Properties

Label 363.2.f.c.215.1
Level $363$
Weight $2$
Character 363.215
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 215.1
Root \(1.37924 - 1.04771i\) of defining polynomial
Character \(\chi\) \(=\) 363.215
Dual form 363.2.f.c.233.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.37924 + 1.04771i) q^{3} +(1.61803 - 1.17557i) q^{4} +(3.15430 + 1.02489i) q^{5} +(0.804606 - 2.89009i) q^{9} +O(q^{10})\) \(q+(-1.37924 + 1.04771i) q^{3} +(1.61803 - 1.17557i) q^{4} +(3.15430 + 1.02489i) q^{5} +(0.804606 - 2.89009i) q^{9} +(-1.00000 + 3.31662i) q^{12} +(-5.42433 + 1.89122i) q^{15} +(1.23607 - 3.80423i) q^{16} +(6.30860 - 2.04979i) q^{20} +3.31662i q^{23} +(4.85410 + 3.52671i) q^{25} +(1.91823 + 4.82912i) q^{27} +(1.54508 + 4.75528i) q^{31} +(-2.09562 - 5.62213i) q^{36} +(5.66312 - 4.11450i) q^{37} +(5.50000 - 8.29156i) q^{45} +(-3.89893 + 5.36641i) q^{47} +(2.28089 + 6.54198i) q^{48} +(2.16312 - 6.65740i) q^{49} +(-12.6172 + 4.09957i) q^{53} +(-1.94946 - 2.68321i) q^{59} +(-6.55348 + 9.43673i) q^{60} +(-2.47214 - 7.60845i) q^{64} -13.0000 q^{67} +(-3.47486 - 4.57442i) q^{69} +(-15.7715 - 5.12447i) q^{71} +(-10.3899 + 0.221511i) q^{75} +(7.79785 - 10.7328i) q^{80} +(-7.70522 - 4.65077i) q^{81} -16.5831i q^{89} +(3.89893 + 5.36641i) q^{92} +(-7.11320 - 4.93987i) q^{93} +(5.25329 + 16.1680i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{3} + 4 q^{4} + 5 q^{9} + O(q^{10}) \) \( 8 q - q^{3} + 4 q^{4} + 5 q^{9} - 8 q^{12} - 11 q^{15} - 8 q^{16} + 12 q^{25} + 8 q^{27} - 10 q^{31} - 10 q^{36} + 14 q^{37} + 44 q^{45} - 4 q^{48} - 14 q^{49} + 22 q^{60} + 16 q^{64} - 104 q^{67} - 11 q^{69} + 6 q^{75} - 7 q^{81} - 5 q^{93} - 34 q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(e\left(\frac{9}{10}\right)\)

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 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(3\) −1.37924 + 1.04771i −0.796305 + 0.604896i
\(4\) 1.61803 1.17557i 0.809017 0.587785i
\(5\) 3.15430 + 1.02489i 1.41064 + 0.458346i 0.912617 0.408815i \(-0.134058\pi\)
0.498027 + 0.867161i \(0.334058\pi\)
\(6\) 0 0
\(7\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(8\) 0 0
\(9\) 0.804606 2.89009i 0.268202 0.963363i
\(10\) 0 0
\(11\) 0 0
\(12\) −1.00000 + 3.31662i −0.288675 + 0.957427i
\(13\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(14\) 0 0
\(15\) −5.42433 + 1.89122i −1.40055 + 0.488310i
\(16\) 1.23607 3.80423i 0.309017 0.951057i
\(17\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(18\) 0 0
\(19\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(20\) 6.30860 2.04979i 1.41064 0.458346i
\(21\) 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
\(25\) 4.85410 + 3.52671i 0.970820 + 0.705342i
\(26\) 0 0
\(27\) 1.91823 + 4.82912i 0.369163 + 0.929364i
\(28\) 0 0
\(29\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(30\) 0 0
\(31\) 1.54508 + 4.75528i 0.277505 + 0.854074i 0.988546 + 0.150923i \(0.0482244\pi\)
−0.711040 + 0.703151i \(0.751776\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −2.09562 5.62213i −0.349270 0.937022i
\(37\) 5.66312 4.11450i 0.931011 0.676419i −0.0152291 0.999884i \(-0.504848\pi\)
0.946240 + 0.323465i \(0.104848\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(42\) 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
\(47\) −3.89893 + 5.36641i −0.568717 + 0.782772i −0.992402 0.123038i \(-0.960736\pi\)
0.423685 + 0.905810i \(0.360736\pi\)
\(48\) 2.28089 + 6.54198i 0.329218 + 0.944254i
\(49\) 2.16312 6.65740i 0.309017 0.951057i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −12.6172 + 4.09957i −1.73310 + 0.563120i −0.993892 0.110353i \(-0.964802\pi\)
−0.739212 + 0.673473i \(0.764802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.94946 2.68321i −0.253798 0.349324i 0.663039 0.748585i \(-0.269267\pi\)
−0.916837 + 0.399262i \(0.869267\pi\)
\(60\) −6.55348 + 9.43673i −0.846051 + 1.21828i
\(61\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −2.47214 7.60845i −0.309017 0.951057i
\(65\) 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
\(69\) −3.47486 4.57442i −0.418324 0.550696i
\(70\) 0 0
\(71\) −15.7715 5.12447i −1.87173 0.608162i −0.990876 0.134777i \(-0.956968\pi\)
−0.880855 0.473386i \(-0.843032\pi\)
\(72\) 0 0
\(73\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(74\) 0 0
\(75\) −10.3899 + 0.221511i −1.19973 + 0.0255779i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(80\) 7.79785 10.7328i 0.871826 1.19997i
\(81\) −7.70522 4.65077i −0.856135 0.516752i
\(82\) 0 0
\(83\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(84\) 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
\(92\) 3.89893 + 5.36641i 0.406491 + 0.559487i
\(93\) −7.11320 4.93987i −0.737605 0.512241i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 5.25329 + 16.1680i 0.533391 + 1.64161i 0.747101 + 0.664711i \(0.231445\pi\)
−0.213710 + 0.976897i \(0.568555\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 12.0000 1.20000
\(101\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(102\) 0 0
\(103\) 3.23607 2.35114i 0.318859 0.231665i −0.416829 0.908985i \(-0.636859\pi\)
0.735689 + 0.677320i \(0.236859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(108\) 8.78073 + 5.55867i 0.844926 + 0.534883i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) −3.50000 + 11.6082i −0.332205 + 1.10180i
\(112\) 0 0
\(113\) 1.94946 2.68321i 0.183390 0.252415i −0.707417 0.706796i \(-0.750140\pi\)
0.890807 + 0.454382i \(0.150140\pi\)
\(114\) 0 0
\(115\) −3.39919 + 10.4616i −0.316976 + 0.975551i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) 8.09017 + 5.87785i 0.726519 + 0.527847i
\(125\) 1.94946 + 2.68321i 0.174365 + 0.239993i
\(126\) 0 0
\(127\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(128\) 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
\(135\) 1.10133 + 17.1985i 0.0947878 + 1.48021i
\(136\) 0 0
\(137\) 22.0801 + 7.17425i 1.88643 + 0.612938i 0.982816 + 0.184588i \(0.0590949\pi\)
0.903613 + 0.428350i \(0.140905\pi\)
\(138\) 0 0
\(139\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(140\) 0 0
\(141\) −0.244889 11.4865i −0.0206234 0.967339i
\(142\) 0 0
\(143\) 0 0
\(144\) −10.0000 6.63325i −0.833333 0.552771i
\(145\) 0 0
\(146\) 0 0
\(147\) 3.99156 + 11.4485i 0.329218 + 0.944254i
\(148\) 4.32624 13.3148i 0.355615 1.09447i
\(149\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(150\) 0 0
\(151\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 16.5831i 1.33199i
\(156\) 0 0
\(157\) −18.6074 13.5191i −1.48503 1.07894i −0.975892 0.218255i \(-0.929964\pi\)
−0.509140 0.860684i \(-0.670036\pi\)
\(158\) 0 0
\(159\) 13.1070 18.8735i 1.03945 1.49676i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4.94427 15.2169i −0.387265 1.19188i −0.934824 0.355112i \(-0.884443\pi\)
0.547558 0.836768i \(-0.315557\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(168\) 0 0
\(169\) −10.5172 + 7.64121i −0.809017 + 0.587785i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.50000 + 1.65831i 0.413405 + 0.124646i
\(178\) 0 0
\(179\) −9.74732 + 13.4160i −0.728549 + 1.00276i 0.270648 + 0.962678i \(0.412762\pi\)
−0.999196 + 0.0400827i \(0.987238\pi\)
\(180\) −0.848129 19.8817i −0.0632158 1.48189i
\(181\) −7.72542 + 23.7764i −0.574226 + 1.76729i 0.0645725 + 0.997913i \(0.479432\pi\)
−0.638799 + 0.769374i \(0.720568\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 22.0801 7.17425i 1.62336 0.527462i
\(186\) 0 0
\(187\) 0 0
\(188\) 13.2665i 0.967559i
\(189\) 0 0
\(190\) 0 0
\(191\) −13.6462 18.7824i −0.987407 1.35905i −0.932742 0.360545i \(-0.882591\pi\)
−0.0546656 0.998505i \(-0.517409\pi\)
\(192\) 11.3811 + 7.90380i 0.821362 + 0.570408i
\(193\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −4.32624 13.3148i −0.309017 0.951057i
\(197\) 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
\(201\) 17.9301 13.6202i 1.26469 0.960697i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 9.58534 + 2.66858i 0.666227 + 0.185479i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(212\) −15.5957 + 21.4656i −1.07112 + 1.47427i
\(213\) 27.1216 9.45608i 1.85834 0.647920i
\(214\) 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
\(223\) 0.809017 + 0.587785i 0.0541758 + 0.0393610i 0.614544 0.788883i \(-0.289340\pi\)
−0.560368 + 0.828244i \(0.689340\pi\)
\(224\) 0 0
\(225\) 14.0981 11.1912i 0.939877 0.746078i
\(226\) 0 0
\(227\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(228\) 0 0
\(229\) 1.54508 + 4.75528i 0.102102 + 0.314238i 0.989039 0.147652i \(-0.0471715\pi\)
−0.886937 + 0.461890i \(0.847172\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(234\) 0 0
\(235\) −17.7984 + 12.9313i −1.16104 + 0.843543i
\(236\) −6.30860 2.04979i −0.410655 0.133430i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(240\) 0.489779 + 22.9730i 0.0316151 + 1.48290i
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 15.5000 1.65831i 0.994325 0.106381i
\(244\) 0 0
\(245\) 13.6462 18.7824i 0.871826 1.19997i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 15.7715 5.12447i 0.995488 0.323453i 0.234427 0.972134i \(-0.424679\pi\)
0.761061 + 0.648680i \(0.224679\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −12.9443 9.40456i −0.809017 0.587785i
\(257\) 15.5957 + 21.4656i 0.972833 + 1.33899i 0.940603 + 0.339510i \(0.110261\pi\)
0.0322308 + 0.999480i \(0.489739\pi\)
\(258\) 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
\(267\) 17.3743 + 22.8721i 1.06329 + 1.39975i
\(268\) −21.0344 + 15.2824i −1.28488 + 0.933522i
\(269\) 12.6172 + 4.09957i 0.769284 + 0.249955i 0.667258 0.744826i \(-0.267468\pi\)
0.102025 + 0.994782i \(0.467468\pi\)
\(270\) 0 0
\(271\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) −11.0000 3.31662i −0.662122 0.199637i
\(277\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(278\) 0 0
\(279\) 14.9864 0.639301i 0.897211 0.0382740i
\(280\) 0 0
\(281\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(282\) 0 0
\(283\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(284\) −31.5430 + 10.2489i −1.87173 + 0.608162i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 13.7533 + 9.99235i 0.809017 + 0.587785i
\(290\) 0 0
\(291\) −24.1849 16.7956i −1.41774 0.984574i
\(292\) 0 0
\(293\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(294\) 0 0
\(295\) −3.39919 10.4616i −0.197908 0.609099i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −16.5509 + 12.5725i −0.955566 + 0.725875i
\(301\) 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
\(311\) 19.4946 26.8321i 1.10544 1.52151i 0.277469 0.960735i \(-0.410505\pi\)
0.827970 0.560772i \(-0.189495\pi\)
\(312\) 0 0
\(313\) −5.87132 + 18.0701i −0.331867 + 1.02138i 0.636378 + 0.771377i \(0.280432\pi\)
−0.968245 + 0.250004i \(0.919568\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −22.0801 + 7.17425i −1.24014 + 0.402946i −0.854378 0.519652i \(-0.826062\pi\)
−0.385763 + 0.922598i \(0.626062\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 26.5330i 1.48324i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −17.9346 + 1.53293i −0.996367 + 0.0851627i
\(325\) 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
\(333\) −7.33468 19.6775i −0.401938 1.07832i
\(334\) 0 0
\(335\) −41.0059 13.3236i −2.24039 0.727947i
\(336\) 0 0
\(337\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(338\) 0 0
\(339\) 0.122445 + 5.74326i 0.00665028 + 0.311931i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −6.27245 17.9905i −0.337698 0.968573i
\(346\) 0 0
\(347\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(348\) 0 0
\(349\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(350\) 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
\(355\) −44.4959 32.3282i −2.36160 1.71580i
\(356\) −19.4946 26.8321i −1.03321 1.42210i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(360\) 0 0
\(361\) 5.87132 + 18.0701i 0.309017 + 0.951057i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 29.9336 21.7481i 1.56252 1.13524i 0.628620 0.777713i \(-0.283620\pi\)
0.933903 0.357526i \(-0.116380\pi\)
\(368\) 12.6172 + 4.09957i 0.657717 + 0.213705i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −17.3166 + 0.369185i −0.897822 + 0.0191413i
\(373\) 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
\(379\) −7.72542 + 23.7764i −0.396828 + 1.22131i 0.530700 + 0.847560i \(0.321929\pi\)
−0.927528 + 0.373753i \(0.878071\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −3.15430 + 1.02489i −0.161177 + 0.0523696i −0.388494 0.921451i \(-0.627005\pi\)
0.227317 + 0.973821i \(0.427005\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 27.5066 + 19.9847i 1.39643 + 1.01457i
\(389\) 21.4441 + 29.5153i 1.08726 + 1.49648i 0.851270 + 0.524727i \(0.175833\pi\)
0.235988 + 0.971756i \(0.424167\pi\)
\(390\) 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
\(400\) 19.4164 14.1068i 0.970820 0.705342i
\(401\) −25.2344 8.19915i −1.26014 0.409446i −0.398599 0.917125i \(-0.630503\pi\)
−0.861546 + 0.507679i \(0.830503\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −19.5380 22.5669i −0.970851 1.12136i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(410\) 0 0
\(411\) −37.9703 + 13.2385i −1.87294 + 0.653008i
\(412\) 2.47214 7.60845i 0.121793 0.374842i
\(413\) 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
\(421\) 8.09017 + 5.87785i 0.394291 + 0.286469i 0.767211 0.641394i \(-0.221644\pi\)
−0.372921 + 0.927863i \(0.621644\pi\)
\(422\) 0 0
\(423\) 12.3723 + 15.5861i 0.601562 + 0.757822i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(432\) 20.7421 1.32826i 0.997956 0.0639059i
\(433\) −23.4615 + 17.0458i −1.12749 + 0.819168i −0.985327 0.170676i \(-0.945405\pi\)
−0.142160 + 0.989844i \(0.545405\pi\)
\(434\) 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
\(443\) −21.4441 + 29.5153i −1.01884 + 1.40231i −0.105825 + 0.994385i \(0.533748\pi\)
−0.913014 + 0.407928i \(0.866252\pi\)
\(444\) 7.98312 + 22.8969i 0.378862 + 1.08664i
\(445\) 16.9959 52.3081i 0.805685 2.47964i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 15.7715 5.12447i 0.744303 0.241839i 0.0877747 0.996140i \(-0.472024\pi\)
0.656528 + 0.754302i \(0.272024\pi\)
\(450\) 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 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 6.79837 + 20.9232i 0.316976 + 0.975551i
\(461\) 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
\(465\) −17.3743 22.8721i −0.805714 1.06067i
\(466\) 0 0
\(467\) 41.0059 + 13.3236i 1.89753 + 0.616543i 0.970212 + 0.242257i \(0.0778878\pi\)
0.927313 + 0.374286i \(0.122112\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 39.8281 0.849124i 1.83518 0.0391256i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.69626 + 39.7633i 0.0776663 + 1.82064i
\(478\) 0 0
\(479\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 56.3826i 2.56020i
\(486\) 0 0
\(487\) 34.7877 + 25.2748i 1.57638 + 1.14531i 0.920699 + 0.390273i \(0.127619\pi\)
0.655683 + 0.755036i \(0.272381\pi\)
\(488\) 0 0
\(489\) 22.7622 + 15.8076i 1.02934 + 0.714844i
\(490\) 0 0
\(491\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 20.0000 0.898027
\(497\) 0 0
\(498\) 0 0
\(499\) 32.3607 23.5114i 1.44866 1.05252i 0.462522 0.886608i \(-0.346945\pi\)
0.986141 0.165907i \(-0.0530552\pi\)
\(500\) 6.30860 + 2.04979i 0.282129 + 0.0916693i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 6.50000 21.5581i 0.288675 0.957427i
\(508\) 0 0
\(509\) 1.94946 2.68321i 0.0864084 0.118931i −0.763624 0.645661i \(-0.776582\pi\)
0.850033 + 0.526730i \(0.176582\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 12.6172 4.09957i 0.555980 0.180649i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −25.3430 34.8817i −1.11030 1.52819i −0.820977 0.570962i \(-0.806570\pi\)
−0.289321 0.957232i \(-0.593430\pi\)
\(522\) 0 0
\(523\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 12.0000 0.521739
\(530\) 0 0
\(531\) −9.32325 + 3.47520i −0.404595 + 0.150811i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −0.612223 28.7163i −0.0264194 1.23920i
\(538\) 0 0
\(539\) 0 0
\(540\) 22.0000 + 26.5330i 0.946729 + 1.14180i
\(541\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(542\) 0 0
\(543\) −14.2556 40.8874i −0.611765 1.75465i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(548\) 44.1602 14.3485i 1.88643 0.612938i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −22.9372 + 33.0285i −0.973630 + 1.40198i
\(556\) 0 0
\(557\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(564\) −13.8994 18.2977i −0.585272 0.770472i
\(565\) 8.89919 6.46564i 0.374392 0.272011i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(570\) 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
\(575\) −11.6968 + 16.0992i −0.487789 + 0.671385i
\(576\) −23.9782 + 1.02288i −0.999091 + 0.0426201i
\(577\) 14.5238 44.6997i 0.604634 1.86087i 0.105344 0.994436i \(-0.466406\pi\)
0.499290 0.866435i \(-0.333594\pi\)
\(578\) 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
\(587\) 3.89893 + 5.36641i 0.160926 + 0.221496i 0.881864 0.471504i \(-0.156289\pi\)
−0.720938 + 0.693000i \(0.756289\pi\)
\(588\) 19.9170 + 13.8316i 0.821362 + 0.570408i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −8.65248 26.6296i −0.355615 1.09447i
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −27.5848 + 20.9542i −1.12897 + 0.857599i
\(598\) 0 0
\(599\) 31.5430 + 10.2489i 1.28881 + 0.418760i 0.871675 0.490084i \(-0.163034\pi\)
0.417136 + 0.908844i \(0.363034\pi\)
\(600\) 0 0
\(601\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(602\) 0 0
\(603\) −10.4599 + 37.5711i −0.425959 + 1.53002i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(614\) 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
\(619\) 0.809017 + 0.587785i 0.0325171 + 0.0236251i 0.603925 0.797041i \(-0.293603\pi\)
−0.571408 + 0.820666i \(0.693603\pi\)
\(620\) 19.4946 + 26.8321i 0.782923 + 1.07760i
\(621\) −16.0164 + 6.36205i −0.642715 + 0.255300i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −5.87132 18.0701i −0.234853 0.722803i
\(626\) 0 0
\(627\) 0 0
\(628\) −46.0000 −1.83560
\(629\) 0 0
\(630\) 0 0
\(631\) 5.66312 4.11450i 0.225445 0.163796i −0.469329 0.883023i \(-0.655504\pi\)
0.694774 + 0.719228i \(0.255504\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) −0.979557 45.9461i −0.0388420 1.82188i
\(637\) 0 0
\(638\) 0 0
\(639\) −27.5000 + 41.4578i −1.08788 + 1.64005i
\(640\) 0 0
\(641\) 13.6462 18.7824i 0.538994 0.741862i −0.449474 0.893294i \(-0.648388\pi\)
0.988468 + 0.151432i \(0.0483884\pi\)
\(642\) 0 0
\(643\) 12.6697 38.9933i 0.499644 1.53775i −0.309948 0.950753i \(-0.600312\pi\)
0.809592 0.586993i \(-0.199688\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −41.0059 + 13.3236i −1.61211 + 0.523805i −0.970061 0.242861i \(-0.921914\pi\)
−0.642046 + 0.766666i \(0.721914\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −25.8885 18.8091i −1.01387 0.736622i
\(653\) −1.94946 2.68321i −0.0762884 0.105002i 0.769167 0.639047i \(-0.220671\pi\)
−0.845456 + 0.534045i \(0.820671\pi\)
\(654\) 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
\(669\) −1.73166 + 0.0369185i −0.0669497 + 0.00142735i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(674\) 0 0
\(675\) −7.71963 + 30.2061i −0.297129 + 1.16263i
\(676\) −8.03444 + 24.7275i −0.309017 + 0.951057i
\(677\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(678\) 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
\(685\) 62.2943 + 45.2595i 2.38014 + 1.72928i
\(686\) 0 0
\(687\) −7.11320 4.93987i −0.271386 0.188468i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 5.25329 + 16.1680i 0.199845 + 0.615058i 0.999886 + 0.0151132i \(0.00481087\pi\)
−0.800041 + 0.599945i \(0.795189\pi\)
\(692\) 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 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 11.0000 36.4829i 0.414284 1.37402i
\(706\) 0 0
\(707\) 0 0
\(708\) 10.8487 3.78243i 0.407717 0.142152i
\(709\) −5.87132 + 18.0701i −0.220502 + 0.678636i 0.778215 + 0.627998i \(0.216125\pi\)
−0.998717 + 0.0506378i \(0.983875\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −15.7715 + 5.12447i −0.590647 + 0.191913i
\(714\) 0 0
\(715\) 0 0
\(716\) 33.1662i 1.23948i
\(717\) 0 0
\(718\) 0 0
\(719\) 9.74732 + 13.4160i 0.363514 + 0.500333i 0.951123 0.308811i \(-0.0999310\pi\)
−0.587610 + 0.809144i \(0.699931\pi\)
\(720\) −24.7446 31.1722i −0.922177 1.16172i
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 15.4508 + 47.5528i 0.574226 + 1.76729i
\(725\) 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
\(729\) −19.6408 + 18.5267i −0.727437 + 0.686175i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(734\) 0 0
\(735\) 0.857113 + 40.2028i 0.0316151 + 1.48290i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(740\) 27.2925 37.5649i 1.00329 1.38091i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −18.6074 13.5191i −0.678993 0.493318i 0.194030 0.980996i \(-0.437844\pi\)
−0.873024 + 0.487678i \(0.837844\pi\)
\(752\) 15.5957 + 21.4656i 0.568717 + 0.782772i
\(753\) −16.3837 + 23.5918i −0.597056 + 0.859734i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 11.7426 + 36.1401i 0.426794 + 1.31354i 0.901266 + 0.433266i \(0.142639\pi\)
−0.474473 + 0.880270i \(0.657361\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −44.1602 14.3485i −1.59766 0.519111i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 27.7065 0.590695i 0.999773 0.0213149i
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −44.0000 13.2665i −1.58462 0.477781i
\(772\) 0 0
\(773\) 7.79785 10.7328i 0.280469 0.386033i −0.645420 0.763828i \(-0.723318\pi\)
0.925889 + 0.377795i \(0.123318\pi\)
\(774\) 0 0
\(775\) −9.27051 + 28.5317i −0.333007 + 1.02489i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −22.6525 16.4580i −0.809017 0.587785i
\(785\) −44.8377 61.7137i −1.60032 2.20266i
\(786\) 0 0
\(787\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 60.6866 46.0993i 2.15233 1.63497i
\(796\) 32.3607 23.5114i 1.14699 0.833340i
\(797\) −53.6231 17.4232i −1.89943 0.617161i −0.966132 0.258048i \(-0.916921\pi\)
−0.933294 0.359113i \(-0.883079\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −47.9267 13.3429i −1.69341 0.471448i
\(802\) 0 0
\(803\) 0 0
\(804\) 13.0000 43.1161i 0.458475 1.52059i
\(805\) 0