Properties

Label 100.6.c.b.49.2
Level $100$
Weight $6$
Character 100.49
Analytic conductor $16.038$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 100.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.0383819813\)
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 49.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 100.49
Dual form 100.6.c.b.49.1

$q$-expansion

\(f(q)\) \(=\) \(q+12.0000i q^{3} -88.0000i q^{7} +99.0000 q^{9} +O(q^{10})\) \(q+12.0000i q^{3} -88.0000i q^{7} +99.0000 q^{9} +540.000 q^{11} +418.000i q^{13} +594.000i q^{17} -836.000 q^{19} +1056.00 q^{21} +4104.00i q^{23} +4104.00i q^{27} +594.000 q^{29} +4256.00 q^{31} +6480.00i q^{33} -298.000i q^{37} -5016.00 q^{39} +17226.0 q^{41} +12100.0i q^{43} -1296.00i q^{47} +9063.00 q^{49} -7128.00 q^{51} -19494.0i q^{53} -10032.0i q^{57} +7668.00 q^{59} -34738.0 q^{61} -8712.00i q^{63} +21812.0i q^{67} -49248.0 q^{69} -46872.0 q^{71} -67562.0i q^{73} -47520.0i q^{77} +76912.0 q^{79} -25191.0 q^{81} -67716.0i q^{83} +7128.00i q^{87} -29754.0 q^{89} +36784.0 q^{91} +51072.0i q^{93} -122398. i q^{97} +53460.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 198 q^{9} + O(q^{10}) \) \( 2 q + 198 q^{9} + 1080 q^{11} - 1672 q^{19} + 2112 q^{21} + 1188 q^{29} + 8512 q^{31} - 10032 q^{39} + 34452 q^{41} + 18126 q^{49} - 14256 q^{51} + 15336 q^{59} - 69476 q^{61} - 98496 q^{69} - 93744 q^{71} + 153824 q^{79} - 50382 q^{81} - 59508 q^{89} + 73568 q^{91} + 106920 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(77\)
\(\chi(n)\) \(1\) \(-1\)

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 \(0.125764\pi\)
−0.922958 + 0.384900i \(0.874236\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 88.0000i − 0.678793i −0.940643 0.339397i \(-0.889777\pi\)
0.940643 0.339397i \(-0.110223\pi\)
\(8\) 0 0
\(9\) 99.0000 0.407407
\(10\) 0 0
\(11\) 540.000 1.34559 0.672794 0.739830i \(-0.265094\pi\)
0.672794 + 0.739830i \(0.265094\pi\)
\(12\) 0 0
\(13\) 418.000i 0.685990i 0.939337 + 0.342995i \(0.111441\pi\)
−0.939337 + 0.342995i \(0.888559\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 594.000i 0.498499i 0.968439 + 0.249249i \(0.0801839\pi\)
−0.968439 + 0.249249i \(0.919816\pi\)
\(18\) 0 0
\(19\) −836.000 −0.531279 −0.265639 0.964072i \(-0.585583\pi\)
−0.265639 + 0.964072i \(0.585583\pi\)
\(20\) 0 0
\(21\) 1056.00 0.522535
\(22\) 0 0
\(23\) 4104.00i 1.61766i 0.588041 + 0.808831i \(0.299899\pi\)
−0.588041 + 0.808831i \(0.700101\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4104.00i 1.08342i
\(28\) 0 0
\(29\) 594.000 0.131157 0.0655785 0.997847i \(-0.479111\pi\)
0.0655785 + 0.997847i \(0.479111\pi\)
\(30\) 0 0
\(31\) 4256.00 0.795422 0.397711 0.917511i \(-0.369805\pi\)
0.397711 + 0.917511i \(0.369805\pi\)
\(32\) 0 0
\(33\) 6480.00i 1.03583i
\(34\) 0 0
\(35\) 0 0
\(36\) 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 \(-0.204730\pi\)
0.800193 + 0.599742i \(0.204730\pi\)
\(42\) 0 0
\(43\) 12100.0i 0.997963i 0.866613 + 0.498981i \(0.166292\pi\)
−0.866613 + 0.498981i \(0.833708\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 1296.00i − 0.0855777i −0.999084 0.0427888i \(-0.986376\pi\)
0.999084 0.0427888i \(-0.0136243\pi\)
\(48\) 0 0
\(49\) 9063.00 0.539240
\(50\) 0 0
\(51\) −7128.00 −0.383745
\(52\) 0 0
\(53\) − 19494.0i − 0.953260i −0.879104 0.476630i \(-0.841858\pi\)
0.879104 0.476630i \(-0.158142\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 10032.0i − 0.408978i
\(58\) 0 0
\(59\) 7668.00 0.286782 0.143391 0.989666i \(-0.454199\pi\)
0.143391 + 0.989666i \(0.454199\pi\)
\(60\) 0 0
\(61\) −34738.0 −1.19531 −0.597655 0.801754i \(-0.703901\pi\)
−0.597655 + 0.801754i \(0.703901\pi\)
\(62\) 0 0
\(63\) − 8712.00i − 0.276545i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 21812.0i 0.593620i 0.954937 + 0.296810i \(0.0959228\pi\)
−0.954937 + 0.296810i \(0.904077\pi\)
\(68\) 0 0
\(69\) −49248.0 −1.24528
\(70\) 0 0
\(71\) −46872.0 −1.10349 −0.551744 0.834014i \(-0.686037\pi\)
−0.551744 + 0.834014i \(0.686037\pi\)
\(72\) 0 0
\(73\) − 67562.0i − 1.48387i −0.670473 0.741934i \(-0.733909\pi\)
0.670473 0.741934i \(-0.266091\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 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 \(-0.818625\pi\)
0.842006 0.539468i \(-0.181375\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 7128.00i 0.100965i
\(88\) 0 0
\(89\) −29754.0 −0.398172 −0.199086 0.979982i \(-0.563797\pi\)
−0.199086 + 0.979982i \(0.563797\pi\)
\(90\) 0 0
\(91\) 36784.0 0.465646
\(92\) 0 0
\(93\) 51072.0i 0.612316i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 122398.i − 1.32082i −0.750903 0.660412i \(-0.770382\pi\)
0.750903 0.660412i \(-0.229618\pi\)
\(98\) 0 0
\(99\) 53460.0 0.548202
\(100\) 0 0
\(101\) 11286.0 0.110087 0.0550436 0.998484i \(-0.482470\pi\)
0.0550436 + 0.998484i \(0.482470\pi\)
\(102\) 0 0
\(103\) 27256.0i 0.253145i 0.991957 + 0.126572i \(0.0403976\pi\)
−0.991957 + 0.126572i \(0.959602\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 122364.i 1.03322i 0.856220 + 0.516612i \(0.172807\pi\)
−0.856220 + 0.516612i \(0.827193\pi\)
\(108\) 0 0
\(109\) −99902.0 −0.805393 −0.402697 0.915334i \(-0.631927\pi\)
−0.402697 + 0.915334i \(0.631927\pi\)
\(110\) 0 0
\(111\) 3576.00 0.0275480
\(112\) 0 0
\(113\) 29646.0i 0.218409i 0.994019 + 0.109204i \(0.0348303\pi\)
−0.994019 + 0.109204i \(0.965170\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 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
\(125\) 0 0
\(126\) 0 0
\(127\) 336512.i 1.85136i 0.378305 + 0.925681i \(0.376507\pi\)
−0.378305 + 0.925681i \(0.623493\pi\)
\(128\) 0 0
\(129\) −145200. −0.768232
\(130\) 0 0
\(131\) 100980. 0.514111 0.257056 0.966397i \(-0.417248\pi\)
0.257056 + 0.966397i \(0.417248\pi\)
\(132\) 0 0
\(133\) 73568.0i 0.360628i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 317142.i − 1.44362i −0.692092 0.721809i \(-0.743311\pi\)
0.692092 0.721809i \(-0.256689\pi\)
\(138\) 0 0
\(139\) 148324. 0.651140 0.325570 0.945518i \(-0.394444\pi\)
0.325570 + 0.945518i \(0.394444\pi\)
\(140\) 0 0
\(141\) 15552.0 0.0658777
\(142\) 0 0
\(143\) 225720.i 0.923060i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 108756.i 0.415107i
\(148\) 0 0
\(149\) −196614. −0.725519 −0.362759 0.931883i \(-0.618165\pi\)
−0.362759 + 0.931883i \(0.618165\pi\)
\(150\) 0 0
\(151\) 74360.0 0.265398 0.132699 0.991156i \(-0.457636\pi\)
0.132699 + 0.991156i \(0.457636\pi\)
\(152\) 0 0
\(153\) 58806.0i 0.203092i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 120878.i 0.391380i 0.980666 + 0.195690i \(0.0626946\pi\)
−0.980666 + 0.195690i \(0.937305\pi\)
\(158\) 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 \(0.0524773\pi\)
−0.986441 + 0.164116i \(0.947523\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 491832.i − 1.36466i −0.731043 0.682332i \(-0.760966\pi\)
0.731043 0.682332i \(-0.239034\pi\)
\(168\) 0 0
\(169\) 196569. 0.529417
\(170\) 0 0
\(171\) −82764.0 −0.216447
\(172\) 0 0
\(173\) − 707454.i − 1.79714i −0.438826 0.898572i \(-0.644605\pi\)
0.438826 0.898572i \(-0.355395\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 92016.0i 0.220765i
\(178\) 0 0
\(179\) −493668. −1.15160 −0.575801 0.817590i \(-0.695310\pi\)
−0.575801 + 0.817590i \(0.695310\pi\)
\(180\) 0 0
\(181\) −559450. −1.26930 −0.634651 0.772799i \(-0.718856\pi\)
−0.634651 + 0.772799i \(0.718856\pi\)
\(182\) 0 0
\(183\) − 416856.i − 0.920149i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 320760.i 0.670774i
\(188\) 0 0
\(189\) 361152. 0.735420
\(190\) 0 0
\(191\) −724032. −1.43607 −0.718033 0.696009i \(-0.754957\pi\)
−0.718033 + 0.696009i \(0.754957\pi\)
\(192\) 0 0
\(193\) − 7106.00i − 0.0137319i −0.999976 0.00686597i \(-0.997814\pi\)
0.999976 0.00686597i \(-0.00218552\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 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 \(-0.516009\pi\)
−0.0502720 + 0.998736i \(0.516009\pi\)
\(200\) 0 0
\(201\) −261744. −0.456969
\(202\) 0 0
\(203\) − 52272.0i − 0.0890285i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 406296.i 0.659047i
\(208\) 0 0
\(209\) −451440. −0.714882
\(210\) 0 0
\(211\) −339196. −0.524499 −0.262249 0.965000i \(-0.584464\pi\)
−0.262249 + 0.965000i \(0.584464\pi\)
\(212\) 0 0
\(213\) − 562464.i − 0.849465i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 374528.i − 0.539927i
\(218\) 0 0
\(219\) 810744. 1.14228
\(220\) 0 0
\(221\) −248292. −0.341965
\(222\) 0 0
\(223\) − 779360.i − 1.04948i −0.851261 0.524742i \(-0.824162\pi\)
0.851261 0.524742i \(-0.175838\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 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. 0.343692 0.171846 0.985124i \(-0.445027\pi\)
0.171846 + 0.985124i \(0.445027\pi\)
\(230\) 0 0
\(231\) 570240. 0.703117
\(232\) 0 0
\(233\) 153846.i 0.185651i 0.995682 + 0.0928253i \(0.0295898\pi\)
−0.995682 + 0.0928253i \(0.970410\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 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
\(245\) 0 0
\(246\) 0 0
\(247\) − 349448.i − 0.364452i
\(248\) 0 0
\(249\) 812592. 0.830566
\(250\) 0 0
\(251\) 1.34190e6 1.34442 0.672211 0.740359i \(-0.265345\pi\)
0.672211 + 0.740359i \(0.265345\pi\)
\(252\) 0 0
\(253\) 2.21616e6i 2.17671i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 132354.i 0.124998i 0.998045 + 0.0624992i \(0.0199071\pi\)
−0.998045 + 0.0624992i \(0.980093\pi\)
\(258\) 0 0
\(259\) −26224.0 −0.0242912
\(260\) 0 0
\(261\) 58806.0 0.0534343
\(262\) 0 0
\(263\) − 943272.i − 0.840906i −0.907314 0.420453i \(-0.861871\pi\)
0.907314 0.420453i \(-0.138129\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 357048.i − 0.306513i
\(268\) 0 0
\(269\) −967518. −0.815227 −0.407613 0.913155i \(-0.633639\pi\)
−0.407613 + 0.913155i \(0.633639\pi\)
\(270\) 0 0
\(271\) −518320. −0.428721 −0.214360 0.976755i \(-0.568767\pi\)
−0.214360 + 0.976755i \(0.568767\pi\)
\(272\) 0 0
\(273\) 441408.i 0.358454i
\(274\) 0 0
\(275\) 0 0
\(276\) 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 \(-0.523663\pi\)
−0.0742709 + 0.997238i \(0.523663\pi\)
\(282\) 0 0
\(283\) 1.55228e6i 1.15213i 0.817403 + 0.576067i \(0.195413\pi\)
−0.817403 + 0.576067i \(0.804587\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 1.51589e6i − 1.08633i
\(288\) 0 0
\(289\) 1.06702e6 0.751499
\(290\) 0 0
\(291\) 1.46878e6 1.01677
\(292\) 0 0
\(293\) 1.07217e6i 0.729616i 0.931083 + 0.364808i \(0.118865\pi\)
−0.931083 + 0.364808i \(0.881135\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2.21616e6i 1.45784i
\(298\) 0 0
\(299\) −1.71547e6 −1.10970
\(300\) 0 0
\(301\) 1.06480e6 0.677410
\(302\) 0 0
\(303\) 135432.i 0.0847451i
\(304\) 0 0
\(305\) 0 0
\(306\) 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. −0.194871
\(310\) 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.i − 0.337435i −0.985664 0.168717i \(-0.946038\pi\)
0.985664 0.168717i \(-0.0539625\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 2.48287e6i − 1.38773i −0.720105 0.693865i \(-0.755906\pi\)
0.720105 0.693865i \(-0.244094\pi\)
\(318\) 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
\(325\) 0 0
\(326\) 0 0
\(327\) − 1.19882e6i − 0.619992i
\(328\) 0 0
\(329\) −114048. −0.0580895
\(330\) 0 0
\(331\) 377948. 0.189610 0.0948052 0.995496i \(-0.469777\pi\)
0.0948052 + 0.995496i \(0.469777\pi\)
\(332\) 0 0
\(333\) − 29502.0i − 0.0145794i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 639122.i 0.306555i 0.988183 + 0.153278i \(0.0489829\pi\)
−0.988183 + 0.153278i \(0.951017\pi\)
\(338\) 0 0
\(339\) −355752. −0.168131
\(340\) 0 0
\(341\) 2.29824e6 1.07031
\(342\) 0 0
\(343\) − 2.27656e6i − 1.04483i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 2.90466e6i − 1.29501i −0.762063 0.647503i \(-0.775813\pi\)
0.762063 0.647503i \(-0.224187\pi\)
\(348\) 0 0
\(349\) 3.99157e6 1.75420 0.877102 0.480304i \(-0.159474\pi\)
0.877102 + 0.480304i \(0.159474\pi\)
\(350\) 0 0
\(351\) −1.71547e6 −0.743217
\(352\) 0 0
\(353\) − 1.42922e6i − 0.610466i −0.952278 0.305233i \(-0.901266\pi\)
0.952278 0.305233i \(-0.0987344\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 627264.i 0.260483i
\(358\) 0 0
\(359\) −1.16186e6 −0.475794 −0.237897 0.971290i \(-0.576458\pi\)
−0.237897 + 0.971290i \(0.576458\pi\)
\(360\) 0 0
\(361\) −1.77720e6 −0.717743
\(362\) 0 0
\(363\) 1.56659e6i 0.624005i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 1.08923e6i − 0.422139i −0.977471 0.211069i \(-0.932305\pi\)
0.977471 0.211069i \(-0.0676946\pi\)
\(368\) 0 0
\(369\) 1.70537e6 0.652009
\(370\) 0 0
\(371\) −1.71547e6 −0.647066
\(372\) 0 0
\(373\) − 3.50577e6i − 1.30470i −0.757918 0.652350i \(-0.773783\pi\)
0.757918 0.652350i \(-0.226217\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 248292.i 0.0899724i
\(378\) 0 0
\(379\) −4.04385e6 −1.44610 −0.723048 0.690798i \(-0.757260\pi\)
−0.723048 + 0.690798i \(0.757260\pi\)
\(380\) 0 0
\(381\) −4.03814e6 −1.42518
\(382\) 0 0
\(383\) − 5.18746e6i − 1.80700i −0.428591 0.903499i \(-0.640990\pi\)
0.428591 0.903499i \(-0.359010\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.19790e6i 0.406577i
\(388\) 0 0
\(389\) 950346. 0.318425 0.159213 0.987244i \(-0.449104\pi\)
0.159213 + 0.987244i \(0.449104\pi\)
\(390\) 0 0
\(391\) −2.43778e6 −0.806403
\(392\) 0 0
\(393\) 1.21176e6i 0.395763i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 520738.i − 0.165822i −0.996557 0.0829112i \(-0.973578\pi\)
0.996557 0.0829112i \(-0.0264218\pi\)
\(398\) 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
\(405\) 0 0
\(406\) 0 0
\(407\) − 160920.i − 0.0481531i
\(408\) 0 0
\(409\) −2.64051e6 −0.780511 −0.390255 0.920707i \(-0.627613\pi\)
−0.390255 + 0.920707i \(0.627613\pi\)
\(410\) 0 0
\(411\) 3.80570e6 1.11130
\(412\) 0 0
\(413\) − 674784.i − 0.194666i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.77989e6i 0.501248i
\(418\) 0 0
\(419\) 4.98020e6 1.38584 0.692918 0.721016i \(-0.256325\pi\)
0.692918 + 0.721016i \(0.256325\pi\)
\(420\) 0 0
\(421\) −237994. −0.0654426 −0.0327213 0.999465i \(-0.510417\pi\)
−0.0327213 + 0.999465i \(0.510417\pi\)
\(422\) 0 0
\(423\) − 128304.i − 0.0348650i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 3.05694e6i 0.811368i
\(428\) 0 0
\(429\) −2.70864e6 −0.710572
\(430\) 0 0
\(431\) −3.88238e6 −1.00671 −0.503356 0.864079i \(-0.667902\pi\)
−0.503356 + 0.864079i \(0.667902\pi\)
\(432\) 0 0
\(433\) 66958.0i 0.0171626i 0.999963 + 0.00858129i \(0.00273154\pi\)
−0.999963 + 0.00858129i \(0.997268\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 3.43094e6i − 0.859429i
\(438\) 0 0
\(439\) 6.50135e6 1.61006 0.805031 0.593233i \(-0.202149\pi\)
0.805031 + 0.593233i \(0.202149\pi\)
\(440\) 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
\(445\) 0 0
\(446\) 0 0
\(447\) − 2.35937e6i − 0.558505i
\(448\) 0 0
\(449\) −3.77671e6 −0.884092 −0.442046 0.896992i \(-0.645747\pi\)
−0.442046 + 0.896992i \(0.645747\pi\)
\(450\) 0 0
\(451\) 9.30204e6 2.15346
\(452\) 0 0
\(453\) 892320.i 0.204303i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 3.18069e6i − 0.712412i −0.934407 0.356206i \(-0.884070\pi\)
0.934407 0.356206i \(-0.115930\pi\)
\(458\) 0 0
\(459\) −2.43778e6 −0.540085
\(460\) 0 0
\(461\) 6.68547e6 1.46514 0.732571 0.680691i \(-0.238320\pi\)
0.732571 + 0.680691i \(0.238320\pi\)
\(462\) 0 0
\(463\) 4.35122e6i 0.943318i 0.881781 + 0.471659i \(0.156345\pi\)
−0.881781 + 0.471659i \(0.843655\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 7.07994e6i 1.50223i 0.660170 + 0.751117i \(0.270484\pi\)
−0.660170 + 0.751117i \(0.729516\pi\)
\(468\) 0 0
\(469\) 1.91946e6 0.402945
\(470\) 0 0
\(471\) −1.45054e6 −0.301284
\(472\) 0 0
\(473\) 6.53400e6i 1.34285i
\(474\) 0 0
\(475\) 0 0
\(476\) 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
\(485\) 0 0
\(486\) 0 0
\(487\) 2.29710e6i 0.438891i 0.975625 + 0.219446i \(0.0704248\pi\)
−0.975625 + 0.219446i \(0.929575\pi\)
\(488\) 0 0
\(489\) −1.33608e6 −0.252674
\(490\) 0 0
\(491\) 2.82150e6 0.528173 0.264087 0.964499i \(-0.414930\pi\)
0.264087 + 0.964499i \(0.414930\pi\)
\(492\) 0 0
\(493\) 352836.i 0.0653816i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.12474e6i 0.749040i
\(498\) 0 0
\(499\) 4.13628e6 0.743634 0.371817 0.928306i \(-0.378735\pi\)
0.371817 + 0.928306i \(0.378735\pi\)
\(500\) 0 0
\(501\) 5.90198e6 1.05052
\(502\) 0 0
\(503\) − 8.33263e6i − 1.46846i −0.678901 0.734230i \(-0.737543\pi\)
0.678901 0.734230i \(-0.262457\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 2.35883e6i 0.407546i
\(508\) 0 0
\(509\) −4.34101e6 −0.742670 −0.371335 0.928499i \(-0.621100\pi\)
−0.371335 + 0.928499i \(0.621100\pi\)
\(510\) 0 0
\(511\) −5.94546e6 −1.00724
\(512\) 0 0
\(513\) − 3.43094e6i − 0.575599i
\(514\) 0 0
\(515\) 0 0
\(516\) 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 \(-0.683117\pi\)
−0.544070 + 0.839040i \(0.683117\pi\)
\(522\) 0 0
\(523\) 7.72196e6i 1.23445i 0.786787 + 0.617224i \(0.211743\pi\)
−0.786787 + 0.617224i \(0.788257\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.52806e6i 0.396517i
\(528\) 0 0
\(529\) −1.04065e7 −1.61683
\(530\) 0 0
\(531\) 759132. 0.116837
\(532\) 0 0
\(533\) 7.20047e6i 1.09785i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 5.92402e6i − 0.886504i
\(538\) 0 0
\(539\) 4.89402e6 0.725594
\(540\) 0 0
\(541\) −682066. −0.100192 −0.0500960 0.998744i \(-0.515953\pi\)
−0.0500960 + 0.998744i \(0.515953\pi\)
\(542\) 0 0
\(543\) − 6.71340e6i − 0.977109i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 2.15772e6i 0.308337i 0.988045 + 0.154169i \(0.0492699\pi\)
−0.988045 + 0.154169i \(0.950730\pi\)
\(548\) 0 0
\(549\) −3.43906e6 −0.486978
\(550\) 0 0
\(551\) −496584. −0.0696809
\(552\) 0 0
\(553\) − 6.76826e6i − 0.941161i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 2.67597e6i − 0.365463i −0.983163 0.182731i \(-0.941506\pi\)
0.983163 0.182731i \(-0.0584939\pi\)
\(558\) 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 \(0.0759113\pi\)
−0.971698 + 0.236228i \(0.924089\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.21681e6i 0.289581i
\(568\) 0 0
\(569\) 1.29225e7 1.67327 0.836633 0.547764i \(-0.184521\pi\)
0.836633 + 0.547764i \(0.184521\pi\)
\(570\) 0 0
\(571\) −6.08357e6 −0.780851 −0.390426 0.920634i \(-0.627672\pi\)
−0.390426 + 0.920634i \(0.627672\pi\)
\(572\) 0 0
\(573\) − 8.68838e6i − 1.10548i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 1.58241e7i − 1.97869i −0.145579 0.989347i \(-0.546505\pi\)
0.145579 0.989347i \(-0.453495\pi\)
\(578\) 0 0
\(579\) 85272.0 0.0105709
\(580\) 0 0
\(581\) −5.95901e6 −0.732375
\(582\) 0 0
\(583\) − 1.05268e7i − 1.28269i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 4.60220e6i 0.551278i 0.961261 + 0.275639i \(0.0888894\pi\)
−0.961261 + 0.275639i \(0.911111\pi\)
\(588\) 0 0
\(589\) −3.55802e6 −0.422590
\(590\) 0 0
\(591\) 6.36530e6 0.749636
\(592\) 0 0
\(593\) − 8.61122e6i − 1.00561i −0.864401 0.502803i \(-0.832302\pi\)
0.864401 0.502803i \(-0.167698\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 674016.i − 0.0773988i
\(598\) 0 0
\(599\) 7.98228e6 0.908992 0.454496 0.890749i \(-0.349819\pi\)
0.454496 + 0.890749i \(0.349819\pi\)
\(600\) 0 0
\(601\) 1.01740e7 1.14896 0.574481 0.818518i \(-0.305204\pi\)
0.574481 + 0.818518i \(0.305204\pi\)
\(602\) 0 0
\(603\) 2.15939e6i 0.241845i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 9.95843e6i − 1.09703i −0.836140 0.548516i \(-0.815193\pi\)
0.836140 0.548516i \(-0.184807\pi\)
\(608\) 0 0
\(609\) 627264. 0.0685342
\(610\) 0 0
\(611\) 541728. 0.0587054
\(612\) 0 0
\(613\) − 4.19586e6i − 0.450993i −0.974244 0.225497i \(-0.927600\pi\)
0.974244 0.225497i \(-0.0724005\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 9.12551e6i 0.965038i 0.875885 + 0.482519i \(0.160278\pi\)
−0.875885 + 0.482519i \(0.839722\pi\)
\(618\) 0 0
\(619\) −6.45734e6 −0.677372 −0.338686 0.940900i \(-0.609982\pi\)
−0.338686 + 0.940900i \(0.609982\pi\)
\(620\) 0 0
\(621\) −1.68428e7 −1.75261
\(622\) 0 0
\(623\) 2.61835e6i 0.270276i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 5.41728e6i − 0.550316i
\(628\) 0 0
\(629\) 177012. 0.0178392
\(630\) 0 0
\(631\) −1.40514e7 −1.40490 −0.702450 0.711733i \(-0.747910\pi\)
−0.702450 + 0.711733i \(0.747910\pi\)
\(632\) 0 0
\(633\) − 4.07035e6i − 0.403759i
\(634\) 0 0
\(635\) 0 0
\(636\) 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 \(-0.992583\pi\)
0.999729 0.0233004i \(-0.00741743\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.48119e6i 0.233023i 0.993189 + 0.116512i \(0.0371713\pi\)
−0.993189 + 0.116512i \(0.962829\pi\)
\(648\) 0 0
\(649\) 4.14072e6 0.385891
\(650\) 0 0
\(651\) 4.49434e6 0.415636
\(652\) 0 0
\(653\) 5.29130e6i 0.485601i 0.970076 + 0.242800i \(0.0780660\pi\)
−0.970076 + 0.242800i \(0.921934\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 6.68864e6i − 0.604539i
\(658\) 0 0
\(659\) −4.72468e6 −0.423798 −0.211899 0.977292i \(-0.567965\pi\)
−0.211899 + 0.977292i \(0.567965\pi\)
\(660\) 0 0
\(661\) −6.17420e6 −0.549639 −0.274819 0.961496i \(-0.588618\pi\)
−0.274819 + 0.961496i \(0.588618\pi\)
\(662\) 0 0
\(663\) − 2.97950e6i − 0.263245i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 2.43778e6i 0.212168i
\(668\) 0 0
\(669\) 9.35232e6 0.807893
\(670\) 0 0
\(671\) −1.87585e7 −1.60839
\(672\) 0 0
\(673\) 9.40925e6i 0.800787i 0.916343 + 0.400394i \(0.131127\pi\)
−0.916343 + 0.400394i \(0.868873\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 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
\(685\) 0 0
\(686\) 0 0
\(687\) 3.27295e6i 0.264574i
\(688\) 0 0
\(689\) 8.14849e6 0.653927
\(690\) 0 0
\(691\) 2.26556e7 1.80501 0.902506 0.430677i \(-0.141725\pi\)
0.902506 + 0.430677i \(0.141725\pi\)
\(692\) 0 0
\(693\) − 4.70448e6i − 0.372116i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.02322e7i 0.797791i
\(698\) 0 0
\(699\) −1.84615e6 −0.142914
\(700\) 0 0
\(701\) 1.90169e7 1.46166 0.730828 0.682562i \(-0.239134\pi\)
0.730828 + 0.682562i \(0.239134\pi\)
\(702\) 0 0
\(703\) 249128.i 0.0190123i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 993168.i − 0.0747264i
\(708\) 0 0
\(709\) −1.51311e7 −1.13046 −0.565231 0.824933i \(-0.691213\pi\)
−0.565231 + 0.824933i \(0.691213\pi\)
\(710\) 0 0
\(711\) 7.61429e6 0.564879
\(712\) 0 0
\(713\) 1.74666e7i 1.28672i
\(714\) 0 0
\(715\) 0 0
\(716\) 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
\(725\) 0 0
\(726\) 0 0
\(727\) − 7.41230e6i − 0.520136i −0.965590 0.260068i \(-0.916255\pi\)
0.965590 0.260068i \(-0.0837449\pi\)
\(728\) 0 0
\(729\) −1.44612e7 −1.00782
\(730\) 0 0
\(731\) −7.18740e6 −0.497483
\(732\) 0 0
\(733\) 2.77928e6i 0.191061i 0.995426 + 0.0955306i \(0.0304548\pi\)
−0.995426 + 0.0955306i \(0.969545\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.17785e7i 0.798768i
\(738\) 0 0
\(739\) 1.21046e7 0.815342 0.407671 0.913129i \(-0.366341\pi\)
0.407671 + 0.913129i \(0.366341\pi\)
\(740\) 0 0
\(741\) 4.19338e6 0.280555
\(742\) 0 0
\(743\) − 4.46926e6i − 0.297005i −0.988912 0.148502i \(-0.952555\pi\)
0.988912 0.148502i \(-0.0474452\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 6.70388e6i − 0.439567i
\(748\) 0 0
\(749\) 1.07680e7 0.701345
\(750\) 0 0
\(751\) 2.88463e7 1.86634 0.933168 0.359442i \(-0.117033\pi\)
0.933168 + 0.359442i \(0.117033\pi\)
\(752\) 0 0
\(753\) 1.61028e7i 1.03494i
\(754\) 0 0
\(755\) 0 0
\(756\) 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 \(-0.454558\pi\)
0.142274 + 0.989827i \(0.454558\pi\)
\(762\) 0 0
\(763\) 8.79138e6i 0.546696i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.20522e6i 0.196730i
\(768\) 0 0
\(769\) 2.15923e7 1.31669 0.658345 0.752716i \(-0.271257\pi\)
0.658345 + 0.752716i \(0.271257\pi\)
\(770\) 0 0
\(771\) −1.58825e6 −0.0962238
\(772\) 0 0
\(773\) 1.48400e7i 0.893276i 0.894715 + 0.446638i \(0.147379\pi\)
−0.894715 + 0.446638i \(0.852621\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 314688.i − 0.0186994i
\(778\) 0 0
\(779\) −1.44009e7 −0.850251
\(780\) 0 0
\(781\) −2.53109e7 −1.48484
\(782\) 0 0
\(783\) 2.43778e6i 0.142098i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 2.48785e7i − 1.43182i −0.698194 0.715909i \(-0.746013\pi\)
0.698194 0.715909i \(-0.253987\pi\)
\(788\) 0 0
\(789\) 1.13193e7 0.647330
\(790\) 0 0
\(791\) 2.60885e6 0.148254
\(792\) 0 0
\(793\) − 1.45205e7i − 0.819970i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 3.16080e7i 1.76259i 0.472568 + 0.881294i \(0.343327\pi\)
−0.472568 + 0.881294i \(0.656673\pi\)
\(798\) 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
\(805\) 0 0
\(806\) 0 0
\(807\) − 1.16102e7i − 0.627562i
\(808\) 0 0
\(809\) 3.10009e6 0.166534 0.0832669 0.996527i \(-0.473465\pi\)
0.0832669 + 0.996527i \(0.473465\pi\)
\(810\) 0 0
\(811\) 1.87180e6 0.0999328 0.0499664 0.998751i \(-0.484089\pi\)
0.0499664 + 0.998751i \(0.484089\pi\)
\(812\) 0 0
\(813\) − 6.21984e6i − 0.330030i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 1.01156e7i − 0.530196i
\(818\) 0 0
\(819\) 3.64162e6 0.189707
\(820\) 0 0
\(821\) −2.00184e7 −1.03650 −0.518252 0.855228i \(-0.673417\pi\)
−0.518252 + 0.855228i \(0.673417\pi\)
\(822\) 0 0
\(823\) − 1.53118e7i − 0.787999i −0.919111 0.394000i \(-0.871091\pi\)
0.919111 0.394000i \(-0.128909\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 9.59310e6i 0.487748i 0.969807 + 0.243874i \(0.0784183\pi\)
−0.969807 + 0.243874i \(0.921582\pi\)
\(828\) 0 0
\(829\) −2.52209e7 −1.27460 −0.637302 0.770615i \(-0.719949\pi\)
−0.637302 + 0.770615i \(0.719949\pi\)
\(830\) 0 0
\(831\) −2.66727e7 −1.33988
\(832\) 0 0
\(833\) 5.38342e6i 0.268810i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1.74666e7i 0.861778i
\(838\) 0 0
\(839\) 1.77623e7 0.871154 0.435577 0.900151i \(-0.356544\pi\)
0.435577 + 0.900151i \(0.356544\pi\)
\(840\) 0 0
\(841\) −2.01583e7 −0.982798
\(842\) 0 0
\(843\) − 2.35937e6i − 0.114348i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 1.14883e7i − 0.550234i
\(848\) 0 0
\(849\) −1.86273e7 −0.886913
\(850\) 0 0
\(851\) 1.22299e6 0.0578895
\(852\) 0 0
\(853\) 486970.i 0.0229155i 0.999934 + 0.0114578i \(0.00364720\pi\)
−0.999934 + 0.0114578i \(0.996353\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 1.92634e6i − 0.0895945i −0.998996 0.0447972i \(-0.985736\pi\)
0.998996 0.0447972i \(-0.0142642\pi\)
\(858\) 0 0
\(859\) −2.23538e7 −1.03364 −0.516820 0.856094i \(-0.672884\pi\)
−0.516820 + 0.856094i \(0.672884\pi\)
\(860\) 0 0
\(861\) 1.81907e7 0.836258
\(862\) 0 0
\(863\) − 1.85838e7i − 0.849390i −0.905337 0.424695i \(-0.860381\pi\)
0.905337 0.424695i \(-0.139619\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 1.28043e7i 0.578504i
\(868\) 0 0
\(869\) 4.15325e7 1.86569
\(870\) 0 0
\(871\) −9.11742e6 −0.407217
\(872\) 0 0
\(873\) − 1.21174e7i − 0.538114i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 2.91048e7i − 1.27781i −0.769286 0.638905i \(-0.779388\pi\)
0.769286 0.638905i \(-0.220612\pi\)
\(878\) 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 \(-0.887783\pi\)
0.938499 0.345283i \(-0.112217\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 3.45874e7i − 1.47608i −0.674758 0.738039i \(-0.735752\pi\)
0.674758 0.738039i \(-0.264248\pi\)
\(888\) 0 0
\(889\) 2.96131e7 1.25669
\(890\) 0 0
\(891\) −1.36031e7 −0.574044
\(892\) 0 0
\(893\) 1.08346e6i 0.0454656i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 2.05857e7i − 0.854248i
\(898\) 0 0
\(899\) 2.52806e6 0.104325
\(900\) 0 0
\(901\) 1.15794e7 0.475199
\(902\) 0 0
\(903\) 1.27776e7i 0.521471i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.74396e7i 0.703914i 0.936016 + 0.351957i \(0.114484\pi\)
−0.936016 + 0.351957i \(0.885516\pi\)
\(908\) 0 0
\(909\) 1.11731e6 0.0448503
\(910\) 0 0
\(911\) −2.59589e6 −0.103631 −0.0518155 0.998657i \(-0.516501\pi\)
−0.0518155 + 0.998657i \(0.516501\pi\)
\(912\) 0 0
\(913\) − 3.65666e7i − 1.45180i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 8.88624e6i − 0.348975i
\(918\) 0 0
\(919\) 1.76411e7 0.689028 0.344514 0.938781i \(-0.388044\pi\)
0.344514 + 0.938781i \(0.388044\pi\)
\(920\) 0 0
\(921\) −1.90307e7 −0.739275
\(922\) 0 0
\(923\) − 1.95925e7i − 0.756982i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 2.69834e6i 0.103133i
\(928\) 0 0
\(929\) −3.96785e7 −1.50840 −0.754199 0.656646i \(-0.771975\pi\)
−0.754199 + 0.656646i \(0.771975\pi\)
\(930\) 0 0
\(931\) −7.57667e6 −0.286486
\(932\) 0 0
\(933\) − 8.76874e6i − 0.329787i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 3.93413e7i 1.46386i 0.681380 + 0.731930i \(0.261380\pi\)
−0.681380 + 0.731930i \(0.738620\pi\)
\(938\) 0 0
\(939\) 7.01830e6 0.259757
\(940\) 0 0
\(941\) 4.62506e7 1.70272 0.851361 0.524581i \(-0.175778\pi\)
0.851361 + 0.524581i \(0.175778\pi\)
\(942\) 0 0
\(943\) 7.06955e7i 2.58888i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 3.79025e7i − 1.37339i −0.726947 0.686693i \(-0.759062\pi\)
0.726947 0.686693i \(-0.240938\pi\)
\(948\) 0 0
\(949\) 2.82409e7 1.01792
\(950\) 0 0
\(951\) 2.97944e7 1.06828
\(952\) 0 0
\(953\) 2.66462e7i 0.950394i 0.879879 + 0.475197i \(0.157623\pi\)
−0.879879 + 0.475197i \(0.842377\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 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
\(965\) 0 0
\(966\) 0 0
\(967\) 4.09790e7i 1.40927i 0.709568 + 0.704637i \(0.248890\pi\)
−0.709568 + 0.704637i \(0.751110\pi\)
\(968\) 0 0
\(969\) 5.95901e6 0.203875
\(970\) 0 0
\(971\) −2.72034e7 −0.925922 −0.462961 0.886379i \(-0.653213\pi\)
−0.462961 + 0.886379i \(0.653213\pi\)
\(972\) 0 0
\(973\) − 1.30525e7i − 0.441990i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 2.53555e7i 0.849839i 0.905231 + 0.424919i \(0.139698\pi\)
−0.905231 + 0.424919i \(0.860302\pi\)
\(978\) 0 0
\(979\) −1.60672e7 −0.535775
\(980\) 0 0
\(981\) −9.89030e6 −0.328123
\(982\) 0 0
\(983\) − 1.19139e7i − 0.393252i −0.980479 0.196626i \(-0.937002\pi\)
0.980479 0.196626i \(-0.0629984\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 1.36858e6i − 0.0447173i
\(988\) 0 0
\(989\) −4.96584e7 −1.61437
\(990\) 0 0
\(991\) 2.91931e7 0.944268 0.472134 0.881527i \(-0.343484\pi\)
0.472134 + 0.881527i \(0.343484\pi\)
\(992\) 0 0
\(993\) 4.53538e6i 0.145962i
\(994\) 0 0
\(995\) 0 0
\(996\) 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 100.6.c.b.49.2 2
3.2 odd 2 900.6.d.a.649.1 2
4.3 odd 2 400.6.c.f.49.1 2
5.2 odd 4 100.6.a.b.1.1 1
5.3 odd 4 4.6.a.a.1.1 1
5.4 even 2 inner 100.6.c.b.49.1 2
15.2 even 4 900.6.a.h.1.1 1
15.8 even 4 36.6.a.a.1.1 1
15.14 odd 2 900.6.d.a.649.2 2
20.3 even 4 16.6.a.b.1.1 1
20.7 even 4 400.6.a.d.1.1 1
20.19 odd 2 400.6.c.f.49.2 2
35.3 even 12 196.6.e.d.177.1 2
35.13 even 4 196.6.a.e.1.1 1
35.18 odd 12 196.6.e.g.177.1 2
35.23 odd 12 196.6.e.g.165.1 2
35.33 even 12 196.6.e.d.165.1 2
40.3 even 4 64.6.a.b.1.1 1
40.13 odd 4 64.6.a.f.1.1 1
45.13 odd 12 324.6.e.a.217.1 2
45.23 even 12 324.6.e.d.217.1 2
45.38 even 12 324.6.e.d.109.1 2
45.43 odd 12 324.6.e.a.109.1 2
55.43 even 4 484.6.a.a.1.1 1
60.23 odd 4 144.6.a.c.1.1 1
65.8 even 4 676.6.d.a.337.2 2
65.18 even 4 676.6.d.a.337.1 2
65.38 odd 4 676.6.a.a.1.1 1
80.3 even 4 256.6.b.c.129.2 2
80.13 odd 4 256.6.b.g.129.1 2
80.43 even 4 256.6.b.c.129.1 2
80.53 odd 4 256.6.b.g.129.2 2
120.53 even 4 576.6.a.bc.1.1 1
120.83 odd 4 576.6.a.bd.1.1 1
140.83 odd 4 784.6.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4.6.a.a.1.1 1 5.3 odd 4
16.6.a.b.1.1 1 20.3 even 4
36.6.a.a.1.1 1 15.8 even 4
64.6.a.b.1.1 1 40.3 even 4
64.6.a.f.1.1 1 40.13 odd 4
100.6.a.b.1.1 1 5.2 odd 4
100.6.c.b.49.1 2 5.4 even 2 inner
100.6.c.b.49.2 2 1.1 even 1 trivial
144.6.a.c.1.1 1 60.23 odd 4
196.6.a.e.1.1 1 35.13 even 4
196.6.e.d.165.1 2 35.33 even 12
196.6.e.d.177.1 2 35.3 even 12
196.6.e.g.165.1 2 35.23 odd 12
196.6.e.g.177.1 2 35.18 odd 12
256.6.b.c.129.1 2 80.43 even 4
256.6.b.c.129.2 2 80.3 even 4
256.6.b.g.129.1 2 80.13 odd 4
256.6.b.g.129.2 2 80.53 odd 4
324.6.e.a.109.1 2 45.43 odd 12
324.6.e.a.217.1 2 45.13 odd 12
324.6.e.d.109.1 2 45.38 even 12
324.6.e.d.217.1 2 45.23 even 12
400.6.a.d.1.1 1 20.7 even 4
400.6.c.f.49.1 2 4.3 odd 2
400.6.c.f.49.2 2 20.19 odd 2
484.6.a.a.1.1 1 55.43 even 4
576.6.a.bc.1.1 1 120.53 even 4
576.6.a.bd.1.1 1 120.83 odd 4
676.6.a.a.1.1 1 65.38 odd 4
676.6.d.a.337.1 2 65.18 even 4
676.6.d.a.337.2 2 65.8 even 4
784.6.a.d.1.1 1 140.83 odd 4
900.6.a.h.1.1 1 15.2 even 4
900.6.d.a.649.1 2 3.2 odd 2
900.6.d.a.649.2 2 15.14 odd 2