Properties

Label 8280.2.p.a.1241.14
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 1241.14
Character \(\chi\) \(=\) 8280.1241
Dual form 8280.2.p.a.1241.35

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{5} -2.35143i q^{7} +O(q^{10})\) \(q-1.00000 q^{5} -2.35143i q^{7} +1.47885 q^{11} +4.67757 q^{13} -5.61193 q^{17} +3.07752i q^{19} +(-4.17933 - 2.35228i) q^{23} +1.00000 q^{25} -2.05045i q^{29} +8.72555 q^{31} +2.35143i q^{35} +3.93677i q^{37} -6.91846i q^{41} -3.78778i q^{43} +10.9770i q^{47} +1.47077 q^{49} +13.7986 q^{53} -1.47885 q^{55} -7.07225i q^{59} +10.2901i q^{61} -4.67757 q^{65} +1.88107i q^{67} -3.91325i q^{71} -6.95228 q^{73} -3.47742i q^{77} -7.99760i q^{79} +0.219311 q^{83} +5.61193 q^{85} -2.65745 q^{89} -10.9990i q^{91} -3.07752i q^{95} +9.88545i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 48 q^{5}+O(q^{10}) \) 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
\(7\) 2.35143i 0.888758i −0.895839 0.444379i \(-0.853424\pi\)
0.895839 0.444379i \(-0.146576\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.47885 0.445890 0.222945 0.974831i \(-0.428433\pi\)
0.222945 + 0.974831i \(0.428433\pi\)
\(12\) 0 0
\(13\) 4.67757 1.29732 0.648662 0.761077i \(-0.275329\pi\)
0.648662 + 0.761077i \(0.275329\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.61193 −1.36109 −0.680546 0.732705i \(-0.738257\pi\)
−0.680546 + 0.732705i \(0.738257\pi\)
\(18\) 0 0
\(19\) 3.07752i 0.706031i 0.935617 + 0.353016i \(0.114844\pi\)
−0.935617 + 0.353016i \(0.885156\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.17933 2.35228i −0.871451 0.490483i
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.05045i 0.380759i −0.981711 0.190379i \(-0.939028\pi\)
0.981711 0.190379i \(-0.0609718\pi\)
\(30\) 0 0
\(31\) 8.72555 1.56716 0.783578 0.621294i \(-0.213393\pi\)
0.783578 + 0.621294i \(0.213393\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.35143i 0.397465i
\(36\) 0 0
\(37\) 3.93677i 0.647200i 0.946194 + 0.323600i \(0.104893\pi\)
−0.946194 + 0.323600i \(0.895107\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.91846i 1.08048i −0.841510 0.540241i \(-0.818333\pi\)
0.841510 0.540241i \(-0.181667\pi\)
\(42\) 0 0
\(43\) 3.78778i 0.577631i −0.957385 0.288816i \(-0.906739\pi\)
0.957385 0.288816i \(-0.0932615\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.9770i 1.60115i 0.599230 + 0.800577i \(0.295474\pi\)
−0.599230 + 0.800577i \(0.704526\pi\)
\(48\) 0 0
\(49\) 1.47077 0.210110
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 13.7986 1.89539 0.947693 0.319182i \(-0.103408\pi\)
0.947693 + 0.319182i \(0.103408\pi\)
\(54\) 0 0
\(55\) −1.47885 −0.199408
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.07225i 0.920729i −0.887730 0.460365i \(-0.847719\pi\)
0.887730 0.460365i \(-0.152281\pi\)
\(60\) 0 0
\(61\) 10.2901i 1.31751i 0.752359 + 0.658754i \(0.228916\pi\)
−0.752359 + 0.658754i \(0.771084\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.67757 −0.580181
\(66\) 0 0
\(67\) 1.88107i 0.229810i 0.993377 + 0.114905i \(0.0366563\pi\)
−0.993377 + 0.114905i \(0.963344\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.91325i 0.464418i −0.972666 0.232209i \(-0.925405\pi\)
0.972666 0.232209i \(-0.0745953\pi\)
\(72\) 0 0
\(73\) −6.95228 −0.813703 −0.406852 0.913494i \(-0.633373\pi\)
−0.406852 + 0.913494i \(0.633373\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.47742i 0.396288i
\(78\) 0 0
\(79\) 7.99760i 0.899800i −0.893079 0.449900i \(-0.851460\pi\)
0.893079 0.449900i \(-0.148540\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0.219311 0.0240725 0.0120363 0.999928i \(-0.496169\pi\)
0.0120363 + 0.999928i \(0.496169\pi\)
\(84\) 0 0
\(85\) 5.61193 0.608699
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.65745 −0.281690 −0.140845 0.990032i \(-0.544982\pi\)
−0.140845 + 0.990032i \(0.544982\pi\)
\(90\) 0 0
\(91\) 10.9990i 1.15301i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.07752i 0.315747i
\(96\) 0 0
\(97\) 9.88545i 1.00372i 0.864950 + 0.501858i \(0.167350\pi\)
−0.864950 + 0.501858i \(0.832650\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 16.7015i 1.66186i −0.556375 0.830931i \(-0.687808\pi\)
0.556375 0.830931i \(-0.312192\pi\)
\(102\) 0 0
\(103\) 6.41186i 0.631779i −0.948796 0.315890i \(-0.897697\pi\)
0.948796 0.315890i \(-0.102303\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.1797 1.27413 0.637065 0.770810i \(-0.280148\pi\)
0.637065 + 0.770810i \(0.280148\pi\)
\(108\) 0 0
\(109\) 0.188623i 0.0180668i −0.999959 0.00903339i \(-0.997125\pi\)
0.999959 0.00903339i \(-0.00287546\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 7.87587 0.740900 0.370450 0.928852i \(-0.379204\pi\)
0.370450 + 0.928852i \(0.379204\pi\)
\(114\) 0 0
\(115\) 4.17933 + 2.35228i 0.389725 + 0.219351i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 13.1961i 1.20968i
\(120\) 0 0
\(121\) −8.81300 −0.801182
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 3.54279 0.314372 0.157186 0.987569i \(-0.449758\pi\)
0.157186 + 0.987569i \(0.449758\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 12.4354i 1.08649i −0.839576 0.543243i \(-0.817196\pi\)
0.839576 0.543243i \(-0.182804\pi\)
\(132\) 0 0
\(133\) 7.23658 0.627491
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −14.5419 −1.24240 −0.621198 0.783653i \(-0.713354\pi\)
−0.621198 + 0.783653i \(0.713354\pi\)
\(138\) 0 0
\(139\) 20.5864 1.74611 0.873056 0.487620i \(-0.162135\pi\)
0.873056 + 0.487620i \(0.162135\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.91742 0.578464
\(144\) 0 0
\(145\) 2.05045i 0.170281i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −15.5566 −1.27444 −0.637221 0.770681i \(-0.719916\pi\)
−0.637221 + 0.770681i \(0.719916\pi\)
\(150\) 0 0
\(151\) 5.92860 0.482463 0.241231 0.970468i \(-0.422449\pi\)
0.241231 + 0.970468i \(0.422449\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −8.72555 −0.700853
\(156\) 0 0
\(157\) 3.52464i 0.281297i 0.990060 + 0.140649i \(0.0449188\pi\)
−0.990060 + 0.140649i \(0.955081\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −5.53122 + 9.82741i −0.435921 + 0.774508i
\(162\) 0 0
\(163\) −6.33629 −0.496297 −0.248148 0.968722i \(-0.579822\pi\)
−0.248148 + 0.968722i \(0.579822\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 14.3269i 1.10865i 0.832300 + 0.554325i \(0.187024\pi\)
−0.832300 + 0.554325i \(0.812976\pi\)
\(168\) 0 0
\(169\) 8.87965 0.683050
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4.53674i 0.344922i 0.985016 + 0.172461i \(0.0551718\pi\)
−0.985016 + 0.172461i \(0.944828\pi\)
\(174\) 0 0
\(175\) 2.35143i 0.177752i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2.95952i 0.221205i −0.993865 0.110602i \(-0.964722\pi\)
0.993865 0.110602i \(-0.0352780\pi\)
\(180\) 0 0
\(181\) 18.0839i 1.34416i −0.740477 0.672081i \(-0.765401\pi\)
0.740477 0.672081i \(-0.234599\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.93677i 0.289437i
\(186\) 0 0
\(187\) −8.29920 −0.606898
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −15.8720 −1.14846 −0.574229 0.818695i \(-0.694698\pi\)
−0.574229 + 0.818695i \(0.694698\pi\)
\(192\) 0 0
\(193\) 20.4693 1.47341 0.736707 0.676212i \(-0.236380\pi\)
0.736707 + 0.676212i \(0.236380\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.02134i 0.144014i 0.997404 + 0.0720071i \(0.0229404\pi\)
−0.997404 + 0.0720071i \(0.977060\pi\)
\(198\) 0 0
\(199\) 4.21021i 0.298454i −0.988803 0.149227i \(-0.952321\pi\)
0.988803 0.149227i \(-0.0476785\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −4.82149 −0.338402
\(204\) 0 0
\(205\) 6.91846i 0.483206i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4.55119i 0.314812i
\(210\) 0 0
\(211\) −12.1131 −0.833901 −0.416951 0.908929i \(-0.636901\pi\)
−0.416951 + 0.908929i \(0.636901\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.78778i 0.258325i
\(216\) 0 0
\(217\) 20.5175i 1.39282i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −26.2502 −1.76578
\(222\) 0 0
\(223\) −4.13946 −0.277199 −0.138599 0.990349i \(-0.544260\pi\)
−0.138599 + 0.990349i \(0.544260\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 14.9769 0.994055 0.497027 0.867735i \(-0.334425\pi\)
0.497027 + 0.867735i \(0.334425\pi\)
\(228\) 0 0
\(229\) 6.14477i 0.406058i −0.979173 0.203029i \(-0.934921\pi\)
0.979173 0.203029i \(-0.0650785\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 15.5499i 1.01871i −0.860557 0.509354i \(-0.829884\pi\)
0.860557 0.509354i \(-0.170116\pi\)
\(234\) 0 0
\(235\) 10.9770i 0.716058i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 8.52633i 0.551522i −0.961226 0.275761i \(-0.911070\pi\)
0.961226 0.275761i \(-0.0889299\pi\)
\(240\) 0 0
\(241\) 5.89346i 0.379631i −0.981820 0.189816i \(-0.939211\pi\)
0.981820 0.189816i \(-0.0607890\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.47077 −0.0939639
\(246\) 0 0
\(247\) 14.3953i 0.915951i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 21.6293 1.36523 0.682616 0.730777i \(-0.260842\pi\)
0.682616 + 0.730777i \(0.260842\pi\)
\(252\) 0 0
\(253\) −6.18060 3.47866i −0.388571 0.218702i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 19.0632i 1.18913i −0.804047 0.594566i \(-0.797324\pi\)
0.804047 0.594566i \(-0.202676\pi\)
\(258\) 0 0
\(259\) 9.25704 0.575204
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −6.89797 −0.425347 −0.212674 0.977123i \(-0.568217\pi\)
−0.212674 + 0.977123i \(0.568217\pi\)
\(264\) 0 0
\(265\) −13.7986 −0.847643
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 10.7798i 0.657256i 0.944459 + 0.328628i \(0.106586\pi\)
−0.944459 + 0.328628i \(0.893414\pi\)
\(270\) 0 0
\(271\) 7.41048 0.450154 0.225077 0.974341i \(-0.427737\pi\)
0.225077 + 0.974341i \(0.427737\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.47885 0.0891780
\(276\) 0 0
\(277\) 28.6625 1.72216 0.861080 0.508469i \(-0.169788\pi\)
0.861080 + 0.508469i \(0.169788\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −16.7648 −1.00010 −0.500052 0.865996i \(-0.666686\pi\)
−0.500052 + 0.865996i \(0.666686\pi\)
\(282\) 0 0
\(283\) 18.6340i 1.10768i −0.832623 0.553840i \(-0.813162\pi\)
0.832623 0.553840i \(-0.186838\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −16.2683 −0.960286
\(288\) 0 0
\(289\) 14.4937 0.852573
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −7.09931 −0.414746 −0.207373 0.978262i \(-0.566491\pi\)
−0.207373 + 0.978262i \(0.566491\pi\)
\(294\) 0 0
\(295\) 7.07225i 0.411763i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −19.5491 11.0029i −1.13055 0.636316i
\(300\) 0 0
\(301\) −8.90671 −0.513374
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 10.2901i 0.589207i
\(306\) 0 0
\(307\) 30.8599 1.76127 0.880635 0.473796i \(-0.157116\pi\)
0.880635 + 0.473796i \(0.157116\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 18.7527i 1.06337i −0.846942 0.531685i \(-0.821559\pi\)
0.846942 0.531685i \(-0.178441\pi\)
\(312\) 0 0
\(313\) 10.2600i 0.579929i 0.957037 + 0.289964i \(0.0936434\pi\)
−0.957037 + 0.289964i \(0.906357\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.42607i 0.136262i 0.997676 + 0.0681308i \(0.0217035\pi\)
−0.997676 + 0.0681308i \(0.978296\pi\)
\(318\) 0 0
\(319\) 3.03231i 0.169777i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 17.2708i 0.960974i
\(324\) 0 0
\(325\) 4.67757 0.259465
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 25.8116 1.42304
\(330\) 0 0
\(331\) 4.23343 0.232690 0.116345 0.993209i \(-0.462882\pi\)
0.116345 + 0.993209i \(0.462882\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.88107i 0.102774i
\(336\) 0 0
\(337\) 21.5688i 1.17493i −0.809251 0.587463i \(-0.800127\pi\)
0.809251 0.587463i \(-0.199873\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 12.9038 0.698779
\(342\) 0 0
\(343\) 19.9184i 1.07549i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 26.3767i 1.41598i 0.706224 + 0.707988i \(0.250397\pi\)
−0.706224 + 0.707988i \(0.749603\pi\)
\(348\) 0 0
\(349\) −21.3172 −1.14108 −0.570542 0.821268i \(-0.693267\pi\)
−0.570542 + 0.821268i \(0.693267\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 16.8537i 0.897032i −0.893775 0.448516i \(-0.851953\pi\)
0.893775 0.448516i \(-0.148047\pi\)
\(354\) 0 0
\(355\) 3.91325i 0.207694i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −7.81434 −0.412425 −0.206213 0.978507i \(-0.566114\pi\)
−0.206213 + 0.978507i \(0.566114\pi\)
\(360\) 0 0
\(361\) 9.52888 0.501520
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 6.95228 0.363899
\(366\) 0 0
\(367\) 13.7083i 0.715568i −0.933804 0.357784i \(-0.883532\pi\)
0.933804 0.357784i \(-0.116468\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 32.4465i 1.68454i
\(372\) 0 0
\(373\) 23.7328i 1.22884i −0.788980 0.614419i \(-0.789391\pi\)
0.788980 0.614419i \(-0.210609\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 9.59112i 0.493968i
\(378\) 0 0
\(379\) 26.6280i 1.36779i −0.729582 0.683894i \(-0.760285\pi\)
0.729582 0.683894i \(-0.239715\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −21.7247 −1.11008 −0.555040 0.831823i \(-0.687297\pi\)
−0.555040 + 0.831823i \(0.687297\pi\)
\(384\) 0 0
\(385\) 3.47742i 0.177225i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 11.8114 0.598861 0.299430 0.954118i \(-0.403203\pi\)
0.299430 + 0.954118i \(0.403203\pi\)
\(390\) 0 0
\(391\) 23.4541 + 13.2008i 1.18612 + 0.667593i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 7.99760i 0.402403i
\(396\) 0 0
\(397\) −6.82731 −0.342653 −0.171327 0.985214i \(-0.554805\pi\)
−0.171327 + 0.985214i \(0.554805\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 6.11871 0.305554 0.152777 0.988261i \(-0.451178\pi\)
0.152777 + 0.988261i \(0.451178\pi\)
\(402\) 0 0
\(403\) 40.8144 2.03311
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 5.82189i 0.288580i
\(408\) 0 0
\(409\) −18.1145 −0.895706 −0.447853 0.894107i \(-0.647811\pi\)
−0.447853 + 0.894107i \(0.647811\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −16.6299 −0.818305
\(414\) 0 0
\(415\) −0.219311 −0.0107656
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 22.8561 1.11659 0.558296 0.829642i \(-0.311455\pi\)
0.558296 + 0.829642i \(0.311455\pi\)
\(420\) 0 0
\(421\) 22.6095i 1.10192i −0.834532 0.550959i \(-0.814262\pi\)
0.834532 0.550959i \(-0.185738\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −5.61193 −0.272219
\(426\) 0 0
\(427\) 24.1964 1.17094
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5.34963 0.257682 0.128841 0.991665i \(-0.458874\pi\)
0.128841 + 0.991665i \(0.458874\pi\)
\(432\) 0 0
\(433\) 12.2176i 0.587138i −0.955938 0.293569i \(-0.905157\pi\)
0.955938 0.293569i \(-0.0948431\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.23917 12.8620i 0.346297 0.615271i
\(438\) 0 0
\(439\) 3.39359 0.161967 0.0809837 0.996715i \(-0.474194\pi\)
0.0809837 + 0.996715i \(0.474194\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 11.5646i 0.549450i −0.961523 0.274725i \(-0.911413\pi\)
0.961523 0.274725i \(-0.0885869\pi\)
\(444\) 0 0
\(445\) 2.65745 0.125975
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 36.3617i 1.71602i −0.513636 0.858008i \(-0.671702\pi\)
0.513636 0.858008i \(-0.328298\pi\)
\(450\) 0 0
\(451\) 10.2314i 0.481776i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 10.9990i 0.515640i
\(456\) 0 0
\(457\) 25.3964i 1.18799i −0.804467 0.593997i \(-0.797549\pi\)
0.804467 0.593997i \(-0.202451\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 23.5239i 1.09562i −0.836604 0.547808i \(-0.815463\pi\)
0.836604 0.547808i \(-0.184537\pi\)
\(462\) 0 0
\(463\) 14.5555 0.676450 0.338225 0.941065i \(-0.390173\pi\)
0.338225 + 0.941065i \(0.390173\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 11.5490 0.534424 0.267212 0.963638i \(-0.413898\pi\)
0.267212 + 0.963638i \(0.413898\pi\)
\(468\) 0 0
\(469\) 4.42321 0.204245
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 5.60156i 0.257560i
\(474\) 0 0
\(475\) 3.07752i 0.141206i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 30.6337 1.39969 0.699844 0.714295i \(-0.253253\pi\)
0.699844 + 0.714295i \(0.253253\pi\)
\(480\) 0 0
\(481\) 18.4145i 0.839629i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 9.88545i 0.448875i
\(486\) 0 0
\(487\) 16.1826 0.733306 0.366653 0.930358i \(-0.380504\pi\)
0.366653 + 0.930358i \(0.380504\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 32.6496i 1.47346i 0.676189 + 0.736729i \(0.263630\pi\)
−0.676189 + 0.736729i \(0.736370\pi\)
\(492\) 0 0
\(493\) 11.5070i 0.518248i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −9.20175 −0.412755
\(498\) 0 0
\(499\) −15.9644 −0.714663 −0.357332 0.933978i \(-0.616313\pi\)
−0.357332 + 0.933978i \(0.616313\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 34.4127 1.53439 0.767193 0.641416i \(-0.221653\pi\)
0.767193 + 0.641416i \(0.221653\pi\)
\(504\) 0 0
\(505\) 16.7015i 0.743207i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 33.0486i 1.46485i −0.680846 0.732427i \(-0.738388\pi\)
0.680846 0.732427i \(-0.261612\pi\)
\(510\) 0 0
\(511\) 16.3478i 0.723185i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 6.41186i 0.282540i
\(516\) 0 0
\(517\) 16.2333i 0.713939i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 20.5894 0.902037 0.451019 0.892515i \(-0.351061\pi\)
0.451019 + 0.892515i \(0.351061\pi\)
\(522\) 0 0
\(523\) 23.3782i 1.02226i 0.859504 + 0.511129i \(0.170772\pi\)
−0.859504 + 0.511129i \(0.829228\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −48.9672 −2.13304
\(528\) 0 0
\(529\) 11.9336 + 19.6619i 0.518852 + 0.854864i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 32.3616i 1.40173i
\(534\) 0 0
\(535\) −13.1797 −0.569808
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.17504 0.0936858
\(540\) 0 0
\(541\) −34.3436 −1.47655 −0.738274 0.674501i \(-0.764359\pi\)
−0.738274 + 0.674501i \(0.764359\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0.188623i 0.00807971i
\(546\) 0 0
\(547\) −21.7771 −0.931124 −0.465562 0.885015i \(-0.654148\pi\)
−0.465562 + 0.885015i \(0.654148\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 6.31030 0.268828
\(552\) 0 0
\(553\) −18.8058 −0.799705
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −28.1226 −1.19159 −0.595797 0.803135i \(-0.703164\pi\)
−0.595797 + 0.803135i \(0.703164\pi\)
\(558\) 0 0
\(559\) 17.7176i 0.749375i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 18.0761 0.761818 0.380909 0.924613i \(-0.375611\pi\)
0.380909 + 0.924613i \(0.375611\pi\)
\(564\) 0 0
\(565\) −7.87587 −0.331340
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 17.6190 0.738627 0.369314 0.929305i \(-0.379593\pi\)
0.369314 + 0.929305i \(0.379593\pi\)
\(570\) 0 0
\(571\) 2.78326i 0.116476i −0.998303 0.0582379i \(-0.981452\pi\)
0.998303 0.0582379i \(-0.0185482\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4.17933 2.35228i −0.174290 0.0980967i
\(576\) 0 0
\(577\) 23.5675 0.981126 0.490563 0.871406i \(-0.336791\pi\)
0.490563 + 0.871406i \(0.336791\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0.515695i 0.0213946i
\(582\) 0 0
\(583\) 20.4061 0.845134
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 25.7309i 1.06203i 0.847364 + 0.531013i \(0.178188\pi\)
−0.847364 + 0.531013i \(0.821812\pi\)
\(588\) 0 0
\(589\) 26.8531i 1.10646i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 45.7373i 1.87821i 0.343637 + 0.939103i \(0.388341\pi\)
−0.343637 + 0.939103i \(0.611659\pi\)
\(594\) 0 0
\(595\) 13.1961i 0.540986i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 43.9583i 1.79609i 0.439906 + 0.898044i \(0.355012\pi\)
−0.439906 + 0.898044i \(0.644988\pi\)
\(600\) 0 0
\(601\) 24.7342 1.00893 0.504465 0.863432i \(-0.331690\pi\)
0.504465 + 0.863432i \(0.331690\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 8.81300 0.358300
\(606\) 0 0
\(607\) 31.0124 1.25875 0.629376 0.777101i \(-0.283310\pi\)
0.629376 + 0.777101i \(0.283310\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 51.3455i 2.07722i
\(612\) 0 0
\(613\) 10.0359i 0.405347i 0.979246 + 0.202674i \(0.0649630\pi\)
−0.979246 + 0.202674i \(0.935037\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −41.1182 −1.65536 −0.827679 0.561202i \(-0.810339\pi\)
−0.827679 + 0.561202i \(0.810339\pi\)
\(618\) 0 0
\(619\) 41.8690i 1.68286i −0.540369 0.841428i \(-0.681715\pi\)
0.540369 0.841428i \(-0.318285\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 6.24882i 0.250354i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 22.0929i 0.880900i
\(630\) 0 0
\(631\) 44.1153i 1.75620i 0.478476 + 0.878101i \(0.341189\pi\)
−0.478476 + 0.878101i \(0.658811\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −3.54279 −0.140591
\(636\) 0 0
\(637\) 6.87961 0.272580
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −11.6145 −0.458746 −0.229373 0.973339i \(-0.573668\pi\)
−0.229373 + 0.973339i \(0.573668\pi\)
\(642\) 0 0
\(643\) 6.64529i 0.262065i 0.991378 + 0.131032i \(0.0418292\pi\)
−0.991378 + 0.131032i \(0.958171\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.5141i 0.963750i −0.876240 0.481875i \(-0.839956\pi\)
0.876240 0.481875i \(-0.160044\pi\)
\(648\) 0 0
\(649\) 10.4588i 0.410544i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.23591i 0.0483650i 0.999708 + 0.0241825i \(0.00769828\pi\)
−0.999708 + 0.0241825i \(0.992302\pi\)
\(654\) 0 0
\(655\) 12.4354i 0.485891i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −10.7206 −0.417617 −0.208808 0.977957i \(-0.566959\pi\)
−0.208808 + 0.977957i \(0.566959\pi\)
\(660\) 0 0
\(661\) 2.62911i 0.102260i −0.998692 0.0511302i \(-0.983718\pi\)
0.998692 0.0511302i \(-0.0162823\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −7.23658 −0.280622
\(666\) 0 0
\(667\) −4.82322 + 8.56951i −0.186756 + 0.331813i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 15.2175i 0.587463i
\(672\) 0 0
\(673\) −29.5342 −1.13846 −0.569229 0.822179i \(-0.692758\pi\)
−0.569229 + 0.822179i \(0.692758\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −28.6805 −1.10228 −0.551140 0.834413i \(-0.685807\pi\)
−0.551140 + 0.834413i \(0.685807\pi\)
\(678\) 0 0
\(679\) 23.2450 0.892060
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 23.7660i 0.909380i 0.890650 + 0.454690i \(0.150250\pi\)
−0.890650 + 0.454690i \(0.849750\pi\)
\(684\) 0 0
\(685\) 14.5419 0.555617
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 64.5440 2.45893
\(690\) 0 0
\(691\) −14.5310 −0.552787 −0.276393 0.961045i \(-0.589139\pi\)
−0.276393 + 0.961045i \(0.589139\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −20.5864 −0.780885
\(696\) 0 0
\(697\) 38.8259i 1.47064i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −5.63840 −0.212960 −0.106480 0.994315i \(-0.533958\pi\)
−0.106480 + 0.994315i \(0.533958\pi\)
\(702\) 0 0
\(703\) −12.1155 −0.456944
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −39.2725 −1.47699
\(708\) 0 0
\(709\) 29.2398i 1.09812i −0.835782 0.549062i \(-0.814985\pi\)
0.835782 0.549062i \(-0.185015\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −36.4670 20.5249i −1.36570 0.768664i
\(714\) 0 0
\(715\) −6.91742 −0.258697
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 36.1032i 1.34642i −0.739451 0.673210i \(-0.764915\pi\)
0.739451 0.673210i \(-0.235085\pi\)
\(720\) 0 0
\(721\) −15.0771 −0.561499
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.05045i 0.0761518i
\(726\) 0 0
\(727\) 6.45374i 0.239356i −0.992813 0.119678i \(-0.961814\pi\)
0.992813 0.119678i \(-0.0381862\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 21.2568i 0.786210i
\(732\) 0 0
\(733\) 28.3379i 1.04669i −0.852122 0.523343i \(-0.824685\pi\)
0.852122 0.523343i \(-0.175315\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.78182i 0.102470i
\(738\) 0 0
\(739\) 0.291688 0.0107299 0.00536496 0.999986i \(-0.498292\pi\)
0.00536496 + 0.999986i \(0.498292\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 27.5249 1.00979 0.504895 0.863181i \(-0.331531\pi\)
0.504895 + 0.863181i \(0.331531\pi\)
\(744\) 0 0
\(745\) 15.5566 0.569948
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 30.9912i 1.13239i
\(750\) 0 0
\(751\) 38.7724i 1.41483i −0.706800 0.707413i \(-0.749862\pi\)
0.706800 0.707413i \(-0.250138\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −5.92860 −0.215764
\(756\) 0 0
\(757\) 48.5522i 1.76466i 0.470633 + 0.882329i \(0.344026\pi\)
−0.470633 + 0.882329i \(0.655974\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 29.1443i 1.05648i 0.849095 + 0.528241i \(0.177148\pi\)
−0.849095 + 0.528241i \(0.822852\pi\)
\(762\) 0 0
\(763\) −0.443534 −0.0160570
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 33.0810i 1.19448i
\(768\) 0 0
\(769\) 11.7605i 0.424093i −0.977260 0.212046i \(-0.931987\pi\)
0.977260 0.212046i \(-0.0680128\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.51910 −0.0546381 −0.0273191 0.999627i \(-0.508697\pi\)
−0.0273191 + 0.999627i \(0.508697\pi\)
\(774\) 0 0
\(775\) 8.72555 0.313431
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 21.2917 0.762854
\(780\) 0 0
\(781\) 5.78711i 0.207079i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 3.52464i 0.125800i
\(786\) 0 0
\(787\) 54.3428i 1.93711i 0.248799 + 0.968555i \(0.419964\pi\)
−0.248799 + 0.968555i \(0.580036\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 18.5196i 0.658480i
\(792\) 0 0
\(793\) 48.1325i 1.70923i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −16.7650 −0.593848 −0.296924 0.954901i \(-0.595961\pi\)
−0.296924 + 0.954901i \(0.595961\pi\)
\(798\) 0 0
\(799\) 61.6019i 2.17932i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −10.2814 −0.362822
\(804\) 0 0
\(805\) 5.53122 9.82741i 0.194950 0.346371i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 36.7731i 1.29287i −0.762967 0.646437i