Properties

Label 4.6.a.a.1.1
Level $4$
Weight $6$
Character 4.1
Self dual yes
Analytic conductor $0.642$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 4.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.641535279252\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4.1

$q$-expansion

\(f(q)\) \(=\) \(q-12.0000 q^{3} +54.0000 q^{5} -88.0000 q^{7} -99.0000 q^{9} +O(q^{10})\) \(q-12.0000 q^{3} +54.0000 q^{5} -88.0000 q^{7} -99.0000 q^{9} +540.000 q^{11} -418.000 q^{13} -648.000 q^{15} +594.000 q^{17} +836.000 q^{19} +1056.00 q^{21} -4104.00 q^{23} -209.000 q^{25} +4104.00 q^{27} -594.000 q^{29} +4256.00 q^{31} -6480.00 q^{33} -4752.00 q^{35} -298.000 q^{37} +5016.00 q^{39} +17226.0 q^{41} -12100.0 q^{43} -5346.00 q^{45} -1296.00 q^{47} -9063.00 q^{49} -7128.00 q^{51} +19494.0 q^{53} +29160.0 q^{55} -10032.0 q^{57} -7668.00 q^{59} -34738.0 q^{61} +8712.00 q^{63} -22572.0 q^{65} +21812.0 q^{67} +49248.0 q^{69} -46872.0 q^{71} +67562.0 q^{73} +2508.00 q^{75} -47520.0 q^{77} -76912.0 q^{79} -25191.0 q^{81} +67716.0 q^{83} +32076.0 q^{85} +7128.00 q^{87} +29754.0 q^{89} +36784.0 q^{91} -51072.0 q^{93} +45144.0 q^{95} -122398. q^{97} -53460.0 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −12.0000 −0.769800 −0.384900 0.922958i \(-0.625764\pi\)
−0.384900 + 0.922958i \(0.625764\pi\)
\(4\) 0 0
\(5\) 54.0000 0.965981 0.482991 0.875625i \(-0.339550\pi\)
0.482991 + 0.875625i \(0.339550\pi\)
\(6\) 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
\(11\) 540.000 1.34559 0.672794 0.739830i \(-0.265094\pi\)
0.672794 + 0.739830i \(0.265094\pi\)
\(12\) 0 0
\(13\) −418.000 −0.685990 −0.342995 0.939337i \(-0.611441\pi\)
−0.342995 + 0.939337i \(0.611441\pi\)
\(14\) 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
\(19\) 836.000 0.531279 0.265639 0.964072i \(-0.414417\pi\)
0.265639 + 0.964072i \(0.414417\pi\)
\(20\) 0 0
\(21\) 1056.00 0.522535
\(22\) 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
\(27\) 4104.00 1.08342
\(28\) 0 0
\(29\) −594.000 −0.131157 −0.0655785 0.997847i \(-0.520889\pi\)
−0.0655785 + 0.997847i \(0.520889\pi\)
\(30\) 0 0
\(31\) 4256.00 0.795422 0.397711 0.917511i \(-0.369805\pi\)
0.397711 + 0.917511i \(0.369805\pi\)
\(32\) 0 0
\(33\) −6480.00 −1.03583
\(34\) 0 0
\(35\) −4752.00 −0.655702
\(36\) 0 0
\(37\) −298.000 −0.0357859 −0.0178930 0.999840i \(-0.505696\pi\)
−0.0178930 + 0.999840i \(0.505696\pi\)
\(38\) 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
\(43\) −12100.0 −0.997963 −0.498981 0.866613i \(-0.666292\pi\)
−0.498981 + 0.866613i \(0.666292\pi\)
\(44\) 0 0
\(45\) −5346.00 −0.393548
\(46\) 0 0
\(47\) −1296.00 −0.0855777 −0.0427888 0.999084i \(-0.513624\pi\)
−0.0427888 + 0.999084i \(0.513624\pi\)
\(48\) 0 0
\(49\) −9063.00 −0.539240
\(50\) 0 0
\(51\) −7128.00 −0.383745
\(52\) 0 0
\(53\) 19494.0 0.953260 0.476630 0.879104i \(-0.341858\pi\)
0.476630 + 0.879104i \(0.341858\pi\)
\(54\) 0 0
\(55\) 29160.0 1.29981
\(56\) 0 0
\(57\) −10032.0 −0.408978
\(58\) 0 0
\(59\) −7668.00 −0.286782 −0.143391 0.989666i \(-0.545801\pi\)
−0.143391 + 0.989666i \(0.545801\pi\)
\(60\) 0 0
\(61\) −34738.0 −1.19531 −0.597655 0.801754i \(-0.703901\pi\)
−0.597655 + 0.801754i \(0.703901\pi\)
\(62\) 0 0
\(63\) 8712.00 0.276545
\(64\) 0 0
\(65\) −22572.0 −0.662654
\(66\) 0 0
\(67\) 21812.0 0.593620 0.296810 0.954937i \(-0.404077\pi\)
0.296810 + 0.954937i \(0.404077\pi\)
\(68\) 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
\(73\) 67562.0 1.48387 0.741934 0.670473i \(-0.233909\pi\)
0.741934 + 0.670473i \(0.233909\pi\)
\(74\) 0 0
\(75\) 2508.00 0.0514842
\(76\) 0 0
\(77\) −47520.0 −0.913376
\(78\) 0 0
\(79\) −76912.0 −1.38652 −0.693260 0.720687i \(-0.743826\pi\)
−0.693260 + 0.720687i \(0.743826\pi\)
\(80\) 0 0
\(81\) −25191.0 −0.426612
\(82\) 0 0
\(83\) 67716.0 1.07894 0.539468 0.842006i \(-0.318625\pi\)
0.539468 + 0.842006i \(0.318625\pi\)
\(84\) 0 0
\(85\) 32076.0 0.481541
\(86\) 0 0
\(87\) 7128.00 0.100965
\(88\) 0 0
\(89\) 29754.0 0.398172 0.199086 0.979982i \(-0.436203\pi\)
0.199086 + 0.979982i \(0.436203\pi\)
\(90\) 0 0
\(91\) 36784.0 0.465646
\(92\) 0 0
\(93\) −51072.0 −0.612316
\(94\) 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
\(99\) −53460.0 −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
\(103\) −27256.0 −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. 1.03322 0.516612 0.856220i \(-0.327193\pi\)
0.516612 + 0.856220i \(0.327193\pi\)
\(108\) 0 0
\(109\) 99902.0 0.805393 0.402697 0.915334i \(-0.368073\pi\)
0.402697 + 0.915334i \(0.368073\pi\)
\(110\) 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. −1.56263
\(116\) 0 0
\(117\) 41382.0 0.279477
\(118\) 0 0
\(119\) −52272.0 −0.338378
\(120\) 0 0
\(121\) 130549. 0.810607
\(122\) 0 0
\(123\) −206712. −1.23198
\(124\) 0 0
\(125\) −180036. −1.03059
\(126\) 0 0
\(127\) 336512. 1.85136 0.925681 0.378305i \(-0.123493\pi\)
0.925681 + 0.378305i \(0.123493\pi\)
\(128\) 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
\(133\) −73568.0 −0.360628
\(134\) 0 0
\(135\) 221616. 1.04657
\(136\) 0 0
\(137\) −317142. −1.44362 −0.721809 0.692092i \(-0.756689\pi\)
−0.721809 + 0.692092i \(0.756689\pi\)
\(138\) 0 0
\(139\) −148324. −0.651140 −0.325570 0.945518i \(-0.605556\pi\)
−0.325570 + 0.945518i \(0.605556\pi\)
\(140\) 0 0
\(141\) 15552.0 0.0658777
\(142\) 0 0
\(143\) −225720. −0.923060
\(144\) 0 0
\(145\) −32076.0 −0.126695
\(146\) 0 0
\(147\) 108756. 0.415107
\(148\) 0 0
\(149\) 196614. 0.725519 0.362759 0.931883i \(-0.381835\pi\)
0.362759 + 0.931883i \(0.381835\pi\)
\(150\) 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. 0.768362
\(156\) 0 0
\(157\) 120878. 0.391380 0.195690 0.980666i \(-0.437305\pi\)
0.195690 + 0.980666i \(0.437305\pi\)
\(158\) 0 0
\(159\) −233928. −0.733820
\(160\) 0 0
\(161\) 361152. 1.09806
\(162\) 0 0
\(163\) −111340. −0.328233 −0.164116 0.986441i \(-0.552477\pi\)
−0.164116 + 0.986441i \(0.552477\pi\)
\(164\) 0 0
\(165\) −349920. −1.00060
\(166\) 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
\(171\) −82764.0 −0.216447
\(172\) 0 0
\(173\) 707454. 1.79714 0.898572 0.438826i \(-0.144605\pi\)
0.898572 + 0.438826i \(0.144605\pi\)
\(174\) 0 0
\(175\) 18392.0 0.0453977
\(176\) 0 0
\(177\) 92016.0 0.220765
\(178\) 0 0
\(179\) 493668. 1.15160 0.575801 0.817590i \(-0.304690\pi\)
0.575801 + 0.817590i \(0.304690\pi\)
\(180\) 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. 0.920149
\(184\) 0 0
\(185\) −16092.0 −0.0345685
\(186\) 0 0
\(187\) 320760. 0.670774
\(188\) 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
\(193\) 7106.00 0.0137319 0.00686597 0.999976i \(-0.497814\pi\)
0.00686597 + 0.999976i \(0.497814\pi\)
\(194\) 0 0
\(195\) 270864. 0.510111
\(196\) 0 0
\(197\) −530442. −0.973806 −0.486903 0.873456i \(-0.661873\pi\)
−0.486903 + 0.873456i \(0.661873\pi\)
\(198\) 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
\(203\) 52272.0 0.0890285
\(204\) 0 0
\(205\) 930204. 1.54594
\(206\) 0 0
\(207\) 406296. 0.659047
\(208\) 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. 0.849465
\(214\) 0 0
\(215\) −653400. −0.964013
\(216\) 0 0
\(217\) −374528. −0.539927
\(218\) 0 0
\(219\) −810744. −1.14228
\(220\) 0 0
\(221\) −248292. −0.341965
\(222\) 0 0
\(223\) 779360. 1.04948 0.524742 0.851261i \(-0.324162\pi\)
0.524742 + 0.851261i \(0.324162\pi\)
\(224\) 0 0
\(225\) 20691.0 0.0272474
\(226\) 0 0
\(227\) −744876. −0.959443 −0.479722 0.877421i \(-0.659262\pi\)
−0.479722 + 0.877421i \(0.659262\pi\)
\(228\) 0 0
\(229\) −272746. −0.343692 −0.171846 0.985124i \(-0.554973\pi\)
−0.171846 + 0.985124i \(0.554973\pi\)
\(230\) 0 0
\(231\) 570240. 0.703117
\(232\) 0 0
\(233\) −153846. −0.185651 −0.0928253 0.995682i \(-0.529590\pi\)
−0.0928253 + 0.995682i \(0.529590\pi\)
\(234\) 0 0
\(235\) −69984.0 −0.0826664
\(236\) 0 0
\(237\) 922944. 1.06734
\(238\) 0 0
\(239\) 1.15474e6 1.30764 0.653820 0.756650i \(-0.273166\pi\)
0.653820 + 0.756650i \(0.273166\pi\)
\(240\) 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. −0.755017
\(244\) 0 0
\(245\) −489402. −0.520895
\(246\) 0 0
\(247\) −349448. −0.364452
\(248\) 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
\(253\) −2.21616e6 −2.17671
\(254\) 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
\(259\) 26224.0 0.0242912
\(260\) 0 0
\(261\) 58806.0 0.0534343
\(262\) 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. −0.306513
\(268\) 0 0
\(269\) 967518. 0.815227 0.407613 0.913155i \(-0.366361\pi\)
0.407613 + 0.913155i \(0.366361\pi\)
\(270\) 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. −0.358454
\(274\) 0 0
\(275\) −112860. −0.0899929
\(276\) 0 0
\(277\) 2.22273e6 1.74055 0.870275 0.492566i \(-0.163941\pi\)
0.870275 + 0.492566i \(0.163941\pi\)
\(278\) 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
\(283\) −1.55228e6 −1.15213 −0.576067 0.817403i \(-0.695413\pi\)
−0.576067 + 0.817403i \(0.695413\pi\)
\(284\) 0 0
\(285\) −541728. −0.395066
\(286\) 0 0
\(287\) −1.51589e6 −1.08633
\(288\) 0 0
\(289\) −1.06702e6 −0.751499
\(290\) 0 0
\(291\) 1.46878e6 1.01677
\(292\) 0 0
\(293\) −1.07217e6 −0.729616 −0.364808 0.931083i \(-0.618865\pi\)
−0.364808 + 0.931083i \(0.618865\pi\)
\(294\) 0 0
\(295\) −414072. −0.277026
\(296\) 0 0
\(297\) 2.21616e6 1.45784
\(298\) 0 0
\(299\) 1.71547e6 1.10970
\(300\) 0 0
\(301\) 1.06480e6 0.677410
\(302\) 0 0
\(303\) −135432. −0.0847451
\(304\) 0 0
\(305\) −1.87585e6 −1.15465
\(306\) 0 0
\(307\) 1.58589e6 0.960346 0.480173 0.877174i \(-0.340574\pi\)
0.480173 + 0.877174i \(0.340574\pi\)
\(308\) 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. 0.337435 0.168717 0.985664i \(-0.446038\pi\)
0.168717 + 0.985664i \(0.446038\pi\)
\(314\) 0 0
\(315\) 470448. 0.267138
\(316\) 0 0
\(317\) −2.48287e6 −1.38773 −0.693865 0.720105i \(-0.744094\pi\)
−0.693865 + 0.720105i \(0.744094\pi\)
\(318\) 0 0
\(319\) −320760. −0.176483
\(320\) 0 0
\(321\) −1.46837e6 −0.795376
\(322\) 0 0
\(323\) 496584. 0.264842
\(324\) 0 0
\(325\) 87362.0 0.0458790
\(326\) 0 0
\(327\) −1.19882e6 −0.619992
\(328\) 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
\(333\) 29502.0 0.0145794
\(334\) 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. 0.168131
\(340\) 0 0
\(341\) 2.29824e6 1.07031
\(342\) 0 0
\(343\) 2.27656e6 1.04483
\(344\) 0 0
\(345\) 2.65939e6 1.20291
\(346\) 0 0
\(347\) −2.90466e6 −1.29501 −0.647503 0.762063i \(-0.724187\pi\)
−0.647503 + 0.762063i \(0.724187\pi\)
\(348\) 0 0
\(349\) −3.99157e6 −1.75420 −0.877102 0.480304i \(-0.840526\pi\)
−0.877102 + 0.480304i \(0.840526\pi\)
\(350\) 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
\(355\) −2.53109e6 −1.06595
\(356\) 0 0
\(357\) 627264. 0.260483
\(358\) 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
\(363\) −1.56659e6 −0.624005
\(364\) 0 0
\(365\) 3.64835e6 1.43339
\(366\) 0 0
\(367\) −1.08923e6 −0.422139 −0.211069 0.977471i \(-0.567695\pi\)
−0.211069 + 0.977471i \(0.567695\pi\)
\(368\) 0 0
\(369\) −1.70537e6 −0.652009
\(370\) 0 0
\(371\) −1.71547e6 −0.647066
\(372\) 0 0
\(373\) 3.50577e6 1.30470 0.652350 0.757918i \(-0.273783\pi\)
0.652350 + 0.757918i \(0.273783\pi\)
\(374\) 0 0
\(375\) 2.16043e6 0.793346
\(376\) 0 0
\(377\) 248292. 0.0899724
\(378\) 0 0
\(379\) 4.04385e6 1.44610 0.723048 0.690798i \(-0.242740\pi\)
0.723048 + 0.690798i \(0.242740\pi\)
\(380\) 0 0
\(381\) −4.03814e6 −1.42518
\(382\) 0 0
\(383\) 5.18746e6 1.80700 0.903499 0.428591i \(-0.140990\pi\)
0.903499 + 0.428591i \(0.140990\pi\)
\(384\) 0 0
\(385\) −2.56608e6 −0.882304
\(386\) 0 0
\(387\) 1.19790e6 0.406577
\(388\) 0 0
\(389\) −950346. −0.318425 −0.159213 0.987244i \(-0.550896\pi\)
−0.159213 + 0.987244i \(0.550896\pi\)
\(390\) 0 0
\(391\) −2.43778e6 −0.806403
\(392\) 0 0
\(393\) −1.21176e6 −0.395763
\(394\) 0 0
\(395\) −4.15325e6 −1.33935
\(396\) 0 0
\(397\) −520738. −0.165822 −0.0829112 0.996557i \(-0.526422\pi\)
−0.0829112 + 0.996557i \(0.526422\pi\)
\(398\) 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
\(403\) −1.77901e6 −0.545651
\(404\) 0 0
\(405\) −1.36031e6 −0.412099
\(406\) 0 0
\(407\) −160920. −0.0481531
\(408\) 0 0
\(409\) 2.64051e6 0.780511 0.390255 0.920707i \(-0.372387\pi\)
0.390255 + 0.920707i \(0.372387\pi\)
\(410\) 0 0
\(411\) 3.80570e6 1.11130
\(412\) 0 0
\(413\) 674784. 0.194666
\(414\) 0 0
\(415\) 3.65666e6 1.04223
\(416\) 0 0
\(417\) 1.77989e6 0.501248
\(418\) 0 0
\(419\) −4.98020e6 −1.38584 −0.692918 0.721016i \(-0.743675\pi\)
−0.692918 + 0.721016i \(0.743675\pi\)
\(420\) 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. 0.0348650
\(424\) 0 0
\(425\) −124146. −0.0333396
\(426\) 0 0
\(427\) 3.05694e6 0.811368
\(428\) 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
\(433\) −66958.0 −0.0171626 −0.00858129 0.999963i \(-0.502732\pi\)
−0.00858129 + 0.999963i \(0.502732\pi\)
\(434\) 0 0
\(435\) 384912. 0.0975300
\(436\) 0 0
\(437\) −3.43094e6 −0.859429
\(438\) 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
\(443\) −4.60760e6 −1.11549 −0.557745 0.830012i \(-0.688333\pi\)
−0.557745 + 0.830012i \(0.688333\pi\)
\(444\) 0 0
\(445\) 1.60672e6 0.384626
\(446\) 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
\(451\) 9.30204e6 2.15346
\(452\) 0 0
\(453\) −892320. −0.204303
\(454\) 0 0
\(455\) 1.98634e6 0.449805
\(456\) 0 0
\(457\) −3.18069e6 −0.712412 −0.356206 0.934407i \(-0.615930\pi\)
−0.356206 + 0.934407i \(0.615930\pi\)
\(458\) 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
\(463\) −4.35122e6 −0.943318 −0.471659 0.881781i \(-0.656345\pi\)
−0.471659 + 0.881781i \(0.656345\pi\)
\(464\) 0 0
\(465\) −2.75789e6 −0.591486
\(466\) 0 0
\(467\) 7.07994e6 1.50223 0.751117 0.660170i \(-0.229516\pi\)
0.751117 + 0.660170i \(0.229516\pi\)
\(468\) 0 0
\(469\) −1.91946e6 −0.402945
\(470\) 0 0
\(471\) −1.45054e6 −0.301284
\(472\) 0 0
\(473\) −6.53400e6 −1.34285
\(474\) 0 0
\(475\) −174724. −0.0355319
\(476\) 0 0
\(477\) −1.92991e6 −0.388365
\(478\) 0 0
\(479\) 3.22186e6 0.641604 0.320802 0.947146i \(-0.396048\pi\)
0.320802 + 0.947146i \(0.396048\pi\)
\(480\) 0 0
\(481\) 124564. 0.0245488
\(482\) 0 0
\(483\) −4.33382e6 −0.845286
\(484\) 0 0
\(485\) −6.60949e6 −1.27589
\(486\) 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
\(491\) 2.82150e6 0.528173 0.264087 0.964499i \(-0.414930\pi\)
0.264087 + 0.964499i \(0.414930\pi\)
\(492\) 0 0
\(493\) −352836. −0.0653816
\(494\) 0 0
\(495\) −2.88684e6 −0.529553
\(496\) 0 0
\(497\) 4.12474e6 0.749040
\(498\) 0 0
\(499\) −4.13628e6 −0.743634 −0.371817 0.928306i \(-0.621265\pi\)
−0.371817 + 0.928306i \(0.621265\pi\)
\(500\) 0 0
\(501\) 5.90198e6 1.05052
\(502\) 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
\(507\) 2.35883e6 0.407546
\(508\) 0 0
\(509\) 4.34101e6 0.742670 0.371335 0.928499i \(-0.378900\pi\)
0.371335 + 0.928499i \(0.378900\pi\)
\(510\) 0 0
\(511\) −5.94546e6 −1.00724
\(512\) 0 0
\(513\) 3.43094e6 0.575599
\(514\) 0 0
\(515\) −1.47182e6 −0.244533
\(516\) 0 0
\(517\) −699840. −0.115152
\(518\) 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
\(523\) −7.72196e6 −1.23445 −0.617224 0.786787i \(-0.711743\pi\)
−0.617224 + 0.786787i \(0.711743\pi\)
\(524\) 0 0
\(525\) −220704. −0.0349472
\(526\) 0 0
\(527\) 2.52806e6 0.396517
\(528\) 0 0
\(529\) 1.04065e7 1.61683
\(530\) 0 0
\(531\) 759132. 0.116837
\(532\) 0 0
\(533\) −7.20047e6 −1.09785
\(534\) 0 0
\(535\) 6.60766e6 0.998075
\(536\) 0 0
\(537\) −5.92402e6 −0.886504
\(538\) 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
\(543\) 6.71340e6 0.977109
\(544\) 0 0
\(545\) 5.39471e6 0.777995
\(546\) 0 0
\(547\) 2.15772e6 0.308337 0.154169 0.988045i \(-0.450730\pi\)
0.154169 + 0.988045i \(0.450730\pi\)
\(548\) 0 0
\(549\) 3.43906e6 0.486978
\(550\) 0 0
\(551\) −496584. −0.0696809
\(552\) 0 0
\(553\) 6.76826e6 0.941161
\(554\) 0 0
\(555\) 193104. 0.0266109
\(556\) 0 0
\(557\) −2.67597e6 −0.365463 −0.182731 0.983163i \(-0.558494\pi\)
−0.182731 + 0.983163i \(0.558494\pi\)
\(558\) 0 0
\(559\) 5.05780e6 0.684592
\(560\) 0 0
\(561\) −3.84912e6 −0.516362
\(562\) 0 0
\(563\) −3.55331e6 −0.472457 −0.236228 0.971698i \(-0.575911\pi\)
−0.236228 + 0.971698i \(0.575911\pi\)
\(564\) 0 0
\(565\) −1.60088e6 −0.210979
\(566\) 0 0
\(567\) 2.21681e6 0.289581
\(568\) 0 0
\(569\) −1.29225e7 −1.67327 −0.836633 0.547764i \(-0.815479\pi\)
−0.836633 + 0.547764i \(0.815479\pi\)
\(570\) 0 0
\(571\) −6.08357e6 −0.780851 −0.390426 0.920634i \(-0.627672\pi\)
−0.390426 + 0.920634i \(0.627672\pi\)
\(572\) 0 0
\(573\) 8.68838e6 1.10548
\(574\) 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
\(579\) −85272.0 −0.0105709
\(580\) 0 0
\(581\) −5.95901e6 −0.732375
\(582\) 0 0
\(583\) 1.05268e7 1.28269
\(584\) 0 0
\(585\) 2.23463e6 0.269970
\(586\) 0 0
\(587\) 4.60220e6 0.551278 0.275639 0.961261i \(-0.411111\pi\)
0.275639 + 0.961261i \(0.411111\pi\)
\(588\) 0 0
\(589\) 3.55802e6 0.422590
\(590\) 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
\(595\) −2.82269e6 −0.326867
\(596\) 0 0
\(597\) −674016. −0.0773988
\(598\) 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 \(-0.305204\pi\)
0.574481 + 0.818518i \(0.305204\pi\)
\(602\) 0 0
\(603\) −2.15939e6 −0.241845
\(604\) 0 0
\(605\) 7.04965e6 0.783031
\(606\) 0 0
\(607\) −9.95843e6 −1.09703 −0.548516 0.836140i \(-0.684807\pi\)
−0.548516 + 0.836140i \(0.684807\pi\)
\(608\) 0 0
\(609\) −627264. −0.0685342
\(610\) 0 0
\(611\) 541728. 0.0587054
\(612\) 0 0
\(613\) 4.19586e6 0.450993 0.225497 0.974244i \(-0.427600\pi\)
0.225497 + 0.974244i \(0.427600\pi\)
\(614\) 0 0
\(615\) −1.11624e7 −1.19007
\(616\) 0 0
\(617\) 9.12551e6 0.965038 0.482519 0.875885i \(-0.339722\pi\)
0.482519 + 0.875885i \(0.339722\pi\)
\(618\) 0 0
\(619\) 6.45734e6 0.677372 0.338686 0.940900i \(-0.390018\pi\)
0.338686 + 0.940900i \(0.390018\pi\)
\(620\) 0 0
\(621\) −1.68428e7 −1.75261
\(622\) 0 0
\(623\) −2.61835e6 −0.270276
\(624\) 0 0
\(625\) −9.06882e6 −0.928647
\(626\) 0 0
\(627\) −5.41728e6 −0.550316
\(628\) 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
\(633\) 4.07035e6 0.403759
\(634\) 0 0
\(635\) 1.81716e7 1.78838
\(636\) 0 0
\(637\) 3.78833e6 0.369913
\(638\) 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. 0.0466009 0.0233004 0.999729i \(-0.492583\pi\)
0.0233004 + 0.999729i \(0.492583\pi\)
\(644\) 0 0
\(645\) 7.84080e6 0.742098
\(646\) 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
\(651\) 4.49434e6 0.415636
\(652\) 0 0
\(653\) −5.29130e6 −0.485601 −0.242800 0.970076i \(-0.578066\pi\)
−0.242800 + 0.970076i \(0.578066\pi\)
\(654\) 0 0
\(655\) 5.45292e6 0.496622
\(656\) 0 0
\(657\) −6.68864e6 −0.604539
\(658\) 0 0
\(659\) 4.72468e6 0.423798 0.211899 0.977292i \(-0.432035\pi\)
0.211899 + 0.977292i \(0.432035\pi\)
\(660\) 0 0
\(661\) −6.17420e6 −0.549639 −0.274819 0.961496i \(-0.588618\pi\)
−0.274819 + 0.961496i \(0.588618\pi\)
\(662\) 0 0
\(663\) 2.97950e6 0.263245
\(664\) 0 0
\(665\) −3.97267e6 −0.348360
\(666\) 0 0
\(667\) 2.43778e6 0.212168
\(668\) 0 0
\(669\) −9.35232e6 −0.807893
\(670\) 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. −0.0724593
\(676\) 0 0
\(677\) 1.50086e7 1.25854 0.629272 0.777185i \(-0.283353\pi\)
0.629272 + 0.777185i \(0.283353\pi\)
\(678\) 0 0
\(679\) 1.07710e7 0.896567
\(680\) 0 0
\(681\) 8.93851e6 0.738580
\(682\) 0 0
\(683\) −1.29707e7 −1.06393 −0.531963 0.846768i \(-0.678545\pi\)
−0.531963 + 0.846768i \(0.678545\pi\)
\(684\) 0 0
\(685\) −1.71257e7 −1.39451
\(686\) 0 0
\(687\) 3.27295e6 0.264574
\(688\) 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
\(693\) 4.70448e6 0.372116
\(694\) 0 0
\(695\) −8.00950e6 −0.628989
\(696\) 0 0
\(697\) 1.02322e7 0.797791
\(698\) 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. −0.0190123
\(704\) 0 0
\(705\) 839808. 0.0636366
\(706\) 0 0
\(707\) −993168. −0.0747264
\(708\) 0 0
\(709\) 1.51311e7 1.13046 0.565231 0.824933i \(-0.308787\pi\)
0.565231 + 0.824933i \(0.308787\pi\)
\(710\) 0 0
\(711\) 7.61429e6 0.564879
\(712\) 0 0
\(713\) −1.74666e7 −1.28672
\(714\) 0 0
\(715\) −1.21889e7 −0.891659
\(716\) 0 0
\(717\) −1.38568e7 −1.00662
\(718\) 0 0
\(719\) −1.50323e7 −1.08443 −0.542217 0.840238i \(-0.682415\pi\)
−0.542217 + 0.840238i \(0.682415\pi\)
\(720\) 0 0
\(721\) 2.39853e6 0.171833
\(722\) 0 0
\(723\) −7.88489e6 −0.560983
\(724\) 0 0
\(725\) 124146. 0.00877178
\(726\) 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
\(731\) −7.18740e6 −0.497483
\(732\) 0 0
\(733\) −2.77928e6 −0.191061 −0.0955306 0.995426i \(-0.530455\pi\)
−0.0955306 + 0.995426i \(0.530455\pi\)
\(734\) 0 0
\(735\) 5.87282e6 0.400985
\(736\) 0 0
\(737\) 1.17785e7 0.798768
\(738\) 0 0
\(739\) −1.21046e7 −0.815342 −0.407671 0.913129i \(-0.633659\pi\)
−0.407671 + 0.913129i \(0.633659\pi\)
\(740\) 0 0
\(741\) 4.19338e6 0.280555
\(742\) 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
\(747\) −6.70388e6 −0.439567
\(748\) 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
\(753\) −1.61028e7 −1.03494
\(754\) 0 0
\(755\) 4.01544e6 0.256369
\(756\) 0 0
\(757\) 9.60868e6 0.609430 0.304715 0.952444i \(-0.401439\pi\)
0.304715 + 0.952444i \(0.401439\pi\)
\(758\) 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
\(763\) −8.79138e6 −0.546696
\(764\) 0 0
\(765\) −3.17552e6 −0.196183
\(766\) 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
\(771\) −1.58825e6 −0.0962238
\(772\) 0 0
\(773\) −1.48400e7 −0.893276 −0.446638 0.894715i \(-0.647379\pi\)
−0.446638 + 0.894715i \(0.647379\pi\)
\(774\) 0 0
\(775\) −889504. −0.0531978
\(776\) 0 0
\(777\) −314688. −0.0186994
\(778\) 0 0
\(779\) 1.44009e7 0.850251
\(780\) 0 0
\(781\) −2.53109e7 −1.48484
\(782\) 0 0
\(783\) −2.43778e6 −0.142098
\(784\) 0 0
\(785\) 6.52741e6 0.378065
\(786\) 0 0
\(787\) −2.48785e7 −1.43182 −0.715909 0.698194i \(-0.753987\pi\)
−0.715909 + 0.698194i \(0.753987\pi\)
\(788\) 0 0
\(789\) −1.13193e7 −0.647330
\(790\) 0 0
\(791\) 2.60885e6 0.148254
\(792\) 0 0
\(793\) 1.45205e7 0.819970
\(794\) 0 0
\(795\) −1.26321e7 −0.708856
\(796\) 0 0
\(797\) 3.16080e7 1.76259 0.881294 0.472568i \(-0.156673\pi\)
0.881294 + 0.472568i \(0.156673\pi\)
\(798\) 0 0
\(799\) −769824. −0.0426604
\(800\) 0 0
\(801\) −2.94565e6 −0.162218
\(802\) 0 0
\(803\) 3.64835e7 1.99668
\(804\) 0 0
\(805\) 1.95022e7 1.06070
\(806\) 0 0
\(807\) −1.16102e7 −0.627562
\(808\) 0 0
\(809\) −3.10009e6 −0.166534 −0.0832669 0.996527i \(-0.526535\pi\)
−0.0832669 + 0.996527i \(0.526535\pi\)
\(810\) 0 0
\(811\) 1.87180e6 0.0999328 0.0499664 0.998751i \(-0.484089\pi\)
0.0499664 + 0.998751i \(0.484089\pi\)
\(812\) 0 0
\(813\) 6.21984e6 0.330030
\(814\) 0 0
\(815\) −6.01236e6 −0.317067
\(816\) 0 0
\(817\) −1.01156e7 −0.530196
\(818\) 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
\(823\) 1.53118e7 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
\(827\) 9.59310e6 0.487748 0.243874 0.969807i \(-0.421582\pi\)
0.243874 + 0.969807i \(0.421582\pi\)
\(828\) 0 0
\(829\) 2.52209e7 1.27460 0.637302 0.770615i \(-0.280051\pi\)
0.637302 + 0.770615i \(0.280051\pi\)
\(830\) 0 0
\(831\) −2.66727e7 −1.33988
\(832\) 0 0
\(833\) −5.38342e6 −0.268810
\(834\) 0 0
\(835\) −2.65589e7 −1.31824
\(836\) 0 0
\(837\) 1.74666e7 0.861778
\(838\) 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
\(843\) 2.35937e6 0.114348
\(844\) 0 0
\(845\) −1.06147e7 −0.511407
\(846\) 0 0
\(847\) −1.14883e7 −0.550234
\(848\) 0 0
\(849\) 1.86273e7 0.886913
\(850\) 0 0
\(851\) 1.22299e6 0.0578895
\(852\) 0 0
\(853\) −486970. −0.0229155 −0.0114578 0.999934i \(-0.503647\pi\)
−0.0114578 + 0.999934i \(0.503647\pi\)
\(854\) 0 0
\(855\) −4.46926e6 −0.209084
\(856\) 0 0
\(857\) −1.92634e6 −0.0895945 −0.0447972 0.998996i \(-0.514264\pi\)
−0.0447972 + 0.998996i \(0.514264\pi\)
\(858\) 0 0
\(859\) 2.23538e7 1.03364 0.516820 0.856094i \(-0.327116\pi\)
0.516820 + 0.856094i \(0.327116\pi\)
\(860\) 0 0
\(861\) 1.81907e7 0.836258
\(862\) 0 0
\(863\) 1.85838e7 0.849390 0.424695 0.905337i \(-0.360381\pi\)
0.424695 + 0.905337i \(0.360381\pi\)
\(864\) 0 0
\(865\) 3.82025e7 1.73601
\(866\) 0 0
\(867\) 1.28043e7 0.578504
\(868\) 0 0
\(869\) −4.15325e7 −1.86569
\(870\) 0 0
\(871\) −9.11742e6 −0.407217
\(872\) 0 0
\(873\) 1.21174e7 0.538114
\(874\) 0 0
\(875\) 1.58432e7 0.699555
\(876\) 0 0
\(877\) −2.91048e7 −1.27781 −0.638905 0.769286i \(-0.720612\pi\)
−0.638905 + 0.769286i \(0.720612\pi\)
\(878\) 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
\(883\) 1.59995e7 0.690566 0.345283 0.938499i \(-0.387783\pi\)
0.345283 + 0.938499i \(0.387783\pi\)
\(884\) 0 0
\(885\) 4.96886e6 0.213255
\(886\) 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
\(891\) −1.36031e7 −0.574044
\(892\) 0 0
\(893\) −1.08346e6 −0.0454656
\(894\) 0 0
\(895\) 2.66581e7 1.11243
\(896\) 0 0
\(897\) −2.05857e7 −0.854248
\(898\) 0 0
\(899\) −2.52806e6 −0.104325
\(900\) 0 0
\(901\) 1.15794e7 0.475199
\(902\) 0 0
\(903\) −1.27776e7 −0.521471
\(904\) 0 0
\(905\) −3.02103e7 −1.22612
\(906\) 0 0
\(907\) 1.74396e7 0.703914 0.351957 0.936016i \(-0.385516\pi\)
0.351957 + 0.936016i \(0.385516\pi\)
\(908\) 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
\(913\) 3.65666e7 1.45180
\(914\) 0 0
\(915\) 2.25102e7 0.888847
\(916\) 0 0
\(917\) −8.88624e6 −0.348975
\(918\) 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
\(923\) 1.95925e7 0.756982
\(924\) 0 0
\(925\) 62282.0 0.00239336
\(926\) 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
\(931\) −7.57667e6 −0.286486
\(932\) 0 0
\(933\) 8.76874e6 0.329787
\(934\) 0 0
\(935\) 1.73210e7 0.647955
\(936\) 0 0
\(937\) 3.93413e7 1.46386 0.731930 0.681380i \(-0.238620\pi\)
0.731930 + 0.681380i \(0.238620\pi\)
\(938\) 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
\(943\) −7.06955e7 −2.58888
\(944\) 0 0
\(945\) −1.95022e7 −0.710402
\(946\) 0 0
\(947\) −3.79025e7 −1.37339 −0.686693 0.726947i \(-0.740938\pi\)
−0.686693 + 0.726947i \(0.740938\pi\)
\(948\) 0 0
\(949\) −2.82409e7 −1.01792
\(950\) 0 0
\(951\) 2.97944e7 1.06828
\(952\) 0 0
\(953\) −2.66462e7 −0.950394 −0.475197 0.879879i \(-0.657623\pi\)
−0.475197 + 0.879879i \(0.657623\pi\)
\(954\) 0 0
\(955\) −3.90977e7 −1.38721
\(956\) 0 0
\(957\) 3.84912e6 0.135857
\(958\) 0 0
\(959\) 2.79085e7 0.979918
\(960\) 0 0
\(961\) −1.05156e7 −0.367304
\(962\) 0 0
\(963\) −1.21140e7 −0.420943
\(964\) 0 0
\(965\) 383724. 0.0132648
\(966\) 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
\(971\) −2.72034e7 −0.925922 −0.462961 0.886379i \(-0.653213\pi\)
−0.462961 + 0.886379i \(0.653213\pi\)
\(972\) 0 0
\(973\) 1.30525e7 0.441990
\(974\) 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
\(979\) 1.60672e7 0.535775
\(980\) 0 0
\(981\) −9.89030e6 −0.328123
\(982\) 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
\(987\) −1.36858e6 −0.0447173
\(988\) 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
\(993\) −4.53538e6 −0.145962
\(994\) 0 0
\(995\) 3.03307e6 0.0971237
\(996\) 0 0
\(997\) −1.73001e7 −0.551201 −0.275601 0.961272i \(-0.588877\pi\)
−0.275601 + 0.961272i \(0.588877\pi\)
\(998\) 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 4.6.a.a.1.1 1
3.2 odd 2 36.6.a.a.1.1 1
4.3 odd 2 16.6.a.b.1.1 1
5.2 odd 4 100.6.c.b.49.2 2
5.3 odd 4 100.6.c.b.49.1 2
5.4 even 2 100.6.a.b.1.1 1
7.2 even 3 196.6.e.g.165.1 2
7.3 odd 6 196.6.e.d.177.1 2
7.4 even 3 196.6.e.g.177.1 2
7.5 odd 6 196.6.e.d.165.1 2
7.6 odd 2 196.6.a.e.1.1 1
8.3 odd 2 64.6.a.b.1.1 1
8.5 even 2 64.6.a.f.1.1 1
9.2 odd 6 324.6.e.d.109.1 2
9.4 even 3 324.6.e.a.217.1 2
9.5 odd 6 324.6.e.d.217.1 2
9.7 even 3 324.6.e.a.109.1 2
11.10 odd 2 484.6.a.a.1.1 1
12.11 even 2 144.6.a.c.1.1 1
13.5 odd 4 676.6.d.a.337.1 2
13.8 odd 4 676.6.d.a.337.2 2
13.12 even 2 676.6.a.a.1.1 1
15.2 even 4 900.6.d.a.649.1 2
15.8 even 4 900.6.d.a.649.2 2
15.14 odd 2 900.6.a.h.1.1 1
16.3 odd 4 256.6.b.c.129.2 2
16.5 even 4 256.6.b.g.129.2 2
16.11 odd 4 256.6.b.c.129.1 2
16.13 even 4 256.6.b.g.129.1 2
20.3 even 4 400.6.c.f.49.2 2
20.7 even 4 400.6.c.f.49.1 2
20.19 odd 2 400.6.a.d.1.1 1
24.5 odd 2 576.6.a.bc.1.1 1
24.11 even 2 576.6.a.bd.1.1 1
28.27 even 2 784.6.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4.6.a.a.1.1 1 1.1 even 1 trivial
16.6.a.b.1.1 1 4.3 odd 2
36.6.a.a.1.1 1 3.2 odd 2
64.6.a.b.1.1 1 8.3 odd 2
64.6.a.f.1.1 1 8.5 even 2
100.6.a.b.1.1 1 5.4 even 2
100.6.c.b.49.1 2 5.3 odd 4
100.6.c.b.49.2 2 5.2 odd 4
144.6.a.c.1.1 1 12.11 even 2
196.6.a.e.1.1 1 7.6 odd 2
196.6.e.d.165.1 2 7.5 odd 6
196.6.e.d.177.1 2 7.3 odd 6
196.6.e.g.165.1 2 7.2 even 3
196.6.e.g.177.1 2 7.4 even 3
256.6.b.c.129.1 2 16.11 odd 4
256.6.b.c.129.2 2 16.3 odd 4
256.6.b.g.129.1 2 16.13 even 4
256.6.b.g.129.2 2 16.5 even 4
324.6.e.a.109.1 2 9.7 even 3
324.6.e.a.217.1 2 9.4 even 3
324.6.e.d.109.1 2 9.2 odd 6
324.6.e.d.217.1 2 9.5 odd 6
400.6.a.d.1.1 1 20.19 odd 2
400.6.c.f.49.1 2 20.7 even 4
400.6.c.f.49.2 2 20.3 even 4
484.6.a.a.1.1 1 11.10 odd 2
576.6.a.bc.1.1 1 24.5 odd 2
576.6.a.bd.1.1 1 24.11 even 2
676.6.a.a.1.1 1 13.12 even 2
676.6.d.a.337.1 2 13.5 odd 4
676.6.d.a.337.2 2 13.8 odd 4
784.6.a.d.1.1 1 28.27 even 2
900.6.a.h.1.1 1 15.14 odd 2
900.6.d.a.649.1 2 15.2 even 4
900.6.d.a.649.2 2 15.8 even 4