Properties

Label 69.2.c.a.68.6
Level $69$
Weight $2$
Character 69.68
Analytic conductor $0.551$
Analytic rank $0$
Dimension $6$
CM discriminant -23
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 69.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.550967773947\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.8869743.1
Defining polynomial: \(x^{6} - 3 x^{3} + 8\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 68.6
Root \(-0.261988 + 1.38973i\) of defining polynomial
Character \(\chi\) \(=\) 69.68
Dual form 69.2.c.a.68.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.77947i q^{2} +(1.60074 - 0.661546i) q^{3} -5.72545 q^{4} +(1.83875 + 4.44920i) q^{6} -10.3548i q^{8} +(2.12471 - 2.11792i) q^{9} +O(q^{10})\) \(q+2.77947i q^{2} +(1.60074 - 0.661546i) q^{3} -5.72545 q^{4} +(1.83875 + 4.44920i) q^{6} -10.3548i q^{8} +(2.12471 - 2.11792i) q^{9} +(-9.16493 + 3.78765i) q^{12} -2.15352 q^{13} +17.3299 q^{16} +(5.88670 + 5.90557i) q^{18} +4.79583i q^{23} +(-6.85016 - 16.5753i) q^{24} -5.00000 q^{25} -5.98564i q^{26} +(2.00000 - 4.79583i) q^{27} -1.58966i q^{29} -5.29738 q^{31} +27.4583i q^{32} +(-12.1649 + 12.1261i) q^{36} +(-3.44722 + 1.42465i) q^{39} -9.52822i q^{41} -13.3299 q^{46} +7.14860i q^{47} +(27.7405 - 11.4645i) q^{48} +7.00000 q^{49} -13.8973i q^{50} +12.3299 q^{52} +(13.3299 + 5.55894i) q^{54} +4.41841 q^{58} +9.59166i q^{59} -14.7239i q^{62} -41.6597 q^{64} +(3.17267 + 7.67686i) q^{69} +15.0872i q^{71} +(-21.9306 - 22.0009i) q^{72} +17.0553 q^{73} +(-8.00368 + 3.30773i) q^{75} +(-3.95978 - 9.58143i) q^{78} +(0.0288070 - 8.99995i) q^{81} +26.4834 q^{82} +(-1.05163 - 2.54463i) q^{87} -27.4583i q^{92} +(-8.47970 + 3.50446i) q^{93} -19.8693 q^{94} +(18.1649 + 43.9535i) q^{96} +19.4563i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 12q^{4} + 3q^{6} + O(q^{10}) \) \( 6q - 12q^{4} + 3q^{6} - 15q^{12} + 24q^{16} + 21q^{18} - 6q^{24} - 30q^{25} + 12q^{27} - 33q^{36} - 24q^{39} + 69q^{48} + 42q^{49} - 6q^{52} + 30q^{58} - 90q^{64} - 42q^{72} - 51q^{78} + 66q^{82} + 48q^{87} - 6q^{93} - 78q^{94} + 69q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(47\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.77947i 1.96538i 0.185254 + 0.982691i \(0.440689\pi\)
−0.185254 + 0.982691i \(0.559311\pi\)
\(3\) 1.60074 0.661546i 0.924185 0.381944i
\(4\) −5.72545 −2.86272
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 1.83875 + 4.44920i 0.750666 + 1.81638i
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 10.3548i 3.66096i
\(9\) 2.12471 2.11792i 0.708238 0.705974i
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −9.16493 + 3.78765i −2.64569 + 1.09340i
\(13\) −2.15352 −0.597279 −0.298639 0.954366i \(-0.596533\pi\)
−0.298639 + 0.954366i \(0.596533\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 17.3299 4.33247
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 5.88670 + 5.90557i 1.38751 + 1.39196i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.79583i 1.00000i
\(24\) −6.85016 16.5753i −1.39828 3.38341i
\(25\) −5.00000 −1.00000
\(26\) 5.98564i 1.17388i
\(27\) 2.00000 4.79583i 0.384900 0.922958i
\(28\) 0 0
\(29\) 1.58966i 0.295192i −0.989048 0.147596i \(-0.952846\pi\)
0.989048 0.147596i \(-0.0471536\pi\)
\(30\) 0 0
\(31\) −5.29738 −0.951437 −0.475719 0.879598i \(-0.657812\pi\)
−0.475719 + 0.879598i \(0.657812\pi\)
\(32\) 27.4583i 4.85399i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −12.1649 + 12.1261i −2.02749 + 2.02101i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) −3.44722 + 1.42465i −0.551996 + 0.228127i
\(40\) 0 0
\(41\) 9.52822i 1.48806i −0.668148 0.744029i \(-0.732913\pi\)
0.668148 0.744029i \(-0.267087\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −13.3299 −1.96538
\(47\) 7.14860i 1.04273i 0.853334 + 0.521365i \(0.174577\pi\)
−0.853334 + 0.521365i \(0.825423\pi\)
\(48\) 27.7405 11.4645i 4.00400 1.65476i
\(49\) 7.00000 1.00000
\(50\) 13.8973i 1.96538i
\(51\) 0 0
\(52\) 12.3299 1.70984
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 13.3299 + 5.55894i 1.81396 + 0.756476i
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 4.41841 0.580166
\(59\) 9.59166i 1.24873i 0.781133 + 0.624364i \(0.214642\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 14.7239i 1.86994i
\(63\) 0 0
\(64\) −41.6597 −5.20747
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 3.17267 + 7.67686i 0.381944 + 0.924185i
\(70\) 0 0
\(71\) 15.0872i 1.79052i 0.445548 + 0.895258i \(0.353009\pi\)
−0.445548 + 0.895258i \(0.646991\pi\)
\(72\) −21.9306 22.0009i −2.58455 2.59283i
\(73\) 17.0553 1.99617 0.998087 0.0618285i \(-0.0196932\pi\)
0.998087 + 0.0618285i \(0.0196932\pi\)
\(74\) 0 0
\(75\) −8.00368 + 3.30773i −0.924185 + 0.381944i
\(76\) 0 0
\(77\) 0 0
\(78\) −3.95978 9.58143i −0.448357 1.08488i
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0.0288070 8.99995i 0.00320078 0.999995i
\(82\) 26.4834 2.92460
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1.05163 2.54463i −0.112747 0.272812i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 27.4583i 2.86272i
\(93\) −8.47970 + 3.50446i −0.879304 + 0.363396i
\(94\) −19.8693 −2.04936
\(95\) 0 0
\(96\) 18.1649 + 43.9535i 1.85395 + 4.48598i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 19.4563i 1.96538i
\(99\) 0 0
\(100\) 28.6272 2.86272
\(101\) 19.1833i 1.90881i −0.298511 0.954406i \(-0.596490\pi\)
0.298511 0.954406i \(-0.403510\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 22.2992i 2.18662i
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −11.4509 + 27.4583i −1.10186 + 2.64217i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 9.10151i 0.845054i
\(117\) −4.57561 + 4.56099i −0.423015 + 0.421663i
\(118\) −26.6597 −2.45423
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) −6.30336 15.2522i −0.568355 1.37524i
\(124\) 30.3299 2.72370
\(125\) 0 0
\(126\) 0 0
\(127\) 13.9115 1.23444 0.617221 0.786790i \(-0.288258\pi\)
0.617221 + 0.786790i \(0.288258\pi\)
\(128\) 60.8754i 5.38067i
\(129\) 0 0
\(130\) 0 0
\(131\) 18.2665i 1.59595i −0.602691 0.797975i \(-0.705905\pi\)
0.602691 0.797975i \(-0.294095\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) −21.3376 + 8.81833i −1.81638 + 0.750666i
\(139\) 0.990339 0.0839994 0.0419997 0.999118i \(-0.486627\pi\)
0.0419997 + 0.999118i \(0.486627\pi\)
\(140\) 0 0
\(141\) 4.72913 + 11.4430i 0.398265 + 0.963676i
\(142\) −41.9343 −3.51905
\(143\) 0 0
\(144\) 36.8210 36.7033i 3.06842 3.05861i
\(145\) 0 0
\(146\) 47.4047i 3.92324i
\(147\) 11.2052 4.63083i 0.924185 0.381944i
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) −9.19374 22.2460i −0.750666 1.81638i
\(151\) −24.5062 −1.99429 −0.997144 0.0755288i \(-0.975936\pi\)
−0.997144 + 0.0755288i \(0.975936\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 19.7369 8.15678i 1.58021 0.653065i
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 25.0151 + 0.0800681i 1.96537 + 0.00629074i
\(163\) 20.1992 1.58212 0.791061 0.611738i \(-0.209529\pi\)
0.791061 + 0.611738i \(0.209529\pi\)
\(164\) 54.5533i 4.25990i
\(165\) 0 0
\(166\) 0 0
\(167\) 9.59166i 0.742225i 0.928588 + 0.371113i \(0.121024\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) 0 0
\(169\) −8.36235 −0.643258
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 19.1833i 1.45848i −0.684257 0.729241i \(-0.739873\pi\)
0.684257 0.729241i \(-0.260127\pi\)
\(174\) 7.07271 2.92298i 0.536181 0.221591i
\(175\) 0 0
\(176\) 0 0
\(177\) 6.34533 + 15.3537i 0.476944 + 1.15406i
\(178\) 0 0
\(179\) 26.2050i 1.95866i −0.202279 0.979328i \(-0.564835\pi\)
0.202279 0.979328i \(-0.435165\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 49.6597 3.66096
\(185\) 0 0
\(186\) −9.74054 23.5691i −0.714211 1.72817i
\(187\) 0 0
\(188\) 40.9289i 2.98505i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −66.6862 + 27.5598i −4.81266 + 1.98896i
\(193\) −8.44124 −0.607613 −0.303807 0.952734i \(-0.598258\pi\)
−0.303807 + 0.952734i \(0.598258\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −40.0781 −2.86272
\(197\) 23.8254i 1.69749i 0.528802 + 0.848745i \(0.322641\pi\)
−0.528802 + 0.848745i \(0.677359\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 51.7738i 3.66096i
\(201\) 0 0
\(202\) 53.3195 3.75154
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 10.1572 + 10.1898i 0.705974 + 0.708238i
\(208\) −37.3202 −2.58769
\(209\) 0 0
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) 9.98085 + 24.1506i 0.683877 + 1.65477i
\(214\) 0 0
\(215\) 0 0
\(216\) −49.6597 20.7095i −3.37892 1.40911i
\(217\) 0 0
\(218\) 0 0
\(219\) 27.3011 11.2829i 1.84483 0.762427i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) −10.6236 + 10.5896i −0.708238 + 0.705974i
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −16.4606 −1.08069
\(233\) 6.34890i 0.415930i 0.978136 + 0.207965i \(0.0666840\pi\)
−0.978136 + 0.207965i \(0.933316\pi\)
\(234\) −12.6771 12.7178i −0.828730 0.831386i
\(235\) 0 0
\(236\) 54.9166i 3.57476i
\(237\) 0 0
\(238\) 0 0
\(239\) 0.789960i 0.0510982i −0.999674 0.0255491i \(-0.991867\pi\)
0.999674 0.0255491i \(-0.00813342\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 30.5742i 1.96538i
\(243\) −5.90778 14.4256i −0.378984 0.925403i
\(244\) 0 0
\(245\) 0 0
\(246\) 42.3929 17.5200i 2.70287 1.11703i
\(247\) 0 0
\(248\) 54.8531i 3.48318i
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 38.6665i 2.42615i
\(255\) 0 0
\(256\) 85.8817 5.36761
\(257\) 31.7640i 1.98138i 0.136130 + 0.990691i \(0.456534\pi\)
−0.136130 + 0.990691i \(0.543466\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −3.36678 3.37757i −0.208398 0.209066i
\(262\) 50.7711 3.13665
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 17.4668i 1.06497i −0.846440 0.532484i \(-0.821259\pi\)
0.846440 0.532484i \(-0.178741\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) −18.1649 43.9535i −1.09340 2.64569i
\(277\) −27.6501 −1.66133 −0.830666 0.556771i \(-0.812040\pi\)
−0.830666 + 0.556771i \(0.812040\pi\)
\(278\) 2.75262i 0.165091i
\(279\) −11.2554 + 11.2194i −0.673843 + 0.671690i
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) −31.8055 + 13.1445i −1.89399 + 0.782742i
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 86.3807i 5.12575i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 58.1545 + 58.3410i 3.42679 + 3.43777i
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −97.6493 −5.71449
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 12.8712 + 31.1444i 0.750666 + 1.81638i
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 10.3279i 0.597279i
\(300\) 45.8247 18.9383i 2.64569 1.09340i
\(301\) 0 0
\(302\) 68.1143i 3.91954i
\(303\) −12.6907 30.7074i −0.729059 1.76410i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 23.0257i 1.30567i 0.757501 + 0.652834i \(0.226420\pi\)
−0.757501 + 0.652834i \(0.773580\pi\)
\(312\) 14.7520 + 35.6951i 0.835165 + 2.02084i
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 19.1833i 1.07744i −0.842484 0.538721i \(-0.818908\pi\)
0.842484 0.538721i \(-0.181092\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −0.164933 + 51.5288i −0.00916294 + 2.86271i
\(325\) 10.7676 0.597279
\(326\) 56.1430i 3.10947i
\(327\) 0 0
\(328\) −98.6625 −5.44772
\(329\) 0 0
\(330\) 0 0
\(331\) −18.2185 −1.00138 −0.500690 0.865627i \(-0.666920\pi\)
−0.500690 + 0.865627i \(0.666920\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) −26.6597 −1.45876
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 23.2429i 1.26425i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 53.3195 2.86647
\(347\) 9.59166i 0.514907i 0.966291 + 0.257454i \(0.0828835\pi\)
−0.966291 + 0.257454i \(0.917117\pi\)
\(348\) 6.02107 + 14.5691i 0.322763 + 0.780987i
\(349\) 36.2641 1.94118 0.970588 0.240748i \(-0.0773927\pi\)
0.970588 + 0.240748i \(0.0773927\pi\)
\(350\) 0 0
\(351\) −4.30704 + 10.3279i −0.229893 + 0.551263i
\(352\) 0 0
\(353\) 15.8869i 0.845572i 0.906230 + 0.422786i \(0.138948\pi\)
−0.906230 + 0.422786i \(0.861052\pi\)
\(354\) −42.6752 + 17.6367i −2.26816 + 0.937377i
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 72.8361 3.84951
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) −17.6081 + 7.27701i −0.924185 + 0.381944i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 83.1111i 4.33247i
\(369\) −20.1800 20.2447i −1.05053 1.05390i
\(370\) 0 0
\(371\) 0 0
\(372\) 48.5501 20.0646i 2.51721 1.04030i
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 74.0221 3.81740
\(377\) 3.42336i 0.176312i
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 22.2686 9.20307i 1.14085 0.471488i
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −40.2719 97.4454i −2.05512 4.97274i
\(385\) 0 0
\(386\) 23.4622i 1.19419i
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 72.4834i 3.66096i
\(393\) −12.0841 29.2398i −0.609563 1.47495i
\(394\) −66.2220 −3.33622
\(395\) 0 0
\(396\) 0 0
\(397\) 23.3430 1.17155 0.585777 0.810473i \(-0.300790\pi\)
0.585777 + 0.810473i \(0.300790\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −86.6493 −4.33247
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 11.4080 0.568273
\(404\) 109.833i 5.46440i
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 4.13420 0.204423 0.102211 0.994763i \(-0.467408\pi\)
0.102211 + 0.994763i \(0.467408\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) −28.3221 + 28.2316i −1.39196 + 1.38751i
\(415\) 0 0
\(416\) 59.1320i 2.89918i
\(417\) 1.58527 0.655155i 0.0776311 0.0320831i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 11.1179i 0.541210i
\(423\) 15.1402 + 15.1887i 0.736141 + 0.738501i
\(424\) 0 0
\(425\) 0 0
\(426\) −67.1257 + 27.7415i −3.25225 + 1.34408i
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 34.6597 83.1111i 1.66757 3.99869i
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 31.3604 + 75.8825i 1.49846 + 3.62580i
\(439\) 33.1203 1.58075 0.790373 0.612626i \(-0.209887\pi\)
0.790373 + 0.612626i \(0.209887\pi\)
\(440\) 0 0
\(441\) 14.8730 14.8255i 0.708238 0.705974i
\(442\) 0 0
\(443\) 34.1436i 1.62221i −0.584900 0.811105i \(-0.698866\pi\)
0.584900 0.811105i \(-0.301134\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 22.2358i 1.05289i
\(447\) 0 0
\(448\) 0 0
\(449\) 38.3667i 1.81063i 0.424736 + 0.905317i \(0.360367\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) −29.4335 29.5279i −1.38751 1.39196i
\(451\) 0 0
\(452\) 0 0
\(453\) −39.2280 + 16.2120i −1.84309 + 0.761706i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 42.8818i 1.99721i −0.0528331 0.998603i \(-0.516825\pi\)
0.0528331 0.998603i \(-0.483175\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 27.5486i 1.27891i
\(465\) 0 0
\(466\) −17.6466 −0.817461
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 26.1974 26.1137i 1.21098 1.20711i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 99.3195 4.57155
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 2.19567 0.100428
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 62.9799 2.86272
\(485\) 0 0
\(486\) 40.0955 16.4205i 1.81877 0.744848i
\(487\) −43.7150 −1.98092 −0.990459 0.137808i \(-0.955994\pi\)
−0.990459 + 0.137808i \(0.955994\pi\)
\(488\) 0 0
\(489\) 32.3335 13.3627i 1.46217 0.604282i
\(490\) 0 0
\(491\) 40.5022i 1.82784i 0.405894 + 0.913920i \(0.366960\pi\)
−0.405894 + 0.913920i \(0.633040\pi\)
\(492\) 36.0896 + 87.3255i 1.62704 + 3.93694i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −91.8028 −4.12207
\(497\) 0 0
\(498\) 0 0
\(499\) −11.5851 −0.518620 −0.259310 0.965794i \(-0.583495\pi\)
−0.259310 + 0.965794i \(0.583495\pi\)
\(500\) 0 0
\(501\) 6.34533 + 15.3537i 0.283488 + 0.685954i
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −13.3859 + 5.53209i −0.594490 + 0.245689i
\(508\) −79.6493 −3.53387
\(509\) 27.0047i 1.19696i −0.801136 0.598482i \(-0.795771\pi\)
0.801136 0.598482i \(-0.204229\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 116.955i 5.16872i
\(513\) 0 0
\(514\) −88.2870 −3.89417
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −12.6907 30.7074i −0.557058 1.34791i
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 9.38785 9.35785i 0.410895 0.409582i
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 104.584i 4.56876i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 20.3144 + 20.3795i 0.881570 + 0.884396i
\(532\) 0 0
\(533\) 20.5192i 0.888785i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −17.3358 41.9473i −0.748097 1.81016i
\(538\) 48.5484 2.09307
\(539\) 0 0
\(540\) 0 0
\(541\) −40.5712 −1.74429 −0.872146 0.489246i \(-0.837272\pi\)
−0.872146 + 0.489246i \(0.837272\pi\)
\(542\) 44.4715i 1.91021i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −30.7939 −1.31665 −0.658327 0.752732i \(-0.728735\pi\)
−0.658327 + 0.752732i \(0.728735\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 79.4921 32.8522i 3.38341 1.39828i
\(553\) 0 0
\(554\) 76.8525i 3.26515i
\(555\) 0 0
\(556\) −5.67013 −0.240467
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) −31.1841 31.2841i −1.32013 1.32436i
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) −27.0764 65.5164i −1.14012 2.75874i
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 156.224 6.55501
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 23.9792i 1.00000i
\(576\) −88.5150 + 88.2321i −3.68812 + 3.67634i
\(577\) −46.8589 −1.95076 −0.975381 0.220527i \(-0.929222\pi\)
−0.975381 + 0.220527i \(0.929222\pi\)
\(578\) 47.2510i 1.96538i
\(579\) −13.5122 + 5.58427i −0.561548 + 0.232074i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 176.604i 7.30792i
\(585\) 0 0
\(586\) 0 0
\(587\) 48.4408i 1.99937i 0.0251938 + 0.999683i \(0.491980\pi\)
−0.0251938 + 0.999683i \(0.508020\pi\)
\(588\) −64.1545 + 26.5136i −2.64569 + 1.09340i
\(589\) 0 0
\(590\) 0 0
\(591\) 15.7616 + 38.1382i 0.648346 + 1.56880i
\(592\) 0 0
\(593\) 38.3667i 1.57553i 0.615976 + 0.787765i \(0.288762\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 28.7061 1.17388
\(599\) 9.59166i 0.391905i 0.980613 + 0.195952i \(0.0627798\pi\)
−0.980613 + 0.195952i \(0.937220\pi\)
\(600\) 34.2508 + 82.8763i 1.39828 + 3.38341i
\(601\) 42.5519 1.73573 0.867863 0.496803i \(-0.165493\pi\)
0.867863 + 0.496803i \(0.165493\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 140.309 5.70909
\(605\) 0 0
\(606\) 85.3504 35.2733i 3.46712 1.43288i
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 15.3946i 0.622801i
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 55.5894i 2.24340i
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 23.0000 + 9.59166i 0.922958 + 0.384900i
\(622\) −63.9993 −2.56614
\(623\) 0 0
\(624\) −59.7398 + 24.6890i −2.39151 + 0.988353i
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) −6.40294 + 2.64619i −0.254494 + 0.105176i
\(634\) 53.3195 2.11759
\(635\) 0 0
\(636\) 0 0
\(637\) −15.0746 −0.597279
\(638\) 0 0
\(639\) 31.9534 + 32.0559i 1.26406 + 1.26811i
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 8.72852i 0.343153i −0.985171 0.171577i \(-0.945114\pi\)
0.985171 0.171577i \(-0.0548861\pi\)
\(648\) −93.1925 0.298290i −3.66095 0.0117179i
\(649\) 0 0
\(650\) 29.9282i 1.17388i
\(651\) 0 0
\(652\) −115.649 −4.52918
\(653\) 14.2875i 0.559111i 0.960129 + 0.279556i \(0.0901872\pi\)
−0.960129 + 0.279556i \(0.909813\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 165.123i 6.44696i
\(657\) 36.2376 36.1218i 1.41377 1.40925i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 50.6377i 1.96809i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 7.62374 0.295192
\(668\) 54.9166i 2.12479i
\(669\) 12.8059 5.29237i 0.495104 0.204615i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 29.9764 1.15551 0.577753 0.816211i \(-0.303930\pi\)
0.577753 + 0.816211i \(0.303930\pi\)
\(674\) 0 0
\(675\) −10.0000 + 23.9792i −0.384900 + 0.922958i
\(676\) 47.8782 1.84147
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 2.38936i 0.0914263i −0.998955 0.0457131i \(-0.985444\pi\)
0.998955 0.0457131i \(-0.0145560\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 44.0000 1.67384 0.836919 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) 109.833i 4.17523i
\(693\) 0 0
\(694\) −26.6597 −1.01199
\(695\) 0 0
\(696\) −26.3490 + 10.8894i −0.998757 + 0.412763i
\(697\) 0 0
\(698\) 100.795i 3.81515i
\(699\) 4.20009 + 10.1629i 0.158862 + 0.384397i
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) −28.7061 11.9713i −1.08344 0.451827i
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −44.1570 −1.66187
\(707\) 0 0
\(708\) −36.3299 87.9069i −1.36536 3.30375i
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 25.4053i 0.951437i
\(714\) 0 0
\(715\) 0 0
\(716\) 150.036i 5.60709i
\(717\) −0.522595 1.26452i −0.0195167 0.0472243i
\(718\) 0 0
\(719\) 47.9583i 1.78854i −0.447524 0.894272i \(-0.647694\pi\)
0.447524 0.894272i \(-0.352306\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 52.8099i 1.96538i
\(723\) 0 0
\(724\) 0 0
\(725\) 7.94830i 0.295192i
\(726\) −20.2262 48.9412i −0.750666 1.81638i
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) −19.0000 19.1833i −0.703704 0.710494i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −131.685 −4.85399
\(737\) 0 0
\(738\) 56.2696 56.0898i 2.07131 2.06469i
\(739\) −37.4273 −1.37679 −0.688393 0.725338i \(-0.741684\pi\)
−0.688393 + 0.725338i \(0.741684\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 36.2879 + 87.8054i 1.33038 + 3.21910i
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 123.884i 4.51759i
\(753\) 0 0
\(754\) −9.51513 −0.346521
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 50.8204i 1.84224i −0.389281 0.921119i \(-0.627276\pi\)
0.389281 0.921119i \(-0.372724\pi\)
\(762\) 25.5797 + 61.8948i 0.926653 + 2.24221i
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 20.6558i 0.745839i
\(768\) 137.474 56.8148i 4.96067 2.05013i
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 21.0133 + 50.8457i 0.756777 + 1.83116i
\(772\) 48.3299 1.73943
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 26.4869 0.951437
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −7.62374 3.17932i −0.272450 0.113620i
\(784\) 121.309 4.33247
\(785\) 0 0
\(786\) 81.2711 33.5874i 2.89885 1.19802i
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 136.411i 4.85945i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 64.8812i 2.30255i
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 137.291i 4.85399i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 31.7082i 1.11687i
\(807\)