Properties

Label 2394.2.b.e.1709.1
Level $2394$
Weight $2$
Character 2394.1709
Analytic conductor $19.116$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2394.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.1161862439\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1709.1
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 2394.1709
Dual form 2394.2.b.e.1709.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.82843i q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.82843i q^{5} +1.00000 q^{7} +1.00000 q^{8} -2.82843i q^{10} +5.65685i q^{11} +4.24264i q^{13} +1.00000 q^{14} +1.00000 q^{16} +5.65685i q^{17} +(1.00000 + 4.24264i) q^{19} -2.82843i q^{20} +5.65685i q^{22} +5.65685i q^{23} -3.00000 q^{25} +4.24264i q^{26} +1.00000 q^{28} -6.00000 q^{29} -8.48528i q^{31} +1.00000 q^{32} +5.65685i q^{34} -2.82843i q^{35} +(1.00000 + 4.24264i) q^{38} -2.82843i q^{40} -4.00000 q^{43} +5.65685i q^{44} +5.65685i q^{46} +9.89949i q^{47} +1.00000 q^{49} -3.00000 q^{50} +4.24264i q^{52} -6.00000 q^{53} +16.0000 q^{55} +1.00000 q^{56} -6.00000 q^{58} +12.0000 q^{59} +2.00000 q^{61} -8.48528i q^{62} +1.00000 q^{64} +12.0000 q^{65} -12.7279i q^{67} +5.65685i q^{68} -2.82843i q^{70} +6.00000 q^{71} +14.0000 q^{73} +(1.00000 + 4.24264i) q^{76} +5.65685i q^{77} -12.7279i q^{79} -2.82843i q^{80} -15.5563i q^{83} +16.0000 q^{85} -4.00000 q^{86} +5.65685i q^{88} -6.00000 q^{89} +4.24264i q^{91} +5.65685i q^{92} +9.89949i q^{94} +(12.0000 - 2.82843i) q^{95} +4.24264i q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8} + 2 q^{14} + 2 q^{16} + 2 q^{19} - 6 q^{25} + 2 q^{28} - 12 q^{29} + 2 q^{32} + 2 q^{38} - 8 q^{43} + 2 q^{49} - 6 q^{50} - 12 q^{53} + 32 q^{55} + 2 q^{56} - 12 q^{58} + 24 q^{59} + 4 q^{61} + 2 q^{64} + 24 q^{65} + 12 q^{71} + 28 q^{73} + 2 q^{76} + 32 q^{85} - 8 q^{86} - 12 q^{89} + 24 q^{95} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times\).

\(n\) \(533\) \(1009\) \(1711\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 2.82843i 1.26491i −0.774597 0.632456i \(-0.782047\pi\)
0.774597 0.632456i \(-0.217953\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 2.82843i 0.894427i
\(11\) 5.65685i 1.70561i 0.522233 + 0.852803i \(0.325099\pi\)
−0.522233 + 0.852803i \(0.674901\pi\)
\(12\) 0 0
\(13\) 4.24264i 1.17670i 0.808608 + 0.588348i \(0.200222\pi\)
−0.808608 + 0.588348i \(0.799778\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.65685i 1.37199i 0.727607 + 0.685994i \(0.240633\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 1.00000 + 4.24264i 0.229416 + 0.973329i
\(20\) 2.82843i 0.632456i
\(21\) 0 0
\(22\) 5.65685i 1.20605i
\(23\) 5.65685i 1.17954i 0.807573 + 0.589768i \(0.200781\pi\)
−0.807573 + 0.589768i \(0.799219\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 4.24264i 0.832050i
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 8.48528i 1.52400i −0.647576 0.762001i \(-0.724217\pi\)
0.647576 0.762001i \(-0.275783\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 5.65685i 0.970143i
\(35\) 2.82843i 0.478091i
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 1.00000 + 4.24264i 0.162221 + 0.688247i
\(39\) 0 0
\(40\) 2.82843i 0.447214i
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 5.65685i 0.852803i
\(45\) 0 0
\(46\) 5.65685i 0.834058i
\(47\) 9.89949i 1.44399i 0.691898 + 0.721995i \(0.256775\pi\)
−0.691898 + 0.721995i \(0.743225\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −3.00000 −0.424264
\(51\) 0 0
\(52\) 4.24264i 0.588348i
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 16.0000 2.15744
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 8.48528i 1.07763i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) 12.7279i 1.55496i −0.628906 0.777482i \(-0.716497\pi\)
0.628906 0.777482i \(-0.283503\pi\)
\(68\) 5.65685i 0.685994i
\(69\) 0 0
\(70\) 2.82843i 0.338062i
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 1.00000 + 4.24264i 0.114708 + 0.486664i
\(77\) 5.65685i 0.644658i
\(78\) 0 0
\(79\) 12.7279i 1.43200i −0.698099 0.716002i \(-0.745970\pi\)
0.698099 0.716002i \(-0.254030\pi\)
\(80\) 2.82843i 0.316228i
\(81\) 0 0
\(82\) 0 0
\(83\) 15.5563i 1.70753i −0.520658 0.853766i \(-0.674313\pi\)
0.520658 0.853766i \(-0.325687\pi\)
\(84\) 0 0
\(85\) 16.0000 1.73544
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 5.65685i 0.603023i
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 4.24264i 0.444750i
\(92\) 5.65685i 0.589768i
\(93\) 0 0
\(94\) 9.89949i 1.02105i
\(95\) 12.0000 2.82843i 1.23117 0.290191i
\(96\) 0 0
\(97\) 4.24264i 0.430775i 0.976529 + 0.215387i \(0.0691014\pi\)
−0.976529 + 0.215387i \(0.930899\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) −3.00000 −0.300000
\(101\) 11.3137i 1.12576i −0.826540 0.562878i \(-0.809694\pi\)
0.826540 0.562878i \(-0.190306\pi\)
\(102\) 0 0
\(103\) 16.9706i 1.67216i 0.548608 + 0.836080i \(0.315158\pi\)
−0.548608 + 0.836080i \(0.684842\pi\)
\(104\) 4.24264i 0.416025i
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −18.0000 −1.74013 −0.870063 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) 0 0
\(109\) 8.48528i 0.812743i 0.913708 + 0.406371i \(0.133206\pi\)
−0.913708 + 0.406371i \(0.866794\pi\)
\(110\) 16.0000 1.52554
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 16.0000 1.49201
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) 5.65685i 0.518563i
\(120\) 0 0
\(121\) −21.0000 −1.90909
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) 8.48528i 0.762001i
\(125\) 5.65685i 0.505964i
\(126\) 0 0
\(127\) 12.7279i 1.12942i −0.825289 0.564710i \(-0.808988\pi\)
0.825289 0.564710i \(-0.191012\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 12.0000 1.05247
\(131\) 1.41421i 0.123560i 0.998090 + 0.0617802i \(0.0196778\pi\)
−0.998090 + 0.0617802i \(0.980322\pi\)
\(132\) 0 0
\(133\) 1.00000 + 4.24264i 0.0867110 + 0.367884i
\(134\) 12.7279i 1.09952i
\(135\) 0 0
\(136\) 5.65685i 0.485071i
\(137\) 9.89949i 0.845771i 0.906183 + 0.422885i \(0.138983\pi\)
−0.906183 + 0.422885i \(0.861017\pi\)
\(138\) 0 0
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 2.82843i 0.239046i
\(141\) 0 0
\(142\) 6.00000 0.503509
\(143\) −24.0000 −2.00698
\(144\) 0 0
\(145\) 16.9706i 1.40933i
\(146\) 14.0000 1.15865
\(147\) 0 0
\(148\) 0 0
\(149\) 7.07107i 0.579284i −0.957135 0.289642i \(-0.906464\pi\)
0.957135 0.289642i \(-0.0935363\pi\)
\(150\) 0 0
\(151\) 12.7279i 1.03578i −0.855446 0.517892i \(-0.826717\pi\)
0.855446 0.517892i \(-0.173283\pi\)
\(152\) 1.00000 + 4.24264i 0.0811107 + 0.344124i
\(153\) 0 0
\(154\) 5.65685i 0.455842i
\(155\) −24.0000 −1.92773
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 12.7279i 1.01258i
\(159\) 0 0
\(160\) 2.82843i 0.223607i
\(161\) 5.65685i 0.445823i
\(162\) 0 0
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 15.5563i 1.20741i
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) 16.0000 1.22714
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) −3.00000 −0.226779
\(176\) 5.65685i 0.426401i
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) 18.0000 1.34538 0.672692 0.739923i \(-0.265138\pi\)
0.672692 + 0.739923i \(0.265138\pi\)
\(180\) 0 0
\(181\) 4.24264i 0.315353i −0.987491 0.157676i \(-0.949600\pi\)
0.987491 0.157676i \(-0.0504003\pi\)
\(182\) 4.24264i 0.314485i
\(183\) 0 0
\(184\) 5.65685i 0.417029i
\(185\) 0 0
\(186\) 0 0
\(187\) −32.0000 −2.34007
\(188\) 9.89949i 0.721995i
\(189\) 0 0
\(190\) 12.0000 2.82843i 0.870572 0.205196i
\(191\) 14.1421i 1.02329i 0.859197 + 0.511645i \(0.170964\pi\)
−0.859197 + 0.511645i \(0.829036\pi\)
\(192\) 0 0
\(193\) 16.9706i 1.22157i 0.791797 + 0.610784i \(0.209146\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) 4.24264i 0.304604i
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 18.3848i 1.30986i 0.755689 + 0.654931i \(0.227302\pi\)
−0.755689 + 0.654931i \(0.772698\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) 11.3137i 0.796030i
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) 0 0
\(206\) 16.9706i 1.18240i
\(207\) 0 0
\(208\) 4.24264i 0.294174i
\(209\) −24.0000 + 5.65685i −1.66011 + 0.391293i
\(210\) 0 0
\(211\) 4.24264i 0.292075i −0.989279 0.146038i \(-0.953348\pi\)
0.989279 0.146038i \(-0.0466521\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −18.0000 −1.23045
\(215\) 11.3137i 0.771589i
\(216\) 0 0
\(217\) 8.48528i 0.576018i
\(218\) 8.48528i 0.574696i
\(219\) 0 0
\(220\) 16.0000 1.07872
\(221\) −24.0000 −1.61441
\(222\) 0 0
\(223\) 25.4558i 1.70465i −0.523013 0.852325i \(-0.675192\pi\)
0.523013 0.852325i \(-0.324808\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 16.0000 1.05501
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 7.07107i 0.463241i −0.972806 0.231621i \(-0.925597\pi\)
0.972806 0.231621i \(-0.0744028\pi\)
\(234\) 0 0
\(235\) 28.0000 1.82652
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) 5.65685i 0.366679i
\(239\) 14.1421i 0.914779i 0.889267 + 0.457389i \(0.151215\pi\)
−0.889267 + 0.457389i \(0.848785\pi\)
\(240\) 0 0
\(241\) 12.7279i 0.819878i −0.912113 0.409939i \(-0.865550\pi\)
0.912113 0.409939i \(-0.134450\pi\)
\(242\) −21.0000 −1.34993
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 2.82843i 0.180702i
\(246\) 0 0
\(247\) −18.0000 + 4.24264i −1.14531 + 0.269953i
\(248\) 8.48528i 0.538816i
\(249\) 0 0
\(250\) 5.65685i 0.357771i
\(251\) 1.41421i 0.0892644i 0.999003 + 0.0446322i \(0.0142116\pi\)
−0.999003 + 0.0446322i \(0.985788\pi\)
\(252\) 0 0
\(253\) −32.0000 −2.01182
\(254\) 12.7279i 0.798621i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 12.0000 0.744208
\(261\) 0 0
\(262\) 1.41421i 0.0873704i
\(263\) 28.2843i 1.74408i −0.489432 0.872041i \(-0.662796\pi\)
0.489432 0.872041i \(-0.337204\pi\)
\(264\) 0 0
\(265\) 16.9706i 1.04249i
\(266\) 1.00000 + 4.24264i 0.0613139 + 0.260133i
\(267\) 0 0
\(268\) 12.7279i 0.777482i
\(269\) 12.0000 0.731653 0.365826 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) 0 0
\(271\) 26.0000 1.57939 0.789694 0.613501i \(-0.210239\pi\)
0.789694 + 0.613501i \(0.210239\pi\)
\(272\) 5.65685i 0.342997i
\(273\) 0 0
\(274\) 9.89949i 0.598050i
\(275\) 16.9706i 1.02336i
\(276\) 0 0
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 2.00000 0.119952
\(279\) 0 0
\(280\) 2.82843i 0.169031i
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) −24.0000 −1.41915
\(287\) 0 0
\(288\) 0 0
\(289\) −15.0000 −0.882353
\(290\) 16.9706i 0.996546i
\(291\) 0 0
\(292\) 14.0000 0.819288
\(293\) 24.0000 1.40209 0.701047 0.713115i \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) 0 0
\(295\) 33.9411i 1.97613i
\(296\) 0 0
\(297\) 0 0
\(298\) 7.07107i 0.409616i
\(299\) −24.0000 −1.38796
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 12.7279i 0.732410i
\(303\) 0 0
\(304\) 1.00000 + 4.24264i 0.0573539 + 0.243332i
\(305\) 5.65685i 0.323911i
\(306\) 0 0
\(307\) 25.4558i 1.45284i 0.687250 + 0.726421i \(0.258818\pi\)
−0.687250 + 0.726421i \(0.741182\pi\)
\(308\) 5.65685i 0.322329i
\(309\) 0 0
\(310\) −24.0000 −1.36311
\(311\) 1.41421i 0.0801927i 0.999196 + 0.0400963i \(0.0127665\pi\)
−0.999196 + 0.0400963i \(0.987234\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) 12.7279i 0.716002i
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 33.9411i 1.90034i
\(320\) 2.82843i 0.158114i
\(321\) 0 0
\(322\) 5.65685i 0.315244i
\(323\) −24.0000 + 5.65685i −1.33540 + 0.314756i
\(324\) 0 0
\(325\) 12.7279i 0.706018i
\(326\) 8.00000 0.443079
\(327\) 0 0
\(328\) 0 0
\(329\) 9.89949i 0.545777i
\(330\) 0 0
\(331\) 29.6985i 1.63238i 0.577786 + 0.816188i \(0.303917\pi\)
−0.577786 + 0.816188i \(0.696083\pi\)
\(332\) 15.5563i 0.853766i
\(333\) 0 0
\(334\) 12.0000 0.656611
\(335\) −36.0000 −1.96689
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) −5.00000 −0.271964
\(339\) 0 0
\(340\) 16.0000 0.867722
\(341\) 48.0000 2.59935
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 2.82843i 0.151838i −0.997114 0.0759190i \(-0.975811\pi\)
0.997114 0.0759190i \(-0.0241890\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) −3.00000 −0.160357
\(351\) 0 0
\(352\) 5.65685i 0.301511i
\(353\) 22.6274i 1.20434i 0.798369 + 0.602168i \(0.205696\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 0 0
\(355\) 16.9706i 0.900704i
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 18.0000 0.951330
\(359\) 28.2843i 1.49279i −0.665505 0.746393i \(-0.731784\pi\)
0.665505 0.746393i \(-0.268216\pi\)
\(360\) 0 0
\(361\) −17.0000 + 8.48528i −0.894737 + 0.446594i
\(362\) 4.24264i 0.222988i
\(363\) 0 0
\(364\) 4.24264i 0.222375i
\(365\) 39.5980i 2.07265i
\(366\) 0 0
\(367\) −10.0000 −0.521996 −0.260998 0.965339i \(-0.584052\pi\)
−0.260998 + 0.965339i \(0.584052\pi\)
\(368\) 5.65685i 0.294884i
\(369\) 0 0
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) 25.4558i 1.31805i −0.752119 0.659027i \(-0.770968\pi\)
0.752119 0.659027i \(-0.229032\pi\)
\(374\) −32.0000 −1.65468
\(375\) 0 0
\(376\) 9.89949i 0.510527i
\(377\) 25.4558i 1.31104i
\(378\) 0 0
\(379\) 4.24264i 0.217930i 0.994046 + 0.108965i \(0.0347536\pi\)
−0.994046 + 0.108965i \(0.965246\pi\)
\(380\) 12.0000 2.82843i 0.615587 0.145095i
\(381\) 0 0
\(382\) 14.1421i 0.723575i
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) 16.0000 0.815436
\(386\) 16.9706i 0.863779i
\(387\) 0 0
\(388\) 4.24264i 0.215387i
\(389\) 9.89949i 0.501924i 0.967997 + 0.250962i \(0.0807470\pi\)
−0.967997 + 0.250962i \(0.919253\pi\)
\(390\) 0 0
\(391\) −32.0000 −1.61831
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 18.3848i 0.926212i
\(395\) −36.0000 −1.81136
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 8.00000 0.401004
\(399\) 0 0
\(400\) −3.00000 −0.150000
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) 36.0000 1.79329
\(404\) 11.3137i 0.562878i
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) 0 0
\(408\) 0 0
\(409\) 12.7279i 0.629355i 0.949199 + 0.314678i \(0.101896\pi\)
−0.949199 + 0.314678i \(0.898104\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 16.9706i 0.836080i
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) −44.0000 −2.15988
\(416\) 4.24264i 0.208013i
\(417\) 0 0
\(418\) −24.0000 + 5.65685i −1.17388 + 0.276686i
\(419\) 15.5563i 0.759977i −0.924991 0.379989i \(-0.875928\pi\)
0.924991 0.379989i \(-0.124072\pi\)
\(420\) 0 0
\(421\) 16.9706i 0.827095i −0.910483 0.413547i \(-0.864290\pi\)
0.910483 0.413547i \(-0.135710\pi\)
\(422\) 4.24264i 0.206529i
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 16.9706i 0.823193i
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) −18.0000 −0.870063
\(429\) 0 0
\(430\) 11.3137i 0.545595i
\(431\) 6.00000 0.289010 0.144505 0.989504i \(-0.453841\pi\)
0.144505 + 0.989504i \(0.453841\pi\)
\(432\) 0 0
\(433\) 4.24264i 0.203888i −0.994790 0.101944i \(-0.967494\pi\)
0.994790 0.101944i \(-0.0325063\pi\)
\(434\) 8.48528i 0.407307i
\(435\) 0 0
\(436\) 8.48528i 0.406371i
\(437\) −24.0000 + 5.65685i −1.14808 + 0.270604i
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 16.0000 0.762770
\(441\) 0 0
\(442\) −24.0000 −1.14156
\(443\) 28.2843i 1.34383i −0.740630 0.671913i \(-0.765473\pi\)
0.740630 0.671913i \(-0.234527\pi\)
\(444\) 0 0
\(445\) 16.9706i 0.804482i
\(446\) 25.4558i 1.20537i
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) 12.0000 0.562569
\(456\) 0 0
\(457\) 32.0000 1.49690 0.748448 0.663193i \(-0.230799\pi\)
0.748448 + 0.663193i \(0.230799\pi\)
\(458\) 14.0000 0.654177
\(459\) 0 0
\(460\) 16.0000 0.746004
\(461\) 22.6274i 1.05386i 0.849907 + 0.526932i \(0.176658\pi\)
−0.849907 + 0.526932i \(0.823342\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 7.07107i 0.327561i
\(467\) 18.3848i 0.850746i 0.905018 + 0.425373i \(0.139857\pi\)
−0.905018 + 0.425373i \(0.860143\pi\)
\(468\) 0 0
\(469\) 12.7279i 0.587721i
\(470\) 28.0000 1.29154
\(471\) 0 0
\(472\) 12.0000 0.552345
\(473\) 22.6274i 1.04041i
\(474\) 0 0
\(475\) −3.00000 12.7279i −0.137649 0.583997i
\(476\) 5.65685i 0.259281i
\(477\) 0 0
\(478\) 14.1421i 0.646846i
\(479\) 18.3848i 0.840022i 0.907519 + 0.420011i \(0.137974\pi\)
−0.907519 + 0.420011i \(0.862026\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 12.7279i 0.579741i
\(483\) 0 0
\(484\) −21.0000 −0.954545
\(485\) 12.0000 0.544892
\(486\) 0 0
\(487\) 12.7279i 0.576757i 0.957516 + 0.288379i \(0.0931162\pi\)
−0.957516 + 0.288379i \(0.906884\pi\)
\(488\) 2.00000 0.0905357
\(489\) 0 0
\(490\) 2.82843i 0.127775i
\(491\) 36.7696i 1.65939i −0.558219 0.829693i \(-0.688515\pi\)
0.558219 0.829693i \(-0.311485\pi\)
\(492\) 0 0
\(493\) 33.9411i 1.52863i
\(494\) −18.0000 + 4.24264i −0.809858 + 0.190885i
\(495\) 0 0
\(496\) 8.48528i 0.381000i
\(497\) 6.00000 0.269137
\(498\) 0 0
\(499\) −16.0000 −0.716258 −0.358129 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) 5.65685i 0.252982i
\(501\) 0 0
\(502\) 1.41421i 0.0631194i
\(503\) 9.89949i 0.441397i 0.975342 + 0.220698i \(0.0708336\pi\)
−0.975342 + 0.220698i \(0.929166\pi\)
\(504\) 0 0
\(505\) −32.0000 −1.42398
\(506\) −32.0000 −1.42257
\(507\) 0 0
\(508\) 12.7279i 0.564710i
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) 14.0000 0.619324
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −18.0000 −0.793946
\(515\) 48.0000 2.11513
\(516\) 0 0
\(517\) −56.0000 −2.46288
\(518\) 0 0
\(519\) 0 0
\(520\) 12.0000 0.526235
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 0 0
\(523\) 16.9706i 0.742071i −0.928619 0.371035i \(-0.879003\pi\)
0.928619 0.371035i \(-0.120997\pi\)
\(524\) 1.41421i 0.0617802i
\(525\) 0 0
\(526\) 28.2843i 1.23325i
\(527\) 48.0000 2.09091
\(528\) 0 0
\(529\) −9.00000 −0.391304
\(530\) 16.9706i 0.737154i
\(531\) 0 0
\(532\) 1.00000 + 4.24264i 0.0433555 + 0.183942i
\(533\) 0 0
\(534\) 0 0
\(535\) 50.9117i 2.20110i
\(536\) 12.7279i 0.549762i
\(537\) 0 0
\(538\) 12.0000 0.517357
\(539\) 5.65685i 0.243658i
\(540\) 0 0
\(541\) −40.0000 −1.71973 −0.859867 0.510518i \(-0.829454\pi\)
−0.859867 + 0.510518i \(0.829454\pi\)
\(542\) 26.0000 1.11680
\(543\) 0 0
\(544\) 5.65685i 0.242536i
\(545\) 24.0000 1.02805
\(546\) 0 0
\(547\) 12.7279i 0.544207i −0.962268 0.272103i \(-0.912281\pi\)
0.962268 0.272103i \(-0.0877193\pi\)
\(548\) 9.89949i 0.422885i
\(549\) 0 0
\(550\) 16.9706i 0.723627i
\(551\) −6.00000 25.4558i −0.255609 1.08446i
\(552\) 0 0
\(553\) 12.7279i 0.541246i
\(554\) 8.00000 0.339887
\(555\) 0 0
\(556\) 2.00000 0.0848189
\(557\) 35.3553i 1.49805i 0.662540 + 0.749027i \(0.269479\pi\)
−0.662540 + 0.749027i \(0.730521\pi\)
\(558\) 0 0
\(559\) 16.9706i 0.717778i
\(560\) 2.82843i 0.119523i
\(561\) 0 0
\(562\) −18.0000 −0.759284
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 16.9706i 0.713957i
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 6.00000 0.251754
\(569\) 42.0000 1.76073 0.880366 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) −24.0000 −1.00349
\(573\) 0 0
\(574\) 0 0
\(575\) 16.9706i 0.707721i
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) −15.0000 −0.623918
\(579\) 0 0
\(580\) 16.9706i 0.704664i
\(581\) 15.5563i 0.645386i
\(582\) 0 0
\(583\) 33.9411i 1.40570i
\(584\) 14.0000 0.579324
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) 15.5563i 0.642079i −0.947066 0.321040i \(-0.895968\pi\)
0.947066 0.321040i \(-0.104032\pi\)
\(588\) 0 0
\(589\) 36.0000 8.48528i 1.48335 0.349630i
\(590\) 33.9411i 1.39733i
\(591\) 0 0
\(592\) 0 0
\(593\) 39.5980i 1.62609i 0.582198 + 0.813047i \(0.302193\pi\)
−0.582198 + 0.813047i \(0.697807\pi\)
\(594\) 0 0
\(595\) 16.0000 0.655936
\(596\) 7.07107i 0.289642i
\(597\) 0 0
\(598\) −24.0000 −0.981433
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 4.24264i 0.173061i −0.996249 0.0865305i \(-0.972422\pi\)
0.996249 0.0865305i \(-0.0275780\pi\)
\(602\) −4.00000 −0.163028
\(603\) 0 0
\(604\) 12.7279i 0.517892i
\(605\) 59.3970i 2.41483i
\(606\) 0 0
\(607\) 33.9411i 1.37763i 0.724938 + 0.688814i \(0.241868\pi\)
−0.724938 + 0.688814i \(0.758132\pi\)
\(608\) 1.00000 + 4.24264i 0.0405554 + 0.172062i
\(609\) 0 0
\(610\) 5.65685i 0.229039i
\(611\) −42.0000 −1.69914
\(612\) 0 0
\(613\) −16.0000 −0.646234 −0.323117 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) 25.4558i 1.02731i
\(615\) 0 0
\(616\) 5.65685i 0.227921i
\(617\) 32.5269i 1.30948i −0.755852 0.654742i \(-0.772777\pi\)
0.755852 0.654742i \(-0.227223\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) −24.0000 −0.963863
\(621\) 0 0
\(622\) 1.41421i 0.0567048i
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 14.0000 0.559553
\(627\) 0 0
\(628\) 14.0000 0.558661
\(629\) 0 0
\(630\) 0 0
\(631\) −28.0000 −1.11466 −0.557331 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) 12.7279i 0.506290i
\(633\) 0 0
\(634\) 18.0000 0.714871
\(635\) −36.0000 −1.42862
\(636\) 0 0
\(637\) 4.24264i 0.168100i
\(638\) 33.9411i 1.34374i
\(639\) 0 0
\(640\) 2.82843i 0.111803i
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) −22.0000 −0.867595 −0.433798 0.901010i \(-0.642827\pi\)
−0.433798 + 0.901010i \(0.642827\pi\)
\(644\) 5.65685i 0.222911i
\(645\) 0 0
\(646\) −24.0000 + 5.65685i −0.944267 + 0.222566i
\(647\) 15.5563i 0.611583i −0.952098 0.305792i \(-0.901079\pi\)
0.952098 0.305792i \(-0.0989211\pi\)
\(648\) 0 0
\(649\) 67.8823i 2.66461i
\(650\) 12.7279i 0.499230i
\(651\) 0 0
\(652\) 8.00000 0.313304
\(653\) 24.0416i 0.940822i −0.882448 0.470411i \(-0.844106\pi\)
0.882448 0.470411i \(-0.155894\pi\)
\(654\) 0 0
\(655\) 4.00000 0.156293
\(656\) 0 0
\(657\) 0 0
\(658\) 9.89949i 0.385922i
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) 38.1838i 1.48518i −0.669748 0.742588i \(-0.733598\pi\)
0.669748 0.742588i \(-0.266402\pi\)
\(662\) 29.6985i 1.15426i
\(663\) 0 0
\(664\) 15.5563i 0.603703i
\(665\) 12.0000 2.82843i 0.465340 0.109682i
\(666\) 0 0
\(667\) 33.9411i 1.31421i
\(668\) 12.0000 0.464294
\(669\) 0 0
\(670\) −36.0000 −1.39080
\(671\) 11.3137i 0.436761i
\(672\) 0 0
\(673\) 42.4264i 1.63542i 0.575632 + 0.817709i \(0.304756\pi\)
−0.575632 + 0.817709i \(0.695244\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −5.00000 −0.192308
\(677\) 30.0000 1.15299 0.576497 0.817099i \(-0.304419\pi\)
0.576497 + 0.817099i \(0.304419\pi\)
\(678\) 0 0
\(679\) 4.24264i 0.162818i
\(680\) 16.0000 0.613572
\(681\) 0 0
\(682\) 48.0000 1.83801
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 28.0000 1.06983
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) 25.4558i 0.969790i
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) 2.82843i 0.107366i
\(695\) 5.65685i 0.214577i
\(696\) 0 0
\(697\) 0 0
\(698\) −10.0000 −0.378506
\(699\) 0 0
\(700\) −3.00000 −0.113389
\(701\) 7.07107i 0.267071i −0.991044 0.133535i \(-0.957367\pi\)
0.991044 0.133535i \(-0.0426329\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 5.65685i 0.213201i
\(705\) 0 0
\(706\) 22.6274i 0.851594i
\(707\) 11.3137i 0.425496i
\(708\) 0 0
\(709\) −4.00000 −0.150223 −0.0751116 0.997175i \(-0.523931\pi\)
−0.0751116 + 0.997175i \(0.523931\pi\)
\(710\) 16.9706i 0.636894i
\(711\) 0 0
\(712\) −6.00000 −0.224860
\(713\) 48.0000 1.79761
\(714\) 0 0
\(715\) 67.8823i 2.53865i
\(716\) 18.0000 0.672692
\(717\) 0 0
\(718\) 28.2843i 1.05556i
\(719\) 7.07107i 0.263706i −0.991269 0.131853i \(-0.957907\pi\)
0.991269 0.131853i \(-0.0420927\pi\)
\(720\) 0 0
\(721\) 16.9706i 0.632017i
\(722\) −17.0000 + 8.48528i −0.632674 + 0.315789i
\(723\) 0 0
\(724\) 4.24264i 0.157676i
\(725\) 18.0000 0.668503
\(726\) 0 0
\(727\) 2.00000 0.0741759 0.0370879 0.999312i \(-0.488192\pi\)
0.0370879 + 0.999312i \(0.488192\pi\)
\(728\) 4.24264i 0.157243i
\(729\) 0 0
\(730\) 39.5980i 1.46559i
\(731\) 22.6274i 0.836905i
\(732\) 0 0
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) −10.0000 −0.369107
\(735\) 0 0
\(736\) 5.65685i 0.208514i
\(737\) 72.0000 2.65215
\(738\) 0 0
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −6.00000 −0.220267
\(743\) −48.0000 −1.76095 −0.880475 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) 0 0
\(745\) −20.0000 −0.732743
\(746\) 25.4558i 0.932005i
\(747\) 0 0
\(748\) −32.0000 −1.17004
\(749\) −18.0000 −0.657706
\(750\) 0 0
\(751\) 12.7279i 0.464448i 0.972662 + 0.232224i \(0.0746003\pi\)
−0.972662 + 0.232224i \(0.925400\pi\)
\(752\) 9.89949i 0.360997i
\(753\) 0 0
\(754\) 25.4558i 0.927047i
\(755\) −36.0000 −1.31017
\(756\) 0 0
\(757\) 44.0000 1.59921 0.799604 0.600528i \(-0.205043\pi\)
0.799604 + 0.600528i \(0.205043\pi\)
\(758\) 4.24264i 0.154100i
\(759\) 0 0
\(760\) 12.0000 2.82843i 0.435286 0.102598i
\(761\) 2.82843i 0.102530i −0.998685 0.0512652i \(-0.983675\pi\)
0.998685 0.0512652i \(-0.0163254\pi\)
\(762\) 0 0
\(763\) 8.48528i 0.307188i
\(764\) 14.1421i 0.511645i
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 50.9117i 1.83831i
\(768\) 0 0
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 16.0000 0.576600
\(771\) 0 0
\(772\) 16.9706i 0.610784i
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 0 0
\(775\) 25.4558i 0.914401i
\(776\) 4.24264i 0.152302i
\(777\) 0 0
\(778\) 9.89949i 0.354914i
\(779\) 0 0
\(780\) 0 0
\(781\) 33.9411i 1.21451i
\(782\) −32.0000 −1.14432
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 39.5980i 1.41331i
\(786\) 0 0
\(787\) 33.9411i 1.20987i −0.796275 0.604935i \(-0.793199\pi\)
0.796275 0.604935i \(-0.206801\pi\)
\(788\) 18.3848i 0.654931i
\(789\) 0 0
\(790\) −36.0000 −1.28082
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) 8.48528i 0.301321i
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) −48.0000 −1.70025 −0.850124 0.526583i \(-0.823473\pi\)
−0.850124 + 0.526583i \(0.823473\pi\)
\(798\) 0 0
\(799\) −56.0000 −1.98114