Properties

Label 4680.2.l.e.2809.3
Level $4680$
Weight $2$
Character 4680.2809
Analytic conductor $37.370$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4680.l (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(37.3699881460\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5161984.1
Defining polynomial: \(x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1560)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2809.3
Root \(0.432320 + 0.432320i\) of defining polynomial
Character \(\chi\) \(=\) 4680.2809
Dual form 4680.2.l.e.2809.4

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.432320 - 2.19388i) q^{5} +O(q^{10})\) \(q+(-0.432320 - 2.19388i) q^{5} +2.86464 q^{11} +1.00000i q^{13} -5.52311i q^{17} -3.52311 q^{19} +7.52311i q^{23} +(-4.62620 + 1.89692i) q^{25} +6.77551 q^{29} +5.72928 q^{31} -3.72928i q^{37} +10.1170 q^{41} +5.52311i q^{43} -8.65847i q^{47} +7.00000 q^{49} -6.77551i q^{53} +(-1.23844 - 6.28467i) q^{55} +0.593923 q^{59} -5.25240 q^{61} +(2.19388 - 0.432320i) q^{65} -10.5048i q^{67} +2.38776 q^{71} -5.45856i q^{73} -2.47689 q^{79} +8.11704i q^{83} +(-12.1170 + 2.38776i) q^{85} -14.1170 q^{89} +(1.52311 + 7.72928i) q^{95} +6.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + O(q^{10}) \) \( 6 q + 12 q^{11} + 4 q^{19} - 10 q^{25} - 20 q^{29} + 24 q^{31} + 20 q^{41} + 42 q^{49} - 20 q^{55} - 12 q^{59} + 4 q^{61} - 2 q^{65} - 16 q^{71} - 40 q^{79} - 32 q^{85} - 44 q^{89} - 16 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4680\mathbb{Z}\right)^\times\).

\(n\) \(937\) \(1081\) \(2081\) \(2341\) \(3511\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −0.432320 2.19388i −0.193340 0.981132i
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.86464 0.863722 0.431861 0.901940i \(-0.357857\pi\)
0.431861 + 0.901940i \(0.357857\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.52311i 1.33955i −0.742563 0.669776i \(-0.766390\pi\)
0.742563 0.669776i \(-0.233610\pi\)
\(18\) 0 0
\(19\) −3.52311 −0.808258 −0.404129 0.914702i \(-0.632425\pi\)
−0.404129 + 0.914702i \(0.632425\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.52311i 1.56868i 0.620333 + 0.784339i \(0.286998\pi\)
−0.620333 + 0.784339i \(0.713002\pi\)
\(24\) 0 0
\(25\) −4.62620 + 1.89692i −0.925240 + 0.379383i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.77551 1.25818 0.629090 0.777332i \(-0.283428\pi\)
0.629090 + 0.777332i \(0.283428\pi\)
\(30\) 0 0
\(31\) 5.72928 1.02901 0.514505 0.857488i \(-0.327976\pi\)
0.514505 + 0.857488i \(0.327976\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.72928i 0.613090i −0.951856 0.306545i \(-0.900827\pi\)
0.951856 0.306545i \(-0.0991730\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.1170 1.58002 0.790008 0.613097i \(-0.210076\pi\)
0.790008 + 0.613097i \(0.210076\pi\)
\(42\) 0 0
\(43\) 5.52311i 0.842267i 0.906999 + 0.421134i \(0.138368\pi\)
−0.906999 + 0.421134i \(0.861632\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.65847i 1.26297i −0.775389 0.631484i \(-0.782446\pi\)
0.775389 0.631484i \(-0.217554\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.77551i 0.930688i −0.885130 0.465344i \(-0.845931\pi\)
0.885130 0.465344i \(-0.154069\pi\)
\(54\) 0 0
\(55\) −1.23844 6.28467i −0.166992 0.847425i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.593923 0.0773221 0.0386611 0.999252i \(-0.487691\pi\)
0.0386611 + 0.999252i \(0.487691\pi\)
\(60\) 0 0
\(61\) −5.25240 −0.672500 −0.336250 0.941773i \(-0.609159\pi\)
−0.336250 + 0.941773i \(0.609159\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.19388 0.432320i 0.272117 0.0536228i
\(66\) 0 0
\(67\) 10.5048i 1.28336i −0.766971 0.641682i \(-0.778237\pi\)
0.766971 0.641682i \(-0.221763\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.38776 0.283374 0.141687 0.989911i \(-0.454747\pi\)
0.141687 + 0.989911i \(0.454747\pi\)
\(72\) 0 0
\(73\) 5.45856i 0.638877i −0.947607 0.319438i \(-0.896506\pi\)
0.947607 0.319438i \(-0.103494\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −2.47689 −0.278671 −0.139336 0.990245i \(-0.544497\pi\)
−0.139336 + 0.990245i \(0.544497\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.11704i 0.890961i 0.895292 + 0.445480i \(0.146967\pi\)
−0.895292 + 0.445480i \(0.853033\pi\)
\(84\) 0 0
\(85\) −12.1170 + 2.38776i −1.31428 + 0.258988i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −14.1170 −1.49640 −0.748201 0.663472i \(-0.769082\pi\)
−0.748201 + 0.663472i \(0.769082\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.52311 + 7.72928i 0.156268 + 0.793008i
\(96\) 0 0
\(97\) 6.00000i 0.609208i 0.952479 + 0.304604i \(0.0985241\pi\)
−0.952479 + 0.304604i \(0.901476\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 11.7293 1.16711 0.583554 0.812075i \(-0.301662\pi\)
0.583554 + 0.812075i \(0.301662\pi\)
\(102\) 0 0
\(103\) 14.7755i 1.45587i −0.685644 0.727937i \(-0.740479\pi\)
0.685644 0.727937i \(-0.259521\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.72928i 0.167176i −0.996500 0.0835880i \(-0.973362\pi\)
0.996500 0.0835880i \(-0.0266380\pi\)
\(108\) 0 0
\(109\) −12.0279 −1.15206 −0.576032 0.817427i \(-0.695400\pi\)
−0.576032 + 0.817427i \(0.695400\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 7.25240i 0.682248i −0.940018 0.341124i \(-0.889192\pi\)
0.940018 0.341124i \(-0.110808\pi\)
\(114\) 0 0
\(115\) 16.5048 3.25240i 1.53908 0.303288i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.79383 −0.253985
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 6.16160 + 9.32924i 0.551110 + 0.834432i
\(126\) 0 0
\(127\) 17.2524i 1.53090i 0.643494 + 0.765451i \(0.277484\pi\)
−0.643494 + 0.765451i \(0.722516\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 8.77551 0.766720 0.383360 0.923599i \(-0.374767\pi\)
0.383360 + 0.923599i \(0.374767\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 19.4340i 1.66036i −0.557497 0.830179i \(-0.688238\pi\)
0.557497 0.830179i \(-0.311762\pi\)
\(138\) 0 0
\(139\) −7.58767 −0.643577 −0.321789 0.946812i \(-0.604284\pi\)
−0.321789 + 0.946812i \(0.604284\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.86464i 0.239553i
\(144\) 0 0
\(145\) −2.92919 14.8646i −0.243256 1.23444i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.40608 −0.115190 −0.0575952 0.998340i \(-0.518343\pi\)
−0.0575952 + 0.998340i \(0.518343\pi\)
\(150\) 0 0
\(151\) −1.31695 −0.107172 −0.0535858 0.998563i \(-0.517065\pi\)
−0.0535858 + 0.998563i \(0.517065\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.47689 12.5693i −0.198948 1.00959i
\(156\) 0 0
\(157\) 20.7110i 1.65291i 0.562999 + 0.826457i \(0.309647\pi\)
−0.562999 + 0.826457i \(0.690353\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.27072i 0.177856i −0.996038 0.0889282i \(-0.971656\pi\)
0.996038 0.0889282i \(-0.0283442\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.34153i 0.568104i −0.958809 0.284052i \(-0.908321\pi\)
0.958809 0.284052i \(-0.0916789\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 16.0925i 1.22349i −0.791056 0.611743i \(-0.790468\pi\)
0.791056 0.611743i \(-0.209532\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 15.0462 1.12461 0.562304 0.826931i \(-0.309915\pi\)
0.562304 + 0.826931i \(0.309915\pi\)
\(180\) 0 0
\(181\) 5.58767 0.415328 0.207664 0.978200i \(-0.433414\pi\)
0.207664 + 0.978200i \(0.433414\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −8.18159 + 1.61224i −0.601522 + 0.118535i
\(186\) 0 0
\(187\) 15.8217i 1.15700i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −17.5510 −1.26995 −0.634974 0.772534i \(-0.718989\pi\)
−0.634974 + 0.772534i \(0.718989\pi\)
\(192\) 0 0
\(193\) 13.8217i 0.994911i −0.867490 0.497455i \(-0.834268\pi\)
0.867490 0.497455i \(-0.165732\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 26.3511i 1.87744i −0.344682 0.938719i \(-0.612013\pi\)
0.344682 0.938719i \(-0.387987\pi\)
\(198\) 0 0
\(199\) 23.0741 1.63568 0.817841 0.575444i \(-0.195171\pi\)
0.817841 + 0.575444i \(0.195171\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −4.37380 22.1955i −0.305480 1.55020i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −10.0925 −0.698110
\(210\) 0 0
\(211\) 1.52311 0.104856 0.0524278 0.998625i \(-0.483304\pi\)
0.0524278 + 0.998625i \(0.483304\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 12.1170 2.38776i 0.826375 0.162844i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 5.52311 0.371525
\(222\) 0 0
\(223\) 13.3169i 0.891769i −0.895091 0.445884i \(-0.852889\pi\)
0.895091 0.445884i \(-0.147111\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6.80009i 0.451338i 0.974204 + 0.225669i \(0.0724568\pi\)
−0.974204 + 0.225669i \(0.927543\pi\)
\(228\) 0 0
\(229\) −1.52311 −0.100650 −0.0503251 0.998733i \(-0.516026\pi\)
−0.0503251 + 0.998733i \(0.516026\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.7110i 0.701698i −0.936432 0.350849i \(-0.885893\pi\)
0.936432 0.350849i \(-0.114107\pi\)
\(234\) 0 0
\(235\) −18.9956 + 3.74324i −1.23914 + 0.244182i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 4.65847 0.301332 0.150666 0.988585i \(-0.451858\pi\)
0.150666 + 0.988585i \(0.451858\pi\)
\(240\) 0 0
\(241\) 5.22449 0.336539 0.168269 0.985741i \(-0.446182\pi\)
0.168269 + 0.985741i \(0.446182\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.02624 15.3571i −0.193340 0.981132i
\(246\) 0 0
\(247\) 3.52311i 0.224170i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −5.85838 −0.369778 −0.184889 0.982759i \(-0.559193\pi\)
−0.184889 + 0.982759i \(0.559193\pi\)
\(252\) 0 0
\(253\) 21.5510i 1.35490i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.29862i 0.143384i 0.997427 + 0.0716921i \(0.0228399\pi\)
−0.997427 + 0.0716921i \(0.977160\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 31.5789i 1.94724i −0.228175 0.973620i \(-0.573276\pi\)
0.228175 0.973620i \(-0.426724\pi\)
\(264\) 0 0
\(265\) −14.8646 + 2.92919i −0.913128 + 0.179939i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −30.3632 −1.85128 −0.925638 0.378411i \(-0.876471\pi\)
−0.925638 + 0.378411i \(0.876471\pi\)
\(270\) 0 0
\(271\) 28.5972 1.73716 0.868580 0.495550i \(-0.165033\pi\)
0.868580 + 0.495550i \(0.165033\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −13.2524 + 5.43398i −0.799150 + 0.327682i
\(276\) 0 0
\(277\) 1.79383i 0.107781i 0.998547 + 0.0538905i \(0.0171622\pi\)
−0.998547 + 0.0538905i \(0.982838\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −32.2095 −1.92146 −0.960729 0.277489i \(-0.910498\pi\)
−0.960729 + 0.277489i \(0.910498\pi\)
\(282\) 0 0
\(283\) 18.0925i 1.07548i −0.843109 0.537742i \(-0.819277\pi\)
0.843109 0.537742i \(-0.180723\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −13.5048 −0.794400
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 12.6218i 0.737375i −0.929553 0.368688i \(-0.879807\pi\)
0.929553 0.368688i \(-0.120193\pi\)
\(294\) 0 0
\(295\) −0.256765 1.30299i −0.0149494 0.0758632i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −7.52311 −0.435073
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.27072 + 11.5231i 0.130021 + 0.659812i
\(306\) 0 0
\(307\) 20.0000i 1.14146i 0.821138 + 0.570730i \(0.193340\pi\)
−0.821138 + 0.570730i \(0.806660\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 31.8217 1.80445 0.902223 0.431271i \(-0.141935\pi\)
0.902223 + 0.431271i \(0.141935\pi\)
\(312\) 0 0
\(313\) 11.7938i 0.666627i −0.942816 0.333313i \(-0.891833\pi\)
0.942816 0.333313i \(-0.108167\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 11.4340i 0.642197i −0.947046 0.321098i \(-0.895948\pi\)
0.947046 0.321098i \(-0.104052\pi\)
\(318\) 0 0
\(319\) 19.4094 1.08672
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 19.4586i 1.08270i
\(324\) 0 0
\(325\) −1.89692 4.62620i −0.105222 0.256615i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 2.20617 0.121262 0.0606310 0.998160i \(-0.480689\pi\)
0.0606310 + 0.998160i \(0.480689\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −23.0462 + 4.54144i −1.25915 + 0.248125i
\(336\) 0 0
\(337\) 16.8034i 0.915340i −0.889122 0.457670i \(-0.848684\pi\)
0.889122 0.457670i \(-0.151316\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 16.4123 0.888778
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 9.96336i 0.534861i −0.963577 0.267430i \(-0.913825\pi\)
0.963577 0.267430i \(-0.0861746\pi\)
\(348\) 0 0
\(349\) 21.3449 1.14256 0.571282 0.820754i \(-0.306446\pi\)
0.571282 + 0.820754i \(0.306446\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 33.3973i 1.77756i 0.458333 + 0.888781i \(0.348447\pi\)
−0.458333 + 0.888781i \(0.651553\pi\)
\(354\) 0 0
\(355\) −1.03228 5.23844i −0.0547875 0.278028i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 21.9754 1.15982 0.579909 0.814681i \(-0.303088\pi\)
0.579909 + 0.814681i \(0.303088\pi\)
\(360\) 0 0
\(361\) −6.58767 −0.346719
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −11.9754 + 2.35985i −0.626822 + 0.123520i
\(366\) 0 0
\(367\) 29.2803i 1.52842i 0.644968 + 0.764210i \(0.276871\pi\)
−0.644968 + 0.764210i \(0.723129\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 2.41233i 0.124906i −0.998048 0.0624530i \(-0.980108\pi\)
0.998048 0.0624530i \(-0.0198923\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.77551i 0.348957i
\(378\) 0 0
\(379\) −19.5231 −1.00284 −0.501418 0.865205i \(-0.667188\pi\)
−0.501418 + 0.865205i \(0.667188\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 23.7047i 1.21125i −0.795749 0.605627i \(-0.792922\pi\)
0.795749 0.605627i \(-0.207078\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) 0 0
\(391\) 41.5510 2.10133
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1.07081 + 5.43398i 0.0538782 + 0.273413i
\(396\) 0 0
\(397\) 28.8680i 1.44884i 0.689358 + 0.724421i \(0.257893\pi\)
−0.689358 + 0.724421i \(0.742107\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −20.0804 −1.00277 −0.501383 0.865225i \(-0.667175\pi\)
−0.501383 + 0.865225i \(0.667175\pi\)
\(402\) 0 0
\(403\) 5.72928i 0.285396i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 10.6831i 0.529539i
\(408\) 0 0
\(409\) 13.6926 0.677057 0.338529 0.940956i \(-0.390071\pi\)
0.338529 + 0.940956i \(0.390071\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 17.8078 3.50916i 0.874150 0.172258i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −16.3632 −0.799393 −0.399697 0.916647i \(-0.630885\pi\)
−0.399697 + 0.916647i \(0.630885\pi\)
\(420\) 0 0
\(421\) 24.9817 1.21753 0.608766 0.793350i \(-0.291665\pi\)
0.608766 + 0.793350i \(0.291665\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 10.4769 + 25.5510i 0.508204 + 1.23941i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 27.9388 1.34576 0.672882 0.739750i \(-0.265056\pi\)
0.672882 + 0.739750i \(0.265056\pi\)
\(432\) 0 0
\(433\) 26.7110i 1.28365i 0.766852 + 0.641823i \(0.221822\pi\)
−0.766852 + 0.641823i \(0.778178\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 26.5048i 1.26790i
\(438\) 0 0
\(439\) 5.00958 0.239094 0.119547 0.992829i \(-0.461856\pi\)
0.119547 + 0.992829i \(0.461856\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 19.6926i 0.935625i −0.883828 0.467813i \(-0.845042\pi\)
0.883828 0.467813i \(-0.154958\pi\)
\(444\) 0 0
\(445\) 6.10308 + 30.9711i 0.289314 + 1.46817i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 38.7143 1.82704 0.913520 0.406794i \(-0.133353\pi\)
0.913520 + 0.406794i \(0.133353\pi\)
\(450\) 0 0
\(451\) 28.9817 1.36469
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 10.9538i 0.512396i 0.966624 + 0.256198i \(0.0824699\pi\)
−0.966624 + 0.256198i \(0.917530\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0.864641 0.0402703 0.0201352 0.999797i \(-0.493590\pi\)
0.0201352 + 0.999797i \(0.493590\pi\)
\(462\) 0 0
\(463\) 16.0000i 0.743583i −0.928316 0.371792i \(-0.878744\pi\)
0.928316 0.371792i \(-0.121256\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 20.7755i 0.961376i −0.876892 0.480688i \(-0.840387\pi\)
0.876892 0.480688i \(-0.159613\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 15.8217i 0.727484i
\(474\) 0 0
\(475\) 16.2986 6.68305i 0.747832 0.306640i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 20.6585 0.943910 0.471955 0.881623i \(-0.343549\pi\)
0.471955 + 0.881623i \(0.343549\pi\)
\(480\) 0 0
\(481\) 3.72928 0.170041
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 13.1633 2.59392i 0.597713 0.117784i
\(486\) 0 0
\(487\) 16.3632i 0.741486i −0.928735 0.370743i \(-0.879103\pi\)
0.928735 0.370743i \(-0.120897\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −26.7389 −1.20671 −0.603354 0.797473i \(-0.706169\pi\)
−0.603354 + 0.797473i \(0.706169\pi\)
\(492\) 0 0
\(493\) 37.4219i 1.68540i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −4.24281 −0.189934 −0.0949672 0.995480i \(-0.530275\pi\)
−0.0949672 + 0.995480i \(0.530275\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0.889220i 0.0396484i −0.999803 0.0198242i \(-0.993689\pi\)
0.999803 0.0198242i \(-0.00631065\pi\)
\(504\) 0 0
\(505\) −5.07081 25.7326i −0.225648 1.14509i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −11.3694 −0.503941 −0.251971 0.967735i \(-0.581079\pi\)
−0.251971 + 0.967735i \(0.581079\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −32.4157 + 6.38776i −1.42840 + 0.281478i
\(516\) 0 0
\(517\) 24.8034i 1.09085i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −3.18785 −0.139662 −0.0698310 0.997559i \(-0.522246\pi\)
−0.0698310 + 0.997559i \(0.522246\pi\)
\(522\) 0 0
\(523\) 37.1666i 1.62518i 0.582835 + 0.812591i \(0.301944\pi\)
−0.582835 + 0.812591i \(0.698056\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 31.6435i 1.37841i
\(528\) 0 0
\(529\) −33.5972 −1.46075
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 10.1170i 0.438218i
\(534\) 0 0
\(535\) −3.79383 + 0.747604i −0.164022 + 0.0323217i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 20.0525 0.863722
\(540\) 0 0
\(541\) −8.02791 −0.345147 −0.172573 0.984997i \(-0.555208\pi\)
−0.172573 + 0.984997i \(0.555208\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 5.19991 + 26.3878i 0.222740 + 1.13033i
\(546\) 0 0
\(547\) 31.2032i 1.33415i 0.744989 + 0.667077i \(0.232455\pi\)
−0.744989 + 0.667077i \(0.767545\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −23.8709 −1.01693
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 29.2191i 1.23805i 0.785370 + 0.619026i \(0.212472\pi\)
−0.785370 + 0.619026i \(0.787528\pi\)
\(558\) 0 0
\(559\) −5.52311 −0.233603
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 30.5048i 1.28562i 0.766024 + 0.642812i \(0.222232\pi\)
−0.766024 + 0.642812i \(0.777768\pi\)
\(564\) 0 0
\(565\) −15.9109 + 3.13536i −0.669375 + 0.131906i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −12.1416 −0.509003 −0.254502 0.967072i \(-0.581911\pi\)
−0.254502 + 0.967072i \(0.581911\pi\)
\(570\) 0 0
\(571\) −8.85258 −0.370469 −0.185234 0.982694i \(-0.559304\pi\)
−0.185234 + 0.982694i \(0.559304\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −14.2707 34.8034i −0.595130 1.45140i
\(576\) 0 0
\(577\) 30.2341i 1.25866i −0.777138 0.629330i \(-0.783329\pi\)
0.777138 0.629330i \(-0.216671\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 19.4094i 0.803855i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 23.0342i 0.950722i 0.879791 + 0.475361i \(0.157682\pi\)
−0.879791 + 0.475361i \(0.842318\pi\)
\(588\) 0 0
\(589\) −20.1849 −0.831705
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 27.9754i 1.14881i 0.818570 + 0.574406i \(0.194767\pi\)
−0.818570 + 0.574406i \(0.805233\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 22.7389 0.929085 0.464542 0.885551i \(-0.346219\pi\)
0.464542 + 0.885551i \(0.346219\pi\)
\(600\) 0 0
\(601\) −11.2158 −0.457500 −0.228750 0.973485i \(-0.573464\pi\)
−0.228750 + 0.973485i \(0.573464\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.20783 + 6.12933i 0.0491053 + 0.249193i
\(606\) 0 0
\(607\) 23.7293i 0.963142i 0.876407 + 0.481571i \(0.159934\pi\)
−0.876407 + 0.481571i \(0.840066\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.65847 0.350284
\(612\) 0 0
\(613\) 18.2341i 0.736467i 0.929733 + 0.368234i \(0.120037\pi\)
−0.929733 + 0.368234i \(0.879963\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0.0245796i 0.000989536i −1.00000 0.000494768i \(-0.999843\pi\)
1.00000 0.000494768i \(-0.000157490\pi\)
\(618\) 0 0
\(619\) −31.2158 −1.25467 −0.627334 0.778751i \(-0.715854\pi\)
−0.627334 + 0.778751i \(0.715854\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 17.8034 17.5510i 0.712137 0.702041i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −20.5972 −0.821266
\(630\) 0 0
\(631\) 13.4460 0.535279 0.267639 0.963519i \(-0.413756\pi\)
0.267639 + 0.963519i \(0.413756\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 37.8496 7.45856i 1.50202 0.295984i
\(636\) 0 0
\(637\) 7.00000i 0.277350i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −2.54144 −0.100381 −0.0501904 0.998740i \(-0.515983\pi\)
−0.0501904 + 0.998740i \(0.515983\pi\)
\(642\) 0 0
\(643\) 13.9634i 0.550661i −0.961350 0.275330i \(-0.911213\pi\)
0.961350 0.275330i \(-0.0887873\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 30.8034i 1.21101i 0.795843 + 0.605504i \(0.207028\pi\)
−0.795843 + 0.605504i \(0.792972\pi\)
\(648\) 0 0
\(649\) 1.70138 0.0667848
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.00000i 0.0782660i 0.999234 + 0.0391330i \(0.0124596\pi\)
−0.999234 + 0.0391330i \(0.987540\pi\)
\(654\) 0 0
\(655\) −3.79383 19.2524i −0.148237 0.752253i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −4.05581 −0.157992 −0.0789960 0.996875i \(-0.525171\pi\)
−0.0789960 + 0.996875i \(0.525171\pi\)
\(660\) 0 0
\(661\) −32.9325 −1.28093 −0.640463 0.767989i \(-0.721258\pi\)
−0.640463 + 0.767989i \(0.721258\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 50.9729i 1.97368i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −15.0462 −0.580853
\(672\) 0 0
\(673\) 9.42192i 0.363188i −0.983374 0.181594i \(-0.941874\pi\)
0.983374 0.181594i \(-0.0581257\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 42.2341i 1.62319i −0.584222 0.811594i \(-0.698600\pi\)
0.584222 0.811594i \(-0.301400\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 18.7509i 0.717484i −0.933437 0.358742i \(-0.883206\pi\)
0.933437 0.358742i \(-0.116794\pi\)
\(684\) 0 0
\(685\) −42.6358 + 8.40171i −1.62903 + 0.321013i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 6.77551 0.258126
\(690\) 0 0
\(691\) 45.2524 1.72148 0.860741 0.509043i \(-0.170001\pi\)
0.860741 + 0.509043i \(0.170001\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3.28030 + 16.6464i 0.124429 + 0.631434i
\(696\) 0 0
\(697\) 55.8776i 2.11651i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 37.4586 1.41479 0.707395 0.706818i \(-0.249870\pi\)
0.707395 + 0.706818i \(0.249870\pi\)
\(702\) 0 0
\(703\) 13.1387i 0.495535i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −11.8988 −0.446869 −0.223435 0.974719i \(-0.571727\pi\)
−0.223435 + 0.974719i \(0.571727\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 43.1020i 1.61418i
\(714\) 0 0
\(715\) 6.28467 1.23844i 0.235033 0.0463151i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 25.7851 0.961622 0.480811 0.876824i \(-0.340342\pi\)
0.480811 + 0.876824i \(0.340342\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −31.3449 + 12.8526i −1.16412 + 0.477333i
\(726\) 0 0
\(727\) 34.8034i 1.29079i 0.763850 + 0.645394i \(0.223307\pi\)
−0.763850 + 0.645394i \(0.776693\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 30.5048 1.12826
\(732\) 0 0
\(733\) 19.5510i 0.722133i −0.932540 0.361067i \(-0.882413\pi\)
0.932540 0.361067i \(-0.117587\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 30.0925i 1.10847i
\(738\) 0 0
\(739\) −12.2986 −0.452412 −0.226206 0.974079i \(-0.572632\pi\)
−0.226206 + 0.974079i \(0.572632\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 40.9975i 1.50405i 0.659133 + 0.752027i \(0.270924\pi\)
−0.659133 + 0.752027i \(0.729076\pi\)
\(744\) 0 0
\(745\) 0.607876 + 3.08476i 0.0222709 + 0.113017i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 33.5510 1.22429 0.612147 0.790744i \(-0.290306\pi\)
0.612147 + 0.790744i \(0.290306\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0.569343 + 2.88922i 0.0207205 + 0.105149i
\(756\) 0 0
\(757\) 25.1020i 0.912349i 0.889890 + 0.456175i \(0.150781\pi\)
−0.889890 + 0.456175i \(0.849219\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 39.9021 1.44645 0.723226 0.690612i \(-0.242659\pi\)
0.723226 + 0.690612i \(0.242659\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0.593923i 0.0214453i
\(768\) 0 0
\(769\) 15.5510 0.560784 0.280392 0.959886i \(-0.409536\pi\)
0.280392 + 0.959886i \(0.409536\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 27.7972i 0.999794i 0.866085 + 0.499897i \(0.166629\pi\)
−0.866085 + 0.499897i \(0.833371\pi\)
\(774\) 0 0
\(775\) −26.5048 + 10.8680i −0.952080 + 0.390389i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −35.6435 −1.27706
\(780\) 0 0
\(781\) 6.84006 0.244757
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 45.4373 8.95377i 1.62173 0.319574i
\(786\) 0 0
\(787\) 30.6339i 1.09198i −0.837792 0.545990i \(-0.816154\pi\)
0.837792 0.545990i \(-0.183846\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 5.25240i 0.186518i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 18.6464i 0.660490i 0.943895 + 0.330245i \(0.107131\pi\)
−0.943895 + 0.330245i \(0.892869\pi\)
\(798\) 0 0
\(799\) −47.8217 −1.69181
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 15.6368i 0.551812i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\)