Properties

Label 200.6.c.f.49.4
Level $200$
Weight $6$
Character 200.49
Analytic conductor $32.077$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 200.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.0767639626\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{241})\)
Defining polynomial: \( x^{4} + 121x^{2} + 3600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.4
Root \(-7.26209i\) of defining polynomial
Character \(\chi\) \(=\) 200.49
Dual form 200.6.c.f.49.1

$q$-expansion

\(f(q)\) \(=\) \(q+19.5242i q^{3} +35.0483i q^{7} -138.193 q^{9} +O(q^{10})\) \(q+19.5242i q^{3} +35.0483i q^{7} -138.193 q^{9} -426.008 q^{11} -1103.26i q^{13} +109.387i q^{17} -495.926 q^{19} -684.290 q^{21} -2497.37i q^{23} +2046.26i q^{27} +42.4221 q^{29} -7999.56 q^{31} -8317.45i q^{33} -13763.7i q^{37} +21540.2 q^{39} +11863.6 q^{41} +16816.0i q^{43} -13036.0i q^{47} +15578.6 q^{49} -2135.69 q^{51} -17817.3i q^{53} -9682.55i q^{57} +47346.1 q^{59} -22782.9 q^{61} -4843.45i q^{63} -39418.3i q^{67} +48759.1 q^{69} -1616.32 q^{71} -53293.3i q^{73} -14930.9i q^{77} +6516.98 q^{79} -73532.6 q^{81} -46016.4i q^{83} +828.257i q^{87} -113488. q^{89} +38667.3 q^{91} -156185. i q^{93} +107418. i q^{97} +58871.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 56 q^{9} - 400 q^{11} + 1680 q^{19} - 1992 q^{21} + 9360 q^{29} - 10016 q^{31} + 54864 q^{39} - 10668 q^{41} + 63308 q^{49} - 13200 q^{51} + 163552 q^{59} - 93864 q^{61} + 110088 q^{69} + 14896 q^{71} + 216208 q^{79} - 187324 q^{81} - 141980 q^{89} + 104992 q^{91} + 167552 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 19.5242i 1.25248i 0.779632 + 0.626238i \(0.215406\pi\)
−0.779632 + 0.626238i \(0.784594\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 35.0483i 0.270348i 0.990822 + 0.135174i \(0.0431593\pi\)
−0.990822 + 0.135174i \(0.956841\pi\)
\(8\) 0 0
\(9\) −138.193 −0.568697
\(10\) 0 0
\(11\) −426.008 −1.06154 −0.530769 0.847516i \(-0.678097\pi\)
−0.530769 + 0.847516i \(0.678097\pi\)
\(12\) 0 0
\(13\) − 1103.26i − 1.81058i −0.424791 0.905291i \(-0.639653\pi\)
0.424791 0.905291i \(-0.360347\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 109.387i 0.0918000i 0.998946 + 0.0459000i \(0.0146156\pi\)
−0.998946 + 0.0459000i \(0.985384\pi\)
\(18\) 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
\(23\) − 2497.37i − 0.984381i −0.870488 0.492190i \(-0.836196\pi\)
0.870488 0.492190i \(-0.163804\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2046.26i 0.540197i
\(28\) 0 0
\(29\) 42.4221 0.00936694 0.00468347 0.999989i \(-0.498509\pi\)
0.00468347 + 0.999989i \(0.498509\pi\)
\(30\) 0 0
\(31\) −7999.56 −1.49507 −0.747535 0.664222i \(-0.768763\pi\)
−0.747535 + 0.664222i \(0.768763\pi\)
\(32\) 0 0
\(33\) − 8317.45i − 1.32955i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 13763.7i − 1.65284i −0.563054 0.826420i \(-0.690374\pi\)
0.563054 0.826420i \(-0.309626\pi\)
\(38\) 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
\(43\) 16816.0i 1.38692i 0.720494 + 0.693461i \(0.243915\pi\)
−0.720494 + 0.693461i \(0.756085\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 13036.0i − 0.860795i −0.902639 0.430397i \(-0.858373\pi\)
0.902639 0.430397i \(-0.141627\pi\)
\(48\) 0 0
\(49\) 15578.6 0.926912
\(50\) 0 0
\(51\) −2135.69 −0.114977
\(52\) 0 0
\(53\) − 17817.3i − 0.871269i −0.900124 0.435634i \(-0.856524\pi\)
0.900124 0.435634i \(-0.143476\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 9682.55i − 0.394732i
\(58\) 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
\(63\) − 4843.45i − 0.153746i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 39418.3i − 1.07278i −0.843970 0.536390i \(-0.819787\pi\)
0.843970 0.536390i \(-0.180213\pi\)
\(68\) 0 0
\(69\) 48759.1 1.23291
\(70\) 0 0
\(71\) −1616.32 −0.0380523 −0.0190261 0.999819i \(-0.506057\pi\)
−0.0190261 + 0.999819i \(0.506057\pi\)
\(72\) 0 0
\(73\) − 53293.3i − 1.17048i −0.810859 0.585242i \(-0.801000\pi\)
0.810859 0.585242i \(-0.199000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 14930.9i − 0.286984i
\(78\) 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
\(83\) − 46016.4i − 0.733191i −0.930380 0.366595i \(-0.880523\pi\)
0.930380 0.366595i \(-0.119477\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 828.257i 0.0117319i
\(88\) 0 0
\(89\) −113488. −1.51872 −0.759358 0.650673i \(-0.774487\pi\)
−0.759358 + 0.650673i \(0.774487\pi\)
\(90\) 0 0
\(91\) 38667.3 0.489487
\(92\) 0 0
\(93\) − 156185.i − 1.87254i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 107418.i 1.15917i 0.814912 + 0.579585i \(0.196785\pi\)
−0.814912 + 0.579585i \(0.803215\pi\)
\(98\) 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
\(103\) 81026.0i 0.752543i 0.926509 + 0.376272i \(0.122794\pi\)
−0.926509 + 0.376272i \(0.877206\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 137442.i − 1.16054i −0.814423 0.580271i \(-0.802946\pi\)
0.814423 0.580271i \(-0.197054\pi\)
\(108\) 0 0
\(109\) 68693.1 0.553792 0.276896 0.960900i \(-0.410694\pi\)
0.276896 + 0.960900i \(0.410694\pi\)
\(110\) 0 0
\(111\) 268725. 2.07014
\(112\) 0 0
\(113\) 139632.i 1.02870i 0.857579 + 0.514352i \(0.171967\pi\)
−0.857579 + 0.514352i \(0.828033\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 152463.i 1.02967i
\(118\) 0 0
\(119\) −3833.83 −0.0248179
\(120\) 0 0
\(121\) 20431.5 0.126864
\(122\) 0 0
\(123\) 231628.i 1.38047i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 65230.1i − 0.358871i −0.983770 0.179436i \(-0.942573\pi\)
0.983770 0.179436i \(-0.0574271\pi\)
\(128\) 0 0
\(129\) −328319. −1.73709
\(130\) 0 0
\(131\) 118542. 0.603524 0.301762 0.953383i \(-0.402425\pi\)
0.301762 + 0.953383i \(0.402425\pi\)
\(132\) 0 0
\(133\) − 17381.4i − 0.0852031i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 84757.8i 0.385814i 0.981217 + 0.192907i \(0.0617916\pi\)
−0.981217 + 0.192907i \(0.938208\pi\)
\(138\) 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.i 1.92200i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 304160.i 1.16094i
\(148\) 0 0
\(149\) −67628.6 −0.249554 −0.124777 0.992185i \(-0.539822\pi\)
−0.124777 + 0.992185i \(0.539822\pi\)
\(150\) 0 0
\(151\) −65622.3 −0.234212 −0.117106 0.993119i \(-0.537362\pi\)
−0.117106 + 0.993119i \(0.537362\pi\)
\(152\) 0 0
\(153\) − 15116.5i − 0.0522064i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 297885.i − 0.964495i −0.876035 0.482247i \(-0.839821\pi\)
0.876035 0.482247i \(-0.160179\pi\)
\(158\) 0 0
\(159\) 347868. 1.09124
\(160\) 0 0
\(161\) 87528.7 0.266125
\(162\) 0 0
\(163\) 473879.i 1.39701i 0.715607 + 0.698503i \(0.246150\pi\)
−0.715607 + 0.698503i \(0.753850\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 267673.i 0.742701i 0.928493 + 0.371351i \(0.121105\pi\)
−0.928493 + 0.371351i \(0.878895\pi\)
\(168\) 0 0
\(169\) −845883. −2.27821
\(170\) 0 0
\(171\) 68533.7 0.179231
\(172\) 0 0
\(173\) − 703284.i − 1.78655i −0.449509 0.893276i \(-0.648401\pi\)
0.449509 0.893276i \(-0.351599\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 924393.i 2.21780i
\(178\) 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.i − 0.981872i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 46599.6i − 0.0974492i
\(188\) 0 0
\(189\) −71718.1 −0.146041
\(190\) 0 0
\(191\) −80607.9 −0.159880 −0.0799400 0.996800i \(-0.525473\pi\)
−0.0799400 + 0.996800i \(0.525473\pi\)
\(192\) 0 0
\(193\) 26549.3i 0.0513049i 0.999671 + 0.0256525i \(0.00816633\pi\)
−0.999671 + 0.0256525i \(0.991834\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 703676.i − 1.29184i −0.763407 0.645918i \(-0.776475\pi\)
0.763407 0.645918i \(-0.223525\pi\)
\(198\) 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
\(203\) 1486.83i 0.00253233i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 345120.i 0.559815i
\(208\) 0 0
\(209\) 211268. 0.334556
\(210\) 0 0
\(211\) −1.00808e6 −1.55879 −0.779395 0.626533i \(-0.784474\pi\)
−0.779395 + 0.626533i \(0.784474\pi\)
\(212\) 0 0
\(213\) − 31557.2i − 0.0476596i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 280371.i − 0.404189i
\(218\) 0 0
\(219\) 1.04051e6 1.46600
\(220\) 0 0
\(221\) 120682. 0.166211
\(222\) 0 0
\(223\) − 141067.i − 0.189961i −0.995479 0.0949805i \(-0.969721\pi\)
0.995479 0.0949805i \(-0.0302789\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 1.14301e6i − 1.47226i −0.676842 0.736129i \(-0.736652\pi\)
0.676842 0.736129i \(-0.263348\pi\)
\(228\) 0 0
\(229\) 1.29731e6 1.63476 0.817380 0.576099i \(-0.195426\pi\)
0.817380 + 0.576099i \(0.195426\pi\)
\(230\) 0 0
\(231\) 291513. 0.359441
\(232\) 0 0
\(233\) 626232.i 0.755693i 0.925868 + 0.377846i \(0.123335\pi\)
−0.925868 + 0.377846i \(0.876665\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 127239.i 0.147146i
\(238\) 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.i − 1.01949i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 547134.i 0.570626i
\(248\) 0 0
\(249\) 898431. 0.918304
\(250\) 0 0
\(251\) 864085. 0.865710 0.432855 0.901464i \(-0.357506\pi\)
0.432855 + 0.901464i \(0.357506\pi\)
\(252\) 0 0
\(253\) 1.06390e6i 1.04496i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 1.00660e6i − 0.950660i −0.879808 0.475330i \(-0.842329\pi\)
0.879808 0.475330i \(-0.157671\pi\)
\(258\) 0 0
\(259\) 482395. 0.446841
\(260\) 0 0
\(261\) −5862.46 −0.00532695
\(262\) 0 0
\(263\) − 931373.i − 0.830299i −0.909753 0.415149i \(-0.863729\pi\)
0.909753 0.415149i \(-0.136271\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 2.21577e6i − 1.90216i
\(268\) 0 0
\(269\) 1.96236e6 1.65348 0.826738 0.562587i \(-0.190194\pi\)
0.826738 + 0.562587i \(0.190194\pi\)
\(270\) 0 0
\(271\) 1.12152e6 0.927649 0.463825 0.885927i \(-0.346477\pi\)
0.463825 + 0.885927i \(0.346477\pi\)
\(272\) 0 0
\(273\) 754948.i 0.613070i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 105917.i − 0.0829401i −0.999140 0.0414701i \(-0.986796\pi\)
0.999140 0.0414701i \(-0.0132041\pi\)
\(278\) 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.i − 0.705267i −0.935762 0.352633i \(-0.885286\pi\)
0.935762 0.352633i \(-0.114714\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 415801.i 0.297975i
\(288\) 0 0
\(289\) 1.40789e6 0.991573
\(290\) 0 0
\(291\) −2.09725e6 −1.45183
\(292\) 0 0
\(293\) − 381801.i − 0.259817i −0.991526 0.129909i \(-0.958532\pi\)
0.991526 0.129909i \(-0.0414684\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 871723.i − 0.573440i
\(298\) 0 0
\(299\) −2.75524e6 −1.78230
\(300\) 0 0
\(301\) −589373. −0.374951
\(302\) 0 0
\(303\) − 3.85708e6i − 2.41353i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.61048e6i 0.975238i 0.873057 + 0.487619i \(0.162134\pi\)
−0.873057 + 0.487619i \(0.837866\pi\)
\(308\) 0 0
\(309\) −1.58197e6 −0.942543
\(310\) 0 0
\(311\) −2.15652e6 −1.26431 −0.632153 0.774843i \(-0.717829\pi\)
−0.632153 + 0.774843i \(0.717829\pi\)
\(312\) 0 0
\(313\) − 364108.i − 0.210073i −0.994468 0.105036i \(-0.966504\pi\)
0.994468 0.105036i \(-0.0334959\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.03846e6i 0.580420i 0.956963 + 0.290210i \(0.0937252\pi\)
−0.956963 + 0.290210i \(0.906275\pi\)
\(318\) 0 0
\(319\) −18072.2 −0.00994336
\(320\) 0 0
\(321\) 2.68345e6 1.45355
\(322\) 0 0
\(323\) − 54247.8i − 0.0289318i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1.34118e6i 0.693612i
\(328\) 0 0
\(329\) 456890. 0.232714
\(330\) 0 0
\(331\) −2.82942e6 −1.41948 −0.709738 0.704466i \(-0.751186\pi\)
−0.709738 + 0.704466i \(0.751186\pi\)
\(332\) 0 0
\(333\) 1.90205e6i 0.939966i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 804843.i − 0.386043i −0.981194 0.193022i \(-0.938171\pi\)
0.981194 0.193022i \(-0.0618288\pi\)
\(338\) 0 0
\(339\) −2.72621e6 −1.28843
\(340\) 0 0
\(341\) 3.40787e6 1.58708
\(342\) 0 0
\(343\) 1.13506e6i 0.520936i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 759889.i − 0.338787i −0.985549 0.169393i \(-0.945819\pi\)
0.985549 0.169393i \(-0.0541809\pi\)
\(348\) 0 0
\(349\) −580673. −0.255193 −0.127596 0.991826i \(-0.540726\pi\)
−0.127596 + 0.991826i \(0.540726\pi\)
\(350\) 0 0
\(351\) 2.25755e6 0.978071
\(352\) 0 0
\(353\) − 1.64210e6i − 0.701393i −0.936489 0.350697i \(-0.885945\pi\)
0.936489 0.350697i \(-0.114055\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 74852.3i − 0.0310838i
\(358\) 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.i 0.158894i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 3.83418e6i 1.48596i 0.669314 + 0.742980i \(0.266588\pi\)
−0.669314 + 0.742980i \(0.733412\pi\)
\(368\) 0 0
\(369\) −1.63947e6 −0.626814
\(370\) 0 0
\(371\) 624467. 0.235545
\(372\) 0 0
\(373\) 3.19341e6i 1.18845i 0.804298 + 0.594227i \(0.202542\pi\)
−0.804298 + 0.594227i \(0.797458\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 46802.5i − 0.0169596i
\(378\) 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
\(383\) 1.56341e6i 0.544599i 0.962213 + 0.272299i \(0.0877841\pi\)
−0.962213 + 0.272299i \(0.912216\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 2.32386e6i − 0.788738i
\(388\) 0 0
\(389\) −4.98980e6 −1.67190 −0.835949 0.548808i \(-0.815082\pi\)
−0.835949 + 0.548808i \(0.815082\pi\)
\(390\) 0 0
\(391\) 273179. 0.0903661
\(392\) 0 0
\(393\) 2.31444e6i 0.755899i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 1.37245e6i 0.437038i 0.975833 + 0.218519i \(0.0701226\pi\)
−0.975833 + 0.218519i \(0.929877\pi\)
\(398\) 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
\(403\) 8.82557e6i 2.70695i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 5.86344e6i 1.75455i
\(408\) 0 0
\(409\) 6.03830e6 1.78487 0.892435 0.451176i \(-0.148995\pi\)
0.892435 + 0.451176i \(0.148995\pi\)
\(410\) 0 0
\(411\) −1.65483e6 −0.483223
\(412\) 0 0
\(413\) 1.65940e6i 0.478714i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 3.28657e6i − 0.925557i
\(418\) 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
\(423\) 1.80149e6i 0.489532i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 798504.i − 0.211937i
\(428\) 0 0
\(429\) −9.17628e6 −2.40726
\(430\) 0 0
\(431\) 5.56690e6 1.44351 0.721756 0.692148i \(-0.243335\pi\)
0.721756 + 0.692148i \(0.243335\pi\)
\(432\) 0 0
\(433\) − 2.22941e6i − 0.571441i −0.958313 0.285720i \(-0.907767\pi\)
0.958313 0.285720i \(-0.0922328\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.23851e6i 0.310239i
\(438\) 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
\(443\) − 4.53358e6i − 1.09757i −0.835964 0.548785i \(-0.815091\pi\)
0.835964 0.548785i \(-0.184909\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 1.32039e6i − 0.312561i
\(448\) 0 0
\(449\) 3.39468e6 0.794662 0.397331 0.917675i \(-0.369936\pi\)
0.397331 + 0.917675i \(0.369936\pi\)
\(450\) 0 0
\(451\) −5.05400e6 −1.17002
\(452\) 0 0
\(453\) − 1.28122e6i − 0.293345i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.34100e6i 1.86822i 0.356988 + 0.934109i \(0.383804\pi\)
−0.356988 + 0.934109i \(0.616196\pi\)
\(458\) 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
\(463\) − 4.05943e6i − 0.880061i −0.897983 0.440030i \(-0.854968\pi\)
0.897983 0.440030i \(-0.145032\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 7.98706e6i − 1.69471i −0.531029 0.847353i \(-0.678195\pi\)
0.531029 0.847353i \(-0.321805\pi\)
\(468\) 0 0
\(469\) 1.38155e6 0.290024
\(470\) 0 0
\(471\) 5.81596e6 1.20801
\(472\) 0 0
\(473\) − 7.16375e6i − 1.47227i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 2.46223e6i 0.495488i
\(478\) 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
\(483\) 1.70892e6i 0.333315i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 8.76969e6i 1.67557i 0.546002 + 0.837784i \(0.316149\pi\)
−0.546002 + 0.837784i \(0.683851\pi\)
\(488\) 0 0
\(489\) −9.25209e6 −1.74972
\(490\) 0 0
\(491\) 7.83080e6 1.46589 0.732946 0.680286i \(-0.238145\pi\)
0.732946 + 0.680286i \(0.238145\pi\)
\(492\) 0 0
\(493\) 4640.42i 0 0.000859885i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 56649.2i − 0.0102873i
\(498\) 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
\(503\) 2.07773e6i 0.366158i 0.983098 + 0.183079i \(0.0586064\pi\)
−0.983098 + 0.183079i \(0.941394\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 1.65152e7i − 2.85340i
\(508\) 0 0
\(509\) 6.25093e6 1.06942 0.534712 0.845034i \(-0.320420\pi\)
0.534712 + 0.845034i \(0.320420\pi\)
\(510\) 0 0
\(511\) 1.86784e6 0.316437
\(512\) 0 0
\(513\) − 1.01480e6i − 0.170249i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 5.55344e6i 0.913767i
\(518\) 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
\(523\) 2.10298e6i 0.336187i 0.985771 + 0.168093i \(0.0537610\pi\)
−0.985771 + 0.168093i \(0.946239\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 875046.i − 0.137247i
\(528\) 0 0
\(529\) 199490. 0.0309944
\(530\) 0 0
\(531\) −6.54291e6 −1.00701
\(532\) 0 0
\(533\) − 1.30886e7i − 1.99561i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 1.24041e7i − 1.85623i
\(538\) 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
\(543\) − 1.06974e7i − 1.55697i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 4.31872e6i 0.617145i 0.951201 + 0.308572i \(0.0998512\pi\)
−0.951201 + 0.308572i \(0.900149\pi\)
\(548\) 0 0
\(549\) 3.14845e6 0.445827
\(550\) 0 0
\(551\) −21038.3 −0.00295210
\(552\) 0 0
\(553\) 228409.i 0.0317615i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 4.26837e6i − 0.582940i −0.956580 0.291470i \(-0.905856\pi\)
0.956580 0.291470i \(-0.0941444\pi\)
\(558\) 0 0
\(559\) 1.85524e7 2.51114
\(560\) 0 0
\(561\) 909819. 0.122053
\(562\) 0 0
\(563\) 1.26350e7i 1.67998i 0.542605 + 0.839988i \(0.317438\pi\)
−0.542605 + 0.839988i \(0.682562\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 2.57720e6i − 0.336659i
\(568\) 0 0
\(569\) 8.67409e6 1.12316 0.561582 0.827421i \(-0.310193\pi\)
0.561582 + 0.827421i \(0.310193\pi\)
\(570\) 0 0
\(571\) 3.28781e6 0.422004 0.211002 0.977486i \(-0.432327\pi\)
0.211002 + 0.977486i \(0.432327\pi\)
\(572\) 0 0
\(573\) − 1.57380e6i − 0.200246i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 117668.i − 0.0147135i −0.999973 0.00735677i \(-0.997658\pi\)
0.999973 0.00735677i \(-0.00234175\pi\)
\(578\) 0 0
\(579\) −518352. −0.0642582
\(580\) 0 0
\(581\) 1.61280e6 0.198216
\(582\) 0 0
\(583\) 7.59031e6i 0.924885i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 1.10151e7i − 1.31945i −0.751509 0.659723i \(-0.770674\pi\)
0.751509 0.659723i \(-0.229326\pi\)
\(588\) 0 0
\(589\) 3.96719e6 0.471189
\(590\) 0 0
\(591\) 1.37387e7 1.61799
\(592\) 0 0
\(593\) 5.33879e6i 0.623456i 0.950171 + 0.311728i \(0.100908\pi\)
−0.950171 + 0.311728i \(0.899092\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 1.43697e6i − 0.165011i
\(598\) 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
\(603\) 5.44735e6i 0.610087i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 591991.i 0.0652143i 0.999468 + 0.0326072i \(0.0103810\pi\)
−0.999468 + 0.0326072i \(0.989619\pi\)
\(608\) 0 0
\(609\) −29029.1 −0.00317168
\(610\) 0 0
\(611\) −1.43821e7 −1.55854
\(612\) 0 0
\(613\) 4.01173e6i 0.431201i 0.976482 + 0.215601i \(0.0691710\pi\)
−0.976482 + 0.215601i \(0.930829\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8.24093e6i 0.871491i 0.900070 + 0.435746i \(0.143515\pi\)
−0.900070 + 0.435746i \(0.856485\pi\)
\(618\) 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
\(623\) − 3.97758e6i − 0.410581i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 4.12484e6i 0.419024i
\(628\) 0 0
\(629\) 1.50557e6 0.151731
\(630\) 0 0
\(631\) −1.82360e7 −1.82329 −0.911646 0.410977i \(-0.865187\pi\)
−0.911646 + 0.410977i \(0.865187\pi\)
\(632\) 0 0
\(633\) − 1.96819e7i − 1.95235i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 1.71872e7i − 1.67825i
\(638\) 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
\(643\) − 2.47762e6i − 0.236323i −0.992994 0.118162i \(-0.962300\pi\)
0.992994 0.118162i \(-0.0377001\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4.79533e6i 0.450358i 0.974317 + 0.225179i \(0.0722967\pi\)
−0.974317 + 0.225179i \(0.927703\pi\)
\(648\) 0 0
\(649\) −2.01698e7 −1.87970
\(650\) 0 0
\(651\) 5.47402e6 0.506237
\(652\) 0 0
\(653\) − 1.67125e7i − 1.53376i −0.641789 0.766881i \(-0.721807\pi\)
0.641789 0.766881i \(-0.278193\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 7.36478e6i 0.665650i
\(658\) 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
\(663\) 2.35621e6i 0.208176i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 105944.i − 0.00922063i
\(668\) 0 0
\(669\) 2.75423e6 0.237922
\(670\) 0 0
\(671\) 9.70571e6 0.832187
\(672\) 0 0
\(673\) 1.39915e7i 1.19076i 0.803443 + 0.595381i \(0.202999\pi\)
−0.803443 + 0.595381i \(0.797001\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3.13032e6i 0.262493i 0.991350 + 0.131246i \(0.0418979\pi\)
−0.991350 + 0.131246i \(0.958102\pi\)
\(678\) 0 0
\(679\) −3.76482e6 −0.313379
\(680\) 0 0
\(681\) 2.23162e7 1.84397
\(682\) 0 0
\(683\) − 8.94034e6i − 0.733335i −0.930352 0.366667i \(-0.880499\pi\)
0.930352 0.366667i \(-0.119501\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 2.53288e7i 2.04750i
\(688\) 0 0
\(689\) −1.96571e7 −1.57750
\(690\) 0 0
\(691\) −2.38378e7 −1.89920 −0.949599 0.313466i \(-0.898510\pi\)
−0.949599 + 0.313466i \(0.898510\pi\)
\(692\) 0 0
\(693\) 2.06335e6i 0.163207i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.29772e6i 0.101181i
\(698\) 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
\(703\) 6.82578e6i 0.520912i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 6.92394e6i − 0.520961i
\(708\) 0 0
\(709\) 3.05866e6 0.228515 0.114258 0.993451i \(-0.463551\pi\)
0.114258 + 0.993451i \(0.463551\pi\)
\(710\) 0 0
\(711\) −900603. −0.0668128
\(712\) 0 0
\(713\) 1.99778e7i 1.47172i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 5.27701e6i − 0.383345i
\(718\) 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
\(723\) − 1.35564e7i − 0.964492i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 2.73589e7i 1.91983i 0.280287 + 0.959916i \(0.409570\pi\)
−0.280287 + 0.959916i \(0.590430\pi\)
\(728\) 0 0
\(729\) 453482. 0.0316039
\(730\) 0 0
\(731\) −1.83945e6 −0.127319
\(732\) 0 0
\(733\) 1.54693e7i 1.06344i 0.846921 + 0.531718i \(0.178453\pi\)
−0.846921 + 0.531718i \(0.821547\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.67925e7i 1.13880i
\(738\) 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
\(743\) − 1.55089e7i − 1.03064i −0.856997 0.515322i \(-0.827672\pi\)
0.856997 0.515322i \(-0.172328\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 6.35916e6i 0.416963i
\(748\) 0 0
\(749\) 4.81713e6 0.313750
\(750\) 0 0
\(751\) 2.20221e7 1.42482 0.712409 0.701765i \(-0.247604\pi\)
0.712409 + 0.701765i \(0.247604\pi\)
\(752\) 0 0
\(753\) 1.68706e7i 1.08428i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.75653e6i 0.111408i 0.998447 + 0.0557040i \(0.0177403\pi\)
−0.998447 + 0.0557040i \(0.982260\pi\)
\(758\) 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
\(763\) 2.40758e6i 0.149716i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 5.22349e7i − 3.20606i
\(768\) 0 0
\(769\) 1.12131e7 0.683769 0.341884 0.939742i \(-0.388935\pi\)
0.341884 + 0.939742i \(0.388935\pi\)
\(770\) 0 0
\(771\) 1.96531e7 1.19068
\(772\) 0 0
\(773\) 1.43367e7i 0.862980i 0.902118 + 0.431490i \(0.142012\pi\)
−0.902118 + 0.431490i \(0.857988\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 9.41836e6i 0.559658i
\(778\) 0 0
\(779\) −5.88348e6 −0.347369
\(780\) 0 0
\(781\) 688563. 0.0403939
\(782\) 0 0
\(783\) 86806.8i 0.00505999i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 2.00549e6i − 0.115421i −0.998333 0.0577103i \(-0.981620\pi\)
0.998333 0.0577103i \(-0.0183800\pi\)
\(788\) 0 0
\(789\) 1.81843e7 1.03993
\(790\) 0 0
\(791\) −4.89389e6 −0.278108
\(792\) 0 0
\(793\) 2.51354e7i 1.41940i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 448115.i − 0.0249887i −0.999922 0.0124944i \(-0.996023\pi\)
0.999922 0.0124944i \(-0.00397718\pi\)
\(798\) 0 0
\(799\) 1.42597e6 0.0790210
\(800\) 0 0
\(801\) 1.56834e7 0.863690
\(802\) 0 0
\(803\) 2.27033e7i 1.24251i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 3.83134e7i 2.07094i
\(808\) 0 0
\(809\) −1.28425e7 −0.689889 −0.344944 0.938623i \(-0.612102\pi\)
−0.344944 + 0.938623i \(0.612102\pi\)
\(810\) 0 0
\(811\) 1.78700e7 0.954054 0.477027 0.878889i \(-0.341714\pi\)
0.477027 + 0.878889i \(0.341714\pi\)
\(812\) 0 0
\(813\) 2.18968e7i 1.16186i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 8.33950e6i − 0.437104i
\(818\) 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
\(823\) 3.05819e6i 0.157385i 0.996899 + 0.0786927i \(0.0250746\pi\)
−0.996899 + 0.0786927i \(0.974925\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 1.23961e7i − 0.630261i −0.949048 0.315131i \(-0.897952\pi\)
0.949048 0.315131i \(-0.102048\pi\)
\(828\) 0 0
\(829\) 399873. 0.0202086 0.0101043 0.999949i \(-0.496784\pi\)
0.0101043 + 0.999949i \(0.496784\pi\)
\(830\) 0 0
\(831\) 2.06793e6 0.103881
\(832\) 0 0
\(833\) 1.70409e6i 0.0850905i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 1.63692e7i − 0.807632i
\(838\) 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
\(843\) − 2.05543e7i − 0.996172i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 716092.i 0.0342973i
\(848\) 0 0
\(849\) 1.85521e7 0.883330
\(850\) 0 0
\(851\) −3.43730e7 −1.62702
\(852\) 0 0
\(853\) − 2.14298e7i − 1.00843i −0.863579 0.504214i \(-0.831782\pi\)
0.863579 0.504214i \(-0.168218\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3.49123e6i 0.162377i 0.996699 + 0.0811887i \(0.0258717\pi\)
−0.996699 + 0.0811887i \(0.974128\pi\)
\(858\) 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
\(863\) − 3.28678e7i − 1.50226i −0.660157 0.751128i \(-0.729510\pi\)
0.660157 0.751128i \(-0.270490\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 2.74879e7i 1.24192i
\(868\) 0 0
\(869\) −2.77628e6 −0.124714
\(870\) 0 0
\(871\) −4.34885e7 −1.94236
\(872\) 0 0
\(873\) − 1.48444e7i − 0.659217i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 2.92945e7i 1.28614i 0.765808 + 0.643069i \(0.222339\pi\)
−0.765808 + 0.643069i \(0.777661\pi\)
\(878\) 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
\(883\) − 2.79477e7i − 1.20627i −0.797639 0.603135i \(-0.793918\pi\)
0.797639 0.603135i \(-0.206082\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 2.89567e7i − 1.23578i −0.786266 0.617888i \(-0.787989\pi\)
0.786266 0.617888i \(-0.212011\pi\)
\(888\) 0 0
\(889\) 2.28621e6 0.0970199
\(890\) 0 0
\(891\) 3.13254e7 1.32191
\(892\) 0 0
\(893\) 6.46490e6i 0.271289i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 5.37938e7i − 2.23229i
\(898\) 0 0
\(899\) −339358. −0.0140042
\(900\) 0 0
\(901\) 1.94898e6 0.0799825
\(902\) 0 0
\(903\) − 1.15070e7i − 0.469617i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 1.18508e6i − 0.0478333i −0.999714 0.0239166i \(-0.992386\pi\)
0.999714 0.0239166i \(-0.00761363\pi\)
\(908\) 0 0
\(909\) 2.73007e7 1.09588
\(910\) 0 0
\(911\) −1.14346e7 −0.456485 −0.228242 0.973604i \(-0.573298\pi\)
−0.228242 + 0.973604i \(0.573298\pi\)
\(912\) 0 0
\(913\) 1.96033e7i 0.778310i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4.15471e6i 0.163161i
\(918\) 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
\(923\) 1.78321e6i 0.0688968i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 1.11973e7i − 0.427969i
\(928\) 0 0
\(929\) 2.02580e7 0.770117 0.385058 0.922892i \(-0.374181\pi\)
0.385058 + 0.922892i \(0.374181\pi\)
\(930\) 0 0
\(931\) −7.72584e6 −0.292127
\(932\) 0 0
\(933\) − 4.21043e7i − 1.58351i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 1.96237e7i − 0.730182i −0.930972 0.365091i \(-0.881038\pi\)
0.930972 0.365091i \(-0.118962\pi\)
\(938\) 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
\(943\) − 2.96279e7i − 1.08498i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 3.93605e7i 1.42622i 0.701054 + 0.713108i \(0.252713\pi\)
−0.701054 + 0.713108i \(0.747287\pi\)
\(948\) 0 0
\(949\) −5.87962e7 −2.11926
\(950\) 0 0
\(951\) −2.02751e7 −0.726963
\(952\) 0 0
\(953\) − 5.09953e7i − 1.81886i −0.415861 0.909428i \(-0.636520\pi\)
0.415861 0.909428i \(-0.363480\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 352844.i − 0.0124538i
\(958\) 0 0
\(959\) −2.97062e6 −0.104304
\(960\) 0 0
\(961\) 3.53638e7 1.23524
\(962\) 0 0
\(963\) 1.89936e7i 0.659997i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 5.17740e7i − 1.78051i −0.455458 0.890257i \(-0.650524\pi\)
0.455458 0.890257i \(-0.349476\pi\)
\(968\) 0 0
\(969\) 1.05914e6 0.0362364
\(970\) 0 0
\(971\) −3.69158e7 −1.25651 −0.628253 0.778009i \(-0.716230\pi\)
−0.628253 + 0.778009i \(0.716230\pi\)
\(972\) 0 0
\(973\) − 5.89981e6i − 0.199782i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.14468e7i 1.05400i 0.849866 + 0.526999i \(0.176683\pi\)
−0.849866 + 0.526999i \(0.823317\pi\)
\(978\) 0 0
\(979\) 4.83470e7 1.61218
\(980\) 0 0
\(981\) −9.49293e6 −0.314940
\(982\) 0 0
\(983\) 5.15823e7i 1.70262i 0.524667 + 0.851308i \(0.324190\pi\)
−0.524667 + 0.851308i \(0.675810\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 8.92041e6i 0.291469i
\(988\) 0 0
\(989\) 4.19958e7 1.36526
\(990\) 0 0
\(991\) −1.30562e7 −0.422312 −0.211156 0.977452i \(-0.567723\pi\)
−0.211156 + 0.977452i \(0.567723\pi\)
\(992\) 0 0
\(993\) − 5.52421e7i − 1.77786i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 4.97672e7i − 1.58564i −0.609454 0.792821i \(-0.708611\pi\)
0.609454 0.792821i \(-0.291389\pi\)
\(998\) 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 200.6.c.f.49.4 4
4.3 odd 2 400.6.c.o.49.1 4
5.2 odd 4 200.6.a.f.1.2 yes 2
5.3 odd 4 200.6.a.e.1.1 2
5.4 even 2 inner 200.6.c.f.49.1 4
20.3 even 4 400.6.a.u.1.2 2
20.7 even 4 400.6.a.r.1.1 2
20.19 odd 2 400.6.c.o.49.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.6.a.e.1.1 2 5.3 odd 4
200.6.a.f.1.2 yes 2 5.2 odd 4
200.6.c.f.49.1 4 5.4 even 2 inner
200.6.c.f.49.4 4 1.1 even 1 trivial
400.6.a.r.1.1 2 20.7 even 4
400.6.a.u.1.2 2 20.3 even 4
400.6.c.o.49.1 4 4.3 odd 2
400.6.c.o.49.4 4 20.19 odd 2