Properties

Label 1152.3.m.f.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.f.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 \(-0.593201\pi\)
−0.288635 + 0.957439i \(0.593201\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.82458 6.82458i 0.620416 0.620416i −0.325222 0.945638i \(-0.605439\pi\)
0.945638 + 0.325222i \(0.105439\pi\)
\(12\) 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 \(0.0129112\pi\)
0.0405505 + 0.999177i \(0.487089\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 28.2345 1.22759 0.613794 0.789466i \(-0.289642\pi\)
0.613794 + 0.789466i \(0.289642\pi\)
\(24\) 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 \(0.210256\pi\)
−0.789662 + 0.613541i \(0.789744\pi\)
\(32\) 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 −0.976307 0.216389i \(-0.930572\pi\)
0.216389 + 0.976307i \(0.430572\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 15.8305i 0.336818i 0.985717 + 0.168409i \(0.0538630\pi\)
−0.985717 + 0.168409i \(0.946137\pi\)
\(48\) 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 −0.617481 0.786586i \(-0.711847\pi\)
0.786586 + 0.617481i \(0.211847\pi\)
\(60\) 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 0.541282 0.840841i \(-0.317939\pi\)
−0.840841 + 0.541282i \(0.817939\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −40.0818 −0.564532 −0.282266 0.959336i \(-0.591086\pi\)
−0.282266 + 0.959336i \(0.591086\pi\)
\(72\) 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 \(0.308025\pi\)
−0.567203 + 0.823578i \(0.691975\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.26155 2.26155i −0.0272476 0.0272476i 0.693352 0.720599i \(-0.256133\pi\)
−0.720599 + 0.693352i \(0.756133\pi\)
\(84\) 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 \(-0.456604\pi\)
0.135910 + 0.990721i \(0.456604\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −40.3835 + 40.3835i −0.377416 + 0.377416i −0.870169 0.492753i \(-0.835990\pi\)
0.492753 + 0.870169i \(0.335990\pi\)
\(108\) 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 \(-0.857531\pi\)
0.901497 0.432785i \(-0.142469\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 75.6795 + 75.6795i 0.577706 + 0.577706i 0.934271 0.356565i \(-0.116052\pi\)
−0.356565 + 0.934271i \(0.616052\pi\)
\(132\) 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 −0.194885 0.980826i \(-0.562433\pi\)
0.980826 + 0.194885i \(0.0624334\pi\)
\(140\) 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 \(-0.386355\pi\)
0.349491 + 0.936940i \(0.386355\pi\)
\(152\) 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 0.865465 0.500969i \(-0.167023\pi\)
−0.500969 + 0.865465i \(0.667023\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −65.3894 −0.391553 −0.195777 0.980649i \(-0.562723\pi\)
−0.195777 + 0.980649i \(0.562723\pi\)
\(168\) 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 \(0.0335768\pi\)
0.105289 + 0.994442i \(0.466423\pi\)
\(180\) 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 \(-0.779011\pi\)
0.768530 0.639814i \(-0.220989\pi\)
\(192\) 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 \(-0.640298\pi\)
−0.426626 + 0.904428i \(0.640298\pi\)
\(200\) 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 0.947783 0.318915i \(-0.103319\pi\)
−0.318915 + 0.947783i \(0.603319\pi\)
\(212\) 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 \(-0.981188\pi\)
0.998254 0.0590664i \(-0.0188124\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −70.3362 70.3362i −0.309851 0.309851i 0.535001 0.844852i \(-0.320311\pi\)
−0.844852 + 0.535001i \(0.820311\pi\)
\(228\) 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 \(-0.770819\pi\)
0.751811 0.659379i \(-0.229181\pi\)
\(240\) 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 −0.117344 0.993091i \(-0.537438\pi\)
0.993091 + 0.117344i \(0.0374381\pi\)
\(252\) 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 \(-0.509013\pi\)
−0.0283121 + 0.999599i \(0.509013\pi\)
\(264\) 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 \(-0.972658\pi\)
0.996313 0.0857917i \(-0.0273420\pi\)
\(272\) 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 −0.424614 0.905374i \(-0.639590\pi\)
0.905374 + 0.424614i \(0.139590\pi\)
\(284\) 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 0.182196 0.983262i \(-0.441680\pi\)
−0.983262 + 0.182196i \(0.941680\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 359.964 1.15744 0.578721 0.815526i \(-0.303552\pi\)
0.578721 + 0.815526i \(0.303552\pi\)
\(312\) 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 −0.836613 0.547794i \(-0.815468\pi\)
0.547794 + 0.836613i \(0.315468\pi\)
\(332\) 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 −0.374790 0.927110i \(-0.622285\pi\)
0.927110 + 0.374790i \(0.122285\pi\)
\(348\) 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 \(-0.666048\pi\)
−0.498317 + 0.866995i \(0.666048\pi\)
\(360\) 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 \(-0.652154\pi\)
0.460010 0.887914i \(-0.347846\pi\)
\(368\) 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 −0.121358 0.992609i \(-0.538725\pi\)
0.992609 + 0.121358i \(0.0387248\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 174.284i 0.455049i 0.973772 + 0.227524i \(0.0730631\pi\)
−0.973772 + 0.227524i \(0.926937\pi\)
\(384\) 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 0.742200 0.670179i \(-0.233783\pi\)
−0.670179 + 0.742200i \(0.733783\pi\)
\(420\) 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 \(0.179172\pi\)
−0.845718 + 0.533630i \(0.820828\pi\)
\(432\) 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 \(-0.366924\pi\)
0.405999 + 0.913874i \(0.366924\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 358.752 358.752i 0.809824 0.809824i −0.174783 0.984607i \(-0.555922\pi\)
0.984607 + 0.174783i \(0.0559224\pi\)
\(444\) 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 \(-0.905588\pi\)
0.956334 0.292275i \(-0.0944123\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 230.389 + 230.389i 0.493338 + 0.493338i 0.909356 0.416018i \(-0.136575\pi\)
−0.416018 + 0.909356i \(0.636575\pi\)
\(468\) 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 \(0.205294\pi\)
−0.799129 + 0.601159i \(0.794706\pi\)
\(480\) 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 \(-0.288598\pi\)
0.616381 + 0.787448i \(0.288598\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 79.7182 79.7182i 0.162359 0.162359i −0.621252 0.783611i \(-0.713376\pi\)
0.783611 + 0.621252i \(0.213376\pi\)
\(492\) 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 0.720496 0.693459i \(-0.243914\pi\)
−0.693459 + 0.720496i \(0.743914\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −892.196 −1.77375 −0.886875 0.462009i \(-0.847129\pi\)
−0.886875 + 0.462009i \(0.847129\pi\)
\(504\) 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 −0.961686 0.274155i \(-0.911602\pi\)
0.274155 + 0.961686i \(0.411602\pi\)
\(524\) 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 0.740673 0.671866i \(-0.234507\pi\)
−0.671866 + 0.740673i \(0.734507\pi\)
\(548\) 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 0.570489 0.821306i \(-0.306754\pi\)
−0.821306 + 0.570489i \(0.806754\pi\)
\(564\) 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 −0.138390 0.990378i \(-0.544193\pi\)
0.990378 + 0.138390i \(0.0441926\pi\)
\(572\) 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 −0.915387 0.402575i \(-0.868115\pi\)
0.402575 + 0.915387i \(0.368115\pi\)
\(588\) 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 \(-0.282462\pi\)
0.631446 + 0.775420i \(0.282462\pi\)
\(600\) 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 \(-0.987642\pi\)
0.999246 0.0388139i \(-0.0123579\pi\)
\(608\) 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 −0.913540 0.406749i \(-0.866662\pi\)
0.406749 + 0.913540i \(0.366662\pi\)
\(620\) 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 \(-0.923999\pi\)
−0.971631 + 0.236502i \(0.923999\pi\)
\(632\) 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 \(0.0502803\pi\)
0.157304 + 0.987550i \(0.449720\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 680.082 1.05113 0.525565 0.850753i \(-0.323854\pi\)
0.525565 + 0.850753i \(0.323854\pi\)
\(648\) 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 0.885250 0.465116i \(-0.153988\pi\)
−0.465116 + 0.885250i \(0.653988\pi\)
\(660\) 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 \(0.551599\pi\)
−0.986890 + 0.161394i \(0.948401\pi\)
\(684\) 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 0.775423 0.631442i \(-0.217536\pi\)
−0.631442 + 0.775423i \(0.717536\pi\)
\(692\) 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 \(-0.750931\pi\)
0.709171 0.705036i \(-0.249069\pi\)
\(720\) 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 \(-0.340827\pi\)
0.479474 + 0.877556i \(0.340827\pi\)
\(728\) 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 0.848713 0.528853i \(-0.177378\pi\)
−0.528853 + 0.848713i \(0.677378\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −804.248 −1.08243 −0.541217 0.840883i \(-0.682036\pi\)
−0.541217 + 0.840883i \(0.682036\pi\)
\(744\) 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 \(0.132539\pi\)
−0.914558 + 0.404456i \(0.867461\pi\)
\(752\) 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 0.557303 0.830309i \(-0.311836\pi\)
−0.830309 + 0.557303i \(0.811836\pi\)
\(788\) 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 0