Properties

Label 1152.3.m.c.415.3
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 415.3
Root \(-0.455024 - 1.94755i\) of defining polynomial
Character \(\chi\) \(=\) 1152.415
Dual form 1152.3.m.c.991.3

$q$-expansion

\(f(q)\) \(=\) \(q+(-3.40572 - 3.40572i) q^{5} -12.1303 q^{7} +O(q^{10})\) \(q+(-3.40572 - 3.40572i) q^{5} -12.1303 q^{7} +(-9.81086 + 9.81086i) q^{11} +(7.76859 - 7.76859i) q^{13} -9.73087 q^{17} +(11.2823 + 11.2823i) q^{19} -20.2635 q^{23} -1.80207i q^{25} +(-16.4069 + 16.4069i) q^{29} -26.3542i q^{31} +(41.3125 + 41.3125i) q^{35} +(23.7263 + 23.7263i) q^{37} -24.7452i q^{41} +(29.8844 - 29.8844i) q^{43} +31.3325i q^{47} +98.1448 q^{49} +(36.8742 + 36.8742i) q^{53} +66.8262 q^{55} +(14.1325 - 14.1325i) q^{59} +(42.5199 - 42.5199i) q^{61} -52.9153 q^{65} +(48.7789 + 48.7789i) q^{67} +7.73935 q^{71} -85.4163i q^{73} +(119.009 - 119.009i) q^{77} +105.294i q^{79} +(62.1229 + 62.1229i) q^{83} +(33.1407 + 33.1407i) q^{85} -127.172i q^{89} +(-94.2355 + 94.2355i) q^{91} -76.8489i q^{95} -147.348 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 32q^{11} - 32q^{19} - 128q^{23} + 32q^{29} - 96q^{35} + 96q^{37} + 160q^{43} + 112q^{49} - 160q^{53} + 256q^{55} + 128q^{59} + 32q^{61} + 32q^{65} + 320q^{67} + 512q^{71} + 224q^{77} + 160q^{83} - 160q^{85} - 480q^{91} + O(q^{100}) \)

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
\(5\) −3.40572 3.40572i −0.681145 0.681145i 0.279113 0.960258i \(-0.409960\pi\)
−0.960258 + 0.279113i \(0.909960\pi\)
\(6\) 0 0
\(7\) −12.1303 −1.73290 −0.866452 0.499261i \(-0.833605\pi\)
−0.866452 + 0.499261i \(0.833605\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −9.81086 + 9.81086i −0.891896 + 0.891896i −0.994702 0.102805i \(-0.967218\pi\)
0.102805 + 0.994702i \(0.467218\pi\)
\(12\) 0 0
\(13\) 7.76859 7.76859i 0.597584 0.597584i −0.342085 0.939669i \(-0.611133\pi\)
0.939669 + 0.342085i \(0.111133\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −9.73087 −0.572404 −0.286202 0.958169i \(-0.592393\pi\)
−0.286202 + 0.958169i \(0.592393\pi\)
\(18\) 0 0
\(19\) 11.2823 + 11.2823i 0.593806 + 0.593806i 0.938657 0.344851i \(-0.112071\pi\)
−0.344851 + 0.938657i \(0.612071\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −20.2635 −0.881020 −0.440510 0.897748i \(-0.645202\pi\)
−0.440510 + 0.897748i \(0.645202\pi\)
\(24\) 0 0
\(25\) 1.80207i 0.0720830i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −16.4069 + 16.4069i −0.565754 + 0.565754i −0.930936 0.365182i \(-0.881007\pi\)
0.365182 + 0.930936i \(0.381007\pi\)
\(30\) 0 0
\(31\) 26.3542i 0.850134i −0.905162 0.425067i \(-0.860251\pi\)
0.905162 0.425067i \(-0.139749\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 41.3125 + 41.3125i 1.18036 + 1.18036i
\(36\) 0 0
\(37\) 23.7263 + 23.7263i 0.641250 + 0.641250i 0.950863 0.309613i \(-0.100199\pi\)
−0.309613 + 0.950863i \(0.600199\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 24.7452i 0.603542i −0.953380 0.301771i \(-0.902422\pi\)
0.953380 0.301771i \(-0.0975779\pi\)
\(42\) 0 0
\(43\) 29.8844 29.8844i 0.694987 0.694987i −0.268338 0.963325i \(-0.586474\pi\)
0.963325 + 0.268338i \(0.0864744\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 31.3325i 0.666648i 0.942812 + 0.333324i \(0.108170\pi\)
−0.942812 + 0.333324i \(0.891830\pi\)
\(48\) 0 0
\(49\) 98.1448 2.00295
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 36.8742 + 36.8742i 0.695739 + 0.695739i 0.963489 0.267750i \(-0.0862800\pi\)
−0.267750 + 0.963489i \(0.586280\pi\)
\(54\) 0 0
\(55\) 66.8262 1.21502
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 14.1325 14.1325i 0.239534 0.239534i −0.577123 0.816657i \(-0.695825\pi\)
0.816657 + 0.577123i \(0.195825\pi\)
\(60\) 0 0
\(61\) 42.5199 42.5199i 0.697048 0.697048i −0.266725 0.963773i \(-0.585941\pi\)
0.963773 + 0.266725i \(0.0859414\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −52.9153 −0.814082
\(66\) 0 0
\(67\) 48.7789 + 48.7789i 0.728044 + 0.728044i 0.970230 0.242186i \(-0.0778644\pi\)
−0.242186 + 0.970230i \(0.577864\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.73935 0.109005 0.0545025 0.998514i \(-0.482643\pi\)
0.0545025 + 0.998514i \(0.482643\pi\)
\(72\) 0 0
\(73\) 85.4163i 1.17009i −0.811002 0.585043i \(-0.801077\pi\)
0.811002 0.585043i \(-0.198923\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 119.009 119.009i 1.54557 1.54557i
\(78\) 0 0
\(79\) 105.294i 1.33283i 0.745581 + 0.666416i \(0.232172\pi\)
−0.745581 + 0.666416i \(0.767828\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 62.1229 + 62.1229i 0.748469 + 0.748469i 0.974192 0.225723i \(-0.0724743\pi\)
−0.225723 + 0.974192i \(0.572474\pi\)
\(84\) 0 0
\(85\) 33.1407 + 33.1407i 0.389890 + 0.389890i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 127.172i 1.42890i −0.699685 0.714451i \(-0.746676\pi\)
0.699685 0.714451i \(-0.253324\pi\)
\(90\) 0 0
\(91\) −94.2355 + 94.2355i −1.03555 + 1.03555i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 76.8489i 0.808936i
\(96\) 0 0
\(97\) −147.348 −1.51905 −0.759525 0.650478i \(-0.774569\pi\)
−0.759525 + 0.650478i \(0.774569\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 12.7690 + 12.7690i 0.126426 + 0.126426i 0.767489 0.641063i \(-0.221506\pi\)
−0.641063 + 0.767489i \(0.721506\pi\)
\(102\) 0 0
\(103\) 17.7621 0.172448 0.0862240 0.996276i \(-0.472520\pi\)
0.0862240 + 0.996276i \(0.472520\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 15.8889 15.8889i 0.148494 0.148494i −0.628951 0.777445i \(-0.716515\pi\)
0.777445 + 0.628951i \(0.216515\pi\)
\(108\) 0 0
\(109\) 79.3257 79.3257i 0.727758 0.727758i −0.242414 0.970173i \(-0.577939\pi\)
0.970173 + 0.242414i \(0.0779394\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 167.538 1.48263 0.741317 0.671155i \(-0.234201\pi\)
0.741317 + 0.671155i \(0.234201\pi\)
\(114\) 0 0
\(115\) 69.0118 + 69.0118i 0.600102 + 0.600102i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 118.039 0.991921
\(120\) 0 0
\(121\) 71.5059i 0.590958i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −91.2805 + 91.2805i −0.730244 + 0.730244i
\(126\) 0 0
\(127\) 198.247i 1.56100i 0.625156 + 0.780500i \(0.285035\pi\)
−0.625156 + 0.780500i \(0.714965\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −134.339 134.339i −1.02549 1.02549i −0.999667 0.0258197i \(-0.991780\pi\)
−0.0258197 0.999667i \(-0.508220\pi\)
\(132\) 0 0
\(133\) −136.858 136.858i −1.02901 1.02901i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 255.937i 1.86816i 0.357069 + 0.934078i \(0.383776\pi\)
−0.357069 + 0.934078i \(0.616224\pi\)
\(138\) 0 0
\(139\) −21.7231 + 21.7231i −0.156281 + 0.156281i −0.780917 0.624635i \(-0.785248\pi\)
0.624635 + 0.780917i \(0.285248\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 152.433i 1.06597i
\(144\) 0 0
\(145\) 111.755 0.770722
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −34.2444 34.2444i −0.229828 0.229828i 0.582793 0.812621i \(-0.301960\pi\)
−0.812621 + 0.582793i \(0.801960\pi\)
\(150\) 0 0
\(151\) 14.4645 0.0957913 0.0478956 0.998852i \(-0.484749\pi\)
0.0478956 + 0.998852i \(0.484749\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −89.7550 + 89.7550i −0.579064 + 0.579064i
\(156\) 0 0
\(157\) −31.4652 + 31.4652i −0.200415 + 0.200415i −0.800178 0.599763i \(-0.795262\pi\)
0.599763 + 0.800178i \(0.295262\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 245.802 1.52672
\(162\) 0 0
\(163\) 31.4002 + 31.4002i 0.192640 + 0.192640i 0.796836 0.604196i \(-0.206506\pi\)
−0.604196 + 0.796836i \(0.706506\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 36.4796 0.218441 0.109220 0.994018i \(-0.465165\pi\)
0.109220 + 0.994018i \(0.465165\pi\)
\(168\) 0 0
\(169\) 48.2981i 0.285788i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 97.6419 97.6419i 0.564404 0.564404i −0.366151 0.930555i \(-0.619325\pi\)
0.930555 + 0.366151i \(0.119325\pi\)
\(174\) 0 0
\(175\) 21.8598i 0.124913i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −89.7427 89.7427i −0.501356 0.501356i 0.410503 0.911859i \(-0.365353\pi\)
−0.911859 + 0.410503i \(0.865353\pi\)
\(180\) 0 0
\(181\) 115.497 + 115.497i 0.638108 + 0.638108i 0.950088 0.311981i \(-0.100992\pi\)
−0.311981 + 0.950088i \(0.600992\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 161.610i 0.873569i
\(186\) 0 0
\(187\) 95.4682 95.4682i 0.510525 0.510525i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 62.6278i 0.327894i −0.986469 0.163947i \(-0.947577\pi\)
0.986469 0.163947i \(-0.0524227\pi\)
\(192\) 0 0
\(193\) 223.342 1.15721 0.578607 0.815607i \(-0.303597\pi\)
0.578607 + 0.815607i \(0.303597\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 29.0959 + 29.0959i 0.147695 + 0.147695i 0.777087 0.629393i \(-0.216696\pi\)
−0.629393 + 0.777087i \(0.716696\pi\)
\(198\) 0 0
\(199\) −11.6967 −0.0587776 −0.0293888 0.999568i \(-0.509356\pi\)
−0.0293888 + 0.999568i \(0.509356\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 199.021 199.021i 0.980398 0.980398i
\(204\) 0 0
\(205\) −84.2755 + 84.2755i −0.411100 + 0.411100i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −221.378 −1.05923
\(210\) 0 0
\(211\) −0.215765 0.215765i −0.00102258 0.00102258i 0.706595 0.707618i \(-0.250230\pi\)
−0.707618 + 0.706595i \(0.750230\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −203.556 −0.946773
\(216\) 0 0
\(217\) 319.684i 1.47320i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −75.5951 + 75.5951i −0.342059 + 0.342059i
\(222\) 0 0
\(223\) 371.347i 1.66523i −0.553850 0.832617i \(-0.686842\pi\)
0.553850 0.832617i \(-0.313158\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 209.823 + 209.823i 0.924330 + 0.924330i 0.997332 0.0730018i \(-0.0232579\pi\)
−0.0730018 + 0.997332i \(0.523258\pi\)
\(228\) 0 0
\(229\) −152.751 152.751i −0.667037 0.667037i 0.289992 0.957029i \(-0.406347\pi\)
−0.957029 + 0.289992i \(0.906347\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 272.899i 1.17124i 0.810586 + 0.585619i \(0.199149\pi\)
−0.810586 + 0.585619i \(0.800851\pi\)
\(234\) 0 0
\(235\) 106.710 106.710i 0.454084 0.454084i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 104.650i 0.437866i 0.975740 + 0.218933i \(0.0702576\pi\)
−0.975740 + 0.218933i \(0.929742\pi\)
\(240\) 0 0
\(241\) 148.875 0.617737 0.308869 0.951105i \(-0.400050\pi\)
0.308869 + 0.951105i \(0.400050\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −334.254 334.254i −1.36430 1.36430i
\(246\) 0 0
\(247\) 175.295 0.709698
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −143.712 + 143.712i −0.572558 + 0.572558i −0.932843 0.360284i \(-0.882680\pi\)
0.360284 + 0.932843i \(0.382680\pi\)
\(252\) 0 0
\(253\) 198.802 198.802i 0.785778 0.785778i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −134.023 −0.521489 −0.260745 0.965408i \(-0.583968\pi\)
−0.260745 + 0.965408i \(0.583968\pi\)
\(258\) 0 0
\(259\) −287.807 287.807i −1.11122 1.11122i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 290.386 1.10413 0.552066 0.833801i \(-0.313840\pi\)
0.552066 + 0.833801i \(0.313840\pi\)
\(264\) 0 0
\(265\) 251.166i 0.947798i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −74.2628 + 74.2628i −0.276070 + 0.276070i −0.831538 0.555468i \(-0.812539\pi\)
0.555468 + 0.831538i \(0.312539\pi\)
\(270\) 0 0
\(271\) 70.8329i 0.261376i 0.991424 + 0.130688i \(0.0417186\pi\)
−0.991424 + 0.130688i \(0.958281\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 17.6799 + 17.6799i 0.0642906 + 0.0642906i
\(276\) 0 0
\(277\) 96.6953 + 96.6953i 0.349081 + 0.349081i 0.859767 0.510686i \(-0.170609\pi\)
−0.510686 + 0.859767i \(0.670609\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 138.151i 0.491640i −0.969316 0.245820i \(-0.920943\pi\)
0.969316 0.245820i \(-0.0790572\pi\)
\(282\) 0 0
\(283\) −295.011 + 295.011i −1.04244 + 1.04244i −0.0433821 + 0.999059i \(0.513813\pi\)
−0.999059 + 0.0433821i \(0.986187\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 300.168i 1.04588i
\(288\) 0 0
\(289\) −194.310 −0.672353
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −33.4759 33.4759i −0.114252 0.114252i 0.647669 0.761922i \(-0.275744\pi\)
−0.761922 + 0.647669i \(0.775744\pi\)
\(294\) 0 0
\(295\) −96.2630 −0.326315
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −157.418 + 157.418i −0.526483 + 0.526483i
\(300\) 0 0
\(301\) −362.508 + 362.508i −1.20434 + 1.20434i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −289.622 −0.949582
\(306\) 0 0
\(307\) −92.6638 92.6638i −0.301836 0.301836i 0.539896 0.841732i \(-0.318464\pi\)
−0.841732 + 0.539896i \(0.818464\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −18.5610 −0.0596817 −0.0298408 0.999555i \(-0.509500\pi\)
−0.0298408 + 0.999555i \(0.509500\pi\)
\(312\) 0 0
\(313\) 55.1534i 0.176209i 0.996111 + 0.0881045i \(0.0280809\pi\)
−0.996111 + 0.0881045i \(0.971919\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 62.2977 62.2977i 0.196523 0.196523i −0.601985 0.798507i \(-0.705623\pi\)
0.798507 + 0.601985i \(0.205623\pi\)
\(318\) 0 0
\(319\) 321.931i 1.00919i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −109.787 109.787i −0.339897 0.339897i
\(324\) 0 0
\(325\) −13.9996 13.9996i −0.0430756 0.0430756i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 380.073i 1.15524i
\(330\) 0 0
\(331\) 373.767 373.767i 1.12921 1.12921i 0.138899 0.990307i \(-0.455644\pi\)
0.990307 0.138899i \(-0.0443564\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 332.255i 0.991807i
\(336\) 0 0
\(337\) −519.936 −1.54284 −0.771419 0.636328i \(-0.780453\pi\)
−0.771419 + 0.636328i \(0.780453\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 258.557 + 258.557i 0.758231 + 0.758231i
\(342\) 0 0
\(343\) −596.142 −1.73802
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −122.160 + 122.160i −0.352045 + 0.352045i −0.860870 0.508825i \(-0.830080\pi\)
0.508825 + 0.860870i \(0.330080\pi\)
\(348\) 0 0
\(349\) 279.483 279.483i 0.800810 0.800810i −0.182412 0.983222i \(-0.558391\pi\)
0.983222 + 0.182412i \(0.0583906\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 212.266 0.601320 0.300660 0.953731i \(-0.402793\pi\)
0.300660 + 0.953731i \(0.402793\pi\)
\(354\) 0 0
\(355\) −26.3581 26.3581i −0.0742482 0.0742482i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −435.033 −1.21179 −0.605895 0.795545i \(-0.707185\pi\)
−0.605895 + 0.795545i \(0.707185\pi\)
\(360\) 0 0
\(361\) 106.419i 0.294789i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −290.905 + 290.905i −0.796999 + 0.796999i
\(366\) 0 0
\(367\) 125.535i 0.342058i 0.985266 + 0.171029i \(0.0547091\pi\)
−0.985266 + 0.171029i \(0.945291\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −447.295 447.295i −1.20565 1.20565i
\(372\) 0 0
\(373\) 302.389 + 302.389i 0.810694 + 0.810694i 0.984738 0.174044i \(-0.0556835\pi\)
−0.174044 + 0.984738i \(0.555683\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 254.917i 0.676171i
\(378\) 0 0
\(379\) 189.784 189.784i 0.500751 0.500751i −0.410921 0.911671i \(-0.634793\pi\)
0.911671 + 0.410921i \(0.134793\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 639.916i 1.67080i −0.549644 0.835399i \(-0.685237\pi\)
0.549644 0.835399i \(-0.314763\pi\)
\(384\) 0 0
\(385\) −810.623 −2.10551
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 499.333 + 499.333i 1.28363 + 1.28363i 0.938586 + 0.345046i \(0.112137\pi\)
0.345046 + 0.938586i \(0.387863\pi\)
\(390\) 0 0
\(391\) 197.181 0.504300
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 358.601 358.601i 0.907851 0.907851i
\(396\) 0 0
\(397\) −492.518 + 492.518i −1.24060 + 1.24060i −0.280846 + 0.959753i \(0.590615\pi\)
−0.959753 + 0.280846i \(0.909385\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −705.045 −1.75822 −0.879109 0.476621i \(-0.841862\pi\)
−0.879109 + 0.476621i \(0.841862\pi\)
\(402\) 0 0
\(403\) −204.735 204.735i −0.508026 0.508026i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −465.550 −1.14386
\(408\) 0 0
\(409\) 279.815i 0.684144i 0.939674 + 0.342072i \(0.111129\pi\)
−0.939674 + 0.342072i \(0.888871\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −171.432 + 171.432i −0.415090 + 0.415090i
\(414\) 0 0
\(415\) 423.147i 1.01963i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 573.583 + 573.583i 1.36893 + 1.36893i 0.861965 + 0.506968i \(0.169234\pi\)
0.506968 + 0.861965i \(0.330766\pi\)
\(420\) 0 0
\(421\) 213.341 + 213.341i 0.506749 + 0.506749i 0.913527 0.406778i \(-0.133348\pi\)
−0.406778 + 0.913527i \(0.633348\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 17.5358i 0.0412606i
\(426\) 0 0
\(427\) −515.781 + 515.781i −1.20792 + 1.20792i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 166.900i 0.387239i 0.981077 + 0.193619i \(0.0620227\pi\)
−0.981077 + 0.193619i \(0.937977\pi\)
\(432\) 0 0
\(433\) 233.153 0.538459 0.269230 0.963076i \(-0.413231\pi\)
0.269230 + 0.963076i \(0.413231\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −228.619 228.619i −0.523155 0.523155i
\(438\) 0 0
\(439\) −440.480 −1.00337 −0.501686 0.865050i \(-0.667287\pi\)
−0.501686 + 0.865050i \(0.667287\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 312.524 312.524i 0.705473 0.705473i −0.260107 0.965580i \(-0.583758\pi\)
0.965580 + 0.260107i \(0.0837579\pi\)
\(444\) 0 0
\(445\) −433.114 + 433.114i −0.973290 + 0.973290i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 734.338 1.63550 0.817748 0.575576i \(-0.195222\pi\)
0.817748 + 0.575576i \(0.195222\pi\)
\(450\) 0 0
\(451\) 242.772 + 242.772i 0.538297 + 0.538297i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 641.880 1.41073
\(456\) 0 0
\(457\) 692.749i 1.51586i 0.652335 + 0.757931i \(0.273789\pi\)
−0.652335 + 0.757931i \(0.726211\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 298.447 298.447i 0.647391 0.647391i −0.304971 0.952362i \(-0.598647\pi\)
0.952362 + 0.304971i \(0.0986467\pi\)
\(462\) 0 0
\(463\) 281.830i 0.608705i −0.952560 0.304352i \(-0.901560\pi\)
0.952560 0.304352i \(-0.0984400\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −198.116 198.116i −0.424232 0.424232i 0.462426 0.886658i \(-0.346979\pi\)
−0.886658 + 0.462426i \(0.846979\pi\)
\(468\) 0 0
\(469\) −591.704 591.704i −1.26163 1.26163i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 586.384i 1.23971i
\(474\) 0 0
\(475\) 20.3316 20.3316i 0.0428033 0.0428033i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 917.713i 1.91589i −0.286945 0.957947i \(-0.592640\pi\)
0.286945 0.957947i \(-0.407360\pi\)
\(480\) 0 0
\(481\) 368.639 0.766401
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 501.826 + 501.826i 1.03469 + 1.03469i
\(486\) 0 0
\(487\) 426.183 0.875119 0.437559 0.899190i \(-0.355843\pi\)
0.437559 + 0.899190i \(0.355843\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 266.299 266.299i 0.542361 0.542361i −0.381859 0.924220i \(-0.624716\pi\)
0.924220 + 0.381859i \(0.124716\pi\)
\(492\) 0 0
\(493\) 159.653 159.653i 0.323840 0.323840i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −93.8809 −0.188895
\(498\) 0 0
\(499\) −264.104 264.104i −0.529266 0.529266i 0.391088 0.920353i \(-0.372099\pi\)
−0.920353 + 0.391088i \(0.872099\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −574.766 −1.14268 −0.571338 0.820715i \(-0.693575\pi\)
−0.571338 + 0.820715i \(0.693575\pi\)
\(504\) 0 0
\(505\) 86.9756i 0.172229i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 170.592 170.592i 0.335152 0.335152i −0.519387 0.854539i \(-0.673840\pi\)
0.854539 + 0.519387i \(0.173840\pi\)
\(510\) 0 0
\(511\) 1036.13i 2.02765i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −60.4930 60.4930i −0.117462 0.117462i
\(516\) 0 0
\(517\) −307.398 307.398i −0.594581 0.594581i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 37.1210i 0.0712496i −0.999365 0.0356248i \(-0.988658\pi\)
0.999365 0.0356248i \(-0.0113421\pi\)
\(522\) 0 0
\(523\) −199.555 + 199.555i −0.381558 + 0.381558i −0.871663 0.490105i \(-0.836958\pi\)
0.490105 + 0.871663i \(0.336958\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 256.449i 0.486620i
\(528\) 0 0
\(529\) −118.392 −0.223804
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −192.236 192.236i −0.360667 0.360667i
\(534\) 0 0
\(535\) −108.226 −0.202292
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −962.884 + 962.884i −1.78643 + 1.78643i
\(540\) 0 0
\(541\) −278.121 + 278.121i −0.514086 + 0.514086i −0.915776 0.401690i \(-0.868423\pi\)
0.401690 + 0.915776i \(0.368423\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −540.323 −0.991418
\(546\) 0 0
\(547\) 724.938 + 724.938i 1.32530 + 1.32530i 0.909421 + 0.415876i \(0.136525\pi\)
0.415876 + 0.909421i \(0.363475\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −370.215 −0.671897
\(552\) 0 0
\(553\) 1277.25i 2.30967i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 268.298 268.298i 0.481685 0.481685i −0.423985 0.905669i \(-0.639369\pi\)
0.905669 + 0.423985i \(0.139369\pi\)
\(558\) 0 0
\(559\) 464.320i 0.830625i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 78.4662 + 78.4662i 0.139372 + 0.139372i 0.773350 0.633979i \(-0.218579\pi\)
−0.633979 + 0.773350i \(0.718579\pi\)
\(564\) 0 0
\(565\) −570.587 570.587i −1.00989 1.00989i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 801.999i 1.40949i −0.709461 0.704744i \(-0.751062\pi\)
0.709461 0.704744i \(-0.248938\pi\)
\(570\) 0 0
\(571\) 79.9964 79.9964i 0.140099 0.140099i −0.633579 0.773678i \(-0.718415\pi\)
0.773678 + 0.633579i \(0.218415\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 36.5163i 0.0635066i
\(576\) 0 0
\(577\) −237.186 −0.411068 −0.205534 0.978650i \(-0.565893\pi\)
−0.205534 + 0.978650i \(0.565893\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −753.571 753.571i −1.29702 1.29702i
\(582\) 0 0
\(583\) −723.534 −1.24105
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 267.958 267.958i 0.456487 0.456487i −0.441014 0.897500i \(-0.645381\pi\)
0.897500 + 0.441014i \(0.145381\pi\)
\(588\) 0 0
\(589\) 297.336 297.336i 0.504815 0.504815i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 607.086 1.02375 0.511877 0.859059i \(-0.328950\pi\)
0.511877 + 0.859059i \(0.328950\pi\)
\(594\) 0 0
\(595\) −402.007 402.007i −0.675642 0.675642i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 575.392 0.960587 0.480294 0.877108i \(-0.340530\pi\)
0.480294 + 0.877108i \(0.340530\pi\)
\(600\) 0 0
\(601\) 310.094i 0.515963i 0.966150 + 0.257981i \(0.0830573\pi\)
−0.966150 + 0.257981i \(0.916943\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −243.529 + 243.529i −0.402528 + 0.402528i
\(606\) 0 0
\(607\) 556.510i 0.916820i −0.888741 0.458410i \(-0.848419\pi\)
0.888741 0.458410i \(-0.151581\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 243.409 + 243.409i 0.398378 + 0.398378i
\(612\) 0 0
\(613\) 326.241 + 326.241i 0.532204 + 0.532204i 0.921228 0.389024i \(-0.127188\pi\)
−0.389024 + 0.921228i \(0.627188\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 502.068i 0.813725i −0.913490 0.406862i \(-0.866623\pi\)
0.913490 0.406862i \(-0.133377\pi\)
\(618\) 0 0
\(619\) 304.429 304.429i 0.491808 0.491808i −0.417067 0.908876i \(-0.636942\pi\)
0.908876 + 0.417067i \(0.136942\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1542.64i 2.47615i
\(624\) 0 0
\(625\) 576.701 0.922721
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −230.877 230.877i −0.367054 0.367054i
\(630\) 0 0
\(631\) −8.60592 −0.0136385 −0.00681927 0.999977i \(-0.502171\pi\)
−0.00681927 + 0.999977i \(0.502171\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 675.174 675.174i 1.06327 1.06327i
\(636\) 0 0
\(637\) 762.446 762.446i 1.19693 1.19693i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 445.780 0.695445 0.347722 0.937598i \(-0.386955\pi\)
0.347722 + 0.937598i \(0.386955\pi\)
\(642\) 0 0
\(643\) −118.001 118.001i −0.183517 0.183517i 0.609369 0.792886i \(-0.291423\pi\)
−0.792886 + 0.609369i \(0.791423\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1081.35 1.67132 0.835662 0.549243i \(-0.185084\pi\)
0.835662 + 0.549243i \(0.185084\pi\)
\(648\) 0 0
\(649\) 277.305i 0.427280i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −586.227 + 586.227i −0.897744 + 0.897744i −0.995236 0.0974927i \(-0.968918\pi\)
0.0974927 + 0.995236i \(0.468918\pi\)
\(654\) 0 0
\(655\) 915.041i 1.39701i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −469.999 469.999i −0.713201 0.713201i 0.254003 0.967204i \(-0.418253\pi\)
−0.967204 + 0.254003i \(0.918253\pi\)
\(660\) 0 0
\(661\) 884.745 + 884.745i 1.33849 + 1.33849i 0.897519 + 0.440976i \(0.145368\pi\)
0.440976 + 0.897519i \(0.354632\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 932.202i 1.40181i
\(666\) 0 0
\(667\) 332.460 332.460i 0.498441 0.498441i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 834.314i 1.24339i
\(672\) 0 0
\(673\) 684.329 1.01683 0.508417 0.861111i \(-0.330231\pi\)
0.508417 + 0.861111i \(0.330231\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −383.762 383.762i −0.566857 0.566857i 0.364390 0.931246i \(-0.381278\pi\)
−0.931246 + 0.364390i \(0.881278\pi\)
\(678\) 0 0
\(679\) 1787.38 2.63237
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −903.626 + 903.626i −1.32302 + 1.32302i −0.411709 + 0.911315i \(0.635068\pi\)
−0.911315 + 0.411709i \(0.864932\pi\)
\(684\) 0 0
\(685\) 871.652 871.652i 1.27248 1.27248i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 572.920 0.831524
\(690\) 0 0
\(691\) −63.6870 63.6870i −0.0921665 0.0921665i 0.659520 0.751687i \(-0.270759\pi\)
−0.751687 + 0.659520i \(0.770759\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 147.966 0.212901
\(696\) 0 0
\(697\) 240.793i 0.345470i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 218.312 218.312i 0.311430 0.311430i −0.534033 0.845463i \(-0.679324\pi\)
0.845463 + 0.534033i \(0.179324\pi\)
\(702\) 0 0
\(703\) 535.374i 0.761557i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −154.893 154.893i −0.219084 0.219084i
\(708\) 0 0
\(709\) 822.199 + 822.199i 1.15966 + 1.15966i 0.984548 + 0.175112i \(0.0560288\pi\)
0.175112 + 0.984548i \(0.443971\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 534.026i 0.748985i
\(714\) 0 0
\(715\) 519.145 519.145i 0.726077 0.726077i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 340.913i 0.474149i −0.971491 0.237074i \(-0.923811\pi\)
0.971491 0.237074i \(-0.0761885\pi\)
\(720\) 0 0
\(721\) −215.461 −0.298836
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 29.5664 + 29.5664i 0.0407813 + 0.0407813i
\(726\) 0 0
\(727\) 803.090 1.10466 0.552331 0.833625i \(-0.313738\pi\)
0.552331 + 0.833625i \(0.313738\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −290.802 + 290.802i −0.397813 + 0.397813i
\(732\) 0 0
\(733\) −481.592 + 481.592i −0.657015 + 0.657015i −0.954673 0.297658i \(-0.903795\pi\)
0.297658 + 0.954673i \(0.403795\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −957.127 −1.29868
\(738\) 0 0
\(739\) 173.622 + 173.622i 0.234941 + 0.234941i 0.814752 0.579810i \(-0.196873\pi\)
−0.579810 + 0.814752i \(0.696873\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1316.22 1.77149 0.885744 0.464173i \(-0.153649\pi\)
0.885744 + 0.464173i \(0.153649\pi\)
\(744\) 0 0
\(745\) 233.254i 0.313093i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −192.737 + 192.737i −0.257326 + 0.257326i
\(750\) 0 0
\(751\) 322.977i 0.430062i −0.976607 0.215031i \(-0.931015\pi\)
0.976607 0.215031i \(-0.0689853\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −49.2621 49.2621i −0.0652478 0.0652478i
\(756\) 0 0
\(757\) 80.2744 + 80.2744i 0.106043 + 0.106043i 0.758138 0.652095i \(-0.226110\pi\)
−0.652095 + 0.758138i \(0.726110\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 596.664i 0.784053i −0.919954 0.392027i \(-0.871774\pi\)
0.919954 0.392027i \(-0.128226\pi\)
\(762\) 0 0
\(763\) −962.246 + 962.246i −1.26113 + 1.26113i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 219.580i 0.286284i
\(768\) 0 0
\(769\) 1515.31 1.97050 0.985249 0.171129i \(-0.0547416\pi\)
0.985249 + 0.171129i \(0.0547416\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 607.901 + 607.901i 0.786418 + 0.786418i 0.980905 0.194487i \(-0.0623042\pi\)
−0.194487 + 0.980905i \(0.562304\pi\)
\(774\) 0 0
\(775\) −47.4922 −0.0612802
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 279.184 279.184i 0.358387 0.358387i
\(780\) 0 0
\(781\) −75.9297 + 75.9297i −0.0972211 + 0.0972211i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 214.324 0.273024
\(786\) 0 0
\(787\) 356.009 + 356.009i 0.452362 + 0.452362i 0.896138 0.443776i \(-0.146361\pi\)
−0.443776 + 0.896138i \(0.646361\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −2032.29 −2.56926
\(792\) 0 0
\(793\) 660.640i 0.833089i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 971.380 971.380i 1.21880 1.21880i 0.250742 0.968054i \(-0.419326\pi\)
0.968054 0.250742i \(-0.0806745\pi\)
\(798\) 0 0
\(799\) 304.892i 0.381592i
\(800\) 0 0