Properties

Label 475.2.g.a.18.1
Level $475$
Weight $2$
Character 475.18
Analytic conductor $3.793$
Analytic rank $0$
Dimension $4$
CM discriminant -19
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{19})\)
Defining polynomial: \(x^{4} - 9 x^{2} + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 18.1
Root \(-2.17945 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 475.18
Dual form 475.2.g.a.132.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000i q^{4} +(-3.67945 + 3.67945i) q^{7} +3.00000i q^{9} +O(q^{10})\) \(q-2.00000i q^{4} +(-3.67945 + 3.67945i) q^{7} +3.00000i q^{9} -4.35890 q^{11} -4.00000 q^{16} +(1.32055 - 1.32055i) q^{17} +4.35890i q^{19} +(2.35890 + 2.35890i) q^{23} +(7.35890 + 7.35890i) q^{28} +6.00000 q^{36} +(-6.03835 - 6.03835i) q^{43} +8.71780i q^{44} +(-8.67945 + 8.67945i) q^{47} -20.0767i q^{49} -4.35890 q^{61} +(-11.0383 - 11.0383i) q^{63} +8.00000i q^{64} +(-2.64110 - 2.64110i) q^{68} +(-1.03835 - 1.03835i) q^{73} +8.71780 q^{76} +(16.0383 - 16.0383i) q^{77} -9.00000 q^{81} +(12.3589 + 12.3589i) q^{83} +(4.71780 - 4.71780i) q^{92} -13.0767i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{7} + O(q^{10}) \) \( 4 q - 6 q^{7} - 16 q^{16} + 14 q^{17} - 8 q^{23} + 12 q^{28} + 24 q^{36} + 2 q^{43} - 26 q^{47} - 18 q^{63} - 28 q^{68} + 22 q^{73} + 38 q^{77} - 36 q^{81} + 32 q^{83} - 16 q^{92} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(4\) 2.00000i 1.00000i
\(5\) 0 0
\(6\) 0 0
\(7\) −3.67945 + 3.67945i −1.39070 + 1.39070i −0.566947 + 0.823754i \(0.691875\pi\)
−0.823754 + 0.566947i \(0.808125\pi\)
\(8\) 0 0
\(9\) 3.00000i 1.00000i
\(10\) 0 0
\(11\) −4.35890 −1.31426 −0.657129 0.753778i \(-0.728229\pi\)
−0.657129 + 0.753778i \(0.728229\pi\)
\(12\) 0 0
\(13\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 1.32055 1.32055i 0.320281 0.320281i −0.528594 0.848875i \(-0.677281\pi\)
0.848875 + 0.528594i \(0.177281\pi\)
\(18\) 0 0
\(19\) 4.35890i 1.00000i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.35890 + 2.35890i 0.491864 + 0.491864i 0.908893 0.417029i \(-0.136929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 7.35890 + 7.35890i 1.39070 + 1.39070i
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −6.03835 6.03835i −0.920840 0.920840i 0.0762493 0.997089i \(-0.475706\pi\)
−0.997089 + 0.0762493i \(0.975706\pi\)
\(44\) 8.71780i 1.31426i
\(45\) 0 0
\(46\) 0 0
\(47\) −8.67945 + 8.67945i −1.26603 + 1.26603i −0.317905 + 0.948122i \(0.602979\pi\)
−0.948122 + 0.317905i \(0.897021\pi\)
\(48\) 0 0
\(49\) 20.0767i 2.86810i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −4.35890 −0.558100 −0.279050 0.960277i \(-0.590019\pi\)
−0.279050 + 0.960277i \(0.590019\pi\)
\(62\) 0 0
\(63\) −11.0383 11.0383i −1.39070 1.39070i
\(64\) 8.00000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(68\) −2.64110 2.64110i −0.320281 0.320281i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −1.03835 1.03835i −0.121529 0.121529i 0.643726 0.765256i \(-0.277388\pi\)
−0.765256 + 0.643726i \(0.777388\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 8.71780 1.00000
\(77\) 16.0383 16.0383i 1.82774 1.82774i
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 12.3589 + 12.3589i 1.35657 + 1.35657i 0.878114 + 0.478451i \(0.158802\pi\)
0.478451 + 0.878114i \(0.341198\pi\)
\(84\) 0 0
\(85\) 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
\(91\) 0 0
\(92\) 4.71780 4.71780i 0.491864 0.491864i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(98\) 0 0
\(99\) 13.0767i 1.31426i
\(100\) 0 0
\(101\) 17.4356 1.73491 0.867453 0.497519i \(-0.165755\pi\)
0.867453 + 0.497519i \(0.165755\pi\)
\(102\) 0 0
\(103\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 14.7178 14.7178i 1.39070 1.39070i
\(113\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 9.71780i 0.890829i
\(120\) 0 0
\(121\) 8.00000 0.727273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.00000 0.611593 0.305796 0.952097i \(-0.401077\pi\)
0.305796 + 0.952097i \(0.401077\pi\)
\(132\) 0 0
\(133\) −16.0383 16.0383i −1.39070 1.39070i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −13.6794 + 13.6794i −1.16871 + 1.16871i −0.186203 + 0.982511i \(0.559618\pi\)
−0.982511 + 0.186203i \(0.940382\pi\)
\(138\) 0 0
\(139\) 9.00000i 0.763370i −0.924292 0.381685i \(-0.875344\pi\)
0.924292 0.381685i \(-0.124656\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 12.0000i 1.00000i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 11.0000i 0.901155i 0.892737 + 0.450578i \(0.148782\pi\)
−0.892737 + 0.450578i \(0.851218\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 3.96165 + 3.96165i 0.320281 + 0.320281i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −0.282202 + 0.282202i −0.0225222 + 0.0225222i −0.718278 0.695756i \(-0.755069\pi\)
0.695756 + 0.718278i \(0.255069\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −17.3589 −1.36807
\(162\) 0 0
\(163\) −7.64110 7.64110i −0.598497 0.598497i 0.341415 0.939913i \(-0.389094\pi\)
−0.939913 + 0.341415i \(0.889094\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(168\) 0 0
\(169\) 13.0000i 1.00000i
\(170\) 0 0
\(171\) −13.0767 −1.00000
\(172\) −12.0767 + 12.0767i −0.920840 + 0.920840i
\(173\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 17.4356 1.31426
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −5.75615 + 5.75615i −0.420931 + 0.420931i
\(188\) 17.3589 + 17.3589i 1.26603 + 1.26603i
\(189\) 0 0
\(190\) 0 0
\(191\) 17.0000 1.23008 0.615038 0.788497i \(-0.289140\pi\)
0.615038 + 0.788497i \(0.289140\pi\)
\(192\) 0 0
\(193\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −40.1534 −2.86810
\(197\) 19.7178 19.7178i 1.40483 1.40483i 0.621117 0.783718i \(-0.286679\pi\)
0.783718 0.621117i \(-0.213321\pi\)
\(198\) 0 0
\(199\) 13.0767i 0.926982i −0.886102 0.463491i \(-0.846597\pi\)
0.886102 0.463491i \(-0.153403\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −7.07670 + 7.07670i −0.491864 + 0.491864i
\(208\) 0 0
\(209\) 19.0000i 1.31426i
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(228\) 0 0
\(229\) 21.0000i 1.38772i 0.720110 + 0.693860i \(0.244091\pi\)
−0.720110 + 0.693860i \(0.755909\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 15.7561 + 15.7561i 1.03222 + 1.03222i 0.999463 + 0.0327561i \(0.0104285\pi\)
0.0327561 + 0.999463i \(0.489572\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 30.5123i 1.97368i 0.161712 + 0.986838i \(0.448299\pi\)
−0.161712 + 0.986838i \(0.551701\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 8.71780i 0.558100i
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −23.0000 −1.45175 −0.725874 0.687828i \(-0.758564\pi\)
−0.725874 + 0.687828i \(0.758564\pi\)
\(252\) −22.0767 + 22.0767i −1.39070 + 1.39070i
\(253\) −10.2822 10.2822i −0.646437 0.646437i
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 20.7561 + 20.7561i 1.27988 + 1.27988i 0.940734 + 0.339145i \(0.110138\pi\)
0.339145 + 0.940734i \(0.389862\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −26.1534 −1.58871 −0.794353 0.607457i \(-0.792190\pi\)
−0.794353 + 0.607457i \(0.792190\pi\)
\(272\) −5.28220 + 5.28220i −0.320281 + 0.320281i
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −18.6794 + 18.6794i −1.12234 + 1.12234i −0.130950 + 0.991389i \(0.541803\pi\)
−0.991389 + 0.130950i \(0.958197\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 8.96165 + 8.96165i 0.532715 + 0.532715i 0.921379 0.388664i \(-0.127063\pi\)
−0.388664 + 0.921379i \(0.627063\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 13.5123i 0.794841i
\(290\) 0 0
\(291\) 0 0
\(292\) −2.07670 + 2.07670i −0.121529 + 0.121529i
\(293\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 44.4356 2.56123
\(302\) 0 0
\(303\) 0 0
\(304\) 17.4356i 1.00000i
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(308\) −32.0767 32.0767i −1.82774 1.82774i
\(309\) 0 0
\(310\) 0 0
\(311\) −4.35890 −0.247170 −0.123585 0.992334i \(-0.539439\pi\)
−0.123585 + 0.992334i \(0.539439\pi\)
\(312\) 0 0
\(313\) −14.4356 14.4356i −0.815948 0.815948i 0.169570 0.985518i \(-0.445762\pi\)
−0.985518 + 0.169570i \(0.945762\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 5.75615 + 5.75615i 0.320281 + 0.320281i
\(324\) 18.0000i 1.00000i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 63.8712i 3.52133i
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 24.7178 24.7178i 1.35657 1.35657i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 48.1150 + 48.1150i 2.59797 + 2.59797i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 16.3206 16.3206i 0.876133 0.876133i −0.116999 0.993132i \(-0.537327\pi\)
0.993132 + 0.116999i \(0.0373274\pi\)
\(348\) 0 0
\(349\) 13.0767i 0.699980i −0.936754 0.349990i \(-0.886185\pi\)
0.936754 0.349990i \(-0.113815\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −24.4356 24.4356i −1.30058 1.30058i −0.928003 0.372572i \(-0.878476\pi\)
−0.372572 0.928003i \(-0.621524\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 31.0000i 1.63612i 0.575135 + 0.818059i \(0.304950\pi\)
−0.575135 + 0.818059i \(0.695050\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −27.0767 + 27.0767i −1.41339 + 1.41339i −0.682598 + 0.730794i \(0.739150\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) −9.43560 9.43560i −0.491864 0.491864i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 18.1150 18.1150i 0.920840 0.920840i
\(388\) 0 0
\(389\) 30.5123i 1.54703i 0.633775 + 0.773517i \(0.281504\pi\)
−0.633775 + 0.773517i \(0.718496\pi\)
\(390\) 0 0
\(391\) 6.23009 0.315069
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) −26.1534 −1.31426
\(397\) 23.1150 23.1150i 1.16011 1.16011i 0.175660 0.984451i \(-0.443794\pi\)
0.984451 0.175660i \(-0.0562059\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 34.8712i 1.73491i
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 8.71780i 0.425892i 0.977064 + 0.212946i \(0.0683059\pi\)
−0.977064 + 0.212946i \(0.931694\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) −26.0383 26.0383i −1.26603 1.26603i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 16.0383 16.0383i 0.776150 0.776150i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −10.2822 + 10.2822i −0.491864 + 0.491864i
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 60.2301 2.86810
\(442\) 0 0
\(443\) 0.756146 + 0.756146i 0.0359256 + 0.0359256i 0.724841 0.688916i \(-0.241913\pi\)
−0.688916 + 0.724841i \(0.741913\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −29.4356 29.4356i −1.39070 1.39070i
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 28.1150 28.1150i 1.31517 1.31517i 0.397613 0.917553i \(-0.369839\pi\)
0.917553 0.397613i \(-0.130161\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 37.0000 1.72326 0.861631 0.507535i \(-0.169443\pi\)
0.861631 + 0.507535i \(0.169443\pi\)
\(462\) 0 0
\(463\) 13.9617 + 13.9617i 0.648853 + 0.648853i 0.952716 0.303863i \(-0.0982765\pi\)
−0.303863 + 0.952716i \(0.598276\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −23.6794 + 23.6794i −1.09575 + 1.09575i −0.100853 + 0.994901i \(0.532157\pi\)
−0.994901 + 0.100853i \(0.967843\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 26.3206 + 26.3206i 1.21022 + 1.21022i
\(474\) 0 0
\(475\) 0 0
\(476\) 19.4356 0.890829
\(477\) 0 0
\(478\) 0 0
\(479\) 4.00000i 0.182765i −0.995816 0.0913823i \(-0.970871\pi\)
0.995816 0.0913823i \(-0.0291285\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 16.0000i 0.727273i
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 39.0000i 1.74588i −0.487828 0.872940i \(-0.662211\pi\)
0.487828 0.872940i \(-0.337789\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −17.6411 17.6411i −0.786578 0.786578i 0.194354 0.980932i \(-0.437739\pi\)
−0.980932 + 0.194354i \(0.937739\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 7.64110 0.338022
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 37.8328 37.8328i 1.66389 1.66389i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(524\) 14.0000i 0.611593i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 11.8712i 0.516139i
\(530\) 0 0
\(531\) 0 0
\(532\) −32.0767 + 32.0767i −1.39070 + 1.39070i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 87.5123i 3.76942i
\(540\) 0 0
\(541\) 39.2301 1.68663 0.843317 0.537417i \(-0.180600\pi\)
0.843317 + 0.537417i \(0.180600\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(548\) 27.3589 + 27.3589i 1.16871 + 1.16871i
\(549\) 13.0767i 0.558100i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −18.0000 −0.763370
\(557\) 21.3206 21.3206i 0.903381 0.903381i −0.0923462 0.995727i \(-0.529437\pi\)
0.995727 + 0.0923462i \(0.0294367\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 33.1150 33.1150i 1.39070 1.39070i
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) −26.1534 −1.09449 −0.547243 0.836974i \(-0.684323\pi\)
−0.547243 + 0.836974i \(0.684323\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −24.0000 −1.00000
\(577\) −25.4739 + 25.4739i −1.06049 + 1.06049i −0.0624458 + 0.998048i \(0.519890\pi\)
−0.998048 + 0.0624458i \(0.980110\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −90.9479 −3.77315
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −20.4739 + 20.4739i −0.845050 + 0.845050i −0.989511 0.144460i \(-0.953855\pi\)
0.144460 + 0.989511i \(0.453855\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −34.4356 34.4356i −1.41410 1.41410i −0.715994 0.698106i \(-0.754026\pi\)
−0.698106 0.715994i \(-0.745974\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 22.0000 0.901155
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 7.92330 7.92330i 0.320281 0.320281i
\(613\) −4.24385 4.24385i −0.171408 0.171408i 0.616190 0.787598i \(-0.288675\pi\)
−0.787598 + 0.616190i \(0.788675\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −30.4739 + 30.4739i −1.22683 + 1.22683i −0.261680 + 0.965155i \(0.584277\pi\)
−0.965155 + 0.261680i \(0.915723\pi\)
\(618\) 0 0
\(619\) 24.0000i 0.964641i −0.875995 0.482321i \(-0.839794\pi\)
0.875995 0.482321i \(-0.160206\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0.564404 + 0.564404i 0.0225222 + 0.0225222i
\(629\) 0 0
\(630\) 0 0
\(631\) −47.9479 −1.90878 −0.954388 0.298570i \(-0.903490\pi\)
−0.954388 + 0.298570i \(0.903490\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −31.0383 31.0383i −1.22403 1.22403i −0.966186 0.257847i \(-0.916987\pi\)
−0.257847 0.966186i \(-0.583013\pi\)
\(644\) 34.7178i 1.36807i
\(645\) 0 0
\(646\) 0 0
\(647\) −15.4739 + 15.4739i −0.608344 + 0.608344i −0.942513 0.334169i \(-0.891544\pi\)
0.334169 + 0.942513i \(0.391544\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −15.2822 + 15.2822i −0.598497 + 0.598497i
\(653\) 35.7561 + 35.7561i 1.39925 + 1.39925i 0.802227 + 0.597019i \(0.203648\pi\)
0.597019 + 0.802227i \(0.296352\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 3.11505 3.11505i 0.121529 0.121529i
\(658\) 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 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 19.0000 0.733487
\(672\) 0 0
\(673\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 26.0000 1.00000
\(677\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(684\) 26.1534i 1.00000i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 24.1534 + 24.1534i 0.920840 + 0.920840i
\(689\) 0 0
\(690\) 0 0
\(691\) 39.2301 1.49238 0.746191 0.665731i \(-0.231880\pi\)
0.746191 + 0.665731i \(0.231880\pi\)
\(692\) 0 0
\(693\) 48.1150 + 48.1150i 1.82774 + 1.82774i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 17.4356 0.658533 0.329267 0.944237i \(-0.393198\pi\)
0.329267 + 0.944237i \(0.393198\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 34.8712i 1.31426i
\(705\) 0 0
\(706\) 0 0
\(707\) −64.1534 + 64.1534i −2.41274 + 2.41274i
\(708\) 0 0
\(709\) 52.3068i 1.96442i 0.187779 + 0.982211i \(0.439871\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 49.0000i 1.82739i −0.406399 0.913696i \(-0.633216\pi\)
0.406399 0.913696i \(-0.366784\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 38.1150 38.1150i 1.41361 1.41361i 0.686127 0.727482i \(-0.259309\pi\)
0.727482 0.686127i \(-0.240691\pi\)
\(728\) 0 0
\(729\) 27.0000i 1.00000i
\(730\) 0 0
\(731\) −15.9479 −0.589854
\(732\) 0 0
\(733\) 19.1534 + 19.1534i 0.707447 + 0.707447i 0.965998 0.258551i \(-0.0832450\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 30.5123i 1.12241i 0.827676 + 0.561206i \(0.189663\pi\)
−0.827676 + 0.561206i \(0.810337\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −37.0767 + 37.0767i −1.35657 + 1.35657i
\(748\) 11.5123 + 11.5123i 0.420931 + 0.420931i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 34.7178 34.7178i 1.26603 1.26603i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −10.4739 + 10.4739i −0.380682 + 0.380682i −0.871348 0.490666i \(-0.836754\pi\)
0.490666 + 0.871348i \(0.336754\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −4.35890 −0.158010 −0.0790050 0.996874i \(-0.525174\pi\)
−0.0790050 + 0.996874i \(0.525174\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 34.0000i 1.23008i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 51.0000i 1.83911i 0.392965 + 0.919554i \(0.371449\pi\)
−0.392965 + 0.919554i \(0.628551\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 80.3068i 2.86810i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(788\) −39.4356 39.4356i −1.40483 1.40483i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −26.1534 −0.926982
\(797\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(798\) 0 0
\(799\) 22.9233i 0.810968i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4.52606 + 4.52606i 0.159721 + 0.159721i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 56.6657i 1.99226i −0.0878953 0.996130i \(-0.528014\pi\)
0.0878953 0.996130i \(-0.471986\pi\)