Properties

Label 69.4.c.a.68.2
Level $69$
Weight $4$
Character 69.68
Analytic conductor $4.071$
Analytic rank $0$
Dimension $2$
CM discriminant -23
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 69.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.07113179040\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-23}) \)
Defining polynomial: \(x^{2} - x + 6\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 68.2
Root \(0.500000 - 2.39792i\) of defining polynomial
Character \(\chi\) \(=\) 69.68
Dual form 69.4.c.a.68.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.79583i q^{2} +(2.00000 + 4.79583i) q^{3} -15.0000 q^{4} +(-23.0000 + 9.59166i) q^{6} -33.5708i q^{8} +(-19.0000 + 19.1833i) q^{9} +O(q^{10})\) \(q+4.79583i q^{2} +(2.00000 + 4.79583i) q^{3} -15.0000 q^{4} +(-23.0000 + 9.59166i) q^{6} -33.5708i q^{8} +(-19.0000 + 19.1833i) q^{9} +(-30.0000 - 71.9375i) q^{12} +74.0000 q^{13} +41.0000 q^{16} +(-92.0000 - 91.1208i) q^{18} +110.304i q^{23} +(161.000 - 67.1416i) q^{24} -125.000 q^{25} +354.892i q^{26} +(-130.000 - 52.7541i) q^{27} +134.283i q^{29} +344.000 q^{31} -71.9375i q^{32} +(285.000 - 287.750i) q^{36} +(148.000 + 354.892i) q^{39} +306.933i q^{41} -529.000 q^{46} -642.641i q^{47} +(82.0000 + 196.629i) q^{48} +343.000 q^{49} -599.479i q^{50} -1110.00 q^{52} +(253.000 - 623.458i) q^{54} -644.000 q^{58} -815.291i q^{59} +1649.77i q^{62} +673.000 q^{64} +(-529.000 + 220.608i) q^{69} +220.608i q^{71} +(644.000 + 637.846i) q^{72} +1226.00 q^{73} +(-250.000 - 599.479i) q^{75} +(-1702.00 + 709.783i) q^{78} +(-7.00000 - 728.966i) q^{81} -1472.00 q^{82} +(-644.000 + 268.567i) q^{87} -1654.56i q^{92} +(688.000 + 1649.77i) q^{93} +3082.00 q^{94} +(345.000 - 143.875i) q^{96} +1644.97i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{3} - 30q^{4} - 46q^{6} - 38q^{9} + O(q^{10}) \) \( 2q + 4q^{3} - 30q^{4} - 46q^{6} - 38q^{9} - 60q^{12} + 148q^{13} + 82q^{16} - 184q^{18} + 322q^{24} - 250q^{25} - 260q^{27} + 688q^{31} + 570q^{36} + 296q^{39} - 1058q^{46} + 164q^{48} + 686q^{49} - 2220q^{52} + 506q^{54} - 1288q^{58} + 1346q^{64} - 1058q^{69} + 1288q^{72} + 2452q^{73} - 500q^{75} - 3404q^{78} - 14q^{81} - 2944q^{82} - 1288q^{87} + 1376q^{93} + 6164q^{94} + 690q^{96} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.79583i 1.69558i 0.530330 + 0.847791i \(0.322068\pi\)
−0.530330 + 0.847791i \(0.677932\pi\)
\(3\) 2.00000 + 4.79583i 0.384900 + 0.922958i
\(4\) −15.0000 −1.87500
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −23.0000 + 9.59166i −1.56495 + 0.652630i
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 33.5708i 1.48363i
\(9\) −19.0000 + 19.1833i −0.703704 + 0.710494i
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −30.0000 71.9375i −0.721688 1.73055i
\(13\) 74.0000 1.57876 0.789381 0.613904i \(-0.210402\pi\)
0.789381 + 0.613904i \(0.210402\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 41.0000 0.640625
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −92.0000 91.1208i −1.20470 1.19319i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 110.304i 1.00000i
\(24\) 161.000 67.1416i 1.36933 0.571051i
\(25\) −125.000 −1.00000
\(26\) 354.892i 2.67692i
\(27\) −130.000 52.7541i −0.926612 0.376020i
\(28\) 0 0
\(29\) 134.283i 0.859854i 0.902864 + 0.429927i \(0.141461\pi\)
−0.902864 + 0.429927i \(0.858539\pi\)
\(30\) 0 0
\(31\) 344.000 1.99304 0.996520 0.0833571i \(-0.0265642\pi\)
0.996520 + 0.0833571i \(0.0265642\pi\)
\(32\) 71.9375i 0.397402i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 285.000 287.750i 1.31944 1.33218i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 148.000 + 354.892i 0.607666 + 1.45713i
\(40\) 0 0
\(41\) 306.933i 1.16914i 0.811342 + 0.584572i \(0.198738\pi\)
−0.811342 + 0.584572i \(0.801262\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −529.000 −1.69558
\(47\) 642.641i 1.99444i −0.0744843 0.997222i \(-0.523731\pi\)
0.0744843 0.997222i \(-0.476269\pi\)
\(48\) 82.0000 + 196.629i 0.246577 + 0.591270i
\(49\) 343.000 1.00000
\(50\) 599.479i 1.69558i
\(51\) 0 0
\(52\) −1110.00 −2.96018
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 253.000 623.458i 0.637573 1.57115i
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −644.000 −1.45795
\(59\) 815.291i 1.79902i −0.436905 0.899508i \(-0.643925\pi\)
0.436905 0.899508i \(-0.356075\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 1649.77i 3.37936i
\(63\) 0 0
\(64\) 673.000 1.31445
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) −529.000 + 220.608i −0.922958 + 0.384900i
\(70\) 0 0
\(71\) 220.608i 0.368752i 0.982856 + 0.184376i \(0.0590264\pi\)
−0.982856 + 0.184376i \(0.940974\pi\)
\(72\) 644.000 + 637.846i 1.05411 + 1.04404i
\(73\) 1226.00 1.96565 0.982825 0.184540i \(-0.0590796\pi\)
0.982825 + 0.184540i \(0.0590796\pi\)
\(74\) 0 0
\(75\) −250.000 599.479i −0.384900 0.922958i
\(76\) 0 0
\(77\) 0 0
\(78\) −1702.00 + 709.783i −2.47069 + 1.03035i
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −7.00000 728.966i −0.00960219 0.999954i
\(82\) −1472.00 −1.98238
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −644.000 + 268.567i −0.793610 + 0.330958i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1654.56i 1.87500i
\(93\) 688.000 + 1649.77i 0.767121 + 1.83949i
\(94\) 3082.00 3.38174
\(95\) 0 0
\(96\) 345.000 143.875i 0.366786 0.152960i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 1644.97i 1.69558i
\(99\) 0 0
\(100\) 1875.00 1.87500
\(101\) 1246.92i 1.22844i −0.789133 0.614222i \(-0.789470\pi\)
0.789133 0.614222i \(-0.210530\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 2484.24i 2.34231i
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 1950.00 + 791.312i 1.73740 + 0.705038i
\(109\) 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
\(116\) 2014.25i 1.61223i
\(117\) −1406.00 + 1419.57i −1.11098 + 1.12170i
\(118\) 3910.00 3.05038
\(119\) 0 0
\(120\) 0 0
\(121\) −1331.00 −1.00000
\(122\) 0 0
\(123\) −1472.00 + 613.866i −1.07907 + 0.450004i
\(124\) −5160.00 −3.73695
\(125\) 0 0
\(126\) 0 0
\(127\) −2608.00 −1.82223 −0.911113 0.412158i \(-0.864775\pi\)
−0.911113 + 0.412158i \(0.864775\pi\)
\(128\) 2652.09i 1.83136i
\(129\) 0 0
\(130\) 0 0
\(131\) 1083.86i 0.722879i 0.932396 + 0.361439i \(0.117715\pi\)
−0.932396 + 0.361439i \(0.882285\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) −1058.00 2536.99i −0.652630 1.56495i
\(139\) −412.000 −0.251406 −0.125703 0.992068i \(-0.540119\pi\)
−0.125703 + 0.992068i \(0.540119\pi\)
\(140\) 0 0
\(141\) 3082.00 1285.28i 1.84079 0.767662i
\(142\) −1058.00 −0.625249
\(143\) 0 0
\(144\) −779.000 + 786.516i −0.450810 + 0.455160i
\(145\) 0 0
\(146\) 5879.69i 3.33292i
\(147\) 686.000 + 1644.97i 0.384900 + 0.922958i
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 2875.00 1198.96i 1.56495 0.652630i
\(151\) −3616.00 −1.94878 −0.974390 0.224863i \(-0.927807\pi\)
−0.974390 + 0.224863i \(0.927807\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −2220.00 5323.37i −1.13937 2.73212i
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 3496.00 33.5708i 1.69550 0.0162813i
\(163\) −1636.00 −0.786144 −0.393072 0.919508i \(-0.628588\pi\)
−0.393072 + 0.919508i \(0.628588\pi\)
\(164\) 4604.00i 2.19215i
\(165\) 0 0
\(166\) 0 0
\(167\) 3922.99i 1.81778i −0.417030 0.908892i \(-0.636929\pi\)
0.417030 0.908892i \(-0.363071\pi\)
\(168\) 0 0
\(169\) 3279.00 1.49249
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2896.68i 1.27301i 0.771273 + 0.636505i \(0.219620\pi\)
−0.771273 + 0.636505i \(0.780380\pi\)
\(174\) −1288.00 3088.52i −0.561167 1.34563i
\(175\) 0 0
\(176\) 0 0
\(177\) 3910.00 1630.58i 1.66042 0.692441i
\(178\) 0 0
\(179\) 3922.99i 1.63809i −0.573730 0.819045i \(-0.694504\pi\)
0.573730 0.819045i \(-0.305496\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 3703.00 1.48363
\(185\) 0 0
\(186\) −7912.00 + 3299.53i −3.11901 + 1.30072i
\(187\) 0 0
\(188\) 9639.62i 3.73958i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1346.00 + 3227.59i 0.505933 + 1.21319i
\(193\) 4286.00 1.59851 0.799257 0.600990i \(-0.205227\pi\)
0.799257 + 0.600990i \(0.205227\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −5145.00 −1.87500
\(197\) 556.316i 0.201197i −0.994927 0.100599i \(-0.967924\pi\)
0.994927 0.100599i \(-0.0320758\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 4196.35i 1.48363i
\(201\) 0 0
\(202\) 5980.00 2.08293
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −2116.00 2095.78i −0.710494 0.703704i
\(208\) 3034.00 1.01139
\(209\) 0 0
\(210\) 0 0
\(211\) 2468.00 0.805233 0.402616 0.915369i \(-0.368101\pi\)
0.402616 + 0.915369i \(0.368101\pi\)
\(212\) 0 0
\(213\) −1058.00 + 441.217i −0.340343 + 0.141933i
\(214\) 0 0
\(215\) 0 0
\(216\) −1771.00 + 4364.21i −0.557876 + 1.37475i
\(217\) 0 0
\(218\) 0 0
\(219\) 2452.00 + 5879.69i 0.756579 + 1.81421i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −4840.00 −1.45341 −0.726705 0.686950i \(-0.758949\pi\)
−0.726705 + 0.686950i \(0.758949\pi\)
\(224\) 0 0
\(225\) 2375.00 2397.92i 0.703704 0.710494i
\(226\) 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
\(232\) 4508.00 1.27571
\(233\) 4181.97i 1.17584i −0.808921 0.587918i \(-0.799948\pi\)
0.808921 0.587918i \(-0.200052\pi\)
\(234\) −6808.00 6742.94i −1.90194 1.88376i
\(235\) 0 0
\(236\) 12229.4i 3.37315i
\(237\) 0 0
\(238\) 0 0
\(239\) 565.908i 0.153161i 0.997063 + 0.0765807i \(0.0244003\pi\)
−0.997063 + 0.0765807i \(0.975600\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 6383.25i 1.69558i
\(243\) 3482.00 1491.50i 0.919220 0.393745i
\(244\) 0 0
\(245\) 0 0
\(246\) −2944.00 7059.46i −0.763019 1.82965i
\(247\) 0 0
\(248\) 11548.4i 2.95694i
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 12507.5i 3.08973i
\(255\) 0 0
\(256\) −7335.00 −1.79077
\(257\) 7558.23i 1.83451i 0.398299 + 0.917256i \(0.369601\pi\)
−0.398299 + 0.917256i \(0.630399\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −2576.00 2551.38i −0.610921 0.605083i
\(262\) −5198.00 −1.22570
\(263\) 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
\(269\) 8766.78i 1.98706i 0.113556 + 0.993532i \(0.463776\pi\)
−0.113556 + 0.993532i \(0.536224\pi\)
\(270\) 0 0
\(271\) 8912.00 1.99766 0.998829 0.0483752i \(-0.0154043\pi\)
0.998829 + 0.0483752i \(0.0154043\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 7935.00 3309.12i 1.73055 0.721688i
\(277\) 1838.00 0.398681 0.199341 0.979930i \(-0.436120\pi\)
0.199341 + 0.979930i \(0.436120\pi\)
\(278\) 1975.88i 0.426279i
\(279\) −6536.00 + 6599.06i −1.40251 + 1.41604i
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 6164.00 + 14780.8i 1.30163 + 3.12121i
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 3309.12i 0.691410i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 1380.00 + 1366.81i 0.282352 + 0.279653i
\(289\) −4913.00 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −18390.0 −3.68559
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) −7889.00 + 3289.94i −1.56495 + 0.652630i
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8162.51i 1.57876i
\(300\) 3750.00 + 8992.18i 0.721688 + 1.73055i
\(301\) 0 0
\(302\) 17341.7i 3.30432i
\(303\) 5980.00 2493.83i 1.13380 0.472828i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −10420.0 −1.93714 −0.968568 0.248749i \(-0.919981\pi\)
−0.968568 + 0.248749i \(0.919981\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9275.14i 1.69114i −0.533864 0.845570i \(-0.679261\pi\)
0.533864 0.845570i \(-0.320739\pi\)
\(312\) 11914.0 4968.48i 2.16185 0.901554i
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 11183.9i 1.98154i 0.135542 + 0.990772i \(0.456723\pi\)
−0.135542 + 0.990772i \(0.543277\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 105.000 + 10934.5i 0.0180041 + 1.87491i
\(325\) −9250.00 −1.57876
\(326\) 7845.98i 1.33297i
\(327\) 0 0
\(328\) 10304.0 1.73458
\(329\) 0 0
\(330\) 0 0
\(331\) 12044.0 1.99999 0.999997 0.00238915i \(-0.000760490\pi\)
0.999997 + 0.00238915i \(0.000760490\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 18814.0 3.08220
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 15725.5i 2.53064i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −13892.0 −2.15849
\(347\) 9102.49i 1.40821i −0.710098 0.704103i \(-0.751350\pi\)
0.710098 0.704103i \(-0.248650\pi\)
\(348\) 9660.00 4028.50i 1.48802 0.620547i
\(349\) 9722.00 1.49114 0.745568 0.666429i \(-0.232178\pi\)
0.745568 + 0.666429i \(0.232178\pi\)
\(350\) 0 0
\(351\) −9620.00 3903.81i −1.46290 0.593646i
\(352\) 0 0
\(353\) 12814.5i 1.93214i −0.258283 0.966069i \(-0.583157\pi\)
0.258283 0.966069i \(-0.416843\pi\)
\(354\) 7820.00 + 18751.7i 1.17409 + 2.81537i
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 18814.0 2.77752
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 6859.00 1.00000
\(362\) 0 0
\(363\) −2662.00 6383.25i −0.384900 0.922958i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 4522.47i 0.640625i
\(369\) −5888.00 5831.73i −0.830669 0.822731i
\(370\) 0 0
\(371\) 0 0
\(372\) −10320.0 24746.5i −1.43835 3.44905i
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −21574.0 −2.95903
\(377\) 9936.96i 1.35751i
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −5216.00 12507.5i −0.701375 1.68184i
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −12719.0 + 5304.19i −1.69027 + 0.704891i
\(385\) 0 0
\(386\) 20554.9i 2.71041i
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 11514.8i 1.48363i
\(393\) −5198.00 + 2167.72i −0.667187 + 0.278236i
\(394\) 2668.00 0.341147
\(395\) 0 0
\(396\) 0 0
\(397\) −15082.0 −1.90666 −0.953330 0.301931i \(-0.902369\pi\)
−0.953330 + 0.301931i \(0.902369\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −5125.00 −0.640625
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 25456.0 3.14654
\(404\) 18703.7i 2.30333i
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −5002.00 −0.604726 −0.302363 0.953193i \(-0.597775\pi\)
−0.302363 + 0.953193i \(0.597775\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 10051.0 10148.0i 1.19319 1.20470i
\(415\) 0 0
\(416\) 5323.37i 0.627403i
\(417\) −824.000 1975.88i −0.0967661 0.232037i
\(418\) 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 \(\pi\)
\(422\) 11836.1i 1.36534i
\(423\) 12328.0 + 12210.2i 1.41704 + 1.40350i
\(424\) 0 0
\(425\) 0 0
\(426\) −2116.00 5073.99i −0.240659 0.577079i
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −5330.00 2162.92i −0.593611 0.240888i
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) −28198.0 + 11759.4i −3.07615 + 1.28284i
\(439\) −7288.00 −0.792340 −0.396170 0.918177i \(-0.629661\pi\)
−0.396170 + 0.918177i \(0.629661\pi\)
\(440\) 0 0
\(441\) −6517.00 + 6579.88i −0.703704 + 0.710494i
\(442\) 0 0
\(443\) 5572.76i 0.597674i 0.954304 + 0.298837i \(0.0965987\pi\)
−0.954304 + 0.298837i \(0.903401\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 23211.8i 2.46438i
\(447\) 0 0
\(448\) 0 0
\(449\) 4795.83i 0.504074i 0.967718 + 0.252037i \(0.0811005\pi\)
−0.967718 + 0.252037i \(0.918900\pi\)
\(450\) 11500.0 + 11390.1i 1.20470 + 1.19319i
\(451\) 0 0
\(452\) 0 0
\(453\) −7232.00 17341.7i −0.750086 1.79864i
\(454\) 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
\(461\) 19547.8i 1.97491i −0.157909 0.987454i \(-0.550475\pi\)
0.157909 0.987454i \(-0.449525\pi\)
\(462\) 0 0
\(463\) −11680.0 −1.17239 −0.586194 0.810171i \(-0.699374\pi\)
−0.586194 + 0.810171i \(0.699374\pi\)
\(464\) 5505.61i 0.550844i
\(465\) 0 0
\(466\) 20056.0 1.99373
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 21090.0 21293.5i 2.08309 2.10319i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −27370.0 −2.66908
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −2714.00 −0.259698
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 19965.0 1.87500
\(485\) 0 0
\(486\) 7153.00 + 16699.1i 0.667627 + 1.55861i
\(487\) −19672.0 −1.83044 −0.915219 0.402957i \(-0.867983\pi\)
−0.915219 + 0.402957i \(0.867983\pi\)
\(488\) 0 0
\(489\) −3272.00 7845.98i −0.302587 0.725578i
\(490\) 0 0
\(491\) 6781.31i 0.623291i 0.950198 + 0.311646i \(0.100880\pi\)
−0.950198 + 0.311646i \(0.899120\pi\)
\(492\) 22080.0 9208.00i 2.02326 0.843757i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 14104.0 1.27679
\(497\) 0 0
\(498\) 0 0
\(499\) 15788.0 1.41637 0.708184 0.706028i \(-0.249515\pi\)
0.708184 + 0.706028i \(0.249515\pi\)
\(500\) 0 0
\(501\) 18814.0 7845.98i 1.67774 0.699666i
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 6558.00 + 15725.5i 0.574460 + 1.37751i
\(508\) 39120.0 3.41667
\(509\) 21542.9i 1.87597i 0.346669 + 0.937987i \(0.387313\pi\)
−0.346669 + 0.937987i \(0.612687\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 13960.7i 1.20504i
\(513\) 0 0
\(514\) −36248.0 −3.11057
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −13892.0 + 5793.36i −1.17493 + 0.489982i
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 12236.0 12354.1i 1.02597 1.03587i
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 16257.9i 1.35540i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −12167.0 −1.00000
\(530\) 0 0
\(531\) 15640.0 + 15490.5i 1.27819 + 1.26597i
\(532\) 0 0
\(533\) 22713.1i 1.84580i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 18814.0 7845.98i 1.51189 0.630501i
\(538\) −42044.0 −3.36923
\(539\) 0 0
\(540\) 0 0
\(541\) −934.000 −0.0742251 −0.0371126 0.999311i \(-0.511816\pi\)
−0.0371126 + 0.999311i \(0.511816\pi\)
\(542\) 42740.5i 3.38719i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 21332.0 1.66744 0.833721 0.552186i \(-0.186206\pi\)
0.833721 + 0.552186i \(0.186206\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 7406.00 + 17759.0i 0.571051 + 1.36933i
\(553\) 0 0
\(554\) 8814.74i 0.675997i
\(555\) 0 0
\(556\) 6180.00 0.471386
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) −31648.0 31345.6i −2.40102 2.37807i
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) −46230.0 + 19279.2i −3.45148 + 1.43937i
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 7406.00 0.547093
\(569\) 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
\(575\) 13788.0i 1.00000i
\(576\) −12787.0 + 12910.4i −0.924986 + 0.933910i
\(577\) −21778.0 −1.57128 −0.785641 0.618682i \(-0.787667\pi\)
−0.785641 + 0.618682i \(0.787667\pi\)
\(578\) 23561.9i 1.69558i
\(579\) 8572.00 + 20554.9i 0.615268 + 1.47536i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 41157.8i 2.91631i
\(585\) 0 0
\(586\) 0 0
\(587\) 28362.5i 1.99429i 0.0755174 + 0.997144i \(0.475939\pi\)
−0.0755174 + 0.997144i \(0.524061\pi\)
\(588\) −10290.0 24674.6i −0.721688 1.73055i
\(589\) 0 0
\(590\) 0 0
\(591\) 2668.00 1112.63i 0.185697 0.0774410i
\(592\) 0 0
\(593\) 11778.6i 0.815662i −0.913057 0.407831i \(-0.866285\pi\)
0.913057 0.407831i \(-0.133715\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) −39146.0 −2.67692
\(599\) 16353.8i 1.11552i −0.830002 0.557761i \(-0.811661\pi\)
0.830002 0.557761i \(-0.188339\pi\)
\(600\) −20125.0 + 8392.71i −1.36933 + 0.571051i
\(601\) 326.000 0.0221262 0.0110631 0.999939i \(-0.496478\pi\)
0.0110631 + 0.999939i \(0.496478\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 54240.0 3.65396
\(605\) 0 0
\(606\) 11960.0 + 28679.1i 0.801719 + 1.92245i
\(607\) 8840.00 0.591111 0.295556 0.955326i \(-0.404495\pi\)
0.295556 + 0.955326i \(0.404495\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 47555.5i 3.14875i
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 49972.6i 3.28457i
\(615\) 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
\(621\) 5819.00 14339.5i 0.376020 0.926612i
\(622\) 44482.0 2.86747
\(623\) 0 0
\(624\) 6068.00 + 14550.6i 0.389286 + 0.933475i
\(625\) 15625.0 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
\(633\) 4936.00 + 11836.1i 0.309934 + 0.743196i
\(634\) −53636.0 −3.35987
\(635\) 0 0
\(636\) 0 0
\(637\) 25382.0 1.57876
\(638\) 0 0
\(639\) −4232.00 4191.56i −0.261996 0.259492i
\(640\) 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
\(647\) 16277.1i 0.989052i 0.869163 + 0.494526i \(0.164658\pi\)
−0.869163 + 0.494526i \(0.835342\pi\)
\(648\) −24472.0 + 234.996i −1.48357 + 0.0142461i
\(649\) 0 0
\(650\) 44361.4i 2.67692i
\(651\) 0 0
\(652\) 24540.0 1.47402
\(653\) 25072.6i 1.50255i −0.659988 0.751276i \(-0.729439\pi\)
0.659988 0.751276i \(-0.270561\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 12584.3i 0.748983i
\(657\) −23294.0 + 23518.8i −1.38324 + 1.39658i
\(658\) 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 \(\pi\)
\(662\) 57761.0i 3.39116i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −14812.0 −0.859854
\(668\) 58844.9i 3.40835i
\(669\) −9680.00 23211.8i −0.559418 1.34144i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −33586.0 −1.92369 −0.961846 0.273590i \(-0.911789\pi\)
−0.961846 + 0.273590i \(0.911789\pi\)
\(674\) 0 0
\(675\) 16250.0 + 6594.27i 0.926612 + 0.376020i
\(676\) −49185.0 −2.79842
\(677\) 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
\(683\) 4882.16i 0.273515i 0.990605 + 0.136757i \(0.0436681\pi\)
−0.990605 + 0.136757i \(0.956332\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −6028.00 −0.331861 −0.165930 0.986137i \(-0.553063\pi\)
−0.165930 + 0.986137i \(0.553063\pi\)
\(692\) 43450.2i 2.38689i
\(693\) 0 0
\(694\) 43654.0 2.38773
\(695\) 0 0
\(696\) 9016.00 + 21619.6i 0.491021 + 1.17743i
\(697\) 0 0
\(698\) 46625.1i 2.52835i
\(699\) 20056.0 8363.93i 1.08525 0.452579i
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 18722.0 46135.9i 1.00658 2.48047i
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 61456.0 3.27610
\(707\) 0 0
\(708\) −58650.0 + 24458.7i −3.11328 + 1.29833i
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 37944.6i 1.99304i
\(714\) 0 0
\(715\) 0 0
\(716\) 58844.9i 3.07142i
\(717\) −2714.00 + 1131.82i −0.141361 + 0.0589518i
\(718\) 0 0
\(719\) 6858.04i 0.355719i −0.984056 0.177859i \(-0.943083\pi\)
0.984056 0.177859i \(-0.0569172\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 32894.6i 1.69558i
\(723\) 0 0
\(724\) 0 0
\(725\) 16785.4i 0.859854i
\(726\) 30613.0 12766.5i 1.56495 0.652630i
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 14117.0 + 13716.1i 0.717218 + 0.696849i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 7935.00 0.397402
\(737\) 0 0
\(738\) 27968.0 28237.9i 1.39501 1.40847i
\(739\) 30548.0 1.52060 0.760302 0.649570i \(-0.225051\pi\)
0.760302 + 0.649570i \(0.225051\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 55384.0 23096.7i 2.72913 1.13813i
\(745\) 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 \(\pi\)
\(752\) 26348.3i 1.27769i
\(753\) 0 0
\(754\) −47656.0 −2.30176
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 15231.6i 0.725550i −0.931877 0.362775i \(-0.881829\pi\)
0.931877 0.362775i \(-0.118171\pi\)
\(762\) 59984.0 25015.1i 2.85169 1.18924i
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 60331.6i 2.84022i
\(768\) −14670.0 35177.4i −0.689268 1.65281i
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −36248.0 + 15116.5i −1.69318 + 0.706104i
\(772\) −64290.0 −2.99721
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) −43000.0 −1.99304
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 7084.00 17456.8i 0.323322 0.796751i
\(784\) 14063.0 0.640625
\(785\) 0 0
\(786\) −10396.0 24928.7i −0.471772 1.13127i
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 8344.75i 0.377245i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 72330.7i 3.23290i
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 8992.18i 0.397402i
\(801\) 0 0
\(802\) 0