Properties

Label 8280.2.p.a.1241.23
Level $8280$
Weight $2$
Character 8280.1241
Analytic conductor $66.116$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8280.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(66.1161328736\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1241.23
Character \(\chi\) \(=\) 8280.1241
Dual form 8280.2.p.a.1241.26

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{5} -0.488657i q^{7} +O(q^{10})\) \(q-1.00000 q^{5} -0.488657i q^{7} -0.477618 q^{11} -4.91681 q^{13} +0.0686460 q^{17} -2.26986i q^{19} +(-4.64530 - 1.19213i) q^{23} +1.00000 q^{25} +2.85439i q^{29} -3.06933 q^{31} +0.488657i q^{35} +5.16861i q^{37} -6.04576i q^{41} +2.17648i q^{43} +0.440529i q^{47} +6.76121 q^{49} -5.95717 q^{53} +0.477618 q^{55} +2.25315i q^{59} -3.28117i q^{61} +4.91681 q^{65} -6.36252i q^{67} +8.73638i q^{71} +13.5209 q^{73} +0.233391i q^{77} +9.13107i q^{79} +3.27923 q^{83} -0.0686460 q^{85} +10.3839 q^{89} +2.40263i q^{91} +2.26986i q^{95} -17.5807i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 48 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 48 q - 48 q^{5} - 8 q^{11} + 4 q^{23} + 48 q^{25} + 8 q^{31} - 32 q^{49} + 8 q^{55} - 16 q^{73} - 32 q^{83} - 8 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8280\mathbb{Z}\right)^\times\).

\(n\) \(1657\) \(2071\) \(3961\) \(4141\) \(4601\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\) \(-1\)

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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0.488657i 0.184695i −0.995727 0.0923475i \(-0.970563\pi\)
0.995727 0.0923475i \(-0.0294371\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.477618 −0.144007 −0.0720036 0.997404i \(-0.522939\pi\)
−0.0720036 + 0.997404i \(0.522939\pi\)
\(12\) 0 0
\(13\) −4.91681 −1.36368 −0.681838 0.731503i \(-0.738819\pi\)
−0.681838 + 0.731503i \(0.738819\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.0686460 0.0166491 0.00832455 0.999965i \(-0.497350\pi\)
0.00832455 + 0.999965i \(0.497350\pi\)
\(18\) 0 0
\(19\) 2.26986i 0.520742i −0.965509 0.260371i \(-0.916155\pi\)
0.965509 0.260371i \(-0.0838449\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.64530 1.19213i −0.968612 0.248576i
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.85439i 0.530047i 0.964242 + 0.265023i \(0.0853797\pi\)
−0.964242 + 0.265023i \(0.914620\pi\)
\(30\) 0 0
\(31\) −3.06933 −0.551268 −0.275634 0.961263i \(-0.588888\pi\)
−0.275634 + 0.961263i \(0.588888\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.488657i 0.0825981i
\(36\) 0 0
\(37\) 5.16861i 0.849714i 0.905261 + 0.424857i \(0.139676\pi\)
−0.905261 + 0.424857i \(0.860324\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.04576i 0.944189i −0.881548 0.472095i \(-0.843498\pi\)
0.881548 0.472095i \(-0.156502\pi\)
\(42\) 0 0
\(43\) 2.17648i 0.331911i 0.986133 + 0.165955i \(0.0530708\pi\)
−0.986133 + 0.165955i \(0.946929\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.440529i 0.0642577i 0.999484 + 0.0321289i \(0.0102287\pi\)
−0.999484 + 0.0321289i \(0.989771\pi\)
\(48\) 0 0
\(49\) 6.76121 0.965888
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.95717 −0.818280 −0.409140 0.912472i \(-0.634171\pi\)
−0.409140 + 0.912472i \(0.634171\pi\)
\(54\) 0 0
\(55\) 0.477618 0.0644020
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.25315i 0.293335i 0.989186 + 0.146668i \(0.0468547\pi\)
−0.989186 + 0.146668i \(0.953145\pi\)
\(60\) 0 0
\(61\) 3.28117i 0.420111i −0.977689 0.210056i \(-0.932636\pi\)
0.977689 0.210056i \(-0.0673645\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.91681 0.609855
\(66\) 0 0
\(67\) 6.36252i 0.777305i −0.921384 0.388653i \(-0.872941\pi\)
0.921384 0.388653i \(-0.127059\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.73638i 1.03682i 0.855133 + 0.518409i \(0.173475\pi\)
−0.855133 + 0.518409i \(0.826525\pi\)
\(72\) 0 0
\(73\) 13.5209 1.58250 0.791249 0.611495i \(-0.209431\pi\)
0.791249 + 0.611495i \(0.209431\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.233391i 0.0265974i
\(78\) 0 0
\(79\) 9.13107i 1.02733i 0.857992 + 0.513663i \(0.171712\pi\)
−0.857992 + 0.513663i \(0.828288\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.27923 0.359943 0.179971 0.983672i \(-0.442399\pi\)
0.179971 + 0.983672i \(0.442399\pi\)
\(84\) 0 0
\(85\) −0.0686460 −0.00744571
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.3839 1.10069 0.550346 0.834937i \(-0.314496\pi\)
0.550346 + 0.834937i \(0.314496\pi\)
\(90\) 0 0
\(91\) 2.40263i 0.251864i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.26986i 0.232883i
\(96\) 0 0
\(97\) 17.5807i 1.78505i −0.450995 0.892526i \(-0.648931\pi\)
0.450995 0.892526i \(-0.351069\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 18.3289i 1.82380i 0.410416 + 0.911898i \(0.365384\pi\)
−0.410416 + 0.911898i \(0.634616\pi\)
\(102\) 0 0
\(103\) 12.0722i 1.18951i −0.803909 0.594753i \(-0.797250\pi\)
0.803909 0.594753i \(-0.202750\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −0.830054 −0.0802443 −0.0401222 0.999195i \(-0.512775\pi\)
−0.0401222 + 0.999195i \(0.512775\pi\)
\(108\) 0 0
\(109\) 3.99309i 0.382468i 0.981544 + 0.191234i \(0.0612490\pi\)
−0.981544 + 0.191234i \(0.938751\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −14.1428 −1.33045 −0.665223 0.746645i \(-0.731663\pi\)
−0.665223 + 0.746645i \(0.731663\pi\)
\(114\) 0 0
\(115\) 4.64530 + 1.19213i 0.433177 + 0.111166i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.0335444i 0.00307501i
\(120\) 0 0
\(121\) −10.7719 −0.979262
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 13.7107 1.21663 0.608313 0.793697i \(-0.291847\pi\)
0.608313 + 0.793697i \(0.291847\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 8.63722i 0.754638i −0.926083 0.377319i \(-0.876846\pi\)
0.926083 0.377319i \(-0.123154\pi\)
\(132\) 0 0
\(133\) −1.10918 −0.0961785
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0.331548 0.0283261 0.0141630 0.999900i \(-0.495492\pi\)
0.0141630 + 0.999900i \(0.495492\pi\)
\(138\) 0 0
\(139\) 12.0248 1.01993 0.509967 0.860194i \(-0.329658\pi\)
0.509967 + 0.860194i \(0.329658\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.34836 0.196379
\(144\) 0 0
\(145\) 2.85439i 0.237044i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 22.8306 1.87035 0.935177 0.354180i \(-0.115240\pi\)
0.935177 + 0.354180i \(0.115240\pi\)
\(150\) 0 0
\(151\) 19.0857 1.55318 0.776588 0.630009i \(-0.216949\pi\)
0.776588 + 0.630009i \(0.216949\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.06933 0.246535
\(156\) 0 0
\(157\) 7.13723i 0.569613i 0.958585 + 0.284807i \(0.0919294\pi\)
−0.958585 + 0.284807i \(0.908071\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −0.582542 + 2.26996i −0.0459107 + 0.178898i
\(162\) 0 0
\(163\) −1.07862 −0.0844840 −0.0422420 0.999107i \(-0.513450\pi\)
−0.0422420 + 0.999107i \(0.513450\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 16.2257i 1.25558i 0.778382 + 0.627791i \(0.216041\pi\)
−0.778382 + 0.627791i \(0.783959\pi\)
\(168\) 0 0
\(169\) 11.1750 0.859615
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4.67906i 0.355742i 0.984054 + 0.177871i \(0.0569210\pi\)
−0.984054 + 0.177871i \(0.943079\pi\)
\(174\) 0 0
\(175\) 0.488657i 0.0369390i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 6.05993i 0.452940i 0.974018 + 0.226470i \(0.0727186\pi\)
−0.974018 + 0.226470i \(0.927281\pi\)
\(180\) 0 0
\(181\) 21.8606i 1.62489i −0.583041 0.812443i \(-0.698137\pi\)
0.583041 0.812443i \(-0.301863\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 5.16861i 0.380004i
\(186\) 0 0
\(187\) −0.0327866 −0.00239759
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −19.6665 −1.42302 −0.711510 0.702676i \(-0.751988\pi\)
−0.711510 + 0.702676i \(0.751988\pi\)
\(192\) 0 0
\(193\) 7.08295 0.509842 0.254921 0.966962i \(-0.417951\pi\)
0.254921 + 0.966962i \(0.417951\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 15.8556i 1.12967i 0.825205 + 0.564834i \(0.191060\pi\)
−0.825205 + 0.564834i \(0.808940\pi\)
\(198\) 0 0
\(199\) 5.91295i 0.419158i 0.977792 + 0.209579i \(0.0672093\pi\)
−0.977792 + 0.209579i \(0.932791\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.39482 0.0978970
\(204\) 0 0
\(205\) 6.04576i 0.422254i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.08413i 0.0749907i
\(210\) 0 0
\(211\) 16.3081 1.12270 0.561348 0.827580i \(-0.310283\pi\)
0.561348 + 0.827580i \(0.310283\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2.17648i 0.148435i
\(216\) 0 0
\(217\) 1.49985i 0.101817i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −0.337519 −0.0227040
\(222\) 0 0
\(223\) 3.19461 0.213927 0.106964 0.994263i \(-0.465887\pi\)
0.106964 + 0.994263i \(0.465887\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −28.3014 −1.87843 −0.939216 0.343326i \(-0.888447\pi\)
−0.939216 + 0.343326i \(0.888447\pi\)
\(228\) 0 0
\(229\) 7.96687i 0.526465i −0.964732 0.263233i \(-0.915211\pi\)
0.964732 0.263233i \(-0.0847887\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 20.5271i 1.34478i 0.740198 + 0.672389i \(0.234732\pi\)
−0.740198 + 0.672389i \(0.765268\pi\)
\(234\) 0 0
\(235\) 0.440529i 0.0287369i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 25.8843i 1.67431i −0.546962 0.837157i \(-0.684216\pi\)
0.546962 0.837157i \(-0.315784\pi\)
\(240\) 0 0
\(241\) 10.4182i 0.671097i 0.942023 + 0.335549i \(0.108922\pi\)
−0.942023 + 0.335549i \(0.891078\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −6.76121 −0.431958
\(246\) 0 0
\(247\) 11.1605i 0.710124i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 6.46023 0.407766 0.203883 0.978995i \(-0.434644\pi\)
0.203883 + 0.978995i \(0.434644\pi\)
\(252\) 0 0
\(253\) 2.21868 + 0.569382i 0.139487 + 0.0357967i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 10.2754i 0.640961i 0.947255 + 0.320480i \(0.103844\pi\)
−0.947255 + 0.320480i \(0.896156\pi\)
\(258\) 0 0
\(259\) 2.52568 0.156938
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 28.2722 1.74334 0.871670 0.490094i \(-0.163038\pi\)
0.871670 + 0.490094i \(0.163038\pi\)
\(264\) 0 0
\(265\) 5.95717 0.365946
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 16.8772i 1.02902i 0.857484 + 0.514511i \(0.172027\pi\)
−0.857484 + 0.514511i \(0.827973\pi\)
\(270\) 0 0
\(271\) 17.2612 1.04855 0.524273 0.851550i \(-0.324337\pi\)
0.524273 + 0.851550i \(0.324337\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −0.477618 −0.0288015
\(276\) 0 0
\(277\) −12.0237 −0.722436 −0.361218 0.932481i \(-0.617639\pi\)
−0.361218 + 0.932481i \(0.617639\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 16.0597 0.958043 0.479021 0.877803i \(-0.340992\pi\)
0.479021 + 0.877803i \(0.340992\pi\)
\(282\) 0 0
\(283\) 21.9028i 1.30199i −0.759084 0.650993i \(-0.774353\pi\)
0.759084 0.650993i \(-0.225647\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.95430 −0.174387
\(288\) 0 0
\(289\) −16.9953 −0.999723
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 22.2825 1.30176 0.650880 0.759181i \(-0.274400\pi\)
0.650880 + 0.759181i \(0.274400\pi\)
\(294\) 0 0
\(295\) 2.25315i 0.131183i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 22.8401 + 5.86146i 1.32087 + 0.338977i
\(300\) 0 0
\(301\) 1.06355 0.0613023
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 3.28117i 0.187879i
\(306\) 0 0
\(307\) 14.9419 0.852777 0.426388 0.904540i \(-0.359786\pi\)
0.426388 + 0.904540i \(0.359786\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 20.9265i 1.18663i −0.804969 0.593317i \(-0.797818\pi\)
0.804969 0.593317i \(-0.202182\pi\)
\(312\) 0 0
\(313\) 21.9377i 1.23999i 0.784604 + 0.619997i \(0.212866\pi\)
−0.784604 + 0.619997i \(0.787134\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 28.1562i 1.58141i 0.612199 + 0.790704i \(0.290285\pi\)
−0.612199 + 0.790704i \(0.709715\pi\)
\(318\) 0 0
\(319\) 1.36331i 0.0763306i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0.155817i 0.00866989i
\(324\) 0 0
\(325\) −4.91681 −0.272735
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0.215267 0.0118681
\(330\) 0 0
\(331\) −5.62411 −0.309129 −0.154565 0.987983i \(-0.549397\pi\)
−0.154565 + 0.987983i \(0.549397\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 6.36252i 0.347621i
\(336\) 0 0
\(337\) 18.9823i 1.03403i −0.855975 0.517017i \(-0.827042\pi\)
0.855975 0.517017i \(-0.172958\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1.46597 0.0793866
\(342\) 0 0
\(343\) 6.72452i 0.363090i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 29.2227i 1.56876i −0.620284 0.784378i \(-0.712983\pi\)
0.620284 0.784378i \(-0.287017\pi\)
\(348\) 0 0
\(349\) −7.19871 −0.385338 −0.192669 0.981264i \(-0.561714\pi\)
−0.192669 + 0.981264i \(0.561714\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.256413i 0.0136475i −0.999977 0.00682374i \(-0.997828\pi\)
0.999977 0.00682374i \(-0.00217208\pi\)
\(354\) 0 0
\(355\) 8.73638i 0.463679i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 5.98763 0.316015 0.158008 0.987438i \(-0.449493\pi\)
0.158008 + 0.987438i \(0.449493\pi\)
\(360\) 0 0
\(361\) 13.8477 0.728828
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −13.5209 −0.707714
\(366\) 0 0
\(367\) 24.3065i 1.26879i −0.773010 0.634394i \(-0.781250\pi\)
0.773010 0.634394i \(-0.218750\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.91101i 0.151132i
\(372\) 0 0
\(373\) 13.5468i 0.701426i 0.936483 + 0.350713i \(0.114061\pi\)
−0.936483 + 0.350713i \(0.885939\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 14.0345i 0.722813i
\(378\) 0 0
\(379\) 36.3096i 1.86510i 0.361041 + 0.932550i \(0.382422\pi\)
−0.361041 + 0.932550i \(0.617578\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −14.7485 −0.753613 −0.376807 0.926292i \(-0.622978\pi\)
−0.376807 + 0.926292i \(0.622978\pi\)
\(384\) 0 0
\(385\) 0.233391i 0.0118947i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 7.45233 0.377848 0.188924 0.981992i \(-0.439500\pi\)
0.188924 + 0.981992i \(0.439500\pi\)
\(390\) 0 0
\(391\) −0.318882 0.0818348i −0.0161265 0.00413856i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 9.13107i 0.459434i
\(396\) 0 0
\(397\) 29.8237 1.49681 0.748405 0.663242i \(-0.230820\pi\)
0.748405 + 0.663242i \(0.230820\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 11.8531 0.591913 0.295957 0.955201i \(-0.404362\pi\)
0.295957 + 0.955201i \(0.404362\pi\)
\(402\) 0 0
\(403\) 15.0913 0.751752
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.46862i 0.122365i
\(408\) 0 0
\(409\) −1.78774 −0.0883980 −0.0441990 0.999023i \(-0.514074\pi\)
−0.0441990 + 0.999023i \(0.514074\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1.10102 0.0541775
\(414\) 0 0
\(415\) −3.27923 −0.160971
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −11.8972 −0.581214 −0.290607 0.956842i \(-0.593857\pi\)
−0.290607 + 0.956842i \(0.593857\pi\)
\(420\) 0 0
\(421\) 32.3121i 1.57480i 0.616445 + 0.787398i \(0.288572\pi\)
−0.616445 + 0.787398i \(0.711428\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0.0686460 0.00332982
\(426\) 0 0
\(427\) −1.60337 −0.0775924
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 9.18357 0.442357 0.221178 0.975233i \(-0.429010\pi\)
0.221178 + 0.975233i \(0.429010\pi\)
\(432\) 0 0
\(433\) 1.75229i 0.0842096i −0.999113 0.0421048i \(-0.986594\pi\)
0.999113 0.0421048i \(-0.0134063\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.70597 + 10.5442i −0.129444 + 0.504397i
\(438\) 0 0
\(439\) 25.6757 1.22543 0.612717 0.790303i \(-0.290077\pi\)
0.612717 + 0.790303i \(0.290077\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0.614786i 0.0292093i −0.999893 0.0146047i \(-0.995351\pi\)
0.999893 0.0146047i \(-0.00464898\pi\)
\(444\) 0 0
\(445\) −10.3839 −0.492244
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 4.49807i 0.212277i −0.994351 0.106138i \(-0.966151\pi\)
0.994351 0.106138i \(-0.0338487\pi\)
\(450\) 0 0
\(451\) 2.88756i 0.135970i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2.40263i 0.112637i
\(456\) 0 0
\(457\) 19.5296i 0.913557i −0.889580 0.456779i \(-0.849003\pi\)
0.889580 0.456779i \(-0.150997\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 27.5125i 1.28138i 0.767798 + 0.640692i \(0.221353\pi\)
−0.767798 + 0.640692i \(0.778647\pi\)
\(462\) 0 0
\(463\) 28.4632 1.32279 0.661397 0.750036i \(-0.269964\pi\)
0.661397 + 0.750036i \(0.269964\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −3.63927 −0.168406 −0.0842028 0.996449i \(-0.526834\pi\)
−0.0842028 + 0.996449i \(0.526834\pi\)
\(468\) 0 0
\(469\) −3.10909 −0.143564
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.03953i 0.0477976i
\(474\) 0 0
\(475\) 2.26986i 0.104148i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 2.59859 0.118732 0.0593662 0.998236i \(-0.481092\pi\)
0.0593662 + 0.998236i \(0.481092\pi\)
\(480\) 0 0
\(481\) 25.4130i 1.15874i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 17.5807i 0.798300i
\(486\) 0 0
\(487\) −34.4106 −1.55929 −0.779647 0.626220i \(-0.784601\pi\)
−0.779647 + 0.626220i \(0.784601\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 13.4169i 0.605498i 0.953070 + 0.302749i \(0.0979043\pi\)
−0.953070 + 0.302749i \(0.902096\pi\)
\(492\) 0 0
\(493\) 0.195943i 0.00882481i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.26910 0.191495
\(498\) 0 0
\(499\) 26.6443 1.19276 0.596381 0.802701i \(-0.296605\pi\)
0.596381 + 0.802701i \(0.296605\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −5.45861 −0.243387 −0.121694 0.992568i \(-0.538833\pi\)
−0.121694 + 0.992568i \(0.538833\pi\)
\(504\) 0 0
\(505\) 18.3289i 0.815627i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 13.2896i 0.589050i 0.955644 + 0.294525i \(0.0951613\pi\)
−0.955644 + 0.294525i \(0.904839\pi\)
\(510\) 0 0
\(511\) 6.60707i 0.292279i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 12.0722i 0.531963i
\(516\) 0 0
\(517\) 0.210404i 0.00925358i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −11.2119 −0.491201 −0.245601 0.969371i \(-0.578985\pi\)
−0.245601 + 0.969371i \(0.578985\pi\)
\(522\) 0 0
\(523\) 21.8051i 0.953471i −0.879047 0.476736i \(-0.841820\pi\)
0.879047 0.476736i \(-0.158180\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −0.210697 −0.00917813
\(528\) 0 0
\(529\) 20.1577 + 11.0756i 0.876420 + 0.481547i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 29.7258i 1.28757i
\(534\) 0 0
\(535\) 0.830054 0.0358863
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −3.22928 −0.139095
\(540\) 0 0
\(541\) 11.8423 0.509139 0.254570 0.967054i \(-0.418066\pi\)
0.254570 + 0.967054i \(0.418066\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 3.99309i 0.171045i
\(546\) 0 0
\(547\) 27.7019 1.18445 0.592224 0.805773i \(-0.298250\pi\)
0.592224 + 0.805773i \(0.298250\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 6.47907 0.276018
\(552\) 0 0
\(553\) 4.46196 0.189742
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.79968 0.203369 0.101685 0.994817i \(-0.467577\pi\)
0.101685 + 0.994817i \(0.467577\pi\)
\(558\) 0 0
\(559\) 10.7014i 0.452619i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 33.9764 1.43193 0.715967 0.698134i \(-0.245986\pi\)
0.715967 + 0.698134i \(0.245986\pi\)
\(564\) 0 0
\(565\) 14.1428 0.594993
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −10.5365 −0.441712 −0.220856 0.975306i \(-0.570885\pi\)
−0.220856 + 0.975306i \(0.570885\pi\)
\(570\) 0 0
\(571\) 16.1162i 0.674442i −0.941425 0.337221i \(-0.890513\pi\)
0.941425 0.337221i \(-0.109487\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4.64530 1.19213i −0.193722 0.0497152i
\(576\) 0 0
\(577\) 1.93736 0.0806534 0.0403267 0.999187i \(-0.487160\pi\)
0.0403267 + 0.999187i \(0.487160\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.60242i 0.0664796i
\(582\) 0 0
\(583\) 2.84525 0.117838
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 33.6865i 1.39039i 0.718821 + 0.695195i \(0.244682\pi\)
−0.718821 + 0.695195i \(0.755318\pi\)
\(588\) 0 0
\(589\) 6.96696i 0.287069i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 5.23620i 0.215025i 0.994204 + 0.107512i \(0.0342885\pi\)
−0.994204 + 0.107512i \(0.965711\pi\)
\(594\) 0 0
\(595\) 0.0335444i 0.00137519i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 24.4592i 0.999376i 0.866206 + 0.499688i \(0.166552\pi\)
−0.866206 + 0.499688i \(0.833448\pi\)
\(600\) 0 0
\(601\) 9.64776 0.393540 0.196770 0.980450i \(-0.436955\pi\)
0.196770 + 0.980450i \(0.436955\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 10.7719 0.437939
\(606\) 0 0
\(607\) 39.5325 1.60458 0.802288 0.596937i \(-0.203616\pi\)
0.802288 + 0.596937i \(0.203616\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.16599i 0.0876268i
\(612\) 0 0
\(613\) 20.9202i 0.844959i 0.906372 + 0.422480i \(0.138840\pi\)
−0.906372 + 0.422480i \(0.861160\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −34.5230 −1.38985 −0.694923 0.719084i \(-0.744561\pi\)
−0.694923 + 0.719084i \(0.744561\pi\)
\(618\) 0 0
\(619\) 4.97391i 0.199918i −0.994992 0.0999591i \(-0.968129\pi\)
0.994992 0.0999591i \(-0.0318712\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 5.07417i 0.203292i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0.354804i 0.0141470i
\(630\) 0 0
\(631\) 21.8330i 0.869157i 0.900634 + 0.434579i \(0.143103\pi\)
−0.900634 + 0.434579i \(0.856897\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −13.7107 −0.544091
\(636\) 0 0
\(637\) −33.2436 −1.31716
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1.18190 −0.0466822 −0.0233411 0.999728i \(-0.507430\pi\)
−0.0233411 + 0.999728i \(0.507430\pi\)
\(642\) 0 0
\(643\) 44.4715i 1.75378i 0.480688 + 0.876892i \(0.340387\pi\)
−0.480688 + 0.876892i \(0.659613\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.14925i 0.0451817i 0.999745 + 0.0225909i \(0.00719151\pi\)
−0.999745 + 0.0225909i \(0.992808\pi\)
\(648\) 0 0
\(649\) 1.07614i 0.0422424i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3.67210i 0.143701i 0.997415 + 0.0718503i \(0.0228904\pi\)
−0.997415 + 0.0718503i \(0.977110\pi\)
\(654\) 0 0
\(655\) 8.63722i 0.337484i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −12.6641 −0.493325 −0.246663 0.969101i \(-0.579334\pi\)
−0.246663 + 0.969101i \(0.579334\pi\)
\(660\) 0 0
\(661\) 23.7090i 0.922173i 0.887355 + 0.461086i \(0.152540\pi\)
−0.887355 + 0.461086i \(0.847460\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.10918 0.0430123
\(666\) 0 0
\(667\) 3.40280 13.2595i 0.131757 0.513410i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1.56715i 0.0604991i
\(672\) 0 0
\(673\) −5.96633 −0.229985 −0.114993 0.993366i \(-0.536684\pi\)
−0.114993 + 0.993366i \(0.536684\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −8.25186 −0.317145 −0.158572 0.987347i \(-0.550689\pi\)
−0.158572 + 0.987347i \(0.550689\pi\)
\(678\) 0 0
\(679\) −8.59095 −0.329690
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 4.06441i 0.155520i −0.996972 0.0777601i \(-0.975223\pi\)
0.996972 0.0777601i \(-0.0247768\pi\)
\(684\) 0 0
\(685\) −0.331548 −0.0126678
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 29.2903 1.11587
\(690\) 0 0
\(691\) 17.8924 0.680659 0.340330 0.940306i \(-0.389461\pi\)
0.340330 + 0.940306i \(0.389461\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −12.0248 −0.456128
\(696\) 0 0
\(697\) 0.415018i 0.0157199i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 22.9103 0.865310 0.432655 0.901559i \(-0.357577\pi\)
0.432655 + 0.901559i \(0.357577\pi\)
\(702\) 0 0
\(703\) 11.7320 0.442482
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8.95656 0.336846
\(708\) 0 0
\(709\) 48.7002i 1.82898i −0.404614 0.914488i \(-0.632594\pi\)
0.404614 0.914488i \(-0.367406\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 14.2580 + 3.65903i 0.533965 + 0.137032i
\(714\) 0 0
\(715\) −2.34836 −0.0878235
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 34.9310i 1.30270i −0.758775 0.651352i \(-0.774202\pi\)
0.758775 0.651352i \(-0.225798\pi\)
\(720\) 0 0
\(721\) −5.89915 −0.219696
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.85439i 0.106009i
\(726\) 0 0
\(727\) 38.9966i 1.44630i −0.690690 0.723151i \(-0.742693\pi\)
0.690690 0.723151i \(-0.257307\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0.149407i 0.00552602i
\(732\) 0 0
\(733\) 10.2487i 0.378545i 0.981925 + 0.189273i \(0.0606130\pi\)
−0.981925 + 0.189273i \(0.939387\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.03885i 0.111938i
\(738\) 0 0
\(739\) −23.3080 −0.857398 −0.428699 0.903447i \(-0.641028\pi\)
−0.428699 + 0.903447i \(0.641028\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 37.0143 1.35792 0.678961 0.734174i \(-0.262431\pi\)
0.678961 + 0.734174i \(0.262431\pi\)
\(744\) 0 0
\(745\) −22.8306 −0.836448
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0.405612i 0.0148207i
\(750\) 0 0
\(751\) 30.2486i 1.10379i −0.833915 0.551893i \(-0.813906\pi\)
0.833915 0.551893i \(-0.186094\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −19.0857 −0.694601
\(756\) 0 0
\(757\) 1.53142i 0.0556603i −0.999613 0.0278302i \(-0.991140\pi\)
0.999613 0.0278302i \(-0.00885976\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 6.10735i 0.221391i 0.993854 + 0.110696i \(0.0353079\pi\)
−0.993854 + 0.110696i \(0.964692\pi\)
\(762\) 0 0
\(763\) 1.95125 0.0706400
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 11.0783i 0.400014i
\(768\) 0 0
\(769\) 23.2168i 0.837221i 0.908166 + 0.418610i \(0.137483\pi\)
−0.908166 + 0.418610i \(0.862517\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 17.0419 0.612955 0.306477 0.951878i \(-0.400850\pi\)
0.306477 + 0.951878i \(0.400850\pi\)
\(774\) 0 0
\(775\) −3.06933 −0.110254
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −13.7230 −0.491679
\(780\) 0 0
\(781\) 4.17265i 0.149309i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 7.13723i 0.254739i
\(786\) 0 0
\(787\) 6.41481i 0.228663i 0.993443 + 0.114332i \(0.0364726\pi\)
−0.993443 + 0.114332i \(0.963527\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 6.91100i 0.245727i
\(792\) 0 0
\(793\) 16.1329i 0.572896i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −54.2270 −1.92082 −0.960410 0.278592i \(-0.910132\pi\)
−0.960410 + 0.278592i \(0.910132\pi\)
\(798\) 0 0
\(799\) 0.0302405i 0.00106983i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −6.45781 −0.227891
\(804\) 0 0
\(805\) 0.582542 2.26996i 0.0205319 0.0800056i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.11751i 0.0392897i 0.999807 + 0.0196448i \(0.00625355\pi\)
−0.999807 +