Properties

Label 5520.2.be.b.1471.6
Level $5520$
Weight $2$
Character 5520.1471
Analytic conductor $44.077$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.be (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 4 x^{15} + 8 x^{14} + 28 x^{13} + 373 x^{12} - 920 x^{11} + 1088 x^{10} - 168 x^{9} + 16460 x^{8} - 45408 x^{7} + 62624 x^{6} - 18048 x^{5} + 2160 x^{4} - 1664 x^{3} + 6272 x^{2} - 896 x + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1471.6
Root \(3.68710 + 3.68710i\) of defining polynomial
Character \(\chi\) \(=\) 5520.1471
Dual form 5520.2.be.b.1471.14

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} -1.00000i q^{5} +1.58474 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} -1.00000i q^{5} +1.58474 q^{7} -1.00000 q^{9} -2.94482 q^{11} -1.23011 q^{13} -1.00000 q^{15} -1.25392i q^{17} +5.26714 q^{19} -1.58474i q^{21} +(1.83865 - 4.42937i) q^{23} -1.00000 q^{25} +1.00000i q^{27} +7.54806 q^{29} -0.690205i q^{31} +2.94482i q^{33} -1.58474i q^{35} +4.83340i q^{37} +1.23011i q^{39} +6.98783 q^{41} +4.06681 q^{43} +1.00000i q^{45} +8.95368i q^{47} -4.48861 q^{49} -1.25392 q^{51} -10.8852i q^{53} +2.94482i q^{55} -5.26714i q^{57} +2.33680i q^{59} -5.28658i q^{61} -1.58474 q^{63} +1.23011i q^{65} -9.49359 q^{67} +(-4.42937 - 1.83865i) q^{69} -13.5262i q^{71} -9.51828 q^{73} +1.00000i q^{75} -4.66677 q^{77} -3.71307 q^{79} +1.00000 q^{81} +16.5130 q^{83} -1.25392 q^{85} -7.54806i q^{87} -12.2406i q^{89} -1.94940 q^{91} -0.690205 q^{93} -5.26714i q^{95} -7.70591i q^{97} +2.94482 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 8q^{7} - 16q^{9} + O(q^{10}) \) \( 16q + 8q^{7} - 16q^{9} - 8q^{11} + 8q^{13} - 16q^{15} - 12q^{23} - 16q^{25} - 4q^{29} + 4q^{41} + 20q^{49} + 4q^{51} - 8q^{63} + 16q^{67} + 40q^{73} + 24q^{77} - 32q^{79} + 16q^{81} + 4q^{85} + 48q^{91} + 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5520\mathbb{Z}\right)^\times\).

\(n\) \(1201\) \(1381\) \(1841\) \(4417\) \(4831\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 1.58474 0.598974 0.299487 0.954100i \(-0.403184\pi\)
0.299487 + 0.954100i \(0.403184\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −2.94482 −0.887898 −0.443949 0.896052i \(-0.646423\pi\)
−0.443949 + 0.896052i \(0.646423\pi\)
\(12\) 0 0
\(13\) −1.23011 −0.341172 −0.170586 0.985343i \(-0.554566\pi\)
−0.170586 + 0.985343i \(0.554566\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 1.25392i 0.304120i −0.988371 0.152060i \(-0.951409\pi\)
0.988371 0.152060i \(-0.0485906\pi\)
\(18\) 0 0
\(19\) 5.26714 1.20836 0.604182 0.796846i \(-0.293500\pi\)
0.604182 + 0.796846i \(0.293500\pi\)
\(20\) 0 0
\(21\) 1.58474i 0.345818i
\(22\) 0 0
\(23\) 1.83865 4.42937i 0.383386 0.923588i
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 7.54806 1.40164 0.700820 0.713338i \(-0.252818\pi\)
0.700820 + 0.713338i \(0.252818\pi\)
\(30\) 0 0
\(31\) 0.690205i 0.123965i −0.998077 0.0619823i \(-0.980258\pi\)
0.998077 0.0619823i \(-0.0197422\pi\)
\(32\) 0 0
\(33\) 2.94482i 0.512628i
\(34\) 0 0
\(35\) 1.58474i 0.267869i
\(36\) 0 0
\(37\) 4.83340i 0.794606i 0.917688 + 0.397303i \(0.130054\pi\)
−0.917688 + 0.397303i \(0.869946\pi\)
\(38\) 0 0
\(39\) 1.23011i 0.196976i
\(40\) 0 0
\(41\) 6.98783 1.09132 0.545658 0.838008i \(-0.316280\pi\)
0.545658 + 0.838008i \(0.316280\pi\)
\(42\) 0 0
\(43\) 4.06681 0.620183 0.310091 0.950707i \(-0.399640\pi\)
0.310091 + 0.950707i \(0.399640\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) 8.95368i 1.30603i 0.757346 + 0.653014i \(0.226496\pi\)
−0.757346 + 0.653014i \(0.773504\pi\)
\(48\) 0 0
\(49\) −4.48861 −0.641230
\(50\) 0 0
\(51\) −1.25392 −0.175583
\(52\) 0 0
\(53\) 10.8852i 1.49520i −0.664148 0.747601i \(-0.731206\pi\)
0.664148 0.747601i \(-0.268794\pi\)
\(54\) 0 0
\(55\) 2.94482i 0.397080i
\(56\) 0 0
\(57\) 5.26714i 0.697649i
\(58\) 0 0
\(59\) 2.33680i 0.304226i 0.988363 + 0.152113i \(0.0486077\pi\)
−0.988363 + 0.152113i \(0.951392\pi\)
\(60\) 0 0
\(61\) 5.28658i 0.676877i −0.940989 0.338438i \(-0.890101\pi\)
0.940989 0.338438i \(-0.109899\pi\)
\(62\) 0 0
\(63\) −1.58474 −0.199658
\(64\) 0 0
\(65\) 1.23011i 0.152577i
\(66\) 0 0
\(67\) −9.49359 −1.15983 −0.579913 0.814678i \(-0.696913\pi\)
−0.579913 + 0.814678i \(0.696913\pi\)
\(68\) 0 0
\(69\) −4.42937 1.83865i −0.533234 0.221348i
\(70\) 0 0
\(71\) 13.5262i 1.60527i −0.596473 0.802633i \(-0.703432\pi\)
0.596473 0.802633i \(-0.296568\pi\)
\(72\) 0 0
\(73\) −9.51828 −1.11403 −0.557015 0.830502i \(-0.688054\pi\)
−0.557015 + 0.830502i \(0.688054\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) −4.66677 −0.531828
\(78\) 0 0
\(79\) −3.71307 −0.417753 −0.208876 0.977942i \(-0.566981\pi\)
−0.208876 + 0.977942i \(0.566981\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 16.5130 1.81253 0.906267 0.422705i \(-0.138919\pi\)
0.906267 + 0.422705i \(0.138919\pi\)
\(84\) 0 0
\(85\) −1.25392 −0.136006
\(86\) 0 0
\(87\) 7.54806i 0.809237i
\(88\) 0 0
\(89\) 12.2406i 1.29750i −0.761003 0.648748i \(-0.775293\pi\)
0.761003 0.648748i \(-0.224707\pi\)
\(90\) 0 0
\(91\) −1.94940 −0.204353
\(92\) 0 0
\(93\) −0.690205 −0.0715710
\(94\) 0 0
\(95\) 5.26714i 0.540397i
\(96\) 0 0
\(97\) 7.70591i 0.782417i −0.920302 0.391208i \(-0.872057\pi\)
0.920302 0.391208i \(-0.127943\pi\)
\(98\) 0 0
\(99\) 2.94482 0.295966
\(100\) 0 0
\(101\) −9.43609 −0.938926 −0.469463 0.882952i \(-0.655552\pi\)
−0.469463 + 0.882952i \(0.655552\pi\)
\(102\) 0 0
\(103\) 7.07692 0.697310 0.348655 0.937251i \(-0.386639\pi\)
0.348655 + 0.937251i \(0.386639\pi\)
\(104\) 0 0
\(105\) −1.58474 −0.154654
\(106\) 0 0
\(107\) 6.20879 0.600227 0.300113 0.953904i \(-0.402976\pi\)
0.300113 + 0.953904i \(0.402976\pi\)
\(108\) 0 0
\(109\) 5.46182i 0.523147i −0.965184 0.261574i \(-0.915759\pi\)
0.965184 0.261574i \(-0.0842414\pi\)
\(110\) 0 0
\(111\) 4.83340 0.458766
\(112\) 0 0
\(113\) 1.98548i 0.186778i −0.995630 0.0933891i \(-0.970230\pi\)
0.995630 0.0933891i \(-0.0297700\pi\)
\(114\) 0 0
\(115\) −4.42937 1.83865i −0.413041 0.171455i
\(116\) 0 0
\(117\) 1.23011 0.113724
\(118\) 0 0
\(119\) 1.98713i 0.182160i
\(120\) 0 0
\(121\) −2.32801 −0.211638
\(122\) 0 0
\(123\) 6.98783i 0.630071i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 4.35495i 0.386439i 0.981156 + 0.193220i \(0.0618929\pi\)
−0.981156 + 0.193220i \(0.938107\pi\)
\(128\) 0 0
\(129\) 4.06681i 0.358063i
\(130\) 0 0
\(131\) 1.92541i 0.168224i 0.996456 + 0.0841118i \(0.0268053\pi\)
−0.996456 + 0.0841118i \(0.973195\pi\)
\(132\) 0 0
\(133\) 8.34702 0.723779
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 11.9282i 1.01910i −0.860442 0.509548i \(-0.829813\pi\)
0.860442 0.509548i \(-0.170187\pi\)
\(138\) 0 0
\(139\) 1.31703i 0.111709i −0.998439 0.0558545i \(-0.982212\pi\)
0.998439 0.0558545i \(-0.0177883\pi\)
\(140\) 0 0
\(141\) 8.95368 0.754036
\(142\) 0 0
\(143\) 3.62246 0.302925
\(144\) 0 0
\(145\) 7.54806i 0.626832i
\(146\) 0 0
\(147\) 4.48861i 0.370214i
\(148\) 0 0
\(149\) 7.50615i 0.614928i 0.951560 + 0.307464i \(0.0994804\pi\)
−0.951560 + 0.307464i \(0.900520\pi\)
\(150\) 0 0
\(151\) 4.78698i 0.389559i −0.980847 0.194780i \(-0.937601\pi\)
0.980847 0.194780i \(-0.0623992\pi\)
\(152\) 0 0
\(153\) 1.25392i 0.101373i
\(154\) 0 0
\(155\) −0.690205 −0.0554386
\(156\) 0 0
\(157\) 6.01182i 0.479795i −0.970798 0.239898i \(-0.922886\pi\)
0.970798 0.239898i \(-0.0771139\pi\)
\(158\) 0 0
\(159\) −10.8852 −0.863255
\(160\) 0 0
\(161\) 2.91378 7.01939i 0.229638 0.553206i
\(162\) 0 0
\(163\) 9.41003i 0.737051i 0.929618 + 0.368525i \(0.120137\pi\)
−0.929618 + 0.368525i \(0.879863\pi\)
\(164\) 0 0
\(165\) 2.94482 0.229254
\(166\) 0 0
\(167\) 2.39043i 0.184977i −0.995714 0.0924884i \(-0.970518\pi\)
0.995714 0.0924884i \(-0.0294821\pi\)
\(168\) 0 0
\(169\) −11.4868 −0.883602
\(170\) 0 0
\(171\) −5.26714 −0.402788
\(172\) 0 0
\(173\) 7.24943 0.551164 0.275582 0.961278i \(-0.411129\pi\)
0.275582 + 0.961278i \(0.411129\pi\)
\(174\) 0 0
\(175\) −1.58474 −0.119795
\(176\) 0 0
\(177\) 2.33680 0.175645
\(178\) 0 0
\(179\) 12.2088i 0.912532i −0.889843 0.456266i \(-0.849187\pi\)
0.889843 0.456266i \(-0.150813\pi\)
\(180\) 0 0
\(181\) 16.9328i 1.25860i −0.777161 0.629302i \(-0.783341\pi\)
0.777161 0.629302i \(-0.216659\pi\)
\(182\) 0 0
\(183\) −5.28658 −0.390795
\(184\) 0 0
\(185\) 4.83340 0.355359
\(186\) 0 0
\(187\) 3.69256i 0.270027i
\(188\) 0 0
\(189\) 1.58474i 0.115273i
\(190\) 0 0
\(191\) −6.46417 −0.467731 −0.233865 0.972269i \(-0.575138\pi\)
−0.233865 + 0.972269i \(0.575138\pi\)
\(192\) 0 0
\(193\) −21.5952 −1.55445 −0.777227 0.629220i \(-0.783375\pi\)
−0.777227 + 0.629220i \(0.783375\pi\)
\(194\) 0 0
\(195\) 1.23011 0.0880901
\(196\) 0 0
\(197\) −4.11055 −0.292865 −0.146432 0.989221i \(-0.546779\pi\)
−0.146432 + 0.989221i \(0.546779\pi\)
\(198\) 0 0
\(199\) −3.25603 −0.230814 −0.115407 0.993318i \(-0.536817\pi\)
−0.115407 + 0.993318i \(0.536817\pi\)
\(200\) 0 0
\(201\) 9.49359i 0.669626i
\(202\) 0 0
\(203\) 11.9617 0.839546
\(204\) 0 0
\(205\) 6.98783i 0.488051i
\(206\) 0 0
\(207\) −1.83865 + 4.42937i −0.127795 + 0.307863i
\(208\) 0 0
\(209\) −15.5108 −1.07290
\(210\) 0 0
\(211\) 18.1011i 1.24613i 0.782169 + 0.623066i \(0.214113\pi\)
−0.782169 + 0.623066i \(0.785887\pi\)
\(212\) 0 0
\(213\) −13.5262 −0.926801
\(214\) 0 0
\(215\) 4.06681i 0.277354i
\(216\) 0 0
\(217\) 1.09379i 0.0742516i
\(218\) 0 0
\(219\) 9.51828i 0.643186i
\(220\) 0 0
\(221\) 1.54246i 0.103757i
\(222\) 0 0
\(223\) 2.72334i 0.182368i −0.995834 0.0911842i \(-0.970935\pi\)
0.995834 0.0911842i \(-0.0290652\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 21.7426 1.44311 0.721554 0.692358i \(-0.243428\pi\)
0.721554 + 0.692358i \(0.243428\pi\)
\(228\) 0 0
\(229\) 1.81560i 0.119978i 0.998199 + 0.0599891i \(0.0191066\pi\)
−0.998199 + 0.0599891i \(0.980893\pi\)
\(230\) 0 0
\(231\) 4.66677i 0.307051i
\(232\) 0 0
\(233\) −25.1093 −1.64496 −0.822482 0.568792i \(-0.807411\pi\)
−0.822482 + 0.568792i \(0.807411\pi\)
\(234\) 0 0
\(235\) 8.95368 0.584074
\(236\) 0 0
\(237\) 3.71307i 0.241190i
\(238\) 0 0
\(239\) 25.5463i 1.65245i 0.563338 + 0.826226i \(0.309517\pi\)
−0.563338 + 0.826226i \(0.690483\pi\)
\(240\) 0 0
\(241\) 17.7500i 1.14338i −0.820471 0.571689i \(-0.806289\pi\)
0.820471 0.571689i \(-0.193711\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 4.48861i 0.286767i
\(246\) 0 0
\(247\) −6.47916 −0.412259
\(248\) 0 0
\(249\) 16.5130i 1.04647i
\(250\) 0 0
\(251\) 18.8514 1.18989 0.594946 0.803766i \(-0.297174\pi\)
0.594946 + 0.803766i \(0.297174\pi\)
\(252\) 0 0
\(253\) −5.41451 + 13.0437i −0.340407 + 0.820052i
\(254\) 0 0
\(255\) 1.25392i 0.0785233i
\(256\) 0 0
\(257\) 18.1245 1.13058 0.565289 0.824893i \(-0.308765\pi\)
0.565289 + 0.824893i \(0.308765\pi\)
\(258\) 0 0
\(259\) 7.65967i 0.475948i
\(260\) 0 0
\(261\) −7.54806 −0.467213
\(262\) 0 0
\(263\) −29.0810 −1.79321 −0.896606 0.442829i \(-0.853975\pi\)
−0.896606 + 0.442829i \(0.853975\pi\)
\(264\) 0 0
\(265\) −10.8852 −0.668675
\(266\) 0 0
\(267\) −12.2406 −0.749110
\(268\) 0 0
\(269\) −21.2290 −1.29435 −0.647176 0.762340i \(-0.724050\pi\)
−0.647176 + 0.762340i \(0.724050\pi\)
\(270\) 0 0
\(271\) 9.18826i 0.558147i 0.960270 + 0.279073i \(0.0900273\pi\)
−0.960270 + 0.279073i \(0.909973\pi\)
\(272\) 0 0
\(273\) 1.94940i 0.117983i
\(274\) 0 0
\(275\) 2.94482 0.177580
\(276\) 0 0
\(277\) 23.9814 1.44090 0.720451 0.693505i \(-0.243935\pi\)
0.720451 + 0.693505i \(0.243935\pi\)
\(278\) 0 0
\(279\) 0.690205i 0.0413215i
\(280\) 0 0
\(281\) 18.5638i 1.10742i −0.832709 0.553711i \(-0.813211\pi\)
0.832709 0.553711i \(-0.186789\pi\)
\(282\) 0 0
\(283\) 12.8548 0.764138 0.382069 0.924134i \(-0.375212\pi\)
0.382069 + 0.924134i \(0.375212\pi\)
\(284\) 0 0
\(285\) −5.26714 −0.311998
\(286\) 0 0
\(287\) 11.0739 0.653670
\(288\) 0 0
\(289\) 15.4277 0.907511
\(290\) 0 0
\(291\) −7.70591 −0.451728
\(292\) 0 0
\(293\) 25.3576i 1.48141i −0.671831 0.740705i \(-0.734492\pi\)
0.671831 0.740705i \(-0.265508\pi\)
\(294\) 0 0
\(295\) 2.33680 0.136054
\(296\) 0 0
\(297\) 2.94482i 0.170876i
\(298\) 0 0
\(299\) −2.26175 + 5.44862i −0.130800 + 0.315102i
\(300\) 0 0
\(301\) 6.44483 0.371474
\(302\) 0 0
\(303\) 9.43609i 0.542089i
\(304\) 0 0
\(305\) −5.28658 −0.302709
\(306\) 0 0
\(307\) 12.7630i 0.728422i 0.931316 + 0.364211i \(0.118661\pi\)
−0.931316 + 0.364211i \(0.881339\pi\)
\(308\) 0 0
\(309\) 7.07692i 0.402592i
\(310\) 0 0
\(311\) 14.7842i 0.838333i −0.907909 0.419166i \(-0.862322\pi\)
0.907909 0.419166i \(-0.137678\pi\)
\(312\) 0 0
\(313\) 1.48723i 0.0840632i 0.999116 + 0.0420316i \(0.0133830\pi\)
−0.999116 + 0.0420316i \(0.986617\pi\)
\(314\) 0 0
\(315\) 1.58474i 0.0892898i
\(316\) 0 0
\(317\) 0.771575 0.0433360 0.0216680 0.999765i \(-0.493102\pi\)
0.0216680 + 0.999765i \(0.493102\pi\)
\(318\) 0 0
\(319\) −22.2277 −1.24451
\(320\) 0 0
\(321\) 6.20879i 0.346541i
\(322\) 0 0
\(323\) 6.60455i 0.367487i
\(324\) 0 0
\(325\) 1.23011 0.0682343
\(326\) 0 0
\(327\) −5.46182 −0.302039
\(328\) 0 0
\(329\) 14.1892i 0.782277i
\(330\) 0 0
\(331\) 33.8360i 1.85980i −0.367817 0.929898i \(-0.619895\pi\)
0.367817 0.929898i \(-0.380105\pi\)
\(332\) 0 0
\(333\) 4.83340i 0.264869i
\(334\) 0 0
\(335\) 9.49359i 0.518690i
\(336\) 0 0
\(337\) 25.3943i 1.38331i −0.722226 0.691657i \(-0.756881\pi\)
0.722226 0.691657i \(-0.243119\pi\)
\(338\) 0 0
\(339\) −1.98548 −0.107836
\(340\) 0 0
\(341\) 2.03253i 0.110068i
\(342\) 0 0
\(343\) −18.2064 −0.983054
\(344\) 0 0
\(345\) −1.83865 + 4.42937i −0.0989898 + 0.238469i
\(346\) 0 0
\(347\) 7.66039i 0.411231i −0.978633 0.205616i \(-0.934080\pi\)
0.978633 0.205616i \(-0.0659197\pi\)
\(348\) 0 0
\(349\) 15.8624 0.849093 0.424546 0.905406i \(-0.360434\pi\)
0.424546 + 0.905406i \(0.360434\pi\)
\(350\) 0 0
\(351\) 1.23011i 0.0656585i
\(352\) 0 0
\(353\) −26.7036 −1.42129 −0.710643 0.703552i \(-0.751596\pi\)
−0.710643 + 0.703552i \(0.751596\pi\)
\(354\) 0 0
\(355\) −13.5262 −0.717897
\(356\) 0 0
\(357\) −1.98713 −0.105170
\(358\) 0 0
\(359\) 17.3015 0.913137 0.456569 0.889688i \(-0.349078\pi\)
0.456569 + 0.889688i \(0.349078\pi\)
\(360\) 0 0
\(361\) 8.74271 0.460143
\(362\) 0 0
\(363\) 2.32801i 0.122189i
\(364\) 0 0
\(365\) 9.51828i 0.498210i
\(366\) 0 0
\(367\) 19.1050 0.997271 0.498635 0.866812i \(-0.333835\pi\)
0.498635 + 0.866812i \(0.333835\pi\)
\(368\) 0 0
\(369\) −6.98783 −0.363772
\(370\) 0 0
\(371\) 17.2502i 0.895587i
\(372\) 0 0
\(373\) 29.5261i 1.52880i 0.644740 + 0.764402i \(0.276966\pi\)
−0.644740 + 0.764402i \(0.723034\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −9.28496 −0.478200
\(378\) 0 0
\(379\) −29.9649 −1.53920 −0.769598 0.638529i \(-0.779543\pi\)
−0.769598 + 0.638529i \(0.779543\pi\)
\(380\) 0 0
\(381\) 4.35495 0.223111
\(382\) 0 0
\(383\) 6.78342 0.346616 0.173308 0.984868i \(-0.444554\pi\)
0.173308 + 0.984868i \(0.444554\pi\)
\(384\) 0 0
\(385\) 4.66677i 0.237841i
\(386\) 0 0
\(387\) −4.06681 −0.206728
\(388\) 0 0
\(389\) 8.20216i 0.415866i −0.978143 0.207933i \(-0.933326\pi\)
0.978143 0.207933i \(-0.0666736\pi\)
\(390\) 0 0
\(391\) −5.55407 2.30552i −0.280881 0.116595i
\(392\) 0 0
\(393\) 1.92541 0.0971240
\(394\) 0 0
\(395\) 3.71307i 0.186825i
\(396\) 0 0
\(397\) −28.2572 −1.41819 −0.709095 0.705113i \(-0.750896\pi\)
−0.709095 + 0.705113i \(0.750896\pi\)
\(398\) 0 0
\(399\) 8.34702i 0.417874i
\(400\) 0 0
\(401\) 17.1193i 0.854899i −0.904039 0.427449i \(-0.859412\pi\)
0.904039 0.427449i \(-0.140588\pi\)
\(402\) 0 0
\(403\) 0.849030i 0.0422932i
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 14.2335i 0.705529i
\(408\) 0 0
\(409\) 22.8130 1.12803 0.564016 0.825764i \(-0.309256\pi\)
0.564016 + 0.825764i \(0.309256\pi\)
\(410\) 0 0
\(411\) −11.9282 −0.588375
\(412\) 0 0
\(413\) 3.70322i 0.182223i
\(414\) 0 0
\(415\) 16.5130i 0.810590i
\(416\) 0 0
\(417\) −1.31703 −0.0644952
\(418\) 0 0
\(419\) 4.54342 0.221960 0.110980 0.993823i \(-0.464601\pi\)
0.110980 + 0.993823i \(0.464601\pi\)
\(420\) 0 0
\(421\) 2.02280i 0.0985853i −0.998784 0.0492926i \(-0.984303\pi\)
0.998784 0.0492926i \(-0.0156967\pi\)
\(422\) 0 0
\(423\) 8.95368i 0.435343i
\(424\) 0 0
\(425\) 1.25392i 0.0608239i
\(426\) 0 0
\(427\) 8.37783i 0.405432i
\(428\) 0 0
\(429\) 3.62246i 0.174894i
\(430\) 0 0
\(431\) 10.5534 0.508339 0.254169 0.967160i \(-0.418198\pi\)
0.254169 + 0.967160i \(0.418198\pi\)
\(432\) 0 0
\(433\) 33.9550i 1.63177i −0.578211 0.815887i \(-0.696249\pi\)
0.578211 0.815887i \(-0.303751\pi\)
\(434\) 0 0
\(435\) −7.54806 −0.361902
\(436\) 0 0
\(437\) 9.68444 23.3301i 0.463270 1.11603i
\(438\) 0 0
\(439\) 2.99231i 0.142815i 0.997447 + 0.0714076i \(0.0227491\pi\)
−0.997447 + 0.0714076i \(0.977251\pi\)
\(440\) 0 0
\(441\) 4.48861 0.213743
\(442\) 0 0
\(443\) 20.9670i 0.996174i −0.867127 0.498087i \(-0.834036\pi\)
0.867127 0.498087i \(-0.165964\pi\)
\(444\) 0 0
\(445\) −12.2406 −0.580258
\(446\) 0 0
\(447\) 7.50615 0.355029
\(448\) 0 0
\(449\) 11.8249 0.558053 0.279026 0.960283i \(-0.409988\pi\)
0.279026 + 0.960283i \(0.409988\pi\)
\(450\) 0 0
\(451\) −20.5779 −0.968977
\(452\) 0 0
\(453\) −4.78698 −0.224912
\(454\) 0 0
\(455\) 1.94940i 0.0913894i
\(456\) 0 0
\(457\) 10.8771i 0.508810i 0.967098 + 0.254405i \(0.0818796\pi\)
−0.967098 + 0.254405i \(0.918120\pi\)
\(458\) 0 0
\(459\) 1.25392 0.0585278
\(460\) 0 0
\(461\) 15.2626 0.710849 0.355425 0.934705i \(-0.384336\pi\)
0.355425 + 0.934705i \(0.384336\pi\)
\(462\) 0 0
\(463\) 38.1730i 1.77405i 0.461721 + 0.887025i \(0.347232\pi\)
−0.461721 + 0.887025i \(0.652768\pi\)
\(464\) 0 0
\(465\) 0.690205i 0.0320075i
\(466\) 0 0
\(467\) 4.21856 0.195212 0.0976059 0.995225i \(-0.468882\pi\)
0.0976059 + 0.995225i \(0.468882\pi\)
\(468\) 0 0
\(469\) −15.0448 −0.694706
\(470\) 0 0
\(471\) −6.01182 −0.277010
\(472\) 0 0
\(473\) −11.9760 −0.550659
\(474\) 0 0
\(475\) −5.26714 −0.241673
\(476\) 0 0
\(477\) 10.8852i 0.498401i
\(478\) 0 0
\(479\) −0.347762 −0.0158897 −0.00794483 0.999968i \(-0.502529\pi\)
−0.00794483 + 0.999968i \(0.502529\pi\)
\(480\) 0 0
\(481\) 5.94562i 0.271097i
\(482\) 0 0
\(483\) −7.01939 2.91378i −0.319393 0.132582i
\(484\) 0 0
\(485\) −7.70591 −0.349907
\(486\) 0 0
\(487\) 12.4417i 0.563789i −0.959445 0.281894i \(-0.909037\pi\)
0.959445 0.281894i \(-0.0909628\pi\)
\(488\) 0 0
\(489\) 9.41003 0.425536
\(490\) 0 0
\(491\) 4.12588i 0.186198i 0.995657 + 0.0930991i \(0.0296774\pi\)
−0.995657 + 0.0930991i \(0.970323\pi\)
\(492\) 0 0
\(493\) 9.46464i 0.426266i
\(494\) 0 0
\(495\) 2.94482i 0.132360i
\(496\) 0 0
\(497\) 21.4355i 0.961514i
\(498\) 0 0
\(499\) 24.3075i 1.08815i −0.839035 0.544077i \(-0.816880\pi\)
0.839035 0.544077i \(-0.183120\pi\)
\(500\) 0 0
\(501\) −2.39043 −0.106796
\(502\) 0 0
\(503\) −11.9762 −0.533991 −0.266995 0.963698i \(-0.586031\pi\)
−0.266995 + 0.963698i \(0.586031\pi\)
\(504\) 0 0
\(505\) 9.43609i 0.419900i
\(506\) 0 0
\(507\) 11.4868i 0.510148i
\(508\) 0 0
\(509\) 43.4081 1.92403 0.962015 0.272997i \(-0.0880150\pi\)
0.962015 + 0.272997i \(0.0880150\pi\)
\(510\) 0 0
\(511\) −15.0840 −0.667276
\(512\) 0 0
\(513\) 5.26714i 0.232550i
\(514\) 0 0
\(515\) 7.07692i 0.311846i
\(516\) 0 0
\(517\) 26.3670i 1.15962i
\(518\) 0 0
\(519\) 7.24943i 0.318215i
\(520\) 0 0
\(521\) 30.1561i 1.32116i 0.750754 + 0.660582i \(0.229690\pi\)
−0.750754 + 0.660582i \(0.770310\pi\)
\(522\) 0 0
\(523\) −25.0503 −1.09537 −0.547686 0.836684i \(-0.684491\pi\)
−0.547686 + 0.836684i \(0.684491\pi\)
\(524\) 0 0
\(525\) 1.58474i 0.0691636i
\(526\) 0 0
\(527\) −0.865460 −0.0377000
\(528\) 0 0
\(529\) −16.2387 16.2882i −0.706031 0.708181i
\(530\) 0 0
\(531\) 2.33680i 0.101409i
\(532\) 0 0
\(533\) −8.59581 −0.372326
\(534\) 0 0
\(535\) 6.20879i 0.268430i
\(536\) 0 0
\(537\) −12.2088 −0.526851
\(538\) 0 0
\(539\) 13.2182 0.569346
\(540\) 0 0
\(541\) 38.9949 1.67652 0.838260 0.545271i \(-0.183573\pi\)
0.838260 + 0.545271i \(0.183573\pi\)
\(542\) 0 0
\(543\) −16.9328 −0.726656
\(544\) 0 0
\(545\) −5.46182 −0.233959
\(546\) 0 0
\(547\) 35.9994i 1.53922i 0.638512 + 0.769612i \(0.279550\pi\)
−0.638512 + 0.769612i \(0.720450\pi\)
\(548\) 0 0
\(549\) 5.28658i 0.225626i
\(550\) 0 0
\(551\) 39.7567 1.69369
\(552\) 0 0
\(553\) −5.88424 −0.250223
\(554\) 0 0
\(555\) 4.83340i 0.205166i
\(556\) 0 0
\(557\) 22.1601i 0.938954i 0.882945 + 0.469477i \(0.155558\pi\)
−0.882945 + 0.469477i \(0.844442\pi\)
\(558\) 0 0
\(559\) −5.00263 −0.211589
\(560\) 0 0
\(561\) 3.69256 0.155900
\(562\) 0 0
\(563\) −2.91544 −0.122871 −0.0614356 0.998111i \(-0.519568\pi\)
−0.0614356 + 0.998111i \(0.519568\pi\)
\(564\) 0 0
\(565\) −1.98548 −0.0835297
\(566\) 0 0
\(567\) 1.58474 0.0665527
\(568\) 0 0
\(569\) 14.9836i 0.628144i −0.949399 0.314072i \(-0.898307\pi\)
0.949399 0.314072i \(-0.101693\pi\)
\(570\) 0 0
\(571\) −3.15642 −0.132092 −0.0660461 0.997817i \(-0.521038\pi\)
−0.0660461 + 0.997817i \(0.521038\pi\)
\(572\) 0 0
\(573\) 6.46417i 0.270044i
\(574\) 0 0
\(575\) −1.83865 + 4.42937i −0.0766772 + 0.184718i
\(576\) 0 0
\(577\) −6.25189 −0.260270 −0.130135 0.991496i \(-0.541541\pi\)
−0.130135 + 0.991496i \(0.541541\pi\)
\(578\) 0 0
\(579\) 21.5952i 0.897465i
\(580\) 0 0
\(581\) 26.1687 1.08566
\(582\) 0 0
\(583\) 32.0551i 1.32759i
\(584\) 0 0
\(585\) 1.23011i 0.0508589i
\(586\) 0 0
\(587\) 3.22099i 0.132945i −0.997788 0.0664723i \(-0.978826\pi\)
0.997788 0.0664723i \(-0.0211744\pi\)
\(588\) 0 0
\(589\) 3.63541i 0.149794i
\(590\) 0 0
\(591\) 4.11055i 0.169085i
\(592\) 0 0
\(593\) −18.3093 −0.751873 −0.375937 0.926645i \(-0.622679\pi\)
−0.375937 + 0.926645i \(0.622679\pi\)
\(594\) 0 0
\(595\) −1.98713 −0.0814643
\(596\) 0 0
\(597\) 3.25603i 0.133261i
\(598\) 0 0
\(599\) 5.79992i 0.236978i −0.992955 0.118489i \(-0.962195\pi\)
0.992955 0.118489i \(-0.0378051\pi\)
\(600\) 0 0
\(601\) −12.1653 −0.496233 −0.248116 0.968730i \(-0.579812\pi\)
−0.248116 + 0.968730i \(0.579812\pi\)
\(602\) 0 0
\(603\) 9.49359 0.386609
\(604\) 0 0
\(605\) 2.32801i 0.0946472i
\(606\) 0 0
\(607\) 33.5799i 1.36296i 0.731834 + 0.681482i \(0.238664\pi\)
−0.731834 + 0.681482i \(0.761336\pi\)
\(608\) 0 0
\(609\) 11.9617i 0.484712i
\(610\) 0 0
\(611\) 11.0140i 0.445580i
\(612\) 0 0
\(613\) 24.5388i 0.991113i 0.868576 + 0.495556i \(0.165036\pi\)
−0.868576 + 0.495556i \(0.834964\pi\)
\(614\) 0 0
\(615\) −6.98783 −0.281776
\(616\) 0 0
\(617\) 10.9240i 0.439782i 0.975524 + 0.219891i \(0.0705701\pi\)
−0.975524 + 0.219891i \(0.929430\pi\)
\(618\) 0 0
\(619\) 29.8447 1.19956 0.599780 0.800165i \(-0.295255\pi\)
0.599780 + 0.800165i \(0.295255\pi\)
\(620\) 0 0
\(621\) 4.42937 + 1.83865i 0.177745 + 0.0737826i
\(622\) 0 0
\(623\) 19.3981i 0.777167i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 15.5108i 0.619441i
\(628\) 0 0
\(629\) 6.06068 0.241655
\(630\) 0 0
\(631\) −4.26198 −0.169667 −0.0848333 0.996395i \(-0.527036\pi\)
−0.0848333 + 0.996395i \(0.527036\pi\)
\(632\) 0 0
\(633\) 18.1011 0.719455
\(634\) 0 0
\(635\) 4.35495 0.172821
\(636\) 0 0
\(637\) 5.52149 0.218769
\(638\) 0 0
\(639\) 13.5262i 0.535089i
\(640\) 0 0
\(641\) 43.5638i 1.72067i −0.509731 0.860334i \(-0.670255\pi\)
0.509731 0.860334i \(-0.329745\pi\)
\(642\) 0 0
\(643\) −11.2217 −0.442542 −0.221271 0.975212i \(-0.571021\pi\)
−0.221271 + 0.975212i \(0.571021\pi\)
\(644\) 0 0
\(645\) −4.06681 −0.160131
\(646\) 0 0
\(647\) 36.7686i 1.44552i 0.691098 + 0.722761i \(0.257127\pi\)
−0.691098 + 0.722761i \(0.742873\pi\)
\(648\) 0 0
\(649\) 6.88147i 0.270121i
\(650\) 0 0
\(651\) −1.09379 −0.0428692
\(652\) 0 0
\(653\) −31.9401 −1.24991 −0.624957 0.780659i \(-0.714883\pi\)
−0.624957 + 0.780659i \(0.714883\pi\)
\(654\) 0 0
\(655\) 1.92541 0.0752319
\(656\) 0 0
\(657\) 9.51828 0.371344
\(658\) 0 0
\(659\) 43.5902 1.69803 0.849016 0.528367i \(-0.177196\pi\)
0.849016 + 0.528367i \(0.177196\pi\)
\(660\) 0 0
\(661\) 10.4393i 0.406042i 0.979174 + 0.203021i \(0.0650759\pi\)
−0.979174 + 0.203021i \(0.934924\pi\)
\(662\) 0 0
\(663\) 1.54246 0.0599041
\(664\) 0 0
\(665\) 8.34702i 0.323684i
\(666\) 0 0
\(667\) 13.8783 33.4332i 0.537369 1.29454i
\(668\) 0 0
\(669\) −2.72334 −0.105290
\(670\) 0 0
\(671\) 15.5680i 0.600997i
\(672\) 0 0
\(673\) 2.83519 0.109288 0.0546442 0.998506i \(-0.482598\pi\)
0.0546442 + 0.998506i \(0.482598\pi\)
\(674\) 0 0
\(675\) 1.00000i 0.0384900i
\(676\) 0 0
\(677\) 36.3830i 1.39831i 0.714969 + 0.699156i \(0.246441\pi\)
−0.714969 + 0.699156i \(0.753559\pi\)
\(678\) 0 0
\(679\) 12.2118i 0.468647i
\(680\) 0 0
\(681\) 21.7426i 0.833179i
\(682\) 0 0
\(683\) 2.81755i 0.107811i 0.998546 + 0.0539053i \(0.0171669\pi\)
−0.998546 + 0.0539053i \(0.982833\pi\)
\(684\) 0 0
\(685\) −11.9282 −0.455754
\(686\) 0 0
\(687\) 1.81560 0.0692695
\(688\) 0 0
\(689\) 13.3901i 0.510120i
\(690\) 0 0
\(691\) 49.1990i 1.87162i −0.352506 0.935809i \(-0.614670\pi\)
0.352506 0.935809i \(-0.385330\pi\)
\(692\) 0 0
\(693\) 4.66677 0.177276
\(694\) 0 0
\(695\) −1.31703 −0.0499578
\(696\) 0 0
\(697\) 8.76216i 0.331890i
\(698\) 0 0
\(699\) 25.1093i 0.949720i
\(700\) 0 0
\(701\) 15.5212i 0.586228i 0.956078 + 0.293114i \(0.0946915\pi\)
−0.956078 + 0.293114i \(0.905308\pi\)
\(702\) 0 0
\(703\) 25.4582i 0.960173i
\(704\) 0 0
\(705\) 8.95368i 0.337215i
\(706\) 0 0
\(707\) −14.9537 −0.562392
\(708\) 0 0
\(709\) 1.54150i 0.0578923i −0.999581 0.0289462i \(-0.990785\pi\)
0.999581 0.0289462i \(-0.00921514\pi\)
\(710\) 0 0
\(711\) 3.71307 0.139251
\(712\) 0 0
\(713\) −3.05718 1.26905i −0.114492 0.0475263i
\(714\) 0 0
\(715\) 3.62246i 0.135472i
\(716\) 0 0
\(717\) 25.5463 0.954044
\(718\) 0 0
\(719\) 2.00464i 0.0747606i 0.999301 + 0.0373803i \(0.0119013\pi\)
−0.999301 + 0.0373803i \(0.988099\pi\)
\(720\) 0 0
\(721\) 11.2151 0.417671
\(722\) 0 0
\(723\) −17.7500 −0.660129
\(724\) 0 0
\(725\) −7.54806 −0.280328
\(726\) 0 0
\(727\) −17.3038 −0.641761 −0.320881 0.947120i \(-0.603979\pi\)
−0.320881 + 0.947120i \(0.603979\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 5.09944i 0.188610i
\(732\) 0 0
\(733\) 26.4881i 0.978358i 0.872183 + 0.489179i \(0.162704\pi\)
−0.872183 + 0.489179i \(0.837296\pi\)
\(734\) 0 0
\(735\) 4.48861 0.165565
\(736\) 0 0
\(737\) 27.9569 1.02981
\(738\) 0 0
\(739\) 7.11302i 0.261657i −0.991405 0.130828i \(-0.958236\pi\)
0.991405 0.130828i \(-0.0417637\pi\)
\(740\) 0 0
\(741\) 6.47916i 0.238018i
\(742\) 0 0
\(743\) 38.7611 1.42201 0.711003 0.703189i \(-0.248241\pi\)
0.711003 + 0.703189i \(0.248241\pi\)
\(744\) 0 0
\(745\) 7.50615 0.275004
\(746\) 0 0
\(747\) −16.5130 −0.604178
\(748\) 0 0
\(749\) 9.83930 0.359520
\(750\) 0 0
\(751\) 26.4017 0.963412 0.481706 0.876333i \(-0.340017\pi\)
0.481706 + 0.876333i \(0.340017\pi\)
\(752\) 0 0
\(753\) 18.8514i 0.686984i
\(754\) 0 0
\(755\) −4.78698 −0.174216
\(756\) 0 0
\(757\) 19.6012i 0.712419i 0.934406 + 0.356210i \(0.115931\pi\)
−0.934406 + 0.356210i \(0.884069\pi\)
\(758\) 0 0
\(759\) 13.0437 + 5.41451i 0.473457 + 0.196534i
\(760\) 0 0
\(761\) −1.90461 −0.0690421 −0.0345211 0.999404i \(-0.510991\pi\)
−0.0345211 + 0.999404i \(0.510991\pi\)
\(762\) 0 0
\(763\) 8.65554i 0.313352i
\(764\) 0 0
\(765\) 1.25392 0.0453355
\(766\) 0 0
\(767\) 2.87453i 0.103793i
\(768\) 0 0
\(769\) 24.1496i 0.870858i 0.900223 + 0.435429i \(0.143403\pi\)
−0.900223 + 0.435429i \(0.856597\pi\)
\(770\) 0 0
\(771\) 18.1245i 0.652739i
\(772\) 0 0
\(773\) 35.9153i 1.29179i 0.763428 + 0.645893i \(0.223515\pi\)
−0.763428 + 0.645893i \(0.776485\pi\)
\(774\) 0 0
\(775\) 0.690205i 0.0247929i
\(776\) 0 0
\(777\) 7.65967 0.274789
\(778\) 0 0
\(779\) 36.8058 1.31871
\(780\) 0 0
\(781\) 39.8323i 1.42531i
\(782\) 0 0
\(783\) 7.54806i 0.269746i
\(784\) 0 0
\(785\) −6.01182 −0.214571
\(786\) 0 0
\(787\) −25.6536 −0.914452 −0.457226 0.889351i \(-0.651157\pi\)
−0.457226 + 0.889351i \(0.651157\pi\)
\(788\) 0 0
\(789\) 29.0810i 1.03531i
\(790\) 0 0
\(791\) 3.14646i 0.111875i
\(792\) 0 0
\(793\) 6.50308i 0.230931i
\(794\) 0 0
\(795\) 10.8852i 0.386059i
\(796\) 0 0
\(797\) 11.6082i 0.411185i 0.978638 + 0.205592i \(0.0659121\pi\)
−0.978638 + 0.205592i \(0.934088\pi\)
\(798\) 0 0
\(799\) 11.2272 0.397189
\(800\) 0 0
\(801\) 12.2406i 0.432499i
\(802\) 0 0
\(803\) 28.0297 0.989145
\(804\) 0 0
\(805\)