Properties

Label 33.2.d.a.32.2
Level 33
Weight 2
Character 33.32
Analytic conductor 0.264
Analytic rank 0
Dimension 2
CM discriminant -11
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 33 = 3 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 33.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.263506326670\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 32.2
Root \(0.500000 + 1.65831i\)
Character \(\chi\) = 33.32
Dual form 33.2.d.a.32.1

$q$-expansion

\(f(q)\) \(=\) \(q+(0.500000 + 1.65831i) q^{3} -2.00000 q^{4} -3.31662i q^{5} +(-2.50000 + 1.65831i) q^{9} +O(q^{10})\) \(q+(0.500000 + 1.65831i) q^{3} -2.00000 q^{4} -3.31662i q^{5} +(-2.50000 + 1.65831i) q^{9} +3.31662i q^{11} +(-1.00000 - 3.31662i) q^{12} +(5.50000 - 1.65831i) q^{15} +4.00000 q^{16} +6.63325i q^{20} -3.31662i q^{23} -6.00000 q^{25} +(-4.00000 - 3.31662i) q^{27} +5.00000 q^{31} +(-5.50000 + 1.65831i) q^{33} +(5.00000 - 3.31662i) q^{36} -7.00000 q^{37} -6.63325i q^{44} +(5.50000 + 8.29156i) q^{45} +6.63325i q^{47} +(2.00000 + 6.63325i) q^{48} +7.00000 q^{49} -13.2665i q^{53} +11.0000 q^{55} -3.31662i q^{59} +(-11.0000 + 3.31662i) q^{60} -8.00000 q^{64} -13.0000 q^{67} +(5.50000 - 1.65831i) q^{69} +16.5831i q^{71} +(-3.00000 - 9.94987i) q^{75} -13.2665i q^{80} +(3.50000 - 8.29156i) q^{81} +16.5831i q^{89} +6.63325i q^{92} +(2.50000 + 8.29156i) q^{93} +17.0000 q^{97} +(-5.50000 - 8.29156i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} - 4q^{4} - 5q^{9} + O(q^{10}) \) \( 2q + q^{3} - 4q^{4} - 5q^{9} - 2q^{12} + 11q^{15} + 8q^{16} - 12q^{25} - 8q^{27} + 10q^{31} - 11q^{33} + 10q^{36} - 14q^{37} + 11q^{45} + 4q^{48} + 14q^{49} + 22q^{55} - 22q^{60} - 16q^{64} - 26q^{67} + 11q^{69} - 6q^{75} + 7q^{81} + 5q^{93} + 34q^{97} - 11q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

\(n\) \(13\) \(23\)
\(\chi(n)\) \(-1\) \(-1\)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0.500000 + 1.65831i 0.288675 + 0.957427i
\(4\) −2.00000 −1.00000
\(5\) 3.31662i 1.48324i −0.670820 0.741620i \(-0.734058\pi\)
0.670820 0.741620i \(-0.265942\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) −2.50000 + 1.65831i −0.833333 + 0.552771i
\(10\) 0 0
\(11\) 3.31662i 1.00000i
\(12\) −1.00000 3.31662i −0.288675 0.957427i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 5.50000 1.65831i 1.42009 0.428174i
\(16\) 4.00000 1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 6.63325i 1.48324i
\(21\) 0 0
\(22\) 0 0
\(23\) 3.31662i 0.691564i −0.938315 0.345782i \(-0.887614\pi\)
0.938315 0.345782i \(-0.112386\pi\)
\(24\) 0 0
\(25\) −6.00000 −1.20000
\(26\) 0 0
\(27\) −4.00000 3.31662i −0.769800 0.638285i
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 0 0
\(33\) −5.50000 + 1.65831i −0.957427 + 0.288675i
\(34\) 0 0
\(35\) 0 0
\(36\) 5.00000 3.31662i 0.833333 0.552771i
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 6.63325i 1.00000i
\(45\) 5.50000 + 8.29156i 0.819892 + 1.23603i
\(46\) 0 0
\(47\) 6.63325i 0.967559i 0.875190 + 0.483779i \(0.160736\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 2.00000 + 6.63325i 0.288675 + 0.957427i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 13.2665i 1.82229i −0.412082 0.911147i \(-0.635198\pi\)
0.412082 0.911147i \(-0.364802\pi\)
\(54\) 0 0
\(55\) 11.0000 1.48324
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.31662i 0.431788i −0.976417 0.215894i \(-0.930733\pi\)
0.976417 0.215894i \(-0.0692665\pi\)
\(60\) −11.0000 + 3.31662i −1.42009 + 0.428174i
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −13.0000 −1.58820 −0.794101 0.607785i \(-0.792058\pi\)
−0.794101 + 0.607785i \(0.792058\pi\)
\(68\) 0 0
\(69\) 5.50000 1.65831i 0.662122 0.199637i
\(70\) 0 0
\(71\) 16.5831i 1.96805i 0.178017 + 0.984027i \(0.443032\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −3.00000 9.94987i −0.346410 1.14891i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 13.2665i 1.48324i
\(81\) 3.50000 8.29156i 0.388889 0.921285i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 16.5831i 1.75781i 0.476999 + 0.878904i \(0.341725\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.63325i 0.691564i
\(93\) 2.50000 + 8.29156i 0.259238 + 0.859795i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 17.0000 1.72609 0.863044 0.505128i \(-0.168555\pi\)
0.863044 + 0.505128i \(0.168555\pi\)
\(98\) 0 0
\(99\) −5.50000 8.29156i −0.552771 0.833333i
\(100\) 12.0000 1.20000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 8.00000 + 6.63325i 0.769800 + 0.638285i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) −3.50000 11.6082i −0.332205 1.10180i
\(112\) 0 0
\(113\) 3.31662i 0.312002i −0.987757 0.156001i \(-0.950140\pi\)
0.987757 0.156001i \(-0.0498603\pi\)
\(114\) 0 0
\(115\) −11.0000 −1.02576
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −10.0000 −0.898027
\(125\) 3.31662i 0.296648i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 11.0000 3.31662i 0.957427 0.288675i
\(133\) 0 0
\(134\) 0 0
\(135\) −11.0000 + 13.2665i −0.946729 + 1.14180i
\(136\) 0 0
\(137\) 23.2164i 1.98351i −0.128154 0.991754i \(-0.540905\pi\)
0.128154 0.991754i \(-0.459095\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) −11.0000 + 3.31662i −0.926367 + 0.279310i
\(142\) 0 0
\(143\) 0 0
\(144\) −10.0000 + 6.63325i −0.833333 + 0.552771i
\(145\) 0 0
\(146\) 0 0
\(147\) 3.50000 + 11.6082i 0.288675 + 0.957427i
\(148\) 14.0000 1.15079
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 16.5831i 1.33199i
\(156\) 0 0
\(157\) 23.0000 1.83560 0.917800 0.397043i \(-0.129964\pi\)
0.917800 + 0.397043i \(0.129964\pi\)
\(158\) 0 0
\(159\) 22.0000 6.63325i 1.74471 0.526051i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 0 0
\(165\) 5.50000 + 18.2414i 0.428174 + 1.42009i
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 13.2665i 1.00000i
\(177\) 5.50000 1.65831i 0.413405 0.124646i
\(178\) 0 0
\(179\) 16.5831i 1.23948i 0.784807 + 0.619740i \(0.212762\pi\)
−0.784807 + 0.619740i \(0.787238\pi\)
\(180\) −11.0000 16.5831i −0.819892 1.23603i
\(181\) −25.0000 −1.85824 −0.929118 0.369784i \(-0.879432\pi\)
−0.929118 + 0.369784i \(0.879432\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 23.2164i 1.70690i
\(186\) 0 0
\(187\) 0 0
\(188\) 13.2665i 0.967559i
\(189\) 0 0
\(190\) 0 0
\(191\) 23.2164i 1.67988i −0.542681 0.839939i \(-0.682591\pi\)
0.542681 0.839939i \(-0.317409\pi\)
\(192\) −4.00000 13.2665i −0.288675 0.957427i
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −14.0000 −1.00000
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) −6.50000 21.5581i −0.458475 1.52059i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 5.50000 + 8.29156i 0.382276 + 0.576303i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 26.5330i 1.82229i
\(213\) −27.5000 + 8.29156i −1.88427 + 0.568128i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) −22.0000 −1.48324
\(221\) 0 0
\(222\) 0 0
\(223\) −1.00000 −0.0669650 −0.0334825 0.999439i \(-0.510660\pi\)
−0.0334825 + 0.999439i \(0.510660\pi\)
\(224\) 0 0
\(225\) 15.0000 9.94987i 1.00000 0.663325i
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 5.00000 0.330409 0.165205 0.986259i \(-0.447172\pi\)
0.165205 + 0.986259i \(0.447172\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 22.0000 1.43512
\(236\) 6.63325i 0.431788i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 22.0000 6.63325i 1.42009 0.428174i
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 15.5000 + 1.65831i 0.994325 + 0.106381i
\(244\) 0 0
\(245\) 23.2164i 1.48324i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 16.5831i 1.04672i 0.852112 + 0.523359i \(0.175321\pi\)
−0.852112 + 0.523359i \(0.824679\pi\)
\(252\) 0 0
\(253\) 11.0000 0.691564
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 26.5330i 1.65508i 0.561405 + 0.827541i \(0.310261\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −44.0000 −2.70290
\(266\) 0 0
\(267\) −27.5000 + 8.29156i −1.68297 + 0.507435i
\(268\) 26.0000 1.58820
\(269\) 13.2665i 0.808873i −0.914566 0.404436i \(-0.867468\pi\)
0.914566 0.404436i \(-0.132532\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 19.8997i 1.20000i
\(276\) −11.0000 + 3.31662i −0.662122 + 0.199637i
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) −12.5000 + 8.29156i −0.748355 + 0.496403i
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 33.1662i 1.96805i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 8.50000 + 28.1913i 0.498279 + 1.65260i
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) −11.0000 −0.640445
\(296\) 0 0
\(297\) 11.0000 13.2665i 0.638285 0.769800i
\(298\) 0 0
\(299\) 0 0
\(300\) 6.00000 + 19.8997i 0.346410 + 1.14891i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) −2.00000 6.63325i −0.113776 0.377352i
\(310\) 0 0
\(311\) 33.1662i 1.88069i −0.340229 0.940343i \(-0.610505\pi\)
0.340229 0.940343i \(-0.389495\pi\)
\(312\) 0 0
\(313\) −19.0000 −1.07394 −0.536972 0.843600i \(-0.680432\pi\)
−0.536972 + 0.843600i \(0.680432\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 23.2164i 1.30396i −0.758236 0.651981i \(-0.773938\pi\)
0.758236 0.651981i \(-0.226062\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 26.5330i 1.48324i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −7.00000 + 16.5831i −0.388889 + 0.921285i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 35.0000 1.92377 0.961887 0.273447i \(-0.0881639\pi\)
0.961887 + 0.273447i \(0.0881639\pi\)
\(332\) 0 0
\(333\) 17.5000 11.6082i 0.958994 0.636125i
\(334\) 0 0
\(335\) 43.1161i 2.35569i
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 5.50000 1.65831i 0.298719 0.0900672i
\(340\) 0 0
\(341\) 16.5831i 0.898027i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −5.50000 18.2414i −0.296110 0.982086i
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 36.4829i 1.94179i 0.239511 + 0.970894i \(0.423013\pi\)
−0.239511 + 0.970894i \(0.576987\pi\)
\(354\) 0 0
\(355\) 55.0000 2.91910
\(356\) 33.1662i 1.75781i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) −5.50000 18.2414i −0.288675 0.957427i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −37.0000 −1.93138 −0.965692 0.259690i \(-0.916380\pi\)
−0.965692 + 0.259690i \(0.916380\pi\)
\(368\) 13.2665i 0.691564i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −5.00000 16.5831i −0.259238 0.859795i
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) −5.50000 + 1.65831i −0.284019 + 0.0856349i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −25.0000 −1.28416 −0.642082 0.766636i \(-0.721929\pi\)
−0.642082 + 0.766636i \(0.721929\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3.31662i 0.169472i −0.996403 0.0847358i \(-0.972995\pi\)
0.996403 0.0847358i \(-0.0270046\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −34.0000 −1.72609
\(389\) 36.4829i 1.84976i 0.380265 + 0.924878i \(0.375833\pi\)
−0.380265 + 0.924878i \(0.624167\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 11.0000 + 16.5831i 0.552771 + 0.833333i
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −24.0000 −1.20000
\(401\) 26.5330i 1.32499i 0.749064 + 0.662497i \(0.230503\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −27.5000 11.6082i −1.36649 0.576815i
\(406\) 0 0
\(407\) 23.2164i 1.15079i
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 38.5000 11.6082i 1.89906 0.572590i
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 33.1662i 1.62028i −0.586238 0.810139i \(-0.699392\pi\)
0.586238 0.810139i \(-0.300608\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 0 0
\(423\) −11.0000 16.5831i −0.534838 0.806299i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −16.0000 13.2665i −0.769800 0.638285i
\(433\) 29.0000 1.39365 0.696826 0.717241i \(-0.254595\pi\)
0.696826 + 0.717241i \(0.254595\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −17.5000 + 11.6082i −0.833333 + 0.552771i
\(442\) 0 0
\(443\) 36.4829i 1.73335i 0.498870 + 0.866677i \(0.333748\pi\)
−0.498870 + 0.866677i \(0.666252\pi\)
\(444\) 7.00000 + 23.2164i 0.332205 + 1.10180i
\(445\) 55.0000 2.60725
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 16.5831i 0.782606i 0.920262 + 0.391303i \(0.127976\pi\)
−0.920262 + 0.391303i \(0.872024\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 6.63325i 0.312002i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 22.0000 1.02576
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −31.0000 −1.44069 −0.720346 0.693615i \(-0.756017\pi\)
−0.720346 + 0.693615i \(0.756017\pi\)
\(464\) 0 0
\(465\) 27.5000 8.29156i 1.27528 0.384512i
\(466\) 0 0
\(467\) 43.1161i 1.99518i −0.0694117 0.997588i \(-0.522112\pi\)
0.0694117 0.997588i \(-0.477888\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 11.5000 + 38.1412i 0.529892 + 1.75745i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 22.0000 + 33.1662i 1.00731 + 1.51858i
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 22.0000 1.00000
\(485\) 56.3826i 2.56020i
\(486\) 0 0
\(487\) −43.0000 −1.94852 −0.974258 0.225436i \(-0.927619\pi\)
−0.974258 + 0.225436i \(0.927619\pi\)
\(488\) 0 0
\(489\) −8.00000 26.5330i −0.361773 1.19986i
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −27.5000 + 18.2414i −1.23603 + 0.819892i
\(496\) 20.0000 0.898027
\(497\) 0 0
\(498\) 0 0
\(499\) −40.0000 −1.79065 −0.895323 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) 6.63325i 0.296648i
\(501\) 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
\(507\) 6.50000 + 21.5581i 0.288675 + 0.957427i
\(508\) 0 0
\(509\) 3.31662i 0.147007i −0.997295 0.0735034i \(-0.976582\pi\)
0.997295 0.0735034i \(-0.0234180\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 13.2665i 0.584592i
\(516\) 0 0
\(517\) −22.0000 −0.967559
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 43.1161i 1.88895i −0.328581 0.944476i \(-0.606570\pi\)
0.328581 0.944476i \(-0.393430\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −22.0000 + 6.63325i −0.957427 + 0.288675i
\(529\) 12.0000 0.521739
\(530\) 0 0
\(531\) 5.50000 + 8.29156i 0.238680 + 0.359823i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −27.5000 + 8.29156i −1.18671 + 0.357807i
\(538\) 0 0
\(539\) 23.2164i 1.00000i
\(540\) 22.0000 26.5330i 0.946729 1.14180i
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) −12.5000 41.4578i −0.536426 1.77912i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 46.4327i 1.98351i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −38.5000 + 11.6082i −1.63423 + 0.492740i
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 22.0000 6.63325i 0.926367 0.279310i
\(565\) −11.0000 −0.462773
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 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
\(573\) 38.5000 11.6082i 1.60836 0.484939i
\(574\) 0 0
\(575\) 19.8997i 0.829877i
\(576\) 20.0000 13.2665i 0.833333 0.552771i
\(577\) 47.0000 1.95664 0.978318 0.207109i \(-0.0664056\pi\)
0.978318 + 0.207109i \(0.0664056\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 44.0000 1.82229
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.63325i 0.273784i 0.990586 + 0.136892i \(0.0437113\pi\)
−0.990586 + 0.136892i \(0.956289\pi\)
\(588\) −7.00000 23.2164i −0.288675 0.957427i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −28.0000 −1.15079
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 10.0000 + 33.1662i 0.409273 + 1.35740i
\(598\) 0 0
\(599\) 33.1662i 1.35514i −0.735460 0.677568i \(-0.763034\pi\)
0.735460 0.677568i \(-0.236966\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 32.5000 21.5581i 1.32350 0.877912i
\(604\) 0 0
\(605\) 36.4829i 1.48324i
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 26.5330i 1.06818i 0.845428 + 0.534089i \(0.179345\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 0 0
\(619\) −1.00000 −0.0401934 −0.0200967 0.999798i \(-0.506397\pi\)
−0.0200967 + 0.999798i \(0.506397\pi\)
\(620\) 33.1662i 1.33199i
\(621\) −11.0000 + 13.2665i −0.441415 + 0.532366i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 0 0
\(628\) −46.0000 −1.83560
\(629\) 0 0
\(630\) 0 0
\(631\) −7.00000 −0.278666 −0.139333 0.990246i \(-0.544496\pi\)
−0.139333 + 0.990246i \(0.544496\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) −44.0000 + 13.2665i −1.74471 + 0.526051i
\(637\) 0 0
\(638\) 0 0
\(639\) −27.5000 41.4578i −1.08788 1.64005i
\(640\) 0 0
\(641\) 23.2164i 0.916992i −0.888697 0.458496i \(-0.848388\pi\)
0.888697 0.458496i \(-0.151612\pi\)
\(642\) 0 0
\(643\) 41.0000 1.61688 0.808441 0.588577i \(-0.200312\pi\)
0.808441 + 0.588577i \(0.200312\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 43.1161i 1.69507i −0.530740 0.847535i \(-0.678086\pi\)
0.530740 0.847535i \(-0.321914\pi\)
\(648\) 0 0
\(649\) 11.0000 0.431788
\(650\) 0 0
\(651\) 0 0
\(652\) 32.0000 1.25322
\(653\) 3.31662i 0.129790i −0.997892 0.0648948i \(-0.979329\pi\)
0.997892 0.0648948i \(-0.0206712\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) −11.0000 36.4829i −0.428174 1.42009i
\(661\) −13.0000 −0.505641 −0.252821 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −0.500000 1.65831i −0.0193311 0.0641141i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 24.0000 + 19.8997i 0.923760 + 0.765942i
\(676\) −26.0000 −1.00000
\(677\) 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
\(683\) 46.4327i 1.77670i 0.459167 + 0.888350i \(0.348148\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) 0 0
\(685\) −77.0000 −2.94202
\(686\) 0 0
\(687\) 2.50000 + 8.29156i 0.0953809 + 0.316343i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 17.0000 0.646710 0.323355 0.946278i \(-0.395189\pi\)
0.323355 + 0.946278i \(0.395189\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 26.5330i 1.00000i
\(705\) 11.0000 + 36.4829i 0.414284 + 1.37402i
\(706\) 0 0
\(707\) 0 0
\(708\) −11.0000 + 3.31662i −0.413405 + 0.124646i
\(709\) −19.0000 −0.713560 −0.356780 0.934188i \(-0.616125\pi\)
−0.356780 + 0.934188i \(0.616125\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 16.5831i 0.621043i
\(714\) 0 0
\(715\) 0 0
\(716\) 33.1662i 1.23948i
\(717\) 0 0
\(718\) 0 0
\(719\) 16.5831i 0.618446i 0.950990 + 0.309223i \(0.100069\pi\)
−0.950990 + 0.309223i \(0.899931\pi\)
\(720\) 22.0000 + 33.1662i 0.819892 + 1.23603i
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 50.0000 1.85824
\(725\) 0 0
\(726\) 0 0
\(727\) 53.0000 1.96566 0.982831 0.184510i \(-0.0590699\pi\)
0.982831 + 0.184510i \(0.0590699\pi\)
\(728\) 0 0
\(729\) 5.00000 + 26.5330i 0.185185 + 0.982704i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 38.5000 11.6082i 1.42009 0.428174i
\(736\) 0 0
\(737\) 43.1161i 1.58820i
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 46.4327i 1.70690i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 23.0000 0.839282 0.419641 0.907690i \(-0.362156\pi\)
0.419641 + 0.907690i \(0.362156\pi\)
\(752\) 26.5330i 0.967559i
\(753\) −27.5000 + 8.29156i −1.00216 + 0.302161i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 0 0
\(759\) 5.50000 + 18.2414i 0.199637 + 0.662122i
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 46.4327i 1.67988i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 8.00000 + 26.5330i 0.288675 + 0.957427i
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −44.0000 + 13.2665i −1.58462 + 0.477781i
\(772\) 0 0
\(773\) 13.2665i 0.477163i −0.971123 0.238581i \(-0.923318\pi\)
0.971123 0.238581i \(-0.0766824\pi\)
\(774\) 0 0
\(775\) −30.0000 −1.07763
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −55.0000 −1.96805
\(782\) 0 0
\(783\) 0 0
\(784\) 28.0000 1.00000
\(785\) 76.2824i 2.72263i
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −22.0000 72.9657i −0.780260 2.58783i
\(796\) −40.0000 −1.41776
\(797\) 56.3826i 1.99717i 0.0531327 + 0.998587i \(0.483079\pi\)
−0.0531327 + 0.998587i \(0.516921\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −27.5000 41.4578i −0.971665 1.46484i
\(802\) 0 0
\(803\) 0 0
\(804\) 13.0000 + 43.1161i 0.458475 + 1.52059i
\(805\) 0 0
\(806\) 0 0
\(807\) 22.0000 6.63325i 0.774437 0.233501i
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 53.0660i 1.85882i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0