Properties

Label 105.4.g.a.104.2
Level $105$
Weight $4$
Character 105.104
Analytic conductor $6.195$
Analytic rank $0$
Dimension $4$
CM discriminant -35
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 105.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.19520055060\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-5}, \sqrt{7})\)
Defining polynomial: \(x^{4} - x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 104.2
Root \(1.32288 + 1.11803i\) of defining polynomial
Character \(\chi\) \(=\) 105.104
Dual form 105.4.g.a.104.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-2.64575 + 4.47214i) q^{3} -8.00000 q^{4} -11.1803i q^{5} +18.5203 q^{7} +(-13.0000 - 23.6643i) q^{9} +O(q^{10})\) \(q+(-2.64575 + 4.47214i) q^{3} -8.00000 q^{4} -11.1803i q^{5} +18.5203 q^{7} +(-13.0000 - 23.6643i) q^{9} +11.8322i q^{11} +(21.1660 - 35.7771i) q^{12} +84.6640 q^{13} +(50.0000 + 29.5804i) q^{15} +64.0000 q^{16} -102.859i q^{17} +89.4427i q^{20} +(-49.0000 + 82.8251i) q^{21} -125.000 q^{25} +(140.225 + 4.47214i) q^{27} -148.162 q^{28} -307.636i q^{29} +(-52.9150 - 31.3050i) q^{33} -207.063i q^{35} +(104.000 + 189.315i) q^{36} +(-224.000 + 378.629i) q^{39} -94.6573i q^{44} +(-264.575 + 145.344i) q^{45} +178.885i q^{47} +(-169.328 + 286.217i) q^{48} +343.000 q^{49} +(460.000 + 272.140i) q^{51} -677.312 q^{52} +132.288 q^{55} +(-400.000 - 236.643i) q^{60} +(-240.763 - 438.269i) q^{63} -512.000 q^{64} -946.573i q^{65} +822.873i q^{68} +863.748i q^{71} +1132.38 q^{73} +(330.719 - 559.017i) q^{75} +219.135i q^{77} +236.000 q^{79} -715.542i q^{80} +(-391.000 + 615.272i) q^{81} -1511.58i q^{83} +(392.000 - 662.601i) q^{84} -1150.00 q^{85} +(1375.79 + 813.929i) q^{87} +1568.00 q^{91} -963.053 q^{97} +(280.000 - 153.818i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 32q^{4} - 52q^{9} + O(q^{10}) \) \( 4q - 32q^{4} - 52q^{9} + 200q^{15} + 256q^{16} - 196q^{21} - 500q^{25} + 416q^{36} - 896q^{39} + 1372q^{49} + 1840q^{51} - 1600q^{60} - 2048q^{64} + 944q^{79} - 1564q^{81} + 1568q^{84} - 4600q^{85} + 6272q^{91} + 1120q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\) \(71\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(0.5\pi\)
\(3\) −2.64575 + 4.47214i −0.509175 + 0.860663i
\(4\) −8.00000 −1.00000
\(5\) 11.1803i 1.00000i
\(6\) 0 0
\(7\) 18.5203 1.00000
\(8\) 0 0
\(9\) −13.0000 23.6643i −0.481481 0.876456i
\(10\) 0 0
\(11\) 11.8322i 0.324321i 0.986764 + 0.162160i \(0.0518462\pi\)
−0.986764 + 0.162160i \(0.948154\pi\)
\(12\) 21.1660 35.7771i 0.509175 0.860663i
\(13\) 84.6640 1.80628 0.903138 0.429351i \(-0.141258\pi\)
0.903138 + 0.429351i \(0.141258\pi\)
\(14\) 0 0
\(15\) 50.0000 + 29.5804i 0.860663 + 0.509175i
\(16\) 64.0000 1.00000
\(17\) 102.859i 1.46747i −0.679435 0.733735i \(-0.737775\pi\)
0.679435 0.733735i \(-0.262225\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 89.4427i 1.00000i
\(21\) −49.0000 + 82.8251i −0.509175 + 0.860663i
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −125.000 −1.00000
\(26\) 0 0
\(27\) 140.225 + 4.47214i 0.999492 + 0.0318764i
\(28\) −148.162 −1.00000
\(29\) 307.636i 1.96988i −0.172889 0.984941i \(-0.555310\pi\)
0.172889 0.984941i \(-0.444690\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) −52.9150 31.3050i −0.279131 0.165136i
\(34\) 0 0
\(35\) 207.063i 1.00000i
\(36\) 104.000 + 189.315i 0.481481 + 0.876456i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) −224.000 + 378.629i −0.919710 + 1.55459i
\(40\) 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 \(\pi\)
\(44\) 94.6573i 0.324321i
\(45\) −264.575 + 145.344i −0.876456 + 0.481481i
\(46\) 0 0
\(47\) 178.885i 0.555173i 0.960701 + 0.277586i \(0.0895345\pi\)
−0.960701 + 0.277586i \(0.910466\pi\)
\(48\) −169.328 + 286.217i −0.509175 + 0.860663i
\(49\) 343.000 1.00000
\(50\) 0 0
\(51\) 460.000 + 272.140i 1.26300 + 0.747200i
\(52\) −677.312 −1.80628
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 132.288 0.324321
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −400.000 236.643i −0.860663 0.509175i
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −240.763 438.269i −0.481481 0.876456i
\(64\) −512.000 −1.00000
\(65\) 946.573i 1.80628i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 822.873i 1.46747i
\(69\) 0 0
\(70\) 0 0
\(71\) 863.748i 1.44377i 0.692011 + 0.721887i \(0.256725\pi\)
−0.692011 + 0.721887i \(0.743275\pi\)
\(72\) 0 0
\(73\) 1132.38 1.81555 0.907776 0.419456i \(-0.137779\pi\)
0.907776 + 0.419456i \(0.137779\pi\)
\(74\) 0 0
\(75\) 330.719 559.017i 0.509175 0.860663i
\(76\) 0 0
\(77\) 219.135i 0.324321i
\(78\) 0 0
\(79\) 236.000 0.336102 0.168051 0.985778i \(-0.446253\pi\)
0.168051 + 0.985778i \(0.446253\pi\)
\(80\) 715.542i 1.00000i
\(81\) −391.000 + 615.272i −0.536351 + 0.843995i
\(82\) 0 0
\(83\) 1511.58i 1.99901i −0.0314901 0.999504i \(-0.510025\pi\)
0.0314901 0.999504i \(-0.489975\pi\)
\(84\) 392.000 662.601i 0.509175 0.860663i
\(85\) −1150.00 −1.46747
\(86\) 0 0
\(87\) 1375.79 + 813.929i 1.69541 + 1.00302i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 1568.00 1.80628
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −963.053 −1.00807 −0.504037 0.863682i \(-0.668153\pi\)
−0.504037 + 0.863682i \(0.668153\pi\)
\(98\) 0 0
\(99\) 280.000 153.818i 0.284253 0.156155i
\(100\) 1000.00 1.00000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 799.017 0.764364 0.382182 0.924087i \(-0.375173\pi\)
0.382182 + 0.924087i \(0.375173\pi\)
\(104\) 0 0
\(105\) 926.013 + 547.837i 0.860663 + 0.509175i
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −1121.80 35.7771i −0.999492 0.0318764i
\(109\) −2266.00 −1.99122 −0.995612 0.0935765i \(-0.970170\pi\)
−0.995612 + 0.0935765i \(0.970170\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1185.30 1.00000
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2461.09i 1.96988i
\(117\) −1100.63 2003.52i −0.869688 1.58312i
\(118\) 0 0
\(119\) 1904.98i 1.46747i
\(120\) 0 0
\(121\) 1191.00 0.894816
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1397.54i 1.00000i
\(126\) 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 \(0.5\pi\)
\(132\) 423.320 + 250.440i 0.279131 + 0.165136i
\(133\) 0 0
\(134\) 0 0
\(135\) 50.0000 1567.76i 0.0318764 0.999492i
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 1656.50i 1.00000i
\(141\) −800.000 473.286i −0.477817 0.282680i
\(142\) 0 0
\(143\) 1001.76i 0.585813i
\(144\) −832.000 1514.52i −0.481481 0.876456i
\(145\) −3439.48 −1.96988
\(146\) 0 0
\(147\) −907.493 + 1533.94i −0.509175 + 0.860663i
\(148\) 0 0
\(149\) 2674.07i 1.47026i 0.677928 + 0.735128i \(0.262878\pi\)
−0.677928 + 0.735128i \(0.737122\pi\)
\(150\) 0 0
\(151\) −2788.00 −1.50254 −0.751272 0.659992i \(-0.770559\pi\)
−0.751272 + 0.659992i \(0.770559\pi\)
\(152\) 0 0
\(153\) −2434.09 + 1337.17i −1.28617 + 0.706560i
\(154\) 0 0
\(155\) 0 0
\(156\) 1792.00 3029.03i 0.919710 1.55459i
\(157\) −486.818 −0.247467 −0.123734 0.992315i \(-0.539487\pi\)
−0.123734 + 0.992315i \(0.539487\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) −350.000 + 591.608i −0.165136 + 0.279131i
\(166\) 0 0
\(167\) 2558.06i 1.18532i −0.805452 0.592661i \(-0.798077\pi\)
0.805452 0.592661i \(-0.201923\pi\)
\(168\) 0 0
\(169\) 4971.00 2.26263
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4405.05i 1.93590i 0.251150 + 0.967948i \(0.419191\pi\)
−0.251150 + 0.967948i \(0.580809\pi\)
\(174\) 0 0
\(175\) −2315.03 −1.00000
\(176\) 757.258i 0.324321i
\(177\) 0 0
\(178\) 0 0
\(179\) 4697.37i 1.96144i 0.195419 + 0.980720i \(0.437393\pi\)
−0.195419 + 0.980720i \(0.562607\pi\)
\(180\) 2116.60 1162.76i 0.876456 0.481481i
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1217.05 0.475931
\(188\) 1431.08i 0.555173i
\(189\) 2597.00 + 82.8251i 0.999492 + 0.0318764i
\(190\) 0 0
\(191\) 3182.85i 1.20577i −0.797826 0.602887i \(-0.794017\pi\)
0.797826 0.602887i \(-0.205983\pi\)
\(192\) 1354.62 2289.73i 0.509175 0.860663i
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 4233.20 + 2504.40i 1.55459 + 0.919710i
\(196\) −2744.00 −1.00000
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 5697.50i 1.96988i
\(204\) −3680.00 2177.12i −1.26300 0.747200i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 5418.50 1.80628
\(209\) 0 0
\(210\) 0 0
\(211\) −2392.00 −0.780436 −0.390218 0.920722i \(-0.627600\pi\)
−0.390218 + 0.920722i \(0.627600\pi\)
\(212\) 0 0
\(213\) −3862.80 2285.26i −1.24260 0.735134i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −2996.00 + 5064.16i −0.924433 + 1.56258i
\(220\) −1058.30 −0.324321
\(221\) 8708.47i 2.65066i
\(222\) 0 0
\(223\) 5180.38 1.55562 0.777812 0.628498i \(-0.216330\pi\)
0.777812 + 0.628498i \(0.216330\pi\)
\(224\) 0 0
\(225\) 1625.00 + 2958.04i 0.481481 + 0.876456i
\(226\) 0 0
\(227\) 4767.30i 1.39391i 0.717117 + 0.696953i \(0.245461\pi\)
−0.717117 + 0.696953i \(0.754539\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) −980.000 579.776i −0.279131 0.165136i
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 2000.00 0.555173
\(236\) 0 0
\(237\) −624.397 + 1055.42i −0.171135 + 0.289271i
\(238\) 0 0
\(239\) 4673.70i 1.26492i −0.774592 0.632462i \(-0.782045\pi\)
0.774592 0.632462i \(-0.217955\pi\)
\(240\) 3200.00 + 1893.15i 0.860663 + 0.509175i
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) −1717.09 3376.46i −0.453299 0.891359i
\(244\) 0 0
\(245\) 3834.86i 1.00000i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 6760.00 + 3999.27i 1.72047 + 1.01785i
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 1926.11 + 3506.15i 0.481481 + 0.876456i
\(253\) 0 0
\(254\) 0 0
\(255\) 3042.61 5142.96i 0.747200 1.26300i
\(256\) 4096.00 1.00000
\(257\) 3358.57i 0.815183i 0.913164 + 0.407592i \(0.133631\pi\)
−0.913164 + 0.407592i \(0.866369\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 7572.58i 1.80628i
\(261\) −7280.00 + 3999.27i −1.72652 + 0.948462i
\(262\) 0 0
\(263\) 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
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 6582.98i 1.46747i
\(273\) −4148.54 + 7012.31i −0.919710 + 1.55459i
\(274\) 0 0
\(275\) 1479.02i 0.324321i
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4780.19i 1.01481i −0.861707 0.507406i \(-0.830604\pi\)
0.861707 0.507406i \(-0.169396\pi\)
\(282\) 0 0
\(283\) 2227.72 0.467931 0.233965 0.972245i \(-0.424830\pi\)
0.233965 + 0.972245i \(0.424830\pi\)
\(284\) 6909.98i 1.44377i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −5667.00 −1.15347
\(290\) 0 0
\(291\) 2548.00 4306.91i 0.513287 0.867613i
\(292\) −9059.05 −1.81555
\(293\) 6739.51i 1.34378i 0.740653 + 0.671888i \(0.234516\pi\)
−0.740653 + 0.671888i \(0.765484\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −52.9150 + 1659.16i −0.0103382 + 0.324156i
\(298\) 0 0
\(299\) 0 0
\(300\) −2645.75 + 4472.14i −0.509175 + 0.860663i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −9011.43 −1.67527 −0.837637 0.546227i \(-0.816064\pi\)
−0.837637 + 0.546227i \(0.816064\pi\)
\(308\) 1753.08i 0.324321i
\(309\) −2114.00 + 3573.31i −0.389195 + 0.657860i
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −2677.50 −0.483518 −0.241759 0.970336i \(-0.577724\pi\)
−0.241759 + 0.970336i \(0.577724\pi\)
\(314\) 0 0
\(315\) −4900.00 + 2691.82i −0.876456 + 0.481481i
\(316\) −1888.00 −0.336102
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 3640.00 0.638874
\(320\) 5724.33i 1.00000i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 3128.00 4922.18i 0.536351 0.843995i
\(325\) −10583.0 −1.80628
\(326\) 0 0
\(327\) 5995.27 10133.9i 1.01388 1.71377i
\(328\) 0 0
\(329\) 3313.00i 0.555173i
\(330\) 0 0
\(331\) −7432.00 −1.23414 −0.617069 0.786909i \(-0.711680\pi\)
−0.617069 + 0.786909i \(0.711680\pi\)
\(332\) 12092.7i 1.99901i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) −3136.00 + 5300.81i −0.509175 + 0.860663i
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 9200.00 1.46747
\(341\) 0 0
\(342\) 0 0
\(343\) 6352.45 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) −11006.3 6511.43i −1.69541 1.00302i
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 11872.0 + 378.629i 1.80536 + 0.0575776i
\(352\) 0 0
\(353\) 6220.74i 0.937951i −0.883211 0.468975i \(-0.844623\pi\)
0.883211 0.468975i \(-0.155377\pi\)
\(354\) 0 0
\(355\) 9656.99 1.44377
\(356\) 0 0
\(357\) 8519.32 + 5040.10i 1.26300 + 0.747200i
\(358\) 0 0
\(359\) 11086.7i 1.62990i 0.579529 + 0.814952i \(0.303237\pi\)
−0.579529 + 0.814952i \(0.696763\pi\)
\(360\) 0 0
\(361\) 6859.00 1.00000
\(362\) 0 0
\(363\) −3151.09 + 5326.31i −0.455618 + 0.770135i
\(364\) −12544.0 −1.80628
\(365\) 12660.4i 1.81555i
\(366\) 0 0
\(367\) 14038.4 1.99672 0.998360 0.0572477i \(-0.0182325\pi\)
0.998360 + 0.0572477i \(0.0182325\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) −6250.00 3697.55i −0.860663 0.509175i
\(376\) 0 0
\(377\) 26045.7i 3.55815i
\(378\) 0 0
\(379\) 14096.0 1.91046 0.955228 0.295870i \(-0.0956097\pi\)
0.955228 + 0.295870i \(0.0956097\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 4686.80i 0.625285i 0.949871 + 0.312643i \(0.101214\pi\)
−0.949871 + 0.312643i \(0.898786\pi\)
\(384\) 0 0
\(385\) 2450.00 0.324321
\(386\) 0 0
\(387\) 0 0
\(388\) 7704.43 1.00807
\(389\) 6697.00i 0.872883i −0.899733 0.436442i \(-0.856239\pi\)
0.899733 0.436442i \(-0.143761\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2638.56i 0.336102i
\(396\) −2240.00 + 1230.54i −0.284253 + 0.156155i
\(397\) 275.158 0.0347854 0.0173927 0.999849i \(-0.494463\pi\)
0.0173927 + 0.999849i \(0.494463\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −8000.00 −1.00000
\(401\) 9702.37i 1.20826i 0.796885 + 0.604131i \(0.206480\pi\)
−0.796885 + 0.604131i \(0.793520\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 6878.95 + 4371.51i 0.843995 + 0.536351i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −6392.14 −0.764364
\(413\) 0 0
\(414\) 0 0
\(415\) −16900.0 −1.99901
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) −7408.10 4382.69i −0.860663 0.509175i
\(421\) −3922.00 −0.454030 −0.227015 0.973891i \(-0.572897\pi\)
−0.227015 + 0.973891i \(0.572897\pi\)
\(422\) 0 0
\(423\) 4233.20 2325.51i 0.486585 0.267305i
\(424\) 0 0
\(425\) 12857.4i 1.46747i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −4480.00 2650.40i −0.504188 0.298281i
\(430\) 0 0
\(431\) 17476.1i 1.95312i 0.215249 + 0.976559i \(0.430944\pi\)
−0.215249 + 0.976559i \(0.569056\pi\)
\(432\) 8974.39 + 286.217i 0.999492 + 0.0318764i
\(433\) 12562.0 1.39421 0.697105 0.716970i \(-0.254471\pi\)
0.697105 + 0.716970i \(0.254471\pi\)
\(434\) 0 0
\(435\) 9100.00 15381.8i 1.00302 1.69541i
\(436\) 18128.0 1.99122
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −4459.00 8116.86i −0.481481 0.876456i
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −11958.8 7074.92i −1.26540 0.748618i
\(448\) −9482.37 −1.00000
\(449\) 15239.8i 1.60181i 0.598793 + 0.800904i \(0.295647\pi\)
−0.598793 + 0.800904i \(0.704353\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 7376.35 12468.3i 0.765058 1.29318i
\(454\) 0 0
\(455\) 17530.8i 1.80628i
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 460.000 14423.4i 0.0467777 1.46673i
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 19688.7i 1.96988i
\(465\) 0 0
\(466\) 0 0
\(467\) 17164.1i 1.70077i 0.526164 + 0.850383i \(0.323630\pi\)
−0.526164 + 0.850383i \(0.676370\pi\)
\(468\) 8805.06 + 16028.1i 0.869688 + 1.58312i
\(469\) 0 0
\(470\) 0 0
\(471\) 1288.00 2177.12i 0.126004 0.212986i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 15239.8i 1.46747i
\(477\) 0 0
\(478\) 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
\(484\) −9528.00 −0.894816
\(485\) 10767.3i 1.00807i
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 17689.1i 1.62586i 0.582362 + 0.812930i \(0.302129\pi\)
−0.582362 + 0.812930i \(0.697871\pi\)
\(492\) 0 0
\(493\) −31643.2 −2.89075
\(494\) 0 0
\(495\) −1719.74 3130.50i −0.156155 0.284253i
\(496\) 0 0
\(497\) 15996.8i 1.44377i
\(498\) 0 0
\(499\) 7544.00 0.676785 0.338393 0.941005i \(-0.390117\pi\)
0.338393 + 0.941005i \(0.390117\pi\)
\(500\) 11180.3i 1.00000i
\(501\) 11440.0 + 6768.00i 1.02016 + 0.603536i
\(502\) 0 0
\(503\) 2432.84i 0.215656i 0.994170 + 0.107828i \(0.0343896\pi\)
−0.994170 + 0.107828i \(0.965610\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −13152.0 + 22231.0i −1.15208 + 1.94736i
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 20972.0 1.81555
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 8933.28i 0.764364i
\(516\) 0 0
\(517\) −2116.60 −0.180054
\(518\) 0 0
\(519\) −19700.0 11654.7i −1.66615 0.985710i
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) −7296.98 −0.610086 −0.305043 0.952339i \(-0.598671\pi\)
−0.305043 + 0.952339i \(0.598671\pi\)
\(524\) 0 0
\(525\) 6125.00 10353.1i 0.509175 0.860663i
\(526\) 0 0
\(527\) 0 0
\(528\) −3386.56 2003.52i −0.279131 0.165136i
\(529\) −12167.0 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −21007.3 12428.1i −1.68814 0.998716i
\(538\) 0 0
\(539\) 4058.43i 0.324321i
\(540\) −400.000 + 12542.1i −0.0318764 + 0.999492i
\(541\) −9718.00 −0.772291 −0.386146 0.922438i \(-0.626194\pi\)
−0.386146 + 0.922438i \(0.626194\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 25334.7i 1.99122i
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 4370.78 0.336102
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 13252.0i 1.00000i
\(561\) −3220.00 + 5442.79i −0.242332 + 0.409617i
\(562\) 0 0
\(563\) 13908.3i 1.04115i −0.853816 0.520574i \(-0.825718\pi\)
0.853816 0.520574i \(-0.174282\pi\)
\(564\) 6400.00 + 3786.29i 0.477817 + 0.282680i
\(565\) 0 0
\(566\) 0 0
\(567\) −7241.42 + 11395.0i −0.536351 + 0.843995i
\(568\) 0 0
\(569\) 25226.2i 1.85859i −0.369343 0.929293i \(-0.620417\pi\)
0.369343 0.929293i \(-0.379583\pi\)
\(570\) 0 0
\(571\) −22048.0 −1.61590 −0.807951 0.589250i \(-0.799423\pi\)
−0.807951 + 0.589250i \(0.799423\pi\)
\(572\) 8014.07i 0.585813i
\(573\) 14234.1 + 8421.03i 1.03777 + 0.613951i
\(574\) 0 0
\(575\) 0 0
\(576\) 6656.00 + 12116.1i 0.481481 + 0.876456i
\(577\) 18848.3 1.35991 0.679954 0.733255i \(-0.262000\pi\)
0.679954 + 0.733255i \(0.262000\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 27515.8 1.96988
\(581\) 27994.9i 1.99901i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −22400.0 + 12305.4i −1.58312 + 0.869688i
\(586\) 0 0
\(587\) 15035.3i 1.05720i −0.848872 0.528598i \(-0.822718\pi\)
0.848872 0.528598i \(-0.177282\pi\)
\(588\) 7259.94 12271.5i 0.509175 0.860663i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1104.62i 0.0764944i 0.999268 + 0.0382472i \(0.0121774\pi\)
−0.999268 + 0.0382472i \(0.987823\pi\)
\(594\) 0 0
\(595\) −21298.3 −1.46747
\(596\) 21392.5i 1.47026i
\(597\) 0 0
\(598\) 0 0
\(599\) 27273.1i 1.86035i 0.367116 + 0.930175i \(0.380345\pi\)
−0.367116 + 0.930175i \(0.619655\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 22304.0 1.50254
\(605\) 13315.8i 0.894816i
\(606\) 0 0
\(607\) 9085.51 0.607528 0.303764 0.952747i \(-0.401757\pi\)
0.303764 + 0.952747i \(0.401757\pi\)
\(608\) 0 0
\(609\) 25480.0 + 15074.2i 1.69541 + 1.00302i
\(610\) 0 0
\(611\) 15145.2i 1.00280i
\(612\) 19472.7 10697.3i 1.28617 0.706560i
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 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
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −14336.0 + 24232.3i −0.919710 + 1.55459i
\(625\) 15625.0 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 3894.55 0.247467
\(629\) 0 0
\(630\) 0 0
\(631\) 14852.0 0.937003 0.468501 0.883463i \(-0.344794\pi\)
0.468501 + 0.883463i \(0.344794\pi\)
\(632\) 0 0
\(633\) 6328.64 10697.3i 0.397379 0.671693i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 29039.8 1.80628
\(638\) 0 0
\(639\) 20440.0 11228.7i 1.26541 0.695151i
\(640\) 0 0
\(641\) 15003.2i 0.924477i −0.886756 0.462239i \(-0.847046\pi\)
0.886756 0.462239i \(-0.152954\pi\)
\(642\) 0 0
\(643\) −30061.0 −1.84369 −0.921844 0.387562i \(-0.873317\pi\)
−0.921844 + 0.387562i \(0.873317\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 28997.3i 1.76198i 0.473133 + 0.880991i \(0.343123\pi\)
−0.473133 + 0.880991i \(0.656877\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −14721.0 26797.0i −0.874154 1.59125i
\(658\) 0 0
\(659\) 11489.0i 0.679133i −0.940582 0.339567i \(-0.889720\pi\)
0.940582 0.339567i \(-0.110280\pi\)
\(660\) 2800.00 4732.86i 0.165136 0.279131i
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 38945.5 + 23040.4i 2.28132 + 1.34965i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 20464.5i 1.18532i
\(669\) −13706.0 + 23167.4i −0.792085 + 1.33887i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −17528.1 559.017i −0.999492 0.0318764i
\(676\) −39768.0 −2.26263
\(677\) 31577.8i 1.79266i −0.443386 0.896331i \(-0.646223\pi\)
0.443386 0.896331i \(-0.353777\pi\)
\(678\) 0 0
\(679\) −17836.0 −1.00807
\(680\) 0 0
\(681\) −21320.0 12613.1i −1.19968 0.709742i
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 35240.4i 1.93590i
\(693\) 5185.67 2848.75i 0.284253 0.156155i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 18520.3 1.00000
\(701\) 16068.1i 0.865739i −0.901457 0.432869i \(-0.857501\pi\)
0.901457 0.432869i \(-0.142499\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 6058.07i 0.324321i
\(705\) −5291.50 + 8944.27i −0.282680 + 0.477817i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 2126.00 0.112614 0.0563072 0.998413i \(-0.482067\pi\)
0.0563072 + 0.998413i \(0.482067\pi\)
\(710\) 0 0
\(711\) −3068.00 5584.78i −0.161827 0.294579i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 11200.0 0.585813
\(716\) 37578.9i 1.96144i
\(717\) 20901.4 + 12365.5i 1.08867 + 0.644068i
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) −16932.8 + 9302.04i −0.876456 + 0.481481i
\(721\) 14798.0 0.764364
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 38454.5i 1.96988i
\(726\) 0 0
\(727\) −11392.6 −0.581194 −0.290597 0.956845i \(-0.593854\pi\)
−0.290597 + 0.956845i \(0.593854\pi\)
\(728\) 0 0
\(729\) 19643.0 + 1254.21i 0.997968 + 0.0637204i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −16488.3 −0.830846 −0.415423 0.909628i \(-0.636366\pi\)
−0.415423 + 0.909628i \(0.636366\pi\)
\(734\) 0 0
\(735\) 17150.0 + 10146.1i 0.860663 + 0.509175i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −23056.0 −1.14767 −0.573835 0.818971i \(-0.694545\pi\)
−0.573835 + 0.818971i \(0.694545\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 29897.0 1.47026
\(746\) 0 0
\(747\) −35770.6 + 19650.6i −1.75204 + 0.962485i
\(748\) −9736.36 −0.475931
\(749\) 0 0
\(750\) 0 0
\(751\) 27092.0 1.31638 0.658190 0.752852i \(-0.271322\pi\)
0.658190 + 0.752852i \(0.271322\pi\)
\(752\) 11448.7i 0.555173i
\(753\) 0 0
\(754\) 0 0
\(755\) 31170.8i 1.50254i
\(756\) −20776.0 662.601i −0.999492 0.0318764i
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) −41966.9 −1.99122
\(764\) 25462.8i 1.20577i
\(765\) 14950.0 + 27214.0i 0.706560 + 1.28617i
\(766\) 0 0
\(767\) 0 0
\(768\) −10837.0 + 18317.9i −0.509175 + 0.860663i
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −15020.0 8885.95i −0.701598 0.415071i
\(772\) 0 0
\(773\) 5174.26i 0.240757i −0.992728 0.120379i \(-0.961589\pi\)
0.992728 0.120379i \(-0.0384108\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) −33865.6 20035.2i −1.55459 0.919710i
\(781\) −10220.0 −0.468246
\(782\) 0 0
\(783\) 1375.79 43138.2i 0.0627928 1.96888i
\(784\) 21952.0 1.00000
\(785\) 5442.79i 0.247467i
\(786\) 0 0
\(787\) −15202.5 −0.688577 −0.344289 0.938864i \(-0.611880\pi\)
−0.344289 + 0.938864i \(0.611880\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 41031.8i 1.82362i 0.410616 + 0.911808i \(0.365314\pi\)
−0.410616 + 0.911808i \(0.634686\pi\)
\(798\) 0 0
\(799\) 18400.0 0.814700
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0