Properties

Label 21.22.c.a.20.1
Level $21$
Weight $22$
Character 21.20
Analytic conductor $58.690$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 21.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(58.6902423003\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 20.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 21.20
Dual form 21.22.c.a.20.2

$q$-expansion

\(f(q)\) \(=\) \(q-102276. i q^{3} +2.09715e6 q^{4} +(-5.61992e8 - 4.92657e8i) q^{7} -1.04604e10 q^{9} +O(q^{10})\) \(q-102276. i q^{3} +2.09715e6 q^{4} +(-5.61992e8 - 4.92657e8i) q^{7} -1.04604e10 q^{9} -2.14488e11i q^{12} +9.22660e11i q^{13} +4.39805e12 q^{16} +3.99212e13i q^{19} +(-5.03870e13 + 5.74782e13i) q^{21} -4.76837e14 q^{25} +1.06984e15i q^{27} +(-1.17858e15 - 1.03318e15i) q^{28} -1.25663e15i q^{31} -2.19370e16 q^{36} +5.77763e16 q^{37} +9.43658e16 q^{39} -2.65258e17 q^{43} -4.49814e17i q^{48} +(7.31231e16 + 5.53739e17i) q^{49} +1.93496e18i q^{52} +4.08297e18 q^{57} +2.49134e18i q^{61} +(5.87863e18 + 5.15337e18i) q^{63} +9.22337e18 q^{64} +6.94433e18 q^{67} +6.21644e19i q^{73} +4.87689e19i q^{75} +8.37208e19i q^{76} +1.68068e20 q^{79} +1.09419e20 q^{81} +(-1.05669e20 + 1.20540e20i) q^{84} +(4.54555e20 - 5.18527e20i) q^{91} -1.28523e20 q^{93} +9.10134e20i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4194304q^{4} - 1123983020q^{7} - 20920706406q^{9} + O(q^{10}) \) \( 2q + 4194304q^{4} - 1123983020q^{7} - 20920706406q^{9} + 8796093022208q^{16} - 100773945863478q^{21} - 953674316406250q^{25} - 2357163238359040q^{28} - 43873901280755712q^{36} + 115552646878006580q^{37} + 188731681059069432q^{39} - 530516888827641040q^{43} + 146246101081752386q^{49} + 8165948250222754380q^{57} + 11757259383374613060q^{63} + 18446744073709551616q^{64} + 13888664740533843040q^{67} + 336136578094166930392q^{79} + 218837978263024718418q^{81} - 211338282115484614656q^{84} + 909110612264755817016q^{91} - 257046033705514549380q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) 102276.i 1.00000i
\(4\) 2.09715e6 1.00000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −5.61992e8 4.92657e8i −0.751970 0.659198i
\(8\) 0 0
\(9\) −1.04604e10 −1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 2.14488e11i 1.00000i
\(13\) 9.22660e11i 1.85625i 0.372269 + 0.928125i \(0.378580\pi\)
−0.372269 + 0.928125i \(0.621420\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.39805e12 1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 3.99212e13i 1.49379i 0.664940 + 0.746897i \(0.268457\pi\)
−0.664940 + 0.746897i \(0.731543\pi\)
\(20\) 0 0
\(21\) −5.03870e13 + 5.74782e13i −0.659198 + 0.751970i
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −4.76837e14 −1.00000
\(26\) 0 0
\(27\) 1.06984e15i 1.00000i
\(28\) −1.17858e15 1.03318e15i −0.751970 0.659198i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 1.25663e15i 0.275366i −0.990476 0.137683i \(-0.956035\pi\)
0.990476 0.137683i \(-0.0439655\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −2.19370e16 −1.00000
\(37\) 5.77763e16 1.97529 0.987647 0.156694i \(-0.0500838\pi\)
0.987647 + 0.156694i \(0.0500838\pi\)
\(38\) 0 0
\(39\) 9.43658e16 1.85625
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −2.65258e17 −1.87176 −0.935881 0.352317i \(-0.885394\pi\)
−0.935881 + 0.352317i \(0.885394\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 4.49814e17i 1.00000i
\(49\) 7.31231e16 + 5.53739e17i 0.130917 + 0.991393i
\(50\) 0 0
\(51\) 0 0
\(52\) 1.93496e18i 1.85625i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.08297e18 1.49379
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 2.49134e18i 0.447166i 0.974685 + 0.223583i \(0.0717754\pi\)
−0.974685 + 0.223583i \(0.928225\pi\)
\(62\) 0 0
\(63\) 5.87863e18 + 5.15337e18i 0.751970 + 0.659198i
\(64\) 9.22337e18 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 6.94433e18 0.465420 0.232710 0.972546i \(-0.425241\pi\)
0.232710 + 0.972546i \(0.425241\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 6.21644e19i 1.69298i 0.532404 + 0.846490i \(0.321289\pi\)
−0.532404 + 0.846490i \(0.678711\pi\)
\(74\) 0 0
\(75\) 4.87689e19i 1.00000i
\(76\) 8.37208e19i 1.49379i
\(77\) 0 0
\(78\) 0 0
\(79\) 1.68068e20 1.99711 0.998553 0.0537677i \(-0.0171231\pi\)
0.998553 + 0.0537677i \(0.0171231\pi\)
\(80\) 0 0
\(81\) 1.09419e20 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −1.05669e20 + 1.20540e20i −0.659198 + 0.751970i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 4.54555e20 5.18527e20i 1.22364 1.39584i
\(92\) 0 0
\(93\) −1.28523e20 −0.275366
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.10134e20i 1.25315i 0.779362 + 0.626573i \(0.215543\pi\)
−0.779362 + 0.626573i \(0.784457\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1.00000e21 −1.00000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 1.21098e21i 0.887864i −0.896060 0.443932i \(-0.853583\pi\)
0.896060 0.443932i \(-0.146417\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 2.24362e21i 1.00000i
\(109\) −4.46449e21 −1.80632 −0.903158 0.429309i \(-0.858757\pi\)
−0.903158 + 0.429309i \(0.858757\pi\)
\(110\) 0 0
\(111\) 5.90912e21i 1.97529i
\(112\) −2.47166e21 2.16673e21i −0.751970 0.659198i
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 9.65135e21i 1.85625i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 7.40025e21 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 2.63535e21i 0.275366i
\(125\) 0 0
\(126\) 0 0
\(127\) −1.97512e22 −1.60566 −0.802832 0.596205i \(-0.796674\pi\)
−0.802832 + 0.596205i \(0.796674\pi\)
\(128\) 0 0
\(129\) 2.71295e22i 1.87176i
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 1.96675e22 2.24354e22i 0.984705 1.12329i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 6.34187e22i 1.99785i 0.0463902 + 0.998923i \(0.485228\pi\)
−0.0463902 + 0.998923i \(0.514772\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −4.60051e22 −1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) 5.66341e22 7.47872e21i 0.991393 0.130917i
\(148\) 1.21166e23 1.97529
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −4.35403e22 −0.574954 −0.287477 0.957787i \(-0.592817\pi\)
−0.287477 + 0.957787i \(0.592817\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 1.97900e23 1.85625
\(157\) 2.26431e23i 1.98605i 0.117920 + 0.993023i \(0.462377\pi\)
−0.117920 + 0.993023i \(0.537623\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1.34412e23 0.795188 0.397594 0.917561i \(-0.369845\pi\)
0.397594 + 0.917561i \(0.369845\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −6.04237e23 −2.44566
\(170\) 0 0
\(171\) 4.17590e23i 1.49379i
\(172\) −5.56287e23 −1.87176
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 2.67978e23 + 2.34917e23i 0.751970 + 0.659198i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 7.38548e23i 1.45463i −0.686302 0.727317i \(-0.740767\pi\)
0.686302 0.727317i \(-0.259233\pi\)
\(182\) 0 0
\(183\) 2.54804e23 0.447166
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 5.27066e23 6.01242e23i 0.659198 0.751970i
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 9.43328e23i 1.00000i
\(193\) −1.98909e24 −1.99665 −0.998326 0.0578425i \(-0.981578\pi\)
−0.998326 + 0.0578425i \(0.981578\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.53350e23 + 1.16127e24i 0.130917 + 0.991393i
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 1.45180e24i 1.05669i −0.849029 0.528346i \(-0.822812\pi\)
0.849029 0.528346i \(-0.177188\pi\)
\(200\) 0 0
\(201\) 7.10238e23i 0.465420i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 4.05790e24i 1.85625i
\(209\) 0 0
\(210\) 0 0
\(211\) −1.58903e24 −0.625412 −0.312706 0.949850i \(-0.601236\pi\)
−0.312706 + 0.949850i \(0.601236\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −6.19089e23 + 7.06216e23i −0.181520 + 0.207067i
\(218\) 0 0
\(219\) 6.35792e24 1.69298
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 8.39856e24i 1.84929i 0.380836 + 0.924643i \(0.375636\pi\)
−0.380836 + 0.924643i \(0.624364\pi\)
\(224\) 0 0
\(225\) 4.98789e24 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 8.56262e24 1.49379
\(229\) 2.25524e24i 0.375769i 0.982191 + 0.187884i \(0.0601630\pi\)
−0.982191 + 0.187884i \(0.939837\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.71893e25i 1.99711i
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 1.89120e25i 1.84314i 0.388212 + 0.921570i \(0.373093\pi\)
−0.388212 + 0.921570i \(0.626907\pi\)
\(242\) 0 0
\(243\) 1.11909e25i 1.00000i
\(244\) 5.22471e24i 0.447166i
\(245\) 0 0
\(246\) 0 0
\(247\) −3.68337e25 −2.77285
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 1.23284e25 + 1.08074e25i 0.751970 + 0.659198i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.93428e25 1.00000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) −3.24698e25 2.84639e25i −1.48536 1.30211i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 1.45633e25 0.465420
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 5.65070e25i 1.60666i −0.595532 0.803332i \(-0.703059\pi\)
0.595532 0.803332i \(-0.296941\pi\)
\(272\) 0 0
\(273\) −5.30328e25 4.64900e25i −1.39584 1.22364i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −2.19618e24 −0.0496169 −0.0248085 0.999692i \(-0.507898\pi\)
−0.0248085 + 0.999692i \(0.507898\pi\)
\(278\) 0 0
\(279\) 1.31448e25i 0.275366i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 3.39377e25i 0.612244i −0.951992 0.306122i \(-0.900968\pi\)
0.951992 0.306122i \(-0.0990315\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −6.90919e25 −1.00000
\(290\) 0 0
\(291\) 9.30847e25 1.25315
\(292\) 1.30368e26i 1.69298i
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 1.02276e26i 1.00000i
\(301\) 1.49073e26 + 1.30682e26i 1.40751 + 1.23386i
\(302\) 0 0
\(303\) 0 0
\(304\) 1.75575e26i 1.49379i
\(305\) 0 0
\(306\) 0 0
\(307\) 2.24533e26i 1.72317i −0.507614 0.861585i \(-0.669472\pi\)
0.507614 0.861585i \(-0.330528\pi\)
\(308\) 0 0
\(309\) −1.23854e26 −0.887864
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 1.48464e25i 0.0929834i 0.998919 + 0.0464917i \(0.0148041\pi\)
−0.998919 + 0.0464917i \(0.985196\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 3.52465e26 1.99711
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 2.29468e26 1.00000
\(325\) 4.39959e26i 1.85625i
\(326\) 0 0
\(327\) 4.56610e26i 1.80632i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 3.13448e26 1.09137 0.545684 0.837991i \(-0.316270\pi\)
0.545684 + 0.837991i \(0.316270\pi\)
\(332\) 0 0
\(333\) −6.04361e26 −1.97529
\(334\) 0 0
\(335\) 0 0
\(336\) −2.21604e26 + 2.52792e26i −0.659198 + 0.751970i
\(337\) −1.86905e26 −0.538898 −0.269449 0.963015i \(-0.586842\pi\)
−0.269449 + 0.963015i \(0.586842\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 2.31709e26 3.47221e26i 0.555079 0.831798i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 2.10349e26i 0.420024i −0.977699 0.210012i \(-0.932650\pi\)
0.977699 0.210012i \(-0.0673503\pi\)
\(350\) 0 0
\(351\) −9.87100e26 −1.85625
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −8.79492e26 −1.23142
\(362\) 0 0
\(363\) 7.56867e26i 1.00000i
\(364\) 9.53272e26 1.08743e27i 1.22364 1.39584i
\(365\) 0 0
\(366\) 0 0
\(367\) 9.94502e26i 1.17115i −0.810619 0.585574i \(-0.800869\pi\)
0.810619 0.585574i \(-0.199131\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −2.69532e26 −0.275366
\(373\) −1.60839e27 −1.59753 −0.798767 0.601641i \(-0.794514\pi\)
−0.798767 + 0.601641i \(0.794514\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 2.22265e27 1.86707 0.933534 0.358488i \(-0.116708\pi\)
0.933534 + 0.358488i \(0.116708\pi\)
\(380\) 0 0
\(381\) 2.02007e27i 1.60566i
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2.77470e27 1.87176
\(388\) 1.90869e27i 1.25315i
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 3.34962e27i 1.72860i 0.502973 + 0.864302i \(0.332239\pi\)
−0.502973 + 0.864302i \(0.667761\pi\)
\(398\) 0 0
\(399\) −2.29460e27 2.01151e27i −1.12329 0.984705i
\(400\) −2.09715e27 −1.00000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 1.15944e27 0.511148
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 4.62099e27i 1.74438i 0.489171 + 0.872188i \(0.337299\pi\)
−0.489171 + 0.872188i \(0.662701\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 2.53961e27i 0.887864i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 6.48621e27 1.99785
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 4.53662e27 1.26407 0.632034 0.774940i \(-0.282220\pi\)
0.632034 + 0.774940i \(0.282220\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.22738e27 1.40011e27i 0.294771 0.336255i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 4.70521e27i 1.00000i
\(433\) 5.13143e27i 1.06443i −0.846611 0.532213i \(-0.821361\pi\)
0.846611 0.532213i \(-0.178639\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −9.36271e27 −1.80632
\(437\) 0 0
\(438\) 0 0
\(439\) 1.04995e28i 1.88490i 0.334341 + 0.942452i \(0.391486\pi\)
−0.334341 + 0.942452i \(0.608514\pi\)
\(440\) 0 0
\(441\) −7.64893e26 5.79230e27i −0.130917 0.991393i
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 1.23923e28i 1.97529i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −5.18346e27 4.54396e27i −0.751970 0.659198i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 4.45312e27i 0.574954i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.64801e28 −1.94017 −0.970083 0.242775i \(-0.921942\pi\)
−0.970083 + 0.242775i \(0.921942\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −1.29245e28 −1.32682 −0.663411 0.748255i \(-0.730892\pi\)
−0.663411 + 0.748255i \(0.730892\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 2.02403e28i 1.85625i
\(469\) −3.90266e27 3.42118e27i −0.349982 0.306804i
\(470\) 0 0
\(471\) 2.31584e28 1.98605
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.90359e28i 1.49379i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 5.33079e28i 3.66664i
\(482\) 0 0
\(483\) 0 0
\(484\) 1.55194e28 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 2.61774e27 0.158078 0.0790391 0.996872i \(-0.474815\pi\)
0.0790391 + 0.996872i \(0.474815\pi\)
\(488\) 0 0
\(489\) 1.37472e28i 0.795188i
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 5.52672e27i 0.275366i
\(497\) 0 0
\(498\) 0 0
\(499\) 3.72846e28 1.74371 0.871855 0.489765i \(-0.162917\pi\)
0.871855 + 0.489765i \(0.162917\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 6.17988e28i 2.44566i
\(508\) −4.14213e28 −1.60566
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 3.06258e28 3.49359e28i 1.11601 1.27307i
\(512\) 0 0
\(513\) −4.27094e28 −1.49379
\(514\) 0 0
\(515\) 0 0
\(516\) 5.68948e28i 1.87176i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 5.96873e28i 1.70457i 0.523081 + 0.852283i \(0.324783\pi\)
−0.523081 + 0.852283i \(0.675217\pi\)
\(524\) 0 0
\(525\) 2.40264e28 2.74077e28i 0.659198 0.751970i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 3.94716e28 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 4.12457e28 4.70504e28i 0.984705 1.12329i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 9.99080e28 1.99999 0.999997 0.00231282i \(-0.000736194\pi\)
0.999997 + 0.00231282i \(0.000736194\pi\)
\(542\) 0 0
\(543\) −7.55356e28 −1.45463
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −4.86895e28 −0.868097 −0.434048 0.900890i \(-0.642915\pi\)
−0.434048 + 0.900890i \(0.642915\pi\)
\(548\) 0 0
\(549\) 2.60603e28i 0.447166i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −9.44530e28 8.28001e28i −1.50176 1.31649i
\(554\) 0 0
\(555\) 0 0
\(556\) 1.32999e29i 1.99785i
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 2.44743e29i 3.47446i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −6.14925e28 5.39061e28i −0.751970 0.659198i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −1.36364e29 −1.54889 −0.774443 0.632643i \(-0.781970\pi\)
−0.774443 + 0.632643i \(0.781970\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −9.64797e28 −1.00000
\(577\) 2.72218e28i 0.277058i −0.990358 0.138529i \(-0.955763\pi\)
0.990358 0.138529i \(-0.0442374\pi\)
\(578\) 0 0
\(579\) 2.03436e29i 1.99665i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 1.18770e29 1.56840e28i 0.991393 0.130917i
\(589\) 5.01662e28 0.411340
\(590\) 0 0
\(591\) 0 0
\(592\) 2.54103e29 1.97529
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.48484e29 −1.05669
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 2.03707e29i 1.35152i −0.737119 0.675762i \(-0.763815\pi\)
0.737119 0.675762i \(-0.236185\pi\)
\(602\) 0 0
\(603\) −7.26402e28 −0.465420
\(604\) −9.13106e28 −0.574954
\(605\) 0 0
\(606\) 0 0
\(607\) 2.66798e29i 1.59478i 0.603461 + 0.797392i \(0.293788\pi\)
−0.603461 + 0.797392i \(0.706212\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 3.35322e29 1.80770 0.903851 0.427848i \(-0.140728\pi\)
0.903851 + 0.427848i \(0.140728\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 2.08457e29i 1.01453i −0.861790 0.507264i \(-0.830657\pi\)
0.861790 0.507264i \(-0.169343\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 4.15025e29 1.85625
\(625\) 2.27374e29 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 4.74860e29i 1.98605i
\(629\) 0 0
\(630\) 0 0
\(631\) −2.04089e29 −0.811914 −0.405957 0.913892i \(-0.633062\pi\)
−0.405957 + 0.913892i \(0.633062\pi\)
\(632\) 0 0
\(633\) 1.62520e29i 0.625412i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −5.10912e29 + 6.74677e28i −1.84027 + 0.243014i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 5.96033e29i 1.94561i 0.231629 + 0.972804i \(0.425594\pi\)
−0.231629 + 0.972804i \(0.574406\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 7.22288e28 + 6.33178e28i 0.207067 + 0.181520i
\(652\) 2.81883e29 0.795188
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 6.50262e29i 1.69298i
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 1.50579e29i 0.367832i −0.982942 0.183916i \(-0.941123\pi\)
0.982942 0.183916i \(-0.0588774\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 8.58970e29 1.84929
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −8.12894e29 −1.64390 −0.821951 0.569559i \(-0.807114\pi\)
−0.821951 + 0.569559i \(0.807114\pi\)
\(674\) 0 0
\(675\) 5.10140e29i 1.00000i
\(676\) −1.26718e30 −2.44566
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 4.48384e29 5.11487e29i 0.826072 0.942329i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 8.75749e29i 1.49379i
\(685\) 0 0
\(686\) 0 0
\(687\) 2.30657e29 0.375769
\(688\) −1.16662e30 −1.87176
\(689\) 0 0
\(690\) 0 0
\(691\) 1.30194e30i 1.99560i −0.0663314 0.997798i \(-0.521129\pi\)
0.0663314 0.997798i \(-0.478871\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 5.61992e29 + 4.92657e29i 0.751970 + 0.659198i
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 2.30650e30i 2.95068i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 7.91584e29 0.926214 0.463107 0.886302i \(-0.346735\pi\)
0.463107 + 0.886302i \(0.346735\pi\)
\(710\) 0 0
\(711\) −1.75805e30 −1.99711
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −5.96599e29 + 6.80561e29i −0.585278 + 0.667647i
\(722\) 0 0
\(723\) 1.93424e30 1.84314
\(724\) 1.54885e30i 1.45463i
\(725\) 0 0
\(726\) 0 0
\(727\) 8.23125e29i 0.740209i 0.928990 + 0.370105i \(0.120678\pi\)
−0.928990 + 0.370105i \(0.879322\pi\)
\(728\) 0 0
\(729\) −1.14456e30 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 5.34362e29 0.447166
\(733\) 2.21458e30i 1.82684i −0.407020 0.913419i \(-0.633432\pi\)
0.407020 0.913419i \(-0.366568\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 2.73020e28 0.0206742 0.0103371 0.999947i \(-0.496710\pi\)
0.0103371 + 0.999947i \(0.496710\pi\)
\(740\) 0 0
\(741\) 3.76720e30i 2.77285i
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 2.79025e30 1.78412 0.892059 0.451918i \(-0.149260\pi\)
0.892059 + 0.451918i \(0.149260\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 1.10534e30 1.26090e30i 0.659198 0.751970i
\(757\) 2.68402e30 1.57862 0.789311 0.613993i \(-0.210438\pi\)
0.789311 + 0.613993i \(0.210438\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 2.50901e30 + 2.19946e30i 1.35829 + 1.19072i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 1.97830e30i 1.00000i
\(769\) 2.91440e30i 1.45319i 0.687066 + 0.726595i \(0.258898\pi\)
−0.687066 + 0.726595i \(0.741102\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −4.17142e30 −1.99665
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 5.99208e29i 0.275366i
\(776\) 0 0
\(777\) −2.91117e30 + 3.32088e30i −1.30211 + 1.48536i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 3.21599e29 + 2.43537e30i 0.130917 + 0.991393i
\(785\) 0 0
\(786\) 0 0
\(787\) 4.92409e30i 1.92571i −0.270018 0.962855i \(-0.587030\pi\)
0.270018 0.962855i \(-0.412970\pi\)
\(788\) 0 0