Properties

Label 400.6.a.u.1.2
Level $400$
Weight $6$
Character 400.1
Self dual yes
Analytic conductor $64.154$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1535279252\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{241}) \)
Defining polynomial: \( x^{2} - x - 60 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 200)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(8.26209\) of defining polynomial
Character \(\chi\) \(=\) 400.1

$q$-expansion

\(f(q)\) \(=\) \(q+19.5242 q^{3} -35.0483 q^{7} +138.193 q^{9} +O(q^{10})\) \(q+19.5242 q^{3} -35.0483 q^{7} +138.193 q^{9} +426.008 q^{11} +1103.26 q^{13} +109.387 q^{17} -495.926 q^{19} -684.290 q^{21} -2497.37 q^{23} -2046.26 q^{27} -42.4221 q^{29} +7999.56 q^{31} +8317.45 q^{33} -13763.7 q^{37} +21540.2 q^{39} +11863.6 q^{41} +16816.0 q^{43} +13036.0 q^{47} -15578.6 q^{49} +2135.69 q^{51} +17817.3 q^{53} -9682.55 q^{57} +47346.1 q^{59} -22782.9 q^{61} -4843.45 q^{63} +39418.3 q^{67} -48759.1 q^{69} +1616.32 q^{71} +53293.3 q^{73} -14930.9 q^{77} +6516.98 q^{79} -73532.6 q^{81} -46016.4 q^{83} -828.257 q^{87} +113488. q^{89} -38667.3 q^{91} +156185. q^{93} +107418. q^{97} +58871.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{3} - 8 q^{7} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{3} - 8 q^{7} + 28 q^{9} + 200 q^{11} + 592 q^{13} - 278 q^{17} + 840 q^{19} - 996 q^{21} - 1952 q^{23} + 2024 q^{27} - 4680 q^{29} + 5008 q^{31} + 10922 q^{33} - 12500 q^{37} + 27432 q^{39} - 5334 q^{41} + 224 q^{43} + 26072 q^{47} - 31654 q^{49} + 6600 q^{51} + 46812 q^{53} - 25078 q^{57} + 81776 q^{59} - 46932 q^{61} - 7824 q^{63} + 68808 q^{67} - 55044 q^{69} - 7448 q^{71} + 108822 q^{73} - 21044 q^{77} + 108104 q^{79} - 93662 q^{81} + 27224 q^{83} + 52616 q^{87} + 70990 q^{89} - 52496 q^{91} + 190660 q^{93} + 96852 q^{97} + 83776 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 19.5242 1.25248 0.626238 0.779632i \(-0.284594\pi\)
0.626238 + 0.779632i \(0.284594\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −35.0483 −0.270348 −0.135174 0.990822i \(-0.543159\pi\)
−0.135174 + 0.990822i \(0.543159\pi\)
\(8\) 0 0
\(9\) 138.193 0.568697
\(10\) 0 0
\(11\) 426.008 1.06154 0.530769 0.847516i \(-0.321903\pi\)
0.530769 + 0.847516i \(0.321903\pi\)
\(12\) 0 0
\(13\) 1103.26 1.81058 0.905291 0.424791i \(-0.139653\pi\)
0.905291 + 0.424791i \(0.139653\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 109.387 0.0918000 0.0459000 0.998946i \(-0.485384\pi\)
0.0459000 + 0.998946i \(0.485384\pi\)
\(18\) 0 0
\(19\) −495.926 −0.315161 −0.157581 0.987506i \(-0.550369\pi\)
−0.157581 + 0.987506i \(0.550369\pi\)
\(20\) 0 0
\(21\) −684.290 −0.338604
\(22\) 0 0
\(23\) −2497.37 −0.984381 −0.492190 0.870488i \(-0.663804\pi\)
−0.492190 + 0.870488i \(0.663804\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −2046.26 −0.540197
\(28\) 0 0
\(29\) −42.4221 −0.00936694 −0.00468347 0.999989i \(-0.501491\pi\)
−0.00468347 + 0.999989i \(0.501491\pi\)
\(30\) 0 0
\(31\) 7999.56 1.49507 0.747535 0.664222i \(-0.231237\pi\)
0.747535 + 0.664222i \(0.231237\pi\)
\(32\) 0 0
\(33\) 8317.45 1.32955
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −13763.7 −1.65284 −0.826420 0.563054i \(-0.809626\pi\)
−0.826420 + 0.563054i \(0.809626\pi\)
\(38\) 0 0
\(39\) 21540.2 2.26771
\(40\) 0 0
\(41\) 11863.6 1.10219 0.551097 0.834441i \(-0.314210\pi\)
0.551097 + 0.834441i \(0.314210\pi\)
\(42\) 0 0
\(43\) 16816.0 1.38692 0.693461 0.720494i \(-0.256085\pi\)
0.693461 + 0.720494i \(0.256085\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 13036.0 0.860795 0.430397 0.902639i \(-0.358373\pi\)
0.430397 + 0.902639i \(0.358373\pi\)
\(48\) 0 0
\(49\) −15578.6 −0.926912
\(50\) 0 0
\(51\) 2135.69 0.114977
\(52\) 0 0
\(53\) 17817.3 0.871269 0.435634 0.900124i \(-0.356524\pi\)
0.435634 + 0.900124i \(0.356524\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −9682.55 −0.394732
\(58\) 0 0
\(59\) 47346.1 1.77074 0.885368 0.464891i \(-0.153906\pi\)
0.885368 + 0.464891i \(0.153906\pi\)
\(60\) 0 0
\(61\) −22782.9 −0.783944 −0.391972 0.919977i \(-0.628207\pi\)
−0.391972 + 0.919977i \(0.628207\pi\)
\(62\) 0 0
\(63\) −4843.45 −0.153746
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 39418.3 1.07278 0.536390 0.843970i \(-0.319787\pi\)
0.536390 + 0.843970i \(0.319787\pi\)
\(68\) 0 0
\(69\) −48759.1 −1.23291
\(70\) 0 0
\(71\) 1616.32 0.0380523 0.0190261 0.999819i \(-0.493943\pi\)
0.0190261 + 0.999819i \(0.493943\pi\)
\(72\) 0 0
\(73\) 53293.3 1.17048 0.585242 0.810859i \(-0.301000\pi\)
0.585242 + 0.810859i \(0.301000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −14930.9 −0.286984
\(78\) 0 0
\(79\) 6516.98 0.117484 0.0587420 0.998273i \(-0.481291\pi\)
0.0587420 + 0.998273i \(0.481291\pi\)
\(80\) 0 0
\(81\) −73532.6 −1.24528
\(82\) 0 0
\(83\) −46016.4 −0.733191 −0.366595 0.930380i \(-0.619477\pi\)
−0.366595 + 0.930380i \(0.619477\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −828.257 −0.0117319
\(88\) 0 0
\(89\) 113488. 1.51872 0.759358 0.650673i \(-0.225513\pi\)
0.759358 + 0.650673i \(0.225513\pi\)
\(90\) 0 0
\(91\) −38667.3 −0.489487
\(92\) 0 0
\(93\) 156185. 1.87254
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 107418. 1.15917 0.579585 0.814912i \(-0.303215\pi\)
0.579585 + 0.814912i \(0.303215\pi\)
\(98\) 0 0
\(99\) 58871.4 0.603694
\(100\) 0 0
\(101\) −197554. −1.92700 −0.963502 0.267703i \(-0.913736\pi\)
−0.963502 + 0.267703i \(0.913736\pi\)
\(102\) 0 0
\(103\) 81026.0 0.752543 0.376272 0.926509i \(-0.377206\pi\)
0.376272 + 0.926509i \(0.377206\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 137442. 1.16054 0.580271 0.814423i \(-0.302946\pi\)
0.580271 + 0.814423i \(0.302946\pi\)
\(108\) 0 0
\(109\) −68693.1 −0.553792 −0.276896 0.960900i \(-0.589306\pi\)
−0.276896 + 0.960900i \(0.589306\pi\)
\(110\) 0 0
\(111\) −268725. −2.07014
\(112\) 0 0
\(113\) −139632. −1.02870 −0.514352 0.857579i \(-0.671967\pi\)
−0.514352 + 0.857579i \(0.671967\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 152463. 1.02967
\(118\) 0 0
\(119\) −3833.83 −0.0248179
\(120\) 0 0
\(121\) 20431.5 0.126864
\(122\) 0 0
\(123\) 231628. 1.38047
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 65230.1 0.358871 0.179436 0.983770i \(-0.442573\pi\)
0.179436 + 0.983770i \(0.442573\pi\)
\(128\) 0 0
\(129\) 328319. 1.73709
\(130\) 0 0
\(131\) −118542. −0.603524 −0.301762 0.953383i \(-0.597575\pi\)
−0.301762 + 0.953383i \(0.597575\pi\)
\(132\) 0 0
\(133\) 17381.4 0.0852031
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 84757.8 0.385814 0.192907 0.981217i \(-0.438208\pi\)
0.192907 + 0.981217i \(0.438208\pi\)
\(138\) 0 0
\(139\) −168334. −0.738982 −0.369491 0.929234i \(-0.620468\pi\)
−0.369491 + 0.929234i \(0.620468\pi\)
\(140\) 0 0
\(141\) 254517. 1.07813
\(142\) 0 0
\(143\) 469996. 1.92200
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −304160. −1.16094
\(148\) 0 0
\(149\) 67628.6 0.249554 0.124777 0.992185i \(-0.460178\pi\)
0.124777 + 0.992185i \(0.460178\pi\)
\(150\) 0 0
\(151\) 65622.3 0.234212 0.117106 0.993119i \(-0.462638\pi\)
0.117106 + 0.993119i \(0.462638\pi\)
\(152\) 0 0
\(153\) 15116.5 0.0522064
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −297885. −0.964495 −0.482247 0.876035i \(-0.660179\pi\)
−0.482247 + 0.876035i \(0.660179\pi\)
\(158\) 0 0
\(159\) 347868. 1.09124
\(160\) 0 0
\(161\) 87528.7 0.266125
\(162\) 0 0
\(163\) 473879. 1.39701 0.698503 0.715607i \(-0.253850\pi\)
0.698503 + 0.715607i \(0.253850\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −267673. −0.742701 −0.371351 0.928493i \(-0.621105\pi\)
−0.371351 + 0.928493i \(0.621105\pi\)
\(168\) 0 0
\(169\) 845883. 2.27821
\(170\) 0 0
\(171\) −68533.7 −0.179231
\(172\) 0 0
\(173\) 703284. 1.78655 0.893276 0.449509i \(-0.148401\pi\)
0.893276 + 0.449509i \(0.148401\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 924393. 2.21780
\(178\) 0 0
\(179\) −635322. −1.48204 −0.741022 0.671481i \(-0.765659\pi\)
−0.741022 + 0.671481i \(0.765659\pi\)
\(180\) 0 0
\(181\) −547906. −1.24311 −0.621555 0.783370i \(-0.713499\pi\)
−0.621555 + 0.783370i \(0.713499\pi\)
\(182\) 0 0
\(183\) −444818. −0.981872
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 46599.6 0.0974492
\(188\) 0 0
\(189\) 71718.1 0.146041
\(190\) 0 0
\(191\) 80607.9 0.159880 0.0799400 0.996800i \(-0.474527\pi\)
0.0799400 + 0.996800i \(0.474527\pi\)
\(192\) 0 0
\(193\) −26549.3 −0.0513049 −0.0256525 0.999671i \(-0.508166\pi\)
−0.0256525 + 0.999671i \(0.508166\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −703676. −1.29184 −0.645918 0.763407i \(-0.723525\pi\)
−0.645918 + 0.763407i \(0.723525\pi\)
\(198\) 0 0
\(199\) −73599.5 −0.131747 −0.0658737 0.997828i \(-0.520983\pi\)
−0.0658737 + 0.997828i \(0.520983\pi\)
\(200\) 0 0
\(201\) 769610. 1.34363
\(202\) 0 0
\(203\) 1486.83 0.00253233
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −345120. −0.559815
\(208\) 0 0
\(209\) −211268. −0.334556
\(210\) 0 0
\(211\) 1.00808e6 1.55879 0.779395 0.626533i \(-0.215526\pi\)
0.779395 + 0.626533i \(0.215526\pi\)
\(212\) 0 0
\(213\) 31557.2 0.0476596
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −280371. −0.404189
\(218\) 0 0
\(219\) 1.04051e6 1.46600
\(220\) 0 0
\(221\) 120682. 0.166211
\(222\) 0 0
\(223\) −141067. −0.189961 −0.0949805 0.995479i \(-0.530279\pi\)
−0.0949805 + 0.995479i \(0.530279\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.14301e6 1.47226 0.736129 0.676842i \(-0.236652\pi\)
0.736129 + 0.676842i \(0.236652\pi\)
\(228\) 0 0
\(229\) −1.29731e6 −1.63476 −0.817380 0.576099i \(-0.804574\pi\)
−0.817380 + 0.576099i \(0.804574\pi\)
\(230\) 0 0
\(231\) −291513. −0.359441
\(232\) 0 0
\(233\) −626232. −0.755693 −0.377846 0.925868i \(-0.623335\pi\)
−0.377846 + 0.925868i \(0.623335\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 127239. 0.147146
\(238\) 0 0
\(239\) −270281. −0.306070 −0.153035 0.988221i \(-0.548905\pi\)
−0.153035 + 0.988221i \(0.548905\pi\)
\(240\) 0 0
\(241\) −694339. −0.770068 −0.385034 0.922902i \(-0.625810\pi\)
−0.385034 + 0.922902i \(0.625810\pi\)
\(242\) 0 0
\(243\) −938421. −1.01949
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −547134. −0.570626
\(248\) 0 0
\(249\) −898431. −0.918304
\(250\) 0 0
\(251\) −864085. −0.865710 −0.432855 0.901464i \(-0.642494\pi\)
−0.432855 + 0.901464i \(0.642494\pi\)
\(252\) 0 0
\(253\) −1.06390e6 −1.04496
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.00660e6 −0.950660 −0.475330 0.879808i \(-0.657671\pi\)
−0.475330 + 0.879808i \(0.657671\pi\)
\(258\) 0 0
\(259\) 482395. 0.446841
\(260\) 0 0
\(261\) −5862.46 −0.00532695
\(262\) 0 0
\(263\) −931373. −0.830299 −0.415149 0.909753i \(-0.636271\pi\)
−0.415149 + 0.909753i \(0.636271\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 2.21577e6 1.90216
\(268\) 0 0
\(269\) −1.96236e6 −1.65348 −0.826738 0.562587i \(-0.809806\pi\)
−0.826738 + 0.562587i \(0.809806\pi\)
\(270\) 0 0
\(271\) −1.12152e6 −0.927649 −0.463825 0.885927i \(-0.653523\pi\)
−0.463825 + 0.885927i \(0.653523\pi\)
\(272\) 0 0
\(273\) −754948. −0.613070
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −105917. −0.0829401 −0.0414701 0.999140i \(-0.513204\pi\)
−0.0414701 + 0.999140i \(0.513204\pi\)
\(278\) 0 0
\(279\) 1.10549e6 0.850242
\(280\) 0 0
\(281\) −1.05276e6 −0.795362 −0.397681 0.917524i \(-0.630185\pi\)
−0.397681 + 0.917524i \(0.630185\pi\)
\(282\) 0 0
\(283\) −950210. −0.705267 −0.352633 0.935762i \(-0.614714\pi\)
−0.352633 + 0.935762i \(0.614714\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −415801. −0.297975
\(288\) 0 0
\(289\) −1.40789e6 −0.991573
\(290\) 0 0
\(291\) 2.09725e6 1.45183
\(292\) 0 0
\(293\) 381801. 0.259817 0.129909 0.991526i \(-0.458532\pi\)
0.129909 + 0.991526i \(0.458532\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −871723. −0.573440
\(298\) 0 0
\(299\) −2.75524e6 −1.78230
\(300\) 0 0
\(301\) −589373. −0.374951
\(302\) 0 0
\(303\) −3.85708e6 −2.41353
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −1.61048e6 −0.975238 −0.487619 0.873057i \(-0.662134\pi\)
−0.487619 + 0.873057i \(0.662134\pi\)
\(308\) 0 0
\(309\) 1.58197e6 0.942543
\(310\) 0 0
\(311\) 2.15652e6 1.26431 0.632153 0.774843i \(-0.282171\pi\)
0.632153 + 0.774843i \(0.282171\pi\)
\(312\) 0 0
\(313\) 364108. 0.210073 0.105036 0.994468i \(-0.466504\pi\)
0.105036 + 0.994468i \(0.466504\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.03846e6 0.580420 0.290210 0.956963i \(-0.406275\pi\)
0.290210 + 0.956963i \(0.406275\pi\)
\(318\) 0 0
\(319\) −18072.2 −0.00994336
\(320\) 0 0
\(321\) 2.68345e6 1.45355
\(322\) 0 0
\(323\) −54247.8 −0.0289318
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −1.34118e6 −0.693612
\(328\) 0 0
\(329\) −456890. −0.232714
\(330\) 0 0
\(331\) 2.82942e6 1.41948 0.709738 0.704466i \(-0.248814\pi\)
0.709738 + 0.704466i \(0.248814\pi\)
\(332\) 0 0
\(333\) −1.90205e6 −0.939966
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −804843. −0.386043 −0.193022 0.981194i \(-0.561829\pi\)
−0.193022 + 0.981194i \(0.561829\pi\)
\(338\) 0 0
\(339\) −2.72621e6 −1.28843
\(340\) 0 0
\(341\) 3.40787e6 1.58708
\(342\) 0 0
\(343\) 1.13506e6 0.520936
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 759889. 0.338787 0.169393 0.985549i \(-0.445819\pi\)
0.169393 + 0.985549i \(0.445819\pi\)
\(348\) 0 0
\(349\) 580673. 0.255193 0.127596 0.991826i \(-0.459274\pi\)
0.127596 + 0.991826i \(0.459274\pi\)
\(350\) 0 0
\(351\) −2.25755e6 −0.978071
\(352\) 0 0
\(353\) 1.64210e6 0.701393 0.350697 0.936489i \(-0.385945\pi\)
0.350697 + 0.936489i \(0.385945\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −74852.3 −0.0310838
\(358\) 0 0
\(359\) 2.88181e6 1.18013 0.590065 0.807356i \(-0.299102\pi\)
0.590065 + 0.807356i \(0.299102\pi\)
\(360\) 0 0
\(361\) −2.23016e6 −0.900673
\(362\) 0 0
\(363\) 398909. 0.158894
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −3.83418e6 −1.48596 −0.742980 0.669314i \(-0.766588\pi\)
−0.742980 + 0.669314i \(0.766588\pi\)
\(368\) 0 0
\(369\) 1.63947e6 0.626814
\(370\) 0 0
\(371\) −624467. −0.235545
\(372\) 0 0
\(373\) −3.19341e6 −1.18845 −0.594227 0.804298i \(-0.702542\pi\)
−0.594227 + 0.804298i \(0.702542\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −46802.5 −0.0169596
\(378\) 0 0
\(379\) −3.47338e6 −1.24209 −0.621046 0.783774i \(-0.713292\pi\)
−0.621046 + 0.783774i \(0.713292\pi\)
\(380\) 0 0
\(381\) 1.27356e6 0.449478
\(382\) 0 0
\(383\) 1.56341e6 0.544599 0.272299 0.962213i \(-0.412216\pi\)
0.272299 + 0.962213i \(0.412216\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2.32386e6 0.788738
\(388\) 0 0
\(389\) 4.98980e6 1.67190 0.835949 0.548808i \(-0.184918\pi\)
0.835949 + 0.548808i \(0.184918\pi\)
\(390\) 0 0
\(391\) −273179. −0.0903661
\(392\) 0 0
\(393\) −2.31444e6 −0.755899
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 1.37245e6 0.437038 0.218519 0.975833i \(-0.429877\pi\)
0.218519 + 0.975833i \(0.429877\pi\)
\(398\) 0 0
\(399\) 339357. 0.106715
\(400\) 0 0
\(401\) 5.44247e6 1.69019 0.845094 0.534617i \(-0.179544\pi\)
0.845094 + 0.534617i \(0.179544\pi\)
\(402\) 0 0
\(403\) 8.82557e6 2.70695
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −5.86344e6 −1.75455
\(408\) 0 0
\(409\) −6.03830e6 −1.78487 −0.892435 0.451176i \(-0.851005\pi\)
−0.892435 + 0.451176i \(0.851005\pi\)
\(410\) 0 0
\(411\) 1.65483e6 0.483223
\(412\) 0 0
\(413\) −1.65940e6 −0.478714
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −3.28657e6 −0.925557
\(418\) 0 0
\(419\) −770366. −0.214369 −0.107184 0.994239i \(-0.534184\pi\)
−0.107184 + 0.994239i \(0.534184\pi\)
\(420\) 0 0
\(421\) 2.00249e6 0.550636 0.275318 0.961353i \(-0.411217\pi\)
0.275318 + 0.961353i \(0.411217\pi\)
\(422\) 0 0
\(423\) 1.80149e6 0.489532
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 798504. 0.211937
\(428\) 0 0
\(429\) 9.17628e6 2.40726
\(430\) 0 0
\(431\) −5.56690e6 −1.44351 −0.721756 0.692148i \(-0.756665\pi\)
−0.721756 + 0.692148i \(0.756665\pi\)
\(432\) 0 0
\(433\) 2.22941e6 0.571441 0.285720 0.958313i \(-0.407767\pi\)
0.285720 + 0.958313i \(0.407767\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.23851e6 0.310239
\(438\) 0 0
\(439\) −543061. −0.134489 −0.0672446 0.997737i \(-0.521421\pi\)
−0.0672446 + 0.997737i \(0.521421\pi\)
\(440\) 0 0
\(441\) −2.15286e6 −0.527132
\(442\) 0 0
\(443\) −4.53358e6 −1.09757 −0.548785 0.835964i \(-0.684909\pi\)
−0.548785 + 0.835964i \(0.684909\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 1.32039e6 0.312561
\(448\) 0 0
\(449\) −3.39468e6 −0.794662 −0.397331 0.917675i \(-0.630064\pi\)
−0.397331 + 0.917675i \(0.630064\pi\)
\(450\) 0 0
\(451\) 5.05400e6 1.17002
\(452\) 0 0
\(453\) 1.28122e6 0.293345
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.34100e6 1.86822 0.934109 0.356988i \(-0.116196\pi\)
0.934109 + 0.356988i \(0.116196\pi\)
\(458\) 0 0
\(459\) −223834. −0.0495900
\(460\) 0 0
\(461\) −3.18453e6 −0.697900 −0.348950 0.937141i \(-0.613462\pi\)
−0.348950 + 0.937141i \(0.613462\pi\)
\(462\) 0 0
\(463\) −4.05943e6 −0.880061 −0.440030 0.897983i \(-0.645032\pi\)
−0.440030 + 0.897983i \(0.645032\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 7.98706e6 1.69471 0.847353 0.531029i \(-0.178195\pi\)
0.847353 + 0.531029i \(0.178195\pi\)
\(468\) 0 0
\(469\) −1.38155e6 −0.290024
\(470\) 0 0
\(471\) −5.81596e6 −1.20801
\(472\) 0 0
\(473\) 7.16375e6 1.47227
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 2.46223e6 0.495488
\(478\) 0 0
\(479\) −5.27553e6 −1.05058 −0.525288 0.850924i \(-0.676042\pi\)
−0.525288 + 0.850924i \(0.676042\pi\)
\(480\) 0 0
\(481\) −1.51849e7 −2.99260
\(482\) 0 0
\(483\) 1.70892e6 0.333315
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −8.76969e6 −1.67557 −0.837784 0.546002i \(-0.816149\pi\)
−0.837784 + 0.546002i \(0.816149\pi\)
\(488\) 0 0
\(489\) 9.25209e6 1.74972
\(490\) 0 0
\(491\) −7.83080e6 −1.46589 −0.732946 0.680286i \(-0.761855\pi\)
−0.732946 + 0.680286i \(0.761855\pi\)
\(492\) 0 0
\(493\) −4640.42 −0.000859885 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −56649.2 −0.0102873
\(498\) 0 0
\(499\) 6.54560e6 1.17679 0.588394 0.808574i \(-0.299760\pi\)
0.588394 + 0.808574i \(0.299760\pi\)
\(500\) 0 0
\(501\) −5.22610e6 −0.930216
\(502\) 0 0
\(503\) 2.07773e6 0.366158 0.183079 0.983098i \(-0.441394\pi\)
0.183079 + 0.983098i \(0.441394\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.65152e7 2.85340
\(508\) 0 0
\(509\) −6.25093e6 −1.06942 −0.534712 0.845034i \(-0.679580\pi\)
−0.534712 + 0.845034i \(0.679580\pi\)
\(510\) 0 0
\(511\) −1.86784e6 −0.316437
\(512\) 0 0
\(513\) 1.01480e6 0.170249
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 5.55344e6 0.913767
\(518\) 0 0
\(519\) 1.37310e7 2.23761
\(520\) 0 0
\(521\) 2.19897e6 0.354916 0.177458 0.984128i \(-0.443213\pi\)
0.177458 + 0.984128i \(0.443213\pi\)
\(522\) 0 0
\(523\) 2.10298e6 0.336187 0.168093 0.985771i \(-0.446239\pi\)
0.168093 + 0.985771i \(0.446239\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 875046. 0.137247
\(528\) 0 0
\(529\) −199490. −0.0309944
\(530\) 0 0
\(531\) 6.54291e6 1.00701
\(532\) 0 0
\(533\) 1.30886e7 1.99561
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −1.24041e7 −1.85623
\(538\) 0 0
\(539\) −6.63661e6 −0.983953
\(540\) 0 0
\(541\) −1.83729e6 −0.269889 −0.134944 0.990853i \(-0.543086\pi\)
−0.134944 + 0.990853i \(0.543086\pi\)
\(542\) 0 0
\(543\) −1.06974e7 −1.55697
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −4.31872e6 −0.617145 −0.308572 0.951201i \(-0.599851\pi\)
−0.308572 + 0.951201i \(0.599851\pi\)
\(548\) 0 0
\(549\) −3.14845e6 −0.445827
\(550\) 0 0
\(551\) 21038.3 0.00295210
\(552\) 0 0
\(553\) −228409. −0.0317615
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −4.26837e6 −0.582940 −0.291470 0.956580i \(-0.594144\pi\)
−0.291470 + 0.956580i \(0.594144\pi\)
\(558\) 0 0
\(559\) 1.85524e7 2.51114
\(560\) 0 0
\(561\) 909819. 0.122053
\(562\) 0 0
\(563\) 1.26350e7 1.67998 0.839988 0.542605i \(-0.182562\pi\)
0.839988 + 0.542605i \(0.182562\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.57720e6 0.336659
\(568\) 0 0
\(569\) −8.67409e6 −1.12316 −0.561582 0.827421i \(-0.689807\pi\)
−0.561582 + 0.827421i \(0.689807\pi\)
\(570\) 0 0
\(571\) −3.28781e6 −0.422004 −0.211002 0.977486i \(-0.567673\pi\)
−0.211002 + 0.977486i \(0.567673\pi\)
\(572\) 0 0
\(573\) 1.57380e6 0.200246
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −117668. −0.0147135 −0.00735677 0.999973i \(-0.502342\pi\)
−0.00735677 + 0.999973i \(0.502342\pi\)
\(578\) 0 0
\(579\) −518352. −0.0642582
\(580\) 0 0
\(581\) 1.61280e6 0.198216
\(582\) 0 0
\(583\) 7.59031e6 0.924885
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.10151e7 1.31945 0.659723 0.751509i \(-0.270674\pi\)
0.659723 + 0.751509i \(0.270674\pi\)
\(588\) 0 0
\(589\) −3.96719e6 −0.471189
\(590\) 0 0
\(591\) −1.37387e7 −1.61799
\(592\) 0 0
\(593\) −5.33879e6 −0.623456 −0.311728 0.950171i \(-0.600908\pi\)
−0.311728 + 0.950171i \(0.600908\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.43697e6 −0.165011
\(598\) 0 0
\(599\) 7.17784e6 0.817385 0.408692 0.912672i \(-0.365985\pi\)
0.408692 + 0.912672i \(0.365985\pi\)
\(600\) 0 0
\(601\) −809127. −0.0913756 −0.0456878 0.998956i \(-0.514548\pi\)
−0.0456878 + 0.998956i \(0.514548\pi\)
\(602\) 0 0
\(603\) 5.44735e6 0.610087
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −591991. −0.0652143 −0.0326072 0.999468i \(-0.510381\pi\)
−0.0326072 + 0.999468i \(0.510381\pi\)
\(608\) 0 0
\(609\) 29029.1 0.00317168
\(610\) 0 0
\(611\) 1.43821e7 1.55854
\(612\) 0 0
\(613\) −4.01173e6 −0.431201 −0.215601 0.976482i \(-0.569171\pi\)
−0.215601 + 0.976482i \(0.569171\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8.24093e6 0.871491 0.435746 0.900070i \(-0.356485\pi\)
0.435746 + 0.900070i \(0.356485\pi\)
\(618\) 0 0
\(619\) −1.96985e6 −0.206636 −0.103318 0.994648i \(-0.532946\pi\)
−0.103318 + 0.994648i \(0.532946\pi\)
\(620\) 0 0
\(621\) 5.11027e6 0.531759
\(622\) 0 0
\(623\) −3.97758e6 −0.410581
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −4.12484e6 −0.419024
\(628\) 0 0
\(629\) −1.50557e6 −0.151731
\(630\) 0 0
\(631\) 1.82360e7 1.82329 0.911646 0.410977i \(-0.134813\pi\)
0.911646 + 0.410977i \(0.134813\pi\)
\(632\) 0 0
\(633\) 1.96819e7 1.95235
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −1.71872e7 −1.67825
\(638\) 0 0
\(639\) 223364. 0.0216402
\(640\) 0 0
\(641\) 7.65378e6 0.735752 0.367876 0.929875i \(-0.380085\pi\)
0.367876 + 0.929875i \(0.380085\pi\)
\(642\) 0 0
\(643\) −2.47762e6 −0.236323 −0.118162 0.992994i \(-0.537700\pi\)
−0.118162 + 0.992994i \(0.537700\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −4.79533e6 −0.450358 −0.225179 0.974317i \(-0.572297\pi\)
−0.225179 + 0.974317i \(0.572297\pi\)
\(648\) 0 0
\(649\) 2.01698e7 1.87970
\(650\) 0 0
\(651\) −5.47402e6 −0.506237
\(652\) 0 0
\(653\) 1.67125e7 1.53376 0.766881 0.641789i \(-0.221807\pi\)
0.766881 + 0.641789i \(0.221807\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 7.36478e6 0.665650
\(658\) 0 0
\(659\) −3.28380e6 −0.294553 −0.147277 0.989095i \(-0.547051\pi\)
−0.147277 + 0.989095i \(0.547051\pi\)
\(660\) 0 0
\(661\) −1.98880e7 −1.77047 −0.885234 0.465145i \(-0.846002\pi\)
−0.885234 + 0.465145i \(0.846002\pi\)
\(662\) 0 0
\(663\) 2.35621e6 0.208176
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 105944. 0.00922063
\(668\) 0 0
\(669\) −2.75423e6 −0.237922
\(670\) 0 0
\(671\) −9.70571e6 −0.832187
\(672\) 0 0
\(673\) −1.39915e7 −1.19076 −0.595381 0.803443i \(-0.702999\pi\)
−0.595381 + 0.803443i \(0.702999\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3.13032e6 0.262493 0.131246 0.991350i \(-0.458102\pi\)
0.131246 + 0.991350i \(0.458102\pi\)
\(678\) 0 0
\(679\) −3.76482e6 −0.313379
\(680\) 0 0
\(681\) 2.23162e7 1.84397
\(682\) 0 0
\(683\) −8.94034e6 −0.733335 −0.366667 0.930352i \(-0.619501\pi\)
−0.366667 + 0.930352i \(0.619501\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −2.53288e7 −2.04750
\(688\) 0 0
\(689\) 1.96571e7 1.57750
\(690\) 0 0
\(691\) 2.38378e7 1.89920 0.949599 0.313466i \(-0.101490\pi\)
0.949599 + 0.313466i \(0.101490\pi\)
\(692\) 0 0
\(693\) −2.06335e6 −0.163207
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.29772e6 0.101181
\(698\) 0 0
\(699\) −1.22267e7 −0.946487
\(700\) 0 0
\(701\) 1.57226e7 1.20845 0.604225 0.796814i \(-0.293483\pi\)
0.604225 + 0.796814i \(0.293483\pi\)
\(702\) 0 0
\(703\) 6.82578e6 0.520912
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 6.92394e6 0.520961
\(708\) 0 0
\(709\) −3.05866e6 −0.228515 −0.114258 0.993451i \(-0.536449\pi\)
−0.114258 + 0.993451i \(0.536449\pi\)
\(710\) 0 0
\(711\) 900603. 0.0668128
\(712\) 0 0
\(713\) −1.99778e7 −1.47172
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −5.27701e6 −0.383345
\(718\) 0 0
\(719\) −2.16146e7 −1.55929 −0.779643 0.626224i \(-0.784600\pi\)
−0.779643 + 0.626224i \(0.784600\pi\)
\(720\) 0 0
\(721\) −2.83983e6 −0.203448
\(722\) 0 0
\(723\) −1.35564e7 −0.964492
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −2.73589e7 −1.91983 −0.959916 0.280287i \(-0.909570\pi\)
−0.959916 + 0.280287i \(0.909570\pi\)
\(728\) 0 0
\(729\) −453482. −0.0316039
\(730\) 0 0
\(731\) 1.83945e6 0.127319
\(732\) 0 0
\(733\) −1.54693e7 −1.06344 −0.531718 0.846921i \(-0.678453\pi\)
−0.531718 + 0.846921i \(0.678453\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.67925e7 1.13880
\(738\) 0 0
\(739\) −5.56327e6 −0.374731 −0.187365 0.982290i \(-0.559995\pi\)
−0.187365 + 0.982290i \(0.559995\pi\)
\(740\) 0 0
\(741\) −1.06823e7 −0.714695
\(742\) 0 0
\(743\) −1.55089e7 −1.03064 −0.515322 0.856997i \(-0.672328\pi\)
−0.515322 + 0.856997i \(0.672328\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −6.35916e6 −0.416963
\(748\) 0 0
\(749\) −4.81713e6 −0.313750
\(750\) 0 0
\(751\) −2.20221e7 −1.42482 −0.712409 0.701765i \(-0.752396\pi\)
−0.712409 + 0.701765i \(0.752396\pi\)
\(752\) 0 0
\(753\) −1.68706e7 −1.08428
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.75653e6 0.111408 0.0557040 0.998447i \(-0.482260\pi\)
0.0557040 + 0.998447i \(0.482260\pi\)
\(758\) 0 0
\(759\) −2.07717e7 −1.30879
\(760\) 0 0
\(761\) −2.16974e6 −0.135814 −0.0679072 0.997692i \(-0.521632\pi\)
−0.0679072 + 0.997692i \(0.521632\pi\)
\(762\) 0 0
\(763\) 2.40758e6 0.149716
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 5.22349e7 3.20606
\(768\) 0 0
\(769\) −1.12131e7 −0.683769 −0.341884 0.939742i \(-0.611065\pi\)
−0.341884 + 0.939742i \(0.611065\pi\)
\(770\) 0 0
\(771\) −1.96531e7 −1.19068
\(772\) 0 0
\(773\) −1.43367e7 −0.862980 −0.431490 0.902118i \(-0.642012\pi\)
−0.431490 + 0.902118i \(0.642012\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 9.41836e6 0.559658
\(778\) 0 0
\(779\) −5.88348e6 −0.347369
\(780\) 0 0
\(781\) 688563. 0.0403939
\(782\) 0 0
\(783\) 86806.8 0.00505999
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 2.00549e6 0.115421 0.0577103 0.998333i \(-0.481620\pi\)
0.0577103 + 0.998333i \(0.481620\pi\)
\(788\) 0 0
\(789\) −1.81843e7 −1.03993
\(790\) 0 0
\(791\) 4.89389e6 0.278108
\(792\) 0 0
\(793\) −2.51354e7 −1.41940
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −448115. −0.0249887 −0.0124944 0.999922i \(-0.503977\pi\)
−0.0124944 + 0.999922i \(0.503977\pi\)
\(798\) 0 0
\(799\) 1.42597e6 0.0790210
\(800\) 0 0
\(801\) 1.56834e7 0.863690
\(802\) 0 0
\(803\) 2.27033e7 1.24251
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −3.83134e7 −2.07094
\(808\) 0 0
\(809\) 1.28425e7 0.689889 0.344944 0.938623i \(-0.387898\pi\)
0.344944 + 0.938623i \(0.387898\pi\)
\(810\) 0 0
\(811\) −1.78700e7 −0.954054 −0.477027 0.878889i \(-0.658286\pi\)
−0.477027 + 0.878889i \(0.658286\pi\)
\(812\) 0 0
\(813\) −2.18968e7 −1.16186
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −8.33950e6 −0.437104
\(818\) 0 0
\(819\) −5.34357e6 −0.278370
\(820\) 0 0
\(821\) 2.73336e7 1.41527 0.707634 0.706580i \(-0.249763\pi\)
0.707634 + 0.706580i \(0.249763\pi\)
\(822\) 0 0
\(823\) 3.05819e6 0.157385 0.0786927 0.996899i \(-0.474925\pi\)
0.0786927 + 0.996899i \(0.474925\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.23961e7 0.630261 0.315131 0.949048i \(-0.397952\pi\)
0.315131 + 0.949048i \(0.397952\pi\)
\(828\) 0 0
\(829\) −399873. −0.0202086 −0.0101043 0.999949i \(-0.503216\pi\)
−0.0101043 + 0.999949i \(0.503216\pi\)
\(830\) 0 0
\(831\) −2.06793e6 −0.103881
\(832\) 0 0
\(833\) −1.70409e6 −0.0850905
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1.63692e7 −0.807632
\(838\) 0 0
\(839\) 2.08411e6 0.102215 0.0511075 0.998693i \(-0.483725\pi\)
0.0511075 + 0.998693i \(0.483725\pi\)
\(840\) 0 0
\(841\) −2.05093e7 −0.999912
\(842\) 0 0
\(843\) −2.05543e7 −0.996172
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −716092. −0.0342973
\(848\) 0 0
\(849\) −1.85521e7 −0.883330
\(850\) 0 0
\(851\) 3.43730e7 1.62702
\(852\) 0 0
\(853\) 2.14298e7 1.00843 0.504214 0.863579i \(-0.331782\pi\)
0.504214 + 0.863579i \(0.331782\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3.49123e6 0.162377 0.0811887 0.996699i \(-0.474128\pi\)
0.0811887 + 0.996699i \(0.474128\pi\)
\(858\) 0 0
\(859\) 568043. 0.0262662 0.0131331 0.999914i \(-0.495819\pi\)
0.0131331 + 0.999914i \(0.495819\pi\)
\(860\) 0 0
\(861\) −8.11816e6 −0.373207
\(862\) 0 0
\(863\) −3.28678e7 −1.50226 −0.751128 0.660157i \(-0.770490\pi\)
−0.751128 + 0.660157i \(0.770490\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −2.74879e7 −1.24192
\(868\) 0 0
\(869\) 2.77628e6 0.124714
\(870\) 0 0
\(871\) 4.34885e7 1.94236
\(872\) 0 0
\(873\) 1.48444e7 0.659217
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 2.92945e7 1.28614 0.643069 0.765808i \(-0.277661\pi\)
0.643069 + 0.765808i \(0.277661\pi\)
\(878\) 0 0
\(879\) 7.45435e6 0.325415
\(880\) 0 0
\(881\) 2.88332e6 0.125156 0.0625782 0.998040i \(-0.480068\pi\)
0.0625782 + 0.998040i \(0.480068\pi\)
\(882\) 0 0
\(883\) −2.79477e7 −1.20627 −0.603135 0.797639i \(-0.706082\pi\)
−0.603135 + 0.797639i \(0.706082\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.89567e7 1.23578 0.617888 0.786266i \(-0.287989\pi\)
0.617888 + 0.786266i \(0.287989\pi\)
\(888\) 0 0
\(889\) −2.28621e6 −0.0970199
\(890\) 0 0
\(891\) −3.13254e7 −1.32191
\(892\) 0 0
\(893\) −6.46490e6 −0.271289
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −5.37938e7 −2.23229
\(898\) 0 0
\(899\) −339358. −0.0140042
\(900\) 0 0
\(901\) 1.94898e6 0.0799825
\(902\) 0 0
\(903\) −1.15070e7 −0.469617
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.18508e6 0.0478333 0.0239166 0.999714i \(-0.492386\pi\)
0.0239166 + 0.999714i \(0.492386\pi\)
\(908\) 0 0
\(909\) −2.73007e7 −1.09588
\(910\) 0 0
\(911\) 1.14346e7 0.456485 0.228242 0.973604i \(-0.426702\pi\)
0.228242 + 0.973604i \(0.426702\pi\)
\(912\) 0 0
\(913\) −1.96033e7 −0.778310
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4.15471e6 0.163161
\(918\) 0 0
\(919\) 1.48904e7 0.581589 0.290795 0.956785i \(-0.406080\pi\)
0.290795 + 0.956785i \(0.406080\pi\)
\(920\) 0 0
\(921\) −3.14434e7 −1.22146
\(922\) 0 0
\(923\) 1.78321e6 0.0688968
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 1.11973e7 0.427969
\(928\) 0 0
\(929\) −2.02580e7 −0.770117 −0.385058 0.922892i \(-0.625819\pi\)
−0.385058 + 0.922892i \(0.625819\pi\)
\(930\) 0 0
\(931\) 7.72584e6 0.292127
\(932\) 0 0
\(933\) 4.21043e7 1.58351
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1.96237e7 −0.730182 −0.365091 0.930972i \(-0.618962\pi\)
−0.365091 + 0.930972i \(0.618962\pi\)
\(938\) 0 0
\(939\) 7.10891e6 0.263111
\(940\) 0 0
\(941\) 7.62854e6 0.280846 0.140423 0.990092i \(-0.455154\pi\)
0.140423 + 0.990092i \(0.455154\pi\)
\(942\) 0 0
\(943\) −2.96279e7 −1.08498
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −3.93605e7 −1.42622 −0.713108 0.701054i \(-0.752713\pi\)
−0.713108 + 0.701054i \(0.752713\pi\)
\(948\) 0 0
\(949\) 5.87962e7 2.11926
\(950\) 0 0
\(951\) 2.02751e7 0.726963
\(952\) 0 0
\(953\) 5.09953e7 1.81886 0.909428 0.415861i \(-0.136520\pi\)
0.909428 + 0.415861i \(0.136520\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −352844. −0.0124538
\(958\) 0 0
\(959\) −2.97062e6 −0.104304
\(960\) 0 0
\(961\) 3.53638e7 1.23524
\(962\) 0 0
\(963\) 1.89936e7 0.659997
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 5.17740e7 1.78051 0.890257 0.455458i \(-0.150524\pi\)
0.890257 + 0.455458i \(0.150524\pi\)
\(968\) 0 0
\(969\) −1.05914e6 −0.0362364
\(970\) 0 0
\(971\) 3.69158e7 1.25651 0.628253 0.778009i \(-0.283770\pi\)
0.628253 + 0.778009i \(0.283770\pi\)
\(972\) 0 0
\(973\) 5.89981e6 0.199782
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.14468e7 1.05400 0.526999 0.849866i \(-0.323317\pi\)
0.526999 + 0.849866i \(0.323317\pi\)
\(978\) 0 0
\(979\) 4.83470e7 1.61218
\(980\) 0 0
\(981\) −9.49293e6 −0.314940
\(982\) 0 0
\(983\) 5.15823e7 1.70262 0.851308 0.524667i \(-0.175810\pi\)
0.851308 + 0.524667i \(0.175810\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −8.92041e6 −0.291469
\(988\) 0 0
\(989\) −4.19958e7 −1.36526
\(990\) 0 0
\(991\) 1.30562e7 0.422312 0.211156 0.977452i \(-0.432277\pi\)
0.211156 + 0.977452i \(0.432277\pi\)
\(992\) 0 0
\(993\) 5.52421e7 1.77786
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −4.97672e7 −1.58564 −0.792821 0.609454i \(-0.791389\pi\)
−0.792821 + 0.609454i \(0.791389\pi\)
\(998\) 0 0
\(999\) 2.81641e7 0.892859
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 400.6.a.u.1.2 2
4.3 odd 2 200.6.a.e.1.1 2
5.2 odd 4 400.6.c.o.49.1 4
5.3 odd 4 400.6.c.o.49.4 4
5.4 even 2 400.6.a.r.1.1 2
20.3 even 4 200.6.c.f.49.1 4
20.7 even 4 200.6.c.f.49.4 4
20.19 odd 2 200.6.a.f.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.6.a.e.1.1 2 4.3 odd 2
200.6.a.f.1.2 yes 2 20.19 odd 2
200.6.c.f.49.1 4 20.3 even 4
200.6.c.f.49.4 4 20.7 even 4
400.6.a.r.1.1 2 5.4 even 2
400.6.a.u.1.2 2 1.1 even 1 trivial
400.6.c.o.49.1 4 5.2 odd 4
400.6.c.o.49.4 4 5.3 odd 4