Properties

Label 1152.3.m.c.991.1
Level $1152$
Weight $3$
Character 1152.991
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 991.1
Root \(-1.87459 - 0.697079i\) of defining polynomial
Character \(\chi\) \(=\) 1152.991
Dual form 1152.3.m.c.415.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-5.24354 + 5.24354i) q^{5} +5.32796 q^{7} +O(q^{10})\) \(q+(-5.24354 + 5.24354i) q^{5} +5.32796 q^{7} +(-12.2863 - 12.2863i) q^{11} +(5.73657 + 5.73657i) q^{13} +23.3997 q^{17} +(11.7492 - 11.7492i) q^{19} +5.80841 q^{23} -29.9894i q^{25} +(18.3914 + 18.3914i) q^{29} +16.9053i q^{31} +(-27.9374 + 27.9374i) q^{35} +(-15.3391 + 15.3391i) q^{37} +29.2351i q^{41} +(33.4099 + 33.4099i) q^{43} -18.2125i q^{47} -20.6128 q^{49} +(-66.9856 + 66.9856i) q^{53} +128.847 q^{55} +(27.1523 + 27.1523i) q^{59} +(-65.2399 - 65.2399i) q^{61} -60.1599 q^{65} +(-37.6951 + 37.6951i) q^{67} +42.6559 q^{71} +106.391i q^{73} +(-65.4607 - 65.4607i) q^{77} +21.2821i q^{79} +(-24.1638 + 24.1638i) q^{83} +(-122.697 + 122.697i) q^{85} -52.8029i q^{89} +(30.5643 + 30.5643i) q^{91} +123.215i q^{95} -21.0222 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 32q^{11} - 32q^{19} - 128q^{23} + 32q^{29} - 96q^{35} + 96q^{37} + 160q^{43} + 112q^{49} - 160q^{53} + 256q^{55} + 128q^{59} + 32q^{61} + 32q^{65} + 320q^{67} + 512q^{71} + 224q^{77} + 160q^{83} - 160q^{85} - 480q^{91} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(5\) −5.24354 + 5.24354i −1.04871 + 1.04871i −0.0499563 + 0.998751i \(0.515908\pi\)
−0.998751 + 0.0499563i \(0.984092\pi\)
\(6\) 0 0
\(7\) 5.32796 0.761138 0.380569 0.924753i \(-0.375728\pi\)
0.380569 + 0.924753i \(0.375728\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −12.2863 12.2863i −1.11693 1.11693i −0.992189 0.124743i \(-0.960189\pi\)
−0.124743 0.992189i \(-0.539811\pi\)
\(12\) 0 0
\(13\) 5.73657 + 5.73657i 0.441275 + 0.441275i 0.892440 0.451165i \(-0.148992\pi\)
−0.451165 + 0.892440i \(0.648992\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 23.3997 1.37645 0.688226 0.725496i \(-0.258390\pi\)
0.688226 + 0.725496i \(0.258390\pi\)
\(18\) 0 0
\(19\) 11.7492 11.7492i 0.618380 0.618380i −0.326736 0.945116i \(-0.605949\pi\)
0.945116 + 0.326736i \(0.105949\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.80841 0.252540 0.126270 0.991996i \(-0.459699\pi\)
0.126270 + 0.991996i \(0.459699\pi\)
\(24\) 0 0
\(25\) 29.9894i 1.19958i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 18.3914 + 18.3914i 0.634185 + 0.634185i 0.949115 0.314930i \(-0.101981\pi\)
−0.314930 + 0.949115i \(0.601981\pi\)
\(30\) 0 0
\(31\) 16.9053i 0.545332i 0.962109 + 0.272666i \(0.0879053\pi\)
−0.962109 + 0.272666i \(0.912095\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −27.9374 + 27.9374i −0.798211 + 0.798211i
\(36\) 0 0
\(37\) −15.3391 + 15.3391i −0.414571 + 0.414571i −0.883327 0.468756i \(-0.844702\pi\)
0.468756 + 0.883327i \(0.344702\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 29.2351i 0.713051i 0.934286 + 0.356526i \(0.116039\pi\)
−0.934286 + 0.356526i \(0.883961\pi\)
\(42\) 0 0
\(43\) 33.4099 + 33.4099i 0.776975 + 0.776975i 0.979315 0.202340i \(-0.0648546\pi\)
−0.202340 + 0.979315i \(0.564855\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 18.2125i 0.387500i −0.981051 0.193750i \(-0.937935\pi\)
0.981051 0.193750i \(-0.0620650\pi\)
\(48\) 0 0
\(49\) −20.6128 −0.420670
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −66.9856 + 66.9856i −1.26388 + 1.26388i −0.314681 + 0.949197i \(0.601898\pi\)
−0.949197 + 0.314681i \(0.898102\pi\)
\(54\) 0 0
\(55\) 128.847 2.34267
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 27.1523 + 27.1523i 0.460209 + 0.460209i 0.898724 0.438515i \(-0.144495\pi\)
−0.438515 + 0.898724i \(0.644495\pi\)
\(60\) 0 0
\(61\) −65.2399 65.2399i −1.06951 1.06951i −0.997397 0.0721103i \(-0.977027\pi\)
−0.0721103 0.997397i \(-0.522973\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −60.1599 −0.925537
\(66\) 0 0
\(67\) −37.6951 + 37.6951i −0.562614 + 0.562614i −0.930049 0.367435i \(-0.880236\pi\)
0.367435 + 0.930049i \(0.380236\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 42.6559 0.600788 0.300394 0.953815i \(-0.402882\pi\)
0.300394 + 0.953815i \(0.402882\pi\)
\(72\) 0 0
\(73\) 106.391i 1.45742i 0.684825 + 0.728708i \(0.259879\pi\)
−0.684825 + 0.728708i \(0.740121\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −65.4607 65.4607i −0.850139 0.850139i
\(78\) 0 0
\(79\) 21.2821i 0.269394i 0.990887 + 0.134697i \(0.0430061\pi\)
−0.990887 + 0.134697i \(0.956994\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −24.1638 + 24.1638i −0.291130 + 0.291130i −0.837527 0.546396i \(-0.815999\pi\)
0.546396 + 0.837527i \(0.315999\pi\)
\(84\) 0 0
\(85\) −122.697 + 122.697i −1.44350 + 1.44350i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 52.8029i 0.593291i −0.954988 0.296645i \(-0.904132\pi\)
0.954988 0.296645i \(-0.0958679\pi\)
\(90\) 0 0
\(91\) 30.5643 + 30.5643i 0.335871 + 0.335871i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 123.215i 1.29700i
\(96\) 0 0
\(97\) −21.0222 −0.216724 −0.108362 0.994112i \(-0.534560\pi\)
−0.108362 + 0.994112i \(0.534560\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.24960 + 3.24960i −0.0321743 + 0.0321743i −0.723011 0.690837i \(-0.757242\pi\)
0.690837 + 0.723011i \(0.257242\pi\)
\(102\) 0 0
\(103\) −105.112 −1.02050 −0.510252 0.860025i \(-0.670448\pi\)
−0.510252 + 0.860025i \(0.670448\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 99.6160 + 99.6160i 0.930991 + 0.930991i 0.997768 0.0667770i \(-0.0212716\pi\)
−0.0667770 + 0.997768i \(0.521272\pi\)
\(108\) 0 0
\(109\) 108.050 + 108.050i 0.991282 + 0.991282i 0.999962 0.00868078i \(-0.00276321\pi\)
−0.00868078 + 0.999962i \(0.502763\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 23.2835 0.206048 0.103024 0.994679i \(-0.467148\pi\)
0.103024 + 0.994679i \(0.467148\pi\)
\(114\) 0 0
\(115\) −30.4566 + 30.4566i −0.264840 + 0.264840i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 124.673 1.04767
\(120\) 0 0
\(121\) 180.904i 1.49508i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 26.1621 + 26.1621i 0.209297 + 0.209297i
\(126\) 0 0
\(127\) 118.180i 0.930550i 0.885166 + 0.465275i \(0.154045\pi\)
−0.885166 + 0.465275i \(0.845955\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 69.2067 69.2067i 0.528296 0.528296i −0.391768 0.920064i \(-0.628137\pi\)
0.920064 + 0.391768i \(0.128137\pi\)
\(132\) 0 0
\(133\) 62.5994 62.5994i 0.470672 0.470672i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 124.474i 0.908572i 0.890856 + 0.454286i \(0.150106\pi\)
−0.890856 + 0.454286i \(0.849894\pi\)
\(138\) 0 0
\(139\) 169.014 + 169.014i 1.21593 + 1.21593i 0.969046 + 0.246881i \(0.0794057\pi\)
0.246881 + 0.969046i \(0.420594\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 140.962i 0.985749i
\(144\) 0 0
\(145\) −192.872 −1.33015
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 146.988 146.988i 0.986495 0.986495i −0.0134145 0.999910i \(-0.504270\pi\)
0.999910 + 0.0134145i \(0.00427011\pi\)
\(150\) 0 0
\(151\) −75.5456 −0.500302 −0.250151 0.968207i \(-0.580480\pi\)
−0.250151 + 0.968207i \(0.580480\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −88.6435 88.6435i −0.571893 0.571893i
\(156\) 0 0
\(157\) 81.5356 + 81.5356i 0.519335 + 0.519335i 0.917370 0.398035i \(-0.130308\pi\)
−0.398035 + 0.917370i \(0.630308\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 30.9470 0.192217
\(162\) 0 0
\(163\) 55.8065 55.8065i 0.342371 0.342371i −0.514887 0.857258i \(-0.672166\pi\)
0.857258 + 0.514887i \(0.172166\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −24.6339 −0.147508 −0.0737540 0.997276i \(-0.523498\pi\)
−0.0737540 + 0.997276i \(0.523498\pi\)
\(168\) 0 0
\(169\) 103.183i 0.610553i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4.88551 + 4.88551i 0.0282399 + 0.0282399i 0.721086 0.692846i \(-0.243643\pi\)
−0.692846 + 0.721086i \(0.743643\pi\)
\(174\) 0 0
\(175\) 159.782i 0.913042i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 229.504 229.504i 1.28215 1.28215i 0.342702 0.939444i \(-0.388658\pi\)
0.939444 0.342702i \(-0.111342\pi\)
\(180\) 0 0
\(181\) −116.607 + 116.607i −0.644238 + 0.644238i −0.951595 0.307356i \(-0.900556\pi\)
0.307356 + 0.951595i \(0.400556\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 160.863i 0.869528i
\(186\) 0 0
\(187\) −287.495 287.495i −1.53740 1.53740i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 94.2316i 0.493359i 0.969097 + 0.246680i \(0.0793395\pi\)
−0.969097 + 0.246680i \(0.920660\pi\)
\(192\) 0 0
\(193\) 84.2667 0.436615 0.218308 0.975880i \(-0.429946\pi\)
0.218308 + 0.975880i \(0.429946\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −56.9578 + 56.9578i −0.289126 + 0.289126i −0.836734 0.547609i \(-0.815538\pi\)
0.547609 + 0.836734i \(0.315538\pi\)
\(198\) 0 0
\(199\) 196.179 0.985827 0.492913 0.870078i \(-0.335932\pi\)
0.492913 + 0.870078i \(0.335932\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 97.9886 + 97.9886i 0.482702 + 0.482702i
\(204\) 0 0
\(205\) −153.295 153.295i −0.747782 0.747782i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −288.708 −1.38138
\(210\) 0 0
\(211\) 177.340 177.340i 0.840475 0.840475i −0.148445 0.988921i \(-0.547427\pi\)
0.988921 + 0.148445i \(0.0474269\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −350.373 −1.62964
\(216\) 0 0
\(217\) 90.0707i 0.415072i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 134.234 + 134.234i 0.607394 + 0.607394i
\(222\) 0 0
\(223\) 377.924i 1.69473i 0.531012 + 0.847364i \(0.321812\pi\)
−0.531012 + 0.847364i \(0.678188\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −103.909 + 103.909i −0.457750 + 0.457750i −0.897916 0.440166i \(-0.854920\pi\)
0.440166 + 0.897916i \(0.354920\pi\)
\(228\) 0 0
\(229\) 101.055 101.055i 0.441290 0.441290i −0.451156 0.892445i \(-0.648988\pi\)
0.892445 + 0.451156i \(0.148988\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 287.259i 1.23287i 0.787405 + 0.616436i \(0.211424\pi\)
−0.787405 + 0.616436i \(0.788576\pi\)
\(234\) 0 0
\(235\) 95.4979 + 95.4979i 0.406374 + 0.406374i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 150.941i 0.631554i 0.948833 + 0.315777i \(0.102265\pi\)
−0.948833 + 0.315777i \(0.897735\pi\)
\(240\) 0 0
\(241\) 37.7817 0.156771 0.0783853 0.996923i \(-0.475024\pi\)
0.0783853 + 0.996923i \(0.475024\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 108.084 108.084i 0.441159 0.441159i
\(246\) 0 0
\(247\) 134.800 0.545751
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −100.915 100.915i −0.402050 0.402050i 0.476905 0.878955i \(-0.341759\pi\)
−0.878955 + 0.476905i \(0.841759\pi\)
\(252\) 0 0
\(253\) −71.3637 71.3637i −0.282070 0.282070i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −241.295 −0.938891 −0.469446 0.882961i \(-0.655546\pi\)
−0.469446 + 0.882961i \(0.655546\pi\)
\(258\) 0 0
\(259\) −81.7263 + 81.7263i −0.315546 + 0.315546i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −118.747 −0.451509 −0.225754 0.974184i \(-0.572485\pi\)
−0.225754 + 0.974184i \(0.572485\pi\)
\(264\) 0 0
\(265\) 702.483i 2.65088i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 7.74853 + 7.74853i 0.0288050 + 0.0288050i 0.721363 0.692558i \(-0.243516\pi\)
−0.692558 + 0.721363i \(0.743516\pi\)
\(270\) 0 0
\(271\) 131.899i 0.486712i −0.969937 0.243356i \(-0.921752\pi\)
0.969937 0.243356i \(-0.0782484\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −368.457 + 368.457i −1.33984 + 1.33984i
\(276\) 0 0
\(277\) 202.352 202.352i 0.730513 0.730513i −0.240208 0.970721i \(-0.577216\pi\)
0.970721 + 0.240208i \(0.0772157\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 68.8493i 0.245015i −0.992468 0.122508i \(-0.960906\pi\)
0.992468 0.122508i \(-0.0390936\pi\)
\(282\) 0 0
\(283\) 206.773 + 206.773i 0.730646 + 0.730646i 0.970748 0.240102i \(-0.0771808\pi\)
−0.240102 + 0.970748i \(0.577181\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 155.764i 0.542730i
\(288\) 0 0
\(289\) 258.545 0.894620
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −361.237 + 361.237i −1.23289 + 1.23289i −0.270043 + 0.962848i \(0.587038\pi\)
−0.962848 + 0.270043i \(0.912962\pi\)
\(294\) 0 0
\(295\) −284.749 −0.965250
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 33.3204 + 33.3204i 0.111439 + 0.111439i
\(300\) 0 0
\(301\) 178.007 + 178.007i 0.591385 + 0.591385i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 684.176 2.24320
\(306\) 0 0
\(307\) −10.9073 + 10.9073i −0.0355286 + 0.0355286i −0.724648 0.689119i \(-0.757998\pi\)
0.689119 + 0.724648i \(0.257998\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −160.251 −0.515278 −0.257639 0.966241i \(-0.582945\pi\)
−0.257639 + 0.966241i \(0.582945\pi\)
\(312\) 0 0
\(313\) 355.500i 1.13578i 0.823103 + 0.567892i \(0.192241\pi\)
−0.823103 + 0.567892i \(0.807759\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 72.5192 + 72.5192i 0.228767 + 0.228767i 0.812178 0.583410i \(-0.198282\pi\)
−0.583410 + 0.812178i \(0.698282\pi\)
\(318\) 0 0
\(319\) 451.922i 1.41668i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 274.928 274.928i 0.851170 0.851170i
\(324\) 0 0
\(325\) 172.036 172.036i 0.529343 0.529343i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 97.0355i 0.294941i
\(330\) 0 0
\(331\) −248.096 248.096i −0.749536 0.749536i 0.224856 0.974392i \(-0.427809\pi\)
−0.974392 + 0.224856i \(0.927809\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 395.312i 1.18003i
\(336\) 0 0
\(337\) −467.271 −1.38656 −0.693280 0.720668i \(-0.743835\pi\)
−0.693280 + 0.720668i \(0.743835\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 207.703 207.703i 0.609098 0.609098i
\(342\) 0 0
\(343\) −370.894 −1.08133
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −292.821 292.821i −0.843863 0.843863i 0.145496 0.989359i \(-0.453522\pi\)
−0.989359 + 0.145496i \(0.953522\pi\)
\(348\) 0 0
\(349\) −346.260 346.260i −0.992150 0.992150i 0.00781941 0.999969i \(-0.497511\pi\)
−0.999969 + 0.00781941i \(0.997511\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −8.01816 −0.0227143 −0.0113572 0.999936i \(-0.503615\pi\)
−0.0113572 + 0.999936i \(0.503615\pi\)
\(354\) 0 0
\(355\) −223.668 + 223.668i −0.630051 + 0.630051i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 590.403 1.64458 0.822289 0.569071i \(-0.192697\pi\)
0.822289 + 0.569071i \(0.192697\pi\)
\(360\) 0 0
\(361\) 84.9121i 0.235213i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −557.867 557.867i −1.52840 1.52840i
\(366\) 0 0
\(367\) 397.100i 1.08202i −0.841017 0.541008i \(-0.818043\pi\)
0.841017 0.541008i \(-0.181957\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −356.897 + 356.897i −0.961986 + 0.961986i
\(372\) 0 0
\(373\) 165.010 165.010i 0.442387 0.442387i −0.450427 0.892814i \(-0.648728\pi\)
0.892814 + 0.450427i \(0.148728\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 211.007i 0.559700i
\(378\) 0 0
\(379\) −206.669 206.669i −0.545300 0.545300i 0.379778 0.925078i \(-0.376000\pi\)
−0.925078 + 0.379778i \(0.876000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 598.414i 1.56244i −0.624257 0.781219i \(-0.714598\pi\)
0.624257 0.781219i \(-0.285402\pi\)
\(384\) 0 0
\(385\) 686.492 1.78310
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 186.696 186.696i 0.479939 0.479939i −0.425173 0.905112i \(-0.639787\pi\)
0.905112 + 0.425173i \(0.139787\pi\)
\(390\) 0 0
\(391\) 135.915 0.347609
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −111.594 111.594i −0.282515 0.282515i
\(396\) 0 0
\(397\) 57.3727 + 57.3727i 0.144516 + 0.144516i 0.775663 0.631147i \(-0.217416\pi\)
−0.631147 + 0.775663i \(0.717416\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 466.082 1.16230 0.581149 0.813797i \(-0.302603\pi\)
0.581149 + 0.813797i \(0.302603\pi\)
\(402\) 0 0
\(403\) −96.9784 + 96.9784i −0.240641 + 0.240641i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 376.921 0.926096
\(408\) 0 0
\(409\) 597.952i 1.46198i −0.682386 0.730992i \(-0.739058\pi\)
0.682386 0.730992i \(-0.260942\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 144.667 + 144.667i 0.350282 + 0.350282i
\(414\) 0 0
\(415\) 253.408i 0.610621i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 4.65301 4.65301i 0.0111050 0.0111050i −0.701532 0.712638i \(-0.747500\pi\)
0.712638 + 0.701532i \(0.247500\pi\)
\(420\) 0 0
\(421\) −34.3754 + 34.3754i −0.0816519 + 0.0816519i −0.746753 0.665101i \(-0.768388\pi\)
0.665101 + 0.746753i \(0.268388\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 701.742i 1.65116i
\(426\) 0 0
\(427\) −347.596 347.596i −0.814042 0.814042i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 423.823i 0.983347i −0.870780 0.491674i \(-0.836385\pi\)
0.870780 0.491674i \(-0.163615\pi\)
\(432\) 0 0
\(433\) 833.377 1.92466 0.962330 0.271885i \(-0.0876472\pi\)
0.962330 + 0.271885i \(0.0876472\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 68.2443 68.2443i 0.156165 0.156165i
\(438\) 0 0
\(439\) −32.3193 −0.0736203 −0.0368102 0.999322i \(-0.511720\pi\)
−0.0368102 + 0.999322i \(0.511720\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −119.527 119.527i −0.269813 0.269813i 0.559212 0.829025i \(-0.311104\pi\)
−0.829025 + 0.559212i \(0.811104\pi\)
\(444\) 0 0
\(445\) 276.874 + 276.874i 0.622189 + 0.622189i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 182.359 0.406146 0.203073 0.979164i \(-0.434907\pi\)
0.203073 + 0.979164i \(0.434907\pi\)
\(450\) 0 0
\(451\) 359.190 359.190i 0.796430 0.796430i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −320.530 −0.704461
\(456\) 0 0
\(457\) 272.942i 0.597246i 0.954371 + 0.298623i \(0.0965274\pi\)
−0.954371 + 0.298623i \(0.903473\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 188.323 + 188.323i 0.408510 + 0.408510i 0.881219 0.472709i \(-0.156724\pi\)
−0.472709 + 0.881219i \(0.656724\pi\)
\(462\) 0 0
\(463\) 116.023i 0.250590i 0.992120 + 0.125295i \(0.0399877\pi\)
−0.992120 + 0.125295i \(0.960012\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 271.914 271.914i 0.582257 0.582257i −0.353266 0.935523i \(-0.614929\pi\)
0.935523 + 0.353266i \(0.114929\pi\)
\(468\) 0 0
\(469\) −200.838 + 200.838i −0.428227 + 0.428227i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 820.966i 1.73566i
\(474\) 0 0
\(475\) −352.352 352.352i −0.741793 0.741793i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 775.808i 1.61964i 0.586678 + 0.809820i \(0.300435\pi\)
−0.586678 + 0.809820i \(0.699565\pi\)
\(480\) 0 0
\(481\) −175.988 −0.365880
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 110.231 110.231i 0.227280 0.227280i
\(486\) 0 0
\(487\) −174.891 −0.359118 −0.179559 0.983747i \(-0.557467\pi\)
−0.179559 + 0.983747i \(0.557467\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 348.578 + 348.578i 0.709934 + 0.709934i 0.966521 0.256587i \(-0.0825980\pi\)
−0.256587 + 0.966521i \(0.582598\pi\)
\(492\) 0 0
\(493\) 430.352 + 430.352i 0.872926 + 0.872926i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 227.269 0.457282
\(498\) 0 0
\(499\) −607.544 + 607.544i −1.21752 + 1.21752i −0.249027 + 0.968496i \(0.580111\pi\)
−0.968496 + 0.249027i \(0.919889\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −130.935 −0.260309 −0.130154 0.991494i \(-0.541547\pi\)
−0.130154 + 0.991494i \(0.541547\pi\)
\(504\) 0 0
\(505\) 34.0789i 0.0674829i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −61.5539 61.5539i −0.120931 0.120931i 0.644051 0.764982i \(-0.277252\pi\)
−0.764982 + 0.644051i \(0.777252\pi\)
\(510\) 0 0
\(511\) 566.849i 1.10929i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 551.159 551.159i 1.07021 1.07021i
\(516\) 0 0
\(517\) −223.763 + 223.763i −0.432811 + 0.432811i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 32.5929i 0.0625584i −0.999511 0.0312792i \(-0.990042\pi\)
0.999511 0.0312792i \(-0.00995810\pi\)
\(522\) 0 0
\(523\) −226.407 226.407i −0.432900 0.432900i 0.456713 0.889614i \(-0.349026\pi\)
−0.889614 + 0.456713i \(0.849026\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 395.578i 0.750623i
\(528\) 0 0
\(529\) −495.262 −0.936224
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −167.709 + 167.709i −0.314652 + 0.314652i
\(534\) 0 0
\(535\) −1044.68 −1.95267
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 253.254 + 253.254i 0.469859 + 0.469859i
\(540\) 0 0
\(541\) −510.912 510.912i −0.944385 0.944385i 0.0541480 0.998533i \(-0.482756\pi\)
−0.998533 + 0.0541480i \(0.982756\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −1133.13 −2.07913
\(546\) 0 0
\(547\) 512.889 512.889i 0.937639 0.937639i −0.0605271 0.998167i \(-0.519278\pi\)
0.998167 + 0.0605271i \(0.0192782\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 432.168 0.784334
\(552\) 0 0
\(553\) 113.390i 0.205046i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 566.691 + 566.691i 1.01740 + 1.01740i 0.999846 + 0.0175529i \(0.00558754\pi\)
0.0175529 + 0.999846i \(0.494412\pi\)
\(558\) 0 0
\(559\) 383.317i 0.685720i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −548.653 + 548.653i −0.974517 + 0.974517i −0.999683 0.0251665i \(-0.991988\pi\)
0.0251665 + 0.999683i \(0.491988\pi\)
\(564\) 0 0
\(565\) −122.088 + 122.088i −0.216085 + 0.216085i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 551.224i 0.968760i 0.874858 + 0.484380i \(0.160955\pi\)
−0.874858 + 0.484380i \(0.839045\pi\)
\(570\) 0 0
\(571\) 458.387 + 458.387i 0.802780 + 0.802780i 0.983529 0.180749i \(-0.0578522\pi\)
−0.180749 + 0.983529i \(0.557852\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 174.191i 0.302941i
\(576\) 0 0
\(577\) −718.488 −1.24521 −0.622607 0.782535i \(-0.713926\pi\)
−0.622607 + 0.782535i \(0.713926\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −128.744 + 128.744i −0.221590 + 0.221590i
\(582\) 0 0
\(583\) 1646.00 2.82333
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −3.02450 3.02450i −0.00515247 0.00515247i 0.704526 0.709678i \(-0.251160\pi\)
−0.709678 + 0.704526i \(0.751160\pi\)
\(588\) 0 0
\(589\) 198.624 + 198.624i 0.337222 + 0.337222i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −576.193 −0.971657 −0.485829 0.874054i \(-0.661482\pi\)
−0.485829 + 0.874054i \(0.661482\pi\)
\(594\) 0 0
\(595\) −653.726 + 653.726i −1.09870 + 1.09870i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −1101.40 −1.83873 −0.919365 0.393406i \(-0.871297\pi\)
−0.919365 + 0.393406i \(0.871297\pi\)
\(600\) 0 0
\(601\) 7.11053i 0.0118312i −0.999983 0.00591558i \(-0.998117\pi\)
0.999983 0.00591558i \(-0.00188300\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −948.578 948.578i −1.56790 1.56790i
\(606\) 0 0
\(607\) 528.384i 0.870485i −0.900313 0.435242i \(-0.856663\pi\)
0.900313 0.435242i \(-0.143337\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 104.477 104.477i 0.170994 0.170994i
\(612\) 0 0
\(613\) 642.364 642.364i 1.04790 1.04790i 0.0491093 0.998793i \(-0.484362\pi\)
0.998793 0.0491093i \(-0.0156383\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1068.16i 1.73122i 0.500717 + 0.865611i \(0.333070\pi\)
−0.500717 + 0.865611i \(0.666930\pi\)
\(618\) 0 0
\(619\) 691.136 + 691.136i 1.11654 + 1.11654i 0.992246 + 0.124290i \(0.0396653\pi\)
0.124290 + 0.992246i \(0.460335\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 281.332i 0.451576i
\(624\) 0 0
\(625\) 475.371 0.760594
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −358.931 + 358.931i −0.570637 + 0.570637i
\(630\) 0 0
\(631\) −486.622 −0.771191 −0.385596 0.922668i \(-0.626004\pi\)
−0.385596 + 0.922668i \(0.626004\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −619.681 619.681i −0.975875 0.975875i
\(636\) 0 0
\(637\) −118.247 118.247i −0.185631 0.185631i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 691.017 1.07803 0.539015 0.842296i \(-0.318797\pi\)
0.539015 + 0.842296i \(0.318797\pi\)
\(642\) 0 0
\(643\) 652.605 652.605i 1.01494 1.01494i 0.0150512 0.999887i \(-0.495209\pi\)
0.999887 0.0150512i \(-0.00479113\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1156.72 1.78782 0.893911 0.448245i \(-0.147951\pi\)
0.893911 + 0.448245i \(0.147951\pi\)
\(648\) 0 0
\(649\) 667.201i 1.02804i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −209.105 209.105i −0.320222 0.320222i 0.528630 0.848852i \(-0.322706\pi\)
−0.848852 + 0.528630i \(0.822706\pi\)
\(654\) 0 0
\(655\) 725.776i 1.10806i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −533.902 + 533.902i −0.810170 + 0.810170i −0.984659 0.174489i \(-0.944173\pi\)
0.174489 + 0.984659i \(0.444173\pi\)
\(660\) 0 0
\(661\) −283.120 + 283.120i −0.428320 + 0.428320i −0.888056 0.459736i \(-0.847944\pi\)
0.459736 + 0.888056i \(0.347944\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 656.484i 0.987195i
\(666\) 0 0
\(667\) 106.825 + 106.825i 0.160157 + 0.160157i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1603.11i 2.38913i
\(672\) 0 0
\(673\) −397.854 −0.591164 −0.295582 0.955317i \(-0.595514\pi\)
−0.295582 + 0.955317i \(0.595514\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 289.959 289.959i 0.428299 0.428299i −0.459749 0.888049i \(-0.652061\pi\)
0.888049 + 0.459749i \(0.152061\pi\)
\(678\) 0 0
\(679\) −112.005 −0.164956
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 150.197 + 150.197i 0.219908 + 0.219908i 0.808460 0.588551i \(-0.200302\pi\)
−0.588551 + 0.808460i \(0.700302\pi\)
\(684\) 0 0
\(685\) −652.686 652.686i −0.952826 0.952826i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −768.535 −1.11544
\(690\) 0 0
\(691\) 791.212 791.212i 1.14502 1.14502i 0.157506 0.987518i \(-0.449655\pi\)
0.987518 0.157506i \(-0.0503453\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1772.46 −2.55030
\(696\) 0 0
\(697\) 684.092i 0.981481i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 900.201 + 900.201i 1.28417 + 1.28417i 0.938274 + 0.345893i \(0.112424\pi\)
0.345893 + 0.938274i \(0.387576\pi\)
\(702\) 0 0
\(703\) 360.445i 0.512724i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −17.3138 + 17.3138i −0.0244891 + 0.0244891i
\(708\) 0 0
\(709\) −128.490 + 128.490i −0.181227 + 0.181227i −0.791891 0.610663i \(-0.790903\pi\)
0.610663 + 0.791891i \(0.290903\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 98.1928i 0.137718i
\(714\) 0 0
\(715\) 739.140 + 739.140i 1.03376 + 1.03376i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1246.14i 1.73315i −0.499045 0.866576i \(-0.666316\pi\)
0.499045 0.866576i \(-0.333684\pi\)
\(720\) 0 0
\(721\) −560.033 −0.776745
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 551.546 551.546i 0.760753 0.760753i
\(726\) 0 0
\(727\) 1130.07 1.55443 0.777216 0.629234i \(-0.216631\pi\)
0.777216 + 0.629234i \(0.216631\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 781.782 + 781.782i 1.06947 + 1.06947i
\(732\) 0 0
\(733\) 708.087 + 708.087i 0.966012 + 0.966012i 0.999441 0.0334292i \(-0.0106428\pi\)
−0.0334292 + 0.999441i \(0.510643\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 926.264 1.25680
\(738\) 0 0
\(739\) −32.7516 + 32.7516i −0.0443188 + 0.0443188i −0.728919 0.684600i \(-0.759977\pi\)
0.684600 + 0.728919i \(0.259977\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 708.128 0.953066 0.476533 0.879157i \(-0.341893\pi\)
0.476533 + 0.879157i \(0.341893\pi\)
\(744\) 0 0
\(745\) 1541.47i 2.06909i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 530.751 + 530.751i 0.708612 + 0.708612i
\(750\) 0 0
\(751\) 1242.37i 1.65429i −0.561990 0.827144i \(-0.689964\pi\)
0.561990 0.827144i \(-0.310036\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 396.127 396.127i 0.524671 0.524671i
\(756\) 0 0
\(757\) 311.304 311.304i 0.411233 0.411233i −0.470935 0.882168i \(-0.656083\pi\)
0.882168 + 0.470935i \(0.156083\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 179.137i 0.235397i −0.993049 0.117699i \(-0.962448\pi\)
0.993049 0.117699i \(-0.0375517\pi\)
\(762\) 0 0
\(763\) 575.685 + 575.685i 0.754502 + 0.754502i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 311.523i 0.406158i
\(768\) 0 0
\(769\) −967.409 −1.25801 −0.629005 0.777402i \(-0.716537\pi\)
−0.629005 + 0.777402i \(0.716537\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −96.7342 + 96.7342i −0.125141 + 0.125141i −0.766904 0.641762i \(-0.778204\pi\)
0.641762 + 0.766904i \(0.278204\pi\)
\(774\) 0 0
\(775\) 506.979 0.654166
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 343.489 + 343.489i 0.440936 + 0.440936i
\(780\) 0 0
\(781\) −524.082 524.082i −0.671039 0.671039i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −855.070 −1.08926
\(786\) 0 0
\(787\) −381.038 + 381.038i −0.484166 + 0.484166i −0.906459 0.422293i \(-0.861225\pi\)
0.422293 + 0.906459i \(0.361225\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 124.054 0.156831
\(792\) 0 0
\(793\) 748.507i 0.943893i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −371.148 371.148i −0.465681 0.465681i 0.434831 0.900512i \(-0.356808\pi\)
−0.900512 + 0.434831i \(0.856808\pi\)
\(798\) 0 0
\(799\) 426.167i 0.533375i
\(800\) 0 0