Properties

Label 4032.2.b.p.3583.6
Level 4032
Weight 2
Character 4032.3583
Analytic conductor 32.196
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4032.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 224)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3583.6
Root \(0.923880 - 0.382683i\)
Character \(\chi\) = 4032.3583
Dual form 4032.2.b.p.3583.4

$q$-expansion

\(f(q)\) \(=\) \(q+1.08239i q^{5} +(1.08239 - 2.41421i) q^{7} +O(q^{10})\) \(q+1.08239i q^{5} +(1.08239 - 2.41421i) q^{7} -2.00000i q^{11} +4.14386i q^{13} +7.39104i q^{17} -4.77791 q^{19} -3.65685i q^{23} +3.82843 q^{25} -7.65685 q^{29} -7.39104 q^{31} +(2.61313 + 1.17157i) q^{35} -3.65685 q^{37} -8.28772i q^{41} +7.65685i q^{43} -3.06147 q^{47} +(-4.65685 - 5.22625i) q^{49} -2.00000 q^{53} +2.16478 q^{55} +5.67459 q^{59} +1.08239i q^{61} -4.48528 q^{65} +4.34315i q^{67} -3.17157i q^{71} -0.896683i q^{73} +(-4.82843 - 2.16478i) q^{77} -7.17157i q^{79} +1.71644 q^{83} -8.00000 q^{85} +5.22625i q^{89} +(10.0042 + 4.48528i) q^{91} -5.17157i q^{95} +11.7206i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{25} - 16q^{29} + 16q^{37} + 8q^{49} - 16q^{53} + 32q^{65} - 16q^{77} - 64q^{85} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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
\(5\) 1.08239i 0.484061i 0.970269 + 0.242030i \(0.0778133\pi\)
−0.970269 + 0.242030i \(0.922187\pi\)
\(6\) 0 0
\(7\) 1.08239 2.41421i 0.409106 0.912487i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000i 0.603023i −0.953463 0.301511i \(-0.902509\pi\)
0.953463 0.301511i \(-0.0974911\pi\)
\(12\) 0 0
\(13\) 4.14386i 1.14930i 0.818399 + 0.574650i \(0.194862\pi\)
−0.818399 + 0.574650i \(0.805138\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.39104i 1.79259i 0.443459 + 0.896295i \(0.353751\pi\)
−0.443459 + 0.896295i \(0.646249\pi\)
\(18\) 0 0
\(19\) −4.77791 −1.09613 −0.548064 0.836436i \(-0.684635\pi\)
−0.548064 + 0.836436i \(0.684635\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.65685i 0.762507i −0.924471 0.381253i \(-0.875493\pi\)
0.924471 0.381253i \(-0.124507\pi\)
\(24\) 0 0
\(25\) 3.82843 0.765685
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −7.65685 −1.42184 −0.710921 0.703272i \(-0.751722\pi\)
−0.710921 + 0.703272i \(0.751722\pi\)
\(30\) 0 0
\(31\) −7.39104 −1.32747 −0.663735 0.747968i \(-0.731030\pi\)
−0.663735 + 0.747968i \(0.731030\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.61313 + 1.17157i 0.441699 + 0.198032i
\(36\) 0 0
\(37\) −3.65685 −0.601183 −0.300592 0.953753i \(-0.597184\pi\)
−0.300592 + 0.953753i \(0.597184\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.28772i 1.29432i −0.762352 0.647162i \(-0.775956\pi\)
0.762352 0.647162i \(-0.224044\pi\)
\(42\) 0 0
\(43\) 7.65685i 1.16766i 0.811876 + 0.583830i \(0.198446\pi\)
−0.811876 + 0.583830i \(0.801554\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.06147 −0.446561 −0.223280 0.974754i \(-0.571677\pi\)
−0.223280 + 0.974754i \(0.571677\pi\)
\(48\) 0 0
\(49\) −4.65685 5.22625i −0.665265 0.746607i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 2.16478 0.291899
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.67459 0.738769 0.369385 0.929277i \(-0.379569\pi\)
0.369385 + 0.929277i \(0.379569\pi\)
\(60\) 0 0
\(61\) 1.08239i 0.138586i 0.997596 + 0.0692931i \(0.0220744\pi\)
−0.997596 + 0.0692931i \(0.977926\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.48528 −0.556331
\(66\) 0 0
\(67\) 4.34315i 0.530600i 0.964166 + 0.265300i \(0.0854709\pi\)
−0.964166 + 0.265300i \(0.914529\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.17157i 0.376396i −0.982131 0.188198i \(-0.939735\pi\)
0.982131 0.188198i \(-0.0602647\pi\)
\(72\) 0 0
\(73\) 0.896683i 0.104949i −0.998622 0.0524744i \(-0.983289\pi\)
0.998622 0.0524744i \(-0.0167108\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.82843 2.16478i −0.550250 0.246700i
\(78\) 0 0
\(79\) 7.17157i 0.806865i −0.915009 0.403432i \(-0.867817\pi\)
0.915009 0.403432i \(-0.132183\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.71644 0.188404 0.0942020 0.995553i \(-0.469970\pi\)
0.0942020 + 0.995553i \(0.469970\pi\)
\(84\) 0 0
\(85\) −8.00000 −0.867722
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.22625i 0.553982i 0.960873 + 0.276991i \(0.0893372\pi\)
−0.960873 + 0.276991i \(0.910663\pi\)
\(90\) 0 0
\(91\) 10.0042 + 4.48528i 1.04872 + 0.470185i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.17157i 0.530592i
\(96\) 0 0
\(97\) 11.7206i 1.19005i 0.803708 + 0.595024i \(0.202857\pi\)
−0.803708 + 0.595024i \(0.797143\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 8.47343i 0.843138i 0.906796 + 0.421569i \(0.138520\pi\)
−0.906796 + 0.421569i \(0.861480\pi\)
\(102\) 0 0
\(103\) −11.3492 −1.11827 −0.559134 0.829077i \(-0.688866\pi\)
−0.559134 + 0.829077i \(0.688866\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.6569i 1.12691i 0.826147 + 0.563455i \(0.190528\pi\)
−0.826147 + 0.563455i \(0.809472\pi\)
\(108\) 0 0
\(109\) −17.3137 −1.65835 −0.829176 0.558987i \(-0.811190\pi\)
−0.829176 + 0.558987i \(0.811190\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −3.17157 −0.298356 −0.149178 0.988810i \(-0.547663\pi\)
−0.149178 + 0.988810i \(0.547663\pi\)
\(114\) 0 0
\(115\) 3.95815 0.369099
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 17.8435 + 8.00000i 1.63571 + 0.733359i
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.55582i 0.854699i
\(126\) 0 0
\(127\) 7.65685i 0.679436i −0.940527 0.339718i \(-0.889668\pi\)
0.940527 0.339718i \(-0.110332\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.50981 0.306653 0.153327 0.988176i \(-0.451001\pi\)
0.153327 + 0.988176i \(0.451001\pi\)
\(132\) 0 0
\(133\) −5.17157 + 11.5349i −0.448432 + 1.00020i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) −10.0042 −0.848542 −0.424271 0.905535i \(-0.639470\pi\)
−0.424271 + 0.905535i \(0.639470\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 8.28772 0.693054
\(144\) 0 0
\(145\) 8.28772i 0.688258i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 19.6569i 1.59965i 0.600232 + 0.799826i \(0.295075\pi\)
−0.600232 + 0.799826i \(0.704925\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.00000i 0.642575i
\(156\) 0 0
\(157\) 8.47343i 0.676253i 0.941101 + 0.338127i \(0.109793\pi\)
−0.941101 + 0.338127i \(0.890207\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −8.82843 3.95815i −0.695778 0.311946i
\(162\) 0 0
\(163\) 5.31371i 0.416202i 0.978107 + 0.208101i \(0.0667282\pi\)
−0.978107 + 0.208101i \(0.933272\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −15.6788 −1.21326 −0.606629 0.794985i \(-0.707479\pi\)
−0.606629 + 0.794985i \(0.707479\pi\)
\(168\) 0 0
\(169\) −4.17157 −0.320890
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 10.6382i 0.808808i 0.914580 + 0.404404i \(0.132521\pi\)
−0.914580 + 0.404404i \(0.867479\pi\)
\(174\) 0 0
\(175\) 4.14386 9.24264i 0.313246 0.698678i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2.00000i 0.149487i −0.997203 0.0747435i \(-0.976186\pi\)
0.997203 0.0747435i \(-0.0238138\pi\)
\(180\) 0 0
\(181\) 11.5349i 0.857382i −0.903451 0.428691i \(-0.858975\pi\)
0.903451 0.428691i \(-0.141025\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.95815i 0.291009i
\(186\) 0 0
\(187\) 14.7821 1.08097
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0.828427i 0.0599429i −0.999551 0.0299714i \(-0.990458\pi\)
0.999551 0.0299714i \(-0.00954164\pi\)
\(192\) 0 0
\(193\) 6.48528 0.466821 0.233410 0.972378i \(-0.425011\pi\)
0.233410 + 0.972378i \(0.425011\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −7.65685 −0.545528 −0.272764 0.962081i \(-0.587938\pi\)
−0.272764 + 0.962081i \(0.587938\pi\)
\(198\) 0 0
\(199\) 0.896683 0.0635642 0.0317821 0.999495i \(-0.489882\pi\)
0.0317821 + 0.999495i \(0.489882\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −8.28772 + 18.4853i −0.581684 + 1.29741i
\(204\) 0 0
\(205\) 8.97056 0.626531
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 9.55582i 0.660990i
\(210\) 0 0
\(211\) 22.9706i 1.58136i −0.612230 0.790679i \(-0.709728\pi\)
0.612230 0.790679i \(-0.290272\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −8.28772 −0.565218
\(216\) 0 0
\(217\) −8.00000 + 17.8435i −0.543075 + 1.21130i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −30.6274 −2.06022
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −29.1158 −1.93248 −0.966242 0.257637i \(-0.917056\pi\)
−0.966242 + 0.257637i \(0.917056\pi\)
\(228\) 0 0
\(229\) 29.3784i 1.94138i −0.240332 0.970691i \(-0.577256\pi\)
0.240332 0.970691i \(-0.422744\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 3.31371i 0.216163i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 19.6569i 1.27150i −0.771897 0.635748i \(-0.780692\pi\)
0.771897 0.635748i \(-0.219308\pi\)
\(240\) 0 0
\(241\) 16.0502i 1.03388i −0.856021 0.516941i \(-0.827071\pi\)
0.856021 0.516941i \(-0.172929\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 5.65685 5.04054i 0.361403 0.322028i
\(246\) 0 0
\(247\) 19.7990i 1.25978i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 2.61313 0.164939 0.0824695 0.996594i \(-0.473719\pi\)
0.0824695 + 0.996594i \(0.473719\pi\)
\(252\) 0 0
\(253\) −7.31371 −0.459809
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 14.7821i 0.922080i 0.887379 + 0.461040i \(0.152524\pi\)
−0.887379 + 0.461040i \(0.847476\pi\)
\(258\) 0 0
\(259\) −3.95815 + 8.82843i −0.245948 + 0.548572i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 24.1421i 1.48867i 0.667808 + 0.744334i \(0.267233\pi\)
−0.667808 + 0.744334i \(0.732767\pi\)
\(264\) 0 0
\(265\) 2.16478i 0.132982i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 25.0489i 1.52726i −0.645656 0.763628i \(-0.723416\pi\)
0.645656 0.763628i \(-0.276584\pi\)
\(270\) 0 0
\(271\) 25.2346 1.53289 0.766446 0.642309i \(-0.222023\pi\)
0.766446 + 0.642309i \(0.222023\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 7.65685i 0.461726i
\(276\) 0 0
\(277\) −6.97056 −0.418821 −0.209410 0.977828i \(-0.567154\pi\)
−0.209410 + 0.977828i \(0.567154\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −16.6274 −0.991909 −0.495954 0.868349i \(-0.665182\pi\)
−0.495954 + 0.868349i \(0.665182\pi\)
\(282\) 0 0
\(283\) −2.61313 −0.155334 −0.0776671 0.996979i \(-0.524747\pi\)
−0.0776671 + 0.996979i \(0.524747\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −20.0083 8.97056i −1.18105 0.529516i
\(288\) 0 0
\(289\) −37.6274 −2.21338
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 8.47343i 0.495023i 0.968885 + 0.247511i \(0.0796128\pi\)
−0.968885 + 0.247511i \(0.920387\pi\)
\(294\) 0 0
\(295\) 6.14214i 0.357609i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 15.1535 0.876349
\(300\) 0 0
\(301\) 18.4853 + 8.28772i 1.06547 + 0.477696i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.17157 −0.0670841
\(306\) 0 0
\(307\) −1.71644 −0.0979626 −0.0489813 0.998800i \(-0.515597\pi\)
−0.0489813 + 0.998800i \(0.515597\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3.95815 −0.224446 −0.112223 0.993683i \(-0.535797\pi\)
−0.112223 + 0.993683i \(0.535797\pi\)
\(312\) 0 0
\(313\) 6.49435i 0.367083i 0.983012 + 0.183541i \(0.0587561\pi\)
−0.983012 + 0.183541i \(0.941244\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 16.3431 0.917923 0.458961 0.888456i \(-0.348222\pi\)
0.458961 + 0.888456i \(0.348222\pi\)
\(318\) 0 0
\(319\) 15.3137i 0.857403i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 35.3137i 1.96491i
\(324\) 0 0
\(325\) 15.8645i 0.880002i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3.31371 + 7.39104i −0.182691 + 0.407481i
\(330\) 0 0
\(331\) 5.31371i 0.292068i 0.989280 + 0.146034i \(0.0466509\pi\)
−0.989280 + 0.146034i \(0.953349\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.70099 −0.256842
\(336\) 0 0
\(337\) 3.17157 0.172767 0.0863833 0.996262i \(-0.472469\pi\)
0.0863833 + 0.996262i \(0.472469\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 14.7821i 0.800494i
\(342\) 0 0
\(343\) −17.6578 + 5.58579i −0.953433 + 0.301604i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 8.34315i 0.447884i 0.974603 + 0.223942i \(0.0718925\pi\)
−0.974603 + 0.223942i \(0.928107\pi\)
\(348\) 0 0
\(349\) 15.8645i 0.849205i 0.905380 + 0.424603i \(0.139586\pi\)
−0.905380 + 0.424603i \(0.860414\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 12.2459i 0.651782i 0.945407 + 0.325891i \(0.105664\pi\)
−0.945407 + 0.325891i \(0.894336\pi\)
\(354\) 0 0
\(355\) 3.43289 0.182199
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 18.9706i 1.00123i 0.865671 + 0.500614i \(0.166892\pi\)
−0.865671 + 0.500614i \(0.833108\pi\)
\(360\) 0 0
\(361\) 3.82843 0.201496
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0.970563 0.0508016
\(366\) 0 0
\(367\) −4.32957 −0.226002 −0.113001 0.993595i \(-0.536046\pi\)
−0.113001 + 0.993595i \(0.536046\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2.16478 + 4.82843i −0.112390 + 0.250679i
\(372\) 0 0
\(373\) 26.9706 1.39648 0.698241 0.715862i \(-0.253966\pi\)
0.698241 + 0.715862i \(0.253966\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 31.7289i 1.63412i
\(378\) 0 0
\(379\) 17.3137i 0.889345i −0.895693 0.444673i \(-0.853320\pi\)
0.895693 0.444673i \(-0.146680\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 13.5140 0.690532 0.345266 0.938505i \(-0.387789\pi\)
0.345266 + 0.938505i \(0.387789\pi\)
\(384\) 0 0
\(385\) 2.34315 5.22625i 0.119418 0.266354i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 8.34315 0.423014 0.211507 0.977376i \(-0.432163\pi\)
0.211507 + 0.977376i \(0.432163\pi\)
\(390\) 0 0
\(391\) 27.0279 1.36686
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 7.76245 0.390571
\(396\) 0 0
\(397\) 14.0711i 0.706208i 0.935584 + 0.353104i \(0.114874\pi\)
−0.935584 + 0.353104i \(0.885126\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 20.8284 1.04012 0.520061 0.854129i \(-0.325909\pi\)
0.520061 + 0.854129i \(0.325909\pi\)
\(402\) 0 0
\(403\) 30.6274i 1.52566i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 7.31371i 0.362527i
\(408\) 0 0
\(409\) 12.6173i 0.623885i 0.950101 + 0.311942i \(0.100980\pi\)
−0.950101 + 0.311942i \(0.899020\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 6.14214 13.6997i 0.302235 0.674117i
\(414\) 0 0
\(415\) 1.85786i 0.0911990i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −5.14933 −0.251561 −0.125781 0.992058i \(-0.540144\pi\)
−0.125781 + 0.992058i \(0.540144\pi\)
\(420\) 0 0
\(421\) −1.31371 −0.0640262 −0.0320131 0.999487i \(-0.510192\pi\)
−0.0320131 + 0.999487i \(0.510192\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 28.2960i 1.37256i
\(426\) 0 0
\(427\) 2.61313 + 1.17157i 0.126458 + 0.0566964i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 23.6569i 1.13951i 0.821814 + 0.569755i \(0.192962\pi\)
−0.821814 + 0.569755i \(0.807038\pi\)
\(432\) 0 0
\(433\) 28.2960i 1.35982i 0.733295 + 0.679911i \(0.237981\pi\)
−0.733295 + 0.679911i \(0.762019\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 17.4721i 0.835805i
\(438\) 0 0
\(439\) 39.6452 1.89216 0.946082 0.323928i \(-0.105003\pi\)
0.946082 + 0.323928i \(0.105003\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 10.0000i 0.475114i −0.971374 0.237557i \(-0.923653\pi\)
0.971374 0.237557i \(-0.0763467\pi\)
\(444\) 0 0
\(445\) −5.65685 −0.268161
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.31371 0.0619977 0.0309989 0.999519i \(-0.490131\pi\)
0.0309989 + 0.999519i \(0.490131\pi\)
\(450\) 0 0
\(451\) −16.5754 −0.780507
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −4.85483 + 10.8284i −0.227598 + 0.507644i
\(456\) 0 0
\(457\) −25.1127 −1.17472 −0.587361 0.809325i \(-0.699833\pi\)
−0.587361 + 0.809325i \(0.699833\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 32.4399i 1.51088i −0.655220 0.755438i \(-0.727424\pi\)
0.655220 0.755438i \(-0.272576\pi\)
\(462\) 0 0
\(463\) 12.1421i 0.564293i 0.959371 + 0.282146i \(0.0910464\pi\)
−0.959371 + 0.282146i \(0.908954\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −24.7862 −1.14697 −0.573485 0.819216i \(-0.694409\pi\)
−0.573485 + 0.819216i \(0.694409\pi\)
\(468\) 0 0
\(469\) 10.4853 + 4.70099i 0.484165 + 0.217071i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 15.3137 0.704125
\(474\) 0 0
\(475\) −18.2919 −0.839289
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −30.8322 −1.40876 −0.704381 0.709822i \(-0.748775\pi\)
−0.704381 + 0.709822i \(0.748775\pi\)
\(480\) 0 0
\(481\) 15.1535i 0.690940i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −12.6863 −0.576055
\(486\) 0 0
\(487\) 26.2843i 1.19105i 0.803335 + 0.595527i \(0.203057\pi\)
−0.803335 + 0.595527i \(0.796943\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 26.2843i 1.18619i 0.805132 + 0.593096i \(0.202095\pi\)
−0.805132 + 0.593096i \(0.797905\pi\)
\(492\) 0 0
\(493\) 56.5921i 2.54878i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −7.65685 3.43289i −0.343457 0.153986i
\(498\) 0 0
\(499\) 5.31371i 0.237874i 0.992902 + 0.118937i \(0.0379487\pi\)
−0.992902 + 0.118937i \(0.962051\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 31.7289 1.41472 0.707362 0.706852i \(-0.249885\pi\)
0.707362 + 0.706852i \(0.249885\pi\)
\(504\) 0 0
\(505\) −9.17157 −0.408130
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 25.9456i 1.15002i 0.818148 + 0.575008i \(0.195001\pi\)
−0.818148 + 0.575008i \(0.804999\pi\)
\(510\) 0 0
\(511\) −2.16478 0.970563i −0.0957644 0.0429352i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 12.2843i 0.541310i
\(516\) 0 0
\(517\) 6.12293i 0.269286i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 12.6173i 0.552773i 0.961047 + 0.276387i \(0.0891371\pi\)
−0.961047 + 0.276387i \(0.910863\pi\)
\(522\) 0 0
\(523\) −1.34502 −0.0588138 −0.0294069 0.999568i \(-0.509362\pi\)
−0.0294069 + 0.999568i \(0.509362\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 54.6274i 2.37961i
\(528\) 0 0
\(529\) 9.62742 0.418583
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 34.3431 1.48757
\(534\) 0 0
\(535\) −12.6173 −0.545493
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −10.4525 + 9.31371i −0.450221 + 0.401170i
\(540\) 0 0
\(541\) −2.68629 −0.115493 −0.0577463 0.998331i \(-0.518391\pi\)
−0.0577463 + 0.998331i \(0.518391\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 18.7402i 0.802743i
\(546\) 0 0
\(547\) 10.6863i 0.456913i −0.973554 0.228456i \(-0.926632\pi\)
0.973554 0.228456i \(-0.0733678\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 36.5838 1.55852
\(552\) 0 0
\(553\) −17.3137 7.76245i −0.736254 0.330093i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 17.3137 0.733605 0.366803 0.930299i \(-0.380452\pi\)
0.366803 + 0.930299i \(0.380452\pi\)
\(558\) 0 0
\(559\) −31.7289 −1.34199
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 25.6829 1.08241 0.541203 0.840892i \(-0.317969\pi\)
0.541203 + 0.840892i \(0.317969\pi\)
\(564\) 0 0
\(565\) 3.43289i 0.144423i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −36.1421 −1.51516 −0.757579 0.652744i \(-0.773618\pi\)
−0.757579 + 0.652744i \(0.773618\pi\)
\(570\) 0 0
\(571\) 19.6569i 0.822614i −0.911497 0.411307i \(-0.865072\pi\)
0.911497 0.411307i \(-0.134928\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 14.0000i 0.583840i
\(576\) 0 0
\(577\) 2.53620i 0.105583i 0.998606 + 0.0527917i \(0.0168120\pi\)
−0.998606 + 0.0527917i \(0.983188\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.85786 4.14386i 0.0770772 0.171916i
\(582\) 0 0
\(583\) 4.00000i 0.165663i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 33.4454 1.38044 0.690219 0.723600i \(-0.257514\pi\)
0.690219 + 0.723600i \(0.257514\pi\)
\(588\) 0 0
\(589\) 35.3137 1.45508
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 25.2346i 1.03626i −0.855302 0.518130i \(-0.826628\pi\)
0.855302 0.518130i \(-0.173372\pi\)
\(594\) 0 0
\(595\) −8.65914 + 19.3137i −0.354990 + 0.791785i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 36.8284i 1.50477i 0.658724 + 0.752384i \(0.271096\pi\)
−0.658724 + 0.752384i \(0.728904\pi\)
\(600\) 0 0
\(601\) 13.1426i 0.536096i 0.963406 + 0.268048i \(0.0863786\pi\)
−0.963406 + 0.268048i \(0.913621\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 7.57675i 0.308039i
\(606\) 0 0
\(607\) −41.8100 −1.69702 −0.848508 0.529182i \(-0.822499\pi\)
−0.848508 + 0.529182i \(0.822499\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 12.6863i 0.513232i
\(612\) 0 0
\(613\) −17.3137 −0.699294 −0.349647 0.936881i \(-0.613698\pi\)
−0.349647 + 0.936881i \(0.613698\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 49.1127 1.97720 0.988601 0.150557i \(-0.0481066\pi\)
0.988601 + 0.150557i \(0.0481066\pi\)
\(618\) 0 0
\(619\) 22.6215 0.909233 0.454616 0.890687i \(-0.349776\pi\)
0.454616 + 0.890687i \(0.349776\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 12.6173 + 5.65685i 0.505501 + 0.226637i
\(624\) 0 0
\(625\) 8.79899 0.351960
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 27.0279i 1.07767i
\(630\) 0 0
\(631\) 9.51472i 0.378775i −0.981902 0.189387i \(-0.939350\pi\)
0.981902 0.189387i \(-0.0606502\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 8.28772 0.328888
\(636\) 0 0
\(637\) 21.6569 19.2974i 0.858076 0.764589i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0.142136 0.00561402 0.00280701 0.999996i \(-0.499106\pi\)
0.00280701 + 0.999996i \(0.499106\pi\)
\(642\) 0 0
\(643\) 26.5796 1.04820 0.524099 0.851658i \(-0.324402\pi\)
0.524099 + 0.851658i \(0.324402\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 40.9133 1.60847 0.804235 0.594312i \(-0.202576\pi\)
0.804235 + 0.594312i \(0.202576\pi\)
\(648\) 0 0
\(649\) 11.3492i 0.445495i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 25.3137 0.990602 0.495301 0.868721i \(-0.335058\pi\)
0.495301 + 0.868721i \(0.335058\pi\)
\(654\) 0 0
\(655\) 3.79899i 0.148439i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 50.9706i 1.98553i −0.120068 0.992766i \(-0.538311\pi\)
0.120068 0.992766i \(-0.461689\pi\)
\(660\) 0 0
\(661\) 6.68006i 0.259824i −0.991526 0.129912i \(-0.958530\pi\)
0.991526 0.129912i \(-0.0414695\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −12.4853 5.59767i −0.484158 0.217068i
\(666\) 0 0
\(667\) 28.0000i 1.08416i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 2.16478 0.0835706
\(672\) 0 0
\(673\) 29.3137 1.12996 0.564980 0.825104i \(-0.308884\pi\)
0.564980 + 0.825104i \(0.308884\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 22.3588i 0.859319i −0.902991 0.429660i \(-0.858634\pi\)
0.902991 0.429660i \(-0.141366\pi\)
\(678\) 0 0
\(679\) 28.2960 + 12.6863i 1.08590 + 0.486855i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 32.3431i 1.23758i 0.785558 + 0.618788i \(0.212376\pi\)
−0.785558 + 0.618788i \(0.787624\pi\)
\(684\) 0 0
\(685\) 2.16478i 0.0827122i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 8.28772i 0.315737i
\(690\) 0 0
\(691\) −10.3756 −0.394706 −0.197353 0.980333i \(-0.563234\pi\)
−0.197353 + 0.980333i \(0.563234\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 10.8284i 0.410746i
\(696\) 0 0
\(697\) 61.2548 2.32019
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −13.3137 −0.502852 −0.251426 0.967877i \(-0.580899\pi\)
−0.251426 + 0.967877i \(0.580899\pi\)
\(702\) 0 0
\(703\) 17.4721 0.658974
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 20.4567 + 9.17157i 0.769352 + 0.344932i
\(708\) 0 0
\(709\) 18.0000 0.676004 0.338002 0.941145i \(-0.390249\pi\)
0.338002 + 0.941145i \(0.390249\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 27.0279i 1.01220i
\(714\) 0 0
\(715\) 8.97056i 0.335480i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −20.3797 −0.760036 −0.380018 0.924979i \(-0.624082\pi\)
−0.380018 + 0.924979i \(0.624082\pi\)
\(720\) 0 0
\(721\) −12.2843 + 27.3994i −0.457490 + 1.02041i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −29.3137 −1.08868
\(726\) 0 0
\(727\) −13.1426 −0.487430 −0.243715 0.969847i \(-0.578366\pi\)
−0.243715 + 0.969847i \(0.578366\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −56.5921 −2.09313
\(732\) 0 0
\(733\) 22.3588i 0.825842i −0.910767 0.412921i \(-0.864509\pi\)
0.910767 0.412921i \(-0.135491\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.68629 0.319964
\(738\) 0 0
\(739\) 28.6274i 1.05308i −0.850151 0.526538i \(-0.823490\pi\)
0.850151 0.526538i \(-0.176510\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 35.6569i 1.30812i −0.756441 0.654062i \(-0.773064\pi\)
0.756441 0.654062i \(-0.226936\pi\)
\(744\) 0 0
\(745\) 10.8239i 0.396558i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 28.1421 + 12.6173i 1.02829 + 0.461026i
\(750\) 0 0
\(751\) 18.9706i 0.692246i −0.938189 0.346123i \(-0.887498\pi\)
0.938189 0.346123i \(-0.112502\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −21.2764 −0.774328
\(756\) 0 0
\(757\) −26.6863 −0.969930 −0.484965 0.874534i \(-0.661168\pi\)
−0.484965 + 0.874534i \(0.661168\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 33.5223i 1.21518i −0.794250 0.607591i \(-0.792136\pi\)
0.794250 0.607591i \(-0.207864\pi\)
\(762\) 0 0
\(763\) −18.7402 + 41.7990i −0.678442 + 1.51323i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 23.5147i 0.849067i
\(768\) 0 0
\(769\) 11.7206i 0.422656i −0.977415 0.211328i \(-0.932221\pi\)
0.977415 0.211328i \(-0.0677788\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 13.3283i 0.479384i 0.970849 + 0.239692i \(0.0770465\pi\)
−0.970849 + 0.239692i \(0.922953\pi\)
\(774\) 0 0
\(775\) −28.2960 −1.01642
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 39.5980i 1.41874i
\(780\) 0 0
\(781\) −6.34315 −0.226976
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −9.17157 −0.327347
\(786\) 0 0
\(787\) −25.6829 −0.915497 −0.457749 0.889082i \(-0.651344\pi\)
−0.457749 + 0.889082i \(0.651344\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.43289 + 7.65685i −0.122059 + 0.272246i
\(792\) 0 0
\(793\) −4.48528 −0.159277
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 11.0096i 0.389981i −0.980805 0.194991i \(-0.937532\pi\)
0.980805 0.194991i \(-0.0624676\pi\)
\(798\) 0 0
\(799\) 22.6274i 0.800500i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1.79337 −0.0632865
\(804\) 0 0
\(805\) 4.28427 9.55582i 0.151001 0.336798i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 34.4853 1.21244 0.606219 0.795298i \(-0.292686\pi\)
0.606219 + 0.795298i