Properties

Label 5520.2.be.c.1471.18
Level $5520$
Weight $2$
Character 5520.1471
Analytic conductor $44.077$
Analytic rank $0$
Dimension $32$
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: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1471.18
Character \(\chi\) \(=\) 5520.1471
Dual form 5520.2.be.c.1471.17

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +1.00000i q^{5} -0.143131 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +1.00000i q^{5} -0.143131 q^{7} -1.00000 q^{9} -4.80659 q^{11} +4.78151 q^{13} -1.00000 q^{15} -4.86743i q^{17} -0.379340 q^{19} -0.143131i q^{21} +(-1.15365 + 4.65501i) q^{23} -1.00000 q^{25} -1.00000i q^{27} +8.39933 q^{29} -2.55983i q^{31} -4.80659i q^{33} -0.143131i q^{35} +5.56282i q^{37} +4.78151i q^{39} -5.17202 q^{41} -1.03505 q^{43} -1.00000i q^{45} +8.93486i q^{47} -6.97951 q^{49} +4.86743 q^{51} +13.3413i q^{53} -4.80659i q^{55} -0.379340i q^{57} +6.57340i q^{59} -7.49837i q^{61} +0.143131 q^{63} +4.78151i q^{65} -4.96888 q^{67} +(-4.65501 - 1.15365i) q^{69} -6.77385i q^{71} -8.04529 q^{73} -1.00000i q^{75} +0.687974 q^{77} +1.53132 q^{79} +1.00000 q^{81} -14.3973 q^{83} +4.86743 q^{85} +8.39933i q^{87} +12.6001i q^{89} -0.684383 q^{91} +2.55983 q^{93} -0.379340i q^{95} -5.67311i q^{97} +4.80659 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q - 8q^{7} - 32q^{9} + O(q^{10}) \) \( 32q - 8q^{7} - 32q^{9} + 8q^{11} - 8q^{13} - 32q^{15} - 32q^{25} + 4q^{29} + 20q^{41} + 52q^{49} - 4q^{51} + 8q^{63} + 32q^{67} - 40q^{73} - 24q^{77} + 32q^{79} + 32q^{81} - 4q^{85} - 48q^{91} - 8q^{99} + O(q^{100}) \)

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
\(7\) −0.143131 −0.0540986 −0.0270493 0.999634i \(-0.508611\pi\)
−0.0270493 + 0.999634i \(0.508611\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −4.80659 −1.44924 −0.724621 0.689148i \(-0.757985\pi\)
−0.724621 + 0.689148i \(0.757985\pi\)
\(12\) 0 0
\(13\) 4.78151 1.32615 0.663075 0.748553i \(-0.269251\pi\)
0.663075 + 0.748553i \(0.269251\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 4.86743i 1.18053i −0.807211 0.590263i \(-0.799024\pi\)
0.807211 0.590263i \(-0.200976\pi\)
\(18\) 0 0
\(19\) −0.379340 −0.0870266 −0.0435133 0.999053i \(-0.513855\pi\)
−0.0435133 + 0.999053i \(0.513855\pi\)
\(20\) 0 0
\(21\) 0.143131i 0.0312338i
\(22\) 0 0
\(23\) −1.15365 + 4.65501i −0.240552 + 0.970636i
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 8.39933 1.55972 0.779858 0.625956i \(-0.215291\pi\)
0.779858 + 0.625956i \(0.215291\pi\)
\(30\) 0 0
\(31\) 2.55983i 0.459759i −0.973219 0.229879i \(-0.926167\pi\)
0.973219 0.229879i \(-0.0738332\pi\)
\(32\) 0 0
\(33\) 4.80659i 0.836720i
\(34\) 0 0
\(35\) 0.143131i 0.0241936i
\(36\) 0 0
\(37\) 5.56282i 0.914521i 0.889333 + 0.457261i \(0.151169\pi\)
−0.889333 + 0.457261i \(0.848831\pi\)
\(38\) 0 0
\(39\) 4.78151i 0.765654i
\(40\) 0 0
\(41\) −5.17202 −0.807734 −0.403867 0.914818i \(-0.632334\pi\)
−0.403867 + 0.914818i \(0.632334\pi\)
\(42\) 0 0
\(43\) −1.03505 −0.157843 −0.0789215 0.996881i \(-0.525148\pi\)
−0.0789215 + 0.996881i \(0.525148\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) 8.93486i 1.30328i 0.758527 + 0.651642i \(0.225919\pi\)
−0.758527 + 0.651642i \(0.774081\pi\)
\(48\) 0 0
\(49\) −6.97951 −0.997073
\(50\) 0 0
\(51\) 4.86743 0.681577
\(52\) 0 0
\(53\) 13.3413i 1.83256i 0.400534 + 0.916282i \(0.368825\pi\)
−0.400534 + 0.916282i \(0.631175\pi\)
\(54\) 0 0
\(55\) 4.80659i 0.648121i
\(56\) 0 0
\(57\) 0.379340i 0.0502448i
\(58\) 0 0
\(59\) 6.57340i 0.855783i 0.903830 + 0.427892i \(0.140743\pi\)
−0.903830 + 0.427892i \(0.859257\pi\)
\(60\) 0 0
\(61\) 7.49837i 0.960068i −0.877250 0.480034i \(-0.840624\pi\)
0.877250 0.480034i \(-0.159376\pi\)
\(62\) 0 0
\(63\) 0.143131 0.0180329
\(64\) 0 0
\(65\) 4.78151i 0.593073i
\(66\) 0 0
\(67\) −4.96888 −0.607046 −0.303523 0.952824i \(-0.598163\pi\)
−0.303523 + 0.952824i \(0.598163\pi\)
\(68\) 0 0
\(69\) −4.65501 1.15365i −0.560397 0.138883i
\(70\) 0 0
\(71\) 6.77385i 0.803908i −0.915660 0.401954i \(-0.868331\pi\)
0.915660 0.401954i \(-0.131669\pi\)
\(72\) 0 0
\(73\) −8.04529 −0.941631 −0.470815 0.882232i \(-0.656040\pi\)
−0.470815 + 0.882232i \(0.656040\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) 0.687974 0.0784019
\(78\) 0 0
\(79\) 1.53132 0.172287 0.0861433 0.996283i \(-0.472546\pi\)
0.0861433 + 0.996283i \(0.472546\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −14.3973 −1.58031 −0.790157 0.612905i \(-0.790001\pi\)
−0.790157 + 0.612905i \(0.790001\pi\)
\(84\) 0 0
\(85\) 4.86743 0.527947
\(86\) 0 0
\(87\) 8.39933i 0.900503i
\(88\) 0 0
\(89\) 12.6001i 1.33561i 0.744337 + 0.667804i \(0.232766\pi\)
−0.744337 + 0.667804i \(0.767234\pi\)
\(90\) 0 0
\(91\) −0.684383 −0.0717429
\(92\) 0 0
\(93\) 2.55983 0.265442
\(94\) 0 0
\(95\) 0.379340i 0.0389195i
\(96\) 0 0
\(97\) 5.67311i 0.576017i −0.957628 0.288009i \(-0.907007\pi\)
0.957628 0.288009i \(-0.0929932\pi\)
\(98\) 0 0
\(99\) 4.80659 0.483081
\(100\) 0 0
\(101\) −17.3244 −1.72384 −0.861921 0.507043i \(-0.830739\pi\)
−0.861921 + 0.507043i \(0.830739\pi\)
\(102\) 0 0
\(103\) −12.2942 −1.21138 −0.605691 0.795700i \(-0.707103\pi\)
−0.605691 + 0.795700i \(0.707103\pi\)
\(104\) 0 0
\(105\) 0.143131 0.0139682
\(106\) 0 0
\(107\) −1.02774 −0.0993557 −0.0496779 0.998765i \(-0.515819\pi\)
−0.0496779 + 0.998765i \(0.515819\pi\)
\(108\) 0 0
\(109\) 16.9971i 1.62802i −0.580849 0.814011i \(-0.697279\pi\)
0.580849 0.814011i \(-0.302721\pi\)
\(110\) 0 0
\(111\) −5.56282 −0.527999
\(112\) 0 0
\(113\) 0.637571i 0.0599776i −0.999550 0.0299888i \(-0.990453\pi\)
0.999550 0.0299888i \(-0.00954716\pi\)
\(114\) 0 0
\(115\) −4.65501 1.15365i −0.434082 0.107578i
\(116\) 0 0
\(117\) −4.78151 −0.442050
\(118\) 0 0
\(119\) 0.696682i 0.0638648i
\(120\) 0 0
\(121\) 12.1033 1.10030
\(122\) 0 0
\(123\) 5.17202i 0.466345i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 3.74112i 0.331970i 0.986128 + 0.165985i \(0.0530804\pi\)
−0.986128 + 0.165985i \(0.946920\pi\)
\(128\) 0 0
\(129\) 1.03505i 0.0911308i
\(130\) 0 0
\(131\) 11.9550i 1.04451i −0.852790 0.522255i \(-0.825091\pi\)
0.852790 0.522255i \(-0.174909\pi\)
\(132\) 0 0
\(133\) 0.0542955 0.00470802
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 8.16863i 0.697893i −0.937143 0.348947i \(-0.886539\pi\)
0.937143 0.348947i \(-0.113461\pi\)
\(138\) 0 0
\(139\) 0.870322i 0.0738198i −0.999319 0.0369099i \(-0.988249\pi\)
0.999319 0.0369099i \(-0.0117514\pi\)
\(140\) 0 0
\(141\) −8.93486 −0.752451
\(142\) 0 0
\(143\) −22.9827 −1.92191
\(144\) 0 0
\(145\) 8.39933i 0.697526i
\(146\) 0 0
\(147\) 6.97951i 0.575661i
\(148\) 0 0
\(149\) 11.4050i 0.934337i 0.884168 + 0.467168i \(0.154726\pi\)
−0.884168 + 0.467168i \(0.845274\pi\)
\(150\) 0 0
\(151\) 5.66516i 0.461024i 0.973069 + 0.230512i \(0.0740401\pi\)
−0.973069 + 0.230512i \(0.925960\pi\)
\(152\) 0 0
\(153\) 4.86743i 0.393509i
\(154\) 0 0
\(155\) 2.55983 0.205610
\(156\) 0 0
\(157\) 5.94964i 0.474833i −0.971408 0.237417i \(-0.923699\pi\)
0.971408 0.237417i \(-0.0763006\pi\)
\(158\) 0 0
\(159\) −13.3413 −1.05803
\(160\) 0 0
\(161\) 0.165123 0.666278i 0.0130135 0.0525100i
\(162\) 0 0
\(163\) 12.0905i 0.946997i −0.880795 0.473499i \(-0.842991\pi\)
0.880795 0.473499i \(-0.157009\pi\)
\(164\) 0 0
\(165\) 4.80659 0.374193
\(166\) 0 0
\(167\) 14.1105i 1.09191i 0.837816 + 0.545953i \(0.183832\pi\)
−0.837816 + 0.545953i \(0.816168\pi\)
\(168\) 0 0
\(169\) 9.86279 0.758676
\(170\) 0 0
\(171\) 0.379340 0.0290089
\(172\) 0 0
\(173\) 3.68070 0.279838 0.139919 0.990163i \(-0.455316\pi\)
0.139919 + 0.990163i \(0.455316\pi\)
\(174\) 0 0
\(175\) 0.143131 0.0108197
\(176\) 0 0
\(177\) −6.57340 −0.494087
\(178\) 0 0
\(179\) 9.27393i 0.693166i −0.938019 0.346583i \(-0.887342\pi\)
0.938019 0.346583i \(-0.112658\pi\)
\(180\) 0 0
\(181\) 5.10162i 0.379200i −0.981861 0.189600i \(-0.939281\pi\)
0.981861 0.189600i \(-0.0607191\pi\)
\(182\) 0 0
\(183\) 7.49837 0.554295
\(184\) 0 0
\(185\) −5.56282 −0.408986
\(186\) 0 0
\(187\) 23.3958i 1.71087i
\(188\) 0 0
\(189\) 0.143131i 0.0104113i
\(190\) 0 0
\(191\) −20.1633 −1.45897 −0.729484 0.683997i \(-0.760240\pi\)
−0.729484 + 0.683997i \(0.760240\pi\)
\(192\) 0 0
\(193\) 1.06748 0.0768392 0.0384196 0.999262i \(-0.487768\pi\)
0.0384196 + 0.999262i \(0.487768\pi\)
\(194\) 0 0
\(195\) −4.78151 −0.342411
\(196\) 0 0
\(197\) 4.31277 0.307272 0.153636 0.988128i \(-0.450902\pi\)
0.153636 + 0.988128i \(0.450902\pi\)
\(198\) 0 0
\(199\) −9.35032 −0.662827 −0.331413 0.943486i \(-0.607525\pi\)
−0.331413 + 0.943486i \(0.607525\pi\)
\(200\) 0 0
\(201\) 4.96888i 0.350478i
\(202\) 0 0
\(203\) −1.20221 −0.0843784
\(204\) 0 0
\(205\) 5.17202i 0.361230i
\(206\) 0 0
\(207\) 1.15365 4.65501i 0.0801840 0.323545i
\(208\) 0 0
\(209\) 1.82333 0.126123
\(210\) 0 0
\(211\) 19.8715i 1.36801i 0.729478 + 0.684004i \(0.239763\pi\)
−0.729478 + 0.684004i \(0.760237\pi\)
\(212\) 0 0
\(213\) 6.77385 0.464136
\(214\) 0 0
\(215\) 1.03505i 0.0705896i
\(216\) 0 0
\(217\) 0.366392i 0.0248723i
\(218\) 0 0
\(219\) 8.04529i 0.543651i
\(220\) 0 0
\(221\) 23.2737i 1.56556i
\(222\) 0 0
\(223\) 13.4715i 0.902116i −0.892495 0.451058i \(-0.851047\pi\)
0.892495 0.451058i \(-0.148953\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −3.56731 −0.236771 −0.118385 0.992968i \(-0.537772\pi\)
−0.118385 + 0.992968i \(0.537772\pi\)
\(228\) 0 0
\(229\) 9.73343i 0.643203i −0.946875 0.321602i \(-0.895779\pi\)
0.946875 0.321602i \(-0.104221\pi\)
\(230\) 0 0
\(231\) 0.687974i 0.0452654i
\(232\) 0 0
\(233\) 19.7008 1.29065 0.645323 0.763910i \(-0.276723\pi\)
0.645323 + 0.763910i \(0.276723\pi\)
\(234\) 0 0
\(235\) −8.93486 −0.582846
\(236\) 0 0
\(237\) 1.53132i 0.0994697i
\(238\) 0 0
\(239\) 9.71151i 0.628185i −0.949392 0.314093i \(-0.898300\pi\)
0.949392 0.314093i \(-0.101700\pi\)
\(240\) 0 0
\(241\) 11.3349i 0.730144i −0.930979 0.365072i \(-0.881044\pi\)
0.930979 0.365072i \(-0.118956\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 6.97951i 0.445905i
\(246\) 0 0
\(247\) −1.81382 −0.115410
\(248\) 0 0
\(249\) 14.3973i 0.912394i
\(250\) 0 0
\(251\) −12.4895 −0.788329 −0.394164 0.919040i \(-0.628966\pi\)
−0.394164 + 0.919040i \(0.628966\pi\)
\(252\) 0 0
\(253\) 5.54511 22.3747i 0.348618 1.40669i
\(254\) 0 0
\(255\) 4.86743i 0.304810i
\(256\) 0 0
\(257\) −4.82467 −0.300954 −0.150477 0.988613i \(-0.548081\pi\)
−0.150477 + 0.988613i \(0.548081\pi\)
\(258\) 0 0
\(259\) 0.796214i 0.0494743i
\(260\) 0 0
\(261\) −8.39933 −0.519905
\(262\) 0 0
\(263\) −6.75865 −0.416756 −0.208378 0.978048i \(-0.566818\pi\)
−0.208378 + 0.978048i \(0.566818\pi\)
\(264\) 0 0
\(265\) −13.3413 −0.819547
\(266\) 0 0
\(267\) −12.6001 −0.771114
\(268\) 0 0
\(269\) 9.49855 0.579137 0.289568 0.957157i \(-0.406488\pi\)
0.289568 + 0.957157i \(0.406488\pi\)
\(270\) 0 0
\(271\) 17.6288i 1.07087i 0.844576 + 0.535435i \(0.179852\pi\)
−0.844576 + 0.535435i \(0.820148\pi\)
\(272\) 0 0
\(273\) 0.684383i 0.0414208i
\(274\) 0 0
\(275\) 4.80659 0.289848
\(276\) 0 0
\(277\) 8.47253 0.509065 0.254533 0.967064i \(-0.418078\pi\)
0.254533 + 0.967064i \(0.418078\pi\)
\(278\) 0 0
\(279\) 2.55983i 0.153253i
\(280\) 0 0
\(281\) 10.6546i 0.635603i 0.948157 + 0.317801i \(0.102945\pi\)
−0.948157 + 0.317801i \(0.897055\pi\)
\(282\) 0 0
\(283\) 9.84325 0.585120 0.292560 0.956247i \(-0.405493\pi\)
0.292560 + 0.956247i \(0.405493\pi\)
\(284\) 0 0
\(285\) 0.379340 0.0224702
\(286\) 0 0
\(287\) 0.740278 0.0436973
\(288\) 0 0
\(289\) −6.69189 −0.393641
\(290\) 0 0
\(291\) 5.67311 0.332564
\(292\) 0 0
\(293\) 29.3775i 1.71625i −0.513439 0.858126i \(-0.671629\pi\)
0.513439 0.858126i \(-0.328371\pi\)
\(294\) 0 0
\(295\) −6.57340 −0.382718
\(296\) 0 0
\(297\) 4.80659i 0.278907i
\(298\) 0 0
\(299\) −5.51617 + 22.2579i −0.319008 + 1.28721i
\(300\) 0 0
\(301\) 0.148148 0.00853909
\(302\) 0 0
\(303\) 17.3244i 0.995260i
\(304\) 0 0
\(305\) 7.49837 0.429355
\(306\) 0 0
\(307\) 15.6395i 0.892591i 0.894886 + 0.446296i \(0.147257\pi\)
−0.894886 + 0.446296i \(0.852743\pi\)
\(308\) 0 0
\(309\) 12.2942i 0.699391i
\(310\) 0 0
\(311\) 16.7393i 0.949201i 0.880201 + 0.474600i \(0.157407\pi\)
−0.880201 + 0.474600i \(0.842593\pi\)
\(312\) 0 0
\(313\) 17.0663i 0.964646i 0.875993 + 0.482323i \(0.160207\pi\)
−0.875993 + 0.482323i \(0.839793\pi\)
\(314\) 0 0
\(315\) 0.143131i 0.00806454i
\(316\) 0 0
\(317\) −29.6407 −1.66479 −0.832394 0.554184i \(-0.813030\pi\)
−0.832394 + 0.554184i \(0.813030\pi\)
\(318\) 0 0
\(319\) −40.3721 −2.26041
\(320\) 0 0
\(321\) 1.02774i 0.0573630i
\(322\) 0 0
\(323\) 1.84641i 0.102737i
\(324\) 0 0
\(325\) −4.78151 −0.265230
\(326\) 0 0
\(327\) 16.9971 0.939939
\(328\) 0 0
\(329\) 1.27886i 0.0705058i
\(330\) 0 0
\(331\) 4.03408i 0.221733i −0.993835 0.110866i \(-0.964637\pi\)
0.993835 0.110866i \(-0.0353626\pi\)
\(332\) 0 0
\(333\) 5.56282i 0.304840i
\(334\) 0 0
\(335\) 4.96888i 0.271479i
\(336\) 0 0
\(337\) 1.54962i 0.0844132i 0.999109 + 0.0422066i \(0.0134388\pi\)
−0.999109 + 0.0422066i \(0.986561\pi\)
\(338\) 0 0
\(339\) 0.637571 0.0346281
\(340\) 0 0
\(341\) 12.3040i 0.666302i
\(342\) 0 0
\(343\) 2.00091 0.108039
\(344\) 0 0
\(345\) 1.15365 4.65501i 0.0621103 0.250617i
\(346\) 0 0
\(347\) 9.87984i 0.530377i −0.964197 0.265189i \(-0.914566\pi\)
0.964197 0.265189i \(-0.0854343\pi\)
\(348\) 0 0
\(349\) 6.41716 0.343503 0.171751 0.985140i \(-0.445057\pi\)
0.171751 + 0.985140i \(0.445057\pi\)
\(350\) 0 0
\(351\) 4.78151i 0.255218i
\(352\) 0 0
\(353\) −6.23647 −0.331934 −0.165967 0.986131i \(-0.553074\pi\)
−0.165967 + 0.986131i \(0.553074\pi\)
\(354\) 0 0
\(355\) 6.77385 0.359519
\(356\) 0 0
\(357\) −0.696682 −0.0368723
\(358\) 0 0
\(359\) −4.19870 −0.221599 −0.110799 0.993843i \(-0.535341\pi\)
−0.110799 + 0.993843i \(0.535341\pi\)
\(360\) 0 0
\(361\) −18.8561 −0.992426
\(362\) 0 0
\(363\) 12.1033i 0.635260i
\(364\) 0 0
\(365\) 8.04529i 0.421110i
\(366\) 0 0
\(367\) 5.21796 0.272375 0.136188 0.990683i \(-0.456515\pi\)
0.136188 + 0.990683i \(0.456515\pi\)
\(368\) 0 0
\(369\) 5.17202 0.269245
\(370\) 0 0
\(371\) 1.90955i 0.0991391i
\(372\) 0 0
\(373\) 24.8539i 1.28689i 0.765493 + 0.643444i \(0.222495\pi\)
−0.765493 + 0.643444i \(0.777505\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 40.1614 2.06842
\(378\) 0 0
\(379\) 14.2322 0.731058 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(380\) 0 0
\(381\) −3.74112 −0.191663
\(382\) 0 0
\(383\) 23.6762 1.20980 0.604898 0.796303i \(-0.293214\pi\)
0.604898 + 0.796303i \(0.293214\pi\)
\(384\) 0 0
\(385\) 0.687974i 0.0350624i
\(386\) 0 0
\(387\) 1.03505 0.0526144
\(388\) 0 0
\(389\) 6.73328i 0.341391i 0.985324 + 0.170695i \(0.0546014\pi\)
−0.985324 + 0.170695i \(0.945399\pi\)
\(390\) 0 0
\(391\) 22.6579 + 5.61530i 1.14586 + 0.283978i
\(392\) 0 0
\(393\) 11.9550 0.603048
\(394\) 0 0
\(395\) 1.53132i 0.0770489i
\(396\) 0 0
\(397\) 9.27378 0.465438 0.232719 0.972544i \(-0.425238\pi\)
0.232719 + 0.972544i \(0.425238\pi\)
\(398\) 0 0
\(399\) 0.0542955i 0.00271817i
\(400\) 0 0
\(401\) 38.6681i 1.93099i 0.260421 + 0.965495i \(0.416139\pi\)
−0.260421 + 0.965495i \(0.583861\pi\)
\(402\) 0 0
\(403\) 12.2398i 0.609709i
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 26.7382i 1.32536i
\(408\) 0 0
\(409\) −23.1274 −1.14358 −0.571789 0.820401i \(-0.693750\pi\)
−0.571789 + 0.820401i \(0.693750\pi\)
\(410\) 0 0
\(411\) 8.16863 0.402929
\(412\) 0 0
\(413\) 0.940859i 0.0462967i
\(414\) 0 0
\(415\) 14.3973i 0.706738i
\(416\) 0 0
\(417\) 0.870322 0.0426199
\(418\) 0 0
\(419\) −2.95569 −0.144395 −0.0721974 0.997390i \(-0.523001\pi\)
−0.0721974 + 0.997390i \(0.523001\pi\)
\(420\) 0 0
\(421\) 4.13502i 0.201528i −0.994910 0.100764i \(-0.967871\pi\)
0.994910 0.100764i \(-0.0321288\pi\)
\(422\) 0 0
\(423\) 8.93486i 0.434428i
\(424\) 0 0
\(425\) 4.86743i 0.236105i
\(426\) 0 0
\(427\) 1.07325i 0.0519383i
\(428\) 0 0
\(429\) 22.9827i 1.10962i
\(430\) 0 0
\(431\) −8.46802 −0.407890 −0.203945 0.978982i \(-0.565376\pi\)
−0.203945 + 0.978982i \(0.565376\pi\)
\(432\) 0 0
\(433\) 17.3572i 0.834132i −0.908876 0.417066i \(-0.863058\pi\)
0.908876 0.417066i \(-0.136942\pi\)
\(434\) 0 0
\(435\) −8.39933 −0.402717
\(436\) 0 0
\(437\) 0.437625 1.76583i 0.0209344 0.0844712i
\(438\) 0 0
\(439\) 24.8336i 1.18524i 0.805481 + 0.592622i \(0.201907\pi\)
−0.805481 + 0.592622i \(0.798093\pi\)
\(440\) 0 0
\(441\) 6.97951 0.332358
\(442\) 0 0
\(443\) 26.8894i 1.27756i −0.769391 0.638778i \(-0.779440\pi\)
0.769391 0.638778i \(-0.220560\pi\)
\(444\) 0 0
\(445\) −12.6001 −0.597302
\(446\) 0 0
\(447\) −11.4050 −0.539440
\(448\) 0 0
\(449\) −33.4193 −1.57716 −0.788578 0.614935i \(-0.789182\pi\)
−0.788578 + 0.614935i \(0.789182\pi\)
\(450\) 0 0
\(451\) 24.8598 1.17060
\(452\) 0 0
\(453\) −5.66516 −0.266172
\(454\) 0 0
\(455\) 0.684383i 0.0320844i
\(456\) 0 0
\(457\) 7.03968i 0.329302i −0.986352 0.164651i \(-0.947350\pi\)
0.986352 0.164651i \(-0.0526498\pi\)
\(458\) 0 0
\(459\) −4.86743 −0.227192
\(460\) 0 0
\(461\) −32.0115 −1.49092 −0.745462 0.666548i \(-0.767771\pi\)
−0.745462 + 0.666548i \(0.767771\pi\)
\(462\) 0 0
\(463\) 16.1154i 0.748945i 0.927238 + 0.374472i \(0.122176\pi\)
−0.927238 + 0.374472i \(0.877824\pi\)
\(464\) 0 0
\(465\) 2.55983i 0.118709i
\(466\) 0 0
\(467\) −40.8400 −1.88985 −0.944926 0.327285i \(-0.893866\pi\)
−0.944926 + 0.327285i \(0.893866\pi\)
\(468\) 0 0
\(469\) 0.711203 0.0328403
\(470\) 0 0
\(471\) 5.94964 0.274145
\(472\) 0 0
\(473\) 4.97505 0.228753
\(474\) 0 0
\(475\) 0.379340 0.0174053
\(476\) 0 0
\(477\) 13.3413i 0.610855i
\(478\) 0 0
\(479\) −12.5812 −0.574848 −0.287424 0.957803i \(-0.592799\pi\)
−0.287424 + 0.957803i \(0.592799\pi\)
\(480\) 0 0
\(481\) 26.5986i 1.21279i
\(482\) 0 0
\(483\) 0.666278 + 0.165123i 0.0303167 + 0.00751336i
\(484\) 0 0
\(485\) 5.67311 0.257603
\(486\) 0 0
\(487\) 31.0193i 1.40562i −0.711378 0.702810i \(-0.751928\pi\)
0.711378 0.702810i \(-0.248072\pi\)
\(488\) 0 0
\(489\) 12.0905 0.546749
\(490\) 0 0
\(491\) 35.8601i 1.61835i 0.587571 + 0.809173i \(0.300084\pi\)
−0.587571 + 0.809173i \(0.699916\pi\)
\(492\) 0 0
\(493\) 40.8832i 1.84129i
\(494\) 0 0
\(495\) 4.80659i 0.216040i
\(496\) 0 0
\(497\) 0.969550i 0.0434903i
\(498\) 0 0
\(499\) 10.0149i 0.448330i −0.974551 0.224165i \(-0.928035\pi\)
0.974551 0.224165i \(-0.0719654\pi\)
\(500\) 0 0
\(501\) −14.1105 −0.630413
\(502\) 0 0
\(503\) −38.5120 −1.71716 −0.858582 0.512676i \(-0.828654\pi\)
−0.858582 + 0.512676i \(0.828654\pi\)
\(504\) 0 0
\(505\) 17.3244i 0.770925i
\(506\) 0 0
\(507\) 9.86279i 0.438022i
\(508\) 0 0
\(509\) 31.4747 1.39509 0.697546 0.716540i \(-0.254275\pi\)
0.697546 + 0.716540i \(0.254275\pi\)
\(510\) 0 0
\(511\) 1.15153 0.0509409
\(512\) 0 0
\(513\) 0.379340i 0.0167483i
\(514\) 0 0
\(515\) 12.2942i 0.541746i
\(516\) 0 0
\(517\) 42.9462i 1.88877i
\(518\) 0 0
\(519\) 3.68070i 0.161565i
\(520\) 0 0
\(521\) 21.5620i 0.944647i 0.881425 + 0.472324i \(0.156585\pi\)
−0.881425 + 0.472324i \(0.843415\pi\)
\(522\) 0 0
\(523\) 40.7270 1.78087 0.890434 0.455112i \(-0.150401\pi\)
0.890434 + 0.455112i \(0.150401\pi\)
\(524\) 0 0
\(525\) 0.143131i 0.00624677i
\(526\) 0 0
\(527\) −12.4598 −0.542757
\(528\) 0 0
\(529\) −20.3382 10.7405i −0.884269 0.466977i
\(530\) 0 0
\(531\) 6.57340i 0.285261i
\(532\) 0 0
\(533\) −24.7300 −1.07118
\(534\) 0 0
\(535\) 1.02774i 0.0444332i
\(536\) 0 0
\(537\) 9.27393 0.400200
\(538\) 0 0
\(539\) 33.5477 1.44500
\(540\) 0 0
\(541\) −38.8931 −1.67215 −0.836073 0.548618i \(-0.815154\pi\)
−0.836073 + 0.548618i \(0.815154\pi\)
\(542\) 0 0
\(543\) 5.10162 0.218931
\(544\) 0 0
\(545\) 16.9971 0.728074
\(546\) 0 0
\(547\) 0.658772i 0.0281671i 0.999901 + 0.0140835i \(0.00448308\pi\)
−0.999901 + 0.0140835i \(0.995517\pi\)
\(548\) 0 0
\(549\) 7.49837i 0.320023i
\(550\) 0 0
\(551\) −3.18620 −0.135737
\(552\) 0 0
\(553\) −0.219179 −0.00932046
\(554\) 0 0
\(555\) 5.56282i 0.236128i
\(556\) 0 0
\(557\) 4.89839i 0.207551i −0.994601 0.103776i \(-0.966908\pi\)
0.994601 0.103776i \(-0.0330924\pi\)
\(558\) 0 0
\(559\) −4.94908 −0.209324
\(560\) 0 0
\(561\) −23.3958 −0.987770
\(562\) 0 0
\(563\) −8.22969 −0.346840 −0.173420 0.984848i \(-0.555482\pi\)
−0.173420 + 0.984848i \(0.555482\pi\)
\(564\) 0 0
\(565\) 0.637571 0.0268228
\(566\) 0 0
\(567\) −0.143131 −0.00601095
\(568\) 0 0
\(569\) 4.84217i 0.202994i 0.994836 + 0.101497i \(0.0323632\pi\)
−0.994836 + 0.101497i \(0.967637\pi\)
\(570\) 0 0
\(571\) −18.6944 −0.782335 −0.391167 0.920320i \(-0.627929\pi\)
−0.391167 + 0.920320i \(0.627929\pi\)
\(572\) 0 0
\(573\) 20.1633i 0.842336i
\(574\) 0 0
\(575\) 1.15365 4.65501i 0.0481104 0.194127i
\(576\) 0 0
\(577\) 37.9160 1.57847 0.789233 0.614094i \(-0.210479\pi\)
0.789233 + 0.614094i \(0.210479\pi\)
\(578\) 0 0
\(579\) 1.06748i 0.0443631i
\(580\) 0 0
\(581\) 2.06071 0.0854927
\(582\) 0 0
\(583\) 64.1260i 2.65583i
\(584\) 0 0
\(585\) 4.78151i 0.197691i
\(586\) 0 0
\(587\) 4.52386i 0.186720i 0.995632 + 0.0933599i \(0.0297607\pi\)
−0.995632 + 0.0933599i \(0.970239\pi\)
\(588\) 0 0
\(589\) 0.971046i 0.0400112i
\(590\) 0 0
\(591\) 4.31277i 0.177404i
\(592\) 0 0
\(593\) 21.4620 0.881338 0.440669 0.897670i \(-0.354741\pi\)
0.440669 + 0.897670i \(0.354741\pi\)
\(594\) 0 0
\(595\) −0.696682 −0.0285612
\(596\) 0 0
\(597\) 9.35032i 0.382683i
\(598\) 0 0
\(599\) 43.6864i 1.78498i 0.451069 + 0.892489i \(0.351043\pi\)
−0.451069 + 0.892489i \(0.648957\pi\)
\(600\) 0 0
\(601\) −43.6463 −1.78037 −0.890185 0.455599i \(-0.849425\pi\)
−0.890185 + 0.455599i \(0.849425\pi\)
\(602\) 0 0
\(603\) 4.96888 0.202349
\(604\) 0 0
\(605\) 12.1033i 0.492070i
\(606\) 0 0
\(607\) 13.8775i 0.563269i −0.959522 0.281635i \(-0.909123\pi\)
0.959522 0.281635i \(-0.0908766\pi\)
\(608\) 0 0
\(609\) 1.20221i 0.0487159i
\(610\) 0 0
\(611\) 42.7221i 1.72835i
\(612\) 0 0
\(613\) 18.7574i 0.757606i −0.925477 0.378803i \(-0.876336\pi\)
0.925477 0.378803i \(-0.123664\pi\)
\(614\) 0 0
\(615\) 5.17202 0.208556
\(616\) 0 0
\(617\) 28.7105i 1.15584i 0.816092 + 0.577922i \(0.196136\pi\)
−0.816092 + 0.577922i \(0.803864\pi\)
\(618\) 0 0
\(619\) −0.458137 −0.0184141 −0.00920703 0.999958i \(-0.502931\pi\)
−0.00920703 + 0.999958i \(0.502931\pi\)
\(620\) 0 0
\(621\) 4.65501 + 1.15365i 0.186799 + 0.0462943i
\(622\) 0 0
\(623\) 1.80347i 0.0722545i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 1.82333i 0.0728169i
\(628\) 0 0
\(629\) 27.0766 1.07962
\(630\) 0 0
\(631\) −37.7755 −1.50382 −0.751910 0.659265i \(-0.770867\pi\)
−0.751910 + 0.659265i \(0.770867\pi\)
\(632\) 0 0
\(633\) −19.8715 −0.789820
\(634\) 0 0
\(635\) −3.74112 −0.148462
\(636\) 0 0
\(637\) −33.3726 −1.32227
\(638\) 0 0
\(639\) 6.77385i 0.267969i
\(640\) 0 0
\(641\) 29.5885i 1.16868i 0.811510 + 0.584338i \(0.198646\pi\)
−0.811510 + 0.584338i \(0.801354\pi\)
\(642\) 0 0
\(643\) −47.0949 −1.85724 −0.928620 0.371032i \(-0.879004\pi\)
−0.928620 + 0.371032i \(0.879004\pi\)
\(644\) 0 0
\(645\) 1.03505 0.0407549
\(646\) 0 0
\(647\) 26.6440i 1.04748i 0.851877 + 0.523741i \(0.175464\pi\)
−0.851877 + 0.523741i \(0.824536\pi\)
\(648\) 0 0
\(649\) 31.5956i 1.24024i
\(650\) 0 0
\(651\) −0.366392 −0.0143600
\(652\) 0 0
\(653\) 32.1606 1.25854 0.629272 0.777185i \(-0.283353\pi\)
0.629272 + 0.777185i \(0.283353\pi\)
\(654\) 0 0
\(655\) 11.9550 0.467119
\(656\) 0 0
\(657\) 8.04529 0.313877
\(658\) 0 0
\(659\) 1.56508 0.0609670 0.0304835 0.999535i \(-0.490295\pi\)
0.0304835 + 0.999535i \(0.490295\pi\)
\(660\) 0 0
\(661\) 11.9297i 0.464010i −0.972715 0.232005i \(-0.925471\pi\)
0.972715 0.232005i \(-0.0745285\pi\)
\(662\) 0 0
\(663\) 23.2737 0.903874
\(664\) 0 0
\(665\) 0.0542955i 0.00210549i
\(666\) 0 0
\(667\) −9.68987 + 39.0989i −0.375193 + 1.51392i
\(668\) 0 0
\(669\) 13.4715 0.520837
\(670\) 0 0
\(671\) 36.0416i 1.39137i
\(672\) 0 0
\(673\) 27.9834 1.07868 0.539340 0.842088i \(-0.318674\pi\)
0.539340 + 0.842088i \(0.318674\pi\)
\(674\) 0 0
\(675\) 1.00000i 0.0384900i
\(676\) 0 0
\(677\) 49.5307i 1.90362i 0.306691 + 0.951809i \(0.400778\pi\)
−0.306691 + 0.951809i \(0.599222\pi\)
\(678\) 0 0
\(679\) 0.812001i 0.0311617i
\(680\) 0 0
\(681\) 3.56731i 0.136700i
\(682\) 0 0
\(683\) 18.2648i 0.698884i −0.936958 0.349442i \(-0.886371\pi\)
0.936958 0.349442i \(-0.113629\pi\)
\(684\) 0 0
\(685\) 8.16863 0.312107
\(686\) 0 0
\(687\) 9.73343 0.371354
\(688\) 0 0
\(689\) 63.7913i 2.43026i
\(690\) 0 0
\(691\) 24.6191i 0.936554i 0.883582 + 0.468277i \(0.155125\pi\)
−0.883582 + 0.468277i \(0.844875\pi\)
\(692\) 0 0
\(693\) −0.687974 −0.0261340
\(694\) 0 0
\(695\) 0.870322 0.0330132
\(696\) 0 0
\(697\) 25.1745i 0.953550i
\(698\) 0 0
\(699\) 19.7008i 0.745154i
\(700\) 0 0
\(701\) 35.5895i 1.34420i −0.740461 0.672099i \(-0.765393\pi\)
0.740461 0.672099i \(-0.234607\pi\)
\(702\) 0 0
\(703\) 2.11020i 0.0795877i
\(704\) 0 0
\(705\) 8.93486i 0.336506i
\(706\) 0 0
\(707\) 2.47966 0.0932574
\(708\) 0 0
\(709\) 8.00880i 0.300777i −0.988627 0.150388i \(-0.951948\pi\)
0.988627 0.150388i \(-0.0480524\pi\)
\(710\) 0 0
\(711\) −1.53132 −0.0574289
\(712\) 0 0
\(713\) 11.9160 + 2.95314i 0.446258 + 0.110596i
\(714\) 0 0
\(715\) 22.9827i 0.859506i
\(716\) 0 0
\(717\) 9.71151 0.362683
\(718\) 0 0
\(719\) 40.1847i 1.49864i −0.662210 0.749318i \(-0.730381\pi\)
0.662210 0.749318i \(-0.269619\pi\)
\(720\) 0 0
\(721\) 1.75968 0.0655340
\(722\) 0 0
\(723\) 11.3349 0.421549
\(724\) 0 0
\(725\) −8.39933 −0.311943
\(726\) 0 0
\(727\) 46.1258 1.71071 0.855355 0.518042i \(-0.173339\pi\)
0.855355 + 0.518042i \(0.173339\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 5.03802i 0.186338i
\(732\) 0 0
\(733\) 11.4344i 0.422340i 0.977449 + 0.211170i \(0.0677274\pi\)
−0.977449 + 0.211170i \(0.932273\pi\)
\(734\) 0 0
\(735\) 6.97951 0.257443
\(736\) 0 0
\(737\) 23.8834 0.879756
\(738\) 0 0
\(739\) 10.7005i 0.393625i 0.980441 + 0.196813i \(0.0630591\pi\)
−0.980441 + 0.196813i \(0.936941\pi\)
\(740\) 0 0
\(741\) 1.81382i 0.0666322i
\(742\) 0 0
\(743\) −44.7313 −1.64103 −0.820515 0.571624i \(-0.806313\pi\)
−0.820515 + 0.571624i \(0.806313\pi\)
\(744\) 0 0
\(745\) −11.4050 −0.417848
\(746\) 0 0
\(747\) 14.3973 0.526771
\(748\) 0 0
\(749\) 0.147102 0.00537500
\(750\) 0 0
\(751\) 15.5462 0.567289 0.283644 0.958930i \(-0.408456\pi\)
0.283644 + 0.958930i \(0.408456\pi\)
\(752\) 0 0
\(753\) 12.4895i 0.455142i
\(754\) 0 0
\(755\) −5.66516 −0.206176
\(756\) 0 0
\(757\) 15.3689i 0.558594i −0.960205 0.279297i \(-0.909899\pi\)
0.960205 0.279297i \(-0.0901014\pi\)
\(758\) 0 0
\(759\) 22.3747 + 5.54511i 0.812151 + 0.201275i
\(760\) 0 0
\(761\) −26.4309 −0.958121 −0.479060 0.877782i \(-0.659023\pi\)
−0.479060 + 0.877782i \(0.659023\pi\)
\(762\) 0 0
\(763\) 2.43281i 0.0880737i
\(764\) 0 0
\(765\) −4.86743 −0.175982
\(766\) 0 0
\(767\) 31.4307i 1.13490i
\(768\) 0 0
\(769\) 51.7442i 1.86594i 0.359948 + 0.932972i \(0.382794\pi\)
−0.359948 + 0.932972i \(0.617206\pi\)
\(770\) 0 0
\(771\) 4.82467i 0.173756i
\(772\) 0 0
\(773\) 11.3165i 0.407026i 0.979072 + 0.203513i \(0.0652360\pi\)
−0.979072 + 0.203513i \(0.934764\pi\)
\(774\) 0 0
\(775\) 2.55983i 0.0919517i
\(776\) 0 0
\(777\) 0.796214 0.0285640
\(778\) 0 0
\(779\) 1.96195 0.0702943
\(780\) 0 0
\(781\) 32.5591i 1.16506i
\(782\) 0 0
\(783\) 8.39933i 0.300168i
\(784\) 0 0
\(785\) 5.94964 0.212352
\(786\) 0 0
\(787\) 20.8240 0.742296 0.371148 0.928574i \(-0.378964\pi\)
0.371148 + 0.928574i \(0.378964\pi\)
\(788\) 0 0
\(789\) 6.75865i 0.240614i
\(790\) 0 0
\(791\) 0.0912564i 0.00324470i
\(792\) 0 0
\(793\) 35.8535i 1.27319i
\(794\) 0 0
\(795\) 13.3413i 0.473166i
\(796\) 0 0
\(797\) 34.5644i 1.22433i 0.790729 + 0.612166i \(0.209702\pi\)
−0.790729 + 0.612166i \(0.790298\pi\)
\(798\) 0 0
\(799\) 43.4898 1.53856
\(800\) 0 0
\(801\) 12.6001i 0.445203i
\(802\) 0 0
\(803\) 38.6704 1.36465
\(804\) 0 0
\(805\) 0.666278 + 0.165123i 0.0234832 + 0.00581983i
\(806\) 0 0