2343i q^{41} +(32.6765 - 32.6765i) q^{43} -15.8305i q^{47} -32.6713 q^{49} +(-0.476870 - 0.476870i) q^{53} -45.4079 q^{55} +(-9.97719 + 9.97719i) q^{59} +(-37.9455 + 37.9455i) q^{61} -28.5500 q^{65} +(20.0705 + 20.0705i) q^{67} +40.0818 q^{71} +30.8095i q^{73} +(-27.5773 + 27.5773i) q^{77} -130.125i q^{79} +(2.26155 + 2.26155i) q^{83} +(-100.200 - 100.200i) q^{85} +72.2232i q^{89} +(-17.3391 + 17.3391i) q^{91} -131.441i q^{95} -112.343 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 32q^{11} - 32q^{19} - 128q^{23} + 32q^{29} - 96q^{35} + 96q^{37} + 160q^{43} + 112q^{49} - 160q^{53} + 256q^{55} + 128q^{59} + 32q^{61} + 32q^{65} + 320q^{67} + 512q^{71} + 224q^{77} + 160q^{83} - 160q^{85} - 480q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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 0
\(3\) 0 0
\(4\) 0 0
\(5\) 3.32679 + 3.32679i 0.665359 + 0.665359i 0.956638 0.291279i \(-0.0940809\pi\)
−0.291279 + 0.956638i \(0.594081\pi\)
\(6\) 0 0
\(7\) 4.04088 0.577269 0.288635 0.957439i \(-0.406799\pi\)
0.288635 + 0.957439i \(0.406799\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −6.82458 + 6.82458i −0.620416 + 0.620416i −0.945638 0.325222i \(-0.894561\pi\)
0.325222 + 0.945638i \(0.394561\pi\)
\(12\) 0 0
\(13\) −4.29091 + 4.29091i −0.330070 + 0.330070i −0.852613 0.522543i \(-0.824983\pi\)
0.522543 + 0.852613i \(0.324983\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −30.1192 −1.77172 −0.885859 0.463954i \(-0.846430\pi\)
−0.885859 + 0.463954i \(0.846430\pi\)
\(18\) 0 0
\(19\) −19.7548 19.7548i −1.03973 1.03973i −0.999177 0.0405505i \(-0.987089\pi\)
−0.0405505 0.999177i \(-0.512911\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −28.2345 −1.22759 −0.613794 0.789466i \(-0.710358\pi\)
−0.613794 + 0.789466i \(0.710358\pi\)
\(24\) 0 0
\(25\) 2.86488i 0.114595i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −21.3607 + 21.3607i −0.736575 + 0.736575i −0.971914 0.235338i \(-0.924380\pi\)
0.235338 + 0.971914i \(0.424380\pi\)
\(30\) 0 0
\(31\) 38.0396i 1.22708i −0.789662 0.613541i \(-0.789744\pi\)
0.789662 0.613541i \(-0.210256\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 13.4432 + 13.4432i 0.384091 + 0.384091i
\(36\) 0 0
\(37\) 42.8916 + 42.8916i 1.15923 + 1.15923i 0.984641 + 0.174590i \(0.0558600\pi\)
0.174590 + 0.984641i \(0.444140\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 48.2343i 1.17645i −0.808699 0.588223i \(-0.799828\pi\)
0.808699 0.588223i \(-0.200172\pi\)
\(42\) 0 0
\(43\) 32.6765 32.6765i 0.759918 0.759918i −0.216389 0.976307i \(-0.569428\pi\)
0.976307 + 0.216389i \(0.0694281\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 15.8305i 0.336818i −0.985717 0.168409i \(-0.946137\pi\)
0.985717 0.168409i \(-0.0538630\pi\)
\(48\) 0 0
\(49\) −32.6713 −0.666760
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.476870 0.476870i −0.00899755 0.00899755i 0.702594 0.711591i \(-0.252025\pi\)
−0.711591 + 0.702594i \(0.752025\pi\)
\(54\) 0 0
\(55\) −45.4079 −0.825599
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −9.97719 + 9.97719i −0.169105 + 0.169105i −0.786586 0.617481i \(-0.788153\pi\)
0.617481 + 0.786586i \(0.288153\pi\)
\(60\) 0 0
\(61\) −37.9455 + 37.9455i −0.622057 + 0.622057i −0.946057 0.324000i \(-0.894972\pi\)
0.324000 + 0.946057i \(0.394972\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −28.5500 −0.439230
\(66\) 0 0
\(67\) 20.0705 + 20.0705i 0.299559 + 0.299559i 0.840841 0.541282i \(-0.182061\pi\)
−0.541282 + 0.840841i \(0.682061\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 40.0818 0.564532 0.282266 0.959336i \(-0.408914\pi\)
0.282266 + 0.959336i \(0.408914\pi\)
\(72\) 0 0
\(73\) 30.8095i 0.422049i 0.977481 + 0.211024i \(0.0676799\pi\)
−0.977481 + 0.211024i \(0.932320\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −27.5773 + 27.5773i −0.358147 + 0.358147i
\(78\) 0 0
\(79\) 130.125i 1.64716i −0.567203 0.823578i \(-0.691975\pi\)
0.567203 0.823578i \(-0.308025\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.26155 + 2.26155i 0.0272476 + 0.0272476i 0.720599 0.693352i \(-0.243867\pi\)
−0.693352 + 0.720599i \(0.743867\pi\)
\(84\) 0 0
\(85\) −100.200 100.200i −1.17883 1.17883i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 72.2232i 0.811496i 0.913985 + 0.405748i \(0.132989\pi\)
−0.913985 + 0.405748i \(0.867011\pi\)
\(90\) 0 0
\(91\) −17.3391 + 17.3391i −0.190539 + 0.190539i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 131.441i 1.38358i
\(96\) 0 0
\(97\) −112.343 −1.15817 −0.579085 0.815267i \(-0.696590\pi\)
−0.579085 + 0.815267i \(0.696590\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.61933 1.61933i −0.0160330 0.0160330i 0.699045 0.715078i \(-0.253609\pi\)
−0.715078 + 0.699045i \(0.753609\pi\)
\(102\) 0 0
\(103\) −27.9974 −0.271819 −0.135910 0.990721i \(-0.543396\pi\)
−0.135910 + 0.990721i \(0.543396\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 40.3835 40.3835i 0.377416 0.377416i −0.492753 0.870169i \(-0.664010\pi\)
0.870169 + 0.492753i \(0.164010\pi\)
\(108\) 0 0
\(109\) −36.8336 + 36.8336i −0.337923 + 0.337923i −0.855585 0.517662i \(-0.826802\pi\)
0.517662 + 0.855585i \(0.326802\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 55.5952 0.491993 0.245997 0.969271i \(-0.420885\pi\)
0.245997 + 0.969271i \(0.420885\pi\)
\(114\) 0 0
\(115\) −93.9305 93.9305i −0.816787 0.816787i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −121.708 −1.02276
\(120\) 0 0
\(121\) 27.8503i 0.230167i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 92.7007 92.7007i 0.741606 0.741606i
\(126\) 0 0
\(127\) 109.927i 0.865569i 0.901497 + 0.432785i \(0.142469\pi\)
−0.901497 + 0.432785i \(0.857531\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −75.6795 75.6795i −0.577706 0.577706i 0.356565 0.934271i \(-0.383948\pi\)
−0.934271 + 0.356565i \(0.883948\pi\)
\(132\) 0 0
\(133\) −79.8270 79.8270i −0.600203 0.600203i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.14751i 0.0156752i 0.999969 + 0.00783762i \(0.00249482\pi\)
−0.999969 + 0.00783762i \(0.997505\pi\)
\(138\) 0 0
\(139\) −109.246 + 109.246i −0.785941 + 0.785941i −0.980826 0.194885i \(-0.937567\pi\)
0.194885 + 0.980826i \(0.437567\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 58.5673i 0.409562i
\(144\) 0 0
\(145\) −142.125 −0.980174
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 79.6950 + 79.6950i 0.534866 + 0.534866i 0.922016 0.387151i \(-0.126541\pi\)
−0.387151 + 0.922016i \(0.626541\pi\)
\(150\) 0 0
\(151\) −105.546 −0.698982 −0.349491 0.936940i \(-0.613645\pi\)
−0.349491 + 0.936940i \(0.613645\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 126.550 126.550i 0.816451 0.816451i
\(156\) 0 0
\(157\) −190.060 + 190.060i −1.21057 + 1.21057i −0.239733 + 0.970839i \(0.577060\pi\)
−0.970839 + 0.239733i \(0.922940\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −114.092 −0.708649
\(162\) 0 0
\(163\) −59.4130 59.4130i −0.364497 0.364497i 0.500969 0.865465i \(-0.332977\pi\)
−0.865465 + 0.500969i \(0.832977\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 65.3894 0.391553 0.195777 0.980649i \(-0.437277\pi\)
0.195777 + 0.980649i \(0.437277\pi\)
\(168\) 0 0
\(169\) 132.176i 0.782107i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −212.939 + 212.939i −1.23086 + 1.23086i −0.267228 + 0.963633i \(0.586108\pi\)
−0.963633 + 0.267228i \(0.913892\pi\)
\(174\) 0 0
\(175\) 11.5766i 0.0661522i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −196.852 196.852i −1.09973 1.09973i −0.994442 0.105289i \(-0.966423\pi\)
−0.105289 0.994442i \(-0.533577\pi\)
\(180\) 0 0
\(181\) 27.4330 + 27.4330i 0.151564 + 0.151564i 0.778816 0.627252i \(-0.215821\pi\)
−0.627252 + 0.778816i \(0.715821\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 285.383i 1.54261i
\(186\) 0 0
\(187\) 205.551 205.551i 1.09920 1.09920i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 244.409i 1.27963i 0.768530 + 0.639814i \(0.220989\pi\)
−0.768530 + 0.639814i \(0.779011\pi\)
\(192\) 0 0
\(193\) 255.040 1.32145 0.660726 0.750627i \(-0.270249\pi\)
0.660726 + 0.750627i \(0.270249\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −194.229 194.229i −0.985936 0.985936i 0.0139666 0.999902i \(-0.495554\pi\)
−0.999902 + 0.0139666i \(0.995554\pi\)
\(198\) 0 0
\(199\) 169.797 0.853252 0.426626 0.904428i \(-0.359702\pi\)
0.426626 + 0.904428i \(0.359702\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −86.3160 + 86.3160i −0.425202 + 0.425202i
\(204\) 0 0
\(205\) 160.466 160.466i 0.782759 0.782759i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 269.637 1.29013
\(210\) 0 0
\(211\) −132.691 132.691i −0.628868 0.628868i 0.318915 0.947783i \(-0.396681\pi\)
−0.947783 + 0.318915i \(0.896681\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 217.416 1.01124
\(216\) 0 0
\(217\) 153.713i 0.708357i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 129.239 129.239i 0.584791 0.584791i
\(222\) 0 0
\(223\) 26.3436i 0.118133i 0.998254 + 0.0590664i \(0.0188124\pi\)
−0.998254 + 0.0590664i \(0.981188\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 70.3362 + 70.3362i 0.309851 + 0.309851i 0.844852 0.535001i \(-0.179689\pi\)
−0.535001 + 0.844852i \(0.679689\pi\)
\(228\) 0 0
\(229\) 215.607 + 215.607i 0.941516 + 0.941516i 0.998382 0.0568658i \(-0.0181107\pi\)
−0.0568658 + 0.998382i \(0.518111\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 183.853i 0.789069i −0.918881 0.394534i \(-0.870906\pi\)
0.918881 0.394534i \(-0.129094\pi\)
\(234\) 0 0
\(235\) 52.6647 52.6647i 0.224105 0.224105i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 315.183i 1.31876i 0.751811 + 0.659379i \(0.229181\pi\)
−0.751811 + 0.659379i \(0.770819\pi\)
\(240\) 0 0
\(241\) −327.804 −1.36018 −0.680090 0.733128i \(-0.738059\pi\)
−0.680090 + 0.733128i \(0.738059\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −108.691 108.691i −0.443635 0.443635i
\(246\) 0 0
\(247\) 169.532 0.686366
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −219.813 + 219.813i −0.875747 + 0.875747i −0.993091 0.117344i \(-0.962562\pi\)
0.117344 + 0.993091i \(0.462562\pi\)
\(252\) 0 0
\(253\) 192.689 192.689i 0.761616 0.761616i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −150.042 −0.583823 −0.291911 0.956445i \(-0.594291\pi\)
−0.291911 + 0.956445i \(0.594291\pi\)
\(258\) 0 0
\(259\) 173.320 + 173.320i 0.669188 + 0.669188i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 14.8922 0.0566242 0.0283121 0.999599i \(-0.490987\pi\)
0.0283121 + 0.999599i \(0.490987\pi\)
\(264\) 0 0
\(265\) 3.17290i 0.0119732i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 95.4169 95.4169i 0.354710 0.354710i −0.507149 0.861858i \(-0.669301\pi\)
0.861858 + 0.507149i \(0.169301\pi\)
\(270\) 0 0
\(271\) 46.4991i 0.171583i 0.996313 + 0.0857917i \(0.0273420\pi\)
−0.996313 + 0.0857917i \(0.972658\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 19.5516 + 19.5516i 0.0710967 + 0.0710967i
\(276\) 0 0
\(277\) −30.5071 30.5071i −0.110134 0.110134i 0.649892 0.760026i \(-0.274814\pi\)
−0.760026 + 0.649892i \(0.774814\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 217.239i 0.773093i −0.922270 0.386547i \(-0.873668\pi\)
0.922270 0.386547i \(-0.126332\pi\)
\(282\) 0 0
\(283\) −136.055 + 136.055i −0.480760 + 0.480760i −0.905374 0.424614i \(-0.860410\pi\)
0.424614 + 0.905374i \(0.360410\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 194.909i 0.679126i
\(288\) 0 0
\(289\) 618.167 2.13898
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −56.8362 56.8362i −0.193980 0.193980i 0.603433 0.797414i \(-0.293799\pi\)
−0.797414 + 0.603433i \(0.793799\pi\)
\(294\) 0 0
\(295\) −66.3841 −0.225031
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 121.152 121.152i 0.405190 0.405190i
\(300\) 0 0
\(301\) 132.042 132.042i 0.438677 0.438677i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −252.474 −0.827782
\(306\) 0 0
\(307\) 245.927 + 245.927i 0.801067 + 0.801067i 0.983262 0.182196i \(-0.0583205\pi\)
−0.182196 + 0.983262i \(0.558320\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −359.964 −1.15744 −0.578721 0.815526i \(-0.696448\pi\)
−0.578721 + 0.815526i \(0.696448\pi\)
\(312\) 0 0
\(313\) 131.023i 0.418605i −0.977851 0.209303i \(-0.932881\pi\)
0.977851 0.209303i \(-0.0671194\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 89.0470 89.0470i 0.280905 0.280905i −0.552565 0.833470i \(-0.686351\pi\)
0.833470 + 0.552565i \(0.186351\pi\)
\(318\) 0 0
\(319\) 291.555i 0.913966i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 595.000 + 595.000i 1.84210 + 1.84210i
\(324\) 0 0
\(325\) 12.2929 + 12.2929i 0.0378244 + 0.0378244i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 63.9690i 0.194435i
\(330\) 0 0
\(331\) 95.5992 95.5992i 0.288819 0.288819i −0.547794 0.836613i \(-0.684532\pi\)
0.836613 + 0.547794i \(0.184532\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 133.541i 0.398629i
\(336\) 0 0
\(337\) 583.717 1.73210 0.866050 0.499958i \(-0.166651\pi\)
0.866050 + 0.499958i \(0.166651\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 259.604 + 259.604i 0.761302 + 0.761302i
\(342\) 0 0
\(343\) −330.024 −0.962169
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −191.655 + 191.655i −0.552320 + 0.552320i −0.927110 0.374790i \(-0.877715\pi\)
0.374790 + 0.927110i \(0.377715\pi\)
\(348\) 0 0
\(349\) 19.4781 19.4781i 0.0558112 0.0558112i −0.678650 0.734462i \(-0.737435\pi\)
0.734462 + 0.678650i \(0.237435\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 82.9610 0.235017 0.117509 0.993072i \(-0.462509\pi\)
0.117509 + 0.993072i \(0.462509\pi\)
\(354\) 0 0
\(355\) 133.344 + 133.344i 0.375616 + 0.375616i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 357.792 0.996634 0.498317 0.866995i \(-0.333952\pi\)
0.498317 + 0.866995i \(0.333952\pi\)
\(360\) 0 0
\(361\) 419.507i 1.16207i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −102.497 + 102.497i −0.280814 + 0.280814i
\(366\) 0 0
\(367\) 651.729i 1.77583i 0.460010 + 0.887914i \(0.347846\pi\)
−0.460010 + 0.887914i \(0.652154\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1.92698 1.92698i −0.00519401 0.00519401i
\(372\) 0 0
\(373\) −199.720 199.720i −0.535442 0.535442i 0.386745 0.922187i \(-0.373599\pi\)
−0.922187 + 0.386745i \(0.873599\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 183.314i 0.486243i
\(378\) 0 0
\(379\) −330.204 + 330.204i −0.871251 + 0.871251i −0.992609 0.121358i \(-0.961275\pi\)
0.121358 + 0.992609i \(0.461275\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 174.284i 0.455049i −0.973772 0.227524i \(-0.926937\pi\)
0.973772 0.227524i \(-0.0730631\pi\)
\(384\) 0 0
\(385\) −183.488 −0.476593
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 207.835 + 207.835i 0.534279 + 0.534279i 0.921843 0.387564i \(-0.126683\pi\)
−0.387564 + 0.921843i \(0.626683\pi\)
\(390\) 0 0
\(391\) 850.402 2.17494
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 432.900 432.900i 1.09595 1.09595i
\(396\) 0 0
\(397\) 37.2994 37.2994i 0.0939533 0.0939533i −0.658568 0.752521i \(-0.728838\pi\)
0.752521 + 0.658568i \(0.228838\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 524.704 1.30849 0.654244 0.756284i \(-0.272987\pi\)
0.654244 + 0.756284i \(0.272987\pi\)
\(402\) 0 0
\(403\) 163.224 + 163.224i 0.405023 + 0.405023i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −585.434 −1.43841
\(408\) 0 0
\(409\) 787.357i 1.92508i −0.271141 0.962540i \(-0.587401\pi\)
0.271141 0.962540i \(-0.412599\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −40.3166 + 40.3166i −0.0976190 + 0.0976190i
\(414\) 0 0
\(415\) 15.0475i 0.0362589i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −30.1767 30.1767i −0.0720209 0.0720209i 0.670179 0.742200i \(-0.266217\pi\)
−0.742200 + 0.670179i \(0.766217\pi\)
\(420\) 0 0
\(421\) 261.021 + 261.021i 0.620003 + 0.620003i 0.945532 0.325529i \(-0.105542\pi\)
−0.325529 + 0.945532i \(0.605542\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 86.2878i 0.203030i
\(426\) 0 0
\(427\) −153.333 + 153.333i −0.359094 + 0.359094i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 459.989i 1.06726i −0.845718 0.533630i \(-0.820828\pi\)
0.845718 0.533630i \(-0.179172\pi\)
\(432\) 0 0
\(433\) −445.246 −1.02828 −0.514140 0.857706i \(-0.671889\pi\)
−0.514140 + 0.857706i \(0.671889\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 557.768 + 557.768i 1.27636 + 1.27636i
\(438\) 0 0
\(439\) −356.467 −0.811998 −0.405999 0.913874i \(-0.633076\pi\)
−0.405999 + 0.913874i \(0.633076\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −358.752 + 358.752i −0.809824 + 0.809824i −0.984607 0.174783i \(-0.944078\pi\)
0.174783 + 0.984607i \(0.444078\pi\)
\(444\) 0 0
\(445\) −240.272 + 240.272i −0.539936 + 0.539936i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 44.6564 0.0994576 0.0497288 0.998763i \(-0.484164\pi\)
0.0497288 + 0.998763i \(0.484164\pi\)
\(450\) 0 0
\(451\) 329.179 + 329.179i 0.729886 + 0.729886i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −115.367 −0.253554
\(456\) 0 0
\(457\) 84.2332i 0.184318i −0.995744 0.0921589i \(-0.970623\pi\)
0.995744 0.0921589i \(-0.0293768\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 205.347 205.347i 0.445438 0.445438i −0.448397 0.893835i \(-0.648005\pi\)
0.893835 + 0.448397i \(0.148005\pi\)
\(462\) 0 0
\(463\) 270.647i 0.584550i 0.956334 + 0.292275i \(0.0944123\pi\)
−0.956334 + 0.292275i \(0.905588\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −230.389 230.389i −0.493338 0.493338i 0.416018 0.909356i \(-0.363425\pi\)
−0.909356 + 0.416018i \(0.863425\pi\)
\(468\) 0 0
\(469\) 81.1024 + 81.1024i 0.172926 + 0.172926i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 446.006i 0.942931i
\(474\) 0 0
\(475\) −56.5952 + 56.5952i −0.119148 + 0.119148i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 575.911i 1.20232i −0.799129 0.601159i \(-0.794706\pi\)
0.799129 0.601159i \(-0.205294\pi\)
\(480\) 0 0
\(481\) −368.088 −0.765255
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −373.740 373.740i −0.770599 0.770599i
\(486\) 0 0
\(487\) −600.355 −1.23276 −0.616381 0.787448i \(-0.711402\pi\)
−0.616381 + 0.787448i \(0.711402\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −79.7182 + 79.7182i −0.162359 + 0.162359i −0.783611 0.621252i \(-0.786624\pi\)
0.621252 + 0.783611i \(0.286624\pi\)
\(492\) 0 0
\(493\) 643.367 643.367i 1.30500 1.30500i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 161.966 0.325887
\(498\) 0 0
\(499\) −13.4912 13.4912i −0.0270365 0.0270365i 0.693459 0.720496i \(-0.256086\pi\)
−0.720496 + 0.693459i \(0.756086\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 892.196 1.77375 0.886875 0.462009i \(-0.152871\pi\)
0.886875 + 0.462009i \(0.152871\pi\)
\(504\) 0 0
\(505\) 10.7744i 0.0213354i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −44.9128 + 44.9128i −0.0882374 + 0.0882374i −0.749848 0.661610i \(-0.769873\pi\)
0.661610 + 0.749848i \(0.269873\pi\)
\(510\) 0 0
\(511\) 124.498i 0.243636i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −93.1416 93.1416i −0.180857 0.180857i
\(516\) 0 0
\(517\) 108.036 + 108.036i 0.208967 + 0.208967i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 866.038i 1.66226i −0.556078 0.831130i \(-0.687694\pi\)
0.556078 0.831130i \(-0.312306\pi\)
\(522\) 0 0
\(523\) 359.579 359.579i 0.687531 0.687531i −0.274155 0.961686i \(-0.588398\pi\)
0.961686 + 0.274155i \(0.0883981\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1145.72i 2.17405i
\(528\) 0 0
\(529\) 268.189 0.506973
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 206.969 + 206.969i 0.388310 + 0.388310i
\(534\) 0 0
\(535\) 268.695 0.502234
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 222.968 222.968i 0.413669 0.413669i
\(540\) 0 0
\(541\) 9.41176 9.41176i 0.0173970 0.0173970i −0.698355 0.715752i \(-0.746084\pi\)
0.715752 + 0.698355i \(0.246084\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −245.076 −0.449680
\(546\) 0 0
\(547\) −37.6377 37.6377i −0.0688075 0.0688075i 0.671866 0.740673i \(-0.265493\pi\)
−0.740673 + 0.671866i \(0.765493\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 843.953 1.53168
\(552\) 0 0
\(553\) 525.821i 0.950852i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 369.172 369.172i 0.662786 0.662786i −0.293250 0.956036i \(-0.594737\pi\)
0.956036 + 0.293250i \(0.0947369\pi\)
\(558\) 0 0
\(559\) 280.424i 0.501652i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 141.210 + 141.210i 0.250817 + 0.250817i 0.821306 0.570489i \(-0.193246\pi\)
−0.570489 + 0.821306i \(0.693246\pi\)
\(564\) 0 0
\(565\) 184.954 + 184.954i 0.327352 + 0.327352i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 134.928i 0.237131i 0.992946 + 0.118566i \(0.0378296\pi\)
−0.992946 + 0.118566i \(0.962170\pi\)
\(570\) 0 0
\(571\) −486.485 + 486.485i −0.851988 + 0.851988i −0.990378 0.138390i \(-0.955807\pi\)
0.138390 + 0.990378i \(0.455807\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 80.8885i 0.140676i
\(576\) 0 0
\(577\) −310.050 −0.537349 −0.268674 0.963231i \(-0.586586\pi\)
−0.268674 + 0.963231i \(0.586586\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 9.13868 + 9.13868i 0.0157292 + 0.0157292i
\(582\) 0 0
\(583\) 6.50888 0.0111645
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 301.021 301.021i 0.512812 0.512812i −0.402575 0.915387i \(-0.631885\pi\)
0.915387 + 0.402575i \(0.131885\pi\)
\(588\) 0 0
\(589\) −751.465 + 751.465i −1.27583 + 1.27583i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −6.08782 −0.0102661 −0.00513307 0.999987i \(-0.501634\pi\)
−0.00513307 + 0.999987i \(0.501634\pi\)
\(594\) 0 0
\(595\) −404.898 404.898i −0.680501 0.680501i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −756.472 −1.26289 −0.631446 0.775420i \(-0.717538\pi\)
−0.631446 + 0.775420i \(0.717538\pi\)
\(600\) 0 0
\(601\) 753.072i 1.25303i 0.779409 + 0.626516i \(0.215520\pi\)
−0.779409 + 0.626516i \(0.784480\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −92.6521 + 92.6521i −0.153144 + 0.153144i
\(606\) 0 0
\(607\) 47.1200i 0.0776277i 0.999246 + 0.0388139i \(0.0123579\pi\)
−0.999246 + 0.0388139i \(0.987642\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 67.9271 + 67.9271i 0.111174 + 0.111174i
\(612\) 0 0
\(613\) −637.192 637.192i −1.03947 1.03947i −0.999189 0.0402769i \(-0.987176\pi\)
−0.0402769 0.999189i \(-0.512824\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 514.635i 0.834092i 0.908885 + 0.417046i \(0.136935\pi\)
−0.908885 + 0.417046i \(0.863065\pi\)
\(618\) 0 0
\(619\) 313.704 313.704i 0.506791 0.506791i −0.406749 0.913540i \(-0.633338\pi\)
0.913540 + 0.406749i \(0.133338\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 291.845i 0.468452i
\(624\) 0 0
\(625\) 545.171 0.872273
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1291.86 1291.86i −2.05383 2.05383i
\(630\) 0 0
\(631\) 1226.20 1.94326 0.971631 0.236502i \(-0.0760009\pi\)
0.971631 + 0.236502i \(0.0760009\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −365.706 + 365.706i −0.575914 + 0.575914i
\(636\) 0 0
\(637\) 140.189 140.189i 0.220078 0.220078i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −241.218 −0.376314 −0.188157 0.982139i \(-0.560251\pi\)
−0.188157 + 0.982139i \(0.560251\pi\)
\(642\) 0 0
\(643\) −736.141 736.141i −1.14485 1.14485i −0.987550 0.157304i \(-0.949720\pi\)
−0.157304 0.987550i \(-0.550280\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −680.082 −1.05113 −0.525565 0.850753i \(-0.676146\pi\)
−0.525565 + 0.850753i \(0.676146\pi\)
\(648\) 0 0
\(649\) 136.180i 0.209831i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −716.929 + 716.929i −1.09790 + 1.09790i −0.103244 + 0.994656i \(0.532922\pi\)
−0.994656 + 0.103244i \(0.967078\pi\)
\(654\) 0 0
\(655\) 503.540i 0.768763i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −276.868 276.868i −0.420133 0.420133i 0.465116 0.885250i \(-0.346012\pi\)
−0.885250 + 0.465116i \(0.846012\pi\)
\(660\) 0 0
\(661\) −251.780 251.780i −0.380907 0.380907i 0.490522 0.871429i \(-0.336806\pi\)
−0.871429 + 0.490522i \(0.836806\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 531.136i 0.798700i
\(666\) 0 0
\(667\) 603.109 603.109i 0.904211 0.904211i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 517.924i 0.771869i
\(672\) 0 0
\(673\) 674.332 1.00198 0.500990 0.865453i \(-0.332969\pi\)
0.500990 + 0.865453i \(0.332969\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 109.048 + 109.048i 0.161075 + 0.161075i 0.783043 0.621968i \(-0.213666\pi\)
−0.621968 + 0.783043i \(0.713666\pi\)
\(678\) 0 0
\(679\) −453.963 −0.668576
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 784.278 784.278i 1.14828 1.14828i 0.161394 0.986890i \(-0.448401\pi\)
0.986890 0.161394i \(-0.0515990\pi\)
\(684\) 0 0
\(685\) −7.14432 + 7.14432i −0.0104297 + 0.0104297i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 4.09241 0.00593964
\(690\) 0 0
\(691\) −99.4915 99.4915i −0.143982 0.143982i 0.631442 0.775423i \(-0.282464\pi\)
−0.775423 + 0.631442i \(0.782464\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −726.877 −1.04587
\(696\) 0 0
\(697\) 1452.78i 2.08433i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 177.909 177.909i 0.253794 0.253794i −0.568730 0.822524i \(-0.692565\pi\)
0.822524 + 0.568730i \(0.192565\pi\)
\(702\) 0 0
\(703\) 1694.63i 2.41057i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6.54353 6.54353i −0.00925535 0.00925535i
\(708\) 0 0
\(709\) 208.080 + 208.080i 0.293484 + 0.293484i 0.838455 0.544971i \(-0.183459\pi\)
−0.544971 + 0.838455i \(0.683459\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1074.03i 1.50635i
\(714\) 0 0
\(715\) 194.841 194.841i 0.272506 0.272506i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1013.84i 1.41007i 0.709171 + 0.705036i \(0.249069\pi\)
−0.709171 + 0.705036i \(0.750931\pi\)
\(720\) 0 0
\(721\) −113.134 −0.156913
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 61.1957 + 61.1957i 0.0844079 + 0.0844079i
\(726\) 0 0
\(727\) −697.156 −0.958949 −0.479474 0.877556i \(-0.659173\pi\)
−0.479474 + 0.877556i \(0.659173\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −984.189 + 984.189i −1.34636 + 1.34636i
\(732\) 0 0
\(733\) 39.9608 39.9608i 0.0545168 0.0545168i −0.679323 0.733840i \(-0.737726\pi\)
0.733840 + 0.679323i \(0.237726\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −273.945 −0.371703
\(738\) 0 0
\(739\) −236.377 236.377i −0.319860 0.319860i 0.528853 0.848713i \(-0.322622\pi\)
−0.848713 + 0.528853i \(0.822622\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 804.248 1.08243 0.541217 0.840883i \(-0.317964\pi\)
0.541217 + 0.840883i \(0.317964\pi\)
\(744\) 0 0
\(745\) 530.258i 0.711755i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 163.185 163.185i 0.217870 0.217870i
\(750\) 0 0
\(751\) 607.492i 0.808911i −0.914558 0.404456i \(-0.867461\pi\)
0.914558 0.404456i \(-0.132539\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −351.131 351.131i −0.465074 0.465074i
\(756\) 0 0
\(757\) 11.6797 + 11.6797i 0.0154289 + 0.0154289i 0.714779 0.699350i \(-0.246527\pi\)
−0.699350 + 0.714779i \(0.746527\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 659.125i 0.866130i 0.901363 + 0.433065i \(0.142568\pi\)
−0.901363 + 0.433065i \(0.857432\pi\)
\(762\) 0 0
\(763\) −148.840 + 148.840i −0.195073 + 0.195073i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 85.6224i 0.111633i
\(768\) 0 0
\(769\) −178.802 −0.232512 −0.116256 0.993219i \(-0.537089\pi\)
−0.116256 + 0.993219i \(0.537089\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −91.8171 91.8171i −0.118780 0.118780i 0.645218 0.763998i \(-0.276766\pi\)
−0.763998 + 0.645218i \(0.776766\pi\)
\(774\) 0 0
\(775\) −108.979 −0.140618
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −952.860 + 952.860i −1.22318 + 1.22318i
\(780\) 0 0
\(781\) −273.541 + 273.541i −0.350245 + 0.350245i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1264.58 −1.61093
\(786\) 0 0
\(787\) 214.856 + 214.856i 0.273006 + 0.273006i 0.830309 0.557303i \(-0.188164\pi\)
−0.557303 + 0.830309i \(0.688164\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 224.654 0.284012
\(792\) 0 0
\(793\) 325.641i 0.410645i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −332.028 + 332.028i −0.416598 + 0.416598i −0.884029 0.467432i \(-0.845179\pi\)
0.467432 + 0.884029i \(0.345179\pi\)
\(798\) 0 0
\(799\) 476.801i 0.596747i
\(800\) 0