Properties

Label 224.3.d.b.127.3
Level 224
Weight 3
Character 224.127
Analytic conductor 6.104
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1997017344.2
Defining polynomial: \(x^{8} + 14 x^{6} + 53 x^{4} + 56 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.3
Root \(-1.27733i\) of defining polynomial
Character \(\chi\) \(=\) 224.127
Dual form 224.3.d.b.127.6

$q$-expansion

\(f(q)\) \(=\) \(q-2.55467i q^{3} +9.86836 q^{5} -2.64575i q^{7} +2.47367 q^{9} +O(q^{10})\) \(q-2.55467i q^{3} +9.86836 q^{5} -2.64575i q^{7} +2.47367 q^{9} +13.1537i q^{11} -5.86836 q^{13} -25.2104i q^{15} -0.570700 q^{17} -15.6640i q^{19} -6.75902 q^{21} +16.4817i q^{23} +72.3844 q^{25} -29.3114i q^{27} -29.7367 q^{29} -54.8014i q^{31} +33.6034 q^{33} -26.1092i q^{35} -42.0853 q^{37} +14.9917i q^{39} +0.773275 q^{41} +41.7931i q^{43} +24.4110 q^{45} +58.4528i q^{47} -7.00000 q^{49} +1.45795i q^{51} +5.65139 q^{53} +129.805i q^{55} -40.0163 q^{57} +42.6434i q^{59} -95.9371 q^{61} -6.54471i q^{63} -57.9110 q^{65} +69.8503i q^{67} +42.1053 q^{69} -92.0882i q^{71} +9.97539 q^{73} -184.918i q^{75} +34.8014 q^{77} -20.1780i q^{79} -52.6180 q^{81} +151.307i q^{83} -5.63187 q^{85} +75.9674i q^{87} +5.79743 q^{89} +15.5262i q^{91} -140.000 q^{93} -154.578i q^{95} +103.696 q^{97} +32.5379i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 40q^{9} + O(q^{10}) \) \( 8q - 40q^{9} + 32q^{13} - 16q^{17} + 104q^{25} - 80q^{29} - 176q^{37} + 144q^{41} + 256q^{45} - 56q^{49} + 48q^{53} - 400q^{57} - 192q^{61} - 304q^{65} + 576q^{69} + 272q^{73} - 112q^{77} + 504q^{81} - 160q^{85} - 80q^{89} - 608q^{93} + 528q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) − 2.55467i − 0.851556i −0.904828 0.425778i \(-0.860000\pi\)
0.904828 0.425778i \(-0.140000\pi\)
\(4\) 0 0
\(5\) 9.86836 1.97367 0.986836 0.161727i \(-0.0517064\pi\)
0.986836 + 0.161727i \(0.0517064\pi\)
\(6\) 0 0
\(7\) − 2.64575i − 0.377964i
\(8\) 0 0
\(9\) 2.47367 0.274852
\(10\) 0 0
\(11\) 13.1537i 1.19579i 0.801574 + 0.597896i \(0.203996\pi\)
−0.801574 + 0.597896i \(0.796004\pi\)
\(12\) 0 0
\(13\) −5.86836 −0.451412 −0.225706 0.974195i \(-0.572469\pi\)
−0.225706 + 0.974195i \(0.572469\pi\)
\(14\) 0 0
\(15\) − 25.2104i − 1.68069i
\(16\) 0 0
\(17\) −0.570700 −0.0335706 −0.0167853 0.999859i \(-0.505343\pi\)
−0.0167853 + 0.999859i \(0.505343\pi\)
\(18\) 0 0
\(19\) − 15.6640i − 0.824421i −0.911089 0.412211i \(-0.864757\pi\)
0.911089 0.412211i \(-0.135243\pi\)
\(20\) 0 0
\(21\) −6.75902 −0.321858
\(22\) 0 0
\(23\) 16.4817i 0.716596i 0.933607 + 0.358298i \(0.116643\pi\)
−0.933607 + 0.358298i \(0.883357\pi\)
\(24\) 0 0
\(25\) 72.3844 2.89538
\(26\) 0 0
\(27\) − 29.3114i − 1.08561i
\(28\) 0 0
\(29\) −29.7367 −1.02540 −0.512702 0.858567i \(-0.671355\pi\)
−0.512702 + 0.858567i \(0.671355\pi\)
\(30\) 0 0
\(31\) − 54.8014i − 1.76779i −0.467687 0.883894i \(-0.654913\pi\)
0.467687 0.883894i \(-0.345087\pi\)
\(32\) 0 0
\(33\) 33.6034 1.01828
\(34\) 0 0
\(35\) − 26.1092i − 0.745978i
\(36\) 0 0
\(37\) −42.0853 −1.13744 −0.568721 0.822531i \(-0.692561\pi\)
−0.568721 + 0.822531i \(0.692561\pi\)
\(38\) 0 0
\(39\) 14.9917i 0.384403i
\(40\) 0 0
\(41\) 0.773275 0.0188604 0.00943019 0.999956i \(-0.496998\pi\)
0.00943019 + 0.999956i \(0.496998\pi\)
\(42\) 0 0
\(43\) 41.7931i 0.971933i 0.873977 + 0.485967i \(0.161532\pi\)
−0.873977 + 0.485967i \(0.838468\pi\)
\(44\) 0 0
\(45\) 24.4110 0.542467
\(46\) 0 0
\(47\) 58.4528i 1.24368i 0.783145 + 0.621839i \(0.213614\pi\)
−0.783145 + 0.621839i \(0.786386\pi\)
\(48\) 0 0
\(49\) −7.00000 −0.142857
\(50\) 0 0
\(51\) 1.45795i 0.0285873i
\(52\) 0 0
\(53\) 5.65139 0.106630 0.0533150 0.998578i \(-0.483021\pi\)
0.0533150 + 0.998578i \(0.483021\pi\)
\(54\) 0 0
\(55\) 129.805i 2.36010i
\(56\) 0 0
\(57\) −40.0163 −0.702041
\(58\) 0 0
\(59\) 42.6434i 0.722770i 0.932417 + 0.361385i \(0.117696\pi\)
−0.932417 + 0.361385i \(0.882304\pi\)
\(60\) 0 0
\(61\) −95.9371 −1.57274 −0.786370 0.617756i \(-0.788042\pi\)
−0.786370 + 0.617756i \(0.788042\pi\)
\(62\) 0 0
\(63\) − 6.54471i − 0.103884i
\(64\) 0 0
\(65\) −57.9110 −0.890939
\(66\) 0 0
\(67\) 69.8503i 1.04254i 0.853391 + 0.521271i \(0.174542\pi\)
−0.853391 + 0.521271i \(0.825458\pi\)
\(68\) 0 0
\(69\) 42.1053 0.610222
\(70\) 0 0
\(71\) − 92.0882i − 1.29702i −0.761207 0.648509i \(-0.775393\pi\)
0.761207 0.648509i \(-0.224607\pi\)
\(72\) 0 0
\(73\) 9.97539 0.136649 0.0683246 0.997663i \(-0.478235\pi\)
0.0683246 + 0.997663i \(0.478235\pi\)
\(74\) 0 0
\(75\) − 184.918i − 2.46558i
\(76\) 0 0
\(77\) 34.8014 0.451967
\(78\) 0 0
\(79\) − 20.1780i − 0.255417i −0.991812 0.127709i \(-0.959238\pi\)
0.991812 0.127709i \(-0.0407622\pi\)
\(80\) 0 0
\(81\) −52.6180 −0.649604
\(82\) 0 0
\(83\) 151.307i 1.82298i 0.411321 + 0.911491i \(0.365068\pi\)
−0.411321 + 0.911491i \(0.634932\pi\)
\(84\) 0 0
\(85\) −5.63187 −0.0662573
\(86\) 0 0
\(87\) 75.9674i 0.873189i
\(88\) 0 0
\(89\) 5.79743 0.0651396 0.0325698 0.999469i \(-0.489631\pi\)
0.0325698 + 0.999469i \(0.489631\pi\)
\(90\) 0 0
\(91\) 15.5262i 0.170618i
\(92\) 0 0
\(93\) −140.000 −1.50537
\(94\) 0 0
\(95\) − 154.578i − 1.62714i
\(96\) 0 0
\(97\) 103.696 1.06903 0.534513 0.845160i \(-0.320495\pi\)
0.534513 + 0.845160i \(0.320495\pi\)
\(98\) 0 0
\(99\) 32.5379i 0.328666i
\(100\) 0 0
\(101\) 41.9176 0.415025 0.207513 0.978232i \(-0.433463\pi\)
0.207513 + 0.978232i \(0.433463\pi\)
\(102\) 0 0
\(103\) 18.7378i 0.181920i 0.995855 + 0.0909602i \(0.0289936\pi\)
−0.995855 + 0.0909602i \(0.971006\pi\)
\(104\) 0 0
\(105\) −66.7004 −0.635242
\(106\) 0 0
\(107\) − 64.8549i − 0.606121i −0.952971 0.303060i \(-0.901992\pi\)
0.952971 0.303060i \(-0.0980084\pi\)
\(108\) 0 0
\(109\) −186.432 −1.71039 −0.855195 0.518307i \(-0.826563\pi\)
−0.855195 + 0.518307i \(0.826563\pi\)
\(110\) 0 0
\(111\) 107.514i 0.968595i
\(112\) 0 0
\(113\) −84.7530 −0.750027 −0.375013 0.927019i \(-0.622362\pi\)
−0.375013 + 0.927019i \(0.622362\pi\)
\(114\) 0 0
\(115\) 162.647i 1.41433i
\(116\) 0 0
\(117\) −14.5164 −0.124071
\(118\) 0 0
\(119\) 1.50993i 0.0126885i
\(120\) 0 0
\(121\) −52.0200 −0.429917
\(122\) 0 0
\(123\) − 1.97546i − 0.0160607i
\(124\) 0 0
\(125\) 467.606 3.74085
\(126\) 0 0
\(127\) 93.3874i 0.735334i 0.929958 + 0.367667i \(0.119843\pi\)
−0.929958 + 0.367667i \(0.880157\pi\)
\(128\) 0 0
\(129\) 106.768 0.827656
\(130\) 0 0
\(131\) − 58.0857i − 0.443403i −0.975115 0.221701i \(-0.928839\pi\)
0.975115 0.221701i \(-0.0711610\pi\)
\(132\) 0 0
\(133\) −41.4431 −0.311602
\(134\) 0 0
\(135\) − 289.256i − 2.14263i
\(136\) 0 0
\(137\) 176.420 1.28774 0.643870 0.765135i \(-0.277328\pi\)
0.643870 + 0.765135i \(0.277328\pi\)
\(138\) 0 0
\(139\) − 12.3041i − 0.0885185i −0.999020 0.0442592i \(-0.985907\pi\)
0.999020 0.0442592i \(-0.0140928\pi\)
\(140\) 0 0
\(141\) 149.328 1.05906
\(142\) 0 0
\(143\) − 77.1906i − 0.539795i
\(144\) 0 0
\(145\) −293.452 −2.02381
\(146\) 0 0
\(147\) 17.8827i 0.121651i
\(148\) 0 0
\(149\) 173.782 1.16632 0.583160 0.812358i \(-0.301816\pi\)
0.583160 + 0.812358i \(0.301816\pi\)
\(150\) 0 0
\(151\) 138.433i 0.916776i 0.888752 + 0.458388i \(0.151573\pi\)
−0.888752 + 0.458388i \(0.848427\pi\)
\(152\) 0 0
\(153\) −1.41172 −0.00922695
\(154\) 0 0
\(155\) − 540.800i − 3.48903i
\(156\) 0 0
\(157\) 189.693 1.20824 0.604119 0.796894i \(-0.293525\pi\)
0.604119 + 0.796894i \(0.293525\pi\)
\(158\) 0 0
\(159\) − 14.4374i − 0.0908014i
\(160\) 0 0
\(161\) 43.6065 0.270848
\(162\) 0 0
\(163\) − 17.7685i − 0.109009i −0.998514 0.0545047i \(-0.982642\pi\)
0.998514 0.0545047i \(-0.0173580\pi\)
\(164\) 0 0
\(165\) 331.610 2.00976
\(166\) 0 0
\(167\) 0.0890922i 0 0.000533486i 1.00000 0.000266743i \(8.49069e-5\pi\)
−1.00000 0.000266743i \(0.999915\pi\)
\(168\) 0 0
\(169\) −134.562 −0.796227
\(170\) 0 0
\(171\) − 38.7475i − 0.226594i
\(172\) 0 0
\(173\) 43.2891 0.250226 0.125113 0.992143i \(-0.460071\pi\)
0.125113 + 0.992143i \(0.460071\pi\)
\(174\) 0 0
\(175\) − 191.511i − 1.09435i
\(176\) 0 0
\(177\) 108.940 0.615479
\(178\) 0 0
\(179\) 190.838i 1.06614i 0.846072 + 0.533068i \(0.178961\pi\)
−0.846072 + 0.533068i \(0.821039\pi\)
\(180\) 0 0
\(181\) −207.153 −1.14449 −0.572246 0.820082i \(-0.693928\pi\)
−0.572246 + 0.820082i \(0.693928\pi\)
\(182\) 0 0
\(183\) 245.087i 1.33928i
\(184\) 0 0
\(185\) −415.313 −2.24493
\(186\) 0 0
\(187\) − 7.50683i − 0.0401434i
\(188\) 0 0
\(189\) −77.5507 −0.410321
\(190\) 0 0
\(191\) − 315.489i − 1.65178i −0.563833 0.825889i \(-0.690674\pi\)
0.563833 0.825889i \(-0.309326\pi\)
\(192\) 0 0
\(193\) −30.9403 −0.160312 −0.0801561 0.996782i \(-0.525542\pi\)
−0.0801561 + 0.996782i \(0.525542\pi\)
\(194\) 0 0
\(195\) 147.943i 0.758684i
\(196\) 0 0
\(197\) −264.248 −1.34136 −0.670681 0.741746i \(-0.733998\pi\)
−0.670681 + 0.741746i \(0.733998\pi\)
\(198\) 0 0
\(199\) 131.321i 0.659903i 0.943998 + 0.329951i \(0.107032\pi\)
−0.943998 + 0.329951i \(0.892968\pi\)
\(200\) 0 0
\(201\) 178.444 0.887783
\(202\) 0 0
\(203\) 78.6759i 0.387566i
\(204\) 0 0
\(205\) 7.63095 0.0372242
\(206\) 0 0
\(207\) 40.7703i 0.196958i
\(208\) 0 0
\(209\) 206.040 0.985836
\(210\) 0 0
\(211\) − 247.994i − 1.17533i −0.809105 0.587664i \(-0.800048\pi\)
0.809105 0.587664i \(-0.199952\pi\)
\(212\) 0 0
\(213\) −235.255 −1.10448
\(214\) 0 0
\(215\) 412.430i 1.91828i
\(216\) 0 0
\(217\) −144.991 −0.668161
\(218\) 0 0
\(219\) − 25.4838i − 0.116364i
\(220\) 0 0
\(221\) 3.34907 0.0151542
\(222\) 0 0
\(223\) − 56.7999i − 0.254708i −0.991857 0.127354i \(-0.959352\pi\)
0.991857 0.127354i \(-0.0406484\pi\)
\(224\) 0 0
\(225\) 179.055 0.795800
\(226\) 0 0
\(227\) − 275.095i − 1.21187i −0.795513 0.605937i \(-0.792798\pi\)
0.795513 0.605937i \(-0.207202\pi\)
\(228\) 0 0
\(229\) 6.29210 0.0274764 0.0137382 0.999906i \(-0.495627\pi\)
0.0137382 + 0.999906i \(0.495627\pi\)
\(230\) 0 0
\(231\) − 88.9061i − 0.384875i
\(232\) 0 0
\(233\) −142.049 −0.609653 −0.304827 0.952408i \(-0.598599\pi\)
−0.304827 + 0.952408i \(0.598599\pi\)
\(234\) 0 0
\(235\) 576.833i 2.45461i
\(236\) 0 0
\(237\) −51.5480 −0.217502
\(238\) 0 0
\(239\) − 51.3954i − 0.215043i −0.994203 0.107522i \(-0.965708\pi\)
0.994203 0.107522i \(-0.0342915\pi\)
\(240\) 0 0
\(241\) 93.7048 0.388817 0.194408 0.980921i \(-0.437721\pi\)
0.194408 + 0.980921i \(0.437721\pi\)
\(242\) 0 0
\(243\) − 129.381i − 0.532433i
\(244\) 0 0
\(245\) −69.0785 −0.281953
\(246\) 0 0
\(247\) 91.9220i 0.372154i
\(248\) 0 0
\(249\) 386.540 1.55237
\(250\) 0 0
\(251\) − 12.4430i − 0.0495737i −0.999693 0.0247869i \(-0.992109\pi\)
0.999693 0.0247869i \(-0.00789071\pi\)
\(252\) 0 0
\(253\) −216.796 −0.856900
\(254\) 0 0
\(255\) 14.3876i 0.0564219i
\(256\) 0 0
\(257\) 169.884 0.661029 0.330514 0.943801i \(-0.392778\pi\)
0.330514 + 0.943801i \(0.392778\pi\)
\(258\) 0 0
\(259\) 111.347i 0.429912i
\(260\) 0 0
\(261\) −73.5587 −0.281834
\(262\) 0 0
\(263\) 118.762i 0.451565i 0.974178 + 0.225782i \(0.0724938\pi\)
−0.974178 + 0.225782i \(0.927506\pi\)
\(264\) 0 0
\(265\) 55.7699 0.210452
\(266\) 0 0
\(267\) − 14.8105i − 0.0554700i
\(268\) 0 0
\(269\) −9.37496 −0.0348512 −0.0174256 0.999848i \(-0.505547\pi\)
−0.0174256 + 0.999848i \(0.505547\pi\)
\(270\) 0 0
\(271\) − 203.067i − 0.749324i −0.927161 0.374662i \(-0.877759\pi\)
0.927161 0.374662i \(-0.122241\pi\)
\(272\) 0 0
\(273\) 39.6643 0.145291
\(274\) 0 0
\(275\) 952.124i 3.46227i
\(276\) 0 0
\(277\) 154.419 0.557471 0.278735 0.960368i \(-0.410085\pi\)
0.278735 + 0.960368i \(0.410085\pi\)
\(278\) 0 0
\(279\) − 135.561i − 0.485880i
\(280\) 0 0
\(281\) −217.495 −0.774005 −0.387002 0.922079i \(-0.626489\pi\)
−0.387002 + 0.922079i \(0.626489\pi\)
\(282\) 0 0
\(283\) 431.849i 1.52597i 0.646418 + 0.762983i \(0.276266\pi\)
−0.646418 + 0.762983i \(0.723734\pi\)
\(284\) 0 0
\(285\) −394.896 −1.38560
\(286\) 0 0
\(287\) − 2.04589i − 0.00712855i
\(288\) 0 0
\(289\) −288.674 −0.998873
\(290\) 0 0
\(291\) − 264.908i − 0.910336i
\(292\) 0 0
\(293\) −212.019 −0.723616 −0.361808 0.932253i \(-0.617840\pi\)
−0.361808 + 0.932253i \(0.617840\pi\)
\(294\) 0 0
\(295\) 420.820i 1.42651i
\(296\) 0 0
\(297\) 385.554 1.29816
\(298\) 0 0
\(299\) − 96.7206i − 0.323480i
\(300\) 0 0
\(301\) 110.574 0.367356
\(302\) 0 0
\(303\) − 107.086i − 0.353418i
\(304\) 0 0
\(305\) −946.741 −3.10407
\(306\) 0 0
\(307\) − 192.868i − 0.628234i −0.949384 0.314117i \(-0.898292\pi\)
0.949384 0.314117i \(-0.101708\pi\)
\(308\) 0 0
\(309\) 47.8689 0.154915
\(310\) 0 0
\(311\) − 269.814i − 0.867569i −0.901017 0.433785i \(-0.857178\pi\)
0.901017 0.433785i \(-0.142822\pi\)
\(312\) 0 0
\(313\) −236.490 −0.755558 −0.377779 0.925896i \(-0.623312\pi\)
−0.377779 + 0.925896i \(0.623312\pi\)
\(314\) 0 0
\(315\) − 64.5855i − 0.205033i
\(316\) 0 0
\(317\) −249.012 −0.785526 −0.392763 0.919640i \(-0.628481\pi\)
−0.392763 + 0.919640i \(0.628481\pi\)
\(318\) 0 0
\(319\) − 391.148i − 1.22617i
\(320\) 0 0
\(321\) −165.683 −0.516146
\(322\) 0 0
\(323\) 8.93945i 0.0276763i
\(324\) 0 0
\(325\) −424.778 −1.30701
\(326\) 0 0
\(327\) 476.273i 1.45649i
\(328\) 0 0
\(329\) 154.652 0.470066
\(330\) 0 0
\(331\) − 465.343i − 1.40587i −0.711254 0.702935i \(-0.751872\pi\)
0.711254 0.702935i \(-0.248128\pi\)
\(332\) 0 0
\(333\) −104.105 −0.312628
\(334\) 0 0
\(335\) 689.308i 2.05763i
\(336\) 0 0
\(337\) 626.243 1.85829 0.929144 0.369718i \(-0.120546\pi\)
0.929144 + 0.369718i \(0.120546\pi\)
\(338\) 0 0
\(339\) 216.516i 0.638690i
\(340\) 0 0
\(341\) 720.842 2.11391
\(342\) 0 0
\(343\) 18.5203i 0.0539949i
\(344\) 0 0
\(345\) 415.510 1.20438
\(346\) 0 0
\(347\) − 450.576i − 1.29849i −0.760580 0.649245i \(-0.775085\pi\)
0.760580 0.649245i \(-0.224915\pi\)
\(348\) 0 0
\(349\) −64.7762 −0.185605 −0.0928025 0.995685i \(-0.529583\pi\)
−0.0928025 + 0.995685i \(0.529583\pi\)
\(350\) 0 0
\(351\) 172.010i 0.490057i
\(352\) 0 0
\(353\) 589.179 1.66906 0.834532 0.550960i \(-0.185738\pi\)
0.834532 + 0.550960i \(0.185738\pi\)
\(354\) 0 0
\(355\) − 908.760i − 2.55989i
\(356\) 0 0
\(357\) 3.85737 0.0108050
\(358\) 0 0
\(359\) − 406.643i − 1.13271i −0.824161 0.566355i \(-0.808353\pi\)
0.824161 0.566355i \(-0.191647\pi\)
\(360\) 0 0
\(361\) 115.639 0.320329
\(362\) 0 0
\(363\) 132.894i 0.366099i
\(364\) 0 0
\(365\) 98.4407 0.269701
\(366\) 0 0
\(367\) − 430.196i − 1.17220i −0.810240 0.586098i \(-0.800663\pi\)
0.810240 0.586098i \(-0.199337\pi\)
\(368\) 0 0
\(369\) 1.91283 0.00518381
\(370\) 0 0
\(371\) − 14.9522i − 0.0403023i
\(372\) 0 0
\(373\) 2.60758 0.00699083 0.00349542 0.999994i \(-0.498887\pi\)
0.00349542 + 0.999994i \(0.498887\pi\)
\(374\) 0 0
\(375\) − 1194.58i − 3.18555i
\(376\) 0 0
\(377\) 174.506 0.462880
\(378\) 0 0
\(379\) 359.118i 0.947541i 0.880648 + 0.473771i \(0.157107\pi\)
−0.880648 + 0.473771i \(0.842893\pi\)
\(380\) 0 0
\(381\) 238.574 0.626178
\(382\) 0 0
\(383\) 470.758i 1.22913i 0.788865 + 0.614567i \(0.210669\pi\)
−0.788865 + 0.614567i \(0.789331\pi\)
\(384\) 0 0
\(385\) 343.433 0.892034
\(386\) 0 0
\(387\) 103.382i 0.267138i
\(388\) 0 0
\(389\) −180.270 −0.463418 −0.231709 0.972785i \(-0.574432\pi\)
−0.231709 + 0.972785i \(0.574432\pi\)
\(390\) 0 0
\(391\) − 9.40612i − 0.0240566i
\(392\) 0 0
\(393\) −148.390 −0.377582
\(394\) 0 0
\(395\) − 199.123i − 0.504110i
\(396\) 0 0
\(397\) −248.223 −0.625248 −0.312624 0.949877i \(-0.601208\pi\)
−0.312624 + 0.949877i \(0.601208\pi\)
\(398\) 0 0
\(399\) 105.873i 0.265347i
\(400\) 0 0
\(401\) 369.502 0.921452 0.460726 0.887542i \(-0.347589\pi\)
0.460726 + 0.887542i \(0.347589\pi\)
\(402\) 0 0
\(403\) 321.594i 0.798001i
\(404\) 0 0
\(405\) −519.253 −1.28211
\(406\) 0 0
\(407\) − 553.578i − 1.36014i
\(408\) 0 0
\(409\) 479.494 1.17236 0.586179 0.810182i \(-0.300632\pi\)
0.586179 + 0.810182i \(0.300632\pi\)
\(410\) 0 0
\(411\) − 450.695i − 1.09658i
\(412\) 0 0
\(413\) 112.824 0.273181
\(414\) 0 0
\(415\) 1493.16i 3.59797i
\(416\) 0 0
\(417\) −31.4328 −0.0753785
\(418\) 0 0
\(419\) − 564.644i − 1.34760i −0.738914 0.673800i \(-0.764661\pi\)
0.738914 0.673800i \(-0.235339\pi\)
\(420\) 0 0
\(421\) 544.282 1.29283 0.646415 0.762986i \(-0.276267\pi\)
0.646415 + 0.762986i \(0.276267\pi\)
\(422\) 0 0
\(423\) 144.593i 0.341827i
\(424\) 0 0
\(425\) −41.3098 −0.0971996
\(426\) 0 0
\(427\) 253.826i 0.594440i
\(428\) 0 0
\(429\) −197.196 −0.459665
\(430\) 0 0
\(431\) − 90.3539i − 0.209638i −0.994491 0.104819i \(-0.966574\pi\)
0.994491 0.104819i \(-0.0334263\pi\)
\(432\) 0 0
\(433\) −131.383 −0.303425 −0.151713 0.988425i \(-0.548479\pi\)
−0.151713 + 0.988425i \(0.548479\pi\)
\(434\) 0 0
\(435\) 749.674i 1.72339i
\(436\) 0 0
\(437\) 258.170 0.590777
\(438\) 0 0
\(439\) 128.655i 0.293064i 0.989206 + 0.146532i \(0.0468111\pi\)
−0.989206 + 0.146532i \(0.953189\pi\)
\(440\) 0 0
\(441\) −17.3157 −0.0392646
\(442\) 0 0
\(443\) 193.210i 0.436139i 0.975933 + 0.218070i \(0.0699760\pi\)
−0.975933 + 0.218070i \(0.930024\pi\)
\(444\) 0 0
\(445\) 57.2111 0.128564
\(446\) 0 0
\(447\) − 443.954i − 0.993186i
\(448\) 0 0
\(449\) −31.5046 −0.0701661 −0.0350830 0.999384i \(-0.511170\pi\)
−0.0350830 + 0.999384i \(0.511170\pi\)
\(450\) 0 0
\(451\) 10.1714i 0.0225531i
\(452\) 0 0
\(453\) 353.651 0.780686
\(454\) 0 0
\(455\) 153.218i 0.336743i
\(456\) 0 0
\(457\) −426.401 −0.933043 −0.466521 0.884510i \(-0.654493\pi\)
−0.466521 + 0.884510i \(0.654493\pi\)
\(458\) 0 0
\(459\) 16.7280i 0.0364445i
\(460\) 0 0
\(461\) 430.182 0.933150 0.466575 0.884482i \(-0.345488\pi\)
0.466575 + 0.884482i \(0.345488\pi\)
\(462\) 0 0
\(463\) − 183.454i − 0.396230i −0.980179 0.198115i \(-0.936518\pi\)
0.980179 0.198115i \(-0.0634819\pi\)
\(464\) 0 0
\(465\) −1381.56 −2.97111
\(466\) 0 0
\(467\) − 110.381i − 0.236362i −0.992992 0.118181i \(-0.962294\pi\)
0.992992 0.118181i \(-0.0377063\pi\)
\(468\) 0 0
\(469\) 184.807 0.394044
\(470\) 0 0
\(471\) − 484.604i − 1.02888i
\(472\) 0 0
\(473\) −549.735 −1.16223
\(474\) 0 0
\(475\) − 1133.83i − 2.38701i
\(476\) 0 0
\(477\) 13.9797 0.0293075
\(478\) 0 0
\(479\) − 515.593i − 1.07639i −0.842819 0.538197i \(-0.819106\pi\)
0.842819 0.538197i \(-0.180894\pi\)
\(480\) 0 0
\(481\) 246.972 0.513455
\(482\) 0 0
\(483\) − 111.400i − 0.230642i
\(484\) 0 0
\(485\) 1023.30 2.10991
\(486\) 0 0
\(487\) 59.2995i 0.121765i 0.998145 + 0.0608824i \(0.0193915\pi\)
−0.998145 + 0.0608824i \(0.980609\pi\)
\(488\) 0 0
\(489\) −45.3927 −0.0928276
\(490\) 0 0
\(491\) 556.042i 1.13247i 0.824244 + 0.566234i \(0.191600\pi\)
−0.824244 + 0.566234i \(0.808400\pi\)
\(492\) 0 0
\(493\) 16.9708 0.0344234
\(494\) 0 0
\(495\) 321.096i 0.648678i
\(496\) 0 0
\(497\) −243.643 −0.490227
\(498\) 0 0
\(499\) − 150.429i − 0.301461i −0.988575 0.150730i \(-0.951837\pi\)
0.988575 0.150730i \(-0.0481626\pi\)
\(500\) 0 0
\(501\) 0.227601 0.000454293 0
\(502\) 0 0
\(503\) − 415.893i − 0.826825i −0.910544 0.413412i \(-0.864337\pi\)
0.910544 0.413412i \(-0.135663\pi\)
\(504\) 0 0
\(505\) 413.658 0.819124
\(506\) 0 0
\(507\) 343.762i 0.678032i
\(508\) 0 0
\(509\) 622.234 1.22246 0.611232 0.791452i \(-0.290674\pi\)
0.611232 + 0.791452i \(0.290674\pi\)
\(510\) 0 0
\(511\) − 26.3924i − 0.0516485i
\(512\) 0 0
\(513\) −459.134 −0.894999
\(514\) 0 0
\(515\) 184.911i 0.359051i
\(516\) 0 0
\(517\) −768.871 −1.48718
\(518\) 0 0
\(519\) − 110.589i − 0.213081i
\(520\) 0 0
\(521\) −644.142 −1.23636 −0.618179 0.786037i \(-0.712129\pi\)
−0.618179 + 0.786037i \(0.712129\pi\)
\(522\) 0 0
\(523\) 889.589i 1.70093i 0.526028 + 0.850467i \(0.323681\pi\)
−0.526028 + 0.850467i \(0.676319\pi\)
\(524\) 0 0
\(525\) −489.248 −0.931900
\(526\) 0 0
\(527\) 31.2752i 0.0593457i
\(528\) 0 0
\(529\) 257.353 0.486490
\(530\) 0 0
\(531\) 105.486i 0.198655i
\(532\) 0 0
\(533\) −4.53785 −0.00851380
\(534\) 0 0
\(535\) − 640.012i − 1.19628i
\(536\) 0 0
\(537\) 487.529 0.907875
\(538\) 0 0
\(539\) − 92.0759i − 0.170827i
\(540\) 0 0
\(541\) 832.015 1.53792 0.768961 0.639296i \(-0.220774\pi\)
0.768961 + 0.639296i \(0.220774\pi\)
\(542\) 0 0
\(543\) 529.207i 0.974599i
\(544\) 0 0
\(545\) −1839.78 −3.37575
\(546\) 0 0
\(547\) − 1014.85i − 1.85531i −0.373441 0.927654i \(-0.621822\pi\)
0.373441 0.927654i \(-0.378178\pi\)
\(548\) 0 0
\(549\) −237.316 −0.432270
\(550\) 0 0
\(551\) 465.796i 0.845365i
\(552\) 0 0
\(553\) −53.3859 −0.0965387
\(554\) 0 0
\(555\) 1060.99i 1.91169i
\(556\) 0 0
\(557\) 473.280 0.849694 0.424847 0.905265i \(-0.360328\pi\)
0.424847 + 0.905265i \(0.360328\pi\)
\(558\) 0 0
\(559\) − 245.257i − 0.438742i
\(560\) 0 0
\(561\) −19.1775 −0.0341844
\(562\) 0 0
\(563\) 102.364i 0.181818i 0.995859 + 0.0909092i \(0.0289773\pi\)
−0.995859 + 0.0909092i \(0.971023\pi\)
\(564\) 0 0
\(565\) −836.373 −1.48031
\(566\) 0 0
\(567\) 139.214i 0.245527i
\(568\) 0 0
\(569\) −68.4059 −0.120221 −0.0601106 0.998192i \(-0.519145\pi\)
−0.0601106 + 0.998192i \(0.519145\pi\)
\(570\) 0 0
\(571\) 701.419i 1.22841i 0.789148 + 0.614203i \(0.210522\pi\)
−0.789148 + 0.614203i \(0.789478\pi\)
\(572\) 0 0
\(573\) −805.971 −1.40658
\(574\) 0 0
\(575\) 1193.02i 2.07482i
\(576\) 0 0
\(577\) 44.4710 0.0770727 0.0385364 0.999257i \(-0.487730\pi\)
0.0385364 + 0.999257i \(0.487730\pi\)
\(578\) 0 0
\(579\) 79.0421i 0.136515i
\(580\) 0 0
\(581\) 400.322 0.689022
\(582\) 0 0
\(583\) 74.3367i 0.127507i
\(584\) 0 0
\(585\) −143.253 −0.244876
\(586\) 0 0
\(587\) − 99.0239i − 0.168695i −0.996436 0.0843475i \(-0.973119\pi\)
0.996436 0.0843475i \(-0.0268806\pi\)
\(588\) 0 0
\(589\) −858.410 −1.45740
\(590\) 0 0
\(591\) 675.066i 1.14224i
\(592\) 0 0
\(593\) −443.870 −0.748516 −0.374258 0.927325i \(-0.622103\pi\)
−0.374258 + 0.927325i \(0.622103\pi\)
\(594\) 0 0
\(595\) 14.9005i 0.0250429i
\(596\) 0 0
\(597\) 335.481 0.561945
\(598\) 0 0
\(599\) 969.972i 1.61932i 0.586900 + 0.809659i \(0.300348\pi\)
−0.586900 + 0.809659i \(0.699652\pi\)
\(600\) 0 0
\(601\) −58.0231 −0.0965443 −0.0482722 0.998834i \(-0.515371\pi\)
−0.0482722 + 0.998834i \(0.515371\pi\)
\(602\) 0 0
\(603\) 172.786i 0.286545i
\(604\) 0 0
\(605\) −513.352 −0.848515
\(606\) 0 0
\(607\) 67.3822i 0.111009i 0.998458 + 0.0555043i \(0.0176767\pi\)
−0.998458 + 0.0555043i \(0.982323\pi\)
\(608\) 0 0
\(609\) 200.991 0.330034
\(610\) 0 0
\(611\) − 343.022i − 0.561411i
\(612\) 0 0
\(613\) −521.917 −0.851415 −0.425707 0.904861i \(-0.639975\pi\)
−0.425707 + 0.904861i \(0.639975\pi\)
\(614\) 0 0
\(615\) − 19.4946i − 0.0316985i
\(616\) 0 0
\(617\) 165.257 0.267840 0.133920 0.990992i \(-0.457243\pi\)
0.133920 + 0.990992i \(0.457243\pi\)
\(618\) 0 0
\(619\) 205.392i 0.331812i 0.986142 + 0.165906i \(0.0530549\pi\)
−0.986142 + 0.165906i \(0.946945\pi\)
\(620\) 0 0
\(621\) 483.103 0.777943
\(622\) 0 0
\(623\) − 15.3385i − 0.0246205i
\(624\) 0 0
\(625\) 2804.90 4.48783
\(626\) 0 0
\(627\) − 526.363i − 0.839495i
\(628\) 0 0
\(629\) 24.0181 0.0381846
\(630\) 0 0
\(631\) − 631.682i − 1.00108i −0.865713 0.500540i \(-0.833135\pi\)
0.865713 0.500540i \(-0.166865\pi\)
\(632\) 0 0
\(633\) −633.543 −1.00086
\(634\) 0 0
\(635\) 921.580i 1.45131i
\(636\) 0 0
\(637\) 41.0785 0.0644874
\(638\) 0 0
\(639\) − 227.796i − 0.356488i
\(640\) 0 0
\(641\) 817.936 1.27603 0.638016 0.770023i \(-0.279755\pi\)
0.638016 + 0.770023i \(0.279755\pi\)
\(642\) 0 0
\(643\) − 607.082i − 0.944140i −0.881561 0.472070i \(-0.843507\pi\)
0.881561 0.472070i \(-0.156493\pi\)
\(644\) 0 0
\(645\) 1053.62 1.63352
\(646\) 0 0
\(647\) 621.063i 0.959911i 0.877293 + 0.479956i \(0.159347\pi\)
−0.877293 + 0.479956i \(0.840653\pi\)
\(648\) 0 0
\(649\) −560.919 −0.864282
\(650\) 0 0
\(651\) 370.404i 0.568977i
\(652\) 0 0
\(653\) −850.083 −1.30181 −0.650906 0.759159i \(-0.725611\pi\)
−0.650906 + 0.759159i \(0.725611\pi\)
\(654\) 0 0
\(655\) − 573.211i − 0.875131i
\(656\) 0 0
\(657\) 24.6758 0.0375583
\(658\) 0 0
\(659\) 331.172i 0.502537i 0.967917 + 0.251268i \(0.0808477\pi\)
−0.967917 + 0.251268i \(0.919152\pi\)
\(660\) 0 0
\(661\) 464.323 0.702456 0.351228 0.936290i \(-0.385764\pi\)
0.351228 + 0.936290i \(0.385764\pi\)
\(662\) 0 0
\(663\) − 8.55577i − 0.0129046i
\(664\) 0 0
\(665\) −408.975 −0.615000
\(666\) 0 0
\(667\) − 490.112i − 0.734801i
\(668\) 0 0
\(669\) −145.105 −0.216898
\(670\) 0 0
\(671\) − 1261.93i − 1.88067i
\(672\) 0 0
\(673\) −844.655 −1.25506 −0.627529 0.778593i \(-0.715934\pi\)
−0.627529 + 0.778593i \(0.715934\pi\)
\(674\) 0 0
\(675\) − 2121.69i − 3.14325i
\(676\) 0 0
\(677\) −664.414 −0.981410 −0.490705 0.871326i \(-0.663261\pi\)
−0.490705 + 0.871326i \(0.663261\pi\)
\(678\) 0 0
\(679\) − 274.353i − 0.404054i
\(680\) 0 0
\(681\) −702.778 −1.03198
\(682\) 0 0
\(683\) − 601.901i − 0.881261i −0.897689 0.440631i \(-0.854755\pi\)
0.897689 0.440631i \(-0.145245\pi\)
\(684\) 0 0
\(685\) 1740.98 2.54157
\(686\) 0 0
\(687\) − 16.0742i − 0.0233977i
\(688\) 0 0
\(689\) −33.1643 −0.0481340
\(690\) 0 0
\(691\) 579.981i 0.839336i 0.907678 + 0.419668i \(0.137854\pi\)
−0.907678 + 0.419668i \(0.862146\pi\)
\(692\) 0 0
\(693\) 86.0872 0.124224
\(694\) 0 0
\(695\) − 121.421i − 0.174706i
\(696\) 0 0
\(697\) −0.441308 −0.000633154 0
\(698\) 0 0
\(699\) 362.889i 0.519154i
\(700\) 0 0
\(701\) 863.561 1.23190 0.615949 0.787786i \(-0.288773\pi\)
0.615949 + 0.787786i \(0.288773\pi\)
\(702\) 0 0
\(703\) 659.225i 0.937731i
\(704\) 0 0
\(705\) 1473.62 2.09024
\(706\) 0 0
\(707\) − 110.903i − 0.156865i
\(708\) 0 0
\(709\) −293.022 −0.413290 −0.206645 0.978416i \(-0.566254\pi\)
−0.206645 + 0.978416i \(0.566254\pi\)
\(710\) 0 0
\(711\) − 49.9136i − 0.0702019i
\(712\) 0 0
\(713\) 903.222 1.26679
\(714\) 0 0
\(715\) − 761.744i − 1.06538i
\(716\) 0 0
\(717\) −131.298 −0.183122
\(718\) 0 0
\(719\) − 381.821i − 0.531044i −0.964105 0.265522i \(-0.914456\pi\)
0.964105 0.265522i \(-0.0855443\pi\)
\(720\) 0 0
\(721\) 49.5756 0.0687594
\(722\) 0 0
\(723\) − 239.385i − 0.331099i
\(724\) 0 0
\(725\) −2152.48 −2.96893
\(726\) 0 0
\(727\) − 387.165i − 0.532552i −0.963897 0.266276i \(-0.914207\pi\)
0.963897 0.266276i \(-0.0857932\pi\)
\(728\) 0 0
\(729\) −804.088 −1.10300
\(730\) 0 0
\(731\) − 23.8514i − 0.0326284i
\(732\) 0 0
\(733\) 59.1034 0.0806322 0.0403161 0.999187i \(-0.487164\pi\)
0.0403161 + 0.999187i \(0.487164\pi\)
\(734\) 0 0
\(735\) 176.473i 0.240099i
\(736\) 0 0
\(737\) −918.790 −1.24666
\(738\) 0 0
\(739\) − 1285.33i − 1.73928i −0.493687 0.869640i \(-0.664351\pi\)
0.493687 0.869640i \(-0.335649\pi\)
\(740\) 0 0
\(741\) 234.830 0.316910
\(742\) 0 0
\(743\) 1450.21i 1.95183i 0.218141 + 0.975917i \(0.430001\pi\)
−0.218141 + 0.975917i \(0.569999\pi\)
\(744\) 0 0
\(745\) 1714.94 2.30193
\(746\) 0 0
\(747\) 374.284i 0.501050i
\(748\) 0 0
\(749\) −171.590 −0.229092
\(750\) 0 0
\(751\) 190.150i 0.253195i 0.991954 + 0.126598i \(0.0404057\pi\)
−0.991954 + 0.126598i \(0.959594\pi\)
\(752\) 0 0
\(753\) −31.7878 −0.0422148
\(754\) 0 0
\(755\) 1366.11i 1.80941i
\(756\) 0 0
\(757\) 441.225 0.582859 0.291430 0.956592i \(-0.405869\pi\)
0.291430 + 0.956592i \(0.405869\pi\)
\(758\) 0 0
\(759\) 553.841i 0.729698i
\(760\) 0 0
\(761\) −1044.28 −1.37225 −0.686123 0.727486i \(-0.740689\pi\)
−0.686123 + 0.727486i \(0.740689\pi\)
\(762\) 0 0
\(763\) 493.254i 0.646466i
\(764\) 0 0
\(765\) −13.9314 −0.0182110
\(766\) 0 0
\(767\) − 250.247i − 0.326267i
\(768\) 0 0
\(769\) −1141.60 −1.48452 −0.742262 0.670110i \(-0.766247\pi\)
−0.742262 + 0.670110i \(0.766247\pi\)
\(770\) 0 0
\(771\) − 433.998i − 0.562903i
\(772\) 0 0
\(773\) −211.566 −0.273694 −0.136847 0.990592i \(-0.543697\pi\)
−0.136847 + 0.990592i \(0.543697\pi\)
\(774\) 0 0
\(775\) − 3966.77i − 5.11841i
\(776\) 0 0
\(777\) 284.455 0.366095
\(778\) 0 0
\(779\) − 12.1126i − 0.0155489i
\(780\) 0 0
\(781\) 1211.30 1.55096
\(782\) 0 0
\(783\) 871.625i 1.11319i
\(784\) 0 0
\(785\) 1871.96 2.38466
\(786\) 0 0
\(787\) 523.213i 0.664820i 0.943135 + 0.332410i \(0.107862\pi\)
−0.943135 + 0.332410i \(0.892138\pi\)
\(788\) 0 0
\(789\) 303.396 0.384533
\(790\) 0 0
\(791\) 224.235i 0.283483i
\(792\) 0 0
\(793\) 562.993 0.709953
\(794\) 0 0
\(795\) − 142.474i − 0.179212i
\(796\) 0 0
\(797\) −503.460 −0.631693 −0.315847 0.948810i \(-0.602289\pi\)
−0.315847 + 0.948810i \(0.602289\pi\)
\(798\) 0 0
\(799\) − 33.3590i − 0.0417510i
\(800\) 0 0
\(801\) 14.3409 0.0179038
\(802\) 0 0
\(803\) 131.213i 0.163404i
\(804\) 0 0
\(805\) 430.325 0.534565
\(806\) 0 0
\(807\) 23.9499i 0.0296777i
\(808\) 0 0
\(809\) 100.517 0.124248 0.0621242 0.998068i \(-0.480212\pi\)
0.0621242 + 0.998068i \(0.480212\pi\)
\(810\) 0 0
\(811\) 759.830i 0.936905i 0.883489 + 0.468453i \(0.155188\pi\)
−0.883489 + 0.468453i \(0.844812\pi\)
\(812\) 0 0
\(813\) −518.768 −0.638091
\(814\) 0 0
\(815\) − 175.346i − 0.215149i
\(816\) 0 0
\(817\) 654.648 0.801283
\(818\) 0 0
\(819\) 38.4067i 0.0468946i
\(820\) 0 0
\(821\) −749.705 −0.913160 −0.456580 0.889682i \(-0.650926\pi\)
−0.456580 + 0.889682i \(0.650926\pi\)
\(822\) 0 0
\(823\) 424.800i 0.516160i 0.966124 + 0.258080i \(0.0830899\pi\)
−0.966124 + 0.258080i \(0.916910\pi\)
\(824\) 0 0
\(825\) 2432.36 2.94832
\(826\) 0 0
\(827\) − 575.272i − 0.695613i −0.937566 0.347806i \(-0.886927\pi\)
0.937566 0.347806i \(-0.113073\pi\)
\(828\) 0 0
\(829\) 157.635 0.190151 0.0950756 0.995470i \(-0.469691\pi\)
0.0950756 + 0.995470i \(0.469691\pi\)
\(830\) 0 0
\(831\) − 394.490i − 0.474718i
\(832\) 0 0
\(833\) 3.99490 0.00479580
\(834\) 0 0
\(835\) 0.879193i 0.00105293i
\(836\) 0 0
\(837\) −1606.31 −1.91913
\(838\) 0 0
\(839\) − 759.895i − 0.905716i −0.891583 0.452858i \(-0.850405\pi\)
0.891583 0.452858i \(-0.149595\pi\)
\(840\) 0 0
\(841\) 43.2720 0.0514530
\(842\) 0 0
\(843\) 555.628i 0.659108i
\(844\) 0 0
\(845\) −1327.91 −1.57149
\(846\) 0 0
\(847\) 137.632i 0.162493i
\(848\) 0 0
\(849\) 1103.23 1.29945
\(850\) 0 0
\(851\) − 693.638i − 0.815086i
\(852\) 0 0
\(853\) 315.373 0.369722 0.184861 0.982765i \(-0.440817\pi\)
0.184861 + 0.982765i \(0.440817\pi\)
\(854\) 0 0
\(855\) − 382.375i − 0.447222i
\(856\) 0 0
\(857\) 1222.53 1.42652 0.713260 0.700900i \(-0.247218\pi\)
0.713260 + 0.700900i \(0.247218\pi\)
\(858\) 0 0
\(859\) − 1008.23i − 1.17373i −0.809687 0.586863i \(-0.800363\pi\)
0.809687 0.586863i \(-0.199637\pi\)
\(860\) 0 0
\(861\) −5.22658 −0.00607036
\(862\) 0 0
\(863\) − 1140.31i − 1.32133i −0.750679 0.660667i \(-0.770274\pi\)
0.750679 0.660667i \(-0.229726\pi\)
\(864\) 0 0
\(865\) 427.192 0.493864
\(866\) 0 0
\(867\) 737.467i 0.850597i
\(868\) 0 0
\(869\) 265.415 0.305426
\(870\) 0 0
\(871\) − 409.906i − 0.470616i
\(872\) 0 0
\(873\) 256.508 0.293824
\(874\) 0 0
\(875\) − 1237.17i − 1.41391i
\(876\) 0 0
\(877\) −306.788 −0.349816 −0.174908 0.984585i \(-0.555963\pi\)
−0.174908 + 0.984585i \(0.555963\pi\)
\(878\) 0 0
\(879\) 541.640i 0.616200i
\(880\) 0 0
\(881\) −248.968 −0.282597 −0.141299 0.989967i \(-0.545128\pi\)
−0.141299 + 0.989967i \(0.545128\pi\)
\(882\) 0 0
\(883\) 458.136i 0.518841i 0.965765 + 0.259420i \(0.0835315\pi\)
−0.965765 + 0.259420i \(0.916469\pi\)
\(884\) 0 0
\(885\) 1075.06 1.21475
\(886\) 0 0
\(887\) 312.628i 0.352456i 0.984349 + 0.176228i \(0.0563895\pi\)
−0.984349 + 0.176228i \(0.943610\pi\)
\(888\) 0 0
\(889\) 247.080 0.277930
\(890\) 0 0
\(891\) − 692.121i − 0.776791i
\(892\) 0 0
\(893\) 915.605 1.02531
\(894\) 0 0
\(895\) 1883.26i 2.10420i
\(896\) 0 0
\(897\) −247.089 −0.275462
\(898\) 0 0
\(899\) 1629.61i 1.81270i
\(900\) 0 0
\(901\) −3.22525 −0.00357963
\(902\) 0 0
\(903\) − 282.481i − 0.312825i
\(904\) 0 0
\(905\) −2044.26 −2.25885
\(906\) 0 0
\(907\) 375.701i 0.414224i 0.978317 + 0.207112i \(0.0664064\pi\)
−0.978317 + 0.207112i \(0.933594\pi\)
\(908\) 0 0
\(909\) 103.690 0.114071
\(910\) 0 0
\(911\) 41.9614i 0.0460608i 0.999735 + 0.0230304i \(0.00733145\pi\)
−0.999735 + 0.0230304i \(0.992669\pi\)
\(912\) 0 0
\(913\) −1990.25 −2.17991
\(914\) 0 0
\(915\) 2418.61i 2.64329i
\(916\) 0 0
\(917\) −153.680 −0.167590
\(918\) 0 0
\(919\) 1770.71i 1.92678i 0.268101 + 0.963391i \(0.413604\pi\)
−0.268101 + 0.963391i \(0.586396\pi\)
\(920\) 0 0
\(921\) −492.713 −0.534976
\(922\) 0 0
\(923\) 540.407i 0.585489i
\(924\) 0 0
\(925\) −3046.32 −3.29332
\(926\) 0 0
\(927\) 46.3511i 0.0500012i
\(928\) 0 0
\(929\) 1668.30 1.79581 0.897903 0.440193i \(-0.145090\pi\)
0.897903 + 0.440193i \(0.145090\pi\)
\(930\) 0 0
\(931\) 109.648i 0.117774i
\(932\) 0 0
\(933\) −689.285 −0.738784
\(934\) 0 0
\(935\) − 74.0800i − 0.0792300i
\(936\) 0 0
\(937\) −183.028 −0.195334 −0.0976668 0.995219i \(-0.531138\pi\)
−0.0976668 + 0.995219i \(0.531138\pi\)
\(938\) 0 0
\(939\) 604.152i 0.643400i
\(940\) 0 0
\(941\) 255.475 0.271493 0.135747 0.990744i \(-0.456657\pi\)
0.135747 + 0.990744i \(0.456657\pi\)
\(942\) 0 0
\(943\) 12.7449i 0.0135153i
\(944\) 0 0
\(945\) −765.298 −0.809839
\(946\) 0 0
\(947\) − 187.173i − 0.197648i −0.995105 0.0988242i \(-0.968492\pi\)
0.995105 0.0988242i \(-0.0315082\pi\)
\(948\) 0 0
\(949\) −58.5391 −0.0616851
\(950\) 0 0
\(951\) 636.142i 0.668920i
\(952\) 0 0
\(953\) 362.350 0.380220 0.190110 0.981763i \(-0.439115\pi\)
0.190110 + 0.981763i \(0.439115\pi\)
\(954\) 0 0
\(955\) − 3113.36i − 3.26007i
\(956\) 0 0
\(957\) −999.253 −1.04415
\(958\) 0 0
\(959\) − 466.764i − 0.486720i
\(960\) 0 0
\(961\) −2042.20 −2.12508
\(962\) 0 0
\(963\) − 160.430i − 0.166594i
\(964\) 0 0
\(965\) −305.330 −0.316404
\(966\) 0 0
\(967\) − 1473.76i − 1.52406i −0.647545 0.762028i \(-0.724204\pi\)
0.647545 0.762028i \(-0.275796\pi\)
\(968\) 0 0
\(969\) 22.8373 0.0235680
\(970\) 0 0
\(971\) − 223.833i − 0.230518i −0.993335 0.115259i \(-0.963230\pi\)
0.993335 0.115259i \(-0.0367698\pi\)
\(972\) 0 0
\(973\) −32.5535 −0.0334568
\(974\) 0 0
\(975\) 1085.17i 1.11299i
\(976\) 0 0
\(977\) 825.305 0.844734 0.422367 0.906425i \(-0.361199\pi\)
0.422367 + 0.906425i \(0.361199\pi\)
\(978\) 0 0
\(979\) 76.2576i 0.0778934i
\(980\) 0 0
\(981\) −461.172 −0.470104
\(982\) 0 0
\(983\) 411.945i 0.419069i 0.977801 + 0.209535i \(0.0671949\pi\)
−0.977801 + 0.209535i \(0.932805\pi\)
\(984\) 0 0
\(985\) −2607.69 −2.64741
\(986\) 0 0
\(987\) − 395.084i − 0.400287i
\(988\) 0 0
\(989\) −688.823 −0.696484
\(990\) 0 0
\(991\) 1607.53i 1.62213i 0.584959 + 0.811063i \(0.301111\pi\)
−0.584959 + 0.811063i \(0.698889\pi\)
\(992\) 0 0
\(993\) −1188.80 −1.19718
\(994\) 0 0
\(995\) 1295.92i 1.30243i
\(996\) 0 0
\(997\) 701.364 0.703474 0.351737 0.936099i \(-0.385591\pi\)
0.351737 + 0.936099i \(0.385591\pi\)
\(998\) 0 0
\(999\) 1233.58i 1.23482i
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 224.3.d.b.127.3 8
3.2 odd 2 2016.3.m.c.127.1 8
4.3 odd 2 inner 224.3.d.b.127.6 yes 8
7.6 odd 2 1568.3.d.n.1471.6 8
8.3 odd 2 448.3.d.e.127.3 8
8.5 even 2 448.3.d.e.127.6 8
12.11 even 2 2016.3.m.c.127.2 8
16.3 odd 4 1792.3.g.d.127.3 8
16.5 even 4 1792.3.g.d.127.4 8
16.11 odd 4 1792.3.g.f.127.6 8
16.13 even 4 1792.3.g.f.127.5 8
28.27 even 2 1568.3.d.n.1471.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.3.d.b.127.3 8 1.1 even 1 trivial
224.3.d.b.127.6 yes 8 4.3 odd 2 inner
448.3.d.e.127.3 8 8.3 odd 2
448.3.d.e.127.6 8 8.5 even 2
1568.3.d.n.1471.3 8 28.27 even 2
1568.3.d.n.1471.6 8 7.6 odd 2
1792.3.g.d.127.3 8 16.3 odd 4
1792.3.g.d.127.4 8 16.5 even 4
1792.3.g.f.127.5 8 16.13 even 4
1792.3.g.f.127.6 8 16.11 odd 4
2016.3.m.c.127.1 8 3.2 odd 2
2016.3.m.c.127.2 8 12.11 even 2