Properties

Label 200.6.c.e.49.3
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: \(\Q(i, \sqrt{129})\)
Defining polynomial: \(x^{4} + 65 x^{2} + 1024\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.3
Root \(5.17891i\) of defining polynomial
Character \(\chi\) \(=\) 200.49
Dual form 200.6.c.e.49.2

$q$-expansion

\(f(q)\) \(=\) \(q+16.7156i q^{3} -94.1469i q^{7} -36.4124 q^{9} +O(q^{10})\) \(q+16.7156i q^{3} -94.1469i q^{7} -36.4124 q^{9} +143.706 q^{11} +421.412i q^{13} -1982.11i q^{17} +1317.76 q^{19} +1573.73 q^{21} -4020.02i q^{23} +3453.24i q^{27} +6417.28 q^{29} -2350.64 q^{31} +2402.14i q^{33} +7876.58i q^{37} -7044.18 q^{39} +15081.6 q^{41} +1141.40i q^{43} +21557.3i q^{47} +7943.36 q^{49} +33132.3 q^{51} +9560.44i q^{53} +22027.2i q^{57} -42740.7 q^{59} +32132.1 q^{61} +3428.11i q^{63} +30371.4i q^{67} +67197.2 q^{69} +36006.7 q^{71} -63438.0i q^{73} -13529.5i q^{77} +89922.8 q^{79} -66571.4 q^{81} +38211.2i q^{83} +107269. i q^{87} -5745.69 q^{89} +39674.7 q^{91} -39292.5i q^{93} -178780. i q^{97} -5232.69 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 1236q^{9} + O(q^{10}) \) \( 4q - 1236q^{9} + 1120q^{11} + 2000q^{19} + 5568q^{21} - 2680q^{29} - 4496q^{31} + 41424q^{39} + 46152q^{41} + 45948q^{49} + 171600q^{51} - 125168q^{59} + 28216q^{61} + 224448q^{69} + 94416q^{71} + 131808q^{79} + 142596q^{81} + 110040q^{89} - 2128q^{91} - 494688q^{99} + O(q^{100}) \)

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
\(3\) 16.7156i 1.07231i 0.844120 + 0.536154i \(0.180123\pi\)
−0.844120 + 0.536154i \(0.819877\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 94.1469i − 0.726208i −0.931749 0.363104i \(-0.881717\pi\)
0.931749 0.363104i \(-0.118283\pi\)
\(8\) 0 0
\(9\) −36.4124 −0.149845
\(10\) 0 0
\(11\) 143.706 0.358091 0.179046 0.983841i \(-0.442699\pi\)
0.179046 + 0.983841i \(0.442699\pi\)
\(12\) 0 0
\(13\) 421.412i 0.691590i 0.938310 + 0.345795i \(0.112391\pi\)
−0.938310 + 0.345795i \(0.887609\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 1982.11i − 1.66344i −0.555198 0.831718i \(-0.687358\pi\)
0.555198 0.831718i \(-0.312642\pi\)
\(18\) 0 0
\(19\) 1317.76 0.837439 0.418720 0.908116i \(-0.362479\pi\)
0.418720 + 0.908116i \(0.362479\pi\)
\(20\) 0 0
\(21\) 1573.73 0.778719
\(22\) 0 0
\(23\) − 4020.02i − 1.58456i −0.610157 0.792280i \(-0.708894\pi\)
0.610157 0.792280i \(-0.291106\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3453.24i 0.911628i
\(28\) 0 0
\(29\) 6417.28 1.41695 0.708477 0.705734i \(-0.249383\pi\)
0.708477 + 0.705734i \(0.249383\pi\)
\(30\) 0 0
\(31\) −2350.64 −0.439322 −0.219661 0.975576i \(-0.570495\pi\)
−0.219661 + 0.975576i \(0.570495\pi\)
\(32\) 0 0
\(33\) 2402.14i 0.383984i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7876.58i 0.945874i 0.881096 + 0.472937i \(0.156806\pi\)
−0.881096 + 0.472937i \(0.843194\pi\)
\(38\) 0 0
\(39\) −7044.18 −0.741598
\(40\) 0 0
\(41\) 15081.6 1.40116 0.700582 0.713572i \(-0.252924\pi\)
0.700582 + 0.713572i \(0.252924\pi\)
\(42\) 0 0
\(43\) 1141.40i 0.0941384i 0.998892 + 0.0470692i \(0.0149881\pi\)
−0.998892 + 0.0470692i \(0.985012\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 21557.3i 1.42348i 0.702445 + 0.711738i \(0.252092\pi\)
−0.702445 + 0.711738i \(0.747908\pi\)
\(48\) 0 0
\(49\) 7943.36 0.472622
\(50\) 0 0
\(51\) 33132.3 1.78372
\(52\) 0 0
\(53\) 9560.44i 0.467507i 0.972296 + 0.233754i \(0.0751009\pi\)
−0.972296 + 0.233754i \(0.924899\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 22027.2i 0.897993i
\(58\) 0 0
\(59\) −42740.7 −1.59850 −0.799248 0.601002i \(-0.794768\pi\)
−0.799248 + 0.601002i \(0.794768\pi\)
\(60\) 0 0
\(61\) 32132.1 1.10564 0.552820 0.833301i \(-0.313552\pi\)
0.552820 + 0.833301i \(0.313552\pi\)
\(62\) 0 0
\(63\) 3428.11i 0.108819i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 30371.4i 0.826567i 0.910602 + 0.413283i \(0.135618\pi\)
−0.910602 + 0.413283i \(0.864382\pi\)
\(68\) 0 0
\(69\) 67197.2 1.69914
\(70\) 0 0
\(71\) 36006.7 0.847692 0.423846 0.905734i \(-0.360680\pi\)
0.423846 + 0.905734i \(0.360680\pi\)
\(72\) 0 0
\(73\) − 63438.0i − 1.39329i −0.717414 0.696647i \(-0.754674\pi\)
0.717414 0.696647i \(-0.245326\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 13529.5i − 0.260049i
\(78\) 0 0
\(79\) 89922.8 1.62107 0.810536 0.585689i \(-0.199176\pi\)
0.810536 + 0.585689i \(0.199176\pi\)
\(80\) 0 0
\(81\) −66571.4 −1.12739
\(82\) 0 0
\(83\) 38211.2i 0.608829i 0.952540 + 0.304414i \(0.0984608\pi\)
−0.952540 + 0.304414i \(0.901539\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 107269.i 1.51941i
\(88\) 0 0
\(89\) −5745.69 −0.0768895 −0.0384447 0.999261i \(-0.512240\pi\)
−0.0384447 + 0.999261i \(0.512240\pi\)
\(90\) 0 0
\(91\) 39674.7 0.502238
\(92\) 0 0
\(93\) − 39292.5i − 0.471088i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 178780.i − 1.92926i −0.263613 0.964629i \(-0.584914\pi\)
0.263613 0.964629i \(-0.415086\pi\)
\(98\) 0 0
\(99\) −5232.69 −0.0536583
\(100\) 0 0
\(101\) 152223. 1.48483 0.742417 0.669938i \(-0.233679\pi\)
0.742417 + 0.669938i \(0.233679\pi\)
\(102\) 0 0
\(103\) − 35830.1i − 0.332778i −0.986060 0.166389i \(-0.946789\pi\)
0.986060 0.166389i \(-0.0532108\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 70030.2i − 0.591324i −0.955293 0.295662i \(-0.904460\pi\)
0.955293 0.295662i \(-0.0955403\pi\)
\(108\) 0 0
\(109\) −38466.9 −0.310114 −0.155057 0.987906i \(-0.549556\pi\)
−0.155057 + 0.987906i \(0.549556\pi\)
\(110\) 0 0
\(111\) −131662. −1.01427
\(112\) 0 0
\(113\) 39951.6i 0.294333i 0.989112 + 0.147166i \(0.0470153\pi\)
−0.989112 + 0.147166i \(0.952985\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 15344.6i − 0.103632i
\(118\) 0 0
\(119\) −186610. −1.20800
\(120\) 0 0
\(121\) −140400. −0.871771
\(122\) 0 0
\(123\) 252099.i 1.50248i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 233239.i 1.28319i 0.767043 + 0.641596i \(0.221727\pi\)
−0.767043 + 0.641596i \(0.778273\pi\)
\(128\) 0 0
\(129\) −19079.2 −0.100945
\(130\) 0 0
\(131\) 55237.0 0.281224 0.140612 0.990065i \(-0.455093\pi\)
0.140612 + 0.990065i \(0.455093\pi\)
\(132\) 0 0
\(133\) − 124063.i − 0.608155i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 261520.i − 1.19043i −0.803566 0.595215i \(-0.797067\pi\)
0.803566 0.595215i \(-0.202933\pi\)
\(138\) 0 0
\(139\) −293366. −1.28787 −0.643935 0.765080i \(-0.722699\pi\)
−0.643935 + 0.765080i \(0.722699\pi\)
\(140\) 0 0
\(141\) −360345. −1.52641
\(142\) 0 0
\(143\) 60559.6i 0.247653i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 132778.i 0.506797i
\(148\) 0 0
\(149\) 304505. 1.12364 0.561822 0.827258i \(-0.310101\pi\)
0.561822 + 0.827258i \(0.310101\pi\)
\(150\) 0 0
\(151\) −337909. −1.20603 −0.603015 0.797730i \(-0.706034\pi\)
−0.603015 + 0.797730i \(0.706034\pi\)
\(152\) 0 0
\(153\) 72173.5i 0.249258i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 68385.9i − 0.221420i −0.993853 0.110710i \(-0.964687\pi\)
0.993853 0.110710i \(-0.0353125\pi\)
\(158\) 0 0
\(159\) −159809. −0.501312
\(160\) 0 0
\(161\) −378473. −1.15072
\(162\) 0 0
\(163\) − 404471.i − 1.19239i −0.802840 0.596195i \(-0.796678\pi\)
0.802840 0.596195i \(-0.203322\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 411733.i − 1.14242i −0.820805 0.571209i \(-0.806475\pi\)
0.820805 0.571209i \(-0.193525\pi\)
\(168\) 0 0
\(169\) 193705. 0.521703
\(170\) 0 0
\(171\) −47982.9 −0.125486
\(172\) 0 0
\(173\) − 162247.i − 0.412155i −0.978536 0.206077i \(-0.933930\pi\)
0.978536 0.206077i \(-0.0660698\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 714438.i − 1.71408i
\(178\) 0 0
\(179\) −384396. −0.896698 −0.448349 0.893859i \(-0.647988\pi\)
−0.448349 + 0.893859i \(0.647988\pi\)
\(180\) 0 0
\(181\) 579219. 1.31415 0.657077 0.753823i \(-0.271792\pi\)
0.657077 + 0.753823i \(0.271792\pi\)
\(182\) 0 0
\(183\) 537108.i 1.18559i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 284842.i − 0.595662i
\(188\) 0 0
\(189\) 325112. 0.662031
\(190\) 0 0
\(191\) 142455. 0.282550 0.141275 0.989970i \(-0.454880\pi\)
0.141275 + 0.989970i \(0.454880\pi\)
\(192\) 0 0
\(193\) 267272.i 0.516487i 0.966080 + 0.258244i \(0.0831438\pi\)
−0.966080 + 0.258244i \(0.916856\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 14189.5i − 0.0260496i −0.999915 0.0130248i \(-0.995854\pi\)
0.999915 0.0130248i \(-0.00414604\pi\)
\(198\) 0 0
\(199\) −169198. −0.302875 −0.151437 0.988467i \(-0.548390\pi\)
−0.151437 + 0.988467i \(0.548390\pi\)
\(200\) 0 0
\(201\) −507677. −0.886335
\(202\) 0 0
\(203\) − 604167.i − 1.02900i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 146379.i 0.237439i
\(208\) 0 0
\(209\) 189371. 0.299880
\(210\) 0 0
\(211\) −407591. −0.630259 −0.315129 0.949049i \(-0.602048\pi\)
−0.315129 + 0.949049i \(0.602048\pi\)
\(212\) 0 0
\(213\) 601875.i 0.908987i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 221306.i 0.319039i
\(218\) 0 0
\(219\) 1.06041e6 1.49404
\(220\) 0 0
\(221\) 835287. 1.15042
\(222\) 0 0
\(223\) − 103743.i − 0.139700i −0.997558 0.0698500i \(-0.977748\pi\)
0.997558 0.0698500i \(-0.0222521\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 803726.i 1.03524i 0.855609 + 0.517622i \(0.173183\pi\)
−0.855609 + 0.517622i \(0.826817\pi\)
\(228\) 0 0
\(229\) −1.35955e6 −1.71319 −0.856596 0.515988i \(-0.827425\pi\)
−0.856596 + 0.515988i \(0.827425\pi\)
\(230\) 0 0
\(231\) 226154. 0.278852
\(232\) 0 0
\(233\) − 622827.i − 0.751584i −0.926704 0.375792i \(-0.877371\pi\)
0.926704 0.375792i \(-0.122629\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.50312e6i 1.73829i
\(238\) 0 0
\(239\) 642843. 0.727964 0.363982 0.931406i \(-0.381417\pi\)
0.363982 + 0.931406i \(0.381417\pi\)
\(240\) 0 0
\(241\) 1.11814e6 1.24009 0.620046 0.784566i \(-0.287114\pi\)
0.620046 + 0.784566i \(0.287114\pi\)
\(242\) 0 0
\(243\) − 273644.i − 0.297283i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 555322.i 0.579165i
\(248\) 0 0
\(249\) −638724. −0.652852
\(250\) 0 0
\(251\) 731067. 0.732442 0.366221 0.930528i \(-0.380651\pi\)
0.366221 + 0.930528i \(0.380651\pi\)
\(252\) 0 0
\(253\) − 577702.i − 0.567418i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.76622e6i 1.66806i 0.551720 + 0.834030i \(0.313972\pi\)
−0.551720 + 0.834030i \(0.686028\pi\)
\(258\) 0 0
\(259\) 741555. 0.686901
\(260\) 0 0
\(261\) −233668. −0.212324
\(262\) 0 0
\(263\) 1.00926e6i 0.899735i 0.893095 + 0.449868i \(0.148529\pi\)
−0.893095 + 0.449868i \(0.851471\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 96042.8i − 0.0824492i
\(268\) 0 0
\(269\) 775681. 0.653585 0.326793 0.945096i \(-0.394032\pi\)
0.326793 + 0.945096i \(0.394032\pi\)
\(270\) 0 0
\(271\) 1.21395e6 1.00410 0.502052 0.864837i \(-0.332578\pi\)
0.502052 + 0.864837i \(0.332578\pi\)
\(272\) 0 0
\(273\) 663187.i 0.538554i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 218505.i 0.171104i 0.996334 + 0.0855522i \(0.0272654\pi\)
−0.996334 + 0.0855522i \(0.972735\pi\)
\(278\) 0 0
\(279\) 85592.6 0.0658303
\(280\) 0 0
\(281\) −316219. −0.238903 −0.119452 0.992840i \(-0.538114\pi\)
−0.119452 + 0.992840i \(0.538114\pi\)
\(282\) 0 0
\(283\) − 927934.i − 0.688733i −0.938835 0.344366i \(-0.888094\pi\)
0.938835 0.344366i \(-0.111906\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 1.41989e6i − 1.01754i
\(288\) 0 0
\(289\) −2.50892e6 −1.76702
\(290\) 0 0
\(291\) 2.98842e6 2.06876
\(292\) 0 0
\(293\) 262992.i 0.178967i 0.995988 + 0.0894835i \(0.0285216\pi\)
−0.995988 + 0.0894835i \(0.971478\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 496252.i 0.326446i
\(298\) 0 0
\(299\) 1.69409e6 1.09587
\(300\) 0 0
\(301\) 107459. 0.0683640
\(302\) 0 0
\(303\) 2.54451e6i 1.59220i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 2.15704e6i − 1.30621i −0.757269 0.653104i \(-0.773467\pi\)
0.757269 0.653104i \(-0.226533\pi\)
\(308\) 0 0
\(309\) 598923. 0.356841
\(310\) 0 0
\(311\) −2.33078e6 −1.36647 −0.683235 0.730199i \(-0.739427\pi\)
−0.683235 + 0.730199i \(0.739427\pi\)
\(312\) 0 0
\(313\) − 1.12235e6i − 0.647539i −0.946136 0.323769i \(-0.895050\pi\)
0.946136 0.323769i \(-0.104950\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.27955e6i 0.715169i 0.933881 + 0.357584i \(0.116400\pi\)
−0.933881 + 0.357584i \(0.883600\pi\)
\(318\) 0 0
\(319\) 922203. 0.507399
\(320\) 0 0
\(321\) 1.17060e6 0.634082
\(322\) 0 0
\(323\) − 2.61196e6i − 1.39303i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 642998.i − 0.332537i
\(328\) 0 0
\(329\) 2.02956e6 1.03374
\(330\) 0 0
\(331\) −2.09571e6 −1.05138 −0.525691 0.850675i \(-0.676193\pi\)
−0.525691 + 0.850675i \(0.676193\pi\)
\(332\) 0 0
\(333\) − 286805.i − 0.141735i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 571289.i 0.274019i 0.990570 + 0.137010i \(0.0437491\pi\)
−0.990570 + 0.137010i \(0.956251\pi\)
\(338\) 0 0
\(339\) −667817. −0.315615
\(340\) 0 0
\(341\) −337802. −0.157317
\(342\) 0 0
\(343\) − 2.33017e6i − 1.06943i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 2.24067e6i − 0.998973i −0.866322 0.499486i \(-0.833522\pi\)
0.866322 0.499486i \(-0.166478\pi\)
\(348\) 0 0
\(349\) −130448. −0.0573290 −0.0286645 0.999589i \(-0.509125\pi\)
−0.0286645 + 0.999589i \(0.509125\pi\)
\(350\) 0 0
\(351\) −1.45524e6 −0.630473
\(352\) 0 0
\(353\) 1.95452e6i 0.834839i 0.908714 + 0.417420i \(0.137065\pi\)
−0.908714 + 0.417420i \(0.862935\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 3.11930e6i − 1.29535i
\(358\) 0 0
\(359\) 850543. 0.348305 0.174153 0.984719i \(-0.444281\pi\)
0.174153 + 0.984719i \(0.444281\pi\)
\(360\) 0 0
\(361\) −739600. −0.298696
\(362\) 0 0
\(363\) − 2.34687e6i − 0.934807i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 1.50673e6i 0.583941i 0.956427 + 0.291970i \(0.0943109\pi\)
−0.956427 + 0.291970i \(0.905689\pi\)
\(368\) 0 0
\(369\) −549159. −0.209958
\(370\) 0 0
\(371\) 900086. 0.339507
\(372\) 0 0
\(373\) 2.90602e6i 1.08150i 0.841184 + 0.540749i \(0.181859\pi\)
−0.841184 + 0.540749i \(0.818141\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.70432e6i 0.979952i
\(378\) 0 0
\(379\) −5.16710e6 −1.84777 −0.923887 0.382665i \(-0.875006\pi\)
−0.923887 + 0.382665i \(0.875006\pi\)
\(380\) 0 0
\(381\) −3.89874e6 −1.37598
\(382\) 0 0
\(383\) 3.85088e6i 1.34142i 0.741721 + 0.670708i \(0.234010\pi\)
−0.741721 + 0.670708i \(0.765990\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 41561.1i − 0.0141062i
\(388\) 0 0
\(389\) −5.49855e6 −1.84236 −0.921179 0.389138i \(-0.872773\pi\)
−0.921179 + 0.389138i \(0.872773\pi\)
\(390\) 0 0
\(391\) −7.96814e6 −2.63582
\(392\) 0 0
\(393\) 923321.i 0.301558i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 1.71425e6i − 0.545881i −0.962031 0.272941i \(-0.912004\pi\)
0.962031 0.272941i \(-0.0879963\pi\)
\(398\) 0 0
\(399\) 2.07380e6 0.652130
\(400\) 0 0
\(401\) 3.63329e6 1.12834 0.564169 0.825660i \(-0.309197\pi\)
0.564169 + 0.825660i \(0.309197\pi\)
\(402\) 0 0
\(403\) − 990591.i − 0.303831i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.13191e6i 0.338709i
\(408\) 0 0
\(409\) 1.38246e6 0.408642 0.204321 0.978904i \(-0.434501\pi\)
0.204321 + 0.978904i \(0.434501\pi\)
\(410\) 0 0
\(411\) 4.37148e6 1.27651
\(412\) 0 0
\(413\) 4.02390e6i 1.16084i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 4.90379e6i − 1.38099i
\(418\) 0 0
\(419\) −2.55435e6 −0.710795 −0.355398 0.934715i \(-0.615655\pi\)
−0.355398 + 0.934715i \(0.615655\pi\)
\(420\) 0 0
\(421\) −1.68110e6 −0.462263 −0.231132 0.972923i \(-0.574243\pi\)
−0.231132 + 0.972923i \(0.574243\pi\)
\(422\) 0 0
\(423\) − 784954.i − 0.213301i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 3.02513e6i − 0.802925i
\(428\) 0 0
\(429\) −1.01229e6 −0.265560
\(430\) 0 0
\(431\) −2.32369e6 −0.602539 −0.301269 0.953539i \(-0.597410\pi\)
−0.301269 + 0.953539i \(0.597410\pi\)
\(432\) 0 0
\(433\) − 4.06439e6i − 1.04178i −0.853624 0.520890i \(-0.825600\pi\)
0.853624 0.520890i \(-0.174400\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 5.29744e6i − 1.32697i
\(438\) 0 0
\(439\) −6.39272e6 −1.58316 −0.791579 0.611066i \(-0.790741\pi\)
−0.791579 + 0.611066i \(0.790741\pi\)
\(440\) 0 0
\(441\) −289237. −0.0708202
\(442\) 0 0
\(443\) 3.23515e6i 0.783222i 0.920131 + 0.391611i \(0.128082\pi\)
−0.920131 + 0.391611i \(0.871918\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 5.08999e6i 1.20489i
\(448\) 0 0
\(449\) 2.74812e6 0.643310 0.321655 0.946857i \(-0.395761\pi\)
0.321655 + 0.946857i \(0.395761\pi\)
\(450\) 0 0
\(451\) 2.16732e6 0.501745
\(452\) 0 0
\(453\) − 5.64837e6i − 1.29324i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.37326e6i 1.20350i 0.798684 + 0.601751i \(0.205530\pi\)
−0.798684 + 0.601751i \(0.794470\pi\)
\(458\) 0 0
\(459\) 6.84472e6 1.51644
\(460\) 0 0
\(461\) −8.19928e6 −1.79690 −0.898449 0.439078i \(-0.855305\pi\)
−0.898449 + 0.439078i \(0.855305\pi\)
\(462\) 0 0
\(463\) 3.76963e6i 0.817233i 0.912706 + 0.408617i \(0.133989\pi\)
−0.912706 + 0.408617i \(0.866011\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 6.03412e6i − 1.28033i −0.768237 0.640165i \(-0.778866\pi\)
0.768237 0.640165i \(-0.221134\pi\)
\(468\) 0 0
\(469\) 2.85937e6 0.600259
\(470\) 0 0
\(471\) 1.14311e6 0.237431
\(472\) 0 0
\(473\) 164026.i 0.0337101i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 348119.i − 0.0700537i
\(478\) 0 0
\(479\) 5.26406e6 1.04829 0.524146 0.851629i \(-0.324385\pi\)
0.524146 + 0.851629i \(0.324385\pi\)
\(480\) 0 0
\(481\) −3.31929e6 −0.654157
\(482\) 0 0
\(483\) − 6.32641e6i − 1.23393i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 2.19979e6i 0.420300i 0.977669 + 0.210150i \(0.0673952\pi\)
−0.977669 + 0.210150i \(0.932605\pi\)
\(488\) 0 0
\(489\) 6.76099e6 1.27861
\(490\) 0 0
\(491\) 6.47314e6 1.21174 0.605872 0.795562i \(-0.292824\pi\)
0.605872 + 0.795562i \(0.292824\pi\)
\(492\) 0 0
\(493\) − 1.27198e7i − 2.35701i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 3.38992e6i − 0.615600i
\(498\) 0 0
\(499\) −8.62090e6 −1.54989 −0.774946 0.632028i \(-0.782223\pi\)
−0.774946 + 0.632028i \(0.782223\pi\)
\(500\) 0 0
\(501\) 6.88238e6 1.22502
\(502\) 0 0
\(503\) 8.10557e6i 1.42845i 0.699918 + 0.714223i \(0.253220\pi\)
−0.699918 + 0.714223i \(0.746780\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 3.23789e6i 0.559426i
\(508\) 0 0
\(509\) 6.39610e6 1.09426 0.547131 0.837047i \(-0.315720\pi\)
0.547131 + 0.837047i \(0.315720\pi\)
\(510\) 0 0
\(511\) −5.97250e6 −1.01182
\(512\) 0 0
\(513\) 4.55055e6i 0.763433i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 3.09792e6i 0.509735i
\(518\) 0 0
\(519\) 2.71205e6 0.441957
\(520\) 0 0
\(521\) 7.53647e6 1.21639 0.608197 0.793786i \(-0.291893\pi\)
0.608197 + 0.793786i \(0.291893\pi\)
\(522\) 0 0
\(523\) 1.87780e6i 0.300189i 0.988672 + 0.150095i \(0.0479578\pi\)
−0.988672 + 0.150095i \(0.952042\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.65924e6i 0.730784i
\(528\) 0 0
\(529\) −9.72424e6 −1.51083
\(530\) 0 0
\(531\) 1.55629e6 0.239527
\(532\) 0 0
\(533\) 6.35559e6i 0.969031i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 6.42542e6i − 0.961537i
\(538\) 0 0
\(539\) 1.14151e6 0.169242
\(540\) 0 0
\(541\) 2.44482e6 0.359131 0.179566 0.983746i \(-0.442531\pi\)
0.179566 + 0.983746i \(0.442531\pi\)
\(542\) 0 0
\(543\) 9.68202e6i 1.40918i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.09485e7i 1.56453i 0.622944 + 0.782266i \(0.285936\pi\)
−0.622944 + 0.782266i \(0.714064\pi\)
\(548\) 0 0
\(549\) −1.17001e6 −0.165675
\(550\) 0 0
\(551\) 8.45645e6 1.18661
\(552\) 0 0
\(553\) − 8.46595e6i − 1.17723i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 768938.i 0.105015i 0.998621 + 0.0525077i \(0.0167214\pi\)
−0.998621 + 0.0525077i \(0.983279\pi\)
\(558\) 0 0
\(559\) −481000. −0.0651052
\(560\) 0 0
\(561\) 4.76131e6 0.638733
\(562\) 0 0
\(563\) − 1.47386e7i − 1.95968i −0.199792 0.979838i \(-0.564026\pi\)
0.199792 0.979838i \(-0.435974\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 6.26749e6i 0.818721i
\(568\) 0 0
\(569\) 3.33696e6 0.432086 0.216043 0.976384i \(-0.430685\pi\)
0.216043 + 0.976384i \(0.430685\pi\)
\(570\) 0 0
\(571\) 1.54413e7 1.98195 0.990974 0.134051i \(-0.0427986\pi\)
0.990974 + 0.134051i \(0.0427986\pi\)
\(572\) 0 0
\(573\) 2.38123e6i 0.302981i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 4.05636e6i 0.507221i 0.967306 + 0.253610i \(0.0816181\pi\)
−0.967306 + 0.253610i \(0.918382\pi\)
\(578\) 0 0
\(579\) −4.46762e6 −0.553834
\(580\) 0 0
\(581\) 3.59746e6 0.442136
\(582\) 0 0
\(583\) 1.37389e6i 0.167410i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.22567e7i 1.46818i 0.679052 + 0.734090i \(0.262391\pi\)
−0.679052 + 0.734090i \(0.737609\pi\)
\(588\) 0 0
\(589\) −3.09759e6 −0.367905
\(590\) 0 0
\(591\) 237186. 0.0279332
\(592\) 0 0
\(593\) 8.58993e6i 1.00312i 0.865123 + 0.501560i \(0.167240\pi\)
−0.865123 + 0.501560i \(0.832760\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 2.82825e6i − 0.324775i
\(598\) 0 0
\(599\) −8.85849e6 −1.00877 −0.504386 0.863479i \(-0.668281\pi\)
−0.504386 + 0.863479i \(0.668281\pi\)
\(600\) 0 0
\(601\) −8.50579e6 −0.960569 −0.480284 0.877113i \(-0.659467\pi\)
−0.480284 + 0.877113i \(0.659467\pi\)
\(602\) 0 0
\(603\) − 1.10590e6i − 0.123857i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 3.15481e6i 0.347537i 0.984787 + 0.173768i \(0.0555944\pi\)
−0.984787 + 0.173768i \(0.944406\pi\)
\(608\) 0 0
\(609\) 1.00990e7 1.10341
\(610\) 0 0
\(611\) −9.08453e6 −0.984463
\(612\) 0 0
\(613\) 3.20689e6i 0.344694i 0.985036 + 0.172347i \(0.0551350\pi\)
−0.985036 + 0.172347i \(0.944865\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.31916e6i 0.139504i 0.997564 + 0.0697519i \(0.0222207\pi\)
−0.997564 + 0.0697519i \(0.977779\pi\)
\(618\) 0 0
\(619\) 3.36061e6 0.352526 0.176263 0.984343i \(-0.443599\pi\)
0.176263 + 0.984343i \(0.443599\pi\)
\(620\) 0 0
\(621\) 1.38821e7 1.44453
\(622\) 0 0
\(623\) 540939.i 0.0558377i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 3.16545e6i 0.321563i
\(628\) 0 0
\(629\) 1.56123e7 1.57340
\(630\) 0 0
\(631\) −1.46547e7 −1.46522 −0.732610 0.680648i \(-0.761698\pi\)
−0.732610 + 0.680648i \(0.761698\pi\)
\(632\) 0 0
\(633\) − 6.81315e6i − 0.675832i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 3.34743e6i 0.326861i
\(638\) 0 0
\(639\) −1.31109e6 −0.127023
\(640\) 0 0
\(641\) −5.67545e6 −0.545576 −0.272788 0.962074i \(-0.587946\pi\)
−0.272788 + 0.962074i \(0.587946\pi\)
\(642\) 0 0
\(643\) − 1.81422e7i − 1.73047i −0.501370 0.865233i \(-0.667170\pi\)
0.501370 0.865233i \(-0.332830\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 472390.i − 0.0443649i −0.999754 0.0221825i \(-0.992939\pi\)
0.999754 0.0221825i \(-0.00706148\pi\)
\(648\) 0 0
\(649\) −6.14210e6 −0.572407
\(650\) 0 0
\(651\) −3.69927e6 −0.342108
\(652\) 0 0
\(653\) 9.79442e6i 0.898867i 0.893314 + 0.449434i \(0.148374\pi\)
−0.893314 + 0.449434i \(0.851626\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2.30993e6i 0.208778i
\(658\) 0 0
\(659\) 9.83251e6 0.881964 0.440982 0.897516i \(-0.354630\pi\)
0.440982 + 0.897516i \(0.354630\pi\)
\(660\) 0 0
\(661\) 1.84120e6 0.163907 0.0819535 0.996636i \(-0.473884\pi\)
0.0819535 + 0.996636i \(0.473884\pi\)
\(662\) 0 0
\(663\) 1.39624e7i 1.23360i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 2.57976e7i − 2.24525i
\(668\) 0 0
\(669\) 1.73413e6 0.149802
\(670\) 0 0
\(671\) 4.61758e6 0.395920
\(672\) 0 0
\(673\) − 1.02990e7i − 0.876510i −0.898851 0.438255i \(-0.855597\pi\)
0.898851 0.438255i \(-0.144403\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3.63461e6i 0.304779i 0.988320 + 0.152390i \(0.0486969\pi\)
−0.988320 + 0.152390i \(0.951303\pi\)
\(678\) 0 0
\(679\) −1.68316e7 −1.40104
\(680\) 0 0
\(681\) −1.34348e7 −1.11010
\(682\) 0 0
\(683\) − 9.78443e6i − 0.802571i −0.915953 0.401286i \(-0.868563\pi\)
0.915953 0.401286i \(-0.131437\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 2.27257e7i − 1.83707i
\(688\) 0 0
\(689\) −4.02889e6 −0.323323
\(690\) 0 0
\(691\) 4.67949e6 0.372824 0.186412 0.982472i \(-0.440314\pi\)
0.186412 + 0.982472i \(0.440314\pi\)
\(692\) 0 0
\(693\) 492641.i 0.0389671i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 2.98935e7i − 2.33075i
\(698\) 0 0
\(699\) 1.04110e7 0.805930
\(700\) 0 0
\(701\) −565147. −0.0434376 −0.0217188 0.999764i \(-0.506914\pi\)
−0.0217188 + 0.999764i \(0.506914\pi\)
\(702\) 0 0
\(703\) 1.03795e7i 0.792112i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 1.43314e7i − 1.07830i
\(708\) 0 0
\(709\) 1.00907e6 0.0753889 0.0376944 0.999289i \(-0.487999\pi\)
0.0376944 + 0.999289i \(0.487999\pi\)
\(710\) 0 0
\(711\) −3.27431e6 −0.242910
\(712\) 0 0
\(713\) 9.44964e6i 0.696132i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 1.07455e7i 0.780602i
\(718\) 0 0
\(719\) 6.02329e6 0.434522 0.217261 0.976114i \(-0.430288\pi\)
0.217261 + 0.976114i \(0.430288\pi\)
\(720\) 0 0
\(721\) −3.37329e6 −0.241666
\(722\) 0 0
\(723\) 1.86904e7i 1.32976i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 2.30828e6i 0.161977i 0.996715 + 0.0809883i \(0.0258076\pi\)
−0.996715 + 0.0809883i \(0.974192\pi\)
\(728\) 0 0
\(729\) −1.16027e7 −0.808612
\(730\) 0 0
\(731\) 2.26238e6 0.156593
\(732\) 0 0
\(733\) − 1.65224e7i − 1.13583i −0.823088 0.567914i \(-0.807751\pi\)
0.823088 0.567914i \(-0.192249\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.36456e6i 0.295986i
\(738\) 0 0
\(739\) −2.41051e7 −1.62367 −0.811834 0.583888i \(-0.801531\pi\)
−0.811834 + 0.583888i \(0.801531\pi\)
\(740\) 0 0
\(741\) −9.28255e6 −0.621043
\(742\) 0 0
\(743\) − 2.54593e7i − 1.69190i −0.533262 0.845950i \(-0.679034\pi\)
0.533262 0.845950i \(-0.320966\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 1.39136e6i − 0.0912301i
\(748\) 0 0
\(749\) −6.59312e6 −0.429424
\(750\) 0 0
\(751\) −1.25073e7 −0.809215 −0.404608 0.914490i \(-0.632592\pi\)
−0.404608 + 0.914490i \(0.632592\pi\)
\(752\) 0 0
\(753\) 1.22203e7i 0.785404i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 5.34914e6i 0.339269i 0.985507 + 0.169635i \(0.0542588\pi\)
−0.985507 + 0.169635i \(0.945741\pi\)
\(758\) 0 0
\(759\) 9.65666e6 0.608447
\(760\) 0 0
\(761\) −1.80369e7 −1.12901 −0.564507 0.825428i \(-0.690934\pi\)
−0.564507 + 0.825428i \(0.690934\pi\)
\(762\) 0 0
\(763\) 3.62154e6i 0.225207i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 1.80115e7i − 1.10550i
\(768\) 0 0
\(769\) −2.41928e7 −1.47526 −0.737632 0.675203i \(-0.764056\pi\)
−0.737632 + 0.675203i \(0.764056\pi\)
\(770\) 0 0
\(771\) −2.95234e7 −1.78867
\(772\) 0 0
\(773\) − 3.22531e6i − 0.194143i −0.995277 0.0970716i \(-0.969052\pi\)
0.995277 0.0970716i \(-0.0309476\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 1.23956e7i 0.736570i
\(778\) 0 0
\(779\) 1.98740e7 1.17339
\(780\) 0 0
\(781\) 5.17439e6 0.303551
\(782\) 0 0
\(783\) 2.21604e7i 1.29174i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 1.12227e6i 0.0645896i 0.999478 + 0.0322948i \(0.0102815\pi\)
−0.999478 + 0.0322948i \(0.989718\pi\)
\(788\) 0 0
\(789\) −1.68705e7 −0.964793
\(790\) 0 0
\(791\) 3.76132e6 0.213747
\(792\) 0 0
\(793\) 1.35408e7i 0.764650i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.48763e7i 0.829560i 0.909922 + 0.414780i \(0.136141\pi\)
−0.909922 + 0.414780i \(0.863859\pi\)
\(798\) 0 0
\(799\) 4.27291e7 2.36786
\(800\) 0 0
\(801\) 209214. 0.0115215
\(802\) 0 0
\(803\) − 9.11644e6i − 0.498926i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 1.29660e7i 0.700845i
\(808\) 0 0
\(809\) −1.86436e7 −1.00152 −0.500759 0.865587i \(-0.666946\pi\)
−0.500759 + 0.865587i \(0.666946\pi\)
\(810\) 0 0
\(811\) 4.50701e6 0.240623 0.120311 0.992736i \(-0.461611\pi\)
0.120311 + 0.992736i \(0.461611\pi\)
\(812\) 0 0
\(813\) 2.02920e7i 1.07671i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.50409e6i 0.0788352i
\(818\) 0 0
\(819\) −1.44465e6 −0.0752580
\(820\) 0 0
\(821\) −2.26560e7 −1.17307 −0.586537 0.809922i \(-0.699509\pi\)
−0.586537 + 0.809922i \(0.699509\pi\)
\(822\) 0 0
\(823\) − 2.76806e7i − 1.42454i −0.701903 0.712272i \(-0.747666\pi\)
0.701903 0.712272i \(-0.252334\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 7.42673e6i 0.377602i 0.982015 + 0.188801i \(0.0604600\pi\)
−0.982015 + 0.188801i \(0.939540\pi\)
\(828\) 0 0
\(829\) −3.41770e7 −1.72722 −0.863610 0.504160i \(-0.831802\pi\)
−0.863610 + 0.504160i \(0.831802\pi\)
\(830\) 0 0
\(831\) −3.65244e6 −0.183477
\(832\) 0 0
\(833\) − 1.57446e7i − 0.786177i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 8.11734e6i − 0.400498i
\(838\) 0 0
\(839\) −155990. −0.00765051 −0.00382526 0.999993i \(-0.501218\pi\)
−0.00382526 + 0.999993i \(0.501218\pi\)
\(840\) 0 0
\(841\) 2.06703e7 1.00776
\(842\) 0 0
\(843\) − 5.28580e6i − 0.256178i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 1.32182e7i 0.633087i
\(848\) 0 0
\(849\) 1.55110e7 0.738534
\(850\) 0 0
\(851\) 3.16640e7 1.49879
\(852\) 0 0
\(853\) − 2.52778e7i − 1.18951i −0.803908 0.594754i \(-0.797250\pi\)
0.803908 0.594754i \(-0.202750\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 1.40854e6i − 0.0655112i −0.999463 0.0327556i \(-0.989572\pi\)
0.999463 0.0327556i \(-0.0104283\pi\)
\(858\) 0 0
\(859\) 246158. 0.0113823 0.00569116 0.999984i \(-0.498188\pi\)
0.00569116 + 0.999984i \(0.498188\pi\)
\(860\) 0 0
\(861\) 2.37344e7 1.09111
\(862\) 0 0
\(863\) 1.17019e7i 0.534849i 0.963579 + 0.267424i \(0.0861725\pi\)
−0.963579 + 0.267424i \(0.913827\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 4.19381e7i − 1.89479i
\(868\) 0 0
\(869\) 1.29225e7 0.580492
\(870\) 0 0
\(871\) −1.27989e7 −0.571646
\(872\) 0 0
\(873\) 6.50982e6i 0.289090i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 7.85174e6i 0.344720i 0.985034 + 0.172360i \(0.0551393\pi\)
−0.985034 + 0.172360i \(0.944861\pi\)
\(878\) 0 0
\(879\) −4.39607e6 −0.191908
\(880\) 0 0
\(881\) 105461. 0.00457777 0.00228888 0.999997i \(-0.499271\pi\)
0.00228888 + 0.999997i \(0.499271\pi\)
\(882\) 0 0
\(883\) 1.43760e7i 0.620491i 0.950656 + 0.310245i \(0.100411\pi\)
−0.950656 + 0.310245i \(0.899589\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 2.71725e7i − 1.15963i −0.814748 0.579816i \(-0.803125\pi\)
0.814748 0.579816i \(-0.196875\pi\)
\(888\) 0 0
\(889\) 2.19587e7 0.931864
\(890\) 0 0
\(891\) −9.56672e6 −0.403709
\(892\) 0 0
\(893\) 2.84075e7i 1.19208i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 2.83177e7i 1.17511i
\(898\) 0 0
\(899\) −1.50847e7 −0.622499
\(900\) 0 0
\(901\) 1.89499e7 0.777668
\(902\) 0 0
\(903\) 1.79625e6i 0.0733073i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 5.25290e6i 0.212022i 0.994365 + 0.106011i \(0.0338079\pi\)
−0.994365 + 0.106011i \(0.966192\pi\)
\(908\) 0 0
\(909\) −5.54282e6 −0.222495
\(910\) 0 0
\(911\) 2.68057e7 1.07012 0.535059 0.844815i \(-0.320289\pi\)
0.535059 + 0.844815i \(0.320289\pi\)
\(912\) 0 0
\(913\) 5.49118e6i 0.218016i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 5.20039e6i − 0.204227i
\(918\) 0 0
\(919\) 1.25954e7 0.491951 0.245976 0.969276i \(-0.420892\pi\)
0.245976 + 0.969276i \(0.420892\pi\)
\(920\) 0 0
\(921\) 3.60563e7 1.40066
\(922\) 0 0
\(923\) 1.51737e7i 0.586255i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 1.30466e6i 0.0498652i
\(928\) 0 0
\(929\) 3.23322e7 1.22912 0.614562 0.788868i \(-0.289333\pi\)
0.614562 + 0.788868i \(0.289333\pi\)
\(930\) 0 0
\(931\) 1.04675e7 0.395792
\(932\) 0 0
\(933\) − 3.89604e7i − 1.46528i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 2.04882e7i − 0.762350i −0.924503 0.381175i \(-0.875519\pi\)
0.924503 0.381175i \(-0.124481\pi\)
\(938\) 0 0
\(939\) 1.87607e7 0.694361
\(940\) 0 0
\(941\) −5.87135e6 −0.216154 −0.108077 0.994143i \(-0.534469\pi\)
−0.108077 + 0.994143i \(0.534469\pi\)
\(942\) 0 0
\(943\) − 6.06285e7i − 2.22023i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.86718e7i 0.676567i 0.941044 + 0.338284i \(0.109846\pi\)
−0.941044 + 0.338284i \(0.890154\pi\)
\(948\) 0 0
\(949\) 2.67336e7 0.963588
\(950\) 0 0
\(951\) −2.13885e7 −0.766882
\(952\) 0 0
\(953\) 1.55896e7i 0.556037i 0.960576 + 0.278018i \(0.0896776\pi\)
−0.960576 + 0.278018i \(0.910322\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 1.54152e7i 0.544088i
\(958\) 0 0
\(959\) −2.46213e7 −0.864500
\(960\) 0 0
\(961\) −2.31036e7 −0.806996
\(962\) 0 0
\(963\) 2.54997e6i 0.0886071i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 3.24491e7i 1.11593i 0.829865 + 0.557964i \(0.188417\pi\)
−0.829865 + 0.557964i \(0.811583\pi\)
\(968\) 0 0
\(969\) 4.36605e7 1.49375
\(970\) 0 0
\(971\) 3.73661e7 1.27183 0.635916 0.771758i \(-0.280622\pi\)
0.635916 + 0.771758i \(0.280622\pi\)
\(972\) 0 0
\(973\) 2.76195e7i 0.935261i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 5.23874e7i 1.75586i 0.478787 + 0.877931i \(0.341077\pi\)
−0.478787 + 0.877931i \(0.658923\pi\)
\(978\) 0 0
\(979\) −825691. −0.0275335
\(980\) 0 0
\(981\) 1.40067e6 0.0464690
\(982\) 0 0
\(983\) − 4.75140e7i − 1.56833i −0.620551 0.784166i \(-0.713091\pi\)
0.620551 0.784166i \(-0.286909\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 3.39253e7i 1.10849i
\(988\) 0 0
\(989\) 4.58846e6 0.149168
\(990\) 0 0
\(991\) 2.47278e7 0.799835 0.399918 0.916551i \(-0.369039\pi\)
0.399918 + 0.916551i \(0.369039\pi\)
\(992\) 0 0
\(993\) − 3.50311e7i − 1.12741i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 1.25139e7i − 0.398709i −0.979927 0.199355i \(-0.936115\pi\)
0.979927 0.199355i \(-0.0638845\pi\)
\(998\) 0 0
\(999\) −2.71997e7 −0.862285
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.e.49.3 4
4.3 odd 2 400.6.c.l.49.2 4
5.2 odd 4 40.6.a.d.1.2 2
5.3 odd 4 200.6.a.g.1.1 2
5.4 even 2 inner 200.6.c.e.49.2 4
15.2 even 4 360.6.a.l.1.2 2
20.3 even 4 400.6.a.q.1.2 2
20.7 even 4 80.6.a.i.1.1 2
20.19 odd 2 400.6.c.l.49.3 4
40.27 even 4 320.6.a.q.1.2 2
40.37 odd 4 320.6.a.w.1.1 2
60.47 odd 4 720.6.a.z.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.6.a.d.1.2 2 5.2 odd 4
80.6.a.i.1.1 2 20.7 even 4
200.6.a.g.1.1 2 5.3 odd 4
200.6.c.e.49.2 4 5.4 even 2 inner
200.6.c.e.49.3 4 1.1 even 1 trivial
320.6.a.q.1.2 2 40.27 even 4
320.6.a.w.1.1 2 40.37 odd 4
360.6.a.l.1.2 2 15.2 even 4
400.6.a.q.1.2 2 20.3 even 4
400.6.c.l.49.2 4 4.3 odd 2
400.6.c.l.49.3 4 20.19 odd 2
720.6.a.z.1.1 2 60.47 odd 4