Properties

Label 180.3.b.a.89.1
Level $180$
Weight $3$
Character 180.89
Analytic conductor $4.905$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 180.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.90464475849\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{23})\)
Defining polynomial: \( x^{4} + 24x^{2} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 89.1
Root \(-4.09827i\) of defining polynomial
Character \(\chi\) \(=\) 180.89
Dual form 180.3.b.a.89.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-4.79583 - 1.41421i) q^{5} +6.78233i q^{7} +O(q^{10})\) \(q+(-4.79583 - 1.41421i) q^{5} +6.78233i q^{7} +9.89949i q^{11} +20.3470i q^{13} -19.1833 q^{17} +12.0000 q^{19} -9.59166 q^{23} +(21.0000 + 13.5647i) q^{25} -8.48528i q^{29} -38.0000 q^{31} +(9.59166 - 32.5269i) q^{35} +6.78233i q^{37} +69.2965i q^{41} -67.8233i q^{43} +76.7333 q^{47} +3.00000 q^{49} +(14.0000 - 47.4763i) q^{55} -83.4386i q^{59} -70.0000 q^{61} +(28.7750 - 97.5807i) q^{65} -108.517i q^{67} +118.794i q^{71} +13.5647i q^{73} -67.1416 q^{77} +30.0000 q^{79} +134.283 q^{83} +(92.0000 + 27.1293i) q^{85} +32.5269i q^{89} -138.000 q^{91} +(-57.5500 - 16.9706i) q^{95} +94.9526i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 48 q^{19} + 84 q^{25} - 152 q^{31} + 12 q^{49} + 56 q^{55} - 280 q^{61} + 120 q^{79} + 368 q^{85} - 552 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(91\) \(101\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −4.79583 1.41421i −0.959166 0.282843i
\(6\) 0 0
\(7\) 6.78233i 0.968904i 0.874818 + 0.484452i \(0.160981\pi\)
−0.874818 + 0.484452i \(0.839019\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 9.89949i 0.899954i 0.893040 + 0.449977i \(0.148568\pi\)
−0.893040 + 0.449977i \(0.851432\pi\)
\(12\) 0 0
\(13\) 20.3470i 1.56515i 0.622554 + 0.782577i \(0.286095\pi\)
−0.622554 + 0.782577i \(0.713905\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −19.1833 −1.12843 −0.564215 0.825628i \(-0.690821\pi\)
−0.564215 + 0.825628i \(0.690821\pi\)
\(18\) 0 0
\(19\) 12.0000 0.631579 0.315789 0.948829i \(-0.397731\pi\)
0.315789 + 0.948829i \(0.397731\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −9.59166 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 0 0
\(25\) 21.0000 + 13.5647i 0.840000 + 0.542586i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.48528i 0.292596i −0.989241 0.146298i \(-0.953264\pi\)
0.989241 0.146298i \(-0.0467358\pi\)
\(30\) 0 0
\(31\) −38.0000 −1.22581 −0.612903 0.790158i \(-0.709998\pi\)
−0.612903 + 0.790158i \(0.709998\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 9.59166 32.5269i 0.274048 0.929340i
\(36\) 0 0
\(37\) 6.78233i 0.183306i 0.995791 + 0.0916531i \(0.0292151\pi\)
−0.995791 + 0.0916531i \(0.970785\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 69.2965i 1.69016i 0.534642 + 0.845079i \(0.320447\pi\)
−0.534642 + 0.845079i \(0.679553\pi\)
\(42\) 0 0
\(43\) 67.8233i 1.57729i −0.614851 0.788643i \(-0.710784\pi\)
0.614851 0.788643i \(-0.289216\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 76.7333 1.63262 0.816312 0.577612i \(-0.196015\pi\)
0.816312 + 0.577612i \(0.196015\pi\)
\(48\) 0 0
\(49\) 3.00000 0.0612245
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 14.0000 47.4763i 0.254545 0.863206i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 83.4386i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(60\) 0 0
\(61\) −70.0000 −1.14754 −0.573770 0.819016i \(-0.694520\pi\)
−0.573770 + 0.819016i \(0.694520\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 28.7750 97.5807i 0.442692 1.50124i
\(66\) 0 0
\(67\) 108.517i 1.61966i −0.586664 0.809830i \(-0.699559\pi\)
0.586664 0.809830i \(-0.300441\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 118.794i 1.67315i 0.547849 + 0.836577i \(0.315447\pi\)
−0.547849 + 0.836577i \(0.684553\pi\)
\(72\) 0 0
\(73\) 13.5647i 0.185817i 0.995675 + 0.0929086i \(0.0296164\pi\)
−0.995675 + 0.0929086i \(0.970384\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −67.1416 −0.871969
\(78\) 0 0
\(79\) 30.0000 0.379747 0.189873 0.981809i \(-0.439192\pi\)
0.189873 + 0.981809i \(0.439192\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 134.283 1.61787 0.808935 0.587897i \(-0.200044\pi\)
0.808935 + 0.587897i \(0.200044\pi\)
\(84\) 0 0
\(85\) 92.0000 + 27.1293i 1.08235 + 0.319168i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 32.5269i 0.365471i 0.983162 + 0.182735i \(0.0584952\pi\)
−0.983162 + 0.182735i \(0.941505\pi\)
\(90\) 0 0
\(91\) −138.000 −1.51648
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −57.5500 16.9706i −0.605789 0.178638i
\(96\) 0 0
\(97\) 94.9526i 0.978893i 0.872033 + 0.489446i \(0.162801\pi\)
−0.872033 + 0.489446i \(0.837199\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 79.1960i 0.784118i 0.919940 + 0.392059i \(0.128237\pi\)
−0.919940 + 0.392059i \(0.871763\pi\)
\(102\) 0 0
\(103\) 88.1703i 0.856022i 0.903773 + 0.428011i \(0.140786\pi\)
−0.903773 + 0.428011i \(0.859214\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −57.5500 −0.537850 −0.268925 0.963161i \(-0.586668\pi\)
−0.268925 + 0.963161i \(0.586668\pi\)
\(108\) 0 0
\(109\) 74.0000 0.678899 0.339450 0.940624i \(-0.389759\pi\)
0.339450 + 0.940624i \(0.389759\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −172.650 −1.52788 −0.763938 0.645290i \(-0.776737\pi\)
−0.763938 + 0.645290i \(0.776737\pi\)
\(114\) 0 0
\(115\) 46.0000 + 13.5647i 0.400000 + 0.117954i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 130.108i 1.09334i
\(120\) 0 0
\(121\) 23.0000 0.190083
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −81.5291 94.7523i −0.652233 0.758018i
\(126\) 0 0
\(127\) 169.558i 1.33510i 0.744563 + 0.667552i \(0.232658\pi\)
−0.744563 + 0.667552i \(0.767342\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 165.463i 1.26308i 0.775345 + 0.631538i \(0.217576\pi\)
−0.775345 + 0.631538i \(0.782424\pi\)
\(132\) 0 0
\(133\) 81.3880i 0.611940i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 115.100 0.840146 0.420073 0.907490i \(-0.362005\pi\)
0.420073 + 0.907490i \(0.362005\pi\)
\(138\) 0 0
\(139\) 62.0000 0.446043 0.223022 0.974814i \(-0.428408\pi\)
0.223022 + 0.974814i \(0.428408\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −201.425 −1.40857
\(144\) 0 0
\(145\) −12.0000 + 40.6940i −0.0827586 + 0.280648i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 158.392i 1.06303i 0.847048 + 0.531517i \(0.178378\pi\)
−0.847048 + 0.531517i \(0.821622\pi\)
\(150\) 0 0
\(151\) 70.0000 0.463576 0.231788 0.972766i \(-0.425542\pi\)
0.231788 + 0.972766i \(0.425542\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 182.242 + 53.7401i 1.17575 + 0.346710i
\(156\) 0 0
\(157\) 47.4763i 0.302397i −0.988503 0.151198i \(-0.951687\pi\)
0.988503 0.151198i \(-0.0483132\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 65.0538i 0.404061i
\(162\) 0 0
\(163\) 94.9526i 0.582531i −0.956642 0.291266i \(-0.905924\pi\)
0.956642 0.291266i \(-0.0940764\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −201.425 −1.20614 −0.603069 0.797689i \(-0.706055\pi\)
−0.603069 + 0.797689i \(0.706055\pi\)
\(168\) 0 0
\(169\) −245.000 −1.44970
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 268.567 1.55241 0.776204 0.630482i \(-0.217143\pi\)
0.776204 + 0.630482i \(0.217143\pi\)
\(174\) 0 0
\(175\) −92.0000 + 142.429i −0.525714 + 0.813880i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 134.350i 0.750560i −0.926911 0.375280i \(-0.877547\pi\)
0.926911 0.375280i \(-0.122453\pi\)
\(180\) 0 0
\(181\) −22.0000 −0.121547 −0.0607735 0.998152i \(-0.519357\pi\)
−0.0607735 + 0.998152i \(0.519357\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 9.59166 32.5269i 0.0518468 0.175821i
\(186\) 0 0
\(187\) 189.905i 1.01554i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 195.161i 1.02179i −0.859644 0.510894i \(-0.829314\pi\)
0.859644 0.510894i \(-0.170686\pi\)
\(192\) 0 0
\(193\) 271.293i 1.40566i −0.711356 0.702832i \(-0.751919\pi\)
0.711356 0.702832i \(-0.248081\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 163.058 0.827707 0.413853 0.910344i \(-0.364183\pi\)
0.413853 + 0.910344i \(0.364183\pi\)
\(198\) 0 0
\(199\) 294.000 1.47739 0.738693 0.674042i \(-0.235443\pi\)
0.738693 + 0.674042i \(0.235443\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 57.5500 0.283497
\(204\) 0 0
\(205\) 98.0000 332.334i 0.478049 1.62114i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 118.794i 0.568392i
\(210\) 0 0
\(211\) −42.0000 −0.199052 −0.0995261 0.995035i \(-0.531733\pi\)
−0.0995261 + 0.995035i \(0.531733\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −95.9166 + 325.269i −0.446124 + 1.51288i
\(216\) 0 0
\(217\) 257.729i 1.18769i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 390.323i 1.76617i
\(222\) 0 0
\(223\) 196.688i 0.882007i 0.897505 + 0.441004i \(0.145377\pi\)
−0.897505 + 0.441004i \(0.854623\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 268.567 1.18311 0.591556 0.806264i \(-0.298514\pi\)
0.591556 + 0.806264i \(0.298514\pi\)
\(228\) 0 0
\(229\) −422.000 −1.84279 −0.921397 0.388622i \(-0.872951\pi\)
−0.921397 + 0.388622i \(0.872951\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 211.017 0.905651 0.452825 0.891599i \(-0.350416\pi\)
0.452825 + 0.891599i \(0.350416\pi\)
\(234\) 0 0
\(235\) −368.000 108.517i −1.56596 0.461776i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 70.7107i 0.295861i 0.988998 + 0.147930i \(0.0472611\pi\)
−0.988998 + 0.147930i \(0.952739\pi\)
\(240\) 0 0
\(241\) 280.000 1.16183 0.580913 0.813966i \(-0.302696\pi\)
0.580913 + 0.813966i \(0.302696\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −14.3875 4.24264i −0.0587245 0.0173169i
\(246\) 0 0
\(247\) 244.164i 0.988518i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 80.6102i 0.321156i −0.987023 0.160578i \(-0.948664\pi\)
0.987023 0.160578i \(-0.0513358\pi\)
\(252\) 0 0
\(253\) 94.9526i 0.375307i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) −46.0000 −0.177606
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −67.1416 −0.255291 −0.127646 0.991820i \(-0.540742\pi\)
−0.127646 + 0.991820i \(0.540742\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 263.044i 0.977858i 0.872324 + 0.488929i \(0.162612\pi\)
−0.872324 + 0.488929i \(0.837388\pi\)
\(270\) 0 0
\(271\) 322.000 1.18819 0.594096 0.804394i \(-0.297510\pi\)
0.594096 + 0.804394i \(0.297510\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −134.283 + 207.889i −0.488303 + 0.755961i
\(276\) 0 0
\(277\) 33.9116i 0.122425i −0.998125 0.0612124i \(-0.980503\pi\)
0.998125 0.0612124i \(-0.0194967\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 326.683i 1.16257i −0.813699 0.581287i \(-0.802549\pi\)
0.813699 0.581287i \(-0.197451\pi\)
\(282\) 0 0
\(283\) 108.517i 0.383453i 0.981448 + 0.191727i \(0.0614087\pi\)
−0.981448 + 0.191727i \(0.938591\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −469.991 −1.63760
\(288\) 0 0
\(289\) 79.0000 0.273356
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −201.425 −0.687457 −0.343729 0.939069i \(-0.611690\pi\)
−0.343729 + 0.939069i \(0.611690\pi\)
\(294\) 0 0
\(295\) −118.000 + 400.157i −0.400000 + 1.35647i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 195.161i 0.652714i
\(300\) 0 0
\(301\) 460.000 1.52824
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 335.708 + 98.9949i 1.10068 + 0.324574i
\(306\) 0 0
\(307\) 162.776i 0.530215i 0.964219 + 0.265107i \(0.0854074\pi\)
−0.964219 + 0.265107i \(0.914593\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 178.191i 0.572961i −0.958086 0.286481i \(-0.907515\pi\)
0.958086 0.286481i \(-0.0924854\pi\)
\(312\) 0 0
\(313\) 54.2586i 0.173350i 0.996237 + 0.0866751i \(0.0276242\pi\)
−0.996237 + 0.0866751i \(0.972376\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −278.158 −0.877471 −0.438735 0.898616i \(-0.644573\pi\)
−0.438735 + 0.898616i \(0.644573\pi\)
\(318\) 0 0
\(319\) 84.0000 0.263323
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −230.200 −0.712693
\(324\) 0 0
\(325\) −276.000 + 427.287i −0.849231 + 1.31473i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 520.431i 1.58186i
\(330\) 0 0
\(331\) 40.0000 0.120846 0.0604230 0.998173i \(-0.480755\pi\)
0.0604230 + 0.998173i \(0.480755\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −153.467 + 520.431i −0.458109 + 1.55352i
\(336\) 0 0
\(337\) 311.987i 0.925778i 0.886416 + 0.462889i \(0.153187\pi\)
−0.886416 + 0.462889i \(0.846813\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 376.181i 1.10317i
\(342\) 0 0
\(343\) 352.681i 1.02822i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 402.850 1.16095 0.580475 0.814278i \(-0.302867\pi\)
0.580475 + 0.814278i \(0.302867\pi\)
\(348\) 0 0
\(349\) −406.000 −1.16332 −0.581662 0.813431i \(-0.697597\pi\)
−0.581662 + 0.813431i \(0.697597\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −537.133 −1.52162 −0.760812 0.648973i \(-0.775199\pi\)
−0.760812 + 0.648973i \(0.775199\pi\)
\(354\) 0 0
\(355\) 168.000 569.716i 0.473239 1.60483i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 161.220i 0.449082i −0.974465 0.224541i \(-0.927912\pi\)
0.974465 0.224541i \(-0.0720882\pi\)
\(360\) 0 0
\(361\) −217.000 −0.601108
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 19.1833 65.0538i 0.0525571 0.178230i
\(366\) 0 0
\(367\) 250.946i 0.683777i 0.939740 + 0.341889i \(0.111067\pi\)
−0.939740 + 0.341889i \(0.888933\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 101.735i 0.272748i 0.990657 + 0.136374i \(0.0435449\pi\)
−0.990657 + 0.136374i \(0.956455\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 172.650 0.457957
\(378\) 0 0
\(379\) 538.000 1.41953 0.709763 0.704441i \(-0.248802\pi\)
0.709763 + 0.704441i \(0.248802\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 268.567 0.701218 0.350609 0.936522i \(-0.385975\pi\)
0.350609 + 0.936522i \(0.385975\pi\)
\(384\) 0 0
\(385\) 322.000 + 94.9526i 0.836364 + 0.246630i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 494.975i 1.27243i 0.771512 + 0.636214i \(0.219501\pi\)
−0.771512 + 0.636214i \(0.780499\pi\)
\(390\) 0 0
\(391\) 184.000 0.470588
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −143.875 42.4264i −0.364240 0.107409i
\(396\) 0 0
\(397\) 278.076i 0.700442i −0.936667 0.350221i \(-0.886106\pi\)
0.936667 0.350221i \(-0.113894\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 247.487i 0.617175i −0.951196 0.308588i \(-0.900144\pi\)
0.951196 0.308588i \(-0.0998563\pi\)
\(402\) 0 0
\(403\) 773.186i 1.91857i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −67.1416 −0.164967
\(408\) 0 0
\(409\) −242.000 −0.591687 −0.295844 0.955236i \(-0.595601\pi\)
−0.295844 + 0.955236i \(0.595601\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 565.908 1.37024
\(414\) 0 0
\(415\) −644.000 189.905i −1.55181 0.457603i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 790.545i 1.88674i −0.331738 0.943372i \(-0.607635\pi\)
0.331738 0.943372i \(-0.392365\pi\)
\(420\) 0 0
\(421\) 358.000 0.850356 0.425178 0.905110i \(-0.360211\pi\)
0.425178 + 0.905110i \(0.360211\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −402.850 260.215i −0.947882 0.612271i
\(426\) 0 0
\(427\) 474.763i 1.11186i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 14.1421i 0.0328124i 0.999865 + 0.0164062i \(0.00522249\pi\)
−0.999865 + 0.0164062i \(0.994778\pi\)
\(432\) 0 0
\(433\) 257.729i 0.595216i 0.954688 + 0.297608i \(0.0961888\pi\)
−0.954688 + 0.297608i \(0.903811\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −115.100 −0.263387
\(438\) 0 0
\(439\) −690.000 −1.57175 −0.785877 0.618383i \(-0.787788\pi\)
−0.785877 + 0.618383i \(0.787788\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 402.850 0.909368 0.454684 0.890653i \(-0.349752\pi\)
0.454684 + 0.890653i \(0.349752\pi\)
\(444\) 0 0
\(445\) 46.0000 155.994i 0.103371 0.350547i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 80.6102i 0.179533i 0.995963 + 0.0897663i \(0.0286120\pi\)
−0.995963 + 0.0897663i \(0.971388\pi\)
\(450\) 0 0
\(451\) −686.000 −1.52106
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 661.825 + 195.161i 1.45456 + 0.428926i
\(456\) 0 0
\(457\) 339.116i 0.742049i −0.928623 0.371025i \(-0.879007\pi\)
0.928623 0.371025i \(-0.120993\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 644.881i 1.39888i 0.714694 + 0.699438i \(0.246566\pi\)
−0.714694 + 0.699438i \(0.753434\pi\)
\(462\) 0 0
\(463\) 264.511i 0.571298i −0.958334 0.285649i \(-0.907791\pi\)
0.958334 0.285649i \(-0.0922091\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 460.400 0.985867 0.492933 0.870067i \(-0.335925\pi\)
0.492933 + 0.870067i \(0.335925\pi\)
\(468\) 0 0
\(469\) 736.000 1.56930
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 671.416 1.41949
\(474\) 0 0
\(475\) 252.000 + 162.776i 0.530526 + 0.342686i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 79.1960i 0.165336i 0.996577 + 0.0826680i \(0.0263441\pi\)
−0.996577 + 0.0826680i \(0.973656\pi\)
\(480\) 0 0
\(481\) −138.000 −0.286902
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 134.283 455.377i 0.276873 0.938921i
\(486\) 0 0
\(487\) 793.533i 1.62943i −0.579861 0.814715i \(-0.696893\pi\)
0.579861 0.814715i \(-0.303107\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 564.271i 1.14923i 0.818424 + 0.574614i \(0.194848\pi\)
−0.818424 + 0.574614i \(0.805152\pi\)
\(492\) 0 0
\(493\) 162.776i 0.330174i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −805.700 −1.62113
\(498\) 0 0
\(499\) −72.0000 −0.144289 −0.0721443 0.997394i \(-0.522984\pi\)
−0.0721443 + 0.997394i \(0.522984\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −230.200 −0.457654 −0.228827 0.973467i \(-0.573489\pi\)
−0.228827 + 0.973467i \(0.573489\pi\)
\(504\) 0 0
\(505\) 112.000 379.810i 0.221782 0.752100i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 492.146i 0.966889i −0.875375 0.483444i \(-0.839386\pi\)
0.875375 0.483444i \(-0.160614\pi\)
\(510\) 0 0
\(511\) −92.0000 −0.180039
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 124.692 422.850i 0.242120 0.821068i
\(516\) 0 0
\(517\) 759.621i 1.46929i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 69.2965i 0.133007i −0.997786 0.0665033i \(-0.978816\pi\)
0.997786 0.0665033i \(-0.0211843\pi\)
\(522\) 0 0
\(523\) 339.116i 0.648406i −0.945987 0.324203i \(-0.894904\pi\)
0.945987 0.324203i \(-0.105096\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 728.966 1.38324
\(528\) 0 0
\(529\) −437.000 −0.826087
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1409.97 −2.64536
\(534\) 0 0
\(535\) 276.000 + 81.3880i 0.515888 + 0.152127i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 29.6985i 0.0550992i
\(540\) 0 0
\(541\) −590.000 −1.09057 −0.545287 0.838250i \(-0.683579\pi\)
−0.545287 + 0.838250i \(0.683579\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −354.892 104.652i −0.651177 0.192022i
\(546\) 0 0
\(547\) 230.599i 0.421571i 0.977532 + 0.210785i \(0.0676021\pi\)
−0.977532 + 0.210785i \(0.932398\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 101.823i 0.184797i
\(552\) 0 0
\(553\) 203.470i 0.367938i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 690.600 1.23986 0.619928 0.784659i \(-0.287162\pi\)
0.619928 + 0.784659i \(0.287162\pi\)
\(558\) 0 0
\(559\) 1380.00 2.46869
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −19.1833 −0.0340734 −0.0170367 0.999855i \(-0.505423\pi\)
−0.0170367 + 0.999855i \(0.505423\pi\)
\(564\) 0 0
\(565\) 828.000 + 244.164i 1.46549 + 0.432148i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 108.894i 0.191379i 0.995411 + 0.0956893i \(0.0305055\pi\)
−0.995411 + 0.0956893i \(0.969494\pi\)
\(570\) 0 0
\(571\) −704.000 −1.23292 −0.616462 0.787384i \(-0.711435\pi\)
−0.616462 + 0.787384i \(0.711435\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −201.425 130.108i −0.350304 0.226274i
\(576\) 0 0
\(577\) 1125.87i 1.95124i 0.219462 + 0.975621i \(0.429570\pi\)
−0.219462 + 0.975621i \(0.570430\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 910.754i 1.56756i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −978.350 −1.66669 −0.833347 0.552750i \(-0.813578\pi\)
−0.833347 + 0.552750i \(0.813578\pi\)
\(588\) 0 0
\(589\) −456.000 −0.774194
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) −184.000 + 623.974i −0.309244 + 1.04870i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 296.985i 0.495801i −0.968785 0.247901i \(-0.920259\pi\)
0.968785 0.247901i \(-0.0797406\pi\)
\(600\) 0 0
\(601\) 20.0000 0.0332779 0.0166389 0.999862i \(-0.494703\pi\)
0.0166389 + 0.999862i \(0.494703\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −110.304 32.5269i −0.182321 0.0537635i
\(606\) 0 0
\(607\) 6.78233i 0.0111735i −0.999984 0.00558676i \(-0.998222\pi\)
0.999984 0.00558676i \(-0.00177833\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1561.29i 2.55531i
\(612\) 0 0
\(613\) 47.4763i 0.0774491i −0.999250 0.0387246i \(-0.987671\pi\)
0.999250 0.0387246i \(-0.0123295\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 38.3667 0.0621826 0.0310913 0.999517i \(-0.490102\pi\)
0.0310913 + 0.999517i \(0.490102\pi\)
\(618\) 0 0
\(619\) 178.000 0.287561 0.143780 0.989610i \(-0.454074\pi\)
0.143780 + 0.989610i \(0.454074\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −220.608 −0.354106
\(624\) 0 0
\(625\) 257.000 + 569.716i 0.411200 + 0.911545i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 130.108i 0.206848i
\(630\) 0 0
\(631\) −154.000 −0.244057 −0.122029 0.992527i \(-0.538940\pi\)
−0.122029 + 0.992527i \(0.538940\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 239.792 813.173i 0.377625 1.28059i
\(636\) 0 0
\(637\) 61.0410i 0.0958257i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 807.516i 1.25978i 0.776686 + 0.629888i \(0.216899\pi\)
−0.776686 + 0.629888i \(0.783101\pi\)
\(642\) 0 0
\(643\) 1017.35i 1.58219i 0.611692 + 0.791096i \(0.290489\pi\)
−0.611692 + 0.791096i \(0.709511\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −805.700 −1.24529 −0.622643 0.782506i \(-0.713941\pi\)
−0.622643 + 0.782506i \(0.713941\pi\)
\(648\) 0 0
\(649\) 826.000 1.27273
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −67.1416 −0.102820 −0.0514101 0.998678i \(-0.516372\pi\)
−0.0514101 + 0.998678i \(0.516372\pi\)
\(654\) 0 0
\(655\) 234.000 793.533i 0.357252 1.21150i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 524.673i 0.796166i −0.917349 0.398083i \(-0.869676\pi\)
0.917349 0.398083i \(-0.130324\pi\)
\(660\) 0 0
\(661\) 74.0000 0.111952 0.0559758 0.998432i \(-0.482173\pi\)
0.0559758 + 0.998432i \(0.482173\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 115.100 390.323i 0.173083 0.586952i
\(666\) 0 0
\(667\) 81.3880i 0.122021i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 692.965i 1.03273i
\(672\) 0 0
\(673\) 691.798i 1.02793i 0.857811 + 0.513966i \(0.171824\pi\)
−0.857811 + 0.513966i \(0.828176\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −278.158 −0.410869 −0.205434 0.978671i \(-0.565861\pi\)
−0.205434 + 0.978671i \(0.565861\pi\)
\(678\) 0 0
\(679\) −644.000 −0.948454
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 422.033 0.617911 0.308955 0.951077i \(-0.400021\pi\)
0.308955 + 0.951077i \(0.400021\pi\)
\(684\) 0 0
\(685\) −552.000 162.776i −0.805839 0.237629i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −68.0000 −0.0984081 −0.0492041 0.998789i \(-0.515668\pi\)
−0.0492041 + 0.998789i \(0.515668\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −297.342 87.6812i −0.427830 0.126160i
\(696\) 0 0
\(697\) 1329.34i 1.90723i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 130.108i 0.185603i −0.995685 0.0928015i \(-0.970418\pi\)
0.995685 0.0928015i \(-0.0295822\pi\)
\(702\) 0 0
\(703\) 81.3880i 0.115772i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −537.133 −0.759736
\(708\) 0 0
\(709\) −210.000 −0.296192 −0.148096 0.988973i \(-0.547314\pi\)
−0.148096 + 0.988973i \(0.547314\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 364.483 0.511197
\(714\) 0 0
\(715\) 966.000 + 284.858i 1.35105 + 0.398403i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 121.622i 0.169155i 0.996417 + 0.0845774i \(0.0269540\pi\)
−0.996417 + 0.0845774i \(0.973046\pi\)
\(720\) 0 0
\(721\) −598.000 −0.829404
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 115.100 178.191i 0.158759 0.245781i
\(726\) 0 0
\(727\) 20.3470i 0.0279876i −0.999902 0.0139938i \(-0.995545\pi\)
0.999902 0.0139938i \(-0.00445451\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1301.08i 1.77986i
\(732\) 0 0
\(733\) 1417.51i 1.93384i −0.255074 0.966922i \(-0.582100\pi\)
0.255074 0.966922i \(-0.417900\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1074.27 1.45762
\(738\) 0 0
\(739\) 416.000 0.562923 0.281461 0.959573i \(-0.409181\pi\)
0.281461 + 0.959573i \(0.409181\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 805.700 1.08439 0.542194 0.840254i \(-0.317594\pi\)
0.542194 + 0.840254i \(0.317594\pi\)
\(744\) 0 0
\(745\) 224.000 759.621i 0.300671 1.01963i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 390.323i 0.521125i
\(750\) 0 0
\(751\) 86.0000 0.114514 0.0572570 0.998359i \(-0.481765\pi\)
0.0572570 + 0.998359i \(0.481765\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −335.708 98.9949i −0.444647 0.131119i
\(756\) 0 0
\(757\) 47.4763i 0.0627164i 0.999508 + 0.0313582i \(0.00998326\pi\)
−0.999508 + 0.0313582i \(0.990017\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 369.110i 0.485033i 0.970147 + 0.242516i \(0.0779728\pi\)
−0.970147 + 0.242516i \(0.922027\pi\)
\(762\) 0 0
\(763\) 501.892i 0.657788i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1697.72 2.21346
\(768\) 0 0
\(769\) 384.000 0.499350 0.249675 0.968330i \(-0.419676\pi\)
0.249675 + 0.968330i \(0.419676\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 201.425 0.260576 0.130288 0.991476i \(-0.458410\pi\)
0.130288 + 0.991476i \(0.458410\pi\)
\(774\) 0 0
\(775\) −798.000 515.457i −1.02968 0.665106i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 831.558i 1.06747i
\(780\) 0 0
\(781\) −1176.00 −1.50576
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −67.1416 + 227.688i −0.0855308 + 0.290049i
\(786\) 0 0
\(787\) 474.763i 0.603257i 0.953426 + 0.301628i \(0.0975302\pi\)
−0.953426 + 0.301628i \(0.902470\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1170.97i 1.48037i
\(792\) 0 0
\(793\) 1424.29i 1.79608i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 498.766 0.625805 0.312902 0.949785i \(-0.398699\pi\)
0.312902 + 0.949785i \(0.398699\pi\)
\(798\) 0 0
\(799\) −1472.00 −1.84230
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −134.283 −0.167227
\(804\) 0 0
\(805\) −92.0000 + 311.987i −0.114286 + 0.387562i
\(806\)