Properties

Label 256.6.b.c.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: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4)
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.c.129.2

$q$-expansion

\(f(q)\) \(=\) \(q-12.0000i q^{3} +54.0000i q^{5} -88.0000 q^{7} +99.0000 q^{9} +O(q^{10})\) \(q-12.0000i q^{3} +54.0000i q^{5} -88.0000 q^{7} +99.0000 q^{9} -540.000i q^{11} +418.000i q^{13} +648.000 q^{15} +594.000 q^{17} +836.000i q^{19} +1056.00i q^{21} -4104.00 q^{23} +209.000 q^{25} -4104.00i q^{27} +594.000i q^{29} -4256.00 q^{31} -6480.00 q^{33} -4752.00i q^{35} -298.000i q^{37} +5016.00 q^{39} -17226.0 q^{41} +12100.0i q^{43} +5346.00i q^{45} +1296.00 q^{47} -9063.00 q^{49} -7128.00i q^{51} +19494.0i q^{53} +29160.0 q^{55} +10032.0 q^{57} +7668.00i q^{59} +34738.0i q^{61} -8712.00 q^{63} -22572.0 q^{65} +21812.0i q^{67} +49248.0i q^{69} -46872.0 q^{71} -67562.0 q^{73} -2508.00i q^{75} +47520.0i q^{77} +76912.0 q^{79} -25191.0 q^{81} +67716.0i q^{83} +32076.0i q^{85} +7128.00 q^{87} -29754.0 q^{89} -36784.0i q^{91} +51072.0i q^{93} -45144.0 q^{95} -122398. q^{97} -53460.0i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 176 q^{7} + 198 q^{9} + O(q^{10}) \) \( 2 q - 176 q^{7} + 198 q^{9} + 1296 q^{15} + 1188 q^{17} - 8208 q^{23} + 418 q^{25} - 8512 q^{31} - 12960 q^{33} + 10032 q^{39} - 34452 q^{41} + 2592 q^{47} - 18126 q^{49} + 58320 q^{55} + 20064 q^{57} - 17424 q^{63} - 45144 q^{65} - 93744 q^{71} - 135124 q^{73} + 153824 q^{79} - 50382 q^{81} + 14256 q^{87} - 59508 q^{89} - 90288 q^{95} - 244796 q^{97} + O(q^{100}) \)

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
\(3\) − 12.0000i − 0.769800i −0.922958 0.384900i \(-0.874236\pi\)
0.922958 0.384900i \(-0.125764\pi\)
\(4\) 0 0
\(5\) 54.0000i 0.965981i 0.875625 + 0.482991i \(0.160450\pi\)
−0.875625 + 0.482991i \(0.839550\pi\)
\(6\) 0 0
\(7\) −88.0000 −0.678793 −0.339397 0.940643i \(-0.610223\pi\)
−0.339397 + 0.940643i \(0.610223\pi\)
\(8\) 0 0
\(9\) 99.0000 0.407407
\(10\) 0 0
\(11\) − 540.000i − 1.34559i −0.739830 0.672794i \(-0.765094\pi\)
0.739830 0.672794i \(-0.234906\pi\)
\(12\) 0 0
\(13\) 418.000i 0.685990i 0.939337 + 0.342995i \(0.111441\pi\)
−0.939337 + 0.342995i \(0.888559\pi\)
\(14\) 0 0
\(15\) 648.000 0.743613
\(16\) 0 0
\(17\) 594.000 0.498499 0.249249 0.968439i \(-0.419816\pi\)
0.249249 + 0.968439i \(0.419816\pi\)
\(18\) 0 0
\(19\) 836.000i 0.531279i 0.964072 + 0.265639i \(0.0855830\pi\)
−0.964072 + 0.265639i \(0.914417\pi\)
\(20\) 0 0
\(21\) 1056.00i 0.522535i
\(22\) 0 0
\(23\) −4104.00 −1.61766 −0.808831 0.588041i \(-0.799899\pi\)
−0.808831 + 0.588041i \(0.799899\pi\)
\(24\) 0 0
\(25\) 209.000 0.0668800
\(26\) 0 0
\(27\) − 4104.00i − 1.08342i
\(28\) 0 0
\(29\) 594.000i 0.131157i 0.997847 + 0.0655785i \(0.0208893\pi\)
−0.997847 + 0.0655785i \(0.979111\pi\)
\(30\) 0 0
\(31\) −4256.00 −0.795422 −0.397711 0.917511i \(-0.630195\pi\)
−0.397711 + 0.917511i \(0.630195\pi\)
\(32\) 0 0
\(33\) −6480.00 −1.03583
\(34\) 0 0
\(35\) − 4752.00i − 0.655702i
\(36\) 0 0
\(37\) − 298.000i − 0.0357859i −0.999840 0.0178930i \(-0.994304\pi\)
0.999840 0.0178930i \(-0.00569581\pi\)
\(38\) 0 0
\(39\) 5016.00 0.528075
\(40\) 0 0
\(41\) −17226.0 −1.60039 −0.800193 0.599742i \(-0.795270\pi\)
−0.800193 + 0.599742i \(0.795270\pi\)
\(42\) 0 0
\(43\) 12100.0i 0.997963i 0.866613 + 0.498981i \(0.166292\pi\)
−0.866613 + 0.498981i \(0.833708\pi\)
\(44\) 0 0
\(45\) 5346.00i 0.393548i
\(46\) 0 0
\(47\) 1296.00 0.0855777 0.0427888 0.999084i \(-0.486376\pi\)
0.0427888 + 0.999084i \(0.486376\pi\)
\(48\) 0 0
\(49\) −9063.00 −0.539240
\(50\) 0 0
\(51\) − 7128.00i − 0.383745i
\(52\) 0 0
\(53\) 19494.0i 0.953260i 0.879104 + 0.476630i \(0.158142\pi\)
−0.879104 + 0.476630i \(0.841858\pi\)
\(54\) 0 0
\(55\) 29160.0 1.29981
\(56\) 0 0
\(57\) 10032.0 0.408978
\(58\) 0 0
\(59\) 7668.00i 0.286782i 0.989666 + 0.143391i \(0.0458007\pi\)
−0.989666 + 0.143391i \(0.954199\pi\)
\(60\) 0 0
\(61\) 34738.0i 1.19531i 0.801754 + 0.597655i \(0.203901\pi\)
−0.801754 + 0.597655i \(0.796099\pi\)
\(62\) 0 0
\(63\) −8712.00 −0.276545
\(64\) 0 0
\(65\) −22572.0 −0.662654
\(66\) 0 0
\(67\) 21812.0i 0.593620i 0.954937 + 0.296810i \(0.0959228\pi\)
−0.954937 + 0.296810i \(0.904077\pi\)
\(68\) 0 0
\(69\) 49248.0i 1.24528i
\(70\) 0 0
\(71\) −46872.0 −1.10349 −0.551744 0.834014i \(-0.686037\pi\)
−0.551744 + 0.834014i \(0.686037\pi\)
\(72\) 0 0
\(73\) −67562.0 −1.48387 −0.741934 0.670473i \(-0.766091\pi\)
−0.741934 + 0.670473i \(0.766091\pi\)
\(74\) 0 0
\(75\) − 2508.00i − 0.0514842i
\(76\) 0 0
\(77\) 47520.0i 0.913376i
\(78\) 0 0
\(79\) 76912.0 1.38652 0.693260 0.720687i \(-0.256174\pi\)
0.693260 + 0.720687i \(0.256174\pi\)
\(80\) 0 0
\(81\) −25191.0 −0.426612
\(82\) 0 0
\(83\) 67716.0i 1.07894i 0.842006 + 0.539468i \(0.181375\pi\)
−0.842006 + 0.539468i \(0.818625\pi\)
\(84\) 0 0
\(85\) 32076.0i 0.481541i
\(86\) 0 0
\(87\) 7128.00 0.100965
\(88\) 0 0
\(89\) −29754.0 −0.398172 −0.199086 0.979982i \(-0.563797\pi\)
−0.199086 + 0.979982i \(0.563797\pi\)
\(90\) 0 0
\(91\) − 36784.0i − 0.465646i
\(92\) 0 0
\(93\) 51072.0i 0.612316i
\(94\) 0 0
\(95\) −45144.0 −0.513205
\(96\) 0 0
\(97\) −122398. −1.32082 −0.660412 0.750903i \(-0.729618\pi\)
−0.660412 + 0.750903i \(0.729618\pi\)
\(98\) 0 0
\(99\) − 53460.0i − 0.548202i
\(100\) 0 0
\(101\) 11286.0i 0.110087i 0.998484 + 0.0550436i \(0.0175298\pi\)
−0.998484 + 0.0550436i \(0.982470\pi\)
\(102\) 0 0
\(103\) −27256.0 −0.253145 −0.126572 0.991957i \(-0.540398\pi\)
−0.126572 + 0.991957i \(0.540398\pi\)
\(104\) 0 0
\(105\) −57024.0 −0.504759
\(106\) 0 0
\(107\) − 122364.i − 1.03322i −0.856220 0.516612i \(-0.827193\pi\)
0.856220 0.516612i \(-0.172807\pi\)
\(108\) 0 0
\(109\) − 99902.0i − 0.805393i −0.915334 0.402697i \(-0.868073\pi\)
0.915334 0.402697i \(-0.131927\pi\)
\(110\) 0 0
\(111\) −3576.00 −0.0275480
\(112\) 0 0
\(113\) −29646.0 −0.218409 −0.109204 0.994019i \(-0.534830\pi\)
−0.109204 + 0.994019i \(0.534830\pi\)
\(114\) 0 0
\(115\) − 221616.i − 1.56263i
\(116\) 0 0
\(117\) 41382.0i 0.279477i
\(118\) 0 0
\(119\) −52272.0 −0.338378
\(120\) 0 0
\(121\) −130549. −0.810607
\(122\) 0 0
\(123\) 206712.i 1.23198i
\(124\) 0 0
\(125\) 180036.i 1.03059i
\(126\) 0 0
\(127\) −336512. −1.85136 −0.925681 0.378305i \(-0.876507\pi\)
−0.925681 + 0.378305i \(0.876507\pi\)
\(128\) 0 0
\(129\) 145200. 0.768232
\(130\) 0 0
\(131\) 100980.i 0.514111i 0.966397 + 0.257056i \(0.0827524\pi\)
−0.966397 + 0.257056i \(0.917248\pi\)
\(132\) 0 0
\(133\) − 73568.0i − 0.360628i
\(134\) 0 0
\(135\) 221616. 1.04657
\(136\) 0 0
\(137\) 317142. 1.44362 0.721809 0.692092i \(-0.243311\pi\)
0.721809 + 0.692092i \(0.243311\pi\)
\(138\) 0 0
\(139\) 148324.i 0.651140i 0.945518 + 0.325570i \(0.105556\pi\)
−0.945518 + 0.325570i \(0.894444\pi\)
\(140\) 0 0
\(141\) − 15552.0i − 0.0658777i
\(142\) 0 0
\(143\) 225720. 0.923060
\(144\) 0 0
\(145\) −32076.0 −0.126695
\(146\) 0 0
\(147\) 108756.i 0.415107i
\(148\) 0 0
\(149\) 196614.i 0.725519i 0.931883 + 0.362759i \(0.118165\pi\)
−0.931883 + 0.362759i \(0.881835\pi\)
\(150\) 0 0
\(151\) 74360.0 0.265398 0.132699 0.991156i \(-0.457636\pi\)
0.132699 + 0.991156i \(0.457636\pi\)
\(152\) 0 0
\(153\) 58806.0 0.203092
\(154\) 0 0
\(155\) − 229824.i − 0.768362i
\(156\) 0 0
\(157\) − 120878.i − 0.391380i −0.980666 0.195690i \(-0.937305\pi\)
0.980666 0.195690i \(-0.0626946\pi\)
\(158\) 0 0
\(159\) 233928. 0.733820
\(160\) 0 0
\(161\) 361152. 1.09806
\(162\) 0 0
\(163\) − 111340.i − 0.328233i −0.986441 0.164116i \(-0.947523\pi\)
0.986441 0.164116i \(-0.0524773\pi\)
\(164\) 0 0
\(165\) − 349920.i − 1.00060i
\(166\) 0 0
\(167\) −491832. −1.36466 −0.682332 0.731043i \(-0.739034\pi\)
−0.682332 + 0.731043i \(0.739034\pi\)
\(168\) 0 0
\(169\) 196569. 0.529417
\(170\) 0 0
\(171\) 82764.0i 0.216447i
\(172\) 0 0
\(173\) − 707454.i − 1.79714i −0.438826 0.898572i \(-0.644605\pi\)
0.438826 0.898572i \(-0.355395\pi\)
\(174\) 0 0
\(175\) −18392.0 −0.0453977
\(176\) 0 0
\(177\) 92016.0 0.220765
\(178\) 0 0
\(179\) 493668.i 1.15160i 0.817590 + 0.575801i \(0.195310\pi\)
−0.817590 + 0.575801i \(0.804690\pi\)
\(180\) 0 0
\(181\) − 559450.i − 1.26930i −0.772799 0.634651i \(-0.781144\pi\)
0.772799 0.634651i \(-0.218856\pi\)
\(182\) 0 0
\(183\) 416856. 0.920149
\(184\) 0 0
\(185\) 16092.0 0.0345685
\(186\) 0 0
\(187\) − 320760.i − 0.670774i
\(188\) 0 0
\(189\) 361152.i 0.735420i
\(190\) 0 0
\(191\) 724032. 1.43607 0.718033 0.696009i \(-0.245043\pi\)
0.718033 + 0.696009i \(0.245043\pi\)
\(192\) 0 0
\(193\) 7106.00 0.0137319 0.00686597 0.999976i \(-0.497814\pi\)
0.00686597 + 0.999976i \(0.497814\pi\)
\(194\) 0 0
\(195\) 270864.i 0.510111i
\(196\) 0 0
\(197\) − 530442.i − 0.973806i −0.873456 0.486903i \(-0.838127\pi\)
0.873456 0.486903i \(-0.161873\pi\)
\(198\) 0 0
\(199\) 56168.0 0.100544 0.0502720 0.998736i \(-0.483991\pi\)
0.0502720 + 0.998736i \(0.483991\pi\)
\(200\) 0 0
\(201\) 261744. 0.456969
\(202\) 0 0
\(203\) − 52272.0i − 0.0890285i
\(204\) 0 0
\(205\) − 930204.i − 1.54594i
\(206\) 0 0
\(207\) −406296. −0.659047
\(208\) 0 0
\(209\) 451440. 0.714882
\(210\) 0 0
\(211\) − 339196.i − 0.524499i −0.965000 0.262249i \(-0.915536\pi\)
0.965000 0.262249i \(-0.0844643\pi\)
\(212\) 0 0
\(213\) 562464.i 0.849465i
\(214\) 0 0
\(215\) −653400. −0.964013
\(216\) 0 0
\(217\) 374528. 0.539927
\(218\) 0 0
\(219\) 810744.i 1.14228i
\(220\) 0 0
\(221\) 248292.i 0.341965i
\(222\) 0 0
\(223\) −779360. −1.04948 −0.524742 0.851261i \(-0.675838\pi\)
−0.524742 + 0.851261i \(0.675838\pi\)
\(224\) 0 0
\(225\) 20691.0 0.0272474
\(226\) 0 0
\(227\) − 744876.i − 0.959443i −0.877421 0.479722i \(-0.840738\pi\)
0.877421 0.479722i \(-0.159262\pi\)
\(228\) 0 0
\(229\) − 272746.i − 0.343692i −0.985124 0.171846i \(-0.945027\pi\)
0.985124 0.171846i \(-0.0549732\pi\)
\(230\) 0 0
\(231\) 570240. 0.703117
\(232\) 0 0
\(233\) 153846. 0.185651 0.0928253 0.995682i \(-0.470410\pi\)
0.0928253 + 0.995682i \(0.470410\pi\)
\(234\) 0 0
\(235\) 69984.0i 0.0826664i
\(236\) 0 0
\(237\) − 922944.i − 1.06734i
\(238\) 0 0
\(239\) −1.15474e6 −1.30764 −0.653820 0.756650i \(-0.726834\pi\)
−0.653820 + 0.756650i \(0.726834\pi\)
\(240\) 0 0
\(241\) 657074. 0.728738 0.364369 0.931255i \(-0.381285\pi\)
0.364369 + 0.931255i \(0.381285\pi\)
\(242\) 0 0
\(243\) − 694980.i − 0.755017i
\(244\) 0 0
\(245\) − 489402.i − 0.520895i
\(246\) 0 0
\(247\) −349448. −0.364452
\(248\) 0 0
\(249\) 812592. 0.830566
\(250\) 0 0
\(251\) − 1.34190e6i − 1.34442i −0.740359 0.672211i \(-0.765345\pi\)
0.740359 0.672211i \(-0.234655\pi\)
\(252\) 0 0
\(253\) 2.21616e6i 2.17671i
\(254\) 0 0
\(255\) 384912. 0.370690
\(256\) 0 0
\(257\) 132354. 0.124998 0.0624992 0.998045i \(-0.480093\pi\)
0.0624992 + 0.998045i \(0.480093\pi\)
\(258\) 0 0
\(259\) 26224.0i 0.0242912i
\(260\) 0 0
\(261\) 58806.0i 0.0534343i
\(262\) 0 0
\(263\) 943272. 0.840906 0.420453 0.907314i \(-0.361871\pi\)
0.420453 + 0.907314i \(0.361871\pi\)
\(264\) 0 0
\(265\) −1.05268e6 −0.920831
\(266\) 0 0
\(267\) 357048.i 0.306513i
\(268\) 0 0
\(269\) − 967518.i − 0.815227i −0.913155 0.407613i \(-0.866361\pi\)
0.913155 0.407613i \(-0.133639\pi\)
\(270\) 0 0
\(271\) 518320. 0.428721 0.214360 0.976755i \(-0.431233\pi\)
0.214360 + 0.976755i \(0.431233\pi\)
\(272\) 0 0
\(273\) −441408. −0.358454
\(274\) 0 0
\(275\) − 112860.i − 0.0899929i
\(276\) 0 0
\(277\) 2.22273e6i 1.74055i 0.492566 + 0.870275i \(0.336059\pi\)
−0.492566 + 0.870275i \(0.663941\pi\)
\(278\) 0 0
\(279\) −421344. −0.324061
\(280\) 0 0
\(281\) 196614. 0.148542 0.0742709 0.997238i \(-0.476337\pi\)
0.0742709 + 0.997238i \(0.476337\pi\)
\(282\) 0 0
\(283\) 1.55228e6i 1.15213i 0.817403 + 0.576067i \(0.195413\pi\)
−0.817403 + 0.576067i \(0.804587\pi\)
\(284\) 0 0
\(285\) 541728.i 0.395066i
\(286\) 0 0
\(287\) 1.51589e6 1.08633
\(288\) 0 0
\(289\) −1.06702e6 −0.751499
\(290\) 0 0
\(291\) 1.46878e6i 1.01677i
\(292\) 0 0
\(293\) − 1.07217e6i − 0.729616i −0.931083 0.364808i \(-0.881135\pi\)
0.931083 0.364808i \(-0.118865\pi\)
\(294\) 0 0
\(295\) −414072. −0.277026
\(296\) 0 0
\(297\) −2.21616e6 −1.45784
\(298\) 0 0
\(299\) − 1.71547e6i − 1.10970i
\(300\) 0 0
\(301\) − 1.06480e6i − 0.677410i
\(302\) 0 0
\(303\) 135432. 0.0847451
\(304\) 0 0
\(305\) −1.87585e6 −1.15465
\(306\) 0 0
\(307\) 1.58589e6i 0.960346i 0.877174 + 0.480173i \(0.159426\pi\)
−0.877174 + 0.480173i \(0.840574\pi\)
\(308\) 0 0
\(309\) 327072.i 0.194871i
\(310\) 0 0
\(311\) −730728. −0.428405 −0.214203 0.976789i \(-0.568715\pi\)
−0.214203 + 0.976789i \(0.568715\pi\)
\(312\) 0 0
\(313\) −584858. −0.337435 −0.168717 0.985664i \(-0.553962\pi\)
−0.168717 + 0.985664i \(0.553962\pi\)
\(314\) 0 0
\(315\) − 470448.i − 0.267138i
\(316\) 0 0
\(317\) 2.48287e6i 1.38773i 0.720105 + 0.693865i \(0.244094\pi\)
−0.720105 + 0.693865i \(0.755906\pi\)
\(318\) 0 0
\(319\) 320760. 0.176483
\(320\) 0 0
\(321\) −1.46837e6 −0.795376
\(322\) 0 0
\(323\) 496584.i 0.264842i
\(324\) 0 0
\(325\) 87362.0i 0.0458790i
\(326\) 0 0
\(327\) −1.19882e6 −0.619992
\(328\) 0 0
\(329\) −114048. −0.0580895
\(330\) 0 0
\(331\) − 377948.i − 0.189610i −0.995496 0.0948052i \(-0.969777\pi\)
0.995496 0.0948052i \(-0.0302228\pi\)
\(332\) 0 0
\(333\) − 29502.0i − 0.0145794i
\(334\) 0 0
\(335\) −1.17785e6 −0.573426
\(336\) 0 0
\(337\) 639122. 0.306555 0.153278 0.988183i \(-0.451017\pi\)
0.153278 + 0.988183i \(0.451017\pi\)
\(338\) 0 0
\(339\) 355752.i 0.168131i
\(340\) 0 0
\(341\) 2.29824e6i 1.07031i
\(342\) 0 0
\(343\) 2.27656e6 1.04483
\(344\) 0 0
\(345\) −2.65939e6 −1.20291
\(346\) 0 0
\(347\) 2.90466e6i 1.29501i 0.762063 + 0.647503i \(0.224187\pi\)
−0.762063 + 0.647503i \(0.775813\pi\)
\(348\) 0 0
\(349\) 3.99157e6i 1.75420i 0.480304 + 0.877102i \(0.340526\pi\)
−0.480304 + 0.877102i \(0.659474\pi\)
\(350\) 0 0
\(351\) 1.71547e6 0.743217
\(352\) 0 0
\(353\) 1.42922e6 0.610466 0.305233 0.952278i \(-0.401266\pi\)
0.305233 + 0.952278i \(0.401266\pi\)
\(354\) 0 0
\(355\) − 2.53109e6i − 1.06595i
\(356\) 0 0
\(357\) 627264.i 0.260483i
\(358\) 0 0
\(359\) 1.16186e6 0.475794 0.237897 0.971290i \(-0.423542\pi\)
0.237897 + 0.971290i \(0.423542\pi\)
\(360\) 0 0
\(361\) 1.77720e6 0.717743
\(362\) 0 0
\(363\) 1.56659e6i 0.624005i
\(364\) 0 0
\(365\) − 3.64835e6i − 1.43339i
\(366\) 0 0
\(367\) 1.08923e6 0.422139 0.211069 0.977471i \(-0.432305\pi\)
0.211069 + 0.977471i \(0.432305\pi\)
\(368\) 0 0
\(369\) −1.70537e6 −0.652009
\(370\) 0 0
\(371\) − 1.71547e6i − 0.647066i
\(372\) 0 0
\(373\) 3.50577e6i 1.30470i 0.757918 + 0.652350i \(0.226217\pi\)
−0.757918 + 0.652350i \(0.773783\pi\)
\(374\) 0 0
\(375\) 2.16043e6 0.793346
\(376\) 0 0
\(377\) −248292. −0.0899724
\(378\) 0 0
\(379\) − 4.04385e6i − 1.44610i −0.690798 0.723048i \(-0.742740\pi\)
0.690798 0.723048i \(-0.257260\pi\)
\(380\) 0 0
\(381\) 4.03814e6i 1.42518i
\(382\) 0 0
\(383\) −5.18746e6 −1.80700 −0.903499 0.428591i \(-0.859010\pi\)
−0.903499 + 0.428591i \(0.859010\pi\)
\(384\) 0 0
\(385\) −2.56608e6 −0.882304
\(386\) 0 0
\(387\) 1.19790e6i 0.406577i
\(388\) 0 0
\(389\) − 950346.i − 0.318425i −0.987244 0.159213i \(-0.949104\pi\)
0.987244 0.159213i \(-0.0508956\pi\)
\(390\) 0 0
\(391\) −2.43778e6 −0.806403
\(392\) 0 0
\(393\) 1.21176e6 0.395763
\(394\) 0 0
\(395\) 4.15325e6i 1.33935i
\(396\) 0 0
\(397\) 520738.i 0.165822i 0.996557 + 0.0829112i \(0.0264218\pi\)
−0.996557 + 0.0829112i \(0.973578\pi\)
\(398\) 0 0
\(399\) −882816. −0.277612
\(400\) 0 0
\(401\) 764370. 0.237379 0.118690 0.992931i \(-0.462131\pi\)
0.118690 + 0.992931i \(0.462131\pi\)
\(402\) 0 0
\(403\) − 1.77901e6i − 0.545651i
\(404\) 0 0
\(405\) − 1.36031e6i − 0.412099i
\(406\) 0 0
\(407\) −160920. −0.0481531
\(408\) 0 0
\(409\) −2.64051e6 −0.780511 −0.390255 0.920707i \(-0.627613\pi\)
−0.390255 + 0.920707i \(0.627613\pi\)
\(410\) 0 0
\(411\) − 3.80570e6i − 1.11130i
\(412\) 0 0
\(413\) − 674784.i − 0.194666i
\(414\) 0 0
\(415\) −3.65666e6 −1.04223
\(416\) 0 0
\(417\) 1.77989e6 0.501248
\(418\) 0 0
\(419\) − 4.98020e6i − 1.38584i −0.721016 0.692918i \(-0.756325\pi\)
0.721016 0.692918i \(-0.243675\pi\)
\(420\) 0 0
\(421\) − 237994.i − 0.0654426i −0.999465 0.0327213i \(-0.989583\pi\)
0.999465 0.0327213i \(-0.0104174\pi\)
\(422\) 0 0
\(423\) 128304. 0.0348650
\(424\) 0 0
\(425\) 124146. 0.0333396
\(426\) 0 0
\(427\) − 3.05694e6i − 0.811368i
\(428\) 0 0
\(429\) − 2.70864e6i − 0.710572i
\(430\) 0 0
\(431\) 3.88238e6 1.00671 0.503356 0.864079i \(-0.332098\pi\)
0.503356 + 0.864079i \(0.332098\pi\)
\(432\) 0 0
\(433\) −66958.0 −0.0171626 −0.00858129 0.999963i \(-0.502732\pi\)
−0.00858129 + 0.999963i \(0.502732\pi\)
\(434\) 0 0
\(435\) 384912.i 0.0975300i
\(436\) 0 0
\(437\) − 3.43094e6i − 0.859429i
\(438\) 0 0
\(439\) −6.50135e6 −1.61006 −0.805031 0.593233i \(-0.797851\pi\)
−0.805031 + 0.593233i \(0.797851\pi\)
\(440\) 0 0
\(441\) −897237. −0.219690
\(442\) 0 0
\(443\) 4.60760e6i 1.11549i 0.830012 + 0.557745i \(0.188333\pi\)
−0.830012 + 0.557745i \(0.811667\pi\)
\(444\) 0 0
\(445\) − 1.60672e6i − 0.384626i
\(446\) 0 0
\(447\) 2.35937e6 0.558505
\(448\) 0 0
\(449\) 3.77671e6 0.884092 0.442046 0.896992i \(-0.354253\pi\)
0.442046 + 0.896992i \(0.354253\pi\)
\(450\) 0 0
\(451\) 9.30204e6i 2.15346i
\(452\) 0 0
\(453\) − 892320.i − 0.204303i
\(454\) 0 0
\(455\) 1.98634e6 0.449805
\(456\) 0 0
\(457\) 3.18069e6 0.712412 0.356206 0.934407i \(-0.384070\pi\)
0.356206 + 0.934407i \(0.384070\pi\)
\(458\) 0 0
\(459\) − 2.43778e6i − 0.540085i
\(460\) 0 0
\(461\) − 6.68547e6i − 1.46514i −0.680691 0.732571i \(-0.738320\pi\)
0.680691 0.732571i \(-0.261680\pi\)
\(462\) 0 0
\(463\) 4.35122e6 0.943318 0.471659 0.881781i \(-0.343655\pi\)
0.471659 + 0.881781i \(0.343655\pi\)
\(464\) 0 0
\(465\) −2.75789e6 −0.591486
\(466\) 0 0
\(467\) 7.07994e6i 1.50223i 0.660170 + 0.751117i \(0.270484\pi\)
−0.660170 + 0.751117i \(0.729516\pi\)
\(468\) 0 0
\(469\) − 1.91946e6i − 0.402945i
\(470\) 0 0
\(471\) −1.45054e6 −0.301284
\(472\) 0 0
\(473\) 6.53400e6 1.34285
\(474\) 0 0
\(475\) 174724.i 0.0355319i
\(476\) 0 0
\(477\) 1.92991e6i 0.388365i
\(478\) 0 0
\(479\) −3.22186e6 −0.641604 −0.320802 0.947146i \(-0.603952\pi\)
−0.320802 + 0.947146i \(0.603952\pi\)
\(480\) 0 0
\(481\) 124564. 0.0245488
\(482\) 0 0
\(483\) − 4.33382e6i − 0.845286i
\(484\) 0 0
\(485\) − 6.60949e6i − 1.27589i
\(486\) 0 0
\(487\) 2.29710e6 0.438891 0.219446 0.975625i \(-0.429575\pi\)
0.219446 + 0.975625i \(0.429575\pi\)
\(488\) 0 0
\(489\) −1.33608e6 −0.252674
\(490\) 0 0
\(491\) − 2.82150e6i − 0.528173i −0.964499 0.264087i \(-0.914930\pi\)
0.964499 0.264087i \(-0.0850705\pi\)
\(492\) 0 0
\(493\) 352836.i 0.0653816i
\(494\) 0 0
\(495\) 2.88684e6 0.529553
\(496\) 0 0
\(497\) 4.12474e6 0.749040
\(498\) 0 0
\(499\) − 4.13628e6i − 0.743634i −0.928306 0.371817i \(-0.878735\pi\)
0.928306 0.371817i \(-0.121265\pi\)
\(500\) 0 0
\(501\) 5.90198e6i 1.05052i
\(502\) 0 0
\(503\) 8.33263e6 1.46846 0.734230 0.678901i \(-0.237543\pi\)
0.734230 + 0.678901i \(0.237543\pi\)
\(504\) 0 0
\(505\) −609444. −0.106342
\(506\) 0 0
\(507\) − 2.35883e6i − 0.407546i
\(508\) 0 0
\(509\) − 4.34101e6i − 0.742670i −0.928499 0.371335i \(-0.878900\pi\)
0.928499 0.371335i \(-0.121100\pi\)
\(510\) 0 0
\(511\) 5.94546e6 1.00724
\(512\) 0 0
\(513\) 3.43094e6 0.575599
\(514\) 0 0
\(515\) − 1.47182e6i − 0.244533i
\(516\) 0 0
\(517\) − 699840.i − 0.115152i
\(518\) 0 0
\(519\) −8.48945e6 −1.38344
\(520\) 0 0
\(521\) 6.74185e6 1.08814 0.544070 0.839040i \(-0.316883\pi\)
0.544070 + 0.839040i \(0.316883\pi\)
\(522\) 0 0
\(523\) 7.72196e6i 1.23445i 0.786787 + 0.617224i \(0.211743\pi\)
−0.786787 + 0.617224i \(0.788257\pi\)
\(524\) 0 0
\(525\) 220704.i 0.0349472i
\(526\) 0 0
\(527\) −2.52806e6 −0.396517
\(528\) 0 0
\(529\) 1.04065e7 1.61683
\(530\) 0 0
\(531\) 759132.i 0.116837i
\(532\) 0 0
\(533\) − 7.20047e6i − 1.09785i
\(534\) 0 0
\(535\) 6.60766e6 0.998075
\(536\) 0 0
\(537\) 5.92402e6 0.886504
\(538\) 0 0
\(539\) 4.89402e6i 0.725594i
\(540\) 0 0
\(541\) 682066.i 0.100192i 0.998744 + 0.0500960i \(0.0159527\pi\)
−0.998744 + 0.0500960i \(0.984047\pi\)
\(542\) 0 0
\(543\) −6.71340e6 −0.977109
\(544\) 0 0
\(545\) 5.39471e6 0.777995
\(546\) 0 0
\(547\) 2.15772e6i 0.308337i 0.988045 + 0.154169i \(0.0492699\pi\)
−0.988045 + 0.154169i \(0.950730\pi\)
\(548\) 0 0
\(549\) 3.43906e6i 0.486978i
\(550\) 0 0
\(551\) −496584. −0.0696809
\(552\) 0 0
\(553\) −6.76826e6 −0.941161
\(554\) 0 0
\(555\) − 193104.i − 0.0266109i
\(556\) 0 0
\(557\) 2.67597e6i 0.365463i 0.983163 + 0.182731i \(0.0584939\pi\)
−0.983163 + 0.182731i \(0.941506\pi\)
\(558\) 0 0
\(559\) −5.05780e6 −0.684592
\(560\) 0 0
\(561\) −3.84912e6 −0.516362
\(562\) 0 0
\(563\) − 3.55331e6i − 0.472457i −0.971698 0.236228i \(-0.924089\pi\)
0.971698 0.236228i \(-0.0759113\pi\)
\(564\) 0 0
\(565\) − 1.60088e6i − 0.210979i
\(566\) 0 0
\(567\) 2.21681e6 0.289581
\(568\) 0 0
\(569\) 1.29225e7 1.67327 0.836633 0.547764i \(-0.184521\pi\)
0.836633 + 0.547764i \(0.184521\pi\)
\(570\) 0 0
\(571\) 6.08357e6i 0.780851i 0.920634 + 0.390426i \(0.127672\pi\)
−0.920634 + 0.390426i \(0.872328\pi\)
\(572\) 0 0
\(573\) − 8.68838e6i − 1.10548i
\(574\) 0 0
\(575\) −857736. −0.108189
\(576\) 0 0
\(577\) −1.58241e7 −1.97869 −0.989347 0.145579i \(-0.953495\pi\)
−0.989347 + 0.145579i \(0.953495\pi\)
\(578\) 0 0
\(579\) − 85272.0i − 0.0105709i
\(580\) 0 0
\(581\) − 5.95901e6i − 0.732375i
\(582\) 0 0
\(583\) 1.05268e7 1.28269
\(584\) 0 0
\(585\) −2.23463e6 −0.269970
\(586\) 0 0
\(587\) − 4.60220e6i − 0.551278i −0.961261 0.275639i \(-0.911111\pi\)
0.961261 0.275639i \(-0.0888894\pi\)
\(588\) 0 0
\(589\) − 3.55802e6i − 0.422590i
\(590\) 0 0
\(591\) −6.36530e6 −0.749636
\(592\) 0 0
\(593\) 8.61122e6 1.00561 0.502803 0.864401i \(-0.332302\pi\)
0.502803 + 0.864401i \(0.332302\pi\)
\(594\) 0 0
\(595\) − 2.82269e6i − 0.326867i
\(596\) 0 0
\(597\) − 674016.i − 0.0773988i
\(598\) 0 0
\(599\) −7.98228e6 −0.908992 −0.454496 0.890749i \(-0.650181\pi\)
−0.454496 + 0.890749i \(0.650181\pi\)
\(600\) 0 0
\(601\) −1.01740e7 −1.14896 −0.574481 0.818518i \(-0.694796\pi\)
−0.574481 + 0.818518i \(0.694796\pi\)
\(602\) 0 0
\(603\) 2.15939e6i 0.241845i
\(604\) 0 0
\(605\) − 7.04965e6i − 0.783031i
\(606\) 0 0
\(607\) 9.95843e6 1.09703 0.548516 0.836140i \(-0.315193\pi\)
0.548516 + 0.836140i \(0.315193\pi\)
\(608\) 0 0
\(609\) −627264. −0.0685342
\(610\) 0 0
\(611\) 541728.i 0.0587054i
\(612\) 0 0
\(613\) 4.19586e6i 0.450993i 0.974244 + 0.225497i \(0.0724005\pi\)
−0.974244 + 0.225497i \(0.927600\pi\)
\(614\) 0 0
\(615\) −1.11624e7 −1.19007
\(616\) 0 0
\(617\) −9.12551e6 −0.965038 −0.482519 0.875885i \(-0.660278\pi\)
−0.482519 + 0.875885i \(0.660278\pi\)
\(618\) 0 0
\(619\) − 6.45734e6i − 0.677372i −0.940900 0.338686i \(-0.890018\pi\)
0.940900 0.338686i \(-0.109982\pi\)
\(620\) 0 0
\(621\) 1.68428e7i 1.75261i
\(622\) 0 0
\(623\) 2.61835e6 0.270276
\(624\) 0 0
\(625\) −9.06882e6 −0.928647
\(626\) 0 0
\(627\) − 5.41728e6i − 0.550316i
\(628\) 0 0
\(629\) − 177012.i − 0.0178392i
\(630\) 0 0
\(631\) −1.40514e7 −1.40490 −0.702450 0.711733i \(-0.747910\pi\)
−0.702450 + 0.711733i \(0.747910\pi\)
\(632\) 0 0
\(633\) −4.07035e6 −0.403759
\(634\) 0 0
\(635\) − 1.81716e7i − 1.78838i
\(636\) 0 0
\(637\) − 3.78833e6i − 0.369913i
\(638\) 0 0
\(639\) −4.64033e6 −0.449569
\(640\) 0 0
\(641\) 8.47168e6 0.814375 0.407188 0.913345i \(-0.366510\pi\)
0.407188 + 0.913345i \(0.366510\pi\)
\(642\) 0 0
\(643\) 488564.i 0.0466009i 0.999729 + 0.0233004i \(0.00741743\pi\)
−0.999729 + 0.0233004i \(0.992583\pi\)
\(644\) 0 0
\(645\) 7.84080e6i 0.742098i
\(646\) 0 0
\(647\) 2.48119e6 0.233023 0.116512 0.993189i \(-0.462829\pi\)
0.116512 + 0.993189i \(0.462829\pi\)
\(648\) 0 0
\(649\) 4.14072e6 0.385891
\(650\) 0 0
\(651\) − 4.49434e6i − 0.415636i
\(652\) 0 0
\(653\) 5.29130e6i 0.485601i 0.970076 + 0.242800i \(0.0780660\pi\)
−0.970076 + 0.242800i \(0.921934\pi\)
\(654\) 0 0
\(655\) −5.45292e6 −0.496622
\(656\) 0 0
\(657\) −6.68864e6 −0.604539
\(658\) 0 0
\(659\) 4.72468e6i 0.423798i 0.977292 + 0.211899i \(0.0679647\pi\)
−0.977292 + 0.211899i \(0.932035\pi\)
\(660\) 0 0
\(661\) − 6.17420e6i − 0.549639i −0.961496 0.274819i \(-0.911382\pi\)
0.961496 0.274819i \(-0.0886180\pi\)
\(662\) 0 0
\(663\) 2.97950e6 0.263245
\(664\) 0 0
\(665\) 3.97267e6 0.348360
\(666\) 0 0
\(667\) − 2.43778e6i − 0.212168i
\(668\) 0 0
\(669\) 9.35232e6i 0.807893i
\(670\) 0 0
\(671\) 1.87585e7 1.60839
\(672\) 0 0
\(673\) −9.40925e6 −0.800787 −0.400394 0.916343i \(-0.631127\pi\)
−0.400394 + 0.916343i \(0.631127\pi\)
\(674\) 0 0
\(675\) − 857736.i − 0.0724593i
\(676\) 0 0
\(677\) 1.50086e7i 1.25854i 0.777185 + 0.629272i \(0.216647\pi\)
−0.777185 + 0.629272i \(0.783353\pi\)
\(678\) 0 0
\(679\) 1.07710e7 0.896567
\(680\) 0 0
\(681\) −8.93851e6 −0.738580
\(682\) 0 0
\(683\) 1.29707e7i 1.06393i 0.846768 + 0.531963i \(0.178545\pi\)
−0.846768 + 0.531963i \(0.821455\pi\)
\(684\) 0 0
\(685\) 1.71257e7i 1.39451i
\(686\) 0 0
\(687\) −3.27295e6 −0.264574
\(688\) 0 0
\(689\) −8.14849e6 −0.653927
\(690\) 0 0
\(691\) 2.26556e7i 1.80501i 0.430677 + 0.902506i \(0.358275\pi\)
−0.430677 + 0.902506i \(0.641725\pi\)
\(692\) 0 0
\(693\) 4.70448e6i 0.372116i
\(694\) 0 0
\(695\) −8.00950e6 −0.628989
\(696\) 0 0
\(697\) −1.02322e7 −0.797791
\(698\) 0 0
\(699\) − 1.84615e6i − 0.142914i
\(700\) 0 0
\(701\) − 1.90169e7i − 1.46166i −0.682562 0.730828i \(-0.739134\pi\)
0.682562 0.730828i \(-0.260866\pi\)
\(702\) 0 0
\(703\) 249128. 0.0190123
\(704\) 0 0
\(705\) 839808. 0.0636366
\(706\) 0 0
\(707\) − 993168.i − 0.0747264i
\(708\) 0 0
\(709\) 1.51311e7i 1.13046i 0.824933 + 0.565231i \(0.191213\pi\)
−0.824933 + 0.565231i \(0.808787\pi\)
\(710\) 0 0
\(711\) 7.61429e6 0.564879
\(712\) 0 0
\(713\) 1.74666e7 1.28672
\(714\) 0 0
\(715\) 1.21889e7i 0.891659i
\(716\) 0 0
\(717\) 1.38568e7i 1.00662i
\(718\) 0 0
\(719\) 1.50323e7 1.08443 0.542217 0.840238i \(-0.317585\pi\)
0.542217 + 0.840238i \(0.317585\pi\)
\(720\) 0 0
\(721\) 2.39853e6 0.171833
\(722\) 0 0
\(723\) − 7.88489e6i − 0.560983i
\(724\) 0 0
\(725\) 124146.i 0.00877178i
\(726\) 0 0
\(727\) −7.41230e6 −0.520136 −0.260068 0.965590i \(-0.583745\pi\)
−0.260068 + 0.965590i \(0.583745\pi\)
\(728\) 0 0
\(729\) −1.44612e7 −1.00782
\(730\) 0 0
\(731\) 7.18740e6i 0.497483i
\(732\) 0 0
\(733\) 2.77928e6i 0.191061i 0.995426 + 0.0955306i \(0.0304548\pi\)
−0.995426 + 0.0955306i \(0.969545\pi\)
\(734\) 0 0
\(735\) −5.87282e6 −0.400985
\(736\) 0 0
\(737\) 1.17785e7 0.798768
\(738\) 0 0
\(739\) − 1.21046e7i − 0.815342i −0.913129 0.407671i \(-0.866341\pi\)
0.913129 0.407671i \(-0.133659\pi\)
\(740\) 0 0
\(741\) 4.19338e6i 0.280555i
\(742\) 0 0
\(743\) 4.46926e6 0.297005 0.148502 0.988912i \(-0.452555\pi\)
0.148502 + 0.988912i \(0.452555\pi\)
\(744\) 0 0
\(745\) −1.06172e7 −0.700838
\(746\) 0 0
\(747\) 6.70388e6i 0.439567i
\(748\) 0 0
\(749\) 1.07680e7i 0.701345i
\(750\) 0 0
\(751\) −2.88463e7 −1.86634 −0.933168 0.359442i \(-0.882967\pi\)
−0.933168 + 0.359442i \(0.882967\pi\)
\(752\) 0 0
\(753\) −1.61028e7 −1.03494
\(754\) 0 0
\(755\) 4.01544e6i 0.256369i
\(756\) 0 0
\(757\) 9.60868e6i 0.609430i 0.952444 + 0.304715i \(0.0985612\pi\)
−0.952444 + 0.304715i \(0.901439\pi\)
\(758\) 0 0
\(759\) 2.65939e7 1.67563
\(760\) 0 0
\(761\) −4.54588e6 −0.284549 −0.142274 0.989827i \(-0.545442\pi\)
−0.142274 + 0.989827i \(0.545442\pi\)
\(762\) 0 0
\(763\) 8.79138e6i 0.546696i
\(764\) 0 0
\(765\) 3.17552e6i 0.196183i
\(766\) 0 0
\(767\) −3.20522e6 −0.196730
\(768\) 0 0
\(769\) −2.15923e7 −1.31669 −0.658345 0.752716i \(-0.728743\pi\)
−0.658345 + 0.752716i \(0.728743\pi\)
\(770\) 0 0
\(771\) − 1.58825e6i − 0.0962238i
\(772\) 0 0
\(773\) − 1.48400e7i − 0.893276i −0.894715 0.446638i \(-0.852621\pi\)
0.894715 0.446638i \(-0.147379\pi\)
\(774\) 0 0
\(775\) −889504. −0.0531978
\(776\) 0 0
\(777\) 314688. 0.0186994
\(778\) 0 0
\(779\) − 1.44009e7i − 0.850251i
\(780\) 0 0
\(781\) 2.53109e7i 1.48484i
\(782\) 0 0
\(783\) 2.43778e6 0.142098
\(784\) 0 0
\(785\) 6.52741e6 0.378065
\(786\) 0 0
\(787\) − 2.48785e7i − 1.43182i −0.698194 0.715909i \(-0.746013\pi\)
0.698194 0.715909i \(-0.253987\pi\)
\(788\) 0 0
\(789\) − 1.13193e7i − 0.647330i
\(790\) 0 0
\(791\) 2.60885e6 0.148254
\(792\) 0 0
\(793\) −1.45205e7 −0.819970
\(794\) 0 0
\(795\) 1.26321e7i 0.708856i
\(796\) 0 0
\(797\) − 3.16080e7i − 1.76259i −0.472568 0.881294i \(-0.656673\pi\)
0.472568 0.881294i \(-0.343327\pi\)
\(798\) 0 0
\(799\) 769824. 0.0426604
\(800\) 0 0
\(801\) −2.94565e6 −0.162218
\(802\) 0 0
\(803\) 3.64835e7i 1.99668i
\(804\) 0 0
\(805\) 1.95022e7i 1.06070i
\(806\) 0 0
\(807\) −1.16102e7 −0.627562
\(808\) 0 0
\(809\) 3.10009e6 0.166534 0.0832669 0.996527i \(-0.473465\pi\)
0.0832669 + 0.996527i \(0.473465\pi\)
\(810\) 0 0
\(811\) − 1.87180e6i − 0.0999328i −0.998751 0.0499664i \(-0.984089\pi\)
0.998751 0.0499664i \(-0.0159114\pi\)
\(812\) 0 0
\(813\) − 6.21984e6i − 0.330030i
\(814\) 0 0
\(815\) 6.01236e6 0.317067
\(816\) 0 0
\(817\) −1.01156e7 −0.530196
\(818\) 0 0
\(819\) − 3.64162e6i − 0.189707i
\(820\) 0 0
\(821\) − 2.00184e7i − 1.03650i −0.855228 0.518252i \(-0.826583\pi\)
0.855228 0.518252i \(-0.173417\pi\)
\(822\) 0 0
\(823\) 1.53118e7 0.787999 0.394000 0.919111i \(-0.371091\pi\)
0.394000 + 0.919111i \(0.371091\pi\)
\(824\) 0 0
\(825\) −1.35432e6 −0.0692766
\(826\) 0 0
\(827\) − 9.59310e6i − 0.487748i −0.969807 0.243874i \(-0.921582\pi\)
0.969807 0.243874i \(-0.0784183\pi\)
\(828\) 0 0
\(829\) − 2.52209e7i − 1.27460i −0.770615 0.637302i \(-0.780051\pi\)
0.770615 0.637302i \(-0.219949\pi\)
\(830\) 0 0
\(831\) 2.66727e7 1.33988
\(832\) 0 0
\(833\) −5.38342e6 −0.268810
\(834\) 0 0
\(835\) − 2.65589e7i − 1.31824i
\(836\) 0 0
\(837\) 1.74666e7i 0.861778i
\(838\) 0 0
\(839\) −1.77623e7 −0.871154 −0.435577 0.900151i \(-0.643456\pi\)
−0.435577 + 0.900151i \(0.643456\pi\)
\(840\) 0 0
\(841\) 2.01583e7 0.982798
\(842\) 0 0
\(843\) − 2.35937e6i − 0.114348i
\(844\) 0 0
\(845\) 1.06147e7i 0.511407i
\(846\) 0 0
\(847\) 1.14883e7 0.550234
\(848\) 0 0
\(849\) 1.86273e7 0.886913
\(850\) 0 0
\(851\) 1.22299e6i 0.0578895i
\(852\) 0 0
\(853\) − 486970.i − 0.0229155i −0.999934 0.0114578i \(-0.996353\pi\)
0.999934 0.0114578i \(-0.00364720\pi\)
\(854\) 0 0
\(855\) −4.46926e6 −0.209084
\(856\) 0 0
\(857\) 1.92634e6 0.0895945 0.0447972 0.998996i \(-0.485736\pi\)
0.0447972 + 0.998996i \(0.485736\pi\)
\(858\) 0 0
\(859\) − 2.23538e7i − 1.03364i −0.856094 0.516820i \(-0.827116\pi\)
0.856094 0.516820i \(-0.172884\pi\)
\(860\) 0 0
\(861\) − 1.81907e7i − 0.836258i
\(862\) 0 0
\(863\) −1.85838e7 −0.849390 −0.424695 0.905337i \(-0.639619\pi\)
−0.424695 + 0.905337i \(0.639619\pi\)
\(864\) 0 0
\(865\) 3.82025e7 1.73601
\(866\) 0 0
\(867\) 1.28043e7i 0.578504i
\(868\) 0 0
\(869\) − 4.15325e7i − 1.86569i
\(870\) 0 0
\(871\) −9.11742e6 −0.407217
\(872\) 0 0
\(873\) −1.21174e7 −0.538114
\(874\) 0 0
\(875\) − 1.58432e7i − 0.699555i
\(876\) 0 0
\(877\) 2.91048e7i 1.27781i 0.769286 + 0.638905i \(0.220612\pi\)
−0.769286 + 0.638905i \(0.779388\pi\)
\(878\) 0 0
\(879\) −1.28660e7 −0.561659
\(880\) 0 0
\(881\) −3.14696e6 −0.136600 −0.0683001 0.997665i \(-0.521758\pi\)
−0.0683001 + 0.997665i \(0.521758\pi\)
\(882\) 0 0
\(883\) 1.59995e7i 0.690566i 0.938499 + 0.345283i \(0.112217\pi\)
−0.938499 + 0.345283i \(0.887783\pi\)
\(884\) 0 0
\(885\) 4.96886e6i 0.213255i
\(886\) 0 0
\(887\) −3.45874e7 −1.47608 −0.738039 0.674758i \(-0.764248\pi\)
−0.738039 + 0.674758i \(0.764248\pi\)
\(888\) 0 0
\(889\) 2.96131e7 1.25669
\(890\) 0 0
\(891\) 1.36031e7i 0.574044i
\(892\) 0 0
\(893\) 1.08346e6i 0.0454656i
\(894\) 0 0
\(895\) −2.66581e7 −1.11243
\(896\) 0 0
\(897\) −2.05857e7 −0.854248
\(898\) 0 0
\(899\) − 2.52806e6i − 0.104325i
\(900\) 0 0
\(901\) 1.15794e7i 0.475199i
\(902\) 0 0
\(903\) −1.27776e7 −0.521471
\(904\) 0 0
\(905\) 3.02103e7 1.22612
\(906\) 0 0
\(907\) − 1.74396e7i − 0.703914i −0.936016 0.351957i \(-0.885516\pi\)
0.936016 0.351957i \(-0.114484\pi\)
\(908\) 0 0
\(909\) 1.11731e6i 0.0448503i
\(910\) 0 0
\(911\) 2.59589e6 0.103631 0.0518155 0.998657i \(-0.483499\pi\)
0.0518155 + 0.998657i \(0.483499\pi\)
\(912\) 0 0
\(913\) 3.65666e7 1.45180
\(914\) 0 0
\(915\) 2.25102e7i 0.888847i
\(916\) 0 0
\(917\) − 8.88624e6i − 0.348975i
\(918\) 0 0
\(919\) −1.76411e7 −0.689028 −0.344514 0.938781i \(-0.611956\pi\)
−0.344514 + 0.938781i \(0.611956\pi\)
\(920\) 0 0
\(921\) 1.90307e7 0.739275
\(922\) 0 0
\(923\) − 1.95925e7i − 0.756982i
\(924\) 0 0
\(925\) − 62282.0i − 0.00239336i
\(926\) 0 0
\(927\) −2.69834e6 −0.103133
\(928\) 0 0
\(929\) 3.96785e7 1.50840 0.754199 0.656646i \(-0.228025\pi\)
0.754199 + 0.656646i \(0.228025\pi\)
\(930\) 0 0
\(931\) − 7.57667e6i − 0.286486i
\(932\) 0 0
\(933\) 8.76874e6i 0.329787i
\(934\) 0 0
\(935\) 1.73210e7 0.647955
\(936\) 0 0
\(937\) −3.93413e7 −1.46386 −0.731930 0.681380i \(-0.761380\pi\)
−0.731930 + 0.681380i \(0.761380\pi\)
\(938\) 0 0
\(939\) 7.01830e6i 0.259757i
\(940\) 0 0
\(941\) − 4.62506e7i − 1.70272i −0.524581 0.851361i \(-0.675778\pi\)
0.524581 0.851361i \(-0.324222\pi\)
\(942\) 0 0
\(943\) 7.06955e7 2.58888
\(944\) 0 0
\(945\) −1.95022e7 −0.710402
\(946\) 0 0
\(947\) − 3.79025e7i − 1.37339i −0.726947 0.686693i \(-0.759062\pi\)
0.726947 0.686693i \(-0.240938\pi\)
\(948\) 0 0
\(949\) − 2.82409e7i − 1.01792i
\(950\) 0 0
\(951\) 2.97944e7 1.06828
\(952\) 0 0
\(953\) 2.66462e7 0.950394 0.475197 0.879879i \(-0.342377\pi\)
0.475197 + 0.879879i \(0.342377\pi\)
\(954\) 0 0
\(955\) 3.90977e7i 1.38721i
\(956\) 0 0
\(957\) − 3.84912e6i − 0.135857i
\(958\) 0 0
\(959\) −2.79085e7 −0.979918
\(960\) 0 0
\(961\) −1.05156e7 −0.367304
\(962\) 0 0
\(963\) − 1.21140e7i − 0.420943i
\(964\) 0 0
\(965\) 383724.i 0.0132648i
\(966\) 0 0
\(967\) 4.09790e7 1.40927 0.704637 0.709568i \(-0.251110\pi\)
0.704637 + 0.709568i \(0.251110\pi\)
\(968\) 0 0
\(969\) 5.95901e6 0.203875
\(970\) 0 0
\(971\) 2.72034e7i 0.925922i 0.886379 + 0.462961i \(0.153213\pi\)
−0.886379 + 0.462961i \(0.846787\pi\)
\(972\) 0 0
\(973\) − 1.30525e7i − 0.441990i
\(974\) 0 0
\(975\) 1.04834e6 0.0353177
\(976\) 0 0
\(977\) 2.53555e7 0.849839 0.424919 0.905231i \(-0.360302\pi\)
0.424919 + 0.905231i \(0.360302\pi\)
\(978\) 0 0
\(979\) 1.60672e7i 0.535775i
\(980\) 0 0
\(981\) − 9.89030e6i − 0.328123i
\(982\) 0 0
\(983\) 1.19139e7 0.393252 0.196626 0.980479i \(-0.437002\pi\)
0.196626 + 0.980479i \(0.437002\pi\)
\(984\) 0 0
\(985\) 2.86439e7 0.940678
\(986\) 0 0
\(987\) 1.36858e6i 0.0447173i
\(988\) 0 0
\(989\) − 4.96584e7i − 1.61437i
\(990\) 0 0
\(991\) −2.91931e7 −0.944268 −0.472134 0.881527i \(-0.656516\pi\)
−0.472134 + 0.881527i \(0.656516\pi\)
\(992\) 0 0
\(993\) −4.53538e6 −0.145962
\(994\) 0 0
\(995\) 3.03307e6i 0.0971237i
\(996\) 0 0
\(997\) − 1.73001e7i − 0.551201i −0.961272 0.275601i \(-0.911123\pi\)
0.961272 0.275601i \(-0.0888767\pi\)
\(998\) 0 0
\(999\) −1.22299e6 −0.0387713
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.c.129.1 2
4.3 odd 2 256.6.b.g.129.2 2
8.3 odd 2 256.6.b.g.129.1 2
8.5 even 2 inner 256.6.b.c.129.2 2
16.3 odd 4 4.6.a.a.1.1 1
16.5 even 4 64.6.a.b.1.1 1
16.11 odd 4 64.6.a.f.1.1 1
16.13 even 4 16.6.a.b.1.1 1
48.5 odd 4 576.6.a.bd.1.1 1
48.11 even 4 576.6.a.bc.1.1 1
48.29 odd 4 144.6.a.c.1.1 1
48.35 even 4 36.6.a.a.1.1 1
80.3 even 4 100.6.c.b.49.1 2
80.13 odd 4 400.6.c.f.49.2 2
80.19 odd 4 100.6.a.b.1.1 1
80.29 even 4 400.6.a.d.1.1 1
80.67 even 4 100.6.c.b.49.2 2
80.77 odd 4 400.6.c.f.49.1 2
112.3 even 12 196.6.e.d.177.1 2
112.13 odd 4 784.6.a.d.1.1 1
112.19 even 12 196.6.e.d.165.1 2
112.51 odd 12 196.6.e.g.165.1 2
112.67 odd 12 196.6.e.g.177.1 2
112.83 even 4 196.6.a.e.1.1 1
144.67 odd 12 324.6.e.a.217.1 2
144.83 even 12 324.6.e.d.109.1 2
144.115 odd 12 324.6.e.a.109.1 2
144.131 even 12 324.6.e.d.217.1 2
176.131 even 4 484.6.a.a.1.1 1
208.51 odd 4 676.6.a.a.1.1 1
208.83 even 4 676.6.d.a.337.1 2
208.99 even 4 676.6.d.a.337.2 2
240.83 odd 4 900.6.d.a.649.2 2
240.179 even 4 900.6.a.h.1.1 1
240.227 odd 4 900.6.d.a.649.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4.6.a.a.1.1 1 16.3 odd 4
16.6.a.b.1.1 1 16.13 even 4
36.6.a.a.1.1 1 48.35 even 4
64.6.a.b.1.1 1 16.5 even 4
64.6.a.f.1.1 1 16.11 odd 4
100.6.a.b.1.1 1 80.19 odd 4
100.6.c.b.49.1 2 80.3 even 4
100.6.c.b.49.2 2 80.67 even 4
144.6.a.c.1.1 1 48.29 odd 4
196.6.a.e.1.1 1 112.83 even 4
196.6.e.d.165.1 2 112.19 even 12
196.6.e.d.177.1 2 112.3 even 12
196.6.e.g.165.1 2 112.51 odd 12
196.6.e.g.177.1 2 112.67 odd 12
256.6.b.c.129.1 2 1.1 even 1 trivial
256.6.b.c.129.2 2 8.5 even 2 inner
256.6.b.g.129.1 2 8.3 odd 2
256.6.b.g.129.2 2 4.3 odd 2
324.6.e.a.109.1 2 144.115 odd 12
324.6.e.a.217.1 2 144.67 odd 12
324.6.e.d.109.1 2 144.83 even 12
324.6.e.d.217.1 2 144.131 even 12
400.6.a.d.1.1 1 80.29 even 4
400.6.c.f.49.1 2 80.77 odd 4
400.6.c.f.49.2 2 80.13 odd 4
484.6.a.a.1.1 1 176.131 even 4
576.6.a.bc.1.1 1 48.11 even 4
576.6.a.bd.1.1 1 48.5 odd 4
676.6.a.a.1.1 1 208.51 odd 4
676.6.d.a.337.1 2 208.83 even 4
676.6.d.a.337.2 2 208.99 even 4
784.6.a.d.1.1 1 112.13 odd 4
900.6.a.h.1.1 1 240.179 even 4
900.6.d.a.649.1 2 240.227 odd 4
900.6.d.a.649.2 2 240.83 odd 4