Properties

Label 1840.2.m.d.1839.4
Level $1840$
Weight $2$
Character 1840.1839
Analytic conductor $14.692$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{6}, \sqrt{-14})\)
Defining polynomial: \(x^{4} + 4 x^{2} + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1839.4
Root \(-1.22474 + 1.87083i\) of defining polynomial
Character \(\chi\) \(=\) 1840.1839
Dual form 1840.2.m.d.1839.3

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +(1.22474 + 1.87083i) q^{5} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +(1.22474 + 1.87083i) q^{5} -2.00000 q^{9} -4.89898 q^{11} -4.58258i q^{13} +(1.22474 + 1.87083i) q^{15} -2.44949 q^{17} -2.44949 q^{19} +(-3.00000 + 3.74166i) q^{23} +(-2.00000 + 4.58258i) q^{25} -5.00000 q^{27} -3.00000 q^{29} +4.58258i q^{31} -4.89898 q^{33} +4.89898 q^{37} -4.58258i q^{39} -3.00000 q^{41} +11.2250i q^{43} +(-2.44949 - 3.74166i) q^{45} -9.00000 q^{47} +7.00000 q^{49} -2.44949 q^{51} -4.89898 q^{53} +(-6.00000 - 9.16515i) q^{55} -2.44949 q^{57} -9.16515i q^{59} +11.2250i q^{61} +(8.57321 - 5.61249i) q^{65} +(-3.00000 + 3.74166i) q^{69} -13.7477i q^{71} -4.58258i q^{73} +(-2.00000 + 4.58258i) q^{75} -7.34847 q^{79} +1.00000 q^{81} -3.74166i q^{83} +(-3.00000 - 4.58258i) q^{85} -3.00000 q^{87} -3.74166i q^{89} +4.58258i q^{93} +(-3.00000 - 4.58258i) q^{95} +9.79796 q^{97} +9.79796 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} - 8q^{9} + O(q^{10}) \) \( 4q + 4q^{3} - 8q^{9} - 12q^{23} - 8q^{25} - 20q^{27} - 12q^{29} - 12q^{41} - 36q^{47} + 28q^{49} - 24q^{55} - 12q^{69} - 8q^{75} + 4q^{81} - 12q^{85} - 12q^{87} - 12q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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
\(3\) 1.00000 0.577350 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 0 0
\(5\) 1.22474 + 1.87083i 0.547723 + 0.836660i
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −4.89898 −1.47710 −0.738549 0.674200i \(-0.764489\pi\)
−0.738549 + 0.674200i \(0.764489\pi\)
\(12\) 0 0
\(13\) 4.58258i 1.27098i −0.772110 0.635489i \(-0.780799\pi\)
0.772110 0.635489i \(-0.219201\pi\)
\(14\) 0 0
\(15\) 1.22474 + 1.87083i 0.316228 + 0.483046i
\(16\) 0 0
\(17\) −2.44949 −0.594089 −0.297044 0.954864i \(-0.596001\pi\)
−0.297044 + 0.954864i \(0.596001\pi\)
\(18\) 0 0
\(19\) −2.44949 −0.561951 −0.280976 0.959715i \(-0.590658\pi\)
−0.280976 + 0.959715i \(0.590658\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.00000 + 3.74166i −0.625543 + 0.780189i
\(24\) 0 0
\(25\) −2.00000 + 4.58258i −0.400000 + 0.916515i
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) 4.58258i 0.823055i 0.911397 + 0.411527i \(0.135005\pi\)
−0.911397 + 0.411527i \(0.864995\pi\)
\(32\) 0 0
\(33\) −4.89898 −0.852803
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.89898 0.805387 0.402694 0.915335i \(-0.368074\pi\)
0.402694 + 0.915335i \(0.368074\pi\)
\(38\) 0 0
\(39\) 4.58258i 0.733799i
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) 11.2250i 1.71179i 0.517148 + 0.855896i \(0.326994\pi\)
−0.517148 + 0.855896i \(0.673006\pi\)
\(44\) 0 0
\(45\) −2.44949 3.74166i −0.365148 0.557773i
\(46\) 0 0
\(47\) −9.00000 −1.31278 −0.656392 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) −2.44949 −0.342997
\(52\) 0 0
\(53\) −4.89898 −0.672927 −0.336463 0.941697i \(-0.609231\pi\)
−0.336463 + 0.941697i \(0.609231\pi\)
\(54\) 0 0
\(55\) −6.00000 9.16515i −0.809040 1.23583i
\(56\) 0 0
\(57\) −2.44949 −0.324443
\(58\) 0 0
\(59\) 9.16515i 1.19320i −0.802538 0.596601i \(-0.796518\pi\)
0.802538 0.596601i \(-0.203482\pi\)
\(60\) 0 0
\(61\) 11.2250i 1.43721i 0.695418 + 0.718605i \(0.255219\pi\)
−0.695418 + 0.718605i \(0.744781\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.57321 5.61249i 1.06338 0.696143i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) −3.00000 + 3.74166i −0.361158 + 0.450443i
\(70\) 0 0
\(71\) 13.7477i 1.63156i −0.578366 0.815778i \(-0.696309\pi\)
0.578366 0.815778i \(-0.303691\pi\)
\(72\) 0 0
\(73\) 4.58258i 0.536350i −0.963370 0.268175i \(-0.913579\pi\)
0.963370 0.268175i \(-0.0864205\pi\)
\(74\) 0 0
\(75\) −2.00000 + 4.58258i −0.230940 + 0.529150i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −7.34847 −0.826767 −0.413384 0.910557i \(-0.635653\pi\)
−0.413384 + 0.910557i \(0.635653\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.74166i 0.410700i −0.978689 0.205350i \(-0.934167\pi\)
0.978689 0.205350i \(-0.0658333\pi\)
\(84\) 0 0
\(85\) −3.00000 4.58258i −0.325396 0.497050i
\(86\) 0 0
\(87\) −3.00000 −0.321634
\(88\) 0 0
\(89\) 3.74166i 0.396615i −0.980140 0.198307i \(-0.936456\pi\)
0.980140 0.198307i \(-0.0635444\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.58258i 0.475191i
\(94\) 0 0
\(95\) −3.00000 4.58258i −0.307794 0.470162i
\(96\) 0 0
\(97\) 9.79796 0.994832 0.497416 0.867512i \(-0.334282\pi\)
0.497416 + 0.867512i \(0.334282\pi\)
\(98\) 0 0
\(99\) 9.79796 0.984732
\(100\) 0 0
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) 11.2250i 1.10603i 0.833172 + 0.553015i \(0.186523\pi\)
−0.833172 + 0.553015i \(0.813477\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.74166i 0.361720i 0.983509 + 0.180860i \(0.0578880\pi\)
−0.983509 + 0.180860i \(0.942112\pi\)
\(108\) 0 0
\(109\) 11.2250i 1.07516i −0.843214 0.537579i \(-0.819339\pi\)
0.843214 0.537579i \(-0.180661\pi\)
\(110\) 0 0
\(111\) 4.89898 0.464991
\(112\) 0 0
\(113\) 14.6969 1.38257 0.691286 0.722581i \(-0.257045\pi\)
0.691286 + 0.722581i \(0.257045\pi\)
\(114\) 0 0
\(115\) −10.6742 1.02991i −0.995378 0.0960396i
\(116\) 0 0
\(117\) 9.16515i 0.847319i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 13.0000 1.18182
\(122\) 0 0
\(123\) −3.00000 −0.270501
\(124\) 0 0
\(125\) −11.0227 + 1.87083i −0.985901 + 0.167332i
\(126\) 0 0
\(127\) −11.0000 −0.976092 −0.488046 0.872818i \(-0.662290\pi\)
−0.488046 + 0.872818i \(0.662290\pi\)
\(128\) 0 0
\(129\) 11.2250i 0.988304i
\(130\) 0 0
\(131\) 4.58258i 0.400381i −0.979757 0.200191i \(-0.935844\pi\)
0.979757 0.200191i \(-0.0641562\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −6.12372 9.35414i −0.527046 0.805076i
\(136\) 0 0
\(137\) −14.6969 −1.25564 −0.627822 0.778357i \(-0.716053\pi\)
−0.627822 + 0.778357i \(0.716053\pi\)
\(138\) 0 0
\(139\) 22.9129i 1.94344i 0.236126 + 0.971722i \(0.424122\pi\)
−0.236126 + 0.971722i \(0.575878\pi\)
\(140\) 0 0
\(141\) −9.00000 −0.757937
\(142\) 0 0
\(143\) 22.4499i 1.87736i
\(144\) 0 0
\(145\) −3.67423 5.61249i −0.305129 0.466092i
\(146\) 0 0
\(147\) 7.00000 0.577350
\(148\) 0 0
\(149\) 14.9666i 1.22611i −0.790039 0.613057i \(-0.789940\pi\)
0.790039 0.613057i \(-0.210060\pi\)
\(150\) 0 0
\(151\) 13.7477i 1.11877i −0.828907 0.559387i \(-0.811037\pi\)
0.828907 0.559387i \(-0.188963\pi\)
\(152\) 0 0
\(153\) 4.89898 0.396059
\(154\) 0 0
\(155\) −8.57321 + 5.61249i −0.688617 + 0.450806i
\(156\) 0 0
\(157\) 2.44949 0.195491 0.0977453 0.995211i \(-0.468837\pi\)
0.0977453 + 0.995211i \(0.468837\pi\)
\(158\) 0 0
\(159\) −4.89898 −0.388514
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 13.0000 1.01824 0.509119 0.860696i \(-0.329971\pi\)
0.509119 + 0.860696i \(0.329971\pi\)
\(164\) 0 0
\(165\) −6.00000 9.16515i −0.467099 0.713506i
\(166\) 0 0
\(167\) −18.0000 −1.39288 −0.696441 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 0 0
\(169\) −8.00000 −0.615385
\(170\) 0 0
\(171\) 4.89898 0.374634
\(172\) 0 0
\(173\) 18.3303i 1.39363i 0.717252 + 0.696814i \(0.245400\pi\)
−0.717252 + 0.696814i \(0.754600\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 9.16515i 0.688895i
\(178\) 0 0
\(179\) 13.7477i 1.02755i −0.857924 0.513777i \(-0.828246\pi\)
0.857924 0.513777i \(-0.171754\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 11.2250i 0.829774i
\(184\) 0 0
\(185\) 6.00000 + 9.16515i 0.441129 + 0.673835i
\(186\) 0 0
\(187\) 12.0000 0.877527
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −12.2474 −0.886194 −0.443097 0.896474i \(-0.646120\pi\)
−0.443097 + 0.896474i \(0.646120\pi\)
\(192\) 0 0
\(193\) 22.9129i 1.64931i 0.565640 + 0.824653i \(0.308629\pi\)
−0.565640 + 0.824653i \(0.691371\pi\)
\(194\) 0 0
\(195\) 8.57321 5.61249i 0.613941 0.401918i
\(196\) 0 0
\(197\) 4.58258i 0.326495i −0.986585 0.163247i \(-0.947803\pi\)
0.986585 0.163247i \(-0.0521969\pi\)
\(198\) 0 0
\(199\) −19.5959 −1.38912 −0.694559 0.719436i \(-0.744400\pi\)
−0.694559 + 0.719436i \(0.744400\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −3.67423 5.61249i −0.256620 0.391993i
\(206\) 0 0
\(207\) 6.00000 7.48331i 0.417029 0.520126i
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) 18.3303i 1.26191i 0.775819 + 0.630955i \(0.217337\pi\)
−0.775819 + 0.630955i \(0.782663\pi\)
\(212\) 0 0
\(213\) 13.7477i 0.941979i
\(214\) 0 0
\(215\) −21.0000 + 13.7477i −1.43219 + 0.937587i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 4.58258i 0.309662i
\(220\) 0 0
\(221\) 11.2250i 0.755073i
\(222\) 0 0
\(223\) 22.0000 1.47323 0.736614 0.676313i \(-0.236423\pi\)
0.736614 + 0.676313i \(0.236423\pi\)
\(224\) 0 0
\(225\) 4.00000 9.16515i 0.266667 0.611010i
\(226\) 0 0
\(227\) 7.48331i 0.496685i −0.968672 0.248343i \(-0.920114\pi\)
0.968672 0.248343i \(-0.0798858\pi\)
\(228\) 0 0
\(229\) 11.2250i 0.741767i 0.928679 + 0.370884i \(0.120945\pi\)
−0.928679 + 0.370884i \(0.879055\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4.58258i 0.300215i 0.988670 + 0.150107i \(0.0479619\pi\)
−0.988670 + 0.150107i \(0.952038\pi\)
\(234\) 0 0
\(235\) −11.0227 16.8375i −0.719042 1.09835i
\(236\) 0 0
\(237\) −7.34847 −0.477334
\(238\) 0 0
\(239\) 4.58258i 0.296422i 0.988956 + 0.148211i \(0.0473515\pi\)
−0.988956 + 0.148211i \(0.952648\pi\)
\(240\) 0 0
\(241\) 11.2250i 0.723064i −0.932360 0.361532i \(-0.882254\pi\)
0.932360 0.361532i \(-0.117746\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 0 0
\(245\) 8.57321 + 13.0958i 0.547723 + 0.836660i
\(246\) 0 0
\(247\) 11.2250i 0.714228i
\(248\) 0 0
\(249\) 3.74166i 0.237118i
\(250\) 0 0
\(251\) 12.2474 0.773052 0.386526 0.922278i \(-0.373675\pi\)
0.386526 + 0.922278i \(0.373675\pi\)
\(252\) 0 0
\(253\) 14.6969 18.3303i 0.923989 1.15242i
\(254\) 0 0
\(255\) −3.00000 4.58258i −0.187867 0.286972i
\(256\) 0 0
\(257\) 22.9129i 1.42927i −0.699500 0.714633i \(-0.746594\pi\)
0.699500 0.714633i \(-0.253406\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 18.7083i 1.15360i 0.816885 + 0.576801i \(0.195699\pi\)
−0.816885 + 0.576801i \(0.804301\pi\)
\(264\) 0 0
\(265\) −6.00000 9.16515i −0.368577 0.563011i
\(266\) 0 0
\(267\) 3.74166i 0.228986i
\(268\) 0 0
\(269\) −9.00000 −0.548740 −0.274370 0.961624i \(-0.588469\pi\)
−0.274370 + 0.961624i \(0.588469\pi\)
\(270\) 0 0
\(271\) 9.16515i 0.556743i 0.960473 + 0.278372i \(0.0897947\pi\)
−0.960473 + 0.278372i \(0.910205\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 9.79796 22.4499i 0.590839 1.35378i
\(276\) 0 0
\(277\) 22.9129i 1.37670i 0.725378 + 0.688351i \(0.241665\pi\)
−0.725378 + 0.688351i \(0.758335\pi\)
\(278\) 0 0
\(279\) 9.16515i 0.548703i
\(280\) 0 0
\(281\) 14.9666i 0.892834i 0.894825 + 0.446417i \(0.147300\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) 11.2250i 0.667255i −0.942705 0.333628i \(-0.891727\pi\)
0.942705 0.333628i \(-0.108273\pi\)
\(284\) 0 0
\(285\) −3.00000 4.58258i −0.177705 0.271448i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −11.0000 −0.647059
\(290\) 0 0
\(291\) 9.79796 0.574367
\(292\) 0 0
\(293\) 9.79796 0.572403 0.286201 0.958169i \(-0.407607\pi\)
0.286201 + 0.958169i \(0.407607\pi\)
\(294\) 0 0
\(295\) 17.1464 11.2250i 0.998304 0.653543i
\(296\) 0 0
\(297\) 24.4949 1.42134
\(298\) 0 0
\(299\) 17.1464 + 13.7477i 0.991604 + 0.795052i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 12.0000 0.689382
\(304\) 0 0
\(305\) −21.0000 + 13.7477i −1.20246 + 0.787193i
\(306\) 0 0
\(307\) 22.0000 1.25561 0.627803 0.778372i \(-0.283954\pi\)
0.627803 + 0.778372i \(0.283954\pi\)
\(308\) 0 0
\(309\) 11.2250i 0.638566i
\(310\) 0 0
\(311\) 4.58258i 0.259854i 0.991524 + 0.129927i \(0.0414743\pi\)
−0.991524 + 0.129927i \(0.958526\pi\)
\(312\) 0 0
\(313\) −29.3939 −1.66144 −0.830720 0.556690i \(-0.812071\pi\)
−0.830720 + 0.556690i \(0.812071\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.3303i 1.02953i −0.857331 0.514766i \(-0.827879\pi\)
0.857331 0.514766i \(-0.172121\pi\)
\(318\) 0 0
\(319\) 14.6969 0.822871
\(320\) 0 0
\(321\) 3.74166i 0.208839i
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) 0 0
\(325\) 21.0000 + 9.16515i 1.16487 + 0.508391i
\(326\) 0 0
\(327\) 11.2250i 0.620742i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 4.58258i 0.251881i 0.992038 + 0.125941i \(0.0401949\pi\)
−0.992038 + 0.125941i \(0.959805\pi\)
\(332\) 0 0
\(333\) −9.79796 −0.536925
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −31.8434 −1.73462 −0.867309 0.497770i \(-0.834153\pi\)
−0.867309 + 0.497770i \(0.834153\pi\)
\(338\) 0 0
\(339\) 14.6969 0.798228
\(340\) 0 0
\(341\) 22.4499i 1.21573i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −10.6742 1.02991i −0.574681 0.0554485i
\(346\) 0 0
\(347\) 6.00000 0.322097 0.161048 0.986947i \(-0.448512\pi\)
0.161048 + 0.986947i \(0.448512\pi\)
\(348\) 0 0
\(349\) 1.00000 0.0535288 0.0267644 0.999642i \(-0.491480\pi\)
0.0267644 + 0.999642i \(0.491480\pi\)
\(350\) 0 0
\(351\) 22.9129i 1.22300i
\(352\) 0 0
\(353\) 4.58258i 0.243906i −0.992536 0.121953i \(-0.961084\pi\)
0.992536 0.121953i \(-0.0389157\pi\)
\(354\) 0 0
\(355\) 25.7196 16.8375i 1.36506 0.893639i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 12.2474 0.646396 0.323198 0.946331i \(-0.395242\pi\)
0.323198 + 0.946331i \(0.395242\pi\)
\(360\) 0 0
\(361\) −13.0000 −0.684211
\(362\) 0 0
\(363\) 13.0000 0.682323
\(364\) 0 0
\(365\) 8.57321 5.61249i 0.448743 0.293771i
\(366\) 0 0
\(367\) 33.6749i 1.75782i −0.476991 0.878908i \(-0.658273\pi\)
0.476991 0.878908i \(-0.341727\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −22.0454 −1.14147 −0.570734 0.821135i \(-0.693341\pi\)
−0.570734 + 0.821135i \(0.693341\pi\)
\(374\) 0 0
\(375\) −11.0227 + 1.87083i −0.569210 + 0.0966092i
\(376\) 0 0
\(377\) 13.7477i 0.708044i
\(378\) 0 0
\(379\) −19.5959 −1.00657 −0.503287 0.864119i \(-0.667876\pi\)
−0.503287 + 0.864119i \(0.667876\pi\)
\(380\) 0 0
\(381\) −11.0000 −0.563547
\(382\) 0 0
\(383\) 26.1916i 1.33833i 0.743115 + 0.669164i \(0.233348\pi\)
−0.743115 + 0.669164i \(0.766652\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 22.4499i 1.14119i
\(388\) 0 0
\(389\) 14.9666i 0.758838i −0.925225 0.379419i \(-0.876124\pi\)
0.925225 0.379419i \(-0.123876\pi\)
\(390\) 0 0
\(391\) 7.34847 9.16515i 0.371628 0.463502i
\(392\) 0 0
\(393\) 4.58258i 0.231160i
\(394\) 0 0
\(395\) −9.00000 13.7477i −0.452839 0.691723i
\(396\) 0 0
\(397\) 13.7477i 0.689979i 0.938607 + 0.344989i \(0.112117\pi\)
−0.938607 + 0.344989i \(0.887883\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 26.1916i 1.30795i 0.756518 + 0.653973i \(0.226899\pi\)
−0.756518 + 0.653973i \(0.773101\pi\)
\(402\) 0 0
\(403\) 21.0000 1.04608
\(404\) 0 0
\(405\) 1.22474 + 1.87083i 0.0608581 + 0.0929622i
\(406\) 0 0
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) 1.00000 0.0494468 0.0247234 0.999694i \(-0.492129\pi\)
0.0247234 + 0.999694i \(0.492129\pi\)
\(410\) 0 0
\(411\) −14.6969 −0.724947
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 7.00000 4.58258i 0.343616 0.224950i
\(416\) 0 0
\(417\) 22.9129i 1.12205i
\(418\) 0 0
\(419\) 26.9444 1.31632 0.658160 0.752878i \(-0.271335\pi\)
0.658160 + 0.752878i \(0.271335\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 18.0000 0.875190
\(424\) 0 0
\(425\) 4.89898 11.2250i 0.237635 0.544491i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 22.4499i 1.08389i
\(430\) 0 0
\(431\) −31.8434 −1.53384 −0.766921 0.641742i \(-0.778212\pi\)
−0.766921 + 0.641742i \(0.778212\pi\)
\(432\) 0 0
\(433\) 7.34847 0.353145 0.176572 0.984288i \(-0.443499\pi\)
0.176572 + 0.984288i \(0.443499\pi\)
\(434\) 0 0
\(435\) −3.67423 5.61249i −0.176166 0.269098i
\(436\) 0 0
\(437\) 7.34847 9.16515i 0.351525 0.438429i
\(438\) 0 0
\(439\) 4.58258i 0.218714i 0.994003 + 0.109357i \(0.0348792\pi\)
−0.994003 + 0.109357i \(0.965121\pi\)
\(440\) 0 0
\(441\) −14.0000 −0.666667
\(442\) 0 0
\(443\) 33.0000 1.56788 0.783939 0.620838i \(-0.213208\pi\)
0.783939 + 0.620838i \(0.213208\pi\)
\(444\) 0 0
\(445\) 7.00000 4.58258i 0.331832 0.217235i
\(446\) 0 0
\(447\) 14.9666i 0.707897i
\(448\) 0 0
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 0 0
\(451\) 14.6969 0.692052
\(452\) 0 0
\(453\) 13.7477i 0.645925i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 12.2474 0.572911 0.286456 0.958093i \(-0.407523\pi\)
0.286456 + 0.958093i \(0.407523\pi\)
\(458\) 0 0
\(459\) 12.2474 0.571662
\(460\) 0 0
\(461\) 15.0000 0.698620 0.349310 0.937007i \(-0.386416\pi\)
0.349310 + 0.937007i \(0.386416\pi\)
\(462\) 0 0
\(463\) 14.0000 0.650635 0.325318 0.945605i \(-0.394529\pi\)
0.325318 + 0.945605i \(0.394529\pi\)
\(464\) 0 0
\(465\) −8.57321 + 5.61249i −0.397573 + 0.260273i
\(466\) 0 0
\(467\) 7.48331i 0.346287i −0.984897 0.173143i \(-0.944608\pi\)
0.984897 0.173143i \(-0.0553924\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 2.44949 0.112867
\(472\) 0 0
\(473\) 54.9909i 2.52848i
\(474\) 0 0
\(475\) 4.89898 11.2250i 0.224781 0.515037i
\(476\) 0 0
\(477\) 9.79796 0.448618
\(478\) 0 0
\(479\) −31.8434 −1.45496 −0.727480 0.686129i \(-0.759309\pi\)
−0.727480 + 0.686129i \(0.759309\pi\)
\(480\) 0 0
\(481\) 22.4499i 1.02363i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 12.0000 + 18.3303i 0.544892 + 0.832336i
\(486\) 0 0
\(487\) −1.00000 −0.0453143 −0.0226572 0.999743i \(-0.507213\pi\)
−0.0226572 + 0.999743i \(0.507213\pi\)
\(488\) 0 0
\(489\) 13.0000 0.587880
\(490\) 0 0
\(491\) 4.58258i 0.206809i −0.994639 0.103404i \(-0.967026\pi\)
0.994639 0.103404i \(-0.0329736\pi\)
\(492\) 0 0
\(493\) 7.34847 0.330958
\(494\) 0 0
\(495\) 12.0000 + 18.3303i 0.539360 + 0.823886i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 22.9129i 1.02572i 0.858472 + 0.512861i \(0.171414\pi\)
−0.858472 + 0.512861i \(0.828586\pi\)
\(500\) 0 0
\(501\) −18.0000 −0.804181
\(502\) 0 0
\(503\) 14.9666i 0.667329i 0.942692 + 0.333665i \(0.108285\pi\)
−0.942692 + 0.333665i \(0.891715\pi\)
\(504\) 0 0
\(505\) 14.6969 + 22.4499i 0.654005 + 0.999009i
\(506\) 0 0
\(507\) −8.00000 −0.355292
\(508\) 0 0
\(509\) −9.00000 −0.398918 −0.199459 0.979906i \(-0.563918\pi\)
−0.199459 + 0.979906i \(0.563918\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 12.2474 0.540738
\(514\) 0 0
\(515\) −21.0000 + 13.7477i −0.925371 + 0.605797i
\(516\) 0 0
\(517\) 44.0908 1.93911
\(518\) 0 0
\(519\) 18.3303i 0.804611i
\(520\) 0 0
\(521\) 7.48331i 0.327850i −0.986473 0.163925i \(-0.947584\pi\)
0.986473 0.163925i \(-0.0524155\pi\)
\(522\) 0 0
\(523\) 22.4499i 0.981668i 0.871253 + 0.490834i \(0.163308\pi\)
−0.871253 + 0.490834i \(0.836692\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 11.2250i 0.488967i
\(528\) 0 0
\(529\) −5.00000 22.4499i −0.217391 0.976085i
\(530\) 0 0
\(531\) 18.3303i 0.795467i
\(532\) 0 0
\(533\) 13.7477i 0.595480i
\(534\) 0 0
\(535\) −7.00000 + 4.58258i −0.302636 + 0.198122i
\(536\) 0 0
\(537\) 13.7477i 0.593258i
\(538\) 0 0
\(539\) −34.2929 −1.47710
\(540\) 0 0
\(541\) 13.0000 0.558914 0.279457 0.960158i \(-0.409846\pi\)
0.279457 + 0.960158i \(0.409846\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 21.0000 13.7477i 0.899541 0.588888i
\(546\) 0 0
\(547\) 19.0000 0.812381 0.406191 0.913788i \(-0.366857\pi\)
0.406191 + 0.913788i \(0.366857\pi\)
\(548\) 0 0
\(549\) 22.4499i 0.958140i
\(550\) 0 0
\(551\) 7.34847 0.313055
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 6.00000 + 9.16515i 0.254686 + 0.389039i
\(556\) 0 0
\(557\) −36.7423 −1.55682 −0.778412 0.627754i \(-0.783974\pi\)
−0.778412 + 0.627754i \(0.783974\pi\)
\(558\) 0 0
\(559\) 51.4393 2.17565
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 0 0
\(563\) 29.9333i 1.26154i 0.775971 + 0.630768i \(0.217260\pi\)
−0.775971 + 0.630768i \(0.782740\pi\)
\(564\) 0 0
\(565\) 18.0000 + 27.4955i 0.757266 + 1.15674i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 18.7083i 0.784292i 0.919903 + 0.392146i \(0.128267\pi\)
−0.919903 + 0.392146i \(0.871733\pi\)
\(570\) 0 0
\(571\) 31.8434 1.33260 0.666302 0.745682i \(-0.267876\pi\)
0.666302 + 0.745682i \(0.267876\pi\)
\(572\) 0 0
\(573\) −12.2474 −0.511645
\(574\) 0 0
\(575\) −11.1464 21.2310i −0.464838 0.885396i
\(576\) 0 0
\(577\) 22.9129i 0.953876i 0.878937 + 0.476938i \(0.158253\pi\)
−0.878937 + 0.476938i \(0.841747\pi\)
\(578\) 0 0
\(579\) 22.9129i 0.952227i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 24.0000 0.993978
\(584\) 0 0
\(585\) −17.1464 + 11.2250i −0.708918 + 0.464095i
\(586\) 0 0
\(587\) −3.00000 −0.123823 −0.0619116 0.998082i \(-0.519720\pi\)
−0.0619116 + 0.998082i \(0.519720\pi\)
\(588\) 0 0
\(589\) 11.2250i 0.462517i
\(590\) 0 0
\(591\) 4.58258i 0.188502i
\(592\) 0 0
\(593\) 36.6606i 1.50547i 0.658323 + 0.752735i \(0.271266\pi\)
−0.658323 + 0.752735i \(0.728734\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −19.5959 −0.802008
\(598\) 0 0
\(599\) 45.8258i 1.87239i −0.351481 0.936195i \(-0.614322\pi\)
0.351481 0.936195i \(-0.385678\pi\)
\(600\) 0 0
\(601\) −11.0000 −0.448699 −0.224350 0.974509i \(-0.572026\pi\)
−0.224350 + 0.974509i \(0.572026\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 15.9217 + 24.3208i 0.647308 + 0.988780i
\(606\) 0 0
\(607\) −4.00000 −0.162355 −0.0811775 0.996700i \(-0.525868\pi\)
−0.0811775 + 0.996700i \(0.525868\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 41.2432i 1.66852i
\(612\) 0 0
\(613\) −22.0454 −0.890406 −0.445203 0.895430i \(-0.646868\pi\)
−0.445203 + 0.895430i \(0.646868\pi\)
\(614\) 0 0
\(615\) −3.67423 5.61249i −0.148159 0.226317i
\(616\) 0 0
\(617\) 7.34847 0.295838 0.147919 0.988999i \(-0.452742\pi\)
0.147919 + 0.988999i \(0.452742\pi\)
\(618\) 0 0
\(619\) −14.6969 −0.590720 −0.295360 0.955386i \(-0.595440\pi\)
−0.295360 + 0.955386i \(0.595440\pi\)
\(620\) 0 0
\(621\) 15.0000 18.7083i 0.601929 0.750738i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −17.0000 18.3303i −0.680000 0.733212i
\(626\) 0 0
\(627\) 12.0000 0.479234
\(628\) 0 0
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −7.34847 −0.292538 −0.146269 0.989245i \(-0.546726\pi\)
−0.146269 + 0.989245i \(0.546726\pi\)
\(632\) 0 0
\(633\) 18.3303i 0.728564i
\(634\) 0 0
\(635\) −13.4722 20.5791i −0.534628 0.816657i
\(636\) 0 0
\(637\) 32.0780i 1.27098i
\(638\) 0 0
\(639\) 27.4955i 1.08770i
\(640\) 0 0
\(641\) 48.6415i 1.92123i −0.277890 0.960613i \(-0.589635\pi\)
0.277890 0.960613i \(-0.410365\pi\)
\(642\) 0 0
\(643\) 11.2250i 0.442670i −0.975198 0.221335i \(-0.928959\pi\)
0.975198 0.221335i \(-0.0710414\pi\)
\(644\) 0 0
\(645\) −21.0000 + 13.7477i −0.826874 + 0.541316i
\(646\) 0 0
\(647\) −33.0000 −1.29736 −0.648682 0.761060i \(-0.724679\pi\)
−0.648682 + 0.761060i \(0.724679\pi\)
\(648\) 0 0
\(649\) 44.8999i 1.76247i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 50.4083i 1.97263i 0.164870 + 0.986315i \(0.447279\pi\)
−0.164870 + 0.986315i \(0.552721\pi\)
\(654\) 0 0
\(655\) 8.57321 5.61249i 0.334983 0.219298i
\(656\) 0 0
\(657\) 9.16515i 0.357567i
\(658\) 0 0
\(659\) −41.6413 −1.62212 −0.811058 0.584966i \(-0.801108\pi\)
−0.811058 + 0.584966i \(0.801108\pi\)
\(660\) 0 0
\(661\) 22.4499i 0.873202i −0.899655 0.436601i \(-0.856182\pi\)
0.899655 0.436601i \(-0.143818\pi\)
\(662\) 0 0
\(663\) 11.2250i 0.435942i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 9.00000 11.2250i 0.348481 0.434633i
\(668\) 0 0
\(669\) 22.0000 0.850569
\(670\) 0 0
\(671\) 54.9909i 2.12290i
\(672\) 0 0
\(673\) 13.7477i 0.529936i −0.964257 0.264968i \(-0.914639\pi\)
0.964257 0.264968i \(-0.0853614\pi\)
\(674\) 0 0
\(675\) 10.0000 22.9129i 0.384900 0.881917i
\(676\) 0 0
\(677\) 4.89898 0.188283 0.0941415 0.995559i \(-0.469989\pi\)
0.0941415 + 0.995559i \(0.469989\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 7.48331i 0.286761i
\(682\) 0 0
\(683\) 3.00000 0.114792 0.0573959 0.998351i \(-0.481720\pi\)
0.0573959 + 0.998351i \(0.481720\pi\)
\(684\) 0 0
\(685\) −18.0000 27.4955i −0.687745 1.05055i
\(686\) 0 0
\(687\) 11.2250i 0.428259i
\(688\) 0 0
\(689\) 22.4499i 0.855275i
\(690\) 0 0
\(691\) 9.16515i 0.348659i −0.984687 0.174329i \(-0.944224\pi\)
0.984687 0.174329i \(-0.0557758\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −42.8661 + 28.0624i −1.62600 + 1.06447i
\(696\) 0 0
\(697\) 7.34847 0.278343
\(698\) 0 0
\(699\) 4.58258i 0.173329i
\(700\) 0 0
\(701\) 14.9666i 0.565282i −0.959226 0.282641i \(-0.908790\pi\)
0.959226 0.282641i \(-0.0912105\pi\)
\(702\) 0 0
\(703\) −12.0000 −0.452589
\(704\) 0 0
\(705\) −11.0227 16.8375i −0.415139 0.634135i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 44.8999i 1.68625i −0.537717 0.843125i \(-0.680713\pi\)
0.537717 0.843125i \(-0.319287\pi\)
\(710\) 0 0
\(711\) 14.6969 0.551178
\(712\) 0 0
\(713\) −17.1464 13.7477i −0.642139 0.514856i
\(714\) 0 0
\(715\) −42.0000 + 27.4955i −1.57071 + 1.02827i
\(716\) 0 0
\(717\) 4.58258i 0.171139i
\(718\) 0 0
\(719\) 27.4955i 1.02541i −0.858566 0.512704i \(-0.828644\pi\)
0.858566 0.512704i \(-0.171356\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 11.2250i 0.417461i
\(724\) 0 0
\(725\) 6.00000 13.7477i 0.222834 0.510578i
\(726\) 0 0
\(727\) 33.6749i 1.24893i −0.781051 0.624467i \(-0.785316\pi\)
0.781051 0.624467i \(-0.214684\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 27.4955i 1.01696i
\(732\) 0 0
\(733\) 51.4393 1.89995 0.949977 0.312321i \(-0.101106\pi\)
0.949977 + 0.312321i \(0.101106\pi\)
\(734\) 0 0
\(735\) 8.57321 + 13.0958i 0.316228 + 0.483046i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 4.58258i 0.168573i −0.996442 0.0842864i \(-0.973139\pi\)
0.996442 0.0842864i \(-0.0268611\pi\)
\(740\) 0 0
\(741\) 11.2250i 0.412360i
\(742\) 0 0
\(743\) 26.1916i 0.960877i 0.877029 + 0.480438i \(0.159522\pi\)
−0.877029 + 0.480438i \(0.840478\pi\)
\(744\) 0 0
\(745\) 28.0000 18.3303i 1.02584 0.671570i
\(746\) 0 0
\(747\) 7.48331i 0.273800i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 12.2474 0.446916 0.223458 0.974714i \(-0.428265\pi\)
0.223458 + 0.974714i \(0.428265\pi\)
\(752\) 0 0
\(753\) 12.2474 0.446322
\(754\) 0 0
\(755\) 25.7196 16.8375i 0.936034 0.612778i
\(756\) 0 0
\(757\) −48.9898 −1.78056 −0.890282 0.455409i \(-0.849493\pi\)
−0.890282 + 0.455409i \(0.849493\pi\)
\(758\) 0 0
\(759\) 14.6969 18.3303i 0.533465 0.665348i
\(760\) 0 0
\(761\) −45.0000 −1.63125 −0.815624 0.578582i \(-0.803606\pi\)
−0.815624 + 0.578582i \(0.803606\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 6.00000 + 9.16515i 0.216930 + 0.331367i
\(766\) 0 0
\(767\) −42.0000 −1.51653
\(768\) 0 0
\(769\) 33.6749i 1.21435i 0.794569 + 0.607174i \(0.207697\pi\)
−0.794569 + 0.607174i \(0.792303\pi\)
\(770\) 0 0
\(771\) 22.9129i 0.825187i
\(772\) 0 0
\(773\) 17.1464 0.616714 0.308357 0.951271i \(-0.400221\pi\)
0.308357 + 0.951271i \(0.400221\pi\)
\(774\) 0 0
\(775\) −21.0000 9.16515i −0.754342 0.329222i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 7.34847 0.263286
\(780\) 0 0
\(781\) 67.3498i 2.40997i
\(782\) 0 0
\(783\) 15.0000 0.536056
\(784\) 0 0
\(785\) 3.00000 + 4.58258i 0.107075 + 0.163559i
\(786\) 0 0
\(787\) 44.8999i 1.60051i 0.599661 + 0.800254i \(0.295302\pi\)
−0.599661 + 0.800254i \(0.704698\pi\)
\(788\) 0 0
\(789\) 18.7083i 0.666033i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 51.4393 1.82666
\(794\) 0 0
\(795\) −6.00000 9.16515i −0.212798 0.325054i
\(796\) 0 0
\(797\) −7.34847 −0.260296 −0.130148 0.991495i \(-0.541545\pi\)
−0.130148 + 0.991495i \(0.541545\pi\)
\(798\) 0 0
\(799\) 22.0454 0.779910
\(800\) 0 0
\(801\) 7.48331i 0.264410i
\(802\) 0 0
\(803\) 22.4499i 0.792241i
\(804\) 0 0