Properties

Label 76.7.c.b.37.4
Level $76$
Weight $7$
Character 76.37
Analytic conductor $17.484$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 76.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.4841103551\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 5090 x^{6} + 8905881 x^{4} + 5831691048 x^{2} + 827887219200\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8}\cdot 3\cdot 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 37.4
Root \(-13.8790i\) of defining polynomial
Character \(\chi\) \(=\) 76.37
Dual form 76.7.c.b.37.5

$q$-expansion

\(f(q)\) \(=\) \(q-13.8790i q^{3} -18.1155 q^{5} +85.5200 q^{7} +536.374 q^{9} +O(q^{10})\) \(q-13.8790i q^{3} -18.1155 q^{5} +85.5200 q^{7} +536.374 q^{9} -316.255 q^{11} -2456.75i q^{13} +251.424i q^{15} -3224.36 q^{17} +(2503.10 - 6385.95i) q^{19} -1186.93i q^{21} -7826.52 q^{23} -15296.8 q^{25} -17562.1i q^{27} -18986.6i q^{29} -32356.9i q^{31} +4389.29i q^{33} -1549.24 q^{35} -15447.1i q^{37} -34097.1 q^{39} -94131.1i q^{41} +39236.1 q^{43} -9716.67 q^{45} +92363.2 q^{47} -110335. q^{49} +44750.8i q^{51} +130058. i q^{53} +5729.10 q^{55} +(-88630.4 - 34740.4i) q^{57} +180241. i q^{59} +124818. q^{61} +45870.7 q^{63} +44505.1i q^{65} +154005. i q^{67} +108624. i q^{69} +177431. i q^{71} +314866. q^{73} +212304. i q^{75} -27046.1 q^{77} +462999. i q^{79} +147273. q^{81} +601062. q^{83} +58410.8 q^{85} -263514. q^{87} -275459. i q^{89} -210101. i q^{91} -449080. q^{93} +(-45344.8 + 115685. i) q^{95} -852579. i q^{97} -169631. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{5} + 362q^{7} - 4348q^{9} + O(q^{10}) \) \( 8q + 2q^{5} + 362q^{7} - 4348q^{9} + 902q^{11} + 1550q^{17} + 6232q^{19} - 18820q^{23} - 12158q^{25} - 101762q^{35} + 167028q^{39} - 335042q^{43} - 57230q^{45} - 570394q^{47} + 448182q^{49} + 1089198q^{55} + 341316q^{57} - 632014q^{61} + 328174q^{63} - 852938q^{73} + 1850530q^{77} - 1819456q^{81} + 441200q^{83} - 1828374q^{85} + 1483380q^{87} + 2131176q^{93} + 627950q^{95} - 865394q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(-1\) \(1\)

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
\(3\) 13.8790i 0.514036i −0.966407 0.257018i \(-0.917260\pi\)
0.966407 0.257018i \(-0.0827400\pi\)
\(4\) 0 0
\(5\) −18.1155 −0.144924 −0.0724619 0.997371i \(-0.523086\pi\)
−0.0724619 + 0.997371i \(0.523086\pi\)
\(6\) 0 0
\(7\) 85.5200 0.249330 0.124665 0.992199i \(-0.460214\pi\)
0.124665 + 0.992199i \(0.460214\pi\)
\(8\) 0 0
\(9\) 536.374 0.735767
\(10\) 0 0
\(11\) −316.255 −0.237607 −0.118803 0.992918i \(-0.537906\pi\)
−0.118803 + 0.992918i \(0.537906\pi\)
\(12\) 0 0
\(13\) 2456.75i 1.11823i −0.829091 0.559114i \(-0.811141\pi\)
0.829091 0.559114i \(-0.188859\pi\)
\(14\) 0 0
\(15\) 251.424i 0.0744960i
\(16\) 0 0
\(17\) −3224.36 −0.656291 −0.328146 0.944627i \(-0.606424\pi\)
−0.328146 + 0.944627i \(0.606424\pi\)
\(18\) 0 0
\(19\) 2503.10 6385.95i 0.364936 0.931033i
\(20\) 0 0
\(21\) 1186.93i 0.128164i
\(22\) 0 0
\(23\) −7826.52 −0.643258 −0.321629 0.946866i \(-0.604230\pi\)
−0.321629 + 0.946866i \(0.604230\pi\)
\(24\) 0 0
\(25\) −15296.8 −0.978997
\(26\) 0 0
\(27\) 17562.1i 0.892247i
\(28\) 0 0
\(29\) 18986.6i 0.778489i −0.921135 0.389244i \(-0.872736\pi\)
0.921135 0.389244i \(-0.127264\pi\)
\(30\) 0 0
\(31\) 32356.9i 1.08613i −0.839691 0.543065i \(-0.817264\pi\)
0.839691 0.543065i \(-0.182736\pi\)
\(32\) 0 0
\(33\) 4389.29i 0.122138i
\(34\) 0 0
\(35\) −1549.24 −0.0361338
\(36\) 0 0
\(37\) 15447.1i 0.304960i −0.988307 0.152480i \(-0.951274\pi\)
0.988307 0.152480i \(-0.0487259\pi\)
\(38\) 0 0
\(39\) −34097.1 −0.574809
\(40\) 0 0
\(41\) 94131.1i 1.36578i −0.730520 0.682892i \(-0.760722\pi\)
0.730520 0.682892i \(-0.239278\pi\)
\(42\) 0 0
\(43\) 39236.1 0.493493 0.246746 0.969080i \(-0.420639\pi\)
0.246746 + 0.969080i \(0.420639\pi\)
\(44\) 0 0
\(45\) −9716.67 −0.106630
\(46\) 0 0
\(47\) 92363.2 0.889622 0.444811 0.895625i \(-0.353271\pi\)
0.444811 + 0.895625i \(0.353271\pi\)
\(48\) 0 0
\(49\) −110335. −0.937835
\(50\) 0 0
\(51\) 44750.8i 0.337357i
\(52\) 0 0
\(53\) 130058.i 0.873597i 0.899559 + 0.436798i \(0.143888\pi\)
−0.899559 + 0.436798i \(0.856112\pi\)
\(54\) 0 0
\(55\) 5729.10 0.0344349
\(56\) 0 0
\(57\) −88630.4 34740.4i −0.478584 0.187590i
\(58\) 0 0
\(59\) 180241.i 0.877602i 0.898584 + 0.438801i \(0.144597\pi\)
−0.898584 + 0.438801i \(0.855403\pi\)
\(60\) 0 0
\(61\) 124818. 0.549907 0.274953 0.961458i \(-0.411338\pi\)
0.274953 + 0.961458i \(0.411338\pi\)
\(62\) 0 0
\(63\) 45870.7 0.183448
\(64\) 0 0
\(65\) 44505.1i 0.162058i
\(66\) 0 0
\(67\) 154005.i 0.512048i 0.966670 + 0.256024i \(0.0824126\pi\)
−0.966670 + 0.256024i \(0.917587\pi\)
\(68\) 0 0
\(69\) 108624.i 0.330658i
\(70\) 0 0
\(71\) 177431.i 0.495740i 0.968793 + 0.247870i \(0.0797305\pi\)
−0.968793 + 0.247870i \(0.920269\pi\)
\(72\) 0 0
\(73\) 314866. 0.809389 0.404694 0.914452i \(-0.367378\pi\)
0.404694 + 0.914452i \(0.367378\pi\)
\(74\) 0 0
\(75\) 212304.i 0.503240i
\(76\) 0 0
\(77\) −27046.1 −0.0592424
\(78\) 0 0
\(79\) 462999.i 0.939071i 0.882914 + 0.469535i \(0.155579\pi\)
−0.882914 + 0.469535i \(0.844421\pi\)
\(80\) 0 0
\(81\) 147273. 0.277120
\(82\) 0 0
\(83\) 601062. 1.05120 0.525600 0.850732i \(-0.323841\pi\)
0.525600 + 0.850732i \(0.323841\pi\)
\(84\) 0 0
\(85\) 58410.8 0.0951122
\(86\) 0 0
\(87\) −263514. −0.400171
\(88\) 0 0
\(89\) 275459.i 0.390739i −0.980730 0.195369i \(-0.937409\pi\)
0.980730 0.195369i \(-0.0625906\pi\)
\(90\) 0 0
\(91\) 210101.i 0.278807i
\(92\) 0 0
\(93\) −449080. −0.558310
\(94\) 0 0
\(95\) −45344.8 + 115685.i −0.0528879 + 0.134929i
\(96\) 0 0
\(97\) 852579.i 0.934156i −0.884216 0.467078i \(-0.845307\pi\)
0.884216 0.467078i \(-0.154693\pi\)
\(98\) 0 0
\(99\) −169631. −0.174823
\(100\) 0 0
\(101\) −702454. −0.681795 −0.340898 0.940100i \(-0.610731\pi\)
−0.340898 + 0.940100i \(0.610731\pi\)
\(102\) 0 0
\(103\) 955700.i 0.874600i 0.899316 + 0.437300i \(0.144065\pi\)
−0.899316 + 0.437300i \(0.855935\pi\)
\(104\) 0 0
\(105\) 21501.8i 0.0185741i
\(106\) 0 0
\(107\) 406564.i 0.331877i 0.986136 + 0.165938i \(0.0530653\pi\)
−0.986136 + 0.165938i \(0.946935\pi\)
\(108\) 0 0
\(109\) 1.98600e6i 1.53356i −0.641912 0.766779i \(-0.721858\pi\)
0.641912 0.766779i \(-0.278142\pi\)
\(110\) 0 0
\(111\) −214390. −0.156760
\(112\) 0 0
\(113\) 1.34434e6i 0.931692i −0.884866 0.465846i \(-0.845750\pi\)
0.884866 0.465846i \(-0.154250\pi\)
\(114\) 0 0
\(115\) 141781. 0.0932234
\(116\) 0 0
\(117\) 1.31774e6i 0.822755i
\(118\) 0 0
\(119\) −275747. −0.163633
\(120\) 0 0
\(121\) −1.67154e6 −0.943543
\(122\) 0 0
\(123\) −1.30644e6 −0.702062
\(124\) 0 0
\(125\) 560164. 0.286804
\(126\) 0 0
\(127\) 1.59869e6i 0.780464i −0.920717 0.390232i \(-0.872395\pi\)
0.920717 0.390232i \(-0.127605\pi\)
\(128\) 0 0
\(129\) 544557.i 0.253673i
\(130\) 0 0
\(131\) 3.03821e6 1.35146 0.675732 0.737148i \(-0.263828\pi\)
0.675732 + 0.737148i \(0.263828\pi\)
\(132\) 0 0
\(133\) 214065. 546127.i 0.0909893 0.232134i
\(134\) 0 0
\(135\) 318146.i 0.129308i
\(136\) 0 0
\(137\) −312378. −0.121484 −0.0607420 0.998153i \(-0.519347\pi\)
−0.0607420 + 0.998153i \(0.519347\pi\)
\(138\) 0 0
\(139\) 383927. 0.142956 0.0714782 0.997442i \(-0.477228\pi\)
0.0714782 + 0.997442i \(0.477228\pi\)
\(140\) 0 0
\(141\) 1.28191e6i 0.457297i
\(142\) 0 0
\(143\) 776957.i 0.265698i
\(144\) 0 0
\(145\) 343951.i 0.112822i
\(146\) 0 0
\(147\) 1.53134e6i 0.482081i
\(148\) 0 0
\(149\) −874420. −0.264339 −0.132169 0.991227i \(-0.542194\pi\)
−0.132169 + 0.991227i \(0.542194\pi\)
\(150\) 0 0
\(151\) 2.08195e6i 0.604698i 0.953197 + 0.302349i \(0.0977709\pi\)
−0.953197 + 0.302349i \(0.902229\pi\)
\(152\) 0 0
\(153\) −1.72946e6 −0.482878
\(154\) 0 0
\(155\) 586161.i 0.157406i
\(156\) 0 0
\(157\) −359939. −0.0930100 −0.0465050 0.998918i \(-0.514808\pi\)
−0.0465050 + 0.998918i \(0.514808\pi\)
\(158\) 0 0
\(159\) 1.80508e6 0.449060
\(160\) 0 0
\(161\) −669324. −0.160383
\(162\) 0 0
\(163\) 2.96880e6 0.685517 0.342759 0.939424i \(-0.388639\pi\)
0.342759 + 0.939424i \(0.388639\pi\)
\(164\) 0 0
\(165\) 79514.0i 0.0177008i
\(166\) 0 0
\(167\) 8.76081e6i 1.88103i 0.339758 + 0.940513i \(0.389655\pi\)
−0.339758 + 0.940513i \(0.610345\pi\)
\(168\) 0 0
\(169\) −1.20879e6 −0.250433
\(170\) 0 0
\(171\) 1.34260e6 3.42526e6i 0.268508 0.685023i
\(172\) 0 0
\(173\) 4.80027e6i 0.927101i 0.886070 + 0.463551i \(0.153425\pi\)
−0.886070 + 0.463551i \(0.846575\pi\)
\(174\) 0 0
\(175\) −1.30819e6 −0.244093
\(176\) 0 0
\(177\) 2.50156e6 0.451119
\(178\) 0 0
\(179\) 8.65531e6i 1.50912i 0.656231 + 0.754560i \(0.272150\pi\)
−0.656231 + 0.754560i \(0.727850\pi\)
\(180\) 0 0
\(181\) 4.56974e6i 0.770647i 0.922782 + 0.385324i \(0.125910\pi\)
−0.922782 + 0.385324i \(0.874090\pi\)
\(182\) 0 0
\(183\) 1.73235e6i 0.282672i
\(184\) 0 0
\(185\) 279832.i 0.0441960i
\(186\) 0 0
\(187\) 1.01972e6 0.155939
\(188\) 0 0
\(189\) 1.50191e6i 0.222463i
\(190\) 0 0
\(191\) 3.05071e6 0.437826 0.218913 0.975744i \(-0.429749\pi\)
0.218913 + 0.975744i \(0.429749\pi\)
\(192\) 0 0
\(193\) 6.92508e6i 0.963281i −0.876369 0.481641i \(-0.840041\pi\)
0.876369 0.481641i \(-0.159959\pi\)
\(194\) 0 0
\(195\) 617685. 0.0833035
\(196\) 0 0
\(197\) 5.70028e6 0.745585 0.372793 0.927915i \(-0.378400\pi\)
0.372793 + 0.927915i \(0.378400\pi\)
\(198\) 0 0
\(199\) −3.19083e6 −0.404897 −0.202449 0.979293i \(-0.564890\pi\)
−0.202449 + 0.979293i \(0.564890\pi\)
\(200\) 0 0
\(201\) 2.13743e6 0.263211
\(202\) 0 0
\(203\) 1.62373e6i 0.194100i
\(204\) 0 0
\(205\) 1.70523e6i 0.197935i
\(206\) 0 0
\(207\) −4.19795e6 −0.473288
\(208\) 0 0
\(209\) −791615. + 2.01959e6i −0.0867112 + 0.221220i
\(210\) 0 0
\(211\) 3.07525e6i 0.327365i −0.986513 0.163683i \(-0.947663\pi\)
0.986513 0.163683i \(-0.0523373\pi\)
\(212\) 0 0
\(213\) 2.46255e6 0.254828
\(214\) 0 0
\(215\) −710781. −0.0715188
\(216\) 0 0
\(217\) 2.76716e6i 0.270804i
\(218\) 0 0
\(219\) 4.37001e6i 0.416055i
\(220\) 0 0
\(221\) 7.92143e6i 0.733883i
\(222\) 0 0
\(223\) 1.25510e7i 1.13179i −0.824478 0.565893i \(-0.808531\pi\)
0.824478 0.565893i \(-0.191469\pi\)
\(224\) 0 0
\(225\) −8.20482e6 −0.720314
\(226\) 0 0
\(227\) 4.77970e6i 0.408623i −0.978906 0.204312i \(-0.934504\pi\)
0.978906 0.204312i \(-0.0654956\pi\)
\(228\) 0 0
\(229\) −1.74322e7 −1.45159 −0.725796 0.687910i \(-0.758529\pi\)
−0.725796 + 0.687910i \(0.758529\pi\)
\(230\) 0 0
\(231\) 375372.i 0.0304527i
\(232\) 0 0
\(233\) 1.39490e7 1.10274 0.551371 0.834260i \(-0.314105\pi\)
0.551371 + 0.834260i \(0.314105\pi\)
\(234\) 0 0
\(235\) −1.67320e6 −0.128927
\(236\) 0 0
\(237\) 6.42594e6 0.482716
\(238\) 0 0
\(239\) −1.10947e7 −0.812684 −0.406342 0.913721i \(-0.633196\pi\)
−0.406342 + 0.913721i \(0.633196\pi\)
\(240\) 0 0
\(241\) 1.24275e7i 0.887834i −0.896068 0.443917i \(-0.853589\pi\)
0.896068 0.443917i \(-0.146411\pi\)
\(242\) 0 0
\(243\) 1.48468e7i 1.03470i
\(244\) 0 0
\(245\) 1.99878e6 0.135915
\(246\) 0 0
\(247\) −1.56887e7 6.14947e6i −1.04111 0.408082i
\(248\) 0 0
\(249\) 8.34213e6i 0.540354i
\(250\) 0 0
\(251\) −1.16292e7 −0.735408 −0.367704 0.929943i \(-0.619856\pi\)
−0.367704 + 0.929943i \(0.619856\pi\)
\(252\) 0 0
\(253\) 2.47517e6 0.152842
\(254\) 0 0
\(255\) 810682.i 0.0488911i
\(256\) 0 0
\(257\) 1.30041e7i 0.766091i 0.923729 + 0.383046i \(0.125125\pi\)
−0.923729 + 0.383046i \(0.874875\pi\)
\(258\) 0 0
\(259\) 1.32104e6i 0.0760355i
\(260\) 0 0
\(261\) 1.01839e7i 0.572786i
\(262\) 0 0
\(263\) 9.63641e6 0.529722 0.264861 0.964287i \(-0.414674\pi\)
0.264861 + 0.964287i \(0.414674\pi\)
\(264\) 0 0
\(265\) 2.35607e6i 0.126605i
\(266\) 0 0
\(267\) −3.82308e6 −0.200854
\(268\) 0 0
\(269\) 1.24713e7i 0.640700i 0.947299 + 0.320350i \(0.103800\pi\)
−0.947299 + 0.320350i \(0.896200\pi\)
\(270\) 0 0
\(271\) 1.03869e7 0.521889 0.260945 0.965354i \(-0.415966\pi\)
0.260945 + 0.965354i \(0.415966\pi\)
\(272\) 0 0
\(273\) −2.91599e6 −0.143317
\(274\) 0 0
\(275\) 4.83769e6 0.232616
\(276\) 0 0
\(277\) 2.60469e7 1.22551 0.612755 0.790273i \(-0.290061\pi\)
0.612755 + 0.790273i \(0.290061\pi\)
\(278\) 0 0
\(279\) 1.73554e7i 0.799139i
\(280\) 0 0
\(281\) 1.69484e7i 0.763855i 0.924192 + 0.381927i \(0.124740\pi\)
−0.924192 + 0.381927i \(0.875260\pi\)
\(282\) 0 0
\(283\) 6.25463e6 0.275958 0.137979 0.990435i \(-0.455939\pi\)
0.137979 + 0.990435i \(0.455939\pi\)
\(284\) 0 0
\(285\) 1.60558e6 + 629339.i 0.0693582 + 0.0271863i
\(286\) 0 0
\(287\) 8.05010e6i 0.340530i
\(288\) 0 0
\(289\) −1.37411e7 −0.569282
\(290\) 0 0
\(291\) −1.18329e7 −0.480190
\(292\) 0 0
\(293\) 3.85805e7i 1.53379i 0.641775 + 0.766893i \(0.278198\pi\)
−0.641775 + 0.766893i \(0.721802\pi\)
\(294\) 0 0
\(295\) 3.26515e6i 0.127185i
\(296\) 0 0
\(297\) 5.55409e6i 0.212004i
\(298\) 0 0
\(299\) 1.92278e7i 0.719309i
\(300\) 0 0
\(301\) 3.35547e6 0.123042
\(302\) 0 0
\(303\) 9.74934e6i 0.350467i
\(304\) 0 0
\(305\) −2.26114e6 −0.0796946
\(306\) 0 0
\(307\) 1.08507e7i 0.375011i −0.982264 0.187505i \(-0.939960\pi\)
0.982264 0.187505i \(-0.0600402\pi\)
\(308\) 0 0
\(309\) 1.32641e7 0.449576
\(310\) 0 0
\(311\) 4.76110e7 1.58280 0.791400 0.611298i \(-0.209352\pi\)
0.791400 + 0.611298i \(0.209352\pi\)
\(312\) 0 0
\(313\) −1.98712e7 −0.648023 −0.324011 0.946053i \(-0.605032\pi\)
−0.324011 + 0.946053i \(0.605032\pi\)
\(314\) 0 0
\(315\) −830970. −0.0265860
\(316\) 0 0
\(317\) 5.27458e7i 1.65581i 0.560869 + 0.827904i \(0.310467\pi\)
−0.560869 + 0.827904i \(0.689533\pi\)
\(318\) 0 0
\(319\) 6.00459e6i 0.184974i
\(320\) 0 0
\(321\) 5.64268e6 0.170597
\(322\) 0 0
\(323\) −8.07088e6 + 2.05906e7i −0.239504 + 0.611029i
\(324\) 0 0
\(325\) 3.75804e7i 1.09474i
\(326\) 0 0
\(327\) −2.75636e7 −0.788303
\(328\) 0 0
\(329\) 7.89890e6 0.221809
\(330\) 0 0
\(331\) 3.89340e7i 1.07361i −0.843708 0.536803i \(-0.819632\pi\)
0.843708 0.536803i \(-0.180368\pi\)
\(332\) 0 0
\(333\) 8.28544e6i 0.224379i
\(334\) 0 0
\(335\) 2.78988e6i 0.0742080i
\(336\) 0 0
\(337\) 1.68475e7i 0.440197i 0.975478 + 0.220098i \(0.0706378\pi\)
−0.975478 + 0.220098i \(0.929362\pi\)
\(338\) 0 0
\(339\) −1.86580e7 −0.478923
\(340\) 0 0
\(341\) 1.02330e7i 0.258072i
\(342\) 0 0
\(343\) −1.94972e7 −0.483159
\(344\) 0 0
\(345\) 1.96778e6i 0.0479202i
\(346\) 0 0
\(347\) 3.85116e7 0.921729 0.460864 0.887471i \(-0.347539\pi\)
0.460864 + 0.887471i \(0.347539\pi\)
\(348\) 0 0
\(349\) 2.79730e7 0.658056 0.329028 0.944320i \(-0.393279\pi\)
0.329028 + 0.944320i \(0.393279\pi\)
\(350\) 0 0
\(351\) −4.31456e7 −0.997735
\(352\) 0 0
\(353\) 4.77207e7 1.08488 0.542441 0.840094i \(-0.317500\pi\)
0.542441 + 0.840094i \(0.317500\pi\)
\(354\) 0 0
\(355\) 3.21424e6i 0.0718445i
\(356\) 0 0
\(357\) 3.82709e6i 0.0841131i
\(358\) 0 0
\(359\) 3.95639e7 0.855098 0.427549 0.903992i \(-0.359377\pi\)
0.427549 + 0.903992i \(0.359377\pi\)
\(360\) 0 0
\(361\) −3.45149e7 3.19693e7i −0.733643 0.679535i
\(362\) 0 0
\(363\) 2.31993e7i 0.485015i
\(364\) 0 0
\(365\) −5.70395e6 −0.117300
\(366\) 0 0
\(367\) 2.36899e7 0.479254 0.239627 0.970865i \(-0.422975\pi\)
0.239627 + 0.970865i \(0.422975\pi\)
\(368\) 0 0
\(369\) 5.04895e7i 1.00490i
\(370\) 0 0
\(371\) 1.11226e7i 0.217813i
\(372\) 0 0
\(373\) 3.34668e7i 0.644893i −0.946588 0.322446i \(-0.895495\pi\)
0.946588 0.322446i \(-0.104505\pi\)
\(374\) 0 0
\(375\) 7.77449e6i 0.147427i
\(376\) 0 0
\(377\) −4.66452e7 −0.870528
\(378\) 0 0
\(379\) 1.72230e7i 0.316368i −0.987410 0.158184i \(-0.949436\pi\)
0.987410 0.158184i \(-0.0505638\pi\)
\(380\) 0 0
\(381\) −2.21882e7 −0.401187
\(382\) 0 0
\(383\) 3.87861e7i 0.690367i −0.938535 0.345184i \(-0.887817\pi\)
0.938535 0.345184i \(-0.112183\pi\)
\(384\) 0 0
\(385\) 489953. 0.00858563
\(386\) 0 0
\(387\) 2.10452e7 0.363096
\(388\) 0 0
\(389\) −6.82796e7 −1.15996 −0.579979 0.814631i \(-0.696939\pi\)
−0.579979 + 0.814631i \(0.696939\pi\)
\(390\) 0 0
\(391\) 2.52355e7 0.422165
\(392\) 0 0
\(393\) 4.21673e7i 0.694701i
\(394\) 0 0
\(395\) 8.38744e6i 0.136094i
\(396\) 0 0
\(397\) 5.29750e7 0.846641 0.423320 0.905980i \(-0.360864\pi\)
0.423320 + 0.905980i \(0.360864\pi\)
\(398\) 0 0
\(399\) −7.57968e6 2.97100e6i −0.119325 0.0467718i
\(400\) 0 0
\(401\) 8.04949e7i 1.24835i −0.781286 0.624173i \(-0.785436\pi\)
0.781286 0.624173i \(-0.214564\pi\)
\(402\) 0 0
\(403\) −7.94927e7 −1.21454
\(404\) 0 0
\(405\) −2.66792e6 −0.0401613
\(406\) 0 0
\(407\) 4.88523e6i 0.0724605i
\(408\) 0 0
\(409\) 7.86984e7i 1.15026i −0.818062 0.575130i \(-0.804951\pi\)
0.818062 0.575130i \(-0.195049\pi\)
\(410\) 0 0
\(411\) 4.33549e6i 0.0624472i
\(412\) 0 0
\(413\) 1.54142e7i 0.218812i
\(414\) 0 0
\(415\) −1.08885e7 −0.152344
\(416\) 0 0
\(417\) 5.32850e6i 0.0734847i
\(418\) 0 0
\(419\) −1.33352e8 −1.81283 −0.906415 0.422389i \(-0.861192\pi\)
−0.906415 + 0.422389i \(0.861192\pi\)
\(420\) 0 0
\(421\) 8.03244e7i 1.07647i 0.842795 + 0.538234i \(0.180908\pi\)
−0.842795 + 0.538234i \(0.819092\pi\)
\(422\) 0 0
\(423\) 4.95412e7 0.654554
\(424\) 0 0
\(425\) 4.93225e7 0.642507
\(426\) 0 0
\(427\) 1.06745e7 0.137108
\(428\) 0 0
\(429\) 1.07834e7 0.136579
\(430\) 0 0
\(431\) 8.75808e7i 1.09390i 0.837166 + 0.546950i \(0.184211\pi\)
−0.837166 + 0.546950i \(0.815789\pi\)
\(432\) 0 0
\(433\) 1.28661e8i 1.58483i −0.609983 0.792414i \(-0.708824\pi\)
0.609983 0.792414i \(-0.291176\pi\)
\(434\) 0 0
\(435\) 4.77368e6 0.0579943
\(436\) 0 0
\(437\) −1.95905e7 + 4.99798e7i −0.234748 + 0.598894i
\(438\) 0 0
\(439\) 2.59660e7i 0.306911i −0.988156 0.153455i \(-0.950960\pi\)
0.988156 0.153455i \(-0.0490401\pi\)
\(440\) 0 0
\(441\) −5.91810e7 −0.690028
\(442\) 0 0
\(443\) −3.50548e6 −0.0403215 −0.0201607 0.999797i \(-0.506418\pi\)
−0.0201607 + 0.999797i \(0.506418\pi\)
\(444\) 0 0
\(445\) 4.99007e6i 0.0566274i
\(446\) 0 0
\(447\) 1.21360e7i 0.135880i
\(448\) 0 0
\(449\) 8.29521e7i 0.916407i −0.888847 0.458203i \(-0.848493\pi\)
0.888847 0.458203i \(-0.151507\pi\)
\(450\) 0 0
\(451\) 2.97694e7i 0.324519i
\(452\) 0 0
\(453\) 2.88953e7 0.310837
\(454\) 0 0
\(455\) 3.80608e6i 0.0404058i
\(456\) 0 0
\(457\) −1.51174e8 −1.58390 −0.791951 0.610584i \(-0.790935\pi\)
−0.791951 + 0.610584i \(0.790935\pi\)
\(458\) 0 0
\(459\) 5.66265e7i 0.585574i
\(460\) 0 0
\(461\) 1.72984e8 1.76565 0.882824 0.469705i \(-0.155640\pi\)
0.882824 + 0.469705i \(0.155640\pi\)
\(462\) 0 0
\(463\) 1.27212e8 1.28169 0.640847 0.767668i \(-0.278583\pi\)
0.640847 + 0.767668i \(0.278583\pi\)
\(464\) 0 0
\(465\) 8.13531e6 0.0809124
\(466\) 0 0
\(467\) 1.69010e8 1.65944 0.829722 0.558177i \(-0.188499\pi\)
0.829722 + 0.558177i \(0.188499\pi\)
\(468\) 0 0
\(469\) 1.31705e7i 0.127669i
\(470\) 0 0
\(471\) 4.99558e6i 0.0478105i
\(472\) 0 0
\(473\) −1.24086e7 −0.117257
\(474\) 0 0
\(475\) −3.82894e7 + 9.76848e7i −0.357271 + 0.911478i
\(476\) 0 0
\(477\) 6.97600e7i 0.642764i
\(478\) 0 0
\(479\) −8.22265e7 −0.748179 −0.374089 0.927393i \(-0.622045\pi\)
−0.374089 + 0.927393i \(0.622045\pi\)
\(480\) 0 0
\(481\) −3.79497e7 −0.341015
\(482\) 0 0
\(483\) 9.28953e6i 0.0824428i
\(484\) 0 0
\(485\) 1.54449e7i 0.135381i
\(486\) 0 0
\(487\) 2.07542e8i 1.79688i 0.439100 + 0.898438i \(0.355297\pi\)
−0.439100 + 0.898438i \(0.644703\pi\)
\(488\) 0 0
\(489\) 4.12039e7i 0.352380i
\(490\) 0 0
\(491\) −2.56953e6 −0.0217074 −0.0108537 0.999941i \(-0.503455\pi\)
−0.0108537 + 0.999941i \(0.503455\pi\)
\(492\) 0 0
\(493\) 6.12195e7i 0.510915i
\(494\) 0 0
\(495\) 3.07294e6 0.0253360
\(496\) 0 0
\(497\) 1.51739e7i 0.123602i
\(498\) 0 0
\(499\) 2.18130e8 1.75555 0.877774 0.479074i \(-0.159028\pi\)
0.877774 + 0.479074i \(0.159028\pi\)
\(500\) 0 0
\(501\) 1.21591e8 0.966915
\(502\) 0 0
\(503\) −1.23404e8 −0.969672 −0.484836 0.874605i \(-0.661121\pi\)
−0.484836 + 0.874605i \(0.661121\pi\)
\(504\) 0 0
\(505\) 1.27253e7 0.0988084
\(506\) 0 0
\(507\) 1.67768e7i 0.128732i
\(508\) 0 0
\(509\) 1.41684e7i 0.107441i 0.998556 + 0.0537203i \(0.0171079\pi\)
−0.998556 + 0.0537203i \(0.982892\pi\)
\(510\) 0 0
\(511\) 2.69273e7 0.201804
\(512\) 0 0
\(513\) −1.12151e8 4.39596e7i −0.830711 0.325613i
\(514\) 0 0
\(515\) 1.73130e7i 0.126750i
\(516\) 0 0
\(517\) −2.92103e7 −0.211380
\(518\) 0 0
\(519\) 6.66228e7 0.476563
\(520\) 0 0
\(521\) 5.98342e7i 0.423094i −0.977368 0.211547i \(-0.932150\pi\)
0.977368 0.211547i \(-0.0678501\pi\)
\(522\) 0 0
\(523\) 5.41820e7i 0.378747i 0.981905 + 0.189374i \(0.0606458\pi\)
−0.981905 + 0.189374i \(0.939354\pi\)
\(524\) 0 0
\(525\) 1.81563e7i 0.125472i
\(526\) 0 0
\(527\) 1.04330e8i 0.712818i
\(528\) 0 0
\(529\) −8.67814e7 −0.586219
\(530\) 0 0
\(531\) 9.66766e7i 0.645711i
\(532\) 0 0
\(533\) −2.31256e8 −1.52726
\(534\) 0 0
\(535\) 7.36509e6i 0.0480969i
\(536\) 0 0
\(537\) 1.20127e8 0.775742
\(538\) 0 0
\(539\) 3.48940e7 0.222836
\(540\) 0 0
\(541\) −6.50918e7 −0.411088 −0.205544 0.978648i \(-0.565896\pi\)
−0.205544 + 0.978648i \(0.565896\pi\)
\(542\) 0 0
\(543\) 6.34233e7 0.396140
\(544\) 0 0
\(545\) 3.59774e7i 0.222249i
\(546\) 0 0
\(547\) 3.47709e7i 0.212449i 0.994342 + 0.106224i \(0.0338762\pi\)
−0.994342 + 0.106224i \(0.966124\pi\)
\(548\) 0 0
\(549\) 6.69494e7 0.404603
\(550\) 0 0
\(551\) −1.21247e8 4.75252e7i −0.724799 0.284099i
\(552\) 0 0
\(553\) 3.95956e7i 0.234138i
\(554\) 0 0
\(555\) 3.88378e6 0.0227183
\(556\) 0 0
\(557\) 4.97836e7 0.288085 0.144043 0.989571i \(-0.453990\pi\)
0.144043 + 0.989571i \(0.453990\pi\)
\(558\) 0 0
\(559\) 9.63932e7i 0.551837i
\(560\) 0 0
\(561\) 1.41526e7i 0.0801584i
\(562\) 0 0
\(563\) 2.74472e8i 1.53806i −0.639215 0.769028i \(-0.720740\pi\)
0.639215 0.769028i \(-0.279260\pi\)
\(564\) 0 0
\(565\) 2.43533e7i 0.135024i
\(566\) 0 0
\(567\) 1.25948e7 0.0690943
\(568\) 0 0
\(569\) 2.65773e8i 1.44270i −0.692573 0.721348i \(-0.743523\pi\)
0.692573 0.721348i \(-0.256477\pi\)
\(570\) 0 0
\(571\) −2.28689e8 −1.22839 −0.614197 0.789153i \(-0.710520\pi\)
−0.614197 + 0.789153i \(0.710520\pi\)
\(572\) 0 0
\(573\) 4.23408e7i 0.225058i
\(574\) 0 0
\(575\) 1.19721e8 0.629748
\(576\) 0 0
\(577\) 3.19929e8 1.66543 0.832715 0.553701i \(-0.186785\pi\)
0.832715 + 0.553701i \(0.186785\pi\)
\(578\) 0 0
\(579\) −9.61130e7 −0.495161
\(580\) 0 0
\(581\) 5.14029e7 0.262095
\(582\) 0 0
\(583\) 4.11316e7i 0.207572i
\(584\) 0 0
\(585\) 2.38714e7i 0.119237i
\(586\) 0 0
\(587\) 2.80875e7 0.138867 0.0694334 0.997587i \(-0.477881\pi\)
0.0694334 + 0.997587i \(0.477881\pi\)
\(588\) 0 0
\(589\) −2.06630e8 8.09924e7i −1.01122 0.396368i
\(590\) 0 0
\(591\) 7.91140e7i 0.383258i
\(592\) 0 0
\(593\) −3.59440e8 −1.72370 −0.861850 0.507163i \(-0.830694\pi\)
−0.861850 + 0.507163i \(0.830694\pi\)
\(594\) 0 0
\(595\) 4.99529e6 0.0237143
\(596\) 0 0
\(597\) 4.42855e7i 0.208132i
\(598\) 0 0
\(599\) 3.80247e8i 1.76923i −0.466318 0.884617i \(-0.654420\pi\)
0.466318 0.884617i \(-0.345580\pi\)
\(600\) 0 0
\(601\) 1.65922e8i 0.764329i −0.924094 0.382165i \(-0.875179\pi\)
0.924094 0.382165i \(-0.124821\pi\)
\(602\) 0 0
\(603\) 8.26044e7i 0.376748i
\(604\) 0 0
\(605\) 3.02808e7 0.136742
\(606\) 0 0
\(607\) 1.16620e8i 0.521441i 0.965414 + 0.260721i \(0.0839601\pi\)
−0.965414 + 0.260721i \(0.916040\pi\)
\(608\) 0 0
\(609\) −2.25357e7 −0.0997745
\(610\) 0 0
\(611\) 2.26913e8i 0.994800i
\(612\) 0 0
\(613\) 2.26238e8 0.982166 0.491083 0.871113i \(-0.336601\pi\)
0.491083 + 0.871113i \(0.336601\pi\)
\(614\) 0 0
\(615\) 2.36668e7 0.101745
\(616\) 0 0
\(617\) −2.64442e8 −1.12584 −0.562918 0.826513i \(-0.690321\pi\)
−0.562918 + 0.826513i \(0.690321\pi\)
\(618\) 0 0
\(619\) 3.93952e8 1.66101 0.830504 0.557013i \(-0.188053\pi\)
0.830504 + 0.557013i \(0.188053\pi\)
\(620\) 0 0
\(621\) 1.37450e8i 0.573945i
\(622\) 0 0
\(623\) 2.35572e7i 0.0974227i
\(624\) 0 0
\(625\) 2.28865e8 0.937432
\(626\) 0 0
\(627\) 2.80298e7 + 1.09868e7i 0.113715 + 0.0445727i
\(628\) 0 0
\(629\) 4.98071e7i 0.200143i
\(630\) 0 0
\(631\) −4.29945e8 −1.71129 −0.855647 0.517559i \(-0.826841\pi\)
−0.855647 + 0.517559i \(0.826841\pi\)
\(632\) 0 0
\(633\) −4.26813e7 −0.168278
\(634\) 0 0
\(635\) 2.89610e7i 0.113108i
\(636\) 0 0
\(637\) 2.71066e8i 1.04871i
\(638\) 0 0
\(639\) 9.51692e7i 0.364749i
\(640\) 0 0
\(641\) 3.42337e8i 1.29981i −0.760015 0.649906i \(-0.774808\pi\)
0.760015 0.649906i \(-0.225192\pi\)
\(642\) 0 0
\(643\) −3.03299e8 −1.14087 −0.570437 0.821342i \(-0.693226\pi\)
−0.570437 + 0.821342i \(0.693226\pi\)
\(644\) 0 0
\(645\) 9.86491e6i 0.0367632i
\(646\) 0 0
\(647\) −3.03991e8 −1.12240 −0.561199 0.827681i \(-0.689660\pi\)
−0.561199 + 0.827681i \(0.689660\pi\)
\(648\) 0 0
\(649\) 5.70020e7i 0.208524i
\(650\) 0 0
\(651\) −3.84054e7 −0.139203
\(652\) 0 0
\(653\) 2.88590e8 1.03643 0.518217 0.855249i \(-0.326596\pi\)
0.518217 + 0.855249i \(0.326596\pi\)
\(654\) 0 0
\(655\) −5.50387e7 −0.195859
\(656\) 0 0
\(657\) 1.68886e8 0.595521
\(658\) 0 0
\(659\) 5.42749e8i 1.89646i −0.317591 0.948228i \(-0.602874\pi\)
0.317591 0.948228i \(-0.397126\pi\)
\(660\) 0 0
\(661\) 2.52061e8i 0.872774i 0.899759 + 0.436387i \(0.143742\pi\)
−0.899759 + 0.436387i \(0.856258\pi\)
\(662\) 0 0
\(663\) 1.09941e8 0.377242
\(664\) 0 0
\(665\) −3.87789e6 + 9.89335e6i −0.0131865 + 0.0336417i
\(666\) 0 0
\(667\) 1.48599e8i 0.500769i
\(668\) 0 0
\(669\) −1.74195e8 −0.581779
\(670\) 0 0
\(671\) −3.94744e7 −0.130662
\(672\) 0 0
\(673\) 1.26000e8i 0.413356i 0.978409 + 0.206678i \(0.0662652\pi\)
−0.978409 + 0.206678i \(0.933735\pi\)
\(674\) 0 0
\(675\) 2.68644e8i 0.873507i
\(676\) 0 0
\(677\) 5.11022e8i 1.64692i 0.567371 + 0.823462i \(0.307961\pi\)
−0.567371 + 0.823462i \(0.692039\pi\)
\(678\) 0 0
\(679\) 7.29126e7i 0.232913i
\(680\) 0 0
\(681\) −6.63373e7 −0.210047
\(682\) 0 0
\(683\) 5.66185e7i 0.177704i −0.996045 0.0888518i \(-0.971680\pi\)
0.996045 0.0888518i \(-0.0283197\pi\)
\(684\) 0 0
\(685\) 5.65888e6 0.0176059
\(686\) 0 0
\(687\) 2.41940e8i 0.746171i
\(688\) 0 0
\(689\) 3.19521e8 0.976880
\(690\) 0 0
\(691\) 4.08142e8 1.23702 0.618511 0.785776i \(-0.287736\pi\)
0.618511 + 0.785776i \(0.287736\pi\)
\(692\) 0 0
\(693\) −1.45068e7 −0.0435886
\(694\) 0 0
\(695\) −6.95501e6 −0.0207178
\(696\) 0 0
\(697\) 3.03513e8i 0.896352i
\(698\) 0 0
\(699\) 1.93597e8i 0.566849i
\(700\) 0 0
\(701\) 4.90639e8 1.42432 0.712161 0.702016i \(-0.247717\pi\)
0.712161 + 0.702016i \(0.247717\pi\)
\(702\) 0 0
\(703\) −9.86447e7 3.86657e7i −0.283928 0.111291i
\(704\) 0 0
\(705\) 2.32223e7i 0.0662733i
\(706\) 0 0
\(707\) −6.00739e7 −0.169992
\(708\) 0 0
\(709\) 6.65761e6 0.0186801 0.00934006 0.999956i \(-0.497027\pi\)
0.00934006 + 0.999956i \(0.497027\pi\)
\(710\) 0 0
\(711\) 2.48340e8i 0.690937i
\(712\) 0 0
\(713\) 2.53242e8i 0.698662i
\(714\) 0 0
\(715\) 1.40749e7i 0.0385060i
\(716\) 0 0
\(717\) 1.53983e8i 0.417749i
\(718\) 0 0
\(719\) −2.02239e8 −0.544098 −0.272049 0.962283i \(-0.587701\pi\)
−0.272049 + 0.962283i \(0.587701\pi\)
\(720\) 0 0
\(721\) 8.17314e7i 0.218064i
\(722\) 0 0
\(723\) −1.72480e8 −0.456378
\(724\) 0 0
\(725\) 2.90434e8i 0.762138i
\(726\) 0 0
\(727\) 6.15946e8 1.60302 0.801511 0.597980i \(-0.204030\pi\)
0.801511 + 0.597980i \(0.204030\pi\)
\(728\) 0 0
\(729\) −9.86957e7 −0.254751
\(730\) 0 0
\(731\) −1.26511e8 −0.323875
\(732\) 0 0
\(733\) 2.80626e8 0.712551 0.356275 0.934381i \(-0.384046\pi\)
0.356275 + 0.934381i \(0.384046\pi\)
\(734\) 0 0
\(735\) 2.77410e7i 0.0698650i
\(736\) 0 0
\(737\) 4.87048e7i 0.121666i
\(738\) 0 0
\(739\) −1.30355e8 −0.322993 −0.161496 0.986873i \(-0.551632\pi\)
−0.161496 + 0.986873i \(0.551632\pi\)
\(740\) 0 0
\(741\) −8.53483e7 + 2.17743e8i −0.209769 + 0.535166i
\(742\) 0 0
\(743\) 4.17618e8i 1.01815i −0.860721 0.509076i \(-0.829987\pi\)
0.860721 0.509076i \(-0.170013\pi\)
\(744\) 0 0
\(745\) 1.58405e7 0.0383090
\(746\) 0 0
\(747\) 3.22394e8 0.773438
\(748\) 0 0
\(749\) 3.47693e7i 0.0827467i
\(750\) 0 0
\(751\) 1.16726e8i 0.275579i −0.990462 0.137790i \(-0.956000\pi\)
0.990462 0.137790i \(-0.0439998\pi\)
\(752\) 0 0
\(753\) 1.61401e8i 0.378026i
\(754\) 0 0
\(755\) 3.77155e7i 0.0876352i
\(756\) 0 0
\(757\) 5.08428e7 0.117204 0.0586019 0.998281i \(-0.481336\pi\)
0.0586019 + 0.998281i \(0.481336\pi\)
\(758\) 0 0
\(759\) 3.43529e7i 0.0785665i
\(760\) 0 0
\(761\) −3.41887e8 −0.775763 −0.387881 0.921709i \(-0.626793\pi\)
−0.387881 + 0.921709i \(0.626793\pi\)
\(762\) 0 0
\(763\) 1.69843e8i 0.382361i
\(764\) 0 0
\(765\) 3.13300e7 0.0699805
\(766\) 0 0
\(767\) 4.42806e8 0.981359
\(768\) 0 0
\(769\) −4.63630e8 −1.01951 −0.509757 0.860319i \(-0.670265\pi\)
−0.509757 + 0.860319i \(0.670265\pi\)
\(770\) 0 0
\(771\) 1.80483e8 0.393799
\(772\) 0 0
\(773\) 2.27556e8i 0.492663i −0.969186 0.246332i \(-0.920775\pi\)
0.969186 0.246332i \(-0.0792253\pi\)
\(774\) 0 0
\(775\) 4.94958e8i 1.06332i
\(776\) 0 0
\(777\) −1.83347e7 −0.0390850
\(778\) 0 0
\(779\) −6.01117e8 2.35619e8i −1.27159 0.498423i
\(780\) 0 0
\(781\) 5.61132e7i 0.117791i
\(782\) 0 0
\(783\) −3.33444e8 −0.694604
\(784\) 0 0
\(785\) 6.52046e6 0.0134794
\(786\) 0 0
\(787\) 5.56749e8i 1.14218i 0.820887 + 0.571091i \(0.193480\pi\)
−0.820887 + 0.571091i \(0.806520\pi\)
\(788\) 0 0
\(789\) 1.33743e8i 0.272296i
\(790\) 0 0
\(791\) 1.14968e8i 0.232298i
\(792\) 0 0
\(793\) 3.06647e8i 0.614921i
\(794\) 0 0
\(795\) −3.26998e7 −0.0650795
\(796\) 0