Properties

Label 7920.2.a.cg.1.1
Level $7920$
Weight $2$
Character 7920.1
Self dual yes
Analytic conductor $63.242$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.2415184009\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 7920.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{5} -0.828427 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} -0.828427 q^{7} -1.00000 q^{11} -5.65685 q^{13} +1.17157 q^{17} +6.82843 q^{19} -4.00000 q^{23} +1.00000 q^{25} +4.82843 q^{29} -0.828427 q^{35} +11.6569 q^{37} -4.82843 q^{41} +8.82843 q^{43} -4.00000 q^{47} -6.31371 q^{49} -9.31371 q^{53} -1.00000 q^{55} -4.00000 q^{59} -11.6569 q^{61} -5.65685 q^{65} +5.65685 q^{67} +2.34315 q^{71} +11.3137 q^{73} +0.828427 q^{77} -8.48528 q^{79} -10.0000 q^{83} +1.17157 q^{85} -3.65685 q^{89} +4.68629 q^{91} +6.82843 q^{95} +11.6569 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} + 4q^{7} + O(q^{10}) \) \( 2q + 2q^{5} + 4q^{7} - 2q^{11} + 8q^{17} + 8q^{19} - 8q^{23} + 2q^{25} + 4q^{29} + 4q^{35} + 12q^{37} - 4q^{41} + 12q^{43} - 8q^{47} + 10q^{49} + 4q^{53} - 2q^{55} - 8q^{59} - 12q^{61} + 16q^{71} - 4q^{77} - 20q^{83} + 8q^{85} + 4q^{89} + 32q^{91} + 8q^{95} + 12q^{97} + O(q^{100}) \)

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
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −0.828427 −0.313116 −0.156558 0.987669i \(-0.550040\pi\)
−0.156558 + 0.987669i \(0.550040\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −5.65685 −1.56893 −0.784465 0.620174i \(-0.787062\pi\)
−0.784465 + 0.620174i \(0.787062\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.17157 0.284148 0.142074 0.989856i \(-0.454623\pi\)
0.142074 + 0.989856i \(0.454623\pi\)
\(18\) 0 0
\(19\) 6.82843 1.56655 0.783274 0.621676i \(-0.213548\pi\)
0.783274 + 0.621676i \(0.213548\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.82843 0.896616 0.448308 0.893879i \(-0.352027\pi\)
0.448308 + 0.893879i \(0.352027\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.828427 −0.140030
\(36\) 0 0
\(37\) 11.6569 1.91638 0.958188 0.286141i \(-0.0923726\pi\)
0.958188 + 0.286141i \(0.0923726\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.82843 −0.754074 −0.377037 0.926198i \(-0.623057\pi\)
−0.377037 + 0.926198i \(0.623057\pi\)
\(42\) 0 0
\(43\) 8.82843 1.34632 0.673161 0.739496i \(-0.264936\pi\)
0.673161 + 0.739496i \(0.264936\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) −6.31371 −0.901958
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −9.31371 −1.27934 −0.639668 0.768651i \(-0.720928\pi\)
−0.639668 + 0.768651i \(0.720928\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −11.6569 −1.49251 −0.746254 0.665662i \(-0.768149\pi\)
−0.746254 + 0.665662i \(0.768149\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −5.65685 −0.701646
\(66\) 0 0
\(67\) 5.65685 0.691095 0.345547 0.938401i \(-0.387693\pi\)
0.345547 + 0.938401i \(0.387693\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.34315 0.278080 0.139040 0.990287i \(-0.455598\pi\)
0.139040 + 0.990287i \(0.455598\pi\)
\(72\) 0 0
\(73\) 11.3137 1.32417 0.662085 0.749429i \(-0.269672\pi\)
0.662085 + 0.749429i \(0.269672\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.828427 0.0944080
\(78\) 0 0
\(79\) −8.48528 −0.954669 −0.477334 0.878722i \(-0.658397\pi\)
−0.477334 + 0.878722i \(0.658397\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −10.0000 −1.09764 −0.548821 0.835940i \(-0.684923\pi\)
−0.548821 + 0.835940i \(0.684923\pi\)
\(84\) 0 0
\(85\) 1.17157 0.127075
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.65685 −0.387626 −0.193813 0.981039i \(-0.562085\pi\)
−0.193813 + 0.981039i \(0.562085\pi\)
\(90\) 0 0
\(91\) 4.68629 0.491257
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.82843 0.700582
\(96\) 0 0
\(97\) 11.6569 1.18357 0.591787 0.806094i \(-0.298423\pi\)
0.591787 + 0.806094i \(0.298423\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0.828427 0.0824316 0.0412158 0.999150i \(-0.486877\pi\)
0.0412158 + 0.999150i \(0.486877\pi\)
\(102\) 0 0
\(103\) 3.31371 0.326509 0.163255 0.986584i \(-0.447801\pi\)
0.163255 + 0.986584i \(0.447801\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 17.3137 1.67378 0.836890 0.547372i \(-0.184372\pi\)
0.836890 + 0.547372i \(0.184372\pi\)
\(108\) 0 0
\(109\) 17.3137 1.65835 0.829176 0.558987i \(-0.188810\pi\)
0.829176 + 0.558987i \(0.188810\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 18.9706 1.78460 0.892300 0.451442i \(-0.149090\pi\)
0.892300 + 0.451442i \(0.149090\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.970563 −0.0889713
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 14.4853 1.28536 0.642680 0.766134i \(-0.277822\pi\)
0.642680 + 0.766134i \(0.277822\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.31371 0.289520 0.144760 0.989467i \(-0.453759\pi\)
0.144760 + 0.989467i \(0.453759\pi\)
\(132\) 0 0
\(133\) −5.65685 −0.490511
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 13.3137 1.13747 0.568733 0.822522i \(-0.307434\pi\)
0.568733 + 0.822522i \(0.307434\pi\)
\(138\) 0 0
\(139\) −0.485281 −0.0411610 −0.0205805 0.999788i \(-0.506551\pi\)
−0.0205805 + 0.999788i \(0.506551\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 5.65685 0.473050
\(144\) 0 0
\(145\) 4.82843 0.400979
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.51472 −0.124091 −0.0620453 0.998073i \(-0.519762\pi\)
−0.0620453 + 0.998073i \(0.519762\pi\)
\(150\) 0 0
\(151\) −16.4853 −1.34155 −0.670777 0.741659i \(-0.734039\pi\)
−0.670777 + 0.741659i \(0.734039\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.31371 0.261157
\(162\) 0 0
\(163\) 7.31371 0.572854 0.286427 0.958102i \(-0.407532\pi\)
0.286427 + 0.958102i \(0.407532\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −13.3137 −1.03025 −0.515123 0.857116i \(-0.672254\pi\)
−0.515123 + 0.857116i \(0.672254\pi\)
\(168\) 0 0
\(169\) 19.0000 1.46154
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2.82843 −0.215041 −0.107521 0.994203i \(-0.534291\pi\)
−0.107521 + 0.994203i \(0.534291\pi\)
\(174\) 0 0
\(175\) −0.828427 −0.0626232
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −17.6569 −1.31974 −0.659868 0.751382i \(-0.729388\pi\)
−0.659868 + 0.751382i \(0.729388\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 11.6569 0.857029
\(186\) 0 0
\(187\) −1.17157 −0.0856739
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 5.65685 0.409316 0.204658 0.978834i \(-0.434392\pi\)
0.204658 + 0.978834i \(0.434392\pi\)
\(192\) 0 0
\(193\) 13.6569 0.983042 0.491521 0.870866i \(-0.336441\pi\)
0.491521 + 0.870866i \(0.336441\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.48528 0.604551 0.302276 0.953221i \(-0.402254\pi\)
0.302276 + 0.953221i \(0.402254\pi\)
\(198\) 0 0
\(199\) 21.6569 1.53521 0.767607 0.640921i \(-0.221447\pi\)
0.767607 + 0.640921i \(0.221447\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −4.00000 −0.280745
\(204\) 0 0
\(205\) −4.82843 −0.337232
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −6.82843 −0.472332
\(210\) 0 0
\(211\) −1.17157 −0.0806544 −0.0403272 0.999187i \(-0.512840\pi\)
−0.0403272 + 0.999187i \(0.512840\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 8.82843 0.602094
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −6.62742 −0.445808
\(222\) 0 0
\(223\) 6.34315 0.424768 0.212384 0.977186i \(-0.431877\pi\)
0.212384 + 0.977186i \(0.431877\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −14.0000 −0.929213 −0.464606 0.885517i \(-0.653804\pi\)
−0.464606 + 0.885517i \(0.653804\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.8284 1.23349 0.616746 0.787163i \(-0.288451\pi\)
0.616746 + 0.787163i \(0.288451\pi\)
\(234\) 0 0
\(235\) −4.00000 −0.260931
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 17.6569 1.14213 0.571063 0.820906i \(-0.306531\pi\)
0.571063 + 0.820906i \(0.306531\pi\)
\(240\) 0 0
\(241\) −12.3431 −0.795092 −0.397546 0.917582i \(-0.630138\pi\)
−0.397546 + 0.917582i \(0.630138\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −6.31371 −0.403368
\(246\) 0 0
\(247\) −38.6274 −2.45780
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 20.9706 1.32365 0.661825 0.749658i \(-0.269782\pi\)
0.661825 + 0.749658i \(0.269782\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 16.3431 1.01946 0.509729 0.860335i \(-0.329746\pi\)
0.509729 + 0.860335i \(0.329746\pi\)
\(258\) 0 0
\(259\) −9.65685 −0.600048
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 0 0
\(265\) −9.31371 −0.572137
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −20.6274 −1.25768 −0.628838 0.777536i \(-0.716469\pi\)
−0.628838 + 0.777536i \(0.716469\pi\)
\(270\) 0 0
\(271\) 11.7990 0.716738 0.358369 0.933580i \(-0.383333\pi\)
0.358369 + 0.933580i \(0.383333\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −2.34315 −0.140786 −0.0703930 0.997519i \(-0.522425\pi\)
−0.0703930 + 0.997519i \(0.522425\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 11.1716 0.666440 0.333220 0.942849i \(-0.391865\pi\)
0.333220 + 0.942849i \(0.391865\pi\)
\(282\) 0 0
\(283\) 8.82843 0.524796 0.262398 0.964960i \(-0.415487\pi\)
0.262398 + 0.964960i \(0.415487\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.00000 0.236113
\(288\) 0 0
\(289\) −15.6274 −0.919260
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6.82843 0.398921 0.199460 0.979906i \(-0.436081\pi\)
0.199460 + 0.979906i \(0.436081\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 22.6274 1.30858
\(300\) 0 0
\(301\) −7.31371 −0.421555
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −11.6569 −0.667470
\(306\) 0 0
\(307\) 3.17157 0.181011 0.0905056 0.995896i \(-0.471152\pi\)
0.0905056 + 0.995896i \(0.471152\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3.31371 −0.187903 −0.0939516 0.995577i \(-0.529950\pi\)
−0.0939516 + 0.995577i \(0.529950\pi\)
\(312\) 0 0
\(313\) 15.6569 0.884978 0.442489 0.896774i \(-0.354096\pi\)
0.442489 + 0.896774i \(0.354096\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 26.2843 1.47627 0.738136 0.674652i \(-0.235706\pi\)
0.738136 + 0.674652i \(0.235706\pi\)
\(318\) 0 0
\(319\) −4.82843 −0.270340
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) 0 0
\(325\) −5.65685 −0.313786
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3.31371 0.182691
\(330\) 0 0
\(331\) −6.34315 −0.348651 −0.174325 0.984688i \(-0.555774\pi\)
−0.174325 + 0.984688i \(0.555774\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 5.65685 0.309067
\(336\) 0 0
\(337\) 3.31371 0.180509 0.0902546 0.995919i \(-0.471232\pi\)
0.0902546 + 0.995919i \(0.471232\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 11.0294 0.595534
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −29.3137 −1.57364 −0.786821 0.617181i \(-0.788275\pi\)
−0.786821 + 0.617181i \(0.788275\pi\)
\(348\) 0 0
\(349\) 10.9706 0.587241 0.293620 0.955922i \(-0.405140\pi\)
0.293620 + 0.955922i \(0.405140\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 0 0
\(355\) 2.34315 0.124361
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 0 0
\(361\) 27.6274 1.45407
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 11.3137 0.592187
\(366\) 0 0
\(367\) −9.65685 −0.504084 −0.252042 0.967716i \(-0.581102\pi\)
−0.252042 + 0.967716i \(0.581102\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 7.71573 0.400581
\(372\) 0 0
\(373\) 10.6274 0.550267 0.275133 0.961406i \(-0.411278\pi\)
0.275133 + 0.961406i \(0.411278\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −27.3137 −1.40673
\(378\) 0 0
\(379\) 23.3137 1.19754 0.598772 0.800919i \(-0.295655\pi\)
0.598772 + 0.800919i \(0.295655\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 0 0
\(385\) 0.828427 0.0422206
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 23.6569 1.19945 0.599725 0.800206i \(-0.295277\pi\)
0.599725 + 0.800206i \(0.295277\pi\)
\(390\) 0 0
\(391\) −4.68629 −0.236996
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −8.48528 −0.426941
\(396\) 0 0
\(397\) −14.9706 −0.751351 −0.375676 0.926751i \(-0.622589\pi\)
−0.375676 + 0.926751i \(0.622589\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 6.68629 0.333897 0.166949 0.985966i \(-0.446609\pi\)
0.166949 + 0.985966i \(0.446609\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −11.6569 −0.577809
\(408\) 0 0
\(409\) −19.6569 −0.971969 −0.485984 0.873967i \(-0.661539\pi\)
−0.485984 + 0.873967i \(0.661539\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 3.31371 0.163057
\(414\) 0 0
\(415\) −10.0000 −0.490881
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −36.9706 −1.80613 −0.903065 0.429504i \(-0.858689\pi\)
−0.903065 + 0.429504i \(0.858689\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.17157 0.0568296
\(426\) 0 0
\(427\) 9.65685 0.467328
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 21.6569 1.04317 0.521587 0.853198i \(-0.325340\pi\)
0.521587 + 0.853198i \(0.325340\pi\)
\(432\) 0 0
\(433\) −15.6569 −0.752420 −0.376210 0.926534i \(-0.622773\pi\)
−0.376210 + 0.926534i \(0.622773\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −27.3137 −1.30659
\(438\) 0 0
\(439\) 20.4853 0.977709 0.488855 0.872365i \(-0.337415\pi\)
0.488855 + 0.872365i \(0.337415\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) −3.65685 −0.173352
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −30.9706 −1.46159 −0.730796 0.682596i \(-0.760851\pi\)
−0.730796 + 0.682596i \(0.760851\pi\)
\(450\) 0 0
\(451\) 4.82843 0.227362
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 4.68629 0.219697
\(456\) 0 0
\(457\) −23.3137 −1.09057 −0.545285 0.838251i \(-0.683578\pi\)
−0.545285 + 0.838251i \(0.683578\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −0.142136 −0.00661992 −0.00330996 0.999995i \(-0.501054\pi\)
−0.00330996 + 0.999995i \(0.501054\pi\)
\(462\) 0 0
\(463\) −4.97056 −0.231002 −0.115501 0.993307i \(-0.536847\pi\)
−0.115501 + 0.993307i \(0.536847\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −22.6274 −1.04707 −0.523536 0.852004i \(-0.675387\pi\)
−0.523536 + 0.852004i \(0.675387\pi\)
\(468\) 0 0
\(469\) −4.68629 −0.216393
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −8.82843 −0.405932
\(474\) 0 0
\(475\) 6.82843 0.313310
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 36.9706 1.68923 0.844614 0.535376i \(-0.179830\pi\)
0.844614 + 0.535376i \(0.179830\pi\)
\(480\) 0 0
\(481\) −65.9411 −3.00666
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 11.6569 0.529310
\(486\) 0 0
\(487\) 12.9706 0.587752 0.293876 0.955844i \(-0.405055\pi\)
0.293876 + 0.955844i \(0.405055\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 14.3431 0.647297 0.323649 0.946177i \(-0.395090\pi\)
0.323649 + 0.946177i \(0.395090\pi\)
\(492\) 0 0
\(493\) 5.65685 0.254772
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.94113 −0.0870714
\(498\) 0 0
\(499\) 22.3431 1.00022 0.500108 0.865963i \(-0.333294\pi\)
0.500108 + 0.865963i \(0.333294\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 17.3137 0.771980 0.385990 0.922503i \(-0.373860\pi\)
0.385990 + 0.922503i \(0.373860\pi\)
\(504\) 0 0
\(505\) 0.828427 0.0368645
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −18.6863 −0.828255 −0.414128 0.910219i \(-0.635913\pi\)
−0.414128 + 0.910219i \(0.635913\pi\)
\(510\) 0 0
\(511\) −9.37258 −0.414619
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3.31371 0.146019
\(516\) 0 0
\(517\) 4.00000 0.175920
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 32.6274 1.42943 0.714717 0.699414i \(-0.246556\pi\)
0.714717 + 0.699414i \(0.246556\pi\)
\(522\) 0 0
\(523\) 9.51472 0.416050 0.208025 0.978124i \(-0.433297\pi\)
0.208025 + 0.978124i \(0.433297\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 27.3137 1.18309
\(534\) 0 0
\(535\) 17.3137 0.748537
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 6.31371 0.271951
\(540\) 0 0
\(541\) 17.3137 0.744374 0.372187 0.928158i \(-0.378608\pi\)
0.372187 + 0.928158i \(0.378608\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 17.3137 0.741638
\(546\) 0 0
\(547\) 8.14214 0.348133 0.174066 0.984734i \(-0.444309\pi\)
0.174066 + 0.984734i \(0.444309\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 32.9706 1.40459
\(552\) 0 0
\(553\) 7.02944 0.298922
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −5.17157 −0.219127 −0.109563 0.993980i \(-0.534945\pi\)
−0.109563 + 0.993980i \(0.534945\pi\)
\(558\) 0 0
\(559\) −49.9411 −2.11228
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 31.6569 1.33418 0.667089 0.744978i \(-0.267540\pi\)
0.667089 + 0.744978i \(0.267540\pi\)
\(564\) 0 0
\(565\) 18.9706 0.798098
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 35.4558 1.48639 0.743193 0.669077i \(-0.233310\pi\)
0.743193 + 0.669077i \(0.233310\pi\)
\(570\) 0 0
\(571\) 16.4853 0.689888 0.344944 0.938623i \(-0.387898\pi\)
0.344944 + 0.938623i \(0.387898\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.28427 0.343689
\(582\) 0 0
\(583\) 9.31371 0.385734
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 14.6274 0.603738 0.301869 0.953349i \(-0.402389\pi\)
0.301869 + 0.953349i \(0.402389\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 22.8284 0.937451 0.468726 0.883344i \(-0.344713\pi\)
0.468726 + 0.883344i \(0.344713\pi\)
\(594\) 0 0
\(595\) −0.970563 −0.0397892
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 27.3137 1.11601 0.558004 0.829838i \(-0.311567\pi\)
0.558004 + 0.829838i \(0.311567\pi\)
\(600\) 0 0
\(601\) −5.31371 −0.216751 −0.108375 0.994110i \(-0.534565\pi\)
−0.108375 + 0.994110i \(0.534565\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) −1.51472 −0.0614805 −0.0307403 0.999527i \(-0.509786\pi\)
−0.0307403 + 0.999527i \(0.509786\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 22.6274 0.915407
\(612\) 0 0
\(613\) −45.9411 −1.85554 −0.927772 0.373147i \(-0.878279\pi\)
−0.927772 + 0.373147i \(0.878279\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −0.343146 −0.0138145 −0.00690726 0.999976i \(-0.502199\pi\)
−0.00690726 + 0.999976i \(0.502199\pi\)
\(618\) 0 0
\(619\) 14.3431 0.576500 0.288250 0.957555i \(-0.406927\pi\)
0.288250 + 0.957555i \(0.406927\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 3.02944 0.121372
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 13.6569 0.544534
\(630\) 0 0
\(631\) −45.6569 −1.81757 −0.908785 0.417264i \(-0.862989\pi\)
−0.908785 + 0.417264i \(0.862989\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 14.4853 0.574831
\(636\) 0 0
\(637\) 35.7157 1.41511
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −6.97056 −0.275321 −0.137660 0.990479i \(-0.543958\pi\)
−0.137660 + 0.990479i \(0.543958\pi\)
\(642\) 0 0
\(643\) −37.9411 −1.49625 −0.748126 0.663557i \(-0.769046\pi\)
−0.748126 + 0.663557i \(0.769046\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4.68629 0.184237 0.0921186 0.995748i \(-0.470636\pi\)
0.0921186 + 0.995748i \(0.470636\pi\)
\(648\) 0 0
\(649\) 4.00000 0.157014
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −6.97056 −0.272779 −0.136390 0.990655i \(-0.543550\pi\)
−0.136390 + 0.990655i \(0.543550\pi\)
\(654\) 0 0
\(655\) 3.31371 0.129477
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −15.3137 −0.596537 −0.298269 0.954482i \(-0.596409\pi\)
−0.298269 + 0.954482i \(0.596409\pi\)
\(660\) 0 0
\(661\) 9.31371 0.362261 0.181131 0.983459i \(-0.442024\pi\)
0.181131 + 0.983459i \(0.442024\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −5.65685 −0.219363
\(666\) 0 0
\(667\) −19.3137 −0.747830
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 11.6569 0.450008
\(672\) 0 0
\(673\) 18.3431 0.707076 0.353538 0.935420i \(-0.384978\pi\)
0.353538 + 0.935420i \(0.384978\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −29.4558 −1.13208 −0.566040 0.824378i \(-0.691525\pi\)
−0.566040 + 0.824378i \(0.691525\pi\)
\(678\) 0 0
\(679\) −9.65685 −0.370596
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 0 0
\(685\) 13.3137 0.508691
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 52.6863 2.00719
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −0.485281 −0.0184078
\(696\) 0 0
\(697\) −5.65685 −0.214269
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −36.1421 −1.36507 −0.682535 0.730853i \(-0.739122\pi\)
−0.682535 + 0.730853i \(0.739122\pi\)
\(702\) 0 0
\(703\) 79.5980 3.00209
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −0.686292 −0.0258106
\(708\) 0 0
\(709\) 6.68629 0.251109 0.125554 0.992087i \(-0.459929\pi\)
0.125554 + 0.992087i \(0.459929\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 5.65685 0.211554
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −47.5980 −1.77511 −0.887553 0.460706i \(-0.847596\pi\)
−0.887553 + 0.460706i \(0.847596\pi\)
\(720\) 0 0
\(721\) −2.74517 −0.102235
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 4.82843 0.179323
\(726\) 0 0
\(727\) −33.9411 −1.25881 −0.629403 0.777079i \(-0.716701\pi\)
−0.629403 + 0.777079i \(0.716701\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 10.3431 0.382555
\(732\) 0 0
\(733\) −6.34315 −0.234289 −0.117145 0.993115i \(-0.537374\pi\)
−0.117145 + 0.993115i \(0.537374\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5.65685 −0.208373
\(738\) 0 0
\(739\) 15.1127 0.555930 0.277965 0.960591i \(-0.410340\pi\)
0.277965 + 0.960591i \(0.410340\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 36.3431 1.33330 0.666650 0.745371i \(-0.267727\pi\)
0.666650 + 0.745371i \(0.267727\pi\)
\(744\) 0 0
\(745\) −1.51472 −0.0554950
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −14.3431 −0.524087
\(750\) 0 0
\(751\) −20.2843 −0.740184 −0.370092 0.928995i \(-0.620674\pi\)
−0.370092 + 0.928995i \(0.620674\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −16.4853 −0.599961
\(756\) 0 0
\(757\) −36.6274 −1.33125 −0.665623 0.746288i \(-0.731834\pi\)
−0.665623 + 0.746288i \(0.731834\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 28.8284 1.04503 0.522515 0.852630i \(-0.324994\pi\)
0.522515 + 0.852630i \(0.324994\pi\)
\(762\) 0 0
\(763\) −14.3431 −0.519257
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 22.6274 0.817029
\(768\) 0 0
\(769\) 10.6863 0.385358 0.192679 0.981262i \(-0.438282\pi\)
0.192679 + 0.981262i \(0.438282\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −3.65685 −0.131528 −0.0657640 0.997835i \(-0.520948\pi\)
−0.0657640 + 0.997835i \(0.520948\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −32.9706 −1.18129
\(780\) 0 0
\(781\) −2.34315 −0.0838443
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 18.0000 0.642448
\(786\) 0 0
\(787\) 20.1421 0.717990 0.358995 0.933340i \(-0.383120\pi\)
0.358995 + 0.933340i \(0.383120\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −15.7157 −0.558787
\(792\) 0 0
\(793\) 65.9411 2.34164
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −34.9706 −1.23872 −0.619360 0.785107i \(-0.712608\pi\)
−0.619360 + 0.785107i \(0.712608\pi\)
\(798\) 0 0
\(799\) −4.68629 −0.165789
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −11.3137 −0.399252
\(804\) 0 0
\(805\) 3.31371 0.116793
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −28.4264 −0.999419 −0.499710 0.866193i \(-0.666560\pi\)
−0.499710 + 0.866193i \(0.666560\pi\)
\(810\) 0 0
\(811\) −0.485281 −0.0170405 −0.00852027 0.999964i \(-0.502712\pi\)
−0.00852027 + 0.999964i \(0.502712\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 7.31371 0.256188
\(816\) 0 0
\(817\) 60.2843 2.10908
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 12.8284 0.447715 0.223858 0.974622i \(-0.428135\pi\)
0.223858 + 0.974622i \(0.428135\pi\)
\(822\) 0 0
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 41.3137 1.43662 0.718309 0.695724i \(-0.244916\pi\)
0.718309 + 0.695724i \(0.244916\pi\)
\(828\) 0 0
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −7.39697 −0.256290
\(834\) 0 0
\(835\) −13.3137 −0.460740
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −22.6274 −0.781185 −0.390593 0.920564i \(-0.627730\pi\)
−0.390593 + 0.920564i \(0.627730\pi\)
\(840\) 0 0
\(841\) −5.68629 −0.196079
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 19.0000 0.653620
\(846\) 0 0
\(847\) −0.828427 −0.0284651
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −46.6274 −1.59837
\(852\) 0 0
\(853\) −8.68629 −0.297413 −0.148706 0.988881i \(-0.547511\pi\)
−0.148706 + 0.988881i \(0.547511\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 28.4853 0.973039 0.486519 0.873670i \(-0.338266\pi\)
0.486519 + 0.873670i \(0.338266\pi\)
\(858\) 0 0
\(859\) 52.9706 1.80733 0.903666 0.428238i \(-0.140865\pi\)
0.903666 + 0.428238i \(0.140865\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −20.6863 −0.704170 −0.352085 0.935968i \(-0.614527\pi\)
−0.352085 + 0.935968i \(0.614527\pi\)
\(864\) 0 0
\(865\) −2.82843 −0.0961694
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 8.48528 0.287843
\(870\) 0 0
\(871\) −32.0000 −1.08428
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −0.828427 −0.0280059
\(876\) 0 0
\(877\) 2.62742 0.0887216 0.0443608 0.999016i \(-0.485875\pi\)
0.0443608 + 0.999016i \(0.485875\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 46.9706 1.58248 0.791239 0.611507i \(-0.209436\pi\)
0.791239 + 0.611507i \(0.209436\pi\)
\(882\) 0 0
\(883\) 5.37258 0.180802 0.0904009 0.995905i \(-0.471185\pi\)
0.0904009 + 0.995905i \(0.471185\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −15.6569 −0.525706 −0.262853 0.964836i \(-0.584663\pi\)
−0.262853 + 0.964836i \(0.584663\pi\)
\(888\) 0 0
\(889\) −12.0000 −0.402467
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −27.3137 −0.914018
\(894\) 0 0
\(895\) −17.6569 −0.590204
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −10.9117 −0.363521
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −14.0000 −0.465376
\(906\) 0 0
\(907\) −40.9706 −1.36041 −0.680203 0.733024i \(-0.738108\pi\)
−0.680203 + 0.733024i \(0.738108\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −48.9706 −1.62247 −0.811234 0.584722i \(-0.801203\pi\)
−0.811234 + 0.584722i \(0.801203\pi\)
\(912\) 0 0
\(913\) 10.0000 0.330952
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2.74517 −0.0906534
\(918\) 0 0
\(919\) −11.5147 −0.379836 −0.189918 0.981800i \(-0.560822\pi\)
−0.189918 + 0.981800i \(0.560822\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −13.2548 −0.436288
\(924\) 0 0
\(925\) 11.6569 0.383275
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 45.5980 1.49602 0.748011 0.663687i \(-0.231009\pi\)
0.748011 + 0.663687i \(0.231009\pi\)
\(930\) 0 0
\(931\) −43.1127 −1.41296
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.17157 −0.0383145
\(936\) 0 0
\(937\) 11.0294 0.360316 0.180158 0.983638i \(-0.442339\pi\)
0.180158 + 0.983638i \(0.442339\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 34.7696 1.13346 0.566728 0.823905i \(-0.308209\pi\)
0.566728 + 0.823905i \(0.308209\pi\)
\(942\) 0 0
\(943\) 19.3137 0.628941
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 6.62742 0.215362 0.107681 0.994185i \(-0.465657\pi\)
0.107681 + 0.994185i \(0.465657\pi\)
\(948\) 0 0
\(949\) −64.0000 −2.07753
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 11.7990 0.382207 0.191103 0.981570i \(-0.438793\pi\)
0.191103 + 0.981570i \(0.438793\pi\)
\(954\) 0 0
\(955\) 5.65685 0.183052
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −11.0294 −0.356159
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 13.6569 0.439630
\(966\) 0 0
\(967\) −11.4558 −0.368395 −0.184198 0.982889i \(-0.558969\pi\)
−0.184198 + 0.982889i \(0.558969\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −34.6274 −1.11125 −0.555623 0.831434i \(-0.687520\pi\)
−0.555623 + 0.831434i \(0.687520\pi\)
\(972\) 0 0
\(973\) 0.402020 0.0128882
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −2.68629 −0.0859421 −0.0429710 0.999076i \(-0.513682\pi\)
−0.0429710 + 0.999076i \(0.513682\pi\)
\(978\) 0 0
\(979\) 3.65685 0.116874
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −30.6274 −0.976863 −0.488431 0.872602i \(-0.662431\pi\)
−0.488431 + 0.872602i \(0.662431\pi\)
\(984\) 0 0
\(985\) 8.48528 0.270364
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −35.3137 −1.12291
\(990\) 0 0
\(991\) 30.6274 0.972912 0.486456 0.873705i \(-0.338289\pi\)
0.486456 + 0.873705i \(0.338289\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 21.6569 0.686568
\(996\) 0 0
\(997\) −39.3137 −1.24508 −0.622539 0.782589i \(-0.713899\pi\)
−0.622539 + 0.782589i \(0.713899\pi\)
\(998\) 0 0
\(999\) 0 0
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7920.2.a.cg.1.1 2
3.2 odd 2 2640.2.a.bb.1.1 2
4.3 odd 2 495.2.a.d.1.2 2
12.11 even 2 165.2.a.a.1.1 2
20.3 even 4 2475.2.c.m.199.1 4
20.7 even 4 2475.2.c.m.199.4 4
20.19 odd 2 2475.2.a.m.1.1 2
44.43 even 2 5445.2.a.m.1.1 2
60.23 odd 4 825.2.c.e.199.4 4
60.47 odd 4 825.2.c.e.199.1 4
60.59 even 2 825.2.a.g.1.2 2
84.83 odd 2 8085.2.a.ba.1.1 2
132.131 odd 2 1815.2.a.k.1.2 2
660.659 odd 2 9075.2.a.v.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.a.1.1 2 12.11 even 2
495.2.a.d.1.2 2 4.3 odd 2
825.2.a.g.1.2 2 60.59 even 2
825.2.c.e.199.1 4 60.47 odd 4
825.2.c.e.199.4 4 60.23 odd 4
1815.2.a.k.1.2 2 132.131 odd 2
2475.2.a.m.1.1 2 20.19 odd 2
2475.2.c.m.199.1 4 20.3 even 4
2475.2.c.m.199.4 4 20.7 even 4
2640.2.a.bb.1.1 2 3.2 odd 2
5445.2.a.m.1.1 2 44.43 even 2
7920.2.a.cg.1.1 2 1.1 even 1 trivial
8085.2.a.ba.1.1 2 84.83 odd 2
9075.2.a.v.1.1 2 660.659 odd 2