Properties

Label 5520.2.be.b.1471.10
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 1471.10
Root \(-0.373815 + 0.373815i\) of defining polynomial
Character \(\chi\) \(=\) 5520.1471
Dual form 5520.2.be.b.1471.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +1.00000i q^{5} -1.61153 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +1.00000i q^{5} -1.61153 q^{7} -1.00000 q^{9} -1.15035 q^{11} +7.10687 q^{13} -1.00000 q^{15} -1.79275i q^{17} -5.93632 q^{19} -1.61153i q^{21} +(-4.40428 - 1.89798i) q^{23} -1.00000 q^{25} -1.00000i q^{27} -0.0365806 q^{29} +4.39344i q^{31} -1.15035i q^{33} -1.61153i q^{35} -10.1186i q^{37} +7.10687i q^{39} +5.43873 q^{41} +9.60250 q^{43} -1.00000i q^{45} -9.16374i q^{47} -4.40297 q^{49} +1.79275 q^{51} -1.95476i q^{53} -1.15035i q^{55} -5.93632i q^{57} -5.29905i q^{59} +9.53475i q^{61} +1.61153 q^{63} +7.10687i q^{65} +2.33657 q^{67} +(1.89798 - 4.40428i) q^{69} -11.3842i q^{71} -1.61161 q^{73} -1.00000i q^{75} +1.85383 q^{77} +5.33892 q^{79} +1.00000 q^{81} +0.722088 q^{83} +1.79275 q^{85} -0.0365806i q^{87} +2.09024i q^{89} -11.4529 q^{91} -4.39344 q^{93} -5.93632i q^{95} -1.93236i q^{97} +1.15035 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 8q^{7} - 16q^{9} + O(q^{10}) \) \( 16q + 8q^{7} - 16q^{9} - 8q^{11} + 8q^{13} - 16q^{15} - 12q^{23} - 16q^{25} - 4q^{29} + 4q^{41} + 20q^{49} + 4q^{51} - 8q^{63} + 16q^{67} + 40q^{73} + 24q^{77} - 32q^{79} + 16q^{81} + 4q^{85} + 48q^{91} + 8q^{99} + O(q^{100}) \)

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
\(7\) −1.61153 −0.609101 −0.304551 0.952496i \(-0.598506\pi\)
−0.304551 + 0.952496i \(0.598506\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.15035 −0.346844 −0.173422 0.984848i \(-0.555482\pi\)
−0.173422 + 0.984848i \(0.555482\pi\)
\(12\) 0 0
\(13\) 7.10687 1.97109 0.985545 0.169414i \(-0.0541876\pi\)
0.985545 + 0.169414i \(0.0541876\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 1.79275i 0.434805i −0.976082 0.217403i \(-0.930242\pi\)
0.976082 0.217403i \(-0.0697584\pi\)
\(18\) 0 0
\(19\) −5.93632 −1.36188 −0.680942 0.732337i \(-0.738429\pi\)
−0.680942 + 0.732337i \(0.738429\pi\)
\(20\) 0 0
\(21\) 1.61153i 0.351665i
\(22\) 0 0
\(23\) −4.40428 1.89798i −0.918356 0.395756i
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −0.0365806 −0.00679285 −0.00339643 0.999994i \(-0.501081\pi\)
−0.00339643 + 0.999994i \(0.501081\pi\)
\(30\) 0 0
\(31\) 4.39344i 0.789085i 0.918878 + 0.394542i \(0.129097\pi\)
−0.918878 + 0.394542i \(0.870903\pi\)
\(32\) 0 0
\(33\) 1.15035i 0.200251i
\(34\) 0 0
\(35\) 1.61153i 0.272398i
\(36\) 0 0
\(37\) 10.1186i 1.66349i −0.555158 0.831745i \(-0.687342\pi\)
0.555158 0.831745i \(-0.312658\pi\)
\(38\) 0 0
\(39\) 7.10687i 1.13801i
\(40\) 0 0
\(41\) 5.43873 0.849387 0.424693 0.905337i \(-0.360382\pi\)
0.424693 + 0.905337i \(0.360382\pi\)
\(42\) 0 0
\(43\) 9.60250 1.46437 0.732183 0.681107i \(-0.238501\pi\)
0.732183 + 0.681107i \(0.238501\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) 9.16374i 1.33667i −0.743861 0.668334i \(-0.767008\pi\)
0.743861 0.668334i \(-0.232992\pi\)
\(48\) 0 0
\(49\) −4.40297 −0.628995
\(50\) 0 0
\(51\) 1.79275 0.251035
\(52\) 0 0
\(53\) 1.95476i 0.268507i −0.990947 0.134253i \(-0.957136\pi\)
0.990947 0.134253i \(-0.0428636\pi\)
\(54\) 0 0
\(55\) 1.15035i 0.155113i
\(56\) 0 0
\(57\) 5.93632i 0.786284i
\(58\) 0 0
\(59\) 5.29905i 0.689877i −0.938625 0.344938i \(-0.887900\pi\)
0.938625 0.344938i \(-0.112100\pi\)
\(60\) 0 0
\(61\) 9.53475i 1.22080i 0.792093 + 0.610400i \(0.208991\pi\)
−0.792093 + 0.610400i \(0.791009\pi\)
\(62\) 0 0
\(63\) 1.61153 0.203034
\(64\) 0 0
\(65\) 7.10687i 0.881498i
\(66\) 0 0
\(67\) 2.33657 0.285457 0.142729 0.989762i \(-0.454412\pi\)
0.142729 + 0.989762i \(0.454412\pi\)
\(68\) 0 0
\(69\) 1.89798 4.40428i 0.228490 0.530213i
\(70\) 0 0
\(71\) 11.3842i 1.35106i −0.737334 0.675528i \(-0.763916\pi\)
0.737334 0.675528i \(-0.236084\pi\)
\(72\) 0 0
\(73\) −1.61161 −0.188624 −0.0943122 0.995543i \(-0.530065\pi\)
−0.0943122 + 0.995543i \(0.530065\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) 1.85383 0.211263
\(78\) 0 0
\(79\) 5.33892 0.600675 0.300338 0.953833i \(-0.402901\pi\)
0.300338 + 0.953833i \(0.402901\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0.722088 0.0792594 0.0396297 0.999214i \(-0.487382\pi\)
0.0396297 + 0.999214i \(0.487382\pi\)
\(84\) 0 0
\(85\) 1.79275 0.194451
\(86\) 0 0
\(87\) 0.0365806i 0.00392186i
\(88\) 0 0
\(89\) 2.09024i 0.221565i 0.993845 + 0.110782i \(0.0353356\pi\)
−0.993845 + 0.110782i \(0.964664\pi\)
\(90\) 0 0
\(91\) −11.4529 −1.20059
\(92\) 0 0
\(93\) −4.39344 −0.455578
\(94\) 0 0
\(95\) 5.93632i 0.609053i
\(96\) 0 0
\(97\) 1.93236i 0.196201i −0.995177 0.0981005i \(-0.968723\pi\)
0.995177 0.0981005i \(-0.0312767\pi\)
\(98\) 0 0
\(99\) 1.15035 0.115615
\(100\) 0 0
\(101\) 15.0107 1.49362 0.746808 0.665040i \(-0.231585\pi\)
0.746808 + 0.665040i \(0.231585\pi\)
\(102\) 0 0
\(103\) −3.04742 −0.300271 −0.150136 0.988665i \(-0.547971\pi\)
−0.150136 + 0.988665i \(0.547971\pi\)
\(104\) 0 0
\(105\) 1.61153 0.157269
\(106\) 0 0
\(107\) −12.6032 −1.21840 −0.609199 0.793017i \(-0.708509\pi\)
−0.609199 + 0.793017i \(0.708509\pi\)
\(108\) 0 0
\(109\) 15.0300i 1.43961i −0.694174 0.719807i \(-0.744230\pi\)
0.694174 0.719807i \(-0.255770\pi\)
\(110\) 0 0
\(111\) 10.1186 0.960417
\(112\) 0 0
\(113\) 8.39445i 0.789684i 0.918749 + 0.394842i \(0.129201\pi\)
−0.918749 + 0.394842i \(0.870799\pi\)
\(114\) 0 0
\(115\) 1.89798 4.40428i 0.176988 0.410701i
\(116\) 0 0
\(117\) −7.10687 −0.657030
\(118\) 0 0
\(119\) 2.88907i 0.264840i
\(120\) 0 0
\(121\) −9.67669 −0.879699
\(122\) 0 0
\(123\) 5.43873i 0.490394i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 18.1343i 1.60916i 0.593845 + 0.804580i \(0.297609\pi\)
−0.593845 + 0.804580i \(0.702391\pi\)
\(128\) 0 0
\(129\) 9.60250i 0.845453i
\(130\) 0 0
\(131\) 1.77034i 0.154676i −0.997005 0.0773378i \(-0.975358\pi\)
0.997005 0.0773378i \(-0.0246420\pi\)
\(132\) 0 0
\(133\) 9.56656 0.829526
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 17.7574i 1.51712i −0.651603 0.758560i \(-0.725903\pi\)
0.651603 0.758560i \(-0.274097\pi\)
\(138\) 0 0
\(139\) 13.6548i 1.15819i −0.815261 0.579094i \(-0.803406\pi\)
0.815261 0.579094i \(-0.196594\pi\)
\(140\) 0 0
\(141\) 9.16374 0.771726
\(142\) 0 0
\(143\) −8.17540 −0.683661
\(144\) 0 0
\(145\) 0.0365806i 0.00303786i
\(146\) 0 0
\(147\) 4.40297i 0.363151i
\(148\) 0 0
\(149\) 9.38262i 0.768654i −0.923197 0.384327i \(-0.874434\pi\)
0.923197 0.384327i \(-0.125566\pi\)
\(150\) 0 0
\(151\) 13.1263i 1.06820i 0.845421 + 0.534100i \(0.179349\pi\)
−0.845421 + 0.534100i \(0.820651\pi\)
\(152\) 0 0
\(153\) 1.79275i 0.144935i
\(154\) 0 0
\(155\) −4.39344 −0.352889
\(156\) 0 0
\(157\) 2.75711i 0.220041i 0.993929 + 0.110021i \(0.0350917\pi\)
−0.993929 + 0.110021i \(0.964908\pi\)
\(158\) 0 0
\(159\) 1.95476 0.155022
\(160\) 0 0
\(161\) 7.09763 + 3.05865i 0.559372 + 0.241056i
\(162\) 0 0
\(163\) 7.13282i 0.558686i 0.960191 + 0.279343i \(0.0901166\pi\)
−0.960191 + 0.279343i \(0.909883\pi\)
\(164\) 0 0
\(165\) 1.15035 0.0895548
\(166\) 0 0
\(167\) 0.264942i 0.0205018i 0.999947 + 0.0102509i \(0.00326303\pi\)
−0.999947 + 0.0102509i \(0.996737\pi\)
\(168\) 0 0
\(169\) 37.5075 2.88520
\(170\) 0 0
\(171\) 5.93632 0.453961
\(172\) 0 0
\(173\) −9.93924 −0.755666 −0.377833 0.925874i \(-0.623331\pi\)
−0.377833 + 0.925874i \(0.623331\pi\)
\(174\) 0 0
\(175\) 1.61153 0.121820
\(176\) 0 0
\(177\) 5.29905 0.398301
\(178\) 0 0
\(179\) 5.00824i 0.374333i −0.982328 0.187167i \(-0.940070\pi\)
0.982328 0.187167i \(-0.0599304\pi\)
\(180\) 0 0
\(181\) 1.45486i 0.108139i −0.998537 0.0540695i \(-0.982781\pi\)
0.998537 0.0540695i \(-0.0172193\pi\)
\(182\) 0 0
\(183\) −9.53475 −0.704829
\(184\) 0 0
\(185\) 10.1186 0.743936
\(186\) 0 0
\(187\) 2.06229i 0.150810i
\(188\) 0 0
\(189\) 1.61153i 0.117222i
\(190\) 0 0
\(191\) 21.9857 1.59083 0.795416 0.606064i \(-0.207252\pi\)
0.795416 + 0.606064i \(0.207252\pi\)
\(192\) 0 0
\(193\) 13.5269 0.973686 0.486843 0.873490i \(-0.338148\pi\)
0.486843 + 0.873490i \(0.338148\pi\)
\(194\) 0 0
\(195\) −7.10687 −0.508933
\(196\) 0 0
\(197\) 14.6237 1.04190 0.520948 0.853588i \(-0.325579\pi\)
0.520948 + 0.853588i \(0.325579\pi\)
\(198\) 0 0
\(199\) −3.24225 −0.229837 −0.114919 0.993375i \(-0.536661\pi\)
−0.114919 + 0.993375i \(0.536661\pi\)
\(200\) 0 0
\(201\) 2.33657i 0.164809i
\(202\) 0 0
\(203\) 0.0589508 0.00413754
\(204\) 0 0
\(205\) 5.43873i 0.379857i
\(206\) 0 0
\(207\) 4.40428 + 1.89798i 0.306119 + 0.131919i
\(208\) 0 0
\(209\) 6.82885 0.472362
\(210\) 0 0
\(211\) 25.4246i 1.75030i −0.483850 0.875151i \(-0.660762\pi\)
0.483850 0.875151i \(-0.339238\pi\)
\(212\) 0 0
\(213\) 11.3842 0.780033
\(214\) 0 0
\(215\) 9.60250i 0.654885i
\(216\) 0 0
\(217\) 7.08016i 0.480633i
\(218\) 0 0
\(219\) 1.61161i 0.108902i
\(220\) 0 0
\(221\) 12.7408i 0.857040i
\(222\) 0 0
\(223\) 18.2691i 1.22339i −0.791095 0.611694i \(-0.790489\pi\)
0.791095 0.611694i \(-0.209511\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 23.8464 1.58274 0.791372 0.611335i \(-0.209367\pi\)
0.791372 + 0.611335i \(0.209367\pi\)
\(228\) 0 0
\(229\) 3.95960i 0.261658i −0.991405 0.130829i \(-0.958236\pi\)
0.991405 0.130829i \(-0.0417639\pi\)
\(230\) 0 0
\(231\) 1.85383i 0.121973i
\(232\) 0 0
\(233\) 26.6401 1.74525 0.872625 0.488392i \(-0.162416\pi\)
0.872625 + 0.488392i \(0.162416\pi\)
\(234\) 0 0
\(235\) 9.16374 0.597776
\(236\) 0 0
\(237\) 5.33892i 0.346800i
\(238\) 0 0
\(239\) 15.7541i 1.01905i −0.860457 0.509523i \(-0.829822\pi\)
0.860457 0.509523i \(-0.170178\pi\)
\(240\) 0 0
\(241\) 2.31154i 0.148899i −0.997225 0.0744497i \(-0.976280\pi\)
0.997225 0.0744497i \(-0.0237200\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 4.40297i 0.281295i
\(246\) 0 0
\(247\) −42.1886 −2.68440
\(248\) 0 0
\(249\) 0.722088i 0.0457605i
\(250\) 0 0
\(251\) 21.6173 1.36447 0.682235 0.731133i \(-0.261008\pi\)
0.682235 + 0.731133i \(0.261008\pi\)
\(252\) 0 0
\(253\) 5.06647 + 2.18335i 0.318526 + 0.137266i
\(254\) 0 0
\(255\) 1.79275i 0.112266i
\(256\) 0 0
\(257\) 4.71865 0.294341 0.147170 0.989111i \(-0.452983\pi\)
0.147170 + 0.989111i \(0.452983\pi\)
\(258\) 0 0
\(259\) 16.3065i 1.01323i
\(260\) 0 0
\(261\) 0.0365806 0.00226428
\(262\) 0 0
\(263\) 4.44240 0.273930 0.136965 0.990576i \(-0.456265\pi\)
0.136965 + 0.990576i \(0.456265\pi\)
\(264\) 0 0
\(265\) 1.95476 0.120080
\(266\) 0 0
\(267\) −2.09024 −0.127920
\(268\) 0 0
\(269\) −9.10119 −0.554909 −0.277455 0.960739i \(-0.589491\pi\)
−0.277455 + 0.960739i \(0.589491\pi\)
\(270\) 0 0
\(271\) 22.9892i 1.39650i 0.715856 + 0.698248i \(0.246037\pi\)
−0.715856 + 0.698248i \(0.753963\pi\)
\(272\) 0 0
\(273\) 11.4529i 0.693163i
\(274\) 0 0
\(275\) 1.15035 0.0693688
\(276\) 0 0
\(277\) 18.6686 1.12169 0.560843 0.827922i \(-0.310477\pi\)
0.560843 + 0.827922i \(0.310477\pi\)
\(278\) 0 0
\(279\) 4.39344i 0.263028i
\(280\) 0 0
\(281\) 23.4639i 1.39974i −0.714271 0.699869i \(-0.753242\pi\)
0.714271 0.699869i \(-0.246758\pi\)
\(282\) 0 0
\(283\) 13.4760 0.801068 0.400534 0.916282i \(-0.368825\pi\)
0.400534 + 0.916282i \(0.368825\pi\)
\(284\) 0 0
\(285\) 5.93632 0.351637
\(286\) 0 0
\(287\) −8.76468 −0.517363
\(288\) 0 0
\(289\) 13.7861 0.810944
\(290\) 0 0
\(291\) 1.93236 0.113277
\(292\) 0 0
\(293\) 26.8108i 1.56630i 0.621831 + 0.783151i \(0.286389\pi\)
−0.621831 + 0.783151i \(0.713611\pi\)
\(294\) 0 0
\(295\) 5.29905 0.308522
\(296\) 0 0
\(297\) 1.15035i 0.0667502i
\(298\) 0 0
\(299\) −31.3006 13.4887i −1.81016 0.780071i
\(300\) 0 0
\(301\) −15.4747 −0.891948
\(302\) 0 0
\(303\) 15.0107i 0.862340i
\(304\) 0 0
\(305\) −9.53475 −0.545958
\(306\) 0 0
\(307\) 1.71461i 0.0978580i 0.998802 + 0.0489290i \(0.0155808\pi\)
−0.998802 + 0.0489290i \(0.984419\pi\)
\(308\) 0 0
\(309\) 3.04742i 0.173362i
\(310\) 0 0
\(311\) 1.97438i 0.111957i 0.998432 + 0.0559785i \(0.0178278\pi\)
−0.998432 + 0.0559785i \(0.982172\pi\)
\(312\) 0 0
\(313\) 8.62500i 0.487514i −0.969836 0.243757i \(-0.921620\pi\)
0.969836 0.243757i \(-0.0783799\pi\)
\(314\) 0 0
\(315\) 1.61153i 0.0907995i
\(316\) 0 0
\(317\) −22.2686 −1.25073 −0.625366 0.780332i \(-0.715050\pi\)
−0.625366 + 0.780332i \(0.715050\pi\)
\(318\) 0 0
\(319\) 0.0420806 0.00235606
\(320\) 0 0
\(321\) 12.6032i 0.703442i
\(322\) 0 0
\(323\) 10.6423i 0.592154i
\(324\) 0 0
\(325\) −7.10687 −0.394218
\(326\) 0 0
\(327\) 15.0300 0.831162
\(328\) 0 0
\(329\) 14.7676i 0.814167i
\(330\) 0 0
\(331\) 13.5388i 0.744162i 0.928200 + 0.372081i \(0.121356\pi\)
−0.928200 + 0.372081i \(0.878644\pi\)
\(332\) 0 0
\(333\) 10.1186i 0.554497i
\(334\) 0 0
\(335\) 2.33657i 0.127660i
\(336\) 0 0
\(337\) 18.2890i 0.996267i −0.867100 0.498134i \(-0.834019\pi\)
0.867100 0.498134i \(-0.165981\pi\)
\(338\) 0 0
\(339\) −8.39445 −0.455924
\(340\) 0 0
\(341\) 5.05400i 0.273689i
\(342\) 0 0
\(343\) 18.3762 0.992223
\(344\) 0 0
\(345\) 4.40428 + 1.89798i 0.237118 + 0.102184i
\(346\) 0 0
\(347\) 15.4805i 0.831037i 0.909585 + 0.415519i \(0.136400\pi\)
−0.909585 + 0.415519i \(0.863600\pi\)
\(348\) 0 0
\(349\) −26.2209 −1.40357 −0.701785 0.712388i \(-0.747614\pi\)
−0.701785 + 0.712388i \(0.747614\pi\)
\(350\) 0 0
\(351\) 7.10687i 0.379336i
\(352\) 0 0
\(353\) 22.9065 1.21919 0.609596 0.792713i \(-0.291332\pi\)
0.609596 + 0.792713i \(0.291332\pi\)
\(354\) 0 0
\(355\) 11.3842 0.604211
\(356\) 0 0
\(357\) −2.88907 −0.152906
\(358\) 0 0
\(359\) 6.73100 0.355248 0.177624 0.984098i \(-0.443159\pi\)
0.177624 + 0.984098i \(0.443159\pi\)
\(360\) 0 0
\(361\) 16.2398 0.854729
\(362\) 0 0
\(363\) 9.67669i 0.507895i
\(364\) 0 0
\(365\) 1.61161i 0.0843554i
\(366\) 0 0
\(367\) 9.41638 0.491531 0.245765 0.969329i \(-0.420961\pi\)
0.245765 + 0.969329i \(0.420961\pi\)
\(368\) 0 0
\(369\) −5.43873 −0.283129
\(370\) 0 0
\(371\) 3.15015i 0.163548i
\(372\) 0 0
\(373\) 32.7090i 1.69361i 0.531904 + 0.846805i \(0.321477\pi\)
−0.531904 + 0.846805i \(0.678523\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −0.259974 −0.0133893
\(378\) 0 0
\(379\) −12.0214 −0.617496 −0.308748 0.951144i \(-0.599910\pi\)
−0.308748 + 0.951144i \(0.599910\pi\)
\(380\) 0 0
\(381\) −18.1343 −0.929049
\(382\) 0 0
\(383\) −28.7465 −1.46888 −0.734438 0.678676i \(-0.762554\pi\)
−0.734438 + 0.678676i \(0.762554\pi\)
\(384\) 0 0
\(385\) 1.85383i 0.0944798i
\(386\) 0 0
\(387\) −9.60250 −0.488122
\(388\) 0 0
\(389\) 8.14172i 0.412801i 0.978468 + 0.206401i \(0.0661750\pi\)
−0.978468 + 0.206401i \(0.933825\pi\)
\(390\) 0 0
\(391\) −3.40260 + 7.89576i −0.172077 + 0.399306i
\(392\) 0 0
\(393\) 1.77034 0.0893021
\(394\) 0 0
\(395\) 5.33892i 0.268630i
\(396\) 0 0
\(397\) −8.41445 −0.422309 −0.211154 0.977453i \(-0.567722\pi\)
−0.211154 + 0.977453i \(0.567722\pi\)
\(398\) 0 0
\(399\) 9.56656i 0.478927i
\(400\) 0 0
\(401\) 24.7496i 1.23594i −0.786203 0.617968i \(-0.787956\pi\)
0.786203 0.617968i \(-0.212044\pi\)
\(402\) 0 0
\(403\) 31.2236i 1.55536i
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 11.6400i 0.576972i
\(408\) 0 0
\(409\) 21.2723 1.05185 0.525923 0.850532i \(-0.323720\pi\)
0.525923 + 0.850532i \(0.323720\pi\)
\(410\) 0 0
\(411\) 17.7574 0.875910
\(412\) 0 0
\(413\) 8.53957i 0.420205i
\(414\) 0 0
\(415\) 0.722088i 0.0354459i
\(416\) 0 0
\(417\) 13.6548 0.668680
\(418\) 0 0
\(419\) −7.56247 −0.369451 −0.184726 0.982790i \(-0.559140\pi\)
−0.184726 + 0.982790i \(0.559140\pi\)
\(420\) 0 0
\(421\) 26.6686i 1.29975i −0.760042 0.649874i \(-0.774822\pi\)
0.760042 0.649874i \(-0.225178\pi\)
\(422\) 0 0
\(423\) 9.16374i 0.445556i
\(424\) 0 0
\(425\) 1.79275i 0.0869610i
\(426\) 0 0
\(427\) 15.3656i 0.743591i
\(428\) 0 0
\(429\) 8.17540i 0.394712i
\(430\) 0 0
\(431\) −19.3147 −0.930356 −0.465178 0.885217i \(-0.654010\pi\)
−0.465178 + 0.885217i \(0.654010\pi\)
\(432\) 0 0
\(433\) 15.6094i 0.750141i 0.926996 + 0.375071i \(0.122382\pi\)
−0.926996 + 0.375071i \(0.877618\pi\)
\(434\) 0 0
\(435\) 0.0365806 0.00175391
\(436\) 0 0
\(437\) 26.1452 + 11.2670i 1.25069 + 0.538974i
\(438\) 0 0
\(439\) 12.5249i 0.597779i 0.954288 + 0.298889i \(0.0966161\pi\)
−0.954288 + 0.298889i \(0.903384\pi\)
\(440\) 0 0
\(441\) 4.40297 0.209665
\(442\) 0 0
\(443\) 37.7262i 1.79242i −0.443625 0.896212i \(-0.646308\pi\)
0.443625 0.896212i \(-0.353692\pi\)
\(444\) 0 0
\(445\) −2.09024 −0.0990867
\(446\) 0 0
\(447\) 9.38262 0.443783
\(448\) 0 0
\(449\) 9.98001 0.470986 0.235493 0.971876i \(-0.424330\pi\)
0.235493 + 0.971876i \(0.424330\pi\)
\(450\) 0 0
\(451\) −6.25645 −0.294605
\(452\) 0 0
\(453\) −13.1263 −0.616726
\(454\) 0 0
\(455\) 11.4529i 0.536922i
\(456\) 0 0
\(457\) 39.0608i 1.82719i 0.406628 + 0.913594i \(0.366705\pi\)
−0.406628 + 0.913594i \(0.633295\pi\)
\(458\) 0 0
\(459\) −1.79275 −0.0836783
\(460\) 0 0
\(461\) −38.5061 −1.79341 −0.896705 0.442629i \(-0.854046\pi\)
−0.896705 + 0.442629i \(0.854046\pi\)
\(462\) 0 0
\(463\) 7.15257i 0.332408i −0.986091 0.166204i \(-0.946849\pi\)
0.986091 0.166204i \(-0.0531510\pi\)
\(464\) 0 0
\(465\) 4.39344i 0.203741i
\(466\) 0 0
\(467\) −21.2146 −0.981692 −0.490846 0.871246i \(-0.663312\pi\)
−0.490846 + 0.871246i \(0.663312\pi\)
\(468\) 0 0
\(469\) −3.76545 −0.173872
\(470\) 0 0
\(471\) −2.75711 −0.127041
\(472\) 0 0
\(473\) −11.0462 −0.507907
\(474\) 0 0
\(475\) 5.93632 0.272377
\(476\) 0 0
\(477\) 1.95476i 0.0895023i
\(478\) 0 0
\(479\) −29.8774 −1.36513 −0.682566 0.730824i \(-0.739136\pi\)
−0.682566 + 0.730824i \(0.739136\pi\)
\(480\) 0 0
\(481\) 71.9117i 3.27889i
\(482\) 0 0
\(483\) −3.05865 + 7.09763i −0.139174 + 0.322953i
\(484\) 0 0
\(485\) 1.93236 0.0877438
\(486\) 0 0
\(487\) 1.40199i 0.0635301i −0.999495 0.0317650i \(-0.989887\pi\)
0.999495 0.0317650i \(-0.0101128\pi\)
\(488\) 0 0
\(489\) −7.13282 −0.322557
\(490\) 0 0
\(491\) 29.6976i 1.34023i −0.742255 0.670117i \(-0.766244\pi\)
0.742255 0.670117i \(-0.233756\pi\)
\(492\) 0 0
\(493\) 0.0655798i 0.00295357i
\(494\) 0 0
\(495\) 1.15035i 0.0517045i
\(496\) 0 0
\(497\) 18.3460i 0.822930i
\(498\) 0 0
\(499\) 8.06976i 0.361252i 0.983552 + 0.180626i \(0.0578124\pi\)
−0.983552 + 0.180626i \(0.942188\pi\)
\(500\) 0 0
\(501\) −0.264942 −0.0118367
\(502\) 0 0
\(503\) 22.4638 1.00161 0.500807 0.865559i \(-0.333037\pi\)
0.500807 + 0.865559i \(0.333037\pi\)
\(504\) 0 0
\(505\) 15.0107i 0.667965i
\(506\) 0 0
\(507\) 37.5075i 1.66577i
\(508\) 0 0
\(509\) 1.54110 0.0683082 0.0341541 0.999417i \(-0.489126\pi\)
0.0341541 + 0.999417i \(0.489126\pi\)
\(510\) 0 0
\(511\) 2.59715 0.114891
\(512\) 0 0
\(513\) 5.93632i 0.262095i
\(514\) 0 0
\(515\) 3.04742i 0.134285i
\(516\) 0 0
\(517\) 10.5415i 0.463616i
\(518\) 0 0
\(519\) 9.93924i 0.436284i
\(520\) 0 0
\(521\) 2.26345i 0.0991637i −0.998770 0.0495818i \(-0.984211\pi\)
0.998770 0.0495818i \(-0.0157889\pi\)
\(522\) 0 0
\(523\) 3.43328 0.150127 0.0750635 0.997179i \(-0.476084\pi\)
0.0750635 + 0.997179i \(0.476084\pi\)
\(524\) 0 0
\(525\) 1.61153i 0.0703330i
\(526\) 0 0
\(527\) 7.87632 0.343098
\(528\) 0 0
\(529\) 15.7953 + 16.7185i 0.686754 + 0.726890i
\(530\) 0 0
\(531\) 5.29905i 0.229959i
\(532\) 0 0
\(533\) 38.6523 1.67422
\(534\) 0 0
\(535\) 12.6032i 0.544884i
\(536\) 0 0
\(537\) 5.00824 0.216121
\(538\) 0 0
\(539\) 5.06496 0.218163
\(540\) 0 0
\(541\) −5.62058 −0.241647 −0.120824 0.992674i \(-0.538554\pi\)
−0.120824 + 0.992674i \(0.538554\pi\)
\(542\) 0 0
\(543\) 1.45486 0.0624341
\(544\) 0 0
\(545\) 15.0300 0.643815
\(546\) 0 0
\(547\) 24.6582i 1.05431i 0.849770 + 0.527154i \(0.176741\pi\)
−0.849770 + 0.527154i \(0.823259\pi\)
\(548\) 0 0
\(549\) 9.53475i 0.406933i
\(550\) 0 0
\(551\) 0.217154 0.00925108
\(552\) 0 0
\(553\) −8.60383 −0.365872
\(554\) 0 0
\(555\) 10.1186i 0.429511i
\(556\) 0 0
\(557\) 33.0462i 1.40021i −0.714039 0.700106i \(-0.753136\pi\)
0.714039 0.700106i \(-0.246864\pi\)
\(558\) 0 0
\(559\) 68.2436 2.88640
\(560\) 0 0
\(561\) −2.06229 −0.0870700
\(562\) 0 0
\(563\) −15.8634 −0.668563 −0.334281 0.942473i \(-0.608494\pi\)
−0.334281 + 0.942473i \(0.608494\pi\)
\(564\) 0 0
\(565\) −8.39445 −0.353157
\(566\) 0 0
\(567\) −1.61153 −0.0676779
\(568\) 0 0
\(569\) 16.6195i 0.696724i −0.937360 0.348362i \(-0.886738\pi\)
0.937360 0.348362i \(-0.113262\pi\)
\(570\) 0 0
\(571\) −6.96762 −0.291586 −0.145793 0.989315i \(-0.546573\pi\)
−0.145793 + 0.989315i \(0.546573\pi\)
\(572\) 0 0
\(573\) 21.9857i 0.918467i
\(574\) 0 0
\(575\) 4.40428 + 1.89798i 0.183671 + 0.0791513i
\(576\) 0 0
\(577\) 20.7254 0.862811 0.431406 0.902158i \(-0.358018\pi\)
0.431406 + 0.902158i \(0.358018\pi\)
\(578\) 0 0
\(579\) 13.5269i 0.562158i
\(580\) 0 0
\(581\) −1.16367 −0.0482770
\(582\) 0 0
\(583\) 2.24866i 0.0931300i
\(584\) 0 0
\(585\) 7.10687i 0.293833i
\(586\) 0 0
\(587\) 32.8221i 1.35471i −0.735655 0.677357i \(-0.763125\pi\)
0.735655 0.677357i \(-0.236875\pi\)
\(588\) 0 0
\(589\) 26.0808i 1.07464i
\(590\) 0 0
\(591\) 14.6237i 0.601539i
\(592\) 0 0
\(593\) −4.92783 −0.202362 −0.101181 0.994868i \(-0.532262\pi\)
−0.101181 + 0.994868i \(0.532262\pi\)
\(594\) 0 0
\(595\) −2.88907 −0.118440
\(596\) 0 0
\(597\) 3.24225i 0.132696i
\(598\) 0 0
\(599\) 19.6282i 0.801987i 0.916081 + 0.400994i \(0.131335\pi\)
−0.916081 + 0.400994i \(0.868665\pi\)
\(600\) 0 0
\(601\) 11.1852 0.456253 0.228126 0.973632i \(-0.426740\pi\)
0.228126 + 0.973632i \(0.426740\pi\)
\(602\) 0 0
\(603\) −2.33657 −0.0951524
\(604\) 0 0
\(605\) 9.67669i 0.393413i
\(606\) 0 0
\(607\) 12.9158i 0.524238i 0.965036 + 0.262119i \(0.0844213\pi\)
−0.965036 + 0.262119i \(0.915579\pi\)
\(608\) 0 0
\(609\) 0.0589508i 0.00238881i
\(610\) 0 0
\(611\) 65.1254i 2.63469i
\(612\) 0 0
\(613\) 24.6992i 0.997591i −0.866720 0.498795i \(-0.833776\pi\)
0.866720 0.498795i \(-0.166224\pi\)
\(614\) 0 0
\(615\) −5.43873 −0.219311
\(616\) 0 0
\(617\) 44.9940i 1.81139i −0.423928 0.905696i \(-0.639349\pi\)
0.423928 0.905696i \(-0.360651\pi\)
\(618\) 0 0
\(619\) 36.3335 1.46037 0.730183 0.683252i \(-0.239435\pi\)
0.730183 + 0.683252i \(0.239435\pi\)
\(620\) 0 0
\(621\) −1.89798 + 4.40428i −0.0761633 + 0.176738i
\(622\) 0 0
\(623\) 3.36848i 0.134955i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 6.82885i 0.272718i
\(628\) 0 0
\(629\) −18.1401 −0.723294
\(630\) 0 0
\(631\) 17.8111 0.709049 0.354525 0.935047i \(-0.384643\pi\)
0.354525 + 0.935047i \(0.384643\pi\)
\(632\) 0 0
\(633\) 25.4246 1.01054
\(634\) 0 0
\(635\) −18.1343 −0.719638
\(636\) 0 0
\(637\) −31.2913 −1.23981
\(638\) 0 0
\(639\) 11.3842i 0.450352i
\(640\) 0 0
\(641\) 34.0713i 1.34574i −0.739762 0.672868i \(-0.765062\pi\)
0.739762 0.672868i \(-0.234938\pi\)
\(642\) 0 0
\(643\) −24.0327 −0.947756 −0.473878 0.880590i \(-0.657146\pi\)
−0.473878 + 0.880590i \(0.657146\pi\)
\(644\) 0 0
\(645\) −9.60250 −0.378098
\(646\) 0 0
\(647\) 29.5150i 1.16036i −0.814490 0.580178i \(-0.802983\pi\)
0.814490 0.580178i \(-0.197017\pi\)
\(648\) 0 0
\(649\) 6.09577i 0.239280i
\(650\) 0 0
\(651\) 7.08016 0.277493
\(652\) 0 0
\(653\) −20.6782 −0.809202 −0.404601 0.914493i \(-0.632590\pi\)
−0.404601 + 0.914493i \(0.632590\pi\)
\(654\) 0 0
\(655\) 1.77034 0.0691731
\(656\) 0 0
\(657\) 1.61161 0.0628748
\(658\) 0 0
\(659\) 24.6283 0.959382 0.479691 0.877438i \(-0.340749\pi\)
0.479691 + 0.877438i \(0.340749\pi\)
\(660\) 0 0
\(661\) 15.4778i 0.602016i −0.953622 0.301008i \(-0.902677\pi\)
0.953622 0.301008i \(-0.0973231\pi\)
\(662\) 0 0
\(663\) 12.7408 0.494812
\(664\) 0 0
\(665\) 9.56656i 0.370975i
\(666\) 0 0
\(667\) 0.161111 + 0.0694293i 0.00623825 + 0.00268831i
\(668\) 0 0
\(669\) 18.2691 0.706323
\(670\) 0 0
\(671\) 10.9683i 0.423427i
\(672\) 0 0
\(673\) 9.60160 0.370115 0.185057 0.982728i \(-0.440753\pi\)
0.185057 + 0.982728i \(0.440753\pi\)
\(674\) 0 0
\(675\) 1.00000i 0.0384900i
\(676\) 0 0
\(677\) 45.5851i 1.75198i 0.482331 + 0.875989i \(0.339790\pi\)
−0.482331 + 0.875989i \(0.660210\pi\)
\(678\) 0 0
\(679\) 3.11405i 0.119506i
\(680\) 0 0
\(681\) 23.8464i 0.913798i
\(682\) 0 0
\(683\) 27.2853i 1.04404i −0.852932 0.522022i \(-0.825178\pi\)
0.852932 0.522022i \(-0.174822\pi\)
\(684\) 0 0
\(685\) 17.7574 0.678477
\(686\) 0 0
\(687\) 3.95960 0.151068
\(688\) 0 0
\(689\) 13.8922i 0.529251i
\(690\) 0 0
\(691\) 19.1940i 0.730173i −0.930974 0.365087i \(-0.881039\pi\)
0.930974 0.365087i \(-0.118961\pi\)
\(692\) 0 0
\(693\) −1.85383 −0.0704211
\(694\) 0 0
\(695\) 13.6548 0.517957
\(696\) 0 0
\(697\) 9.75027i 0.369318i
\(698\) 0 0
\(699\) 26.6401i 1.00762i
\(700\) 0 0
\(701\) 22.8391i 0.862622i 0.902203 + 0.431311i \(0.141949\pi\)
−0.902203 + 0.431311i \(0.858051\pi\)
\(702\) 0 0
\(703\) 60.0673i 2.26548i
\(704\) 0 0
\(705\) 9.16374i 0.345126i
\(706\) 0 0
\(707\) −24.1901 −0.909764
\(708\) 0 0
\(709\) 1.45458i 0.0546278i 0.999627 + 0.0273139i \(0.00869536\pi\)
−0.999627 + 0.0273139i \(0.991305\pi\)
\(710\) 0 0
\(711\) −5.33892 −0.200225
\(712\) 0 0
\(713\) 8.33866 19.3499i 0.312285 0.724660i
\(714\) 0 0
\(715\) 8.17540i 0.305742i
\(716\) 0 0
\(717\) 15.7541 0.588346
\(718\) 0 0
\(719\) 29.2388i 1.09042i −0.838298 0.545212i \(-0.816449\pi\)
0.838298 0.545212i \(-0.183551\pi\)
\(720\) 0 0
\(721\) 4.91101 0.182896
\(722\) 0 0
\(723\) 2.31154 0.0859671
\(724\) 0 0
\(725\) 0.0365806 0.00135857
\(726\) 0 0
\(727\) −40.7426 −1.51106 −0.755529 0.655115i \(-0.772620\pi\)
−0.755529 + 0.655115i \(0.772620\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 17.2149i 0.636714i
\(732\) 0 0
\(733\) 25.3692i 0.937033i 0.883455 + 0.468516i \(0.155211\pi\)
−0.883455 + 0.468516i \(0.844789\pi\)
\(734\) 0 0
\(735\) 4.40297 0.162406
\(736\) 0 0
\(737\) −2.68787 −0.0990091
\(738\) 0 0
\(739\) 8.67928i 0.319273i 0.987176 + 0.159636i \(0.0510321\pi\)
−0.987176 + 0.159636i \(0.948968\pi\)
\(740\) 0 0
\(741\) 42.1886i 1.54984i
\(742\) 0 0
\(743\) 18.8741 0.692424 0.346212 0.938156i \(-0.387468\pi\)
0.346212 + 0.938156i \(0.387468\pi\)
\(744\) 0 0
\(745\) 9.38262 0.343753
\(746\) 0 0
\(747\) −0.722088 −0.0264198
\(748\) 0 0
\(749\) 20.3105 0.742128
\(750\) 0 0
\(751\) −11.9864 −0.437390 −0.218695 0.975793i \(-0.570180\pi\)
−0.218695 + 0.975793i \(0.570180\pi\)
\(752\) 0 0
\(753\) 21.6173i 0.787777i
\(754\) 0 0
\(755\) −13.1263 −0.477714
\(756\) 0 0
\(757\) 9.99273i 0.363192i 0.983373 + 0.181596i \(0.0581263\pi\)
−0.983373 + 0.181596i \(0.941874\pi\)
\(758\) 0 0
\(759\) −2.18335 + 5.06647i −0.0792504 + 0.183901i
\(760\) 0 0
\(761\) −13.8964 −0.503746 −0.251873 0.967760i \(-0.581046\pi\)
−0.251873 + 0.967760i \(0.581046\pi\)
\(762\) 0 0
\(763\) 24.2213i 0.876871i
\(764\) 0 0
\(765\) −1.79275 −0.0648169
\(766\) 0 0
\(767\) 37.6596i 1.35981i
\(768\) 0 0
\(769\) 8.90960i 0.321288i 0.987012 + 0.160644i \(0.0513572\pi\)
−0.987012 + 0.160644i \(0.948643\pi\)
\(770\) 0 0
\(771\) 4.71865i 0.169938i
\(772\) 0 0
\(773\) 11.6166i 0.417821i 0.977935 + 0.208910i \(0.0669917\pi\)
−0.977935 + 0.208910i \(0.933008\pi\)
\(774\) 0 0
\(775\) 4.39344i 0.157817i
\(776\) 0 0
\(777\) −16.3065 −0.584991
\(778\) 0 0
\(779\) −32.2860 −1.15677
\(780\) 0 0
\(781\) 13.0958i 0.468606i
\(782\) 0 0
\(783\) 0.0365806i 0.00130729i
\(784\) 0 0
\(785\) −2.75711 −0.0984054
\(786\) 0 0
\(787\) 32.1690 1.14670 0.573350 0.819310i \(-0.305643\pi\)
0.573350 + 0.819310i \(0.305643\pi\)
\(788\) 0 0
\(789\) 4.44240i 0.158154i
\(790\) 0 0
\(791\) 13.5279i 0.480998i
\(792\) 0 0
\(793\) 67.7622i 2.40631i
\(794\) 0 0
\(795\) 1.95476i 0.0693282i
\(796\) 0 0
\(797\) 12.5397i 0.444178i −0.975026 0.222089i \(-0.928712\pi\)
0.975026 0.222089i \(-0.0712876\pi\)
\(798\) 0 0
\(799\) −16.4283 −0.581190
\(800\) 0 0
\(801\) 2.09024i 0.0738549i
\(802\) 0 0
\(803\) 1.85392 0.0654233
\(804\) 0 0
\(805\) −3.05865 + 7.09763i