Properties

Label 200.6.c.a.49.1
Level $200$
Weight $6$
Character 200.49
Analytic conductor $32.077$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 200.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.0767639626\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 200.49
Dual form 200.6.c.a.49.2

$q$-expansion

\(f(q)\) \(=\) \(q-20.0000i q^{3} -24.0000i q^{7} -157.000 q^{9} +O(q^{10})\) \(q-20.0000i q^{3} -24.0000i q^{7} -157.000 q^{9} +124.000 q^{11} -478.000i q^{13} -1198.00i q^{17} -3044.00 q^{19} -480.000 q^{21} -184.000i q^{23} -1720.00i q^{27} +3282.00 q^{29} -5728.00 q^{31} -2480.00i q^{33} +10326.0i q^{37} -9560.00 q^{39} -8886.00 q^{41} +9188.00i q^{43} +23664.0i q^{47} +16231.0 q^{49} -23960.0 q^{51} -11686.0i q^{53} +60880.0i q^{57} -16876.0 q^{59} -18482.0 q^{61} +3768.00i q^{63} -15532.0i q^{67} -3680.00 q^{69} -31960.0 q^{71} +4886.00i q^{73} -2976.00i q^{77} -44560.0 q^{79} -72551.0 q^{81} -67364.0i q^{83} -65640.0i q^{87} -71994.0 q^{89} -11472.0 q^{91} +114560. i q^{93} +48866.0i q^{97} -19468.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 314 q^{9} + O(q^{10}) \) \( 2 q - 314 q^{9} + 248 q^{11} - 6088 q^{19} - 960 q^{21} + 6564 q^{29} - 11456 q^{31} - 19120 q^{39} - 17772 q^{41} + 32462 q^{49} - 47920 q^{51} - 33752 q^{59} - 36964 q^{61} - 7360 q^{69} - 63920 q^{71} - 89120 q^{79} - 145102 q^{81} - 143988 q^{89} - 22944 q^{91} - 38936 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) − 20.0000i − 1.28300i −0.767123 0.641500i \(-0.778312\pi\)
0.767123 0.641500i \(-0.221688\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 24.0000i − 0.185125i −0.995707 0.0925627i \(-0.970494\pi\)
0.995707 0.0925627i \(-0.0295059\pi\)
\(8\) 0 0
\(9\) −157.000 −0.646091
\(10\) 0 0
\(11\) 124.000 0.308987 0.154493 0.987994i \(-0.450625\pi\)
0.154493 + 0.987994i \(0.450625\pi\)
\(12\) 0 0
\(13\) − 478.000i − 0.784458i −0.919868 0.392229i \(-0.871704\pi\)
0.919868 0.392229i \(-0.128296\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 1198.00i − 1.00539i −0.864464 0.502695i \(-0.832342\pi\)
0.864464 0.502695i \(-0.167658\pi\)
\(18\) 0 0
\(19\) −3044.00 −1.93446 −0.967232 0.253894i \(-0.918288\pi\)
−0.967232 + 0.253894i \(0.918288\pi\)
\(20\) 0 0
\(21\) −480.000 −0.237516
\(22\) 0 0
\(23\) − 184.000i − 0.0725268i −0.999342 0.0362634i \(-0.988454\pi\)
0.999342 0.0362634i \(-0.0115455\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 1720.00i − 0.454066i
\(28\) 0 0
\(29\) 3282.00 0.724676 0.362338 0.932047i \(-0.381979\pi\)
0.362338 + 0.932047i \(0.381979\pi\)
\(30\) 0 0
\(31\) −5728.00 −1.07053 −0.535265 0.844684i \(-0.679788\pi\)
−0.535265 + 0.844684i \(0.679788\pi\)
\(32\) 0 0
\(33\) − 2480.00i − 0.396430i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10326.0i 1.24002i 0.784595 + 0.620009i \(0.212871\pi\)
−0.784595 + 0.620009i \(0.787129\pi\)
\(38\) 0 0
\(39\) −9560.00 −1.00646
\(40\) 0 0
\(41\) −8886.00 −0.825556 −0.412778 0.910832i \(-0.635442\pi\)
−0.412778 + 0.910832i \(0.635442\pi\)
\(42\) 0 0
\(43\) 9188.00i 0.757792i 0.925439 + 0.378896i \(0.123696\pi\)
−0.925439 + 0.378896i \(0.876304\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 23664.0i 1.56258i 0.624165 + 0.781292i \(0.285439\pi\)
−0.624165 + 0.781292i \(0.714561\pi\)
\(48\) 0 0
\(49\) 16231.0 0.965729
\(50\) 0 0
\(51\) −23960.0 −1.28992
\(52\) 0 0
\(53\) − 11686.0i − 0.571447i −0.958312 0.285724i \(-0.907766\pi\)
0.958312 0.285724i \(-0.0922339\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 60880.0i 2.48192i
\(58\) 0 0
\(59\) −16876.0 −0.631160 −0.315580 0.948899i \(-0.602199\pi\)
−0.315580 + 0.948899i \(0.602199\pi\)
\(60\) 0 0
\(61\) −18482.0 −0.635952 −0.317976 0.948099i \(-0.603003\pi\)
−0.317976 + 0.948099i \(0.603003\pi\)
\(62\) 0 0
\(63\) 3768.00i 0.119608i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 15532.0i − 0.422708i −0.977410 0.211354i \(-0.932213\pi\)
0.977410 0.211354i \(-0.0677873\pi\)
\(68\) 0 0
\(69\) −3680.00 −0.0930519
\(70\) 0 0
\(71\) −31960.0 −0.752421 −0.376210 0.926534i \(-0.622773\pi\)
−0.376210 + 0.926534i \(0.622773\pi\)
\(72\) 0 0
\(73\) 4886.00i 0.107312i 0.998559 + 0.0536558i \(0.0170874\pi\)
−0.998559 + 0.0536558i \(0.982913\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 2976.00i − 0.0572013i
\(78\) 0 0
\(79\) −44560.0 −0.803299 −0.401650 0.915793i \(-0.631563\pi\)
−0.401650 + 0.915793i \(0.631563\pi\)
\(80\) 0 0
\(81\) −72551.0 −1.22866
\(82\) 0 0
\(83\) − 67364.0i − 1.07333i −0.843796 0.536664i \(-0.819684\pi\)
0.843796 0.536664i \(-0.180316\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 65640.0i − 0.929759i
\(88\) 0 0
\(89\) −71994.0 −0.963432 −0.481716 0.876327i \(-0.659986\pi\)
−0.481716 + 0.876327i \(0.659986\pi\)
\(90\) 0 0
\(91\) −11472.0 −0.145223
\(92\) 0 0
\(93\) 114560.i 1.37349i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 48866.0i 0.527324i 0.964615 + 0.263662i \(0.0849303\pi\)
−0.964615 + 0.263662i \(0.915070\pi\)
\(98\) 0 0
\(99\) −19468.0 −0.199633
\(100\) 0 0
\(101\) 51606.0 0.503381 0.251690 0.967808i \(-0.419014\pi\)
0.251690 + 0.967808i \(0.419014\pi\)
\(102\) 0 0
\(103\) − 180424.i − 1.67572i −0.545886 0.837860i \(-0.683807\pi\)
0.545886 0.837860i \(-0.316193\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 65700.0i − 0.554761i −0.960760 0.277381i \(-0.910534\pi\)
0.960760 0.277381i \(-0.0894663\pi\)
\(108\) 0 0
\(109\) 112706. 0.908617 0.454308 0.890844i \(-0.349886\pi\)
0.454308 + 0.890844i \(0.349886\pi\)
\(110\) 0 0
\(111\) 206520. 1.59094
\(112\) 0 0
\(113\) 23502.0i 0.173145i 0.996246 + 0.0865723i \(0.0275913\pi\)
−0.996246 + 0.0865723i \(0.972409\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 75046.0i 0.506831i
\(118\) 0 0
\(119\) −28752.0 −0.186123
\(120\) 0 0
\(121\) −145675. −0.904527
\(122\) 0 0
\(123\) 177720.i 1.05919i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 94592.0i − 0.520409i −0.965553 0.260205i \(-0.916210\pi\)
0.965553 0.260205i \(-0.0837900\pi\)
\(128\) 0 0
\(129\) 183760. 0.972247
\(130\) 0 0
\(131\) 70292.0 0.357872 0.178936 0.983861i \(-0.442735\pi\)
0.178936 + 0.983861i \(0.442735\pi\)
\(132\) 0 0
\(133\) 73056.0i 0.358119i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 277290.i 1.26221i 0.775696 + 0.631107i \(0.217399\pi\)
−0.775696 + 0.631107i \(0.782601\pi\)
\(138\) 0 0
\(139\) 130308. 0.572050 0.286025 0.958222i \(-0.407666\pi\)
0.286025 + 0.958222i \(0.407666\pi\)
\(140\) 0 0
\(141\) 473280. 2.00480
\(142\) 0 0
\(143\) − 59272.0i − 0.242387i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 324620.i − 1.23903i
\(148\) 0 0
\(149\) 401530. 1.48167 0.740836 0.671685i \(-0.234429\pi\)
0.740836 + 0.671685i \(0.234429\pi\)
\(150\) 0 0
\(151\) −75976.0 −0.271165 −0.135583 0.990766i \(-0.543291\pi\)
−0.135583 + 0.990766i \(0.543291\pi\)
\(152\) 0 0
\(153\) 188086.i 0.649573i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 394322.i − 1.27674i −0.769730 0.638369i \(-0.779609\pi\)
0.769730 0.638369i \(-0.220391\pi\)
\(158\) 0 0
\(159\) −233720. −0.733167
\(160\) 0 0
\(161\) −4416.00 −0.0134265
\(162\) 0 0
\(163\) 11724.0i 0.0345626i 0.999851 + 0.0172813i \(0.00550109\pi\)
−0.999851 + 0.0172813i \(0.994499\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 551928.i − 1.53141i −0.643192 0.765705i \(-0.722390\pi\)
0.643192 0.765705i \(-0.277610\pi\)
\(168\) 0 0
\(169\) 142809. 0.384626
\(170\) 0 0
\(171\) 477908. 1.24984
\(172\) 0 0
\(173\) − 432894.i − 1.09968i −0.835270 0.549840i \(-0.814689\pi\)
0.835270 0.549840i \(-0.185311\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 337520.i 0.809779i
\(178\) 0 0
\(179\) −559620. −1.30545 −0.652726 0.757594i \(-0.726375\pi\)
−0.652726 + 0.757594i \(0.726375\pi\)
\(180\) 0 0
\(181\) 604710. 1.37199 0.685995 0.727607i \(-0.259367\pi\)
0.685995 + 0.727607i \(0.259367\pi\)
\(182\) 0 0
\(183\) 369640.i 0.815927i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 148552.i − 0.310652i
\(188\) 0 0
\(189\) −41280.0 −0.0840592
\(190\) 0 0
\(191\) −409152. −0.811524 −0.405762 0.913979i \(-0.632994\pi\)
−0.405762 + 0.913979i \(0.632994\pi\)
\(192\) 0 0
\(193\) − 540866.i − 1.04519i −0.852580 0.522596i \(-0.824963\pi\)
0.852580 0.522596i \(-0.175037\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 629898.i − 1.15639i −0.815898 0.578195i \(-0.803757\pi\)
0.815898 0.578195i \(-0.196243\pi\)
\(198\) 0 0
\(199\) −283048. −0.506673 −0.253336 0.967378i \(-0.581528\pi\)
−0.253336 + 0.967378i \(0.581528\pi\)
\(200\) 0 0
\(201\) −310640. −0.542335
\(202\) 0 0
\(203\) − 78768.0i − 0.134156i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 28888.0i 0.0468588i
\(208\) 0 0
\(209\) −377456. −0.597724
\(210\) 0 0
\(211\) 142756. 0.220744 0.110372 0.993890i \(-0.464796\pi\)
0.110372 + 0.993890i \(0.464796\pi\)
\(212\) 0 0
\(213\) 639200.i 0.965357i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 137472.i 0.198182i
\(218\) 0 0
\(219\) 97720.0 0.137681
\(220\) 0 0
\(221\) −572644. −0.788686
\(222\) 0 0
\(223\) − 889696.i − 1.19806i −0.800726 0.599031i \(-0.795553\pi\)
0.800726 0.599031i \(-0.204447\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.14316e6i 1.47245i 0.676736 + 0.736226i \(0.263394\pi\)
−0.676736 + 0.736226i \(0.736606\pi\)
\(228\) 0 0
\(229\) 695786. 0.876773 0.438386 0.898787i \(-0.355550\pi\)
0.438386 + 0.898787i \(0.355550\pi\)
\(230\) 0 0
\(231\) −59520.0 −0.0733893
\(232\) 0 0
\(233\) 347126.i 0.418887i 0.977821 + 0.209444i \(0.0671653\pi\)
−0.977821 + 0.209444i \(0.932835\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 891200.i 1.03063i
\(238\) 0 0
\(239\) 1.64296e6 1.86051 0.930255 0.366912i \(-0.119585\pi\)
0.930255 + 0.366912i \(0.119585\pi\)
\(240\) 0 0
\(241\) −1.16744e6 −1.29477 −0.647383 0.762165i \(-0.724137\pi\)
−0.647383 + 0.762165i \(0.724137\pi\)
\(242\) 0 0
\(243\) 1.03306e6i 1.12230i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.45503e6i 1.51751i
\(248\) 0 0
\(249\) −1.34728e6 −1.37708
\(250\) 0 0
\(251\) −790612. −0.792098 −0.396049 0.918229i \(-0.629619\pi\)
−0.396049 + 0.918229i \(0.629619\pi\)
\(252\) 0 0
\(253\) − 22816.0i − 0.0224098i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 129790.i − 0.122577i −0.998120 0.0612884i \(-0.980479\pi\)
0.998120 0.0612884i \(-0.0195209\pi\)
\(258\) 0 0
\(259\) 247824. 0.229559
\(260\) 0 0
\(261\) −515274. −0.468206
\(262\) 0 0
\(263\) − 70888.0i − 0.0631951i −0.999501 0.0315975i \(-0.989941\pi\)
0.999501 0.0315975i \(-0.0100595\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 1.43988e6i 1.23608i
\(268\) 0 0
\(269\) −1.79017e6 −1.50839 −0.754197 0.656649i \(-0.771973\pi\)
−0.754197 + 0.656649i \(0.771973\pi\)
\(270\) 0 0
\(271\) −1.77362e6 −1.46702 −0.733511 0.679678i \(-0.762120\pi\)
−0.733511 + 0.679678i \(0.762120\pi\)
\(272\) 0 0
\(273\) 229440.i 0.186321i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 275450.i − 0.215697i −0.994167 0.107848i \(-0.965604\pi\)
0.994167 0.107848i \(-0.0343961\pi\)
\(278\) 0 0
\(279\) 899296. 0.691659
\(280\) 0 0
\(281\) 594170. 0.448895 0.224448 0.974486i \(-0.427942\pi\)
0.224448 + 0.974486i \(0.427942\pi\)
\(282\) 0 0
\(283\) − 1.09243e6i − 0.810824i −0.914134 0.405412i \(-0.867128\pi\)
0.914134 0.405412i \(-0.132872\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 213264.i 0.152831i
\(288\) 0 0
\(289\) −15347.0 −0.0108088
\(290\) 0 0
\(291\) 977320. 0.676557
\(292\) 0 0
\(293\) − 333654.i − 0.227053i −0.993535 0.113527i \(-0.963785\pi\)
0.993535 0.113527i \(-0.0362147\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 213280.i − 0.140300i
\(298\) 0 0
\(299\) −87952.0 −0.0568942
\(300\) 0 0
\(301\) 220512. 0.140287
\(302\) 0 0
\(303\) − 1.03212e6i − 0.645838i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.05997e6i 0.641872i 0.947101 + 0.320936i \(0.103997\pi\)
−0.947101 + 0.320936i \(0.896003\pi\)
\(308\) 0 0
\(309\) −3.60848e6 −2.14995
\(310\) 0 0
\(311\) −1.33649e6 −0.783545 −0.391773 0.920062i \(-0.628138\pi\)
−0.391773 + 0.920062i \(0.628138\pi\)
\(312\) 0 0
\(313\) − 1.64419e6i − 0.948615i −0.880359 0.474308i \(-0.842698\pi\)
0.880359 0.474308i \(-0.157302\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 1.72370e6i − 0.963414i −0.876332 0.481707i \(-0.840017\pi\)
0.876332 0.481707i \(-0.159983\pi\)
\(318\) 0 0
\(319\) 406968. 0.223915
\(320\) 0 0
\(321\) −1.31400e6 −0.711759
\(322\) 0 0
\(323\) 3.64671e6i 1.94489i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 2.25412e6i − 1.16576i
\(328\) 0 0
\(329\) 567936. 0.289274
\(330\) 0 0
\(331\) 2.74963e6 1.37944 0.689722 0.724074i \(-0.257733\pi\)
0.689722 + 0.724074i \(0.257733\pi\)
\(332\) 0 0
\(333\) − 1.62118e6i − 0.801164i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 3.41489e6i − 1.63796i −0.573824 0.818978i \(-0.694541\pi\)
0.573824 0.818978i \(-0.305459\pi\)
\(338\) 0 0
\(339\) 470040. 0.222145
\(340\) 0 0
\(341\) −710272. −0.330780
\(342\) 0 0
\(343\) − 792912.i − 0.363906i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 730764.i 0.325802i 0.986642 + 0.162901i \(0.0520851\pi\)
−0.986642 + 0.162901i \(0.947915\pi\)
\(348\) 0 0
\(349\) 2.29749e6 1.00969 0.504847 0.863209i \(-0.331549\pi\)
0.504847 + 0.863209i \(0.331549\pi\)
\(350\) 0 0
\(351\) −822160. −0.356196
\(352\) 0 0
\(353\) 1.17072e6i 0.500052i 0.968239 + 0.250026i \(0.0804392\pi\)
−0.968239 + 0.250026i \(0.919561\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 575040.i 0.238796i
\(358\) 0 0
\(359\) −3.88654e6 −1.59157 −0.795787 0.605577i \(-0.792942\pi\)
−0.795787 + 0.605577i \(0.792942\pi\)
\(360\) 0 0
\(361\) 6.78984e6 2.74215
\(362\) 0 0
\(363\) 2.91350e6i 1.16051i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 933040.i 0.361606i 0.983519 + 0.180803i \(0.0578696\pi\)
−0.983519 + 0.180803i \(0.942130\pi\)
\(368\) 0 0
\(369\) 1.39510e6 0.533384
\(370\) 0 0
\(371\) −280464. −0.105789
\(372\) 0 0
\(373\) 392218.i 0.145967i 0.997333 + 0.0729836i \(0.0232521\pi\)
−0.997333 + 0.0729836i \(0.976748\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 1.56880e6i − 0.568477i
\(378\) 0 0
\(379\) 4.72930e6 1.69122 0.845608 0.533805i \(-0.179238\pi\)
0.845608 + 0.533805i \(0.179238\pi\)
\(380\) 0 0
\(381\) −1.89184e6 −0.667686
\(382\) 0 0
\(383\) − 1.89734e6i − 0.660920i −0.943820 0.330460i \(-0.892796\pi\)
0.943820 0.330460i \(-0.107204\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 1.44252e6i − 0.489602i
\(388\) 0 0
\(389\) 3.72295e6 1.24742 0.623711 0.781655i \(-0.285624\pi\)
0.623711 + 0.781655i \(0.285624\pi\)
\(390\) 0 0
\(391\) −220432. −0.0729177
\(392\) 0 0
\(393\) − 1.40584e6i − 0.459150i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 3.33808e6i 1.06297i 0.847068 + 0.531484i \(0.178365\pi\)
−0.847068 + 0.531484i \(0.821635\pi\)
\(398\) 0 0
\(399\) 1.46112e6 0.459466
\(400\) 0 0
\(401\) 4.27490e6 1.32759 0.663796 0.747913i \(-0.268944\pi\)
0.663796 + 0.747913i \(0.268944\pi\)
\(402\) 0 0
\(403\) 2.73798e6i 0.839785i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.28042e6i 0.383149i
\(408\) 0 0
\(409\) 2.57319e6 0.760613 0.380306 0.924861i \(-0.375819\pi\)
0.380306 + 0.924861i \(0.375819\pi\)
\(410\) 0 0
\(411\) 5.54580e6 1.61942
\(412\) 0 0
\(413\) 405024.i 0.116844i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 2.60616e6i − 0.733941i
\(418\) 0 0
\(419\) −5.26828e6 −1.46600 −0.732999 0.680230i \(-0.761880\pi\)
−0.732999 + 0.680230i \(0.761880\pi\)
\(420\) 0 0
\(421\) −973354. −0.267649 −0.133824 0.991005i \(-0.542726\pi\)
−0.133824 + 0.991005i \(0.542726\pi\)
\(422\) 0 0
\(423\) − 3.71525e6i − 1.00957i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 443568.i 0.117731i
\(428\) 0 0
\(429\) −1.18544e6 −0.310983
\(430\) 0 0
\(431\) 3.55736e6 0.922433 0.461216 0.887288i \(-0.347413\pi\)
0.461216 + 0.887288i \(0.347413\pi\)
\(432\) 0 0
\(433\) 1.95496e6i 0.501092i 0.968105 + 0.250546i \(0.0806102\pi\)
−0.968105 + 0.250546i \(0.919390\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 560096.i 0.140300i
\(438\) 0 0
\(439\) 3.29681e6 0.816455 0.408228 0.912880i \(-0.366147\pi\)
0.408228 + 0.912880i \(0.366147\pi\)
\(440\) 0 0
\(441\) −2.54827e6 −0.623948
\(442\) 0 0
\(443\) 5.05820e6i 1.22458i 0.790634 + 0.612289i \(0.209751\pi\)
−0.790634 + 0.612289i \(0.790249\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 8.03060e6i − 1.90099i
\(448\) 0 0
\(449\) −2.12730e6 −0.497981 −0.248990 0.968506i \(-0.580099\pi\)
−0.248990 + 0.968506i \(0.580099\pi\)
\(450\) 0 0
\(451\) −1.10186e6 −0.255086
\(452\) 0 0
\(453\) 1.51952e6i 0.347905i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 289130.i 0.0647594i 0.999476 + 0.0323797i \(0.0103086\pi\)
−0.999476 + 0.0323797i \(0.989691\pi\)
\(458\) 0 0
\(459\) −2.06056e6 −0.456513
\(460\) 0 0
\(461\) 2.66870e6 0.584854 0.292427 0.956288i \(-0.405537\pi\)
0.292427 + 0.956288i \(0.405537\pi\)
\(462\) 0 0
\(463\) − 7.58619e6i − 1.64464i −0.569024 0.822321i \(-0.692679\pi\)
0.569024 0.822321i \(-0.307321\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 1.41961e6i − 0.301216i −0.988594 0.150608i \(-0.951877\pi\)
0.988594 0.150608i \(-0.0481231\pi\)
\(468\) 0 0
\(469\) −372768. −0.0782540
\(470\) 0 0
\(471\) −7.88644e6 −1.63806
\(472\) 0 0
\(473\) 1.13931e6i 0.234148i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.83470e6i 0.369207i
\(478\) 0 0
\(479\) 1.88406e6 0.375195 0.187597 0.982246i \(-0.439930\pi\)
0.187597 + 0.982246i \(0.439930\pi\)
\(480\) 0 0
\(481\) 4.93583e6 0.972741
\(482\) 0 0
\(483\) 88320.0i 0.0172263i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 6.01388e6i − 1.14903i −0.818493 0.574516i \(-0.805190\pi\)
0.818493 0.574516i \(-0.194810\pi\)
\(488\) 0 0
\(489\) 234480. 0.0443439
\(490\) 0 0
\(491\) 4.29232e6 0.803504 0.401752 0.915749i \(-0.368401\pi\)
0.401752 + 0.915749i \(0.368401\pi\)
\(492\) 0 0
\(493\) − 3.93184e6i − 0.728581i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 767040.i 0.139292i
\(498\) 0 0
\(499\) −1.34509e6 −0.241825 −0.120912 0.992663i \(-0.538582\pi\)
−0.120912 + 0.992663i \(0.538582\pi\)
\(500\) 0 0
\(501\) −1.10386e7 −1.96480
\(502\) 0 0
\(503\) − 202008.i − 0.0355999i −0.999842 0.0177999i \(-0.994334\pi\)
0.999842 0.0177999i \(-0.00566620\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 2.85618e6i − 0.493476i
\(508\) 0 0
\(509\) −9.78344e6 −1.67377 −0.836887 0.547375i \(-0.815627\pi\)
−0.836887 + 0.547375i \(0.815627\pi\)
\(510\) 0 0
\(511\) 117264. 0.0198661
\(512\) 0 0
\(513\) 5.23568e6i 0.878374i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 2.93434e6i 0.482818i
\(518\) 0 0
\(519\) −8.65788e6 −1.41089
\(520\) 0 0
\(521\) −1.04830e7 −1.69197 −0.845985 0.533207i \(-0.820987\pi\)
−0.845985 + 0.533207i \(0.820987\pi\)
\(522\) 0 0
\(523\) − 6.21017e6i − 0.992772i −0.868102 0.496386i \(-0.834660\pi\)
0.868102 0.496386i \(-0.165340\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.86214e6i 1.07630i
\(528\) 0 0
\(529\) 6.40249e6 0.994740
\(530\) 0 0
\(531\) 2.64953e6 0.407787
\(532\) 0 0
\(533\) 4.24751e6i 0.647614i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 1.11924e7i 1.67489i
\(538\) 0 0
\(539\) 2.01264e6 0.298397
\(540\) 0 0
\(541\) 5.08088e6 0.746355 0.373178 0.927760i \(-0.378268\pi\)
0.373178 + 0.927760i \(0.378268\pi\)
\(542\) 0 0
\(543\) − 1.20942e7i − 1.76026i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 3.34687e6i 0.478267i 0.970987 + 0.239133i \(0.0768633\pi\)
−0.970987 + 0.239133i \(0.923137\pi\)
\(548\) 0 0
\(549\) 2.90167e6 0.410883
\(550\) 0 0
\(551\) −9.99041e6 −1.40186
\(552\) 0 0
\(553\) 1.06944e6i 0.148711i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 7.00377e6i 0.956520i 0.878218 + 0.478260i \(0.158732\pi\)
−0.878218 + 0.478260i \(0.841268\pi\)
\(558\) 0 0
\(559\) 4.39186e6 0.594456
\(560\) 0 0
\(561\) −2.97104e6 −0.398567
\(562\) 0 0
\(563\) 1.29819e7i 1.72610i 0.505116 + 0.863052i \(0.331450\pi\)
−0.505116 + 0.863052i \(0.668550\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.74122e6i 0.227456i
\(568\) 0 0
\(569\) −1.89942e6 −0.245946 −0.122973 0.992410i \(-0.539243\pi\)
−0.122973 + 0.992410i \(0.539243\pi\)
\(570\) 0 0
\(571\) −1.66300e6 −0.213452 −0.106726 0.994288i \(-0.534037\pi\)
−0.106726 + 0.994288i \(0.534037\pi\)
\(572\) 0 0
\(573\) 8.18304e6i 1.04119i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 8.77344e6i 1.09706i 0.836131 + 0.548530i \(0.184812\pi\)
−0.836131 + 0.548530i \(0.815188\pi\)
\(578\) 0 0
\(579\) −1.08173e7 −1.34098
\(580\) 0 0
\(581\) −1.61674e6 −0.198700
\(582\) 0 0
\(583\) − 1.44906e6i − 0.176570i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 5.18393e6i 0.620961i 0.950580 + 0.310480i \(0.100490\pi\)
−0.950580 + 0.310480i \(0.899510\pi\)
\(588\) 0 0
\(589\) 1.74360e7 2.07090
\(590\) 0 0
\(591\) −1.25980e7 −1.48365
\(592\) 0 0
\(593\) − 8.49858e6i − 0.992452i −0.868193 0.496226i \(-0.834719\pi\)
0.868193 0.496226i \(-0.165281\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 5.66096e6i 0.650061i
\(598\) 0 0
\(599\) −1.12471e7 −1.28078 −0.640388 0.768051i \(-0.721227\pi\)
−0.640388 + 0.768051i \(0.721227\pi\)
\(600\) 0 0
\(601\) −3.46439e6 −0.391238 −0.195619 0.980680i \(-0.562672\pi\)
−0.195619 + 0.980680i \(0.562672\pi\)
\(602\) 0 0
\(603\) 2.43852e6i 0.273108i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 999712.i − 0.110129i −0.998483 0.0550647i \(-0.982463\pi\)
0.998483 0.0550647i \(-0.0175365\pi\)
\(608\) 0 0
\(609\) −1.57536e6 −0.172122
\(610\) 0 0
\(611\) 1.13114e7 1.22578
\(612\) 0 0
\(613\) − 9.81340e6i − 1.05480i −0.849619 0.527398i \(-0.823168\pi\)
0.849619 0.527398i \(-0.176832\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 5.34745e6i − 0.565501i −0.959193 0.282751i \(-0.908753\pi\)
0.959193 0.282751i \(-0.0912469\pi\)
\(618\) 0 0
\(619\) 6.82768e6 0.716221 0.358110 0.933679i \(-0.383421\pi\)
0.358110 + 0.933679i \(0.383421\pi\)
\(620\) 0 0
\(621\) −316480. −0.0329319
\(622\) 0 0
\(623\) 1.72786e6i 0.178356i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 7.54912e6i 0.766880i
\(628\) 0 0
\(629\) 1.23705e7 1.24670
\(630\) 0 0
\(631\) −3.60970e6 −0.360909 −0.180455 0.983583i \(-0.557757\pi\)
−0.180455 + 0.983583i \(0.557757\pi\)
\(632\) 0 0
\(633\) − 2.85512e6i − 0.283214i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 7.75842e6i − 0.757573i
\(638\) 0 0
\(639\) 5.01772e6 0.486132
\(640\) 0 0
\(641\) −1.33853e7 −1.28672 −0.643361 0.765563i \(-0.722460\pi\)
−0.643361 + 0.765563i \(0.722460\pi\)
\(642\) 0 0
\(643\) 9.91115e6i 0.945358i 0.881235 + 0.472679i \(0.156713\pi\)
−0.881235 + 0.472679i \(0.843287\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 1.78359e7i − 1.67508i −0.546378 0.837539i \(-0.683994\pi\)
0.546378 0.837539i \(-0.316006\pi\)
\(648\) 0 0
\(649\) −2.09262e6 −0.195020
\(650\) 0 0
\(651\) 2.74944e6 0.254268
\(652\) 0 0
\(653\) 4.32323e6i 0.396758i 0.980125 + 0.198379i \(0.0635677\pi\)
−0.980125 + 0.198379i \(0.936432\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 767102.i − 0.0693330i
\(658\) 0 0
\(659\) −1.97858e7 −1.77476 −0.887382 0.461035i \(-0.847478\pi\)
−0.887382 + 0.461035i \(0.847478\pi\)
\(660\) 0 0
\(661\) 1.57772e7 1.40451 0.702255 0.711925i \(-0.252176\pi\)
0.702255 + 0.711925i \(0.252176\pi\)
\(662\) 0 0
\(663\) 1.14529e7i 1.01188i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 603888.i − 0.0525584i
\(668\) 0 0
\(669\) −1.77939e7 −1.53711
\(670\) 0 0
\(671\) −2.29177e6 −0.196501
\(672\) 0 0
\(673\) − 6.78762e6i − 0.577670i −0.957379 0.288835i \(-0.906732\pi\)
0.957379 0.288835i \(-0.0932679\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 1.49942e7i − 1.25734i −0.777673 0.628669i \(-0.783600\pi\)
0.777673 0.628669i \(-0.216400\pi\)
\(678\) 0 0
\(679\) 1.17278e6 0.0976211
\(680\) 0 0
\(681\) 2.28631e7 1.88916
\(682\) 0 0
\(683\) − 1.15580e7i − 0.948053i −0.880511 0.474026i \(-0.842800\pi\)
0.880511 0.474026i \(-0.157200\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 1.39157e7i − 1.12490i
\(688\) 0 0
\(689\) −5.58591e6 −0.448276
\(690\) 0 0
\(691\) −220156. −0.0175402 −0.00877012 0.999962i \(-0.502792\pi\)
−0.00877012 + 0.999962i \(0.502792\pi\)
\(692\) 0 0
\(693\) 467232.i 0.0369572i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.06454e7i 0.830006i
\(698\) 0 0
\(699\) 6.94252e6 0.537433
\(700\) 0 0
\(701\) 4.78933e6 0.368111 0.184056 0.982916i \(-0.441077\pi\)
0.184056 + 0.982916i \(0.441077\pi\)
\(702\) 0 0
\(703\) − 3.14323e7i − 2.39877i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 1.23854e6i − 0.0931886i
\(708\) 0 0
\(709\) −4.26892e6 −0.318935 −0.159468 0.987203i \(-0.550978\pi\)
−0.159468 + 0.987203i \(0.550978\pi\)
\(710\) 0 0
\(711\) 6.99592e6 0.519004
\(712\) 0 0
\(713\) 1.05395e6i 0.0776421i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 3.28592e7i − 2.38704i
\(718\) 0 0
\(719\) 1.61960e7 1.16838 0.584190 0.811617i \(-0.301412\pi\)
0.584190 + 0.811617i \(0.301412\pi\)
\(720\) 0 0
\(721\) −4.33018e6 −0.310218
\(722\) 0 0
\(723\) 2.33488e7i 1.66119i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 6.53426e6i 0.458522i 0.973365 + 0.229261i \(0.0736310\pi\)
−0.973365 + 0.229261i \(0.926369\pi\)
\(728\) 0 0
\(729\) 3.03131e6 0.211257
\(730\) 0 0
\(731\) 1.10072e7 0.761876
\(732\) 0 0
\(733\) − 1.31617e7i − 0.904800i −0.891815 0.452400i \(-0.850568\pi\)
0.891815 0.452400i \(-0.149432\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 1.92597e6i − 0.130611i
\(738\) 0 0
\(739\) 1.42348e7 0.958825 0.479412 0.877590i \(-0.340850\pi\)
0.479412 + 0.877590i \(0.340850\pi\)
\(740\) 0 0
\(741\) 2.91006e7 1.94696
\(742\) 0 0
\(743\) 2.15835e7i 1.43434i 0.696901 + 0.717168i \(0.254562\pi\)
−0.696901 + 0.717168i \(0.745438\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 1.05761e7i 0.693467i
\(748\) 0 0
\(749\) −1.57680e6 −0.102700
\(750\) 0 0
\(751\) 1.86594e7 1.20725 0.603625 0.797268i \(-0.293722\pi\)
0.603625 + 0.797268i \(0.293722\pi\)
\(752\) 0 0
\(753\) 1.58122e7i 1.01626i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 2.56681e6i − 0.162800i −0.996682 0.0813999i \(-0.974061\pi\)
0.996682 0.0813999i \(-0.0259391\pi\)
\(758\) 0 0
\(759\) −456320. −0.0287518
\(760\) 0 0
\(761\) −2.59586e7 −1.62487 −0.812436 0.583051i \(-0.801859\pi\)
−0.812436 + 0.583051i \(0.801859\pi\)
\(762\) 0 0
\(763\) − 2.70494e6i − 0.168208i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.06673e6i 0.495118i
\(768\) 0 0
\(769\) −5.53267e6 −0.337380 −0.168690 0.985669i \(-0.553954\pi\)
−0.168690 + 0.985669i \(0.553954\pi\)
\(770\) 0 0
\(771\) −2.59580e6 −0.157266
\(772\) 0 0
\(773\) − 8.32940e6i − 0.501378i −0.968068 0.250689i \(-0.919343\pi\)
0.968068 0.250689i \(-0.0806571\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 4.95648e6i − 0.294524i
\(778\) 0 0
\(779\) 2.70490e7 1.59701
\(780\) 0 0
\(781\) −3.96304e6 −0.232488
\(782\) 0 0
\(783\) − 5.64504e6i − 0.329051i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 1.36523e7i − 0.785719i −0.919598 0.392860i \(-0.871486\pi\)
0.919598 0.392860i \(-0.128514\pi\)
\(788\) 0 0
\(789\) −1.41776e6 −0.0810793
\(790\) 0 0
\(791\) 564048. 0.0320535
\(792\) 0 0
\(793\) 8.83440e6i 0.498877i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 8.54626e6i − 0.476574i −0.971195 0.238287i \(-0.923414\pi\)
0.971195 0.238287i \(-0.0765859\pi\)
\(798\) 0 0
\(799\) 2.83495e7 1.57101
\(800\) 0 0
\(801\) 1.13031e7 0.622465
\(802\) 0 0
\(803\) 605864.i 0.0331578i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 3.58035e7i 1.93527i
\(808\) 0 0
\(809\) −7.58484e6 −0.407451 −0.203725 0.979028i \(-0.565305\pi\)
−0.203725 + 0.979028i \(0.565305\pi\)
\(810\) 0 0
\(811\) 6.18473e6 0.330194 0.165097 0.986277i \(-0.447206\pi\)
0.165097 + 0.986277i \(0.447206\pi\)
\(812\) 0 0
\(813\) 3.54723e7i 1.88219i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 2.79683e7i − 1.46592i
\(818\) 0 0
\(819\) 1.80110e6 0.0938273
\(820\) 0 0
\(821\) −2.78102e6 −0.143995 −0.0719973 0.997405i \(-0.522937\pi\)
−0.0719973 + 0.997405i \(0.522937\pi\)
\(822\) 0 0
\(823\) − 1.63895e7i − 0.843461i −0.906721 0.421731i \(-0.861423\pi\)
0.906721 0.421731i \(-0.138577\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.29511e7i 1.16692i 0.812142 + 0.583459i \(0.198301\pi\)
−0.812142 + 0.583459i \(0.801699\pi\)
\(828\) 0 0
\(829\) 3.50136e6 0.176950 0.0884750 0.996078i \(-0.471801\pi\)
0.0884750 + 0.996078i \(0.471801\pi\)
\(830\) 0 0
\(831\) −5.50900e6 −0.276739
\(832\) 0 0
\(833\) − 1.94447e7i − 0.970934i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 9.85216e6i 0.486091i
\(838\) 0 0
\(839\) −5.29668e6 −0.259776 −0.129888 0.991529i \(-0.541462\pi\)
−0.129888 + 0.991529i \(0.541462\pi\)
\(840\) 0 0
\(841\) −9.73962e6 −0.474845
\(842\) 0 0
\(843\) − 1.18834e7i − 0.575933i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 3.49620e6i 0.167451i
\(848\) 0 0
\(849\) −2.18486e7 −1.04029
\(850\) 0 0
\(851\) 1.89998e6 0.0899344
\(852\) 0 0
\(853\) 2.02948e7i 0.955021i 0.878626 + 0.477511i \(0.158461\pi\)
−0.878626 + 0.477511i \(0.841539\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 4.82785e6i − 0.224544i −0.993678 0.112272i \(-0.964187\pi\)
0.993678 0.112272i \(-0.0358128\pi\)
\(858\) 0 0
\(859\) 1.30210e7 0.602092 0.301046 0.953610i \(-0.402664\pi\)
0.301046 + 0.953610i \(0.402664\pi\)
\(860\) 0 0
\(861\) 4.26528e6 0.196083
\(862\) 0 0
\(863\) 3.92387e7i 1.79344i 0.442596 + 0.896721i \(0.354058\pi\)
−0.442596 + 0.896721i \(0.645942\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 306940.i 0.0138677i
\(868\) 0 0
\(869\) −5.52544e6 −0.248209
\(870\) 0 0
\(871\) −7.42430e6 −0.331596
\(872\) 0 0
\(873\) − 7.67196e6i − 0.340699i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.34622e7i 0.591041i 0.955336 + 0.295520i \(0.0954930\pi\)
−0.955336 + 0.295520i \(0.904507\pi\)
\(878\) 0 0
\(879\) −6.67308e6 −0.291309
\(880\) 0 0
\(881\) −917710. −0.0398351 −0.0199175 0.999802i \(-0.506340\pi\)
−0.0199175 + 0.999802i \(0.506340\pi\)
\(882\) 0 0
\(883\) − 2.45488e7i − 1.05957i −0.848133 0.529784i \(-0.822273\pi\)
0.848133 0.529784i \(-0.177727\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 1.61463e7i − 0.689070i −0.938773 0.344535i \(-0.888037\pi\)
0.938773 0.344535i \(-0.111963\pi\)
\(888\) 0 0
\(889\) −2.27021e6 −0.0963410
\(890\) 0 0
\(891\) −8.99632e6 −0.379639
\(892\) 0 0
\(893\) − 7.20332e7i − 3.02276i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 1.75904e6i 0.0729953i
\(898\) 0 0
\(899\) −1.87993e7 −0.775787
\(900\) 0 0
\(901\) −1.39998e7 −0.574527
\(902\) 0 0
\(903\) − 4.41024e6i − 0.179988i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 2.03361e7i − 0.820824i −0.911900 0.410412i \(-0.865385\pi\)
0.911900 0.410412i \(-0.134615\pi\)
\(908\) 0 0
\(909\) −8.10214e6 −0.325230
\(910\) 0 0
\(911\) 1.07726e7 0.430054 0.215027 0.976608i \(-0.431016\pi\)
0.215027 + 0.976608i \(0.431016\pi\)
\(912\) 0 0
\(913\) − 8.35314e6i − 0.331644i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 1.68701e6i − 0.0662512i
\(918\) 0 0
\(919\) −4.18566e7 −1.63484 −0.817419 0.576043i \(-0.804596\pi\)
−0.817419 + 0.576043i \(0.804596\pi\)
\(920\) 0 0
\(921\) 2.11994e7 0.823522
\(922\) 0 0
\(923\) 1.52769e7i 0.590242i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 2.83266e7i 1.08267i
\(928\) 0 0
\(929\) −2.99845e7 −1.13988 −0.569939 0.821687i \(-0.693033\pi\)
−0.569939 + 0.821687i \(0.693033\pi\)
\(930\) 0 0
\(931\) −4.94072e7 −1.86817
\(932\) 0 0
\(933\) 2.67298e7i 1.00529i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.42402e7i 0.529867i 0.964267 + 0.264934i \(0.0853501\pi\)
−0.964267 + 0.264934i \(0.914650\pi\)
\(938\) 0 0
\(939\) −3.28837e7 −1.21707
\(940\) 0 0
\(941\) −4.14546e7 −1.52615 −0.763077 0.646307i \(-0.776313\pi\)
−0.763077 + 0.646307i \(0.776313\pi\)
\(942\) 0 0
\(943\) 1.63502e6i 0.0598749i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 1.54079e7i − 0.558300i −0.960248 0.279150i \(-0.909947\pi\)
0.960248 0.279150i \(-0.0900527\pi\)
\(948\) 0 0
\(949\) 2.33551e6 0.0841813
\(950\) 0 0
\(951\) −3.44740e7 −1.23606
\(952\) 0 0
\(953\) 2.06328e7i 0.735912i 0.929843 + 0.367956i \(0.119942\pi\)
−0.929843 + 0.367956i \(0.880058\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 8.13936e6i − 0.287283i
\(958\) 0 0
\(959\) 6.65496e6 0.233668
\(960\) 0 0
\(961\) 4.18083e6 0.146034
\(962\) 0 0
\(963\) 1.03149e7i 0.358426i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1.18724e7i 0.408294i 0.978940 + 0.204147i \(0.0654421\pi\)
−0.978940 + 0.204147i \(0.934558\pi\)
\(968\) 0 0
\(969\) 7.29342e7 2.49530
\(970\) 0 0
\(971\) 1.53222e6 0.0521523 0.0260761 0.999660i \(-0.491699\pi\)
0.0260761 + 0.999660i \(0.491699\pi\)
\(972\) 0 0
\(973\) − 3.12739e6i − 0.105901i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.74321e7i 0.584269i 0.956377 + 0.292135i \(0.0943655\pi\)
−0.956377 + 0.292135i \(0.905635\pi\)
\(978\) 0 0
\(979\) −8.92726e6 −0.297688
\(980\) 0 0
\(981\) −1.76948e7 −0.587049
\(982\) 0 0
\(983\) − 2.23270e6i − 0.0736963i −0.999321 0.0368482i \(-0.988268\pi\)
0.999321 0.0368482i \(-0.0117318\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 1.13587e7i − 0.371139i
\(988\) 0 0
\(989\) 1.69059e6 0.0549602
\(990\) 0 0
\(991\) 2.22501e7 0.719693 0.359847 0.933011i \(-0.382829\pi\)
0.359847 + 0.933011i \(0.382829\pi\)
\(992\) 0 0
\(993\) − 5.49926e7i − 1.76983i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 5.32662e7i 1.69712i 0.529095 + 0.848562i \(0.322531\pi\)
−0.529095 + 0.848562i \(0.677469\pi\)
\(998\) 0 0
\(999\) 1.77607e7 0.563050
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 200.6.c.a.49.1 2
4.3 odd 2 400.6.c.d.49.2 2
5.2 odd 4 200.6.a.a.1.1 1
5.3 odd 4 8.6.a.a.1.1 1
5.4 even 2 inner 200.6.c.a.49.2 2
15.8 even 4 72.6.a.f.1.1 1
20.3 even 4 16.6.a.a.1.1 1
20.7 even 4 400.6.a.l.1.1 1
20.19 odd 2 400.6.c.d.49.1 2
35.3 even 12 392.6.i.e.177.1 2
35.13 even 4 392.6.a.b.1.1 1
35.18 odd 12 392.6.i.b.177.1 2
35.23 odd 12 392.6.i.b.361.1 2
35.33 even 12 392.6.i.e.361.1 2
40.3 even 4 64.6.a.g.1.1 1
40.13 odd 4 64.6.a.a.1.1 1
55.43 even 4 968.6.a.a.1.1 1
60.23 odd 4 144.6.a.k.1.1 1
80.3 even 4 256.6.b.d.129.1 2
80.13 odd 4 256.6.b.f.129.2 2
80.43 even 4 256.6.b.d.129.2 2
80.53 odd 4 256.6.b.f.129.1 2
120.53 even 4 576.6.a.g.1.1 1
120.83 odd 4 576.6.a.h.1.1 1
140.83 odd 4 784.6.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.6.a.a.1.1 1 5.3 odd 4
16.6.a.a.1.1 1 20.3 even 4
64.6.a.a.1.1 1 40.13 odd 4
64.6.a.g.1.1 1 40.3 even 4
72.6.a.f.1.1 1 15.8 even 4
144.6.a.k.1.1 1 60.23 odd 4
200.6.a.a.1.1 1 5.2 odd 4
200.6.c.a.49.1 2 1.1 even 1 trivial
200.6.c.a.49.2 2 5.4 even 2 inner
256.6.b.d.129.1 2 80.3 even 4
256.6.b.d.129.2 2 80.43 even 4
256.6.b.f.129.1 2 80.53 odd 4
256.6.b.f.129.2 2 80.13 odd 4
392.6.a.b.1.1 1 35.13 even 4
392.6.i.b.177.1 2 35.18 odd 12
392.6.i.b.361.1 2 35.23 odd 12
392.6.i.e.177.1 2 35.3 even 12
392.6.i.e.361.1 2 35.33 even 12
400.6.a.l.1.1 1 20.7 even 4
400.6.c.d.49.1 2 20.19 odd 2
400.6.c.d.49.2 2 4.3 odd 2
576.6.a.g.1.1 1 120.53 even 4
576.6.a.h.1.1 1 120.83 odd 4
784.6.a.l.1.1 1 140.83 odd 4
968.6.a.a.1.1 1 55.43 even 4