Properties

Label 1152.3.m.c.991.4
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 991.4
Root \(-1.25564 + 1.55672i\) of defining polynomial
Character \(\chi\) \(=\) 1152.991
Dual form 1152.3.m.c.415.4

$q$-expansion

\(f(q)\) \(=\) \(q+(0.909023 - 0.909023i) q^{5} +0.654713 q^{7} +O(q^{10})\) \(q+(0.909023 - 0.909023i) q^{5} +0.654713 q^{7} +(13.3760 + 13.3760i) q^{11} +(-8.32795 - 8.32795i) q^{13} +3.93529 q^{17} +(16.8974 - 16.8974i) q^{19} -23.1787 q^{23} +23.3474i q^{25} +(35.6105 + 35.6105i) q^{29} -45.5687i q^{31} +(0.595149 - 0.595149i) q^{35} +(-10.1527 + 10.1527i) q^{37} +28.4661i q^{41} +(22.7354 + 22.7354i) q^{43} +10.7746i q^{47} -48.5714 q^{49} +(41.5142 - 41.5142i) q^{53} +24.3182 q^{55} +(21.0646 + 21.0646i) q^{59} +(68.7531 + 68.7531i) q^{61} -15.1406 q^{65} +(67.8242 - 67.8242i) q^{67} +33.3094 q^{71} -18.6331i q^{73} +(8.75745 + 8.75745i) q^{77} -6.29222i q^{79} +(72.0774 - 72.0774i) q^{83} +(3.57727 - 3.57727i) q^{85} -10.6131i q^{89} +(-5.45242 - 5.45242i) q^{91} -30.7202i q^{95} +143.631 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 32q^{11} - 32q^{19} - 128q^{23} + 32q^{29} - 96q^{35} + 96q^{37} + 160q^{43} + 112q^{49} - 160q^{53} + 256q^{55} + 128q^{59} + 32q^{61} + 32q^{65} + 320q^{67} + 512q^{71} + 224q^{77} + 160q^{83} - 160q^{85} - 480q^{91} + O(q^{100}) \)

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
\(5\) 0.909023 0.909023i 0.181805 0.181805i −0.610337 0.792142i \(-0.708966\pi\)
0.792142 + 0.610337i \(0.208966\pi\)
\(6\) 0 0
\(7\) 0.654713 0.0935305 0.0467652 0.998906i \(-0.485109\pi\)
0.0467652 + 0.998906i \(0.485109\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 13.3760 + 13.3760i 1.21600 + 1.21600i 0.969021 + 0.246980i \(0.0794382\pi\)
0.246980 + 0.969021i \(0.420562\pi\)
\(12\) 0 0
\(13\) −8.32795 8.32795i −0.640612 0.640612i 0.310094 0.950706i \(-0.399639\pi\)
−0.950706 + 0.310094i \(0.899639\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.93529 0.231488 0.115744 0.993279i \(-0.463075\pi\)
0.115744 + 0.993279i \(0.463075\pi\)
\(18\) 0 0
\(19\) 16.8974 16.8974i 0.889336 0.889336i −0.105123 0.994459i \(-0.533524\pi\)
0.994459 + 0.105123i \(0.0335236\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −23.1787 −1.00777 −0.503884 0.863771i \(-0.668096\pi\)
−0.503884 + 0.863771i \(0.668096\pi\)
\(24\) 0 0
\(25\) 23.3474i 0.933894i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 35.6105 + 35.6105i 1.22795 + 1.22795i 0.964739 + 0.263209i \(0.0847809\pi\)
0.263209 + 0.964739i \(0.415219\pi\)
\(30\) 0 0
\(31\) 45.5687i 1.46996i −0.678089 0.734980i \(-0.737192\pi\)
0.678089 0.734980i \(-0.262808\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.595149 0.595149i 0.0170043 0.0170043i
\(36\) 0 0
\(37\) −10.1527 + 10.1527i −0.274398 + 0.274398i −0.830868 0.556470i \(-0.812156\pi\)
0.556470 + 0.830868i \(0.312156\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 28.4661i 0.694295i 0.937811 + 0.347148i \(0.112850\pi\)
−0.937811 + 0.347148i \(0.887150\pi\)
\(42\) 0 0
\(43\) 22.7354 + 22.7354i 0.528730 + 0.528730i 0.920194 0.391464i \(-0.128031\pi\)
−0.391464 + 0.920194i \(0.628031\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.7746i 0.229247i 0.993409 + 0.114623i \(0.0365661\pi\)
−0.993409 + 0.114623i \(0.963434\pi\)
\(48\) 0 0
\(49\) −48.5714 −0.991252
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 41.5142 41.5142i 0.783287 0.783287i −0.197097 0.980384i \(-0.563151\pi\)
0.980384 + 0.197097i \(0.0631514\pi\)
\(54\) 0 0
\(55\) 24.3182 0.442149
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 21.0646 + 21.0646i 0.357027 + 0.357027i 0.862716 0.505689i \(-0.168762\pi\)
−0.505689 + 0.862716i \(0.668762\pi\)
\(60\) 0 0
\(61\) 68.7531 + 68.7531i 1.12710 + 1.12710i 0.990647 + 0.136453i \(0.0435703\pi\)
0.136453 + 0.990647i \(0.456430\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −15.1406 −0.232932
\(66\) 0 0
\(67\) 67.8242 67.8242i 1.01230 1.01230i 0.0123779 0.999923i \(-0.496060\pi\)
0.999923 0.0123779i \(-0.00394012\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 33.3094 0.469147 0.234573 0.972098i \(-0.424631\pi\)
0.234573 + 0.972098i \(0.424631\pi\)
\(72\) 0 0
\(73\) 18.6331i 0.255248i −0.991823 0.127624i \(-0.959265\pi\)
0.991823 0.127624i \(-0.0407351\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.75745 + 8.75745i 0.113733 + 0.113733i
\(78\) 0 0
\(79\) 6.29222i 0.0796483i −0.999207 0.0398242i \(-0.987320\pi\)
0.999207 0.0398242i \(-0.0126798\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 72.0774 72.0774i 0.868402 0.868402i −0.123894 0.992296i \(-0.539538\pi\)
0.992296 + 0.123894i \(0.0395382\pi\)
\(84\) 0 0
\(85\) 3.57727 3.57727i 0.0420855 0.0420855i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.6131i 0.119248i −0.998221 0.0596240i \(-0.981010\pi\)
0.998221 0.0596240i \(-0.0189902\pi\)
\(90\) 0 0
\(91\) −5.45242 5.45242i −0.0599167 0.0599167i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 30.7202i 0.323371i
\(96\) 0 0
\(97\) 143.631 1.48073 0.740366 0.672204i \(-0.234652\pi\)
0.740366 + 0.672204i \(0.234652\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −90.3100 + 90.3100i −0.894159 + 0.894159i −0.994912 0.100753i \(-0.967875\pi\)
0.100753 + 0.994912i \(0.467875\pi\)
\(102\) 0 0
\(103\) 95.1656 0.923938 0.461969 0.886896i \(-0.347143\pi\)
0.461969 + 0.886896i \(0.347143\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −27.2524 27.2524i −0.254695 0.254695i 0.568197 0.822892i \(-0.307641\pi\)
−0.822892 + 0.568197i \(0.807641\pi\)
\(108\) 0 0
\(109\) 132.413 + 132.413i 1.21480 + 1.21480i 0.969430 + 0.245366i \(0.0789082\pi\)
0.245366 + 0.969430i \(0.421092\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −37.9551 −0.335886 −0.167943 0.985797i \(-0.553712\pi\)
−0.167943 + 0.985797i \(0.553712\pi\)
\(114\) 0 0
\(115\) −21.0699 + 21.0699i −0.183217 + 0.183217i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.57649 0.0216512
\(120\) 0 0
\(121\) 236.835i 1.95731i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 43.9488 + 43.9488i 0.351591 + 0.351591i
\(126\) 0 0
\(127\) 96.5399i 0.760157i 0.924954 + 0.380078i \(0.124103\pi\)
−0.924954 + 0.380078i \(0.875897\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 54.5082 54.5082i 0.416093 0.416093i −0.467762 0.883855i \(-0.654939\pi\)
0.883855 + 0.467762i \(0.154939\pi\)
\(132\) 0 0
\(133\) 11.0629 11.0629i 0.0831801 0.0831801i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 25.9333i 0.189294i 0.995511 + 0.0946471i \(0.0301723\pi\)
−0.995511 + 0.0946471i \(0.969828\pi\)
\(138\) 0 0
\(139\) 3.64066 + 3.64066i 0.0261918 + 0.0261918i 0.720081 0.693890i \(-0.244104\pi\)
−0.693890 + 0.720081i \(0.744104\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 222.789i 1.55797i
\(144\) 0 0
\(145\) 64.7415 0.446493
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −18.9718 + 18.9718i −0.127328 + 0.127328i −0.767899 0.640571i \(-0.778698\pi\)
0.640571 + 0.767899i \(0.278698\pi\)
\(150\) 0 0
\(151\) 103.209 0.683503 0.341751 0.939790i \(-0.388980\pi\)
0.341751 + 0.939790i \(0.388980\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −41.4230 41.4230i −0.267245 0.267245i
\(156\) 0 0
\(157\) −88.2067 88.2067i −0.561826 0.561826i 0.368000 0.929826i \(-0.380043\pi\)
−0.929826 + 0.368000i \(0.880043\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −15.1754 −0.0942571
\(162\) 0 0
\(163\) 18.8038 18.8038i 0.115361 0.115361i −0.647070 0.762431i \(-0.724006\pi\)
0.762431 + 0.647070i \(0.224006\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 267.105 1.59943 0.799715 0.600380i \(-0.204984\pi\)
0.799715 + 0.600380i \(0.204984\pi\)
\(168\) 0 0
\(169\) 30.2905i 0.179234i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −153.520 153.520i −0.887396 0.887396i 0.106876 0.994272i \(-0.465915\pi\)
−0.994272 + 0.106876i \(0.965915\pi\)
\(174\) 0 0
\(175\) 15.2858i 0.0873476i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −123.581 + 123.581i −0.690399 + 0.690399i −0.962320 0.271921i \(-0.912341\pi\)
0.271921 + 0.962320i \(0.412341\pi\)
\(180\) 0 0
\(181\) −122.965 + 122.965i −0.679364 + 0.679364i −0.959856 0.280493i \(-0.909502\pi\)
0.280493 + 0.959856i \(0.409502\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 18.4581i 0.0997737i
\(186\) 0 0
\(187\) 52.6385 + 52.6385i 0.281489 + 0.281489i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 193.992i 1.01566i 0.861456 + 0.507832i \(0.169553\pi\)
−0.861456 + 0.507832i \(0.830447\pi\)
\(192\) 0 0
\(193\) 141.555 0.733444 0.366722 0.930331i \(-0.380480\pi\)
0.366722 + 0.930331i \(0.380480\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 28.9507 28.9507i 0.146958 0.146958i −0.629800 0.776758i \(-0.716863\pi\)
0.776758 + 0.629800i \(0.216863\pi\)
\(198\) 0 0
\(199\) 27.6253 0.138821 0.0694104 0.997588i \(-0.477888\pi\)
0.0694104 + 0.997588i \(0.477888\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 23.3147 + 23.3147i 0.114851 + 0.114851i
\(204\) 0 0
\(205\) 25.8763 + 25.8763i 0.126226 + 0.126226i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 452.039 2.16287
\(210\) 0 0
\(211\) 7.35041 7.35041i 0.0348361 0.0348361i −0.689474 0.724310i \(-0.742158\pi\)
0.724310 + 0.689474i \(0.242158\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 41.3340 0.192251
\(216\) 0 0
\(217\) 29.8345i 0.137486i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −32.7729 32.7729i −0.148294 0.148294i
\(222\) 0 0
\(223\) 386.106i 1.73142i −0.500549 0.865708i \(-0.666869\pi\)
0.500549 0.865708i \(-0.333131\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 49.7286 49.7286i 0.219069 0.219069i −0.589037 0.808106i \(-0.700493\pi\)
0.808106 + 0.589037i \(0.200493\pi\)
\(228\) 0 0
\(229\) 191.870 191.870i 0.837861 0.837861i −0.150716 0.988577i \(-0.548158\pi\)
0.988577 + 0.150716i \(0.0481579\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 298.610i 1.28159i 0.767712 + 0.640795i \(0.221395\pi\)
−0.767712 + 0.640795i \(0.778605\pi\)
\(234\) 0 0
\(235\) 9.79435 + 9.79435i 0.0416781 + 0.0416781i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 247.352i 1.03495i 0.855700 + 0.517473i \(0.173127\pi\)
−0.855700 + 0.517473i \(0.826873\pi\)
\(240\) 0 0
\(241\) −220.337 −0.914260 −0.457130 0.889400i \(-0.651123\pi\)
−0.457130 + 0.889400i \(0.651123\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −44.1525 + 44.1525i −0.180214 + 0.180214i
\(246\) 0 0
\(247\) −281.441 −1.13944
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −162.716 162.716i −0.648272 0.648272i 0.304303 0.952575i \(-0.401576\pi\)
−0.952575 + 0.304303i \(0.901576\pi\)
\(252\) 0 0
\(253\) −310.038 310.038i −1.22545 1.22545i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −101.165 −0.393637 −0.196819 0.980440i \(-0.563061\pi\)
−0.196819 + 0.980440i \(0.563061\pi\)
\(258\) 0 0
\(259\) −6.64713 + 6.64713i −0.0256646 + 0.0256646i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −323.635 −1.23055 −0.615276 0.788312i \(-0.710955\pi\)
−0.615276 + 0.788312i \(0.710955\pi\)
\(264\) 0 0
\(265\) 75.4747i 0.284810i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1.51275 + 1.51275i 0.00562361 + 0.00562361i 0.709913 0.704289i \(-0.248734\pi\)
−0.704289 + 0.709913i \(0.748734\pi\)
\(270\) 0 0
\(271\) 166.098i 0.612909i −0.951885 0.306454i \(-0.900857\pi\)
0.951885 0.306454i \(-0.0991427\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −312.294 + 312.294i −1.13562 + 1.13562i
\(276\) 0 0
\(277\) −317.830 + 317.830i −1.14740 + 1.14740i −0.160338 + 0.987062i \(0.551259\pi\)
−0.987062 + 0.160338i \(0.948741\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 402.790i 1.43342i −0.697374 0.716708i \(-0.745648\pi\)
0.697374 0.716708i \(-0.254352\pi\)
\(282\) 0 0
\(283\) −192.406 192.406i −0.679881 0.679881i 0.280092 0.959973i \(-0.409635\pi\)
−0.959973 + 0.280092i \(0.909635\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 18.6371i 0.0649378i
\(288\) 0 0
\(289\) −273.513 −0.946413
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −75.3645 + 75.3645i −0.257217 + 0.257217i −0.823921 0.566704i \(-0.808218\pi\)
0.566704 + 0.823921i \(0.308218\pi\)
\(294\) 0 0
\(295\) 38.2964 0.129818
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 193.031 + 193.031i 0.645588 + 0.645588i
\(300\) 0 0
\(301\) 14.8852 + 14.8852i 0.0494524 + 0.0494524i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 124.996 0.409824
\(306\) 0 0
\(307\) −111.544 + 111.544i −0.363337 + 0.363337i −0.865040 0.501703i \(-0.832707\pi\)
0.501703 + 0.865040i \(0.332707\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −224.484 −0.721813 −0.360906 0.932602i \(-0.617533\pi\)
−0.360906 + 0.932602i \(0.617533\pi\)
\(312\) 0 0
\(313\) 488.339i 1.56019i −0.625661 0.780095i \(-0.715171\pi\)
0.625661 0.780095i \(-0.284829\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 257.361 + 257.361i 0.811863 + 0.811863i 0.984913 0.173050i \(-0.0553621\pi\)
−0.173050 + 0.984913i \(0.555362\pi\)
\(318\) 0 0
\(319\) 952.652i 2.98637i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 66.4962 66.4962i 0.205871 0.205871i
\(324\) 0 0
\(325\) 194.436 194.436i 0.598263 0.598263i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 7.05427i 0.0214416i
\(330\) 0 0
\(331\) −123.553 123.553i −0.373271 0.373271i 0.495396 0.868667i \(-0.335023\pi\)
−0.868667 + 0.495396i \(0.835023\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 123.307i 0.368082i
\(336\) 0 0
\(337\) −246.234 −0.730665 −0.365333 0.930877i \(-0.619045\pi\)
−0.365333 + 0.930877i \(0.619045\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 609.528 609.528i 1.78747 1.78747i
\(342\) 0 0
\(343\) −63.8813 −0.186243
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 123.212 + 123.212i 0.355076 + 0.355076i 0.861994 0.506918i \(-0.169215\pi\)
−0.506918 + 0.861994i \(0.669215\pi\)
\(348\) 0 0
\(349\) −115.371 115.371i −0.330575 0.330575i 0.522230 0.852805i \(-0.325100\pi\)
−0.852805 + 0.522230i \(0.825100\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −650.544 −1.84290 −0.921451 0.388495i \(-0.872995\pi\)
−0.921451 + 0.388495i \(0.872995\pi\)
\(354\) 0 0
\(355\) 30.2790 30.2790i 0.0852930 0.0852930i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 94.4878 0.263197 0.131599 0.991303i \(-0.457989\pi\)
0.131599 + 0.991303i \(0.457989\pi\)
\(360\) 0 0
\(361\) 210.044i 0.581838i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −16.9379 16.9379i −0.0464053 0.0464053i
\(366\) 0 0
\(367\) 131.379i 0.357982i −0.983851 0.178991i \(-0.942717\pi\)
0.983851 0.178991i \(-0.0572832\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 27.1799 27.1799i 0.0732612 0.0732612i
\(372\) 0 0
\(373\) 275.796 275.796i 0.739400 0.739400i −0.233062 0.972462i \(-0.574874\pi\)
0.972462 + 0.233062i \(0.0748745\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 593.125i 1.57328i
\(378\) 0 0
\(379\) −13.0427 13.0427i −0.0344135 0.0344135i 0.689691 0.724104i \(-0.257747\pi\)
−0.724104 + 0.689691i \(0.757747\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 121.974i 0.318470i 0.987241 + 0.159235i \(0.0509027\pi\)
−0.987241 + 0.159235i \(0.949097\pi\)
\(384\) 0 0
\(385\) 15.9214 0.0413544
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −233.267 + 233.267i −0.599659 + 0.599659i −0.940222 0.340563i \(-0.889382\pi\)
0.340563 + 0.940222i \(0.389382\pi\)
\(390\) 0 0
\(391\) −91.2149 −0.233286
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −5.71977 5.71977i −0.0144804 0.0144804i
\(396\) 0 0
\(397\) 83.7693 + 83.7693i 0.211006 + 0.211006i 0.804695 0.593689i \(-0.202329\pi\)
−0.593689 + 0.804695i \(0.702329\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −589.134 −1.46916 −0.734581 0.678521i \(-0.762621\pi\)
−0.734581 + 0.678521i \(0.762621\pi\)
\(402\) 0 0
\(403\) −379.494 + 379.494i −0.941673 + 0.941673i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −271.606 −0.667337
\(408\) 0 0
\(409\) 449.285i 1.09850i 0.835659 + 0.549248i \(0.185086\pi\)
−0.835659 + 0.549248i \(0.814914\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 13.7913 + 13.7913i 0.0333929 + 0.0333929i
\(414\) 0 0
\(415\) 131.040i 0.315759i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 218.639 218.639i 0.521811 0.521811i −0.396307 0.918118i \(-0.629708\pi\)
0.918118 + 0.396307i \(0.129708\pi\)
\(420\) 0 0
\(421\) 61.2101 61.2101i 0.145392 0.145392i −0.630664 0.776056i \(-0.717217\pi\)
0.776056 + 0.630664i \(0.217217\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 91.8787i 0.216185i
\(426\) 0 0
\(427\) 45.0136 + 45.0136i 0.105418 + 0.105418i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 501.119i 1.16269i −0.813657 0.581345i \(-0.802527\pi\)
0.813657 0.581345i \(-0.197473\pi\)
\(432\) 0 0
\(433\) 75.5505 0.174482 0.0872408 0.996187i \(-0.472195\pi\)
0.0872408 + 0.996187i \(0.472195\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −391.659 + 391.659i −0.896245 + 0.896245i
\(438\) 0 0
\(439\) −717.251 −1.63383 −0.816915 0.576758i \(-0.804318\pi\)
−0.816915 + 0.576758i \(0.804318\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −299.093 299.093i −0.675153 0.675153i 0.283746 0.958899i \(-0.408423\pi\)
−0.958899 + 0.283746i \(0.908423\pi\)
\(444\) 0 0
\(445\) −9.64753 9.64753i −0.0216798 0.0216798i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 44.5560 0.0992339 0.0496170 0.998768i \(-0.484200\pi\)
0.0496170 + 0.998768i \(0.484200\pi\)
\(450\) 0 0
\(451\) −380.763 + 380.763i −0.844263 + 0.844263i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −9.91275 −0.0217863
\(456\) 0 0
\(457\) 641.227i 1.40312i −0.712609 0.701562i \(-0.752486\pi\)
0.712609 0.701562i \(-0.247514\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −393.690 393.690i −0.853991 0.853991i 0.136631 0.990622i \(-0.456373\pi\)
−0.990622 + 0.136631i \(0.956373\pi\)
\(462\) 0 0
\(463\) 395.861i 0.854991i 0.904018 + 0.427495i \(0.140604\pi\)
−0.904018 + 0.427495i \(0.859396\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 83.1457 83.1457i 0.178042 0.178042i −0.612460 0.790502i \(-0.709820\pi\)
0.790502 + 0.612460i \(0.209820\pi\)
\(468\) 0 0
\(469\) 44.4054 44.4054i 0.0946810 0.0946810i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 608.217i 1.28587i
\(474\) 0 0
\(475\) 394.509 + 394.509i 0.830546 + 0.830546i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 430.043i 0.897793i 0.893584 + 0.448896i \(0.148183\pi\)
−0.893584 + 0.448896i \(0.851817\pi\)
\(480\) 0 0
\(481\) 169.103 0.351565
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 130.564 130.564i 0.269204 0.269204i
\(486\) 0 0
\(487\) −573.790 −1.17821 −0.589107 0.808055i \(-0.700520\pi\)
−0.589107 + 0.808055i \(0.700520\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 489.133 + 489.133i 0.996197 + 0.996197i 0.999993 0.00379588i \(-0.00120827\pi\)
−0.00379588 + 0.999993i \(0.501208\pi\)
\(492\) 0 0
\(493\) 140.138 + 140.138i 0.284255 + 0.284255i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 21.8081 0.0438795
\(498\) 0 0
\(499\) 260.469 260.469i 0.521982 0.521982i −0.396188 0.918170i \(-0.629667\pi\)
0.918170 + 0.396188i \(0.129667\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −975.416 −1.93920 −0.969598 0.244701i \(-0.921310\pi\)
−0.969598 + 0.244701i \(0.921310\pi\)
\(504\) 0 0
\(505\) 164.188i 0.325124i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −420.191 420.191i −0.825523 0.825523i 0.161371 0.986894i \(-0.448408\pi\)
−0.986894 + 0.161371i \(0.948408\pi\)
\(510\) 0 0
\(511\) 12.1994i 0.0238735i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 86.5077 86.5077i 0.167976 0.167976i
\(516\) 0 0
\(517\) −144.121 + 144.121i −0.278764 + 0.278764i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 396.333i 0.760716i −0.924839 0.380358i \(-0.875801\pi\)
0.924839 0.380358i \(-0.124199\pi\)
\(522\) 0 0
\(523\) 564.600 + 564.600i 1.07954 + 1.07954i 0.996550 + 0.0829913i \(0.0264474\pi\)
0.0829913 + 0.996550i \(0.473553\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 179.326i 0.340278i
\(528\) 0 0
\(529\) 8.25115 0.0155976
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 237.064 237.064i 0.444773 0.444773i
\(534\) 0 0
\(535\) −49.5461 −0.0926095
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −649.691 649.691i −1.20536 1.20536i
\(540\) 0 0
\(541\) 29.5601 + 29.5601i 0.0546398 + 0.0546398i 0.733899 0.679259i \(-0.237699\pi\)
−0.679259 + 0.733899i \(0.737699\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 240.733 0.441711
\(546\) 0 0
\(547\) −138.608 + 138.608i −0.253397 + 0.253397i −0.822362 0.568965i \(-0.807344\pi\)
0.568965 + 0.822362i \(0.307344\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1203.45 2.18412
\(552\) 0 0
\(553\) 4.11960i 0.00744955i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 60.4400 + 60.4400i 0.108510 + 0.108510i 0.759277 0.650767i \(-0.225553\pi\)
−0.650767 + 0.759277i \(0.725553\pi\)
\(558\) 0 0
\(559\) 378.678i 0.677421i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −267.325 + 267.325i −0.474822 + 0.474822i −0.903471 0.428649i \(-0.858990\pi\)
0.428649 + 0.903471i \(0.358990\pi\)
\(564\) 0 0
\(565\) −34.5021 + 34.5021i −0.0610656 + 0.0610656i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 315.715i 0.554859i 0.960746 + 0.277429i \(0.0894825\pi\)
−0.960746 + 0.277429i \(0.910518\pi\)
\(570\) 0 0
\(571\) −670.572 670.572i −1.17438 1.17438i −0.981154 0.193228i \(-0.938104\pi\)
−0.193228 0.981154i \(-0.561896\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 541.161i 0.941149i
\(576\) 0 0
\(577\) 413.628 0.716859 0.358430 0.933557i \(-0.383312\pi\)
0.358430 + 0.933557i \(0.383312\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 47.1900 47.1900i 0.0812220 0.0812220i
\(582\) 0 0
\(583\) 1110.59 1.90495
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 420.085 + 420.085i 0.715647 + 0.715647i 0.967711 0.252064i \(-0.0811093\pi\)
−0.252064 + 0.967711i \(0.581109\pi\)
\(588\) 0 0
\(589\) −769.993 769.993i −1.30729 1.30729i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 740.798 1.24924 0.624619 0.780930i \(-0.285254\pi\)
0.624619 + 0.780930i \(0.285254\pi\)
\(594\) 0 0
\(595\) 2.34209 2.34209i 0.00393628 0.00393628i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 435.161 0.726479 0.363240 0.931696i \(-0.381671\pi\)
0.363240 + 0.931696i \(0.381671\pi\)
\(600\) 0 0
\(601\) 380.001i 0.632280i 0.948712 + 0.316140i \(0.102387\pi\)
−0.948712 + 0.316140i \(0.897613\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 215.288 + 215.288i 0.355849 + 0.355849i
\(606\) 0 0
\(607\) 181.813i 0.299527i −0.988722 0.149763i \(-0.952149\pi\)
0.988722 0.149763i \(-0.0478512\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 89.7303 89.7303i 0.146858 0.146858i
\(612\) 0 0
\(613\) −55.1479 + 55.1479i −0.0899640 + 0.0899640i −0.750657 0.660693i \(-0.770263\pi\)
0.660693 + 0.750657i \(0.270263\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 579.674i 0.939504i −0.882798 0.469752i \(-0.844343\pi\)
0.882798 0.469752i \(-0.155657\pi\)
\(618\) 0 0
\(619\) 91.1070 + 91.1070i 0.147184 + 0.147184i 0.776859 0.629675i \(-0.216812\pi\)
−0.629675 + 0.776859i \(0.716812\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 6.94852i 0.0111533i
\(624\) 0 0
\(625\) −503.783 −0.806053
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −39.9540 + 39.9540i −0.0635199 + 0.0635199i
\(630\) 0 0
\(631\) 693.474 1.09901 0.549504 0.835491i \(-0.314817\pi\)
0.549504 + 0.835491i \(0.314817\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 87.7570 + 87.7570i 0.138200 + 0.138200i
\(636\) 0 0
\(637\) 404.500 + 404.500i 0.635007 + 0.635007i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 218.329 0.340607 0.170304 0.985392i \(-0.445525\pi\)
0.170304 + 0.985392i \(0.445525\pi\)
\(642\) 0 0
\(643\) 887.430 887.430i 1.38014 1.38014i 0.535787 0.844353i \(-0.320015\pi\)
0.844353 0.535787i \(-0.179985\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −223.177 −0.344941 −0.172470 0.985015i \(-0.555175\pi\)
−0.172470 + 0.985015i \(0.555175\pi\)
\(648\) 0 0
\(649\) 563.520i 0.868290i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 539.691 + 539.691i 0.826479 + 0.826479i 0.987028 0.160549i \(-0.0513264\pi\)
−0.160549 + 0.987028i \(0.551326\pi\)
\(654\) 0 0
\(655\) 99.0983i 0.151295i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −625.166 + 625.166i −0.948659 + 0.948659i −0.998745 0.0500862i \(-0.984050\pi\)
0.0500862 + 0.998745i \(0.484050\pi\)
\(660\) 0 0
\(661\) 326.893 326.893i 0.494544 0.494544i −0.415191 0.909734i \(-0.636285\pi\)
0.909734 + 0.415191i \(0.136285\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 20.1129i 0.0302450i
\(666\) 0 0
\(667\) −825.404 825.404i −1.23749 1.23749i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1839.28i 2.74111i
\(672\) 0 0
\(673\) 422.147 0.627262 0.313631 0.949545i \(-0.398455\pi\)
0.313631 + 0.949545i \(0.398455\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −126.017 + 126.017i −0.186140 + 0.186140i −0.794025 0.607885i \(-0.792018\pi\)
0.607885 + 0.794025i \(0.292018\pi\)
\(678\) 0 0
\(679\) 94.0372 0.138494
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −621.906 621.906i −0.910551 0.910551i 0.0857647 0.996315i \(-0.472667\pi\)
−0.996315 + 0.0857647i \(0.972667\pi\)
\(684\) 0 0
\(685\) 23.5740 + 23.5740i 0.0344145 + 0.0344145i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −691.456 −1.00357
\(690\) 0 0
\(691\) −403.376 + 403.376i −0.583758 + 0.583758i −0.935934 0.352176i \(-0.885442\pi\)
0.352176 + 0.935934i \(0.385442\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 6.61889 0.00952359
\(696\) 0 0
\(697\) 112.022i 0.160721i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 466.593 + 466.593i 0.665611 + 0.665611i 0.956697 0.291086i \(-0.0940166\pi\)
−0.291086 + 0.956697i \(0.594017\pi\)
\(702\) 0 0
\(703\) 343.109i 0.488065i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −59.1272 + 59.1272i −0.0836311 + 0.0836311i
\(708\) 0 0
\(709\) −822.764 + 822.764i −1.16046 + 1.16046i −0.176081 + 0.984376i \(0.556342\pi\)
−0.984376 + 0.176081i \(0.943658\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1056.22i 1.48138i
\(714\) 0 0
\(715\) −202.521 202.521i −0.283246 0.283246i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 710.142i 0.987681i −0.869553 0.493840i \(-0.835593\pi\)
0.869553 0.493840i \(-0.164407\pi\)
\(720\) 0 0
\(721\) 62.3062 0.0864164
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −831.411 + 831.411i −1.14677 + 1.14677i
\(726\) 0 0
\(727\) 214.095 0.294490 0.147245 0.989100i \(-0.452959\pi\)
0.147245 + 0.989100i \(0.452959\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 89.4704 + 89.4704i 0.122395 + 0.122395i
\(732\) 0 0
\(733\) 96.1768 + 96.1768i 0.131210 + 0.131210i 0.769662 0.638452i \(-0.220425\pi\)
−0.638452 + 0.769662i \(0.720425\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1814.43 2.46192
\(738\) 0 0
\(739\) 885.341 885.341i 1.19803 1.19803i 0.223268 0.974757i \(-0.428327\pi\)
0.974757 0.223268i \(-0.0716726\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 906.258 1.21973 0.609864 0.792506i \(-0.291224\pi\)
0.609864 + 0.792506i \(0.291224\pi\)
\(744\) 0 0
\(745\) 34.4917i 0.0462976i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −17.8425 17.8425i −0.0238218 0.0238218i
\(750\) 0 0
\(751\) 1147.02i 1.52732i −0.645618 0.763661i \(-0.723400\pi\)
0.645618 0.763661i \(-0.276600\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 93.8192 93.8192i 0.124264 0.124264i
\(756\) 0 0
\(757\) −525.591 + 525.591i −0.694308 + 0.694308i −0.963177 0.268869i \(-0.913350\pi\)
0.268869 + 0.963177i \(0.413350\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 788.107i 1.03562i 0.855495 + 0.517810i \(0.173253\pi\)
−0.855495 + 0.517810i \(0.826747\pi\)
\(762\) 0 0
\(763\) 86.6925 + 86.6925i 0.113621 + 0.113621i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 350.850i 0.457431i
\(768\) 0 0
\(769\) −768.187 −0.998943 −0.499471 0.866330i \(-0.666472\pi\)
−0.499471 + 0.866330i \(0.666472\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 275.915 275.915i 0.356941 0.356941i −0.505743 0.862684i \(-0.668782\pi\)
0.862684 + 0.505743i \(0.168782\pi\)
\(774\) 0 0
\(775\) 1063.91 1.37279
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 481.003 + 481.003i 0.617462 + 0.617462i
\(780\) 0 0
\(781\) 445.547 + 445.547i 0.570483 + 0.570483i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −160.364 −0.204285
\(786\) 0 0
\(787\) 240.824 240.824i 0.306002 0.306002i −0.537354 0.843356i \(-0.680576\pi\)
0.843356 + 0.537354i \(0.180576\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −24.8497 −0.0314156
\(792\) 0 0
\(793\) 1145.14i 1.44407i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −681.033 681.033i −0.854496 0.854496i 0.136187 0.990683i \(-0.456515\pi\)
−0.990683 + 0.136187i \(0.956515\pi\)
\(798\) 0 0
\(799\) 42.4012i 0.0530678i
\(800\) 0 0