Properties

Label 400.6.c.o.49.1
Level $400$
Weight $6$
Character 400.49
Analytic conductor $64.154$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 400.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(64.1535279252\)
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: no (minimal twist has level 200)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(7.26209i\) of defining polynomial
Character \(\chi\) \(=\) 400.49
Dual form 400.6.c.o.49.4

$q$-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 \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(-0.784594\pi\)
0.779632 0.626238i \(-0.215406\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 35.0483i − 0.270348i −0.990822 0.135174i \(-0.956841\pi\)
0.990822 0.135174i \(-0.0431593\pi\)
\(8\) 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
\(13\) − 1103.26i − 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 \(-0.449631\pi\)
0.157581 + 0.987506i \(0.449631\pi\)
\(20\) 0 0
\(21\) −684.290 −0.338604
\(22\) 0 0
\(23\) 2497.37i 0.984381i 0.870488 + 0.492190i \(0.163804\pi\)
−0.870488 + 0.492190i \(0.836196\pi\)
\(24\) 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 \(-0.231237\pi\)
0.747535 + 0.664222i \(0.231237\pi\)
\(32\) 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 \(-0.756085\pi\)
0.720494 0.693461i \(-0.243915\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 13036.0i 0.860795i 0.902639 + 0.430397i \(0.141627\pi\)
−0.902639 + 0.430397i \(0.858373\pi\)
\(48\) 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 \(-0.846094\pi\)
−0.885368 + 0.464891i \(0.846094\pi\)
\(60\) 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 \(0.180213\pi\)
−0.843970 + 0.536390i \(0.819787\pi\)
\(68\) 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
\(73\) − 53293.3i − 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 \(-0.518709\pi\)
−0.0587420 + 0.998273i \(0.518709\pi\)
\(80\) 0 0
\(81\) −73532.6 −1.24528
\(82\) 0 0
\(83\) 46016.4i 0.733191i 0.930380 + 0.366595i \(0.119477\pi\)
−0.930380 + 0.366595i \(0.880523\pi\)
\(84\) 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 \(-0.877206\pi\)
0.926509 0.376272i \(-0.122794\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 137442.i 1.16054i 0.814423 + 0.580271i \(0.197054\pi\)
−0.814423 + 0.580271i \(0.802946\pi\)
\(108\) 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 \(0.0574271\pi\)
−0.983770 + 0.179436i \(0.942573\pi\)
\(128\) 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
\(133\) − 17381.4i − 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 \(-0.379532\pi\)
0.369491 + 0.929234i \(0.379532\pi\)
\(140\) 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 \(-0.462638\pi\)
0.117106 + 0.993119i \(0.462638\pi\)
\(152\) 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 \(-0.753850\pi\)
0.715607 0.698503i \(-0.246150\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 267673.i − 0.742701i −0.928493 0.371351i \(-0.878895\pi\)
0.928493 0.371351i \(-0.121105\pi\)
\(168\) 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 \(-0.234341\pi\)
0.741022 + 0.671481i \(0.234341\pi\)
\(180\) 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 \(-0.474527\pi\)
0.0799400 + 0.996800i \(0.474527\pi\)
\(192\) 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 \(-0.479017\pi\)
0.0658737 + 0.997828i \(0.479017\pi\)
\(200\) 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 \(-0.215526\pi\)
0.779395 + 0.626533i \(0.215526\pi\)
\(212\) 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 \(0.0302789\pi\)
−0.995479 + 0.0949805i \(0.969721\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.14301e6i 1.47226i 0.676842 + 0.736129i \(0.263348\pi\)
−0.676842 + 0.736129i \(0.736652\pi\)
\(228\) 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 \(-0.451095\pi\)
0.153035 + 0.988221i \(0.451095\pi\)
\(240\) 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 \(-0.642494\pi\)
−0.432855 + 0.901464i \(0.642494\pi\)
\(252\) 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 \(0.136271\pi\)
−0.909753 + 0.415149i \(0.863729\pi\)
\(264\) 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 \(-0.653523\pi\)
−0.463825 + 0.885927i \(0.653523\pi\)
\(272\) 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 \(0.114714\pi\)
−0.935762 + 0.352633i \(0.885286\pi\)
\(284\) 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 \(-0.837866\pi\)
0.873057 0.487619i \(-0.162134\pi\)
\(308\) 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.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 \(-0.248814\pi\)
0.709738 + 0.704466i \(0.248814\pi\)
\(332\) 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 \(0.0541809\pi\)
−0.985549 + 0.169393i \(0.945819\pi\)
\(348\) 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 \(-0.700898\pi\)
−0.590065 + 0.807356i \(0.700898\pi\)
\(360\) 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 \(-0.733412\pi\)
0.669314 0.742980i \(-0.266588\pi\)
\(368\) 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 \(-0.286708\pi\)
0.621046 + 0.783774i \(0.286708\pi\)
\(380\) 0 0
\(381\) 1.27356e6 0.449478
\(382\) 0 0
\(383\) − 1.56341e6i − 0.544599i −0.962213 0.272299i \(-0.912216\pi\)
0.962213 0.272299i \(-0.0877841\pi\)
\(384\) 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 \(-0.465816\pi\)
0.107184 + 0.994239i \(0.465816\pi\)
\(420\) 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 \(-0.756665\pi\)
−0.721756 + 0.692148i \(0.756665\pi\)
\(432\) 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 \(-0.478579\pi\)
0.0672446 + 0.997737i \(0.478579\pi\)
\(440\) 0 0
\(441\) −2.15286e6 −0.527132
\(442\) 0 0
\(443\) 4.53358e6i 1.09757i 0.835964 + 0.548785i \(0.184909\pi\)
−0.835964 + 0.548785i \(0.815091\pi\)
\(444\) 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 \(0.145032\pi\)
−0.897983 + 0.440030i \(0.854968\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 7.98706e6i 1.69471i 0.531029 + 0.847353i \(0.321805\pi\)
−0.531029 + 0.847353i \(0.678195\pi\)
\(468\) 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 \(-0.323958\pi\)
0.525288 + 0.850924i \(0.323958\pi\)
\(480\) 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 \(-0.683851\pi\)
0.546002 0.837784i \(-0.316149\pi\)
\(488\) 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
\(493\) 4640.42i 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 \(-0.700240\pi\)
−0.588394 + 0.808574i \(0.700240\pi\)
\(500\) 0 0
\(501\) −5.22610e6 −0.930216
\(502\) 0 0
\(503\) − 2.07773e6i − 0.366158i −0.983098 0.183079i \(-0.941394\pi\)
0.983098 0.183079i \(-0.0586064\pi\)
\(504\) 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 \(-0.946239\pi\)
0.985771 0.168093i \(-0.0537610\pi\)
\(524\) 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 \(-0.900149\pi\)
0.951201 0.308572i \(-0.0998512\pi\)
\(548\) 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 \(-0.682562\pi\)
0.542605 0.839988i \(-0.317438\pi\)
\(564\) 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 \(-0.567673\pi\)
−0.211002 + 0.977486i \(0.567673\pi\)
\(572\) 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 \(0.229326\pi\)
−0.751509 + 0.659723i \(0.770674\pi\)
\(588\) 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 \(-0.634015\pi\)
−0.408692 + 0.912672i \(0.634015\pi\)
\(600\) 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 \(-0.989619\pi\)
0.999468 0.0326072i \(-0.0103810\pi\)
\(608\) 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 \(-0.467054\pi\)
0.103318 + 0.994648i \(0.467054\pi\)
\(620\) 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 \(-0.134813\pi\)
0.911646 + 0.410977i \(0.134813\pi\)
\(632\) 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 \(0.0377001\pi\)
−0.992994 + 0.118162i \(0.962300\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 4.79533e6i − 0.450358i −0.974317 0.225179i \(-0.927703\pi\)
0.974317 0.225179i \(-0.0722967\pi\)
\(648\) 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 \(-0.452949\pi\)
0.147277 + 0.989095i \(0.452949\pi\)
\(660\) 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 \(0.119501\pi\)
−0.930352 + 0.366667i \(0.880499\pi\)
\(684\) 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 \(-0.101490\pi\)
0.949599 + 0.313466i \(0.101490\pi\)
\(692\) 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 \(-0.215400\pi\)
0.779643 + 0.626224i \(0.215400\pi\)
\(720\) 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 \(-0.590430\pi\)
0.280287 0.959916i \(-0.409570\pi\)
\(728\) 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 \(-0.440005\pi\)
0.187365 + 0.982290i \(0.440005\pi\)
\(740\) 0 0
\(741\) −1.06823e7 −0.714695
\(742\) 0 0
\(743\) 1.55089e7i 1.03064i 0.856997 + 0.515322i \(0.172328\pi\)
−0.856997 + 0.515322i \(0.827672\pi\)
\(744\) 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 \(-0.752396\pi\)
−0.712409 + 0.701765i \(0.752396\pi\)
\(752\) 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 \(0.0183800\pi\)
−0.998333 + 0.0577103i \(0.981620\pi\)
\(788\) 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 \(-0.658286\pi\)
−0.477027 + 0.878889i \(0.658286\pi\)
\(812\) 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 \(-0.974925\pi\)
0.996899 0.0786927i \(-0.0250746\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.23961e7i 0.630261i 0.949048 + 0.315131i \(0.102048\pi\)
−0.949048 + 0.315131i \(0.897952\pi\)
\(828\) 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 \(-0.516275\pi\)
−0.0511075 + 0.998693i \(0.516275\pi\)
\(840\) 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 \(-0.504181\pi\)
−0.0131331 + 0.999914i \(0.504181\pi\)
\(860\) 0 0
\(861\) −8.11816e6 −0.373207
\(862\) 0 0
\(863\) 3.28678e7i 1.50226i 0.660157 + 0.751128i \(0.270490\pi\)
−0.660157 + 0.751128i \(0.729510\pi\)
\(864\) 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 \(0.206082\pi\)
−0.797639 + 0.603135i \(0.793918\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.89567e7i 1.23578i 0.786266 + 0.617888i \(0.212011\pi\)
−0.786266 + 0.617888i \(0.787989\pi\)
\(888\) 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 \(0.00761363\pi\)
−0.999714 + 0.0239166i \(0.992386\pi\)
\(908\) 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
\(913\) 1.96033e7i 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 \(-0.593920\pi\)
−0.290795 + 0.956785i \(0.593920\pi\)
\(920\) 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 \(-0.747287\pi\)
0.701054 0.713108i \(-0.252713\pi\)
\(948\) 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 \(0.349476\pi\)
−0.455458 + 0.890257i \(0.650524\pi\)
\(968\) 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
\(973\) − 5.89981e6i − 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 \(-0.675810\pi\)
0.524667 0.851308i \(-0.324190\pi\)
\(984\) 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 \(-0.432277\pi\)
0.211156 + 0.977452i \(0.432277\pi\)
\(992\) 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 400.6.c.o.49.1 4
4.3 odd 2 200.6.c.f.49.4 4
5.2 odd 4 400.6.a.r.1.1 2
5.3 odd 4 400.6.a.u.1.2 2
5.4 even 2 inner 400.6.c.o.49.4 4
20.3 even 4 200.6.a.e.1.1 2
20.7 even 4 200.6.a.f.1.2 yes 2
20.19 odd 2 200.6.c.f.49.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.6.a.e.1.1 2 20.3 even 4
200.6.a.f.1.2 yes 2 20.7 even 4
200.6.c.f.49.1 4 20.19 odd 2
200.6.c.f.49.4 4 4.3 odd 2
400.6.a.r.1.1 2 5.2 odd 4
400.6.a.u.1.2 2 5.3 odd 4
400.6.c.o.49.1 4 1.1 even 1 trivial
400.6.c.o.49.4 4 5.4 even 2 inner