Properties

Label 363.2.f.c.239.2
Level $363$
Weight $2$
Character 363.239
Analytic conductor $2.899$
Analytic rank $0$
Dimension $8$
CM discriminant -11
Inner twists $16$

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Newspace parameters

Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 363.f (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.89856959337\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.228765625.1
Defining polynomial: \(x^{8} - x^{7} - 2 x^{6} + 5 x^{5} + x^{4} + 15 x^{3} - 18 x^{2} - 27 x + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{U}(1)[D_{10}]$

Embedding invariants

Embedding label 239.2
Root \(-1.73166 + 0.0369185i\) of defining polynomial
Character \(\chi\) \(=\) 363.239
Dual form 363.2.f.c.161.2

$q$-expansion

\(f(q)\) \(=\) \(q+(1.73166 - 0.0369185i) q^{3} +(-0.618034 + 1.90211i) q^{4} +(1.94946 - 2.68321i) q^{5} +(2.99727 - 0.127860i) q^{9} +O(q^{10})\) \(q+(1.73166 - 0.0369185i) q^{3} +(-0.618034 + 1.90211i) q^{4} +(1.94946 - 2.68321i) q^{5} +(2.99727 - 0.127860i) q^{9} +(-1.00000 + 3.31662i) q^{12} +(3.27674 - 4.71836i) q^{15} +(-3.23607 - 2.35114i) q^{16} +(3.89893 + 5.36641i) q^{20} +3.31662i q^{23} +(-1.85410 - 5.70634i) q^{25} +(5.18553 - 0.332065i) q^{27} +(-4.04508 + 2.93893i) q^{31} +(-1.60921 + 5.78018i) q^{36} +(-2.16312 + 6.65740i) q^{37} +(5.50000 - 8.29156i) q^{45} +(6.30860 - 2.04979i) q^{47} +(-5.69056 - 3.95190i) q^{48} +(-5.66312 - 4.11450i) q^{49} +(-7.79785 - 10.7328i) q^{53} +(3.15430 + 1.02489i) q^{59} +(6.94972 + 9.14884i) q^{60} +(6.47214 - 4.70228i) q^{64} -13.0000 q^{67} +(0.122445 + 5.74326i) q^{69} +(-9.74732 + 13.4160i) q^{71} +(-3.42134 - 9.81297i) q^{75} +(-12.6172 + 4.09957i) q^{80} +(8.96730 - 0.766464i) q^{81} -16.5831i q^{89} +(-6.30860 - 2.04979i) q^{92} +(-6.89620 + 5.23855i) q^{93} +(-13.7533 + 9.99235i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{3} + 4 q^{4} + 5 q^{9} + O(q^{10}) \) \( 8 q - q^{3} + 4 q^{4} + 5 q^{9} - 8 q^{12} - 11 q^{15} - 8 q^{16} + 12 q^{25} + 8 q^{27} - 10 q^{31} - 10 q^{36} + 14 q^{37} + 44 q^{45} - 4 q^{48} - 14 q^{49} + 22 q^{60} + 16 q^{64} - 104 q^{67} - 11 q^{69} + 6 q^{75} - 7 q^{81} - 5 q^{93} - 34 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{10}\right)\)

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 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(3\) 1.73166 0.0369185i 0.999773 0.0213149i
\(4\) −0.618034 + 1.90211i −0.309017 + 0.951057i
\(5\) 1.94946 2.68321i 0.871826 1.19997i −0.106792 0.994281i \(-0.534058\pi\)
0.978618 0.205685i \(-0.0659421\pi\)
\(6\) 0 0
\(7\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(8\) 0 0
\(9\) 2.99727 0.127860i 0.999091 0.0426201i
\(10\) 0 0
\(11\) 0 0
\(12\) −1.00000 + 3.31662i −0.288675 + 0.957427i
\(13\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(14\) 0 0
\(15\) 3.27674 4.71836i 0.846051 1.21828i
\(16\) −3.23607 2.35114i −0.809017 0.587785i
\(17\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(18\) 0 0
\(19\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(20\) 3.89893 + 5.36641i 0.871826 + 1.19997i
\(21\) 0 0
\(22\) 0 0
\(23\) 3.31662i 0.691564i 0.938315 + 0.345782i \(0.112386\pi\)
−0.938315 + 0.345782i \(0.887614\pi\)
\(24\) 0 0
\(25\) −1.85410 5.70634i −0.370820 1.14127i
\(26\) 0 0
\(27\) 5.18553 0.332065i 0.997956 0.0639059i
\(28\) 0 0
\(29\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(30\) 0 0
\(31\) −4.04508 + 2.93893i −0.726519 + 0.527847i −0.888460 0.458954i \(-0.848224\pi\)
0.161942 + 0.986800i \(0.448224\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −1.60921 + 5.78018i −0.268202 + 0.963363i
\(37\) −2.16312 + 6.65740i −0.355615 + 1.09447i 0.600038 + 0.799972i \(0.295152\pi\)
−0.955652 + 0.294497i \(0.904848\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 5.50000 8.29156i 0.819892 1.23603i
\(46\) 0 0
\(47\) 6.30860 2.04979i 0.920203 0.298992i 0.189653 0.981851i \(-0.439264\pi\)
0.730550 + 0.682859i \(0.239264\pi\)
\(48\) −5.69056 3.95190i −0.821362 0.570408i
\(49\) −5.66312 4.11450i −0.809017 0.587785i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.79785 10.7328i −1.07112 1.47427i −0.868940 0.494918i \(-0.835198\pi\)
−0.202178 0.979349i \(-0.564802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.15430 + 1.02489i 0.410655 + 0.133430i 0.507057 0.861913i \(-0.330733\pi\)
−0.0964021 + 0.995342i \(0.530733\pi\)
\(60\) 6.94972 + 9.14884i 0.897205 + 1.18111i
\(61\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 6.47214 4.70228i 0.809017 0.587785i
\(65\) 0 0
\(66\) 0 0
\(67\) −13.0000 −1.58820 −0.794101 0.607785i \(-0.792058\pi\)
−0.794101 + 0.607785i \(0.792058\pi\)
\(68\) 0 0
\(69\) 0.122445 + 5.74326i 0.0147406 + 0.691407i
\(70\) 0 0
\(71\) −9.74732 + 13.4160i −1.15679 + 1.59219i −0.434378 + 0.900731i \(0.643032\pi\)
−0.722416 + 0.691459i \(0.756968\pi\)
\(72\) 0 0
\(73\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(74\) 0 0
\(75\) −3.42134 9.81297i −0.395062 1.13310i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(80\) −12.6172 + 4.09957i −1.41064 + 0.458346i
\(81\) 8.96730 0.766464i 0.996367 0.0851627i
\(82\) 0 0
\(83\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 16.5831i 1.75781i −0.476999 0.878904i \(-0.658275\pi\)
0.476999 0.878904i \(-0.341725\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −6.30860 2.04979i −0.657717 0.213705i
\(93\) −6.89620 + 5.23855i −0.715103 + 0.543212i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −13.7533 + 9.99235i −1.39643 + 1.01457i −0.401310 + 0.915942i \(0.631445\pi\)
−0.995124 + 0.0986273i \(0.968555\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 12.0000 1.20000
\(101\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(102\) 0 0
\(103\) −1.23607 + 3.80423i −0.121793 + 0.374842i −0.993303 0.115536i \(-0.963141\pi\)
0.871510 + 0.490378i \(0.163141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(108\) −2.57321 + 10.0687i −0.247607 + 0.968860i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) −3.50000 + 11.6082i −0.332205 + 1.10180i
\(112\) 0 0
\(113\) −3.15430 + 1.02489i −0.296731 + 0.0964139i −0.453599 0.891206i \(-0.649860\pi\)
0.156868 + 0.987620i \(0.449860\pi\)
\(114\) 0 0
\(115\) 8.89919 + 6.46564i 0.829853 + 0.602924i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) −3.09017 9.51057i −0.277505 0.854074i
\(125\) −3.15430 1.02489i −0.282129 0.0916693i
\(126\) 0 0
\(127\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 9.21800 14.5612i 0.793359 1.25323i
\(136\) 0 0
\(137\) 13.6462 18.7824i 1.16588 1.60469i 0.479260 0.877673i \(-0.340905\pi\)
0.686617 0.727019i \(-0.259095\pi\)
\(138\) 0 0
\(139\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(140\) 0 0
\(141\) 10.8487 3.78243i 0.913621 0.318538i
\(142\) 0 0
\(143\) 0 0
\(144\) −10.0000 6.63325i −0.833333 0.552771i
\(145\) 0 0
\(146\) 0 0
\(147\) −9.95848 6.91582i −0.821362 0.570408i
\(148\) −11.3262 8.22899i −0.931011 0.676419i
\(149\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(150\) 0 0
\(151\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 16.5831i 1.33199i
\(156\) 0 0
\(157\) 7.10739 + 21.8743i 0.567232 + 1.74576i 0.661226 + 0.750186i \(0.270036\pi\)
−0.0939948 + 0.995573i \(0.529964\pi\)
\(158\) 0 0
\(159\) −13.8994 18.2977i −1.10230 1.45110i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 12.9443 9.40456i 1.01387 0.736622i 0.0488556 0.998806i \(-0.484443\pi\)
0.965018 + 0.262184i \(0.0844426\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(168\) 0 0
\(169\) 4.01722 12.3637i 0.309017 0.951057i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.50000 + 1.65831i 0.413405 + 0.124646i
\(178\) 0 0
\(179\) 15.7715 5.12447i 1.17882 0.383021i 0.346890 0.937906i \(-0.387238\pi\)
0.831927 + 0.554885i \(0.187238\pi\)
\(180\) 12.3723 + 15.5861i 0.922177 + 1.16172i
\(181\) 20.2254 + 14.6946i 1.50334 + 1.09224i 0.969026 + 0.246960i \(0.0794316\pi\)
0.534318 + 0.845283i \(0.320568\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 13.6462 + 18.7824i 1.00329 + 1.38091i
\(186\) 0 0
\(187\) 0 0
\(188\) 13.2665i 0.967559i
\(189\) 0 0
\(190\) 0 0
\(191\) 22.0801 + 7.17425i 1.59766 + 0.519111i 0.966527 0.256565i \(-0.0825907\pi\)
0.631132 + 0.775676i \(0.282591\pi\)
\(192\) 11.0339 8.38168i 0.796305 0.604896i
\(193\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 11.3262 8.22899i 0.809017 0.587785i
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) −22.5115 + 0.479940i −1.58784 + 0.0338524i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0.424064 + 9.94083i 0.0294745 + 0.690936i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(212\) 25.2344 8.19915i 1.73310 0.563120i
\(213\) −16.3837 + 23.5918i −1.12259 + 1.61648i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −0.309017 0.951057i −0.0206933 0.0636875i 0.940177 0.340687i \(-0.110660\pi\)
−0.960870 + 0.277000i \(0.910660\pi\)
\(224\) 0 0
\(225\) −6.28687 16.8664i −0.419124 1.12443i
\(226\) 0 0
\(227\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(228\) 0 0
\(229\) −4.04508 + 2.93893i −0.267307 + 0.194210i −0.713362 0.700796i \(-0.752828\pi\)
0.446055 + 0.895005i \(0.352828\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(234\) 0 0
\(235\) 6.79837 20.9232i 0.443477 1.36488i
\(236\) −3.89893 + 5.36641i −0.253798 + 0.349324i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(240\) −21.6973 + 7.56486i −1.40055 + 0.488310i
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 15.5000 1.65831i 0.994325 0.106381i
\(244\) 0 0
\(245\) −22.0801 + 7.17425i −1.41064 + 0.458346i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 9.74732 + 13.4160i 0.615245 + 0.846812i 0.996996 0.0774530i \(-0.0246788\pi\)
−0.381751 + 0.924265i \(0.624679\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 4.94427 + 15.2169i 0.309017 + 0.951057i
\(257\) −25.2344 8.19915i −1.57408 0.511449i −0.613555 0.789652i \(-0.710261\pi\)
−0.960522 + 0.278203i \(0.910261\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −44.0000 −2.70290
\(266\) 0 0
\(267\) −0.612223 28.7163i −0.0374675 1.75741i
\(268\) 8.03444 24.7275i 0.490782 1.51047i
\(269\) 7.79785 10.7328i 0.475443 0.654392i −0.502178 0.864764i \(-0.667468\pi\)
0.977621 + 0.210373i \(0.0674677\pi\)
\(270\) 0 0
\(271\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) −11.0000 3.31662i −0.662122 0.199637i
\(277\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(278\) 0 0
\(279\) −11.7485 + 9.32597i −0.703362 + 0.558331i
\(280\) 0 0
\(281\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(282\) 0 0
\(283\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(284\) −19.4946 26.8321i −1.15679 1.59219i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −5.25329 16.1680i −0.309017 0.951057i
\(290\) 0 0
\(291\) −23.4471 + 17.8111i −1.37449 + 1.04410i
\(292\) 0 0
\(293\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(294\) 0 0
\(295\) 8.89919 6.46564i 0.518131 0.376444i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 20.7799 0.443021i 1.19973 0.0255779i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) −2.00000 + 6.63325i −0.113776 + 0.377352i
\(310\) 0 0
\(311\) −31.5430 + 10.2489i −1.78864 + 0.581164i −0.999456 0.0329949i \(-0.989495\pi\)
−0.789183 + 0.614159i \(0.789495\pi\)
\(312\) 0 0
\(313\) 15.3713 + 11.1679i 0.868839 + 0.631248i 0.930275 0.366863i \(-0.119568\pi\)
−0.0614365 + 0.998111i \(0.519568\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −13.6462 18.7824i −0.766449 1.05493i −0.996650 0.0817838i \(-0.973938\pi\)
0.230201 0.973143i \(-0.426062\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 26.5330i 1.48324i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −4.08420 + 17.5305i −0.226900 + 0.973918i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 35.0000 1.92377 0.961887 0.273447i \(-0.0881639\pi\)
0.961887 + 0.273447i \(0.0881639\pi\)
\(332\) 0 0
\(333\) −5.63224 + 20.2306i −0.308645 + 1.10863i
\(334\) 0 0
\(335\) −25.3430 + 34.8817i −1.38464 + 1.90579i
\(336\) 0 0
\(337\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(338\) 0 0
\(339\) −5.42433 + 1.89122i −0.294609 + 0.102717i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 15.6490 + 10.8677i 0.842516 + 0.585099i
\(346\) 0 0
\(347\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(348\) 0 0
\(349\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 36.4829i 1.94179i −0.239511 0.970894i \(-0.576987\pi\)
0.239511 0.970894i \(-0.423013\pi\)
\(354\) 0 0
\(355\) 16.9959 + 52.3081i 0.902051 + 2.77623i
\(356\) 31.5430 + 10.2489i 1.67177 + 0.543192i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(360\) 0 0
\(361\) −15.3713 + 11.1679i −0.809017 + 0.587785i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −11.4336 + 35.1891i −0.596831 + 1.83686i −0.0514358 + 0.998676i \(0.516380\pi\)
−0.545395 + 0.838179i \(0.683620\pi\)
\(368\) 7.79785 10.7328i 0.406491 0.559487i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −5.70223 16.3550i −0.295647 0.847965i
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) −5.50000 1.65831i −0.284019 0.0856349i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 20.2254 + 14.6946i 1.03891 + 0.754813i 0.970073 0.242815i \(-0.0780709\pi\)
0.0688378 + 0.997628i \(0.478071\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.94946 2.68321i −0.0996129 0.137105i 0.756301 0.654224i \(-0.227005\pi\)
−0.855914 + 0.517119i \(0.827005\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −10.5066 32.3359i −0.533391 1.64161i
\(389\) −34.6973 11.2738i −1.75922 0.571606i −0.762102 0.647456i \(-0.775833\pi\)
−0.997119 + 0.0758507i \(0.975833\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −7.41641 + 22.8254i −0.370820 + 1.14127i
\(401\) −15.5957 + 21.4656i −0.778812 + 1.07194i 0.216600 + 0.976261i \(0.430503\pi\)
−0.995412 + 0.0956827i \(0.969497\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 15.4248 25.5553i 0.766467 1.26985i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(410\) 0 0
\(411\) 22.9372 33.0285i 1.13141 1.62918i
\(412\) −6.47214 4.70228i −0.318859 0.231665i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 33.1662i 1.62028i 0.586238 + 0.810139i \(0.300608\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) −3.09017 9.51057i −0.150606 0.463517i 0.847084 0.531460i \(-0.178356\pi\)
−0.997689 + 0.0679432i \(0.978356\pi\)
\(422\) 0 0
\(423\) 18.6465 6.95039i 0.906624 0.337940i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(432\) −17.5615 11.1173i −0.844926 0.534883i
\(433\) 8.96149 27.5806i 0.430662 1.32544i −0.466805 0.884360i \(-0.654595\pi\)
0.897467 0.441081i \(-0.145405\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −17.5000 11.6082i −0.833333 0.552771i
\(442\) 0 0
\(443\) 34.6973 11.2738i 1.64852 0.535636i 0.670099 0.742271i \(-0.266252\pi\)
0.978418 + 0.206636i \(0.0662515\pi\)
\(444\) −19.9170 13.8316i −0.945217 0.656421i
\(445\) −44.4959 32.3282i −2.10931 1.53250i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 9.74732 + 13.4160i 0.460004 + 0.633142i 0.974510 0.224346i \(-0.0720244\pi\)
−0.514505 + 0.857487i \(0.672024\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 6.63325i 0.312002i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −17.7984 + 12.9313i −0.829853 + 0.602924i
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −31.0000 −1.44069 −0.720346 0.693615i \(-0.756017\pi\)
−0.720346 + 0.693615i \(0.756017\pi\)
\(464\) 0 0
\(465\) 0.612223 + 28.7163i 0.0283912 + 1.33169i
\(466\) 0 0
\(467\) 25.3430 34.8817i 1.17274 1.61413i 0.530212 0.847865i \(-0.322112\pi\)
0.642523 0.766267i \(-0.277888\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 13.1151 + 37.6164i 0.604313 + 1.73327i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −24.7446 31.1722i −1.13298 1.42728i
\(478\) 0 0
\(479\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 56.3826i 2.56020i
\(486\) 0 0
\(487\) −13.2877 40.8954i −0.602125 1.85315i −0.515465 0.856911i \(-0.672381\pi\)
−0.0866600 0.996238i \(-0.527619\pi\)
\(488\) 0 0
\(489\) 22.0678 16.7634i 0.997942 0.758066i
\(490\) 0 0
\(491\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 20.0000 0.898027
\(497\) 0 0
\(498\) 0 0
\(499\) −12.3607 + 38.0423i −0.553340 + 1.70301i 0.146947 + 0.989144i \(0.453055\pi\)
−0.700287 + 0.713861i \(0.746945\pi\)
\(500\) 3.89893 5.36641i 0.174365 0.239993i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 6.50000 21.5581i 0.288675 0.957427i
\(508\) 0 0
\(509\) −3.15430 + 1.02489i −0.139812 + 0.0454276i −0.378087 0.925770i \(-0.623418\pi\)
0.238275 + 0.971198i \(0.423418\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 7.79785 + 10.7328i 0.343614 + 0.472945i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 41.0059 + 13.3236i 1.79650 + 0.583718i 0.999787 0.0206400i \(-0.00657038\pi\)
0.796713 + 0.604358i \(0.206570\pi\)
\(522\) 0 0
\(523\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 12.0000 0.521739
\(530\) 0 0
\(531\) 9.58534 + 2.66858i 0.415968 + 0.115806i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 27.1216 9.45608i 1.17038 0.408060i
\(538\) 0 0
\(539\) 0 0
\(540\) 22.0000 + 26.5330i 0.946729 + 1.14180i
\(541\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(542\) 0 0
\(543\) 35.5660 + 24.6994i 1.52628 + 1.05995i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(548\) 27.2925 + 37.5649i 1.16588 + 1.60469i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 24.3240 + 32.0210i 1.03250 + 1.35921i
\(556\) 0 0
\(557\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(564\) 0.489779 + 22.9730i 0.0206234 + 0.967339i
\(565\) −3.39919 + 10.4616i −0.143005 + 0.440124i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 38.5000 + 11.6082i 1.60836 + 0.484939i
\(574\) 0 0
\(575\) 18.9258 6.14936i 0.789260 0.256446i
\(576\) 18.7975 14.9216i 0.783230 0.621732i
\(577\) −38.0238 27.6259i −1.58295 1.15008i −0.913212 0.407486i \(-0.866406\pi\)
−0.669740 0.742596i \(-0.733594\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −6.30860 2.04979i −0.260384 0.0846038i 0.175916 0.984405i \(-0.443711\pi\)
−0.436300 + 0.899801i \(0.643711\pi\)
\(588\) 19.3094 14.6679i 0.796305 0.604896i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 22.6525 16.4580i 0.931011 0.676419i
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 34.6331 0.738369i 1.41744 0.0302194i
\(598\) 0 0
\(599\) 19.4946 26.8321i 0.796529 1.09633i −0.196735 0.980457i \(-0.563034\pi\)
0.993264 0.115872i \(-0.0369661\pi\)
\(600\) 0 0
\(601\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(602\) 0 0
\(603\) −38.9646 + 1.66218i −1.58676 + 0.0676893i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 26.5330i 1.06818i −0.845428 0.534089i \(-0.820655\pi\)
0.845428 0.534089i \(-0.179345\pi\)
\(618\) 0 0
\(619\) −0.309017 0.951057i −0.0124204 0.0382262i 0.944654 0.328068i \(-0.106397\pi\)
−0.957075 + 0.289841i \(0.906397\pi\)
\(620\) −31.5430 10.2489i −1.26680 0.411607i
\(621\) 1.10133 + 17.1985i 0.0441950 + 0.690150i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 15.3713 11.1679i 0.614853 0.446717i
\(626\) 0 0
\(627\) 0 0
\(628\) −46.0000 −1.83560
\(629\) 0 0
\(630\) 0 0
\(631\) −2.16312 + 6.65740i −0.0861124 + 0.265027i −0.984836 0.173489i \(-0.944496\pi\)
0.898723 + 0.438516i \(0.144496\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 43.3946 15.1297i 1.72071 0.599933i
\(637\) 0 0
\(638\) 0 0
\(639\) −27.5000 + 41.4578i −1.08788 + 1.64005i
\(640\) 0 0
\(641\) −22.0801 + 7.17425i −0.872111 + 0.283366i −0.710678 0.703518i \(-0.751612\pi\)
−0.161433 + 0.986884i \(0.551612\pi\)
\(642\) 0 0
\(643\) −33.1697 24.0992i −1.30809 0.950379i −0.308086 0.951359i \(-0.599688\pi\)
−1.00000 0.000979141i \(0.999688\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −25.3430 34.8817i −0.996337 1.37134i −0.927546 0.373709i \(-0.878086\pi\)
−0.0687910 0.997631i \(-0.521914\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 9.88854 + 30.4338i 0.387265 + 1.19188i
\(653\) 3.15430 + 1.02489i 0.123437 + 0.0401072i 0.370084 0.928998i \(-0.379329\pi\)
−0.246647 + 0.969105i \(0.579329\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −13.0000 −0.505641 −0.252821 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −0.570223 1.63550i −0.0220461 0.0632319i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(674\) 0 0
\(675\) −11.5094 28.9747i −0.442996 1.11524i
\(676\) 21.0344 + 15.2824i 0.809017 + 0.587785i
\(677\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 46.4327i 1.77670i −0.459167 0.888350i \(-0.651852\pi\)
0.459167 0.888350i \(-0.348148\pi\)
\(684\) 0 0
\(685\) −23.7943 73.2314i −0.909134 2.79803i
\(686\) 0 0
\(687\) −6.89620 + 5.23855i −0.263106 + 0.199863i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −13.7533 + 9.99235i −0.523200 + 0.380127i −0.817808 0.575491i \(-0.804811\pi\)
0.294608 + 0.955618i \(0.404811\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 11.0000 36.4829i 0.414284 1.37402i
\(706\) 0 0
\(707\) 0 0
\(708\) −6.55348 + 9.43673i −0.246295 + 0.354654i
\(709\) 15.3713 + 11.1679i 0.577282 + 0.419420i 0.837743 0.546064i \(-0.183875\pi\)
−0.260461 + 0.965484i \(0.583875\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −9.74732 13.4160i −0.365040 0.502434i
\(714\) 0 0
\(715\) 0 0
\(716\) 33.1662i 1.23948i
\(717\) 0 0
\(718\) 0 0
\(719\) −15.7715 5.12447i −0.588177 0.191110i −0.000216702 1.00000i \(-0.500069\pi\)
−0.587961 + 0.808890i \(0.700069\pi\)
\(720\) −37.2930 + 13.9008i −1.38983 + 0.518052i
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −40.4508 + 29.3893i −1.50334 + 1.09224i
\(725\) 0 0
\(726\) 0 0
\(727\) 53.0000 1.96566 0.982831 0.184510i \(-0.0590699\pi\)
0.982831 + 0.184510i \(0.0590699\pi\)
\(728\) 0 0
\(729\) 26.7795 3.44386i 0.991832 0.127551i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(734\) 0 0
\(735\) −37.9703 + 13.2385i −1.40055 + 0.488310i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(740\) −44.1602 + 14.3485i −1.62336 + 0.527462i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 7.10739 + 21.8743i 0.259352 + 0.798205i 0.992941 + 0.118611i \(0.0378440\pi\)
−0.733588 + 0.679594i \(0.762156\pi\)
\(752\) −25.2344 8.19915i −0.920203 0.298992i
\(753\) 17.3743 + 22.8721i 0.633155 + 0.833506i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −30.7426 + 22.3358i −1.11736 + 0.811810i −0.983807 0.179232i \(-0.942639\pi\)
−0.133554 + 0.991042i \(0.542639\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −27.2925 + 37.5649i −0.987407 + 1.35905i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 9.12357 + 26.1679i 0.329218 + 0.944254i
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −44.0000 13.2665i −1.58462 0.477781i
\(772\) 0 0
\(773\) −12.6172 + 4.09957i −0.453809 + 0.147451i −0.526998 0.849867i \(-0.676682\pi\)
0.0731890 + 0.997318i \(0.476682\pi\)
\(774\) 0 0
\(775\) 24.2705 + 17.6336i 0.871822 + 0.633416i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 8.65248 + 26.6296i 0.309017 + 0.951057i
\(785\) 72.5488 + 23.5725i 2.58938 + 0.841340i
\(786\) 0 0
\(787\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −76.1929 + 1.62441i −2.70228 + 0.0576120i
\(796\) −12.3607 + 38.0423i −0.438113 + 1.34837i
\(797\) −33.1409 + 45.6145i −1.17391 + 1.61575i −0.543970 + 0.839105i \(0.683079\pi\)
−0.629940 + 0.776644i \(0.716921\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −2.12032 49.7042i −0.0749179 1.75621i
\(802\) 0 0
\(803\) 0 0
\(804\) 13.0000 43.1161i 0.458475 1.52059i