Properties

Label 540.3.b.a.269.2
Level $540$
Weight $3$
Character 540.269
Analytic conductor $14.714$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 540 = 2^{2} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 540.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.7139342755\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-11}, \sqrt{-19})\)
Defining polynomial: \( x^{4} + 15x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 269.2
Root \(0.521137i\) of defining polynomial
Character \(\chi\) \(=\) 540.269
Dual form 540.3.b.a.269.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-4.36421 + 2.44002i) q^{5} -2.79549i q^{7} +O(q^{10})\) \(q+(-4.36421 + 2.44002i) q^{5} -2.79549i q^{7} -18.1465i q^{11} +23.0266i q^{13} +5.72842 q^{17} +23.1852 q^{19} +0.271584 q^{23} +(13.0926 - 21.2975i) q^{25} +39.7995i q^{29} +47.3705 q^{31} +(6.82104 + 12.2001i) q^{35} +34.8712i q^{37} +13.2665i q^{41} +46.7158i q^{43} -40.9137 q^{47} +41.1852 q^{49} +91.3705 q^{53} +(44.2779 + 79.1953i) q^{55} -78.8398i q^{59} +31.1852 q^{61} +(-56.1852 - 100.493i) q^{65} +6.91631i q^{67} +81.5870i q^{71} -106.084i q^{73} -50.7284 q^{77} +63.5557 q^{79} -0.284161 q^{83} +(-25.0000 + 13.9774i) q^{85} +28.5210i q^{89} +64.3705 q^{91} +(-101.185 + 56.5724i) q^{95} +92.9138i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{5} - 6 q^{17} + 6 q^{19} + 30 q^{23} + 9 q^{25} + 16 q^{31} - 45 q^{35} - 48 q^{47} + 78 q^{49} + 192 q^{53} + 47 q^{55} + 38 q^{61} - 138 q^{65} - 174 q^{77} - 6 q^{79} + 288 q^{83} - 100 q^{85} + 84 q^{91} - 318 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/540\mathbb{Z}\right)^\times\).

\(n\) \(217\) \(271\) \(461\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(5\) −4.36421 + 2.44002i −0.872842 + 0.488004i
\(6\) 0 0
\(7\) 2.79549i 0.399355i −0.979862 0.199678i \(-0.936010\pi\)
0.979862 0.199678i \(-0.0639895\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 18.1465i 1.64969i −0.565363 0.824843i \(-0.691264\pi\)
0.565363 0.824843i \(-0.308736\pi\)
\(12\) 0 0
\(13\) 23.0266i 1.77127i 0.464378 + 0.885637i \(0.346278\pi\)
−0.464378 + 0.885637i \(0.653722\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.72842 0.336966 0.168483 0.985705i \(-0.446113\pi\)
0.168483 + 0.985705i \(0.446113\pi\)
\(18\) 0 0
\(19\) 23.1852 1.22028 0.610138 0.792295i \(-0.291114\pi\)
0.610138 + 0.792295i \(0.291114\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.271584 0.0118080 0.00590400 0.999983i \(-0.498121\pi\)
0.00590400 + 0.999983i \(0.498121\pi\)
\(24\) 0 0
\(25\) 13.0926 21.2975i 0.523705 0.851900i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 39.7995i 1.37240i 0.727415 + 0.686198i \(0.240722\pi\)
−0.727415 + 0.686198i \(0.759278\pi\)
\(30\) 0 0
\(31\) 47.3705 1.52808 0.764040 0.645169i \(-0.223213\pi\)
0.764040 + 0.645169i \(0.223213\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.82104 + 12.2001i 0.194887 + 0.348574i
\(36\) 0 0
\(37\) 34.8712i 0.942465i 0.882009 + 0.471232i \(0.156191\pi\)
−0.882009 + 0.471232i \(0.843809\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 13.2665i 0.323573i 0.986826 + 0.161787i \(0.0517256\pi\)
−0.986826 + 0.161787i \(0.948274\pi\)
\(42\) 0 0
\(43\) 46.7158i 1.08641i 0.839599 + 0.543207i \(0.182790\pi\)
−0.839599 + 0.543207i \(0.817210\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −40.9137 −0.870504 −0.435252 0.900309i \(-0.643341\pi\)
−0.435252 + 0.900309i \(0.643341\pi\)
\(48\) 0 0
\(49\) 41.1852 0.840515
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 91.3705 1.72397 0.861986 0.506932i \(-0.169221\pi\)
0.861986 + 0.506932i \(0.169221\pi\)
\(54\) 0 0
\(55\) 44.2779 + 79.1953i 0.805052 + 1.43991i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 78.8398i 1.33627i −0.744041 0.668134i \(-0.767093\pi\)
0.744041 0.668134i \(-0.232907\pi\)
\(60\) 0 0
\(61\) 31.1852 0.511234 0.255617 0.966778i \(-0.417721\pi\)
0.255617 + 0.966778i \(0.417721\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −56.1852 100.493i −0.864388 1.54604i
\(66\) 0 0
\(67\) 6.91631i 0.103229i 0.998667 + 0.0516143i \(0.0164366\pi\)
−0.998667 + 0.0516143i \(0.983563\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 81.5870i 1.14911i 0.818465 + 0.574556i \(0.194825\pi\)
−0.818465 + 0.574556i \(0.805175\pi\)
\(72\) 0 0
\(73\) 106.084i 1.45320i −0.687060 0.726601i \(-0.741099\pi\)
0.687060 0.726601i \(-0.258901\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −50.7284 −0.658811
\(78\) 0 0
\(79\) 63.5557 0.804503 0.402252 0.915529i \(-0.368228\pi\)
0.402252 + 0.915529i \(0.368228\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −0.284161 −0.00342363 −0.00171182 0.999999i \(-0.500545\pi\)
−0.00171182 + 0.999999i \(0.500545\pi\)
\(84\) 0 0
\(85\) −25.0000 + 13.9774i −0.294118 + 0.164440i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 28.5210i 0.320461i 0.987080 + 0.160230i \(0.0512237\pi\)
−0.987080 + 0.160230i \(0.948776\pi\)
\(90\) 0 0
\(91\) 64.3705 0.707368
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −101.185 + 56.5724i −1.06511 + 0.595499i
\(96\) 0 0
\(97\) 92.9138i 0.957874i 0.877849 + 0.478937i \(0.158978\pi\)
−0.877849 + 0.478937i \(0.841022\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 46.6675i 0.462055i −0.972947 0.231027i \(-0.925791\pi\)
0.972947 0.231027i \(-0.0742087\pi\)
\(102\) 0 0
\(103\) 11.1820i 0.108563i −0.998526 0.0542813i \(-0.982713\pi\)
0.998526 0.0542813i \(-0.0172868\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 147.297 1.37661 0.688303 0.725424i \(-0.258356\pi\)
0.688303 + 0.725424i \(0.258356\pi\)
\(108\) 0 0
\(109\) −121.556 −1.11519 −0.557595 0.830113i \(-0.688276\pi\)
−0.557595 + 0.830113i \(0.688276\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 97.6295 0.863978 0.431989 0.901879i \(-0.357812\pi\)
0.431989 + 0.901879i \(0.357812\pi\)
\(114\) 0 0
\(115\) −1.18525 + 0.662669i −0.0103065 + 0.00576234i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 16.0137i 0.134569i
\(120\) 0 0
\(121\) −208.297 −1.72146
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −5.17267 + 124.893i −0.0413814 + 0.999143i
\(126\) 0 0
\(127\) 88.6481i 0.698017i −0.937120 0.349008i \(-0.886518\pi\)
0.937120 0.349008i \(-0.113482\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 89.9735i 0.686820i −0.939186 0.343410i \(-0.888418\pi\)
0.939186 0.343410i \(-0.111582\pi\)
\(132\) 0 0
\(133\) 64.8141i 0.487324i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −229.926 −1.67829 −0.839147 0.543905i \(-0.816945\pi\)
−0.839147 + 0.543905i \(0.816945\pi\)
\(138\) 0 0
\(139\) 57.6295 0.414601 0.207300 0.978277i \(-0.433532\pi\)
0.207300 + 0.978277i \(0.433532\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 417.852 2.92205
\(144\) 0 0
\(145\) −97.1115 173.693i −0.669734 1.19788i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 11.1337i 0.0747227i 0.999302 + 0.0373613i \(0.0118953\pi\)
−0.999302 + 0.0373613i \(0.988105\pi\)
\(150\) 0 0
\(151\) −22.1852 −0.146922 −0.0734611 0.997298i \(-0.523404\pi\)
−0.0734611 + 0.997298i \(0.523404\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −206.735 + 115.585i −1.33377 + 0.745709i
\(156\) 0 0
\(157\) 225.337i 1.43527i −0.696419 0.717635i \(-0.745225\pi\)
0.696419 0.717635i \(-0.254775\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0.759209i 0.00471559i
\(162\) 0 0
\(163\) 302.948i 1.85858i 0.369352 + 0.929290i \(0.379580\pi\)
−0.369352 + 0.929290i \(0.620420\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −238.667 −1.42915 −0.714573 0.699561i \(-0.753379\pi\)
−0.714573 + 0.699561i \(0.753379\pi\)
\(168\) 0 0
\(169\) −361.223 −2.13741
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 108.802 0.628914 0.314457 0.949272i \(-0.398178\pi\)
0.314457 + 0.949272i \(0.398178\pi\)
\(174\) 0 0
\(175\) −59.5369 36.6003i −0.340211 0.209144i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 168.054i 0.938849i 0.882973 + 0.469425i \(0.155539\pi\)
−0.882973 + 0.469425i \(0.844461\pi\)
\(180\) 0 0
\(181\) −154.297 −0.852468 −0.426234 0.904613i \(-0.640160\pi\)
−0.426234 + 0.904613i \(0.640160\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −85.0863 152.185i −0.459926 0.822622i
\(186\) 0 0
\(187\) 103.951i 0.555887i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 154.932i 0.811164i −0.914059 0.405582i \(-0.867069\pi\)
0.914059 0.405582i \(-0.132931\pi\)
\(192\) 0 0
\(193\) 64.0066i 0.331640i −0.986156 0.165820i \(-0.946973\pi\)
0.986156 0.165820i \(-0.0530271\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 9.56832 0.0485702 0.0242851 0.999705i \(-0.492269\pi\)
0.0242851 + 0.999705i \(0.492269\pi\)
\(198\) 0 0
\(199\) −117.815 −0.592034 −0.296017 0.955183i \(-0.595658\pi\)
−0.296017 + 0.955183i \(0.595658\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 111.259 0.548074
\(204\) 0 0
\(205\) −32.3705 57.8978i −0.157905 0.282428i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 420.732i 2.01307i
\(210\) 0 0
\(211\) −48.8148 −0.231350 −0.115675 0.993287i \(-0.536903\pi\)
−0.115675 + 0.993287i \(0.536903\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −113.987 203.878i −0.530174 0.948268i
\(216\) 0 0
\(217\) 132.424i 0.610247i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 131.906i 0.596859i
\(222\) 0 0
\(223\) 319.721i 1.43373i −0.697213 0.716864i \(-0.745577\pi\)
0.697213 0.716864i \(-0.254423\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −18.2716 −0.0804916 −0.0402458 0.999190i \(-0.512814\pi\)
−0.0402458 + 0.999190i \(0.512814\pi\)
\(228\) 0 0
\(229\) 134.074 0.585475 0.292737 0.956193i \(-0.405434\pi\)
0.292737 + 0.956193i \(0.405434\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 284.173 1.21963 0.609813 0.792546i \(-0.291245\pi\)
0.609813 + 0.792546i \(0.291245\pi\)
\(234\) 0 0
\(235\) 178.556 99.8301i 0.759812 0.424809i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 206.770i 0.865145i 0.901599 + 0.432572i \(0.142394\pi\)
−0.901599 + 0.432572i \(0.857606\pi\)
\(240\) 0 0
\(241\) −162.667 −0.674968 −0.337484 0.941331i \(-0.609576\pi\)
−0.337484 + 0.941331i \(0.609576\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −179.741 + 100.493i −0.733637 + 0.410174i
\(246\) 0 0
\(247\) 533.877i 2.16144i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 347.387i 1.38401i 0.721893 + 0.692005i \(0.243272\pi\)
−0.721893 + 0.692005i \(0.756728\pi\)
\(252\) 0 0
\(253\) 4.92831i 0.0194795i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 21.0377 0.0818589 0.0409294 0.999162i \(-0.486968\pi\)
0.0409294 + 0.999162i \(0.486968\pi\)
\(258\) 0 0
\(259\) 97.4820 0.376378
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 215.457 0.819227 0.409614 0.912259i \(-0.365663\pi\)
0.409614 + 0.912259i \(0.365663\pi\)
\(264\) 0 0
\(265\) −398.760 + 222.946i −1.50475 + 0.841304i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 180.237i 0.670024i 0.942214 + 0.335012i \(0.108740\pi\)
−0.942214 + 0.335012i \(0.891260\pi\)
\(270\) 0 0
\(271\) 231.741 0.855133 0.427566 0.903984i \(-0.359371\pi\)
0.427566 + 0.903984i \(0.359371\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −386.476 237.586i −1.40537 0.863948i
\(276\) 0 0
\(277\) 105.566i 0.381104i −0.981677 0.190552i \(-0.938972\pi\)
0.981677 0.190552i \(-0.0610278\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 504.597i 1.79572i 0.440284 + 0.897859i \(0.354878\pi\)
−0.440284 + 0.897859i \(0.645122\pi\)
\(282\) 0 0
\(283\) 151.619i 0.535756i −0.963453 0.267878i \(-0.913678\pi\)
0.963453 0.267878i \(-0.0863224\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 37.0863 0.129221
\(288\) 0 0
\(289\) −256.185 −0.886454
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 549.038 1.87385 0.936924 0.349532i \(-0.113659\pi\)
0.936924 + 0.349532i \(0.113659\pi\)
\(294\) 0 0
\(295\) 192.370 + 344.073i 0.652103 + 1.16635i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.25364i 0.0209152i
\(300\) 0 0
\(301\) 130.593 0.433865
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −136.099 + 76.0926i −0.446226 + 0.249484i
\(306\) 0 0
\(307\) 146.401i 0.476876i 0.971158 + 0.238438i \(0.0766355\pi\)
−0.971158 + 0.238438i \(0.923365\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 487.824i 1.56856i −0.620404 0.784282i \(-0.713031\pi\)
0.620404 0.784282i \(-0.286969\pi\)
\(312\) 0 0
\(313\) 109.687i 0.350437i 0.984530 + 0.175218i \(0.0560631\pi\)
−0.984530 + 0.175218i \(0.943937\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −363.568 −1.14690 −0.573452 0.819239i \(-0.694396\pi\)
−0.573452 + 0.819239i \(0.694396\pi\)
\(318\) 0 0
\(319\) 722.223 2.26402
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 132.815 0.411191
\(324\) 0 0
\(325\) 490.408 + 301.478i 1.50895 + 0.927625i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 114.374i 0.347640i
\(330\) 0 0
\(331\) 300.223 0.907018 0.453509 0.891252i \(-0.350172\pi\)
0.453509 + 0.891252i \(0.350172\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −16.8759 30.1842i −0.0503759 0.0901022i
\(336\) 0 0
\(337\) 373.353i 1.10787i −0.832559 0.553937i \(-0.813125\pi\)
0.832559 0.553937i \(-0.186875\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 859.610i 2.52085i
\(342\) 0 0
\(343\) 252.112i 0.735020i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 17.4443 0.0502716 0.0251358 0.999684i \(-0.491998\pi\)
0.0251358 + 0.999684i \(0.491998\pi\)
\(348\) 0 0
\(349\) 234.149 0.670915 0.335457 0.942055i \(-0.391109\pi\)
0.335457 + 0.942055i \(0.391109\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 381.284 1.08013 0.540063 0.841625i \(-0.318401\pi\)
0.540063 + 0.841625i \(0.318401\pi\)
\(354\) 0 0
\(355\) −199.074 356.063i −0.560771 1.00299i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 222.024i 0.618451i −0.950989 0.309226i \(-0.899930\pi\)
0.950989 0.309226i \(-0.100070\pi\)
\(360\) 0 0
\(361\) 176.556 0.489074
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 258.846 + 462.971i 0.709168 + 1.26841i
\(366\) 0 0
\(367\) 449.205i 1.22399i 0.790861 + 0.611995i \(0.209633\pi\)
−0.790861 + 0.611995i \(0.790367\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 255.425i 0.688477i
\(372\) 0 0
\(373\) 322.082i 0.863492i 0.901995 + 0.431746i \(0.142102\pi\)
−0.901995 + 0.431746i \(0.857898\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −916.446 −2.43089
\(378\) 0 0
\(379\) −216.074 −0.570115 −0.285058 0.958510i \(-0.592013\pi\)
−0.285058 + 0.958510i \(0.592013\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −373.977 −0.976440 −0.488220 0.872721i \(-0.662354\pi\)
−0.488220 + 0.872721i \(0.662354\pi\)
\(384\) 0 0
\(385\) 221.389 123.778i 0.575037 0.321502i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 223.108i 0.573543i 0.957999 + 0.286771i \(0.0925820\pi\)
−0.957999 + 0.286771i \(0.907418\pi\)
\(390\) 0 0
\(391\) 1.55575 0.00397889
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −277.370 + 155.077i −0.702204 + 0.392600i
\(396\) 0 0
\(397\) 610.535i 1.53787i 0.639325 + 0.768936i \(0.279214\pi\)
−0.639325 + 0.768936i \(0.720786\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 411.441i 1.02604i 0.858377 + 0.513019i \(0.171473\pi\)
−0.858377 + 0.513019i \(0.828527\pi\)
\(402\) 0 0
\(403\) 1090.78i 2.70665i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 632.791 1.55477
\(408\) 0 0
\(409\) 590.630 1.44408 0.722041 0.691850i \(-0.243204\pi\)
0.722041 + 0.691850i \(0.243204\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −220.396 −0.533646
\(414\) 0 0
\(415\) 1.24014 0.693359i 0.00298829 0.00167074i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 239.556i 0.571733i −0.958269 0.285867i \(-0.907719\pi\)
0.958269 0.285867i \(-0.0922814\pi\)
\(420\) 0 0
\(421\) 593.408 1.40952 0.704760 0.709445i \(-0.251055\pi\)
0.704760 + 0.709445i \(0.251055\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 75.0000 122.001i 0.176471 0.287061i
\(426\) 0 0
\(427\) 87.1780i 0.204164i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 436.566i 1.01291i 0.862265 + 0.506457i \(0.169045\pi\)
−0.862265 + 0.506457i \(0.830955\pi\)
\(432\) 0 0
\(433\) 231.736i 0.535187i −0.963532 0.267593i \(-0.913772\pi\)
0.963532 0.267593i \(-0.0862284\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 6.29674 0.0144090
\(438\) 0 0
\(439\) −419.223 −0.954950 −0.477475 0.878645i \(-0.658448\pi\)
−0.477475 + 0.878645i \(0.658448\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 446.520 1.00795 0.503973 0.863720i \(-0.331871\pi\)
0.503973 + 0.863720i \(0.331871\pi\)
\(444\) 0 0
\(445\) −69.5918 124.472i −0.156386 0.279711i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 495.776i 1.10418i −0.833785 0.552089i \(-0.813831\pi\)
0.833785 0.552089i \(-0.186169\pi\)
\(450\) 0 0
\(451\) 240.741 0.533794
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −280.926 + 157.065i −0.617420 + 0.345198i
\(456\) 0 0
\(457\) 683.363i 1.49532i −0.664080 0.747662i \(-0.731176\pi\)
0.664080 0.747662i \(-0.268824\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 400.128i 0.867956i −0.900923 0.433978i \(-0.857110\pi\)
0.900923 0.433978i \(-0.142890\pi\)
\(462\) 0 0
\(463\) 316.263i 0.683074i 0.939868 + 0.341537i \(0.110947\pi\)
−0.939868 + 0.341537i \(0.889053\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 419.741 0.898803 0.449401 0.893330i \(-0.351637\pi\)
0.449401 + 0.893330i \(0.351637\pi\)
\(468\) 0 0
\(469\) 19.3345 0.0412249
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 847.730 1.79224
\(474\) 0 0
\(475\) 303.556 493.788i 0.639065 1.03955i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 844.970i 1.76403i 0.471222 + 0.882015i \(0.343813\pi\)
−0.471222 + 0.882015i \(0.656187\pi\)
\(480\) 0 0
\(481\) −802.964 −1.66936
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −226.711 405.495i −0.467446 0.836072i
\(486\) 0 0
\(487\) 546.384i 1.12194i −0.827837 0.560969i \(-0.810429\pi\)
0.827837 0.560969i \(-0.189571\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 647.456i 1.31865i 0.751859 + 0.659324i \(0.229157\pi\)
−0.751859 + 0.659324i \(0.770843\pi\)
\(492\) 0 0
\(493\) 227.988i 0.462450i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 228.075 0.458904
\(498\) 0 0
\(499\) 582.815 1.16797 0.583983 0.811766i \(-0.301494\pi\)
0.583983 + 0.811766i \(0.301494\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −752.520 −1.49606 −0.748032 0.663663i \(-0.769001\pi\)
−0.748032 + 0.663663i \(0.769001\pi\)
\(504\) 0 0
\(505\) 113.870 + 203.667i 0.225484 + 0.403301i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 291.538i 0.572767i −0.958115 0.286383i \(-0.907547\pi\)
0.958115 0.286383i \(-0.0924531\pi\)
\(510\) 0 0
\(511\) −296.556 −0.580344
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 27.2842 + 48.8004i 0.0529790 + 0.0947580i
\(516\) 0 0
\(517\) 742.441i 1.43606i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 137.690i 0.264280i 0.991231 + 0.132140i \(0.0421848\pi\)
−0.991231 + 0.132140i \(0.957815\pi\)
\(522\) 0 0
\(523\) 330.987i 0.632862i −0.948616 0.316431i \(-0.897515\pi\)
0.948616 0.316431i \(-0.102485\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 271.358 0.514911
\(528\) 0 0
\(529\) −528.926 −0.999861
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −305.482 −0.573137
\(534\) 0 0
\(535\) −642.834 + 359.407i −1.20156 + 0.671788i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 747.370i 1.38659i
\(540\) 0 0
\(541\) 556.593 1.02882 0.514412 0.857543i \(-0.328010\pi\)
0.514412 + 0.857543i \(0.328010\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 530.495 296.598i 0.973385 0.544217i
\(546\) 0 0
\(547\) 465.170i 0.850402i −0.905099 0.425201i \(-0.860203\pi\)
0.905099 0.425201i \(-0.139797\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 922.761i 1.67470i
\(552\) 0 0
\(553\) 177.669i 0.321283i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1059.37 −1.90192 −0.950962 0.309306i \(-0.899903\pi\)
−0.950962 + 0.309306i \(0.899903\pi\)
\(558\) 0 0
\(559\) −1075.70 −1.92434
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −386.160 −0.685897 −0.342949 0.939354i \(-0.611426\pi\)
−0.342949 + 0.939354i \(0.611426\pi\)
\(564\) 0 0
\(565\) −426.075 + 238.218i −0.754116 + 0.421624i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 686.062i 1.20573i 0.797842 + 0.602866i \(0.205975\pi\)
−0.797842 + 0.602866i \(0.794025\pi\)
\(570\) 0 0
\(571\) −821.631 −1.43893 −0.719467 0.694527i \(-0.755614\pi\)
−0.719467 + 0.694527i \(0.755614\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3.55575 5.78406i 0.00618390 0.0100592i
\(576\) 0 0
\(577\) 183.840i 0.318613i 0.987229 + 0.159306i \(0.0509258\pi\)
−0.987229 + 0.159306i \(0.949074\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0.794370i 0.00136725i
\(582\) 0 0
\(583\) 1658.06i 2.84401i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 320.407 0.545837 0.272919 0.962037i \(-0.412011\pi\)
0.272919 + 0.962037i \(0.412011\pi\)
\(588\) 0 0
\(589\) 1098.30 1.86468
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1061.34 −1.78977 −0.894887 0.446292i \(-0.852744\pi\)
−0.894887 + 0.446292i \(0.852744\pi\)
\(594\) 0 0
\(595\) 39.0738 + 69.8872i 0.0656702 + 0.117457i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 977.056i 1.63115i −0.578655 0.815573i \(-0.696422\pi\)
0.578655 0.815573i \(-0.303578\pi\)
\(600\) 0 0
\(601\) −261.816 −0.435635 −0.217817 0.975990i \(-0.569894\pi\)
−0.217817 + 0.975990i \(0.569894\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 909.050 508.248i 1.50256 0.840079i
\(606\) 0 0
\(607\) 668.806i 1.10182i −0.834564 0.550911i \(-0.814280\pi\)
0.834564 0.550911i \(-0.185720\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 942.101i 1.54190i
\(612\) 0 0
\(613\) 220.119i 0.359086i 0.983750 + 0.179543i \(0.0574618\pi\)
−0.983750 + 0.179543i \(0.942538\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 321.063 0.520361 0.260181 0.965560i \(-0.416218\pi\)
0.260181 + 0.965560i \(0.416218\pi\)
\(618\) 0 0
\(619\) −891.187 −1.43972 −0.719860 0.694119i \(-0.755794\pi\)
−0.719860 + 0.694119i \(0.755794\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 79.7301 0.127978
\(624\) 0 0
\(625\) −282.166 557.680i −0.451466 0.892288i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 199.757i 0.317578i
\(630\) 0 0
\(631\) −583.669 −0.924990 −0.462495 0.886622i \(-0.653046\pi\)
−0.462495 + 0.886622i \(0.653046\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 216.303 + 386.879i 0.340635 + 0.609258i
\(636\) 0 0
\(637\) 948.355i 1.48878i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 696.292i 1.08626i −0.839649 0.543129i \(-0.817239\pi\)
0.839649 0.543129i \(-0.182761\pi\)
\(642\) 0 0
\(643\) 801.664i 1.24676i −0.781920 0.623378i \(-0.785760\pi\)
0.781920 0.623378i \(-0.214240\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 216.221 0.334191 0.167095 0.985941i \(-0.446561\pi\)
0.167095 + 0.985941i \(0.446561\pi\)
\(648\) 0 0
\(649\) −1430.67 −2.20442
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −438.802 −0.671979 −0.335989 0.941866i \(-0.609071\pi\)
−0.335989 + 0.941866i \(0.609071\pi\)
\(654\) 0 0
\(655\) 219.537 + 392.663i 0.335171 + 0.599485i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 366.112i 0.555557i −0.960645 0.277779i \(-0.910402\pi\)
0.960645 0.277779i \(-0.0895982\pi\)
\(660\) 0 0
\(661\) 158.075 0.239146 0.119573 0.992825i \(-0.461847\pi\)
0.119573 + 0.992825i \(0.461847\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 158.148 + 282.862i 0.237816 + 0.425357i
\(666\) 0 0
\(667\) 10.8089i 0.0162052i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 565.904i 0.843374i
\(672\) 0 0
\(673\) 648.575i 0.963708i −0.876252 0.481854i \(-0.839964\pi\)
0.876252 0.481854i \(-0.160036\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −202.345 −0.298885 −0.149443 0.988770i \(-0.547748\pi\)
−0.149443 + 0.988770i \(0.547748\pi\)
\(678\) 0 0
\(679\) 259.739 0.382532
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 173.829 0.254508 0.127254 0.991870i \(-0.459384\pi\)
0.127254 + 0.991870i \(0.459384\pi\)
\(684\) 0 0
\(685\) 1003.45 561.024i 1.46488 0.819013i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2103.95i 3.05363i
\(690\) 0 0
\(691\) −846.743 −1.22539 −0.612694 0.790320i \(-0.709914\pi\)
−0.612694 + 0.790320i \(0.709914\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −251.507 + 140.617i −0.361881 + 0.202327i
\(696\) 0 0
\(697\) 75.9960i 0.109033i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 72.2614i 0.103083i 0.998671 + 0.0515416i \(0.0164135\pi\)
−0.998671 + 0.0515416i \(0.983587\pi\)
\(702\) 0 0
\(703\) 808.497i 1.15007i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −130.459 −0.184524
\(708\) 0 0
\(709\) 287.557 0.405582 0.202791 0.979222i \(-0.434999\pi\)
0.202791 + 0.979222i \(0.434999\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 12.8651 0.0180436
\(714\) 0 0
\(715\) −1823.60 + 1019.57i −2.55048 + 1.42597i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1015.27i 1.41205i 0.708185 + 0.706027i \(0.249514\pi\)
−0.708185 + 0.706027i \(0.750486\pi\)
\(720\) 0 0
\(721\) −31.2590 −0.0433551
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 847.630 + 521.080i 1.16914 + 0.718731i
\(726\) 0 0
\(727\) 311.418i 0.428361i −0.976794 0.214180i \(-0.931292\pi\)
0.976794 0.214180i \(-0.0687080\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 267.608i 0.366084i
\(732\) 0 0
\(733\) 321.047i 0.437990i −0.975726 0.218995i \(-0.929722\pi\)
0.975726 0.218995i \(-0.0702778\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 125.507 0.170295
\(738\) 0 0
\(739\) −657.185 −0.889290 −0.444645 0.895707i \(-0.646670\pi\)
−0.444645 + 0.895707i \(0.646670\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1024.96 −1.37949 −0.689747 0.724051i \(-0.742278\pi\)
−0.689747 + 0.724051i \(0.742278\pi\)
\(744\) 0 0
\(745\) −27.1664 48.5897i −0.0364649 0.0652211i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 411.766i 0.549755i
\(750\) 0 0
\(751\) −63.1475 −0.0840846 −0.0420423 0.999116i \(-0.513386\pi\)
−0.0420423 + 0.999116i \(0.513386\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 96.8210 54.1324i 0.128240 0.0716986i
\(756\) 0 0
\(757\) 1452.21i 1.91837i −0.282781 0.959185i \(-0.591257\pi\)
0.282781 0.959185i \(-0.408743\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 487.824i 0.641030i −0.947243 0.320515i \(-0.896144\pi\)
0.947243 0.320515i \(-0.103856\pi\)
\(762\) 0 0
\(763\) 339.808i 0.445357i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1815.41 2.36690
\(768\) 0 0
\(769\) 677.148 0.880556 0.440278 0.897862i \(-0.354880\pi\)
0.440278 + 0.897862i \(0.354880\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −949.926 −1.22888 −0.614441 0.788963i \(-0.710619\pi\)
−0.614441 + 0.788963i \(0.710619\pi\)
\(774\) 0 0
\(775\) 620.204 1008.87i 0.800263 1.30177i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 307.587i 0.394849i
\(780\) 0 0
\(781\) 1480.52 1.89567
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 549.827 + 983.419i 0.700417 + 1.25276i
\(786\) 0 0
\(787\) 1144.33i 1.45404i −0.686617 0.727020i \(-0.740905\pi\)
0.686617 0.727020i \(-0.259095\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 272.922i 0.345034i
\(792\) 0 0
\(793\) 718.089i 0.905535i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 651.101 0.816939 0.408470 0.912772i \(-0.366063\pi\)
0.408470 + 0.912772i \(0.366063\pi\)
\(798\) 0 0
\(799\) −234.370 −0.293330
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1925.05 −2.39733
\(804\) 0 0
\(805\) 1.85248 + 3.31335i 0.00230122 + 0.00411596i