Properties

Label 256.6.b.f.129.1
Level $256$
Weight $6$
Character 256.129
Analytic conductor $41.058$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 256.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(41.0582578721\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 129.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 256.129
Dual form 256.6.b.f.129.2

$q$-expansion

\(f(q)\) \(=\) \(q-20.0000i q^{3} -74.0000i q^{5} +24.0000 q^{7} -157.000 q^{9} +O(q^{10})\) \(q-20.0000i q^{3} -74.0000i q^{5} +24.0000 q^{7} -157.000 q^{9} +124.000i q^{11} -478.000i q^{13} -1480.00 q^{15} -1198.00 q^{17} -3044.00i q^{19} -480.000i q^{21} -184.000 q^{23} -2351.00 q^{25} -1720.00i q^{27} +3282.00i q^{29} -5728.00 q^{31} +2480.00 q^{33} -1776.00i q^{35} +10326.0i q^{37} -9560.00 q^{39} +8886.00 q^{41} -9188.00i q^{43} +11618.0i q^{45} +23664.0 q^{47} -16231.0 q^{49} +23960.0i q^{51} +11686.0i q^{53} +9176.00 q^{55} -60880.0 q^{57} +16876.0i q^{59} +18482.0i q^{61} -3768.00 q^{63} -35372.0 q^{65} +15532.0i q^{67} +3680.00i q^{69} +31960.0 q^{71} +4886.00 q^{73} +47020.0i q^{75} +2976.00i q^{77} +44560.0 q^{79} -72551.0 q^{81} -67364.0i q^{83} +88652.0i q^{85} +65640.0 q^{87} -71994.0 q^{89} -11472.0i q^{91} +114560. i q^{93} -225256. q^{95} +48866.0 q^{97} -19468.0i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 48 q^{7} - 314 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 48 q^{7} - 314 q^{9} - 2960 q^{15} - 2396 q^{17} - 368 q^{23} - 4702 q^{25} - 11456 q^{31} + 4960 q^{33} - 19120 q^{39} + 17772 q^{41} + 47328 q^{47} - 32462 q^{49} + 18352 q^{55} - 121760 q^{57} - 7536 q^{63} - 70744 q^{65} + 63920 q^{71} + 9772 q^{73} + 89120 q^{79} - 145102 q^{81} + 131280 q^{87} - 143988 q^{89} - 450512 q^{95} + 97732 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(255\)
\(\chi(n)\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 20.0000i − 1.28300i −0.767123 0.641500i \(-0.778312\pi\)
0.767123 0.641500i \(-0.221688\pi\)
\(4\) 0 0
\(5\) − 74.0000i − 1.32375i −0.749613 0.661876i \(-0.769760\pi\)
0.749613 0.661876i \(-0.230240\pi\)
\(6\) 0 0
\(7\) 24.0000 0.185125 0.0925627 0.995707i \(-0.470494\pi\)
0.0925627 + 0.995707i \(0.470494\pi\)
\(8\) 0 0
\(9\) −157.000 −0.646091
\(10\) 0 0
\(11\) 124.000i 0.308987i 0.987994 + 0.154493i \(0.0493745\pi\)
−0.987994 + 0.154493i \(0.950625\pi\)
\(12\) 0 0
\(13\) − 478.000i − 0.784458i −0.919868 0.392229i \(-0.871704\pi\)
0.919868 0.392229i \(-0.128296\pi\)
\(14\) 0 0
\(15\) −1480.00 −1.69837
\(16\) 0 0
\(17\) −1198.00 −1.00539 −0.502695 0.864464i \(-0.667658\pi\)
−0.502695 + 0.864464i \(0.667658\pi\)
\(18\) 0 0
\(19\) − 3044.00i − 1.93446i −0.253894 0.967232i \(-0.581712\pi\)
0.253894 0.967232i \(-0.418288\pi\)
\(20\) 0 0
\(21\) − 480.000i − 0.237516i
\(22\) 0 0
\(23\) −184.000 −0.0725268 −0.0362634 0.999342i \(-0.511546\pi\)
−0.0362634 + 0.999342i \(0.511546\pi\)
\(24\) 0 0
\(25\) −2351.00 −0.752320
\(26\) 0 0
\(27\) − 1720.00i − 0.454066i
\(28\) 0 0
\(29\) 3282.00i 0.724676i 0.932047 + 0.362338i \(0.118021\pi\)
−0.932047 + 0.362338i \(0.881979\pi\)
\(30\) 0 0
\(31\) −5728.00 −1.07053 −0.535265 0.844684i \(-0.679788\pi\)
−0.535265 + 0.844684i \(0.679788\pi\)
\(32\) 0 0
\(33\) 2480.00 0.396430
\(34\) 0 0
\(35\) − 1776.00i − 0.245060i
\(36\) 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 \(-0.364558\pi\)
0.412778 + 0.910832i \(0.364558\pi\)
\(42\) 0 0
\(43\) − 9188.00i − 0.757792i −0.925439 0.378896i \(-0.876304\pi\)
0.925439 0.378896i \(-0.123696\pi\)
\(44\) 0 0
\(45\) 11618.0i 0.855264i
\(46\) 0 0
\(47\) 23664.0 1.56258 0.781292 0.624165i \(-0.214561\pi\)
0.781292 + 0.624165i \(0.214561\pi\)
\(48\) 0 0
\(49\) −16231.0 −0.965729
\(50\) 0 0
\(51\) 23960.0i 1.28992i
\(52\) 0 0
\(53\) 11686.0i 0.571447i 0.958312 + 0.285724i \(0.0922339\pi\)
−0.958312 + 0.285724i \(0.907766\pi\)
\(54\) 0 0
\(55\) 9176.00 0.409022
\(56\) 0 0
\(57\) −60880.0 −2.48192
\(58\) 0 0
\(59\) 16876.0i 0.631160i 0.948899 + 0.315580i \(0.102199\pi\)
−0.948899 + 0.315580i \(0.897801\pi\)
\(60\) 0 0
\(61\) 18482.0i 0.635952i 0.948099 + 0.317976i \(0.103003\pi\)
−0.948099 + 0.317976i \(0.896997\pi\)
\(62\) 0 0
\(63\) −3768.00 −0.119608
\(64\) 0 0
\(65\) −35372.0 −1.03843
\(66\) 0 0
\(67\) 15532.0i 0.422708i 0.977410 + 0.211354i \(0.0677873\pi\)
−0.977410 + 0.211354i \(0.932213\pi\)
\(68\) 0 0
\(69\) 3680.00i 0.0930519i
\(70\) 0 0
\(71\) 31960.0 0.752421 0.376210 0.926534i \(-0.377227\pi\)
0.376210 + 0.926534i \(0.377227\pi\)
\(72\) 0 0
\(73\) 4886.00 0.107312 0.0536558 0.998559i \(-0.482913\pi\)
0.0536558 + 0.998559i \(0.482913\pi\)
\(74\) 0 0
\(75\) 47020.0i 0.965227i
\(76\) 0 0
\(77\) 2976.00i 0.0572013i
\(78\) 0 0
\(79\) 44560.0 0.803299 0.401650 0.915793i \(-0.368437\pi\)
0.401650 + 0.915793i \(0.368437\pi\)
\(80\) 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
\(85\) 88652.0i 1.33089i
\(86\) 0 0
\(87\) 65640.0 0.929759
\(88\) 0 0
\(89\) −71994.0 −0.963432 −0.481716 0.876327i \(-0.659986\pi\)
−0.481716 + 0.876327i \(0.659986\pi\)
\(90\) 0 0
\(91\) − 11472.0i − 0.145223i
\(92\) 0 0
\(93\) 114560.i 1.37349i
\(94\) 0 0
\(95\) −225256. −2.56075
\(96\) 0 0
\(97\) 48866.0 0.527324 0.263662 0.964615i \(-0.415070\pi\)
0.263662 + 0.964615i \(0.415070\pi\)
\(98\) 0 0
\(99\) − 19468.0i − 0.199633i
\(100\) 0 0
\(101\) 51606.0i 0.503381i 0.967808 + 0.251690i \(0.0809865\pi\)
−0.967808 + 0.251690i \(0.919014\pi\)
\(102\) 0 0
\(103\) −180424. −1.67572 −0.837860 0.545886i \(-0.816193\pi\)
−0.837860 + 0.545886i \(0.816193\pi\)
\(104\) 0 0
\(105\) −35520.0 −0.314412
\(106\) 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.i 0.908617i 0.890844 + 0.454308i \(0.150114\pi\)
−0.890844 + 0.454308i \(0.849886\pi\)
\(110\) 0 0
\(111\) 206520. 1.59094
\(112\) 0 0
\(113\) −23502.0 −0.173145 −0.0865723 0.996246i \(-0.527591\pi\)
−0.0865723 + 0.996246i \(0.527591\pi\)
\(114\) 0 0
\(115\) 13616.0i 0.0960075i
\(116\) 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
\(125\) − 57276.0i − 0.327867i
\(126\) 0 0
\(127\) −94592.0 −0.520409 −0.260205 0.965553i \(-0.583790\pi\)
−0.260205 + 0.965553i \(0.583790\pi\)
\(128\) 0 0
\(129\) −183760. −0.972247
\(130\) 0 0
\(131\) − 70292.0i − 0.357872i −0.983861 0.178936i \(-0.942735\pi\)
0.983861 0.178936i \(-0.0572655\pi\)
\(132\) 0 0
\(133\) − 73056.0i − 0.358119i
\(134\) 0 0
\(135\) −127280. −0.601071
\(136\) 0 0
\(137\) −277290. −1.26221 −0.631107 0.775696i \(-0.717399\pi\)
−0.631107 + 0.775696i \(0.717399\pi\)
\(138\) 0 0
\(139\) − 130308.i − 0.572050i −0.958222 0.286025i \(-0.907666\pi\)
0.958222 0.286025i \(-0.0923341\pi\)
\(140\) 0 0
\(141\) − 473280.i − 2.00480i
\(142\) 0 0
\(143\) 59272.0 0.242387
\(144\) 0 0
\(145\) 242868. 0.959291
\(146\) 0 0
\(147\) 324620.i 1.23903i
\(148\) 0 0
\(149\) − 401530.i − 1.48167i −0.671685 0.740836i \(-0.734429\pi\)
0.671685 0.740836i \(-0.265571\pi\)
\(150\) 0 0
\(151\) 75976.0 0.271165 0.135583 0.990766i \(-0.456709\pi\)
0.135583 + 0.990766i \(0.456709\pi\)
\(152\) 0 0
\(153\) 188086. 0.649573
\(154\) 0 0
\(155\) 423872.i 1.41712i
\(156\) 0 0
\(157\) 394322.i 1.27674i 0.769730 + 0.638369i \(0.220391\pi\)
−0.769730 + 0.638369i \(0.779609\pi\)
\(158\) 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
\(165\) − 183520.i − 0.524775i
\(166\) 0 0
\(167\) 551928. 1.53141 0.765705 0.643192i \(-0.222390\pi\)
0.765705 + 0.643192i \(0.222390\pi\)
\(168\) 0 0
\(169\) 142809. 0.384626
\(170\) 0 0
\(171\) 477908.i 1.24984i
\(172\) 0 0
\(173\) − 432894.i − 1.09968i −0.835270 0.549840i \(-0.814689\pi\)
0.835270 0.549840i \(-0.185311\pi\)
\(174\) 0 0
\(175\) −56424.0 −0.139274
\(176\) 0 0
\(177\) 337520. 0.809779
\(178\) 0 0
\(179\) − 559620.i − 1.30545i −0.757594 0.652726i \(-0.773625\pi\)
0.757594 0.652726i \(-0.226375\pi\)
\(180\) 0 0
\(181\) 604710.i 1.37199i 0.727607 + 0.685995i \(0.240633\pi\)
−0.727607 + 0.685995i \(0.759367\pi\)
\(182\) 0 0
\(183\) 369640. 0.815927
\(184\) 0 0
\(185\) 764124. 1.64148
\(186\) 0 0
\(187\) − 148552.i − 0.310652i
\(188\) 0 0
\(189\) − 41280.0i − 0.0840592i
\(190\) 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. 1.04519 0.522596 0.852580i \(-0.324963\pi\)
0.522596 + 0.852580i \(0.324963\pi\)
\(194\) 0 0
\(195\) 707440.i 1.33230i
\(196\) 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
\(205\) − 657564.i − 1.09283i
\(206\) 0 0
\(207\) 28888.0 0.0468588
\(208\) 0 0
\(209\) 377456. 0.597724
\(210\) 0 0
\(211\) − 142756.i − 0.220744i −0.993890 0.110372i \(-0.964796\pi\)
0.993890 0.110372i \(-0.0352042\pi\)
\(212\) 0 0
\(213\) − 639200.i − 0.965357i
\(214\) 0 0
\(215\) −679912. −1.00313
\(216\) 0 0
\(217\) −137472. −0.198182
\(218\) 0 0
\(219\) − 97720.0i − 0.137681i
\(220\) 0 0
\(221\) 572644.i 0.788686i
\(222\) 0 0
\(223\) 889696. 1.19806 0.599031 0.800726i \(-0.295553\pi\)
0.599031 + 0.800726i \(0.295553\pi\)
\(224\) 0 0
\(225\) 369107. 0.486067
\(226\) 0 0
\(227\) − 1.14316e6i − 1.47245i −0.676736 0.736226i \(-0.736606\pi\)
0.676736 0.736226i \(-0.263394\pi\)
\(228\) 0 0
\(229\) − 695786.i − 0.876773i −0.898787 0.438386i \(-0.855550\pi\)
0.898787 0.438386i \(-0.144450\pi\)
\(230\) 0 0
\(231\) 59520.0 0.0733893
\(232\) 0 0
\(233\) 347126. 0.418887 0.209444 0.977821i \(-0.432835\pi\)
0.209444 + 0.977821i \(0.432835\pi\)
\(234\) 0 0
\(235\) − 1.75114e6i − 2.06847i
\(236\) 0 0
\(237\) − 891200.i − 1.03063i
\(238\) 0 0
\(239\) −1.64296e6 −1.86051 −0.930255 0.366912i \(-0.880415\pi\)
−0.930255 + 0.366912i \(0.880415\pi\)
\(240\) 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
\(245\) 1.20109e6i 1.27839i
\(246\) 0 0
\(247\) −1.45503e6 −1.51751
\(248\) 0 0
\(249\) −1.34728e6 −1.37708
\(250\) 0 0
\(251\) − 790612.i − 0.792098i −0.918229 0.396049i \(-0.870381\pi\)
0.918229 0.396049i \(-0.129619\pi\)
\(252\) 0 0
\(253\) − 22816.0i − 0.0224098i
\(254\) 0 0
\(255\) 1.77304e6 1.70753
\(256\) 0 0
\(257\) −129790. −0.122577 −0.0612884 0.998120i \(-0.519521\pi\)
−0.0612884 + 0.998120i \(0.519521\pi\)
\(258\) 0 0
\(259\) 247824.i 0.229559i
\(260\) 0 0
\(261\) − 515274.i − 0.468206i
\(262\) 0 0
\(263\) −70888.0 −0.0631951 −0.0315975 0.999501i \(-0.510059\pi\)
−0.0315975 + 0.999501i \(0.510059\pi\)
\(264\) 0 0
\(265\) 864764. 0.756455
\(266\) 0 0
\(267\) 1.43988e6i 1.23608i
\(268\) 0 0
\(269\) − 1.79017e6i − 1.50839i −0.656649 0.754197i \(-0.728027\pi\)
0.656649 0.754197i \(-0.271973\pi\)
\(270\) 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. −0.186321
\(274\) 0 0
\(275\) − 291524.i − 0.232457i
\(276\) 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 \(-0.572058\pi\)
−0.224448 + 0.974486i \(0.572058\pi\)
\(282\) 0 0
\(283\) 1.09243e6i 0.810824i 0.914134 + 0.405412i \(0.132872\pi\)
−0.914134 + 0.405412i \(0.867128\pi\)
\(284\) 0 0
\(285\) 4.50512e6i 3.28545i
\(286\) 0 0
\(287\) 213264. 0.152831
\(288\) 0 0
\(289\) 15347.0 0.0108088
\(290\) 0 0
\(291\) − 977320.i − 0.676557i
\(292\) 0 0
\(293\) 333654.i 0.227053i 0.993535 + 0.113527i \(0.0362147\pi\)
−0.993535 + 0.113527i \(0.963785\pi\)
\(294\) 0 0
\(295\) 1.24882e6 0.835500
\(296\) 0 0
\(297\) 213280. 0.140300
\(298\) 0 0
\(299\) 87952.0i 0.0568942i
\(300\) 0 0
\(301\) − 220512.i − 0.140287i
\(302\) 0 0
\(303\) 1.03212e6 0.645838
\(304\) 0 0
\(305\) 1.36767e6 0.841843
\(306\) 0 0
\(307\) − 1.05997e6i − 0.641872i −0.947101 0.320936i \(-0.896003\pi\)
0.947101 0.320936i \(-0.103997\pi\)
\(308\) 0 0
\(309\) 3.60848e6i 2.14995i
\(310\) 0 0
\(311\) 1.33649e6 0.783545 0.391773 0.920062i \(-0.371862\pi\)
0.391773 + 0.920062i \(0.371862\pi\)
\(312\) 0 0
\(313\) −1.64419e6 −0.948615 −0.474308 0.880359i \(-0.657302\pi\)
−0.474308 + 0.880359i \(0.657302\pi\)
\(314\) 0 0
\(315\) 278832.i 0.158331i
\(316\) 0 0
\(317\) 1.72370e6i 0.963414i 0.876332 + 0.481707i \(0.159983\pi\)
−0.876332 + 0.481707i \(0.840017\pi\)
\(318\) 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
\(325\) 1.12378e6i 0.590163i
\(326\) 0 0
\(327\) 2.25412e6 1.16576
\(328\) 0 0
\(329\) 567936. 0.289274
\(330\) 0 0
\(331\) 2.74963e6i 1.37944i 0.724074 + 0.689722i \(0.242267\pi\)
−0.724074 + 0.689722i \(0.757733\pi\)
\(332\) 0 0
\(333\) − 1.62118e6i − 0.801164i
\(334\) 0 0
\(335\) 1.14937e6 0.559561
\(336\) 0 0
\(337\) −3.41489e6 −1.63796 −0.818978 0.573824i \(-0.805459\pi\)
−0.818978 + 0.573824i \(0.805459\pi\)
\(338\) 0 0
\(339\) 470040.i 0.222145i
\(340\) 0 0
\(341\) − 710272.i − 0.330780i
\(342\) 0 0
\(343\) −792912. −0.363906
\(344\) 0 0
\(345\) 272320. 0.123178
\(346\) 0 0
\(347\) 730764.i 0.325802i 0.986642 + 0.162901i \(0.0520851\pi\)
−0.986642 + 0.162901i \(0.947915\pi\)
\(348\) 0 0
\(349\) 2.29749e6i 1.00969i 0.863209 + 0.504847i \(0.168451\pi\)
−0.863209 + 0.504847i \(0.831549\pi\)
\(350\) 0 0
\(351\) −822160. −0.356196
\(352\) 0 0
\(353\) −1.17072e6 −0.500052 −0.250026 0.968239i \(-0.580439\pi\)
−0.250026 + 0.968239i \(0.580439\pi\)
\(354\) 0 0
\(355\) − 2.36504e6i − 0.996019i
\(356\) 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
\(365\) − 361564.i − 0.142054i
\(366\) 0 0
\(367\) 933040. 0.361606 0.180803 0.983519i \(-0.442130\pi\)
0.180803 + 0.983519i \(0.442130\pi\)
\(368\) 0 0
\(369\) −1.39510e6 −0.533384
\(370\) 0 0
\(371\) 280464.i 0.105789i
\(372\) 0 0
\(373\) − 392218.i − 0.145967i −0.997333 0.0729836i \(-0.976748\pi\)
0.997333 0.0729836i \(-0.0232521\pi\)
\(374\) 0 0
\(375\) −1.14552e6 −0.420653
\(376\) 0 0
\(377\) 1.56880e6 0.568477
\(378\) 0 0
\(379\) − 4.72930e6i − 1.69122i −0.533805 0.845608i \(-0.679238\pi\)
0.533805 0.845608i \(-0.320762\pi\)
\(380\) 0 0
\(381\) 1.89184e6i 0.667686i
\(382\) 0 0
\(383\) 1.89734e6 0.660920 0.330460 0.943820i \(-0.392796\pi\)
0.330460 + 0.943820i \(0.392796\pi\)
\(384\) 0 0
\(385\) 220224. 0.0757204
\(386\) 0 0
\(387\) 1.44252e6i 0.489602i
\(388\) 0 0
\(389\) − 3.72295e6i − 1.24742i −0.781655 0.623711i \(-0.785624\pi\)
0.781655 0.623711i \(-0.214376\pi\)
\(390\) 0 0
\(391\) 220432. 0.0729177
\(392\) 0 0
\(393\) −1.40584e6 −0.459150
\(394\) 0 0
\(395\) − 3.29744e6i − 1.06337i
\(396\) 0 0
\(397\) − 3.33808e6i − 1.06297i −0.847068 0.531484i \(-0.821635\pi\)
0.847068 0.531484i \(-0.178365\pi\)
\(398\) 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
\(405\) 5.36877e6i 1.62644i
\(406\) 0 0
\(407\) −1.28042e6 −0.383149
\(408\) 0 0
\(409\) 2.57319e6 0.760613 0.380306 0.924861i \(-0.375819\pi\)
0.380306 + 0.924861i \(0.375819\pi\)
\(410\) 0 0
\(411\) 5.54580e6i 1.61942i
\(412\) 0 0
\(413\) 405024.i 0.116844i
\(414\) 0 0
\(415\) −4.98494e6 −1.42082
\(416\) 0 0
\(417\) −2.60616e6 −0.733941
\(418\) 0 0
\(419\) − 5.26828e6i − 1.46600i −0.680230 0.732999i \(-0.738120\pi\)
0.680230 0.732999i \(-0.261880\pi\)
\(420\) 0 0
\(421\) − 973354.i − 0.267649i −0.991005 0.133824i \(-0.957274\pi\)
0.991005 0.133824i \(-0.0427258\pi\)
\(422\) 0 0
\(423\) −3.71525e6 −1.00957
\(424\) 0 0
\(425\) 2.81650e6 0.756375
\(426\) 0 0
\(427\) 443568.i 0.117731i
\(428\) 0 0
\(429\) − 1.18544e6i − 0.310983i
\(430\) 0 0
\(431\) 3.55736e6 0.922433 0.461216 0.887288i \(-0.347413\pi\)
0.461216 + 0.887288i \(0.347413\pi\)
\(432\) 0 0
\(433\) −1.95496e6 −0.501092 −0.250546 0.968105i \(-0.580610\pi\)
−0.250546 + 0.968105i \(0.580610\pi\)
\(434\) 0 0
\(435\) − 4.85736e6i − 1.23077i
\(436\) 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 \(-0.790249\pi\)
0.790634 0.612289i \(-0.209751\pi\)
\(444\) 0 0
\(445\) 5.32756e6i 1.27535i
\(446\) 0 0
\(447\) −8.03060e6 −1.90099
\(448\) 0 0
\(449\) 2.12730e6 0.497981 0.248990 0.968506i \(-0.419901\pi\)
0.248990 + 0.968506i \(0.419901\pi\)
\(450\) 0 0
\(451\) 1.10186e6i 0.255086i
\(452\) 0 0
\(453\) − 1.51952e6i − 0.347905i
\(454\) 0 0
\(455\) −848928. −0.192239
\(456\) 0 0
\(457\) −289130. −0.0647594 −0.0323797 0.999476i \(-0.510309\pi\)
−0.0323797 + 0.999476i \(0.510309\pi\)
\(458\) 0 0
\(459\) 2.06056e6i 0.456513i
\(460\) 0 0
\(461\) − 2.66870e6i − 0.584854i −0.956288 0.292427i \(-0.905537\pi\)
0.956288 0.292427i \(-0.0944629\pi\)
\(462\) 0 0
\(463\) 7.58619e6 1.64464 0.822321 0.569024i \(-0.192679\pi\)
0.822321 + 0.569024i \(0.192679\pi\)
\(464\) 0 0
\(465\) 8.47744e6 1.81816
\(466\) 0 0
\(467\) 1.41961e6i 0.301216i 0.988594 + 0.150608i \(0.0481231\pi\)
−0.988594 + 0.150608i \(0.951877\pi\)
\(468\) 0 0
\(469\) 372768.i 0.0782540i
\(470\) 0 0
\(471\) 7.88644e6 1.63806
\(472\) 0 0
\(473\) 1.13931e6 0.234148
\(474\) 0 0
\(475\) 7.15644e6i 1.45534i
\(476\) 0 0
\(477\) − 1.83470e6i − 0.369207i
\(478\) 0 0
\(479\) −1.88406e6 −0.375195 −0.187597 0.982246i \(-0.560070\pi\)
−0.187597 + 0.982246i \(0.560070\pi\)
\(480\) 0 0
\(481\) 4.93583e6 0.972741
\(482\) 0 0
\(483\) 88320.0i 0.0172263i
\(484\) 0 0
\(485\) − 3.61608e6i − 0.698046i
\(486\) 0 0
\(487\) 6.01388e6 1.14903 0.574516 0.818493i \(-0.305190\pi\)
0.574516 + 0.818493i \(0.305190\pi\)
\(488\) 0 0
\(489\) 234480. 0.0443439
\(490\) 0 0
\(491\) 4.29232e6i 0.803504i 0.915749 + 0.401752i \(0.131599\pi\)
−0.915749 + 0.401752i \(0.868401\pi\)
\(492\) 0 0
\(493\) − 3.93184e6i − 0.728581i
\(494\) 0 0
\(495\) −1.44063e6 −0.264265
\(496\) 0 0
\(497\) 767040. 0.139292
\(498\) 0 0
\(499\) − 1.34509e6i − 0.241825i −0.992663 0.120912i \(-0.961418\pi\)
0.992663 0.120912i \(-0.0385820\pi\)
\(500\) 0 0
\(501\) − 1.10386e7i − 1.96480i
\(502\) 0 0
\(503\) −202008. −0.0355999 −0.0177999 0.999842i \(-0.505666\pi\)
−0.0177999 + 0.999842i \(0.505666\pi\)
\(504\) 0 0
\(505\) 3.81884e6 0.666352
\(506\) 0 0
\(507\) − 2.85618e6i − 0.493476i
\(508\) 0 0
\(509\) − 9.78344e6i − 1.67377i −0.547375 0.836887i \(-0.684373\pi\)
0.547375 0.836887i \(-0.315627\pi\)
\(510\) 0 0
\(511\) 117264. 0.0198661
\(512\) 0 0
\(513\) −5.23568e6 −0.878374
\(514\) 0 0
\(515\) 1.33514e7i 2.21824i
\(516\) 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 \(-0.179013\pi\)
0.845985 + 0.533207i \(0.179013\pi\)
\(522\) 0 0
\(523\) 6.21017e6i 0.992772i 0.868102 + 0.496386i \(0.165340\pi\)
−0.868102 + 0.496386i \(0.834660\pi\)
\(524\) 0 0
\(525\) 1.12848e6i 0.178688i
\(526\) 0 0
\(527\) 6.86214e6 1.07630
\(528\) 0 0
\(529\) −6.40249e6 −0.994740
\(530\) 0 0
\(531\) − 2.64953e6i − 0.407787i
\(532\) 0 0
\(533\) − 4.24751e6i − 0.647614i
\(534\) 0 0
\(535\) −4.86180e6 −0.734366
\(536\) 0 0
\(537\) −1.11924e7 −1.67489
\(538\) 0 0
\(539\) − 2.01264e6i − 0.298397i
\(540\) 0 0
\(541\) − 5.08088e6i − 0.746355i −0.927760 0.373178i \(-0.878268\pi\)
0.927760 0.373178i \(-0.121732\pi\)
\(542\) 0 0
\(543\) 1.20942e7 1.76026
\(544\) 0 0
\(545\) 8.34024e6 1.20278
\(546\) 0 0
\(547\) − 3.34687e6i − 0.478267i −0.970987 0.239133i \(-0.923137\pi\)
0.970987 0.239133i \(-0.0768633\pi\)
\(548\) 0 0
\(549\) − 2.90167e6i − 0.410883i
\(550\) 0 0
\(551\) 9.99041e6 1.40186
\(552\) 0 0
\(553\) 1.06944e6 0.148711
\(554\) 0 0
\(555\) − 1.52825e7i − 2.10601i
\(556\) 0 0
\(557\) − 7.00377e6i − 0.956520i −0.878218 0.478260i \(-0.841268\pi\)
0.878218 0.478260i \(-0.158732\pi\)
\(558\) 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
\(565\) 1.73915e6i 0.229200i
\(566\) 0 0
\(567\) −1.74122e6 −0.227456
\(568\) 0 0
\(569\) −1.89942e6 −0.245946 −0.122973 0.992410i \(-0.539243\pi\)
−0.122973 + 0.992410i \(0.539243\pi\)
\(570\) 0 0
\(571\) − 1.66300e6i − 0.213452i −0.994288 0.106726i \(-0.965963\pi\)
0.994288 0.106726i \(-0.0340368\pi\)
\(572\) 0 0
\(573\) 8.18304e6i 1.04119i
\(574\) 0 0
\(575\) 432584. 0.0545633
\(576\) 0 0
\(577\) 8.77344e6 1.09706 0.548530 0.836131i \(-0.315188\pi\)
0.548530 + 0.836131i \(0.315188\pi\)
\(578\) 0 0
\(579\) − 1.08173e7i − 1.34098i
\(580\) 0 0
\(581\) − 1.61674e6i − 0.198700i
\(582\) 0 0
\(583\) −1.44906e6 −0.176570
\(584\) 0 0
\(585\) 5.55340e6 0.670918
\(586\) 0 0
\(587\) 5.18393e6i 0.620961i 0.950580 + 0.310480i \(0.100490\pi\)
−0.950580 + 0.310480i \(0.899510\pi\)
\(588\) 0 0
\(589\) 1.74360e7i 2.07090i
\(590\) 0 0
\(591\) −1.25980e7 −1.48365
\(592\) 0 0
\(593\) 8.49858e6 0.992452 0.496226 0.868193i \(-0.334719\pi\)
0.496226 + 0.868193i \(0.334719\pi\)
\(594\) 0 0
\(595\) 2.12765e6i 0.246381i
\(596\) 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 \(-0.437328\pi\)
0.195619 + 0.980680i \(0.437328\pi\)
\(602\) 0 0
\(603\) − 2.43852e6i − 0.273108i
\(604\) 0 0
\(605\) − 1.07799e7i − 1.19737i
\(606\) 0 0
\(607\) −999712. −0.110129 −0.0550647 0.998483i \(-0.517537\pi\)
−0.0550647 + 0.998483i \(0.517537\pi\)
\(608\) 0 0
\(609\) 1.57536e6 0.172122
\(610\) 0 0
\(611\) − 1.13114e7i − 1.22578i
\(612\) 0 0
\(613\) 9.81340e6i 1.05480i 0.849619 + 0.527398i \(0.176832\pi\)
−0.849619 + 0.527398i \(0.823168\pi\)
\(614\) 0 0
\(615\) −1.31513e7 −1.40210
\(616\) 0 0
\(617\) 5.34745e6 0.565501 0.282751 0.959193i \(-0.408753\pi\)
0.282751 + 0.959193i \(0.408753\pi\)
\(618\) 0 0
\(619\) − 6.82768e6i − 0.716221i −0.933679 0.358110i \(-0.883421\pi\)
0.933679 0.358110i \(-0.116579\pi\)
\(620\) 0 0
\(621\) 316480.i 0.0329319i
\(622\) 0 0
\(623\) −1.72786e6 −0.178356
\(624\) 0 0
\(625\) −1.15853e7 −1.18633
\(626\) 0 0
\(627\) − 7.54912e6i − 0.766880i
\(628\) 0 0
\(629\) − 1.23705e7i − 1.24670i
\(630\) 0 0
\(631\) 3.60970e6 0.360909 0.180455 0.983583i \(-0.442243\pi\)
0.180455 + 0.983583i \(0.442243\pi\)
\(632\) 0 0
\(633\) −2.85512e6 −0.283214
\(634\) 0 0
\(635\) 6.99981e6i 0.688893i
\(636\) 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
\(645\) 1.35982e7i 1.28701i
\(646\) 0 0
\(647\) 1.78359e7 1.67508 0.837539 0.546378i \(-0.183994\pi\)
0.837539 + 0.546378i \(0.183994\pi\)
\(648\) 0 0
\(649\) −2.09262e6 −0.195020
\(650\) 0 0
\(651\) 2.74944e6i 0.254268i
\(652\) 0 0
\(653\) 4.32323e6i 0.396758i 0.980125 + 0.198379i \(0.0635677\pi\)
−0.980125 + 0.198379i \(0.936432\pi\)
\(654\) 0 0
\(655\) −5.20161e6 −0.473734
\(656\) 0 0
\(657\) −767102. −0.0693330
\(658\) 0 0
\(659\) − 1.97858e7i − 1.77476i −0.461035 0.887382i \(-0.652522\pi\)
0.461035 0.887382i \(-0.347478\pi\)
\(660\) 0 0
\(661\) 1.57772e7i 1.40451i 0.711925 + 0.702255i \(0.247824\pi\)
−0.711925 + 0.702255i \(0.752176\pi\)
\(662\) 0 0
\(663\) 1.14529e7 1.01188
\(664\) 0 0
\(665\) −5.40614e6 −0.474060
\(666\) 0 0
\(667\) − 603888.i − 0.0525584i
\(668\) 0 0
\(669\) − 1.77939e7i − 1.53711i
\(670\) 0 0
\(671\) −2.29177e6 −0.196501
\(672\) 0 0
\(673\) 6.78762e6 0.577670 0.288835 0.957379i \(-0.406732\pi\)
0.288835 + 0.957379i \(0.406732\pi\)
\(674\) 0 0
\(675\) 4.04372e6i 0.341603i
\(676\) 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 \(0.157200\pi\)
−0.880511 + 0.474026i \(0.842800\pi\)
\(684\) 0 0
\(685\) 2.05195e7i 1.67086i
\(686\) 0 0
\(687\) −1.39157e7 −1.12490
\(688\) 0 0
\(689\) 5.58591e6 0.448276
\(690\) 0 0
\(691\) 220156.i 0.0175402i 0.999962 + 0.00877012i \(0.00279165\pi\)
−0.999962 + 0.00877012i \(0.997208\pi\)
\(692\) 0 0
\(693\) − 467232.i − 0.0369572i
\(694\) 0 0
\(695\) −9.64279e6 −0.757253
\(696\) 0 0
\(697\) −1.06454e7 −0.830006
\(698\) 0 0
\(699\) − 6.94252e6i − 0.537433i
\(700\) 0 0
\(701\) − 4.78933e6i − 0.368111i −0.982916 0.184056i \(-0.941077\pi\)
0.982916 0.184056i \(-0.0589227\pi\)
\(702\) 0 0
\(703\) 3.14323e7 2.39877
\(704\) 0 0
\(705\) −3.50227e7 −2.65385
\(706\) 0 0
\(707\) 1.23854e6i 0.0931886i
\(708\) 0 0
\(709\) 4.26892e6i 0.318935i 0.987203 + 0.159468i \(0.0509777\pi\)
−0.987203 + 0.159468i \(0.949022\pi\)
\(710\) 0 0
\(711\) −6.99592e6 −0.519004
\(712\) 0 0
\(713\) 1.05395e6 0.0776421
\(714\) 0 0
\(715\) − 4.38613e6i − 0.320860i
\(716\) 0 0
\(717\) 3.28592e7i 2.38704i
\(718\) 0 0
\(719\) −1.61960e7 −1.16838 −0.584190 0.811617i \(-0.698588\pi\)
−0.584190 + 0.811617i \(0.698588\pi\)
\(720\) 0 0
\(721\) −4.33018e6 −0.310218
\(722\) 0 0
\(723\) 2.33488e7i 1.66119i
\(724\) 0 0
\(725\) − 7.71598e6i − 0.545188i
\(726\) 0 0
\(727\) −6.53426e6 −0.458522 −0.229261 0.973365i \(-0.573631\pi\)
−0.229261 + 0.973365i \(0.573631\pi\)
\(728\) 0 0
\(729\) 3.03131e6 0.211257
\(730\) 0 0
\(731\) 1.10072e7i 0.761876i
\(732\) 0 0
\(733\) − 1.31617e7i − 0.904800i −0.891815 0.452400i \(-0.850568\pi\)
0.891815 0.452400i \(-0.149432\pi\)
\(734\) 0 0
\(735\) 2.40219e7 1.64017
\(736\) 0 0
\(737\) −1.92597e6 −0.130611
\(738\) 0 0
\(739\) 1.42348e7i 0.958825i 0.877590 + 0.479412i \(0.159150\pi\)
−0.877590 + 0.479412i \(0.840850\pi\)
\(740\) 0 0
\(741\) 2.91006e7i 1.94696i
\(742\) 0 0
\(743\) 2.15835e7 1.43434 0.717168 0.696901i \(-0.245438\pi\)
0.717168 + 0.696901i \(0.245438\pi\)
\(744\) 0 0
\(745\) −2.97132e7 −1.96137
\(746\) 0 0
\(747\) 1.05761e7i 0.693467i
\(748\) 0 0
\(749\) − 1.57680e6i − 0.102700i
\(750\) 0 0
\(751\) 1.86594e7 1.20725 0.603625 0.797268i \(-0.293722\pi\)
0.603625 + 0.797268i \(0.293722\pi\)
\(752\) 0 0
\(753\) −1.58122e7 −1.01626
\(754\) 0 0
\(755\) − 5.62222e6i − 0.358956i
\(756\) 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 \(-0.198141\pi\)
0.812436 + 0.583051i \(0.198141\pi\)
\(762\) 0 0
\(763\) 2.70494e6i 0.168208i
\(764\) 0 0
\(765\) − 1.39184e7i − 0.859874i
\(766\) 0 0
\(767\) 8.06673e6 0.495118
\(768\) 0 0
\(769\) 5.53267e6 0.337380 0.168690 0.985669i \(-0.446046\pi\)
0.168690 + 0.985669i \(0.446046\pi\)
\(770\) 0 0
\(771\) 2.59580e6i 0.157266i
\(772\) 0 0
\(773\) 8.32940e6i 0.501378i 0.968068 + 0.250689i \(0.0806571\pi\)
−0.968068 + 0.250689i \(0.919343\pi\)
\(774\) 0 0
\(775\) 1.34665e7 0.805381
\(776\) 0 0
\(777\) 4.95648e6 0.294524
\(778\) 0 0
\(779\) − 2.70490e7i − 1.59701i
\(780\) 0 0
\(781\) 3.96304e6i 0.232488i
\(782\) 0 0
\(783\) 5.64504e6 0.329051
\(784\) 0 0
\(785\) 2.91798e7 1.69009
\(786\) 0 0
\(787\) 1.36523e7i 0.785719i 0.919598 + 0.392860i \(0.128514\pi\)
−0.919598 + 0.392860i \(0.871486\pi\)
\(788\) 0 0
\(789\) 1.41776e6i 0.0810793i
\(790\) 0 0
\(791\) −564048. −0.0320535
\(792\) 0 0
\(793\) 8.83440e6 0.498877
\(794\) 0 0
\(795\) − 1.72953e7i − 0.970532i
\(796\) 0 0
\(797\) 8.54626e6i 0.476574i 0.971195 + 0.238287i \(0.0765859\pi\)
−0.971195 + 0.238287i \(0.923414\pi\)
\(798\) 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
\(805\) 326784.i 0.0177734i
\(806\) 0 0
\(807\) −3.58035e7 −1.93527
\(808\) 0 0
\(809\) −7.58484e6 −0.407451 −0.203725 0.979028i \(-0.565305\pi\)
−0.203725 + 0.979028i \(0.565305\pi\)
\(810\) 0 0
\(811\) 6.18473e6i 0.330194i 0.986277 + 0.165097i \(0.0527937\pi\)
−0.986277 + 0.165097i \(0.947206\pi\)
\(812\) 0 0
\(813\) 3.54723e7i 1.88219i
\(814\) 0 0
\(815\) 867576. 0.0457524
\(816\) 0 0
\(817\) −2.79683e7 −1.46592
\(818\) 0 0
\(819\) 1.80110e6i 0.0938273i
\(820\) 0 0
\(821\) − 2.78102e6i − 0.143995i −0.997405 0.0719973i \(-0.977063\pi\)
0.997405 0.0719973i \(-0.0229373\pi\)
\(822\) 0 0
\(823\) −1.63895e7 −0.843461 −0.421731 0.906721i \(-0.638577\pi\)
−0.421731 + 0.906721i \(0.638577\pi\)
\(824\) 0 0
\(825\) −5.83048e6 −0.298242
\(826\) 0 0
\(827\) 2.29511e7i 1.16692i 0.812142 + 0.583459i \(0.198301\pi\)
−0.812142 + 0.583459i \(0.801699\pi\)
\(828\) 0 0
\(829\) 3.50136e6i 0.176950i 0.996078 + 0.0884750i \(0.0281993\pi\)
−0.996078 + 0.0884750i \(0.971801\pi\)
\(830\) 0 0
\(831\) −5.50900e6 −0.276739
\(832\) 0 0
\(833\) 1.94447e7 0.970934
\(834\) 0 0
\(835\) − 4.08427e7i − 2.02721i
\(836\) 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
\(845\) − 1.05679e7i − 0.509150i
\(846\) 0 0
\(847\) 3.49620e6 0.167451
\(848\) 0 0
\(849\) 2.18486e7 1.04029
\(850\) 0 0
\(851\) − 1.89998e6i − 0.0899344i
\(852\) 0 0
\(853\) − 2.02948e7i − 0.955021i −0.878626 0.477511i \(-0.841539\pi\)
0.878626 0.477511i \(-0.158461\pi\)
\(854\) 0 0
\(855\) 3.53652e7 1.65448
\(856\) 0 0
\(857\) 4.82785e6 0.224544 0.112272 0.993678i \(-0.464187\pi\)
0.112272 + 0.993678i \(0.464187\pi\)
\(858\) 0 0
\(859\) − 1.30210e7i − 0.602092i −0.953610 0.301046i \(-0.902664\pi\)
0.953610 0.301046i \(-0.0973358\pi\)
\(860\) 0 0
\(861\) − 4.26528e6i − 0.196083i
\(862\) 0 0
\(863\) −3.92387e7 −1.79344 −0.896721 0.442596i \(-0.854058\pi\)
−0.896721 + 0.442596i \(0.854058\pi\)
\(864\) 0 0
\(865\) −3.20342e7 −1.45570
\(866\) 0 0
\(867\) − 306940.i − 0.0138677i
\(868\) 0 0
\(869\) 5.52544e6i 0.248209i
\(870\) 0 0
\(871\) 7.42430e6 0.331596
\(872\) 0 0
\(873\) −7.67196e6 −0.340699
\(874\) 0 0
\(875\) − 1.37462e6i − 0.0606965i
\(876\) 0 0
\(877\) − 1.34622e7i − 0.591041i −0.955336 0.295520i \(-0.904507\pi\)
0.955336 0.295520i \(-0.0954930\pi\)
\(878\) 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
\(885\) − 2.49765e7i − 1.07195i
\(886\) 0 0
\(887\) 1.61463e7 0.689070 0.344535 0.938773i \(-0.388037\pi\)
0.344535 + 0.938773i \(0.388037\pi\)
\(888\) 0 0
\(889\) −2.27021e6 −0.0963410
\(890\) 0 0
\(891\) − 8.99632e6i − 0.379639i
\(892\) 0 0
\(893\) − 7.20332e7i − 3.02276i
\(894\) 0 0
\(895\) −4.14119e7 −1.72809
\(896\) 0 0
\(897\) 1.75904e6 0.0729953
\(898\) 0 0
\(899\) − 1.87993e7i − 0.775787i
\(900\) 0 0
\(901\) − 1.39998e7i − 0.574527i
\(902\) 0 0
\(903\) −4.41024e6 −0.179988
\(904\) 0 0
\(905\) 4.47485e7 1.81617
\(906\) 0 0
\(907\) − 2.03361e7i − 0.820824i −0.911900 0.410412i \(-0.865385\pi\)
0.911900 0.410412i \(-0.134615\pi\)
\(908\) 0 0
\(909\) − 8.10214e6i − 0.325230i
\(910\) 0 0
\(911\) 1.07726e7 0.430054 0.215027 0.976608i \(-0.431016\pi\)
0.215027 + 0.976608i \(0.431016\pi\)
\(912\) 0 0
\(913\) 8.35314e6 0.331644
\(914\) 0 0
\(915\) − 2.73534e7i − 1.08009i
\(916\) 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
\(925\) − 2.42764e7i − 0.932890i
\(926\) 0 0
\(927\) 2.83266e7 1.08267
\(928\) 0 0
\(929\) 2.99845e7 1.13988 0.569939 0.821687i \(-0.306967\pi\)
0.569939 + 0.821687i \(0.306967\pi\)
\(930\) 0 0
\(931\) 4.94072e7i 1.86817i
\(932\) 0 0
\(933\) − 2.67298e7i − 1.00529i
\(934\) 0 0
\(935\) −1.09928e7 −0.411227
\(936\) 0 0
\(937\) −1.42402e7 −0.529867 −0.264934 0.964267i \(-0.585350\pi\)
−0.264934 + 0.964267i \(0.585350\pi\)
\(938\) 0 0
\(939\) 3.28837e7i 1.21707i
\(940\) 0 0
\(941\) 4.14546e7i 1.52615i 0.646307 + 0.763077i \(0.276313\pi\)
−0.646307 + 0.763077i \(0.723687\pi\)
\(942\) 0 0
\(943\) −1.63502e6 −0.0598749
\(944\) 0 0
\(945\) −3.05472e6 −0.111274
\(946\) 0 0
\(947\) 1.54079e7i 0.558300i 0.960248 + 0.279150i \(0.0900527\pi\)
−0.960248 + 0.279150i \(0.909947\pi\)
\(948\) 0 0
\(949\) − 2.33551e6i − 0.0841813i
\(950\) 0 0
\(951\) 3.44740e7 1.23606
\(952\) 0 0
\(953\) 2.06328e7 0.735912 0.367956 0.929843i \(-0.380058\pi\)
0.367956 + 0.929843i \(0.380058\pi\)
\(954\) 0 0
\(955\) 3.02772e7i 1.07426i
\(956\) 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
\(965\) − 4.00241e7i − 1.38358i
\(966\) 0 0
\(967\) −1.18724e7 −0.408294 −0.204147 0.978940i \(-0.565442\pi\)
−0.204147 + 0.978940i \(0.565442\pi\)
\(968\) 0 0
\(969\) 7.29342e7 2.49530
\(970\) 0 0
\(971\) 1.53222e6i 0.0521523i 0.999660 + 0.0260761i \(0.00830123\pi\)
−0.999660 + 0.0260761i \(0.991699\pi\)
\(972\) 0 0
\(973\) − 3.12739e6i − 0.105901i
\(974\) 0 0
\(975\) 2.24756e7 0.757180
\(976\) 0 0
\(977\) 1.74321e7 0.584269 0.292135 0.956377i \(-0.405635\pi\)
0.292135 + 0.956377i \(0.405635\pi\)
\(978\) 0 0
\(979\) − 8.92726e6i − 0.297688i
\(980\) 0 0
\(981\) − 1.76948e7i − 0.587049i
\(982\) 0 0
\(983\) −2.23270e6 −0.0736963 −0.0368482 0.999321i \(-0.511732\pi\)
−0.0368482 + 0.999321i \(0.511732\pi\)
\(984\) 0 0
\(985\) −4.66125e7 −1.53078
\(986\) 0 0
\(987\) − 1.13587e7i − 0.371139i
\(988\) 0 0
\(989\) 1.69059e6i 0.0549602i
\(990\) 0 0
\(991\) 2.22501e7 0.719693 0.359847 0.933011i \(-0.382829\pi\)
0.359847 + 0.933011i \(0.382829\pi\)
\(992\) 0 0
\(993\) 5.49926e7 1.76983
\(994\) 0 0
\(995\) 2.09456e7i 0.670709i
\(996\) 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 256.6.b.f.129.1 2
4.3 odd 2 256.6.b.d.129.2 2
8.3 odd 2 256.6.b.d.129.1 2
8.5 even 2 inner 256.6.b.f.129.2 2
16.3 odd 4 16.6.a.a.1.1 1
16.5 even 4 64.6.a.a.1.1 1
16.11 odd 4 64.6.a.g.1.1 1
16.13 even 4 8.6.a.a.1.1 1
48.5 odd 4 576.6.a.g.1.1 1
48.11 even 4 576.6.a.h.1.1 1
48.29 odd 4 72.6.a.f.1.1 1
48.35 even 4 144.6.a.k.1.1 1
80.3 even 4 400.6.c.d.49.1 2
80.13 odd 4 200.6.c.a.49.2 2
80.19 odd 4 400.6.a.l.1.1 1
80.29 even 4 200.6.a.a.1.1 1
80.67 even 4 400.6.c.d.49.2 2
80.77 odd 4 200.6.c.a.49.1 2
112.13 odd 4 392.6.a.b.1.1 1
112.45 odd 12 392.6.i.e.177.1 2
112.61 odd 12 392.6.i.e.361.1 2
112.83 even 4 784.6.a.l.1.1 1
112.93 even 12 392.6.i.b.361.1 2
112.109 even 12 392.6.i.b.177.1 2
176.109 odd 4 968.6.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.6.a.a.1.1 1 16.13 even 4
16.6.a.a.1.1 1 16.3 odd 4
64.6.a.a.1.1 1 16.5 even 4
64.6.a.g.1.1 1 16.11 odd 4
72.6.a.f.1.1 1 48.29 odd 4
144.6.a.k.1.1 1 48.35 even 4
200.6.a.a.1.1 1 80.29 even 4
200.6.c.a.49.1 2 80.77 odd 4
200.6.c.a.49.2 2 80.13 odd 4
256.6.b.d.129.1 2 8.3 odd 2
256.6.b.d.129.2 2 4.3 odd 2
256.6.b.f.129.1 2 1.1 even 1 trivial
256.6.b.f.129.2 2 8.5 even 2 inner
392.6.a.b.1.1 1 112.13 odd 4
392.6.i.b.177.1 2 112.109 even 12
392.6.i.b.361.1 2 112.93 even 12
392.6.i.e.177.1 2 112.45 odd 12
392.6.i.e.361.1 2 112.61 odd 12
400.6.a.l.1.1 1 80.19 odd 4
400.6.c.d.49.1 2 80.3 even 4
400.6.c.d.49.2 2 80.67 even 4
576.6.a.g.1.1 1 48.5 odd 4
576.6.a.h.1.1 1 48.11 even 4
784.6.a.l.1.1 1 112.83 even 4
968.6.a.a.1.1 1 176.109 odd 4