Properties

Label 1440.2.d.e.1009.4
Level 1440
Weight 2
Character 1440.1009
Analytic conductor 11.498
Analytic rank 0
Dimension 6
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.4984578911\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.839056.1
Defining polynomial: \(x^{6} + 6 x^{4} + 8 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1009.4
Root \(-1.32132i\) of defining polynomial
Character \(\chi\) \(=\) 1440.1009
Dual form 1440.2.d.e.1009.3

$q$-expansion

\(f(q)\) \(=\) \(q+(0.254102 + 2.22158i) q^{5} +2.64265i q^{7} +O(q^{10})\) \(q+(0.254102 + 2.22158i) q^{5} +2.64265i q^{7} +1.51363i q^{11} +3.87086 q^{13} +3.31415i q^{17} -7.08582i q^{19} +4.82778i q^{23} +(-4.87086 + 1.12902i) q^{25} +2.18513i q^{29} +7.36266 q^{31} +(-5.87086 + 0.671502i) q^{35} -7.87086 q^{37} -8.72532 q^{41} -1.01641 q^{43} +7.08582i q^{47} +0.0164068 q^{49} -4.50820 q^{53} +(-3.36266 + 0.384617i) q^{55} -6.79893i q^{59} +3.60104i q^{61} +(0.983593 + 8.59945i) q^{65} +1.01641 q^{67} -6.72532 q^{71} +15.5146i q^{73} -4.00000 q^{77} -7.36266 q^{79} +7.74173 q^{83} +(-7.36266 + 0.842131i) q^{85} +14.7581 q^{89} +10.2293i q^{91} +(15.7417 - 1.80052i) q^{95} -11.1444i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + O(q^{10}) \) \( 6q - 8q^{13} + 2q^{25} + 16q^{31} - 4q^{35} - 16q^{37} + 4q^{41} - 6q^{49} - 24q^{53} + 8q^{55} + 12q^{65} + 16q^{71} - 24q^{77} - 16q^{79} - 16q^{83} - 16q^{85} + 20q^{89} + 32q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0.254102 + 2.22158i 0.113638 + 0.993522i
\(6\) 0 0
\(7\) 2.64265i 0.998827i 0.866364 + 0.499414i \(0.166451\pi\)
−0.866364 + 0.499414i \(0.833549\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.51363i 0.456377i 0.973617 + 0.228189i \(0.0732803\pi\)
−0.973617 + 0.228189i \(0.926720\pi\)
\(12\) 0 0
\(13\) 3.87086 1.07358 0.536792 0.843714i \(-0.319636\pi\)
0.536792 + 0.843714i \(0.319636\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.31415i 0.803800i 0.915684 + 0.401900i \(0.131650\pi\)
−0.915684 + 0.401900i \(0.868350\pi\)
\(18\) 0 0
\(19\) 7.08582i 1.62560i −0.582545 0.812799i \(-0.697943\pi\)
0.582545 0.812799i \(-0.302057\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.82778i 1.00666i 0.864094 + 0.503331i \(0.167892\pi\)
−0.864094 + 0.503331i \(0.832108\pi\)
\(24\) 0 0
\(25\) −4.87086 + 1.12902i −0.974173 + 0.225803i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.18513i 0.405769i 0.979203 + 0.202885i \(0.0650316\pi\)
−0.979203 + 0.202885i \(0.934968\pi\)
\(30\) 0 0
\(31\) 7.36266 1.32237 0.661187 0.750222i \(-0.270053\pi\)
0.661187 + 0.750222i \(0.270053\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −5.87086 + 0.671502i −0.992357 + 0.113504i
\(36\) 0 0
\(37\) −7.87086 −1.29396 −0.646981 0.762506i \(-0.723969\pi\)
−0.646981 + 0.762506i \(0.723969\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.72532 −1.36267 −0.681333 0.731973i \(-0.738600\pi\)
−0.681333 + 0.731973i \(0.738600\pi\)
\(42\) 0 0
\(43\) −1.01641 −0.155001 −0.0775003 0.996992i \(-0.524694\pi\)
−0.0775003 + 0.996992i \(0.524694\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.08582i 1.03357i 0.856114 + 0.516786i \(0.172872\pi\)
−0.856114 + 0.516786i \(0.827128\pi\)
\(48\) 0 0
\(49\) 0.0164068 0.00234382
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.50820 −0.619249 −0.309625 0.950859i \(-0.600203\pi\)
−0.309625 + 0.950859i \(0.600203\pi\)
\(54\) 0 0
\(55\) −3.36266 + 0.384617i −0.453421 + 0.0518617i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.79893i 0.885145i −0.896733 0.442573i \(-0.854066\pi\)
0.896733 0.442573i \(-0.145934\pi\)
\(60\) 0 0
\(61\) 3.60104i 0.461065i 0.973065 + 0.230533i \(0.0740469\pi\)
−0.973065 + 0.230533i \(0.925953\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.983593 + 8.59945i 0.122000 + 1.06663i
\(66\) 0 0
\(67\) 1.01641 0.124174 0.0620869 0.998071i \(-0.480224\pi\)
0.0620869 + 0.998071i \(0.480224\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.72532 −0.798149 −0.399074 0.916919i \(-0.630669\pi\)
−0.399074 + 0.916919i \(0.630669\pi\)
\(72\) 0 0
\(73\) 15.5146i 1.81585i 0.419132 + 0.907925i \(0.362334\pi\)
−0.419132 + 0.907925i \(0.637666\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) −7.36266 −0.828364 −0.414182 0.910194i \(-0.635932\pi\)
−0.414182 + 0.910194i \(0.635932\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.74173 0.849765 0.424883 0.905248i \(-0.360315\pi\)
0.424883 + 0.905248i \(0.360315\pi\)
\(84\) 0 0
\(85\) −7.36266 + 0.842131i −0.798593 + 0.0913420i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 14.7581 1.56436 0.782180 0.623053i \(-0.214108\pi\)
0.782180 + 0.623053i \(0.214108\pi\)
\(90\) 0 0
\(91\) 10.2293i 1.07233i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 15.7417 1.80052i 1.61507 0.184729i
\(96\) 0 0
\(97\) 11.1444i 1.13154i −0.824563 0.565769i \(-0.808579\pi\)
0.824563 0.565769i \(-0.191421\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 13.3295i 1.32633i −0.748471 0.663167i \(-0.769212\pi\)
0.748471 0.663167i \(-0.230788\pi\)
\(102\) 0 0
\(103\) 0.958386i 0.0944326i 0.998885 + 0.0472163i \(0.0150350\pi\)
−0.998885 + 0.0472163i \(0.984965\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) 0.769233i 0.0736792i 0.999321 + 0.0368396i \(0.0117291\pi\)
−0.999321 + 0.0368396i \(0.988271\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 14.4585i 1.36014i 0.733146 + 0.680071i \(0.238051\pi\)
−0.733146 + 0.680071i \(0.761949\pi\)
\(114\) 0 0
\(115\) −10.7253 + 1.22675i −1.00014 + 0.114395i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −8.75814 −0.802857
\(120\) 0 0
\(121\) 8.70892 0.791720
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −3.74590 10.5341i −0.335043 0.942203i
\(126\) 0 0
\(127\) 11.5290i 1.02303i 0.859274 + 0.511516i \(0.170916\pi\)
−0.859274 + 0.511516i \(0.829084\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.37270i 0.644156i −0.946713 0.322078i \(-0.895619\pi\)
0.946713 0.322078i \(-0.104381\pi\)
\(132\) 0 0
\(133\) 18.7253 1.62369
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.88792i 0.332167i −0.986112 0.166084i \(-0.946888\pi\)
0.986112 0.166084i \(-0.0531122\pi\)
\(138\) 0 0
\(139\) 14.6291i 1.24083i 0.784275 + 0.620414i \(0.213035\pi\)
−0.784275 + 0.620414i \(0.786965\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 5.85907i 0.489960i
\(144\) 0 0
\(145\) −4.85446 + 0.555246i −0.403141 + 0.0461107i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 11.0715i 0.907010i 0.891254 + 0.453505i \(0.149827\pi\)
−0.891254 + 0.453505i \(0.850173\pi\)
\(150\) 0 0
\(151\) −0.637339 −0.0518659 −0.0259329 0.999664i \(-0.508256\pi\)
−0.0259329 + 0.999664i \(0.508256\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.87086 + 16.3568i 0.150271 + 1.31381i
\(156\) 0 0
\(157\) 0.129135 0.0103061 0.00515306 0.999987i \(-0.498360\pi\)
0.00515306 + 0.999987i \(0.498360\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −12.7581 −1.00548
\(162\) 0 0
\(163\) 19.4835 1.52606 0.763031 0.646362i \(-0.223710\pi\)
0.763031 + 0.646362i \(0.223710\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.80052i 0.139328i −0.997571 0.0696641i \(-0.977807\pi\)
0.997571 0.0696641i \(-0.0221928\pi\)
\(168\) 0 0
\(169\) 1.98359 0.152584
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 23.2335 1.76641 0.883206 0.468985i \(-0.155380\pi\)
0.883206 + 0.468985i \(0.155380\pi\)
\(174\) 0 0
\(175\) −2.98359 12.8720i −0.225538 0.973031i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2.85664i 0.213515i −0.994285 0.106757i \(-0.965953\pi\)
0.994285 0.106757i \(-0.0340468\pi\)
\(180\) 0 0
\(181\) 5.28530i 0.392853i −0.980519 0.196427i \(-0.937066\pi\)
0.980519 0.196427i \(-0.0629337\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.00000 17.4858i −0.147043 1.28558i
\(186\) 0 0
\(187\) −5.01641 −0.366836
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −5.96719 −0.431770 −0.215885 0.976419i \(-0.569264\pi\)
−0.215885 + 0.976419i \(0.569264\pi\)
\(192\) 0 0
\(193\) 14.9409i 1.07547i −0.843115 0.537733i \(-0.819281\pi\)
0.843115 0.537733i \(-0.180719\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3.23353 −0.230379 −0.115190 0.993344i \(-0.536748\pi\)
−0.115190 + 0.993344i \(0.536748\pi\)
\(198\) 0 0
\(199\) −8.12080 −0.575668 −0.287834 0.957680i \(-0.592935\pi\)
−0.287834 + 0.957680i \(0.592935\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −5.77454 −0.405293
\(204\) 0 0
\(205\) −2.21712 19.3840i −0.154850 1.35384i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 10.7253 0.741886
\(210\) 0 0
\(211\) 13.7141i 0.944119i −0.881567 0.472059i \(-0.843511\pi\)
0.881567 0.472059i \(-0.156489\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −0.258271 2.25803i −0.0176139 0.153997i
\(216\) 0 0
\(217\) 19.4569i 1.32082i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 12.8286i 0.862947i
\(222\) 0 0
\(223\) 9.84472i 0.659251i 0.944112 + 0.329626i \(0.106923\pi\)
−0.944112 + 0.329626i \(0.893077\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5.70892 −0.378914 −0.189457 0.981889i \(-0.560673\pi\)
−0.189457 + 0.981889i \(0.560673\pi\)
\(228\) 0 0
\(229\) 0.769233i 0.0508324i −0.999677 0.0254162i \(-0.991909\pi\)
0.999677 0.0254162i \(-0.00809109\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.4008i 1.20548i −0.797939 0.602739i \(-0.794076\pi\)
0.797939 0.602739i \(-0.205924\pi\)
\(234\) 0 0
\(235\) −15.7417 + 1.80052i −1.02688 + 0.117453i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −10.0328 −0.648969 −0.324484 0.945891i \(-0.605191\pi\)
−0.324484 + 0.945891i \(0.605191\pi\)
\(240\) 0 0
\(241\) 10.7581 0.692992 0.346496 0.938051i \(-0.387371\pi\)
0.346496 + 0.938051i \(0.387371\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.00416898 + 0.0364490i 0.000266347 + 0.00232864i
\(246\) 0 0
\(247\) 27.4282i 1.74522i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 12.6580i 0.798966i 0.916741 + 0.399483i \(0.130810\pi\)
−0.916741 + 0.399483i \(0.869190\pi\)
\(252\) 0 0
\(253\) −7.30749 −0.459418
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 13.3110i 0.830316i −0.909749 0.415158i \(-0.863726\pi\)
0.909749 0.415158i \(-0.136274\pi\)
\(258\) 0 0
\(259\) 20.7999i 1.29244i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 18.4256i 1.13617i −0.822969 0.568087i \(-0.807684\pi\)
0.822969 0.568087i \(-0.192316\pi\)
\(264\) 0 0
\(265\) −1.14554 10.0153i −0.0703701 0.615238i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.86940i 0.235921i 0.993018 + 0.117961i \(0.0376357\pi\)
−0.993018 + 0.117961i \(0.962364\pi\)
\(270\) 0 0
\(271\) 17.3955 1.05670 0.528350 0.849027i \(-0.322811\pi\)
0.528350 + 0.849027i \(0.322811\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.70892 7.37270i −0.103052 0.444591i
\(276\) 0 0
\(277\) 0.887271 0.0533110 0.0266555 0.999645i \(-0.491514\pi\)
0.0266555 + 0.999645i \(0.491514\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 13.4835 0.804356 0.402178 0.915562i \(-0.368253\pi\)
0.402178 + 0.915562i \(0.368253\pi\)
\(282\) 0 0
\(283\) 28.4342 1.69024 0.845120 0.534577i \(-0.179529\pi\)
0.845120 + 0.534577i \(0.179529\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 23.0580i 1.36107i
\(288\) 0 0
\(289\) 6.01641 0.353906
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 7.99166 0.466878 0.233439 0.972371i \(-0.425002\pi\)
0.233439 + 0.972371i \(0.425002\pi\)
\(294\) 0 0
\(295\) 15.1044 1.72762i 0.879412 0.100586i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 18.6877i 1.08074i
\(300\) 0 0
\(301\) 2.68601i 0.154819i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −8.00000 + 0.915029i −0.458079 + 0.0523944i
\(306\) 0 0
\(307\) −17.4506 −0.995961 −0.497980 0.867188i \(-0.665925\pi\)
−0.497980 + 0.867188i \(0.665925\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 21.4506 1.21635 0.608177 0.793801i \(-0.291901\pi\)
0.608177 + 0.793801i \(0.291901\pi\)
\(312\) 0 0
\(313\) 7.73879i 0.437422i −0.975790 0.218711i \(-0.929815\pi\)
0.975790 0.218711i \(-0.0701853\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −11.2335 −0.630938 −0.315469 0.948936i \(-0.602162\pi\)
−0.315469 + 0.948936i \(0.602162\pi\)
\(318\) 0 0
\(319\) −3.30749 −0.185184
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 23.4835 1.30665
\(324\) 0 0
\(325\) −18.8545 + 4.37027i −1.04586 + 0.242419i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −18.7253 −1.03236
\(330\) 0 0
\(331\) 8.00084i 0.439766i 0.975526 + 0.219883i \(0.0705676\pi\)
−0.975526 + 0.219883i \(0.929432\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0.258271 + 2.25803i 0.0141108 + 0.123369i
\(336\) 0 0
\(337\) 21.5692i 1.17495i −0.809243 0.587474i \(-0.800123\pi\)
0.809243 0.587474i \(-0.199877\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 11.1444i 0.603501i
\(342\) 0 0
\(343\) 18.5419i 1.00117i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 21.7089 1.16540 0.582698 0.812689i \(-0.301997\pi\)
0.582698 + 0.812689i \(0.301997\pi\)
\(348\) 0 0
\(349\) 24.7422i 1.32442i 0.749318 + 0.662211i \(0.230382\pi\)
−0.749318 + 0.662211i \(0.769618\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 3.31415i 0.176394i 0.996103 + 0.0881972i \(0.0281106\pi\)
−0.996103 + 0.0881972i \(0.971889\pi\)
\(354\) 0 0
\(355\) −1.70892 14.9409i −0.0906998 0.792979i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 16.7581 0.884461 0.442230 0.896902i \(-0.354187\pi\)
0.442230 + 0.896902i \(0.354187\pi\)
\(360\) 0 0
\(361\) −31.2088 −1.64257
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −34.4671 + 3.94229i −1.80409 + 0.206349i
\(366\) 0 0
\(367\) 28.5324i 1.48938i −0.667411 0.744690i \(-0.732597\pi\)
0.667411 0.744690i \(-0.267403\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 11.9136i 0.618523i
\(372\) 0 0
\(373\) 37.5798 1.94581 0.972904 0.231211i \(-0.0742688\pi\)
0.972904 + 0.231211i \(0.0742688\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.45836i 0.435628i
\(378\) 0 0
\(379\) 6.74456i 0.346445i 0.984883 + 0.173222i \(0.0554179\pi\)
−0.984883 + 0.173222i \(0.944582\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 21.8312i 1.11552i −0.830001 0.557762i \(-0.811660\pi\)
0.830001 0.557762i \(-0.188340\pi\)
\(384\) 0 0
\(385\) −1.01641 8.88633i −0.0518009 0.452889i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 8.81344i 0.446859i −0.974720 0.223429i \(-0.928275\pi\)
0.974720 0.223429i \(-0.0717252\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −1.87086 16.3568i −0.0941334 0.822998i
\(396\) 0 0
\(397\) −0.821644 −0.0412372 −0.0206186 0.999787i \(-0.506564\pi\)
−0.0206186 + 0.999787i \(0.506564\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 12.7253 0.635472 0.317736 0.948179i \(-0.397077\pi\)
0.317736 + 0.948179i \(0.397077\pi\)
\(402\) 0 0
\(403\) 28.4999 1.41968
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 11.9136i 0.590535i
\(408\) 0 0
\(409\) −2.25827 −0.111664 −0.0558321 0.998440i \(-0.517781\pi\)
−0.0558321 + 0.998440i \(0.517781\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 17.9672 0.884107
\(414\) 0 0
\(415\) 1.96719 + 17.1989i 0.0965654 + 0.844261i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 33.4579i 1.63453i −0.576264 0.817263i \(-0.695490\pi\)
0.576264 0.817263i \(-0.304510\pi\)
\(420\) 0 0
\(421\) 11.3398i 0.552669i 0.961061 + 0.276335i \(0.0891198\pi\)
−0.961061 + 0.276335i \(0.910880\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.74173 16.1428i −0.181501 0.783040i
\(426\) 0 0
\(427\) −9.51627 −0.460525
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 10.6597 0.513459 0.256730 0.966483i \(-0.417355\pi\)
0.256730 + 0.966483i \(0.417355\pi\)
\(432\) 0 0
\(433\) 26.5132i 1.27414i 0.770805 + 0.637072i \(0.219854\pi\)
−0.770805 + 0.637072i \(0.780146\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 34.2088 1.63643
\(438\) 0 0
\(439\) 32.8789 1.56923 0.784613 0.619986i \(-0.212862\pi\)
0.784613 + 0.619986i \(0.212862\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 5.70892 0.271239 0.135619 0.990761i \(-0.456698\pi\)
0.135619 + 0.990761i \(0.456698\pi\)
\(444\) 0 0
\(445\) 3.75007 + 32.7864i 0.177770 + 1.55423i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −2.00000 −0.0943858 −0.0471929 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 0 0
\(451\) 13.2069i 0.621890i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −22.7253 + 2.59929i −1.06538 + 0.121857i
\(456\) 0 0
\(457\) 3.94229i 0.184413i 0.995740 + 0.0922064i \(0.0293920\pi\)
−0.995740 + 0.0922064i \(0.970608\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 33.8969i 1.57874i −0.613920 0.789369i \(-0.710408\pi\)
0.613920 0.789369i \(-0.289592\pi\)
\(462\) 0 0
\(463\) 22.8688i 1.06280i −0.847120 0.531402i \(-0.821665\pi\)
0.847120 0.531402i \(-0.178335\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −15.7417 −0.728440 −0.364220 0.931313i \(-0.618664\pi\)
−0.364220 + 0.931313i \(0.618664\pi\)
\(468\) 0 0
\(469\) 2.68601i 0.124028i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.53847i 0.0707388i
\(474\) 0 0
\(475\) 8.00000 + 34.5140i 0.367065 + 1.58361i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −20.6925 −0.945465 −0.472732 0.881206i \(-0.656732\pi\)
−0.472732 + 0.881206i \(0.656732\pi\)
\(480\) 0 0
\(481\) −30.4671 −1.38918
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 24.7581 2.83180i 1.12421 0.128586i
\(486\) 0 0
\(487\) 30.8401i 1.39750i 0.715366 + 0.698750i \(0.246260\pi\)
−0.715366 + 0.698750i \(0.753740\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 10.9737i 0.495238i 0.968858 + 0.247619i \(0.0796481\pi\)
−0.968858 + 0.247619i \(0.920352\pi\)
\(492\) 0 0
\(493\) −7.24186 −0.326157
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 17.7727i 0.797213i
\(498\) 0 0
\(499\) 3.71729i 0.166409i −0.996533 0.0832044i \(-0.973485\pi\)
0.996533 0.0832044i \(-0.0265154\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 39.9451i 1.78107i 0.454919 + 0.890533i \(0.349668\pi\)
−0.454919 + 0.890533i \(0.650332\pi\)
\(504\) 0 0
\(505\) 29.6126 3.38705i 1.31774 0.150722i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0.0728979i 0.00323114i 0.999999 + 0.00161557i \(0.000514253\pi\)
−0.999999 + 0.00161557i \(0.999486\pi\)
\(510\) 0 0
\(511\) −40.9997 −1.81372
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −2.12914 + 0.243528i −0.0938209 + 0.0107311i
\(516\) 0 0
\(517\) −10.7253 −0.471699
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 11.9672 0.524292 0.262146 0.965028i \(-0.415570\pi\)
0.262146 + 0.965028i \(0.415570\pi\)
\(522\) 0 0
\(523\) −16.0656 −0.702501 −0.351250 0.936282i \(-0.614243\pi\)
−0.351250 + 0.936282i \(0.614243\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 24.4010i 1.06292i
\(528\) 0 0
\(529\) −0.307491 −0.0133692
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −33.7745 −1.46294
\(534\) 0 0
\(535\) −1.01641 8.88633i −0.0439431 0.384190i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0.0248338i 0.00106967i
\(540\) 0 0
\(541\) 15.8559i 0.681698i −0.940118 0.340849i \(-0.889285\pi\)
0.940118 0.340849i \(-0.110715\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −1.70892 + 0.195463i −0.0732019 + 0.00837274i
\(546\) 0 0
\(547\) −4.95078 −0.211680 −0.105840 0.994383i \(-0.533753\pi\)
−0.105840 + 0.994383i \(0.533753\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 15.4835 0.659618
\(552\) 0 0
\(553\) 19.4569i 0.827393i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.26634 0.0536565 0.0268283 0.999640i \(-0.491459\pi\)
0.0268283 + 0.999640i \(0.491459\pi\)
\(558\) 0 0
\(559\) −3.93437 −0.166406
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −5.70892 −0.240602 −0.120301 0.992737i \(-0.538386\pi\)
−0.120301 + 0.992737i \(0.538386\pi\)
\(564\) 0 0
\(565\) −32.1208 + 3.67393i −1.35133 + 0.154564i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −2.75814 −0.115627 −0.0578135 0.998327i \(-0.518413\pi\)
−0.0578135 + 0.998327i \(0.518413\pi\)
\(570\) 0 0
\(571\) 25.7735i 1.07859i 0.842118 + 0.539294i \(0.181309\pi\)
−0.842118 + 0.539294i \(0.818691\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −5.45065 23.5155i −0.227308 0.980663i
\(576\) 0 0
\(577\) 32.7135i 1.36188i 0.732338 + 0.680941i \(0.238429\pi\)
−0.732338 + 0.680941i \(0.761571\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 20.4587i 0.848769i
\(582\) 0 0
\(583\) 6.82376i 0.282611i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −43.4835 −1.79475 −0.897377 0.441264i \(-0.854530\pi\)
−0.897377 + 0.441264i \(0.854530\pi\)
\(588\) 0 0
\(589\) 52.1705i 2.14965i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 7.83021i 0.321548i −0.986991 0.160774i \(-0.948601\pi\)
0.986991 0.160774i \(-0.0513991\pi\)
\(594\) 0 0
\(595\) −2.22546 19.4569i −0.0912348 0.797656i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 32.7581 1.33846 0.669231 0.743055i \(-0.266624\pi\)
0.669231 + 0.743055i \(0.266624\pi\)
\(600\) 0 0
\(601\) 17.8074 0.726377 0.363189 0.931716i \(-0.381688\pi\)
0.363189 + 0.931716i \(0.381688\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2.21295 + 19.3476i 0.0899692 + 0.786591i
\(606\) 0 0
\(607\) 3.41188i 0.138484i −0.997600 0.0692420i \(-0.977942\pi\)
0.997600 0.0692420i \(-0.0220581\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 27.4282i 1.10963i
\(612\) 0 0
\(613\) 36.6290 1.47943 0.739716 0.672920i \(-0.234960\pi\)
0.739716 + 0.672920i \(0.234960\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 40.3979i 1.62636i 0.582012 + 0.813180i \(0.302266\pi\)
−0.582012 + 0.813180i \(0.697734\pi\)
\(618\) 0 0
\(619\) 24.5172i 0.985430i 0.870191 + 0.492715i \(0.163996\pi\)
−0.870191 + 0.492715i \(0.836004\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 39.0006i 1.56252i
\(624\) 0 0
\(625\) 22.4506 10.9986i 0.898026 0.439943i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 26.0852i 1.04009i
\(630\) 0 0
\(631\) −18.7805 −0.747640 −0.373820 0.927501i \(-0.621952\pi\)
−0.373820 + 0.927501i \(0.621952\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −25.6126 + 2.92953i −1.01640 + 0.116255i
\(636\) 0 0
\(637\) 0.0635083 0.00251629
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 15.5163 0.612856 0.306428 0.951894i \(-0.400866\pi\)
0.306428 + 0.951894i \(0.400866\pi\)
\(642\) 0 0
\(643\) −17.4506 −0.688186 −0.344093 0.938936i \(-0.611814\pi\)
−0.344093 + 0.938936i \(0.611814\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 13.1403i 0.516600i 0.966065 + 0.258300i \(0.0831624\pi\)
−0.966065 + 0.258300i \(0.916838\pi\)
\(648\) 0 0
\(649\) 10.2911 0.403960
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 14.7993 0.579141 0.289570 0.957157i \(-0.406488\pi\)
0.289570 + 0.957157i \(0.406488\pi\)
\(654\) 0 0
\(655\) 16.3791 1.87342i 0.639983 0.0732004i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 7.99614i 0.311485i 0.987798 + 0.155743i \(0.0497771\pi\)
−0.987798 + 0.155743i \(0.950223\pi\)
\(660\) 0 0
\(661\) 0.915029i 0.0355905i 0.999842 + 0.0177953i \(0.00566470\pi\)
−0.999842 + 0.0177953i \(0.994335\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 4.75814 + 41.5999i 0.184513 + 1.61317i
\(666\) 0 0
\(667\) −10.5494 −0.408473
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −5.45065 −0.210420
\(672\) 0 0
\(673\) 34.3978i 1.32594i 0.748647 + 0.662969i \(0.230704\pi\)
−0.748647 + 0.662969i \(0.769296\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −40.1676 −1.54377 −0.771884 0.635764i \(-0.780685\pi\)
−0.771884 + 0.635764i \(0.780685\pi\)
\(678\) 0 0
\(679\) 29.4506 1.13021
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 33.2580 1.27258 0.636291 0.771449i \(-0.280468\pi\)
0.636291 + 0.771449i \(0.280468\pi\)
\(684\) 0 0
\(685\) 8.63734 0.987927i 0.330016 0.0377468i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −17.4506 −0.664817
\(690\) 0 0
\(691\) 50.2241i 1.91062i −0.295611 0.955308i \(-0.595523\pi\)
0.295611 0.955308i \(-0.404477\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −32.4999 + 3.71729i −1.23279 + 0.141005i
\(696\) 0 0
\(697\) 28.9170i 1.09531i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 23.7543i 0.897188i 0.893736 + 0.448594i \(0.148075\pi\)
−0.893736 + 0.448594i \(0.851925\pi\)
\(702\) 0 0
\(703\) 55.7715i 2.10346i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 35.2252 1.32478
\(708\) 0 0
\(709\) 36.3146i 1.36382i −0.731435 0.681911i \(-0.761149\pi\)
0.731435 0.681911i \(-0.238851\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 35.5453i 1.33118i
\(714\) 0 0
\(715\) −13.0164 + 1.48880i −0.486786 + 0.0556779i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −30.7253 −1.14586 −0.572931 0.819604i \(-0.694194\pi\)
−0.572931 + 0.819604i \(0.694194\pi\)
\(720\) 0 0
\(721\) −2.53268 −0.0943219
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −2.46705 10.6435i −0.0916240 0.395289i
\(726\) 0 0
\(727\) 5.47445i 0.203036i 0.994834 + 0.101518i \(0.0323700\pi\)
−0.994834 + 0.101518i \(0.967630\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 3.36852i 0.124589i
\(732\) 0 0
\(733\) −17.1455 −0.633285 −0.316643 0.948545i \(-0.602556\pi\)
−0.316643 + 0.948545i \(0.602556\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.53847i 0.0566701i
\(738\) 0 0
\(739\) 11.6019i 0.426782i −0.976967 0.213391i \(-0.931549\pi\)
0.976967 0.213391i \(-0.0684508\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 23.6613i 0.868048i −0.900901 0.434024i \(-0.857093\pi\)
0.900901 0.434024i \(-0.142907\pi\)
\(744\) 0 0
\(745\) −24.5962 + 2.81328i −0.901135 + 0.103071i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 10.5706i 0.386241i
\(750\) 0 0
\(751\) 11.4283 0.417024 0.208512 0.978020i \(-0.433138\pi\)
0.208512 + 0.978020i \(0.433138\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −0.161949 1.41590i −0.00589392 0.0515299i
\(756\) 0 0
\(757\) −19.1784 −0.697049 −0.348525 0.937300i \(-0.613317\pi\)
−0.348525 + 0.937300i \(0.613317\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −4.03281 −0.146189 −0.0730947 0.997325i \(-0.523288\pi\)
−0.0730947 + 0.997325i \(0.523288\pi\)
\(762\) 0 0
\(763\) −2.03281 −0.0735928
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 26.3177i 0.950279i
\(768\) 0 0
\(769\) 2.95078 0.106408 0.0532039 0.998584i \(-0.483057\pi\)
0.0532039 + 0.998584i \(0.483057\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −45.2663 −1.62812 −0.814059 0.580783i \(-0.802747\pi\)
−0.814059 + 0.580783i \(0.802747\pi\)
\(774\) 0 0
\(775\) −35.8625 + 8.31256i −1.28822 + 0.298596i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 61.8260i 2.21515i
\(780\) 0 0
\(781\) 10.1797i 0.364257i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0.0328135 + 0.286885i 0.00117116 + 0.0102394i
\(786\) 0 0
\(787\) 52.9997 1.88924 0.944618 0.328171i \(-0.106432\pi\)
0.944618 + 0.328171i \(0.106432\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −38.2088 −1.35855
\(792\) 0 0
\(793\) 13.9391i 0.494993i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −16.5738 −0.587075 −0.293538 0.955948i \(-0.594833\pi\)
−0.293538 + 0.955948i \(0.594833\pi\)
\(798\) 0 0
\(799\) −23.4835 −0.830785
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −23.4835 −0.828713
\(804\) 0 0
\(805\) −3.24186 28.3433i −0.114261 0.998969i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −37.5491 −1.32016 −0.660078 0.751197i \(-0.729477\pi\)
−0.660078 + 0.751197i \(0.729477\pi\)
\(810\) 0 0
\(811\) 32.1102i 1.12754i 0.825931 + 0.563771i \(0.190650\pi\)
−0.825931 + 0.563771i \(0.809350\pi\)