Properties

Label 1792.3.g.d.127.3
Level 1792
Weight 3
Character 1792.127
Analytic conductor 48.828
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1792.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.8284633734\)
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: \( 2^{12} \)
Twist minimal: no (minimal twist has level 224)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.3
Root \(1.27733i\) of defining polynomial
Character \(\chi\) \(=\) 1792.127
Dual form 1792.3.g.d.127.4

$q$-expansion

\(f(q)\) \(=\) \(q-2.55467 q^{3} -9.86836i q^{5} -2.64575i q^{7} -2.47367 q^{9} +O(q^{10})\) \(q-2.55467 q^{3} -9.86836i q^{5} -2.64575i q^{7} -2.47367 q^{9} -13.1537 q^{11} -5.86836i q^{13} +25.2104i q^{15} -0.570700 q^{17} -15.6640 q^{19} +6.75902i q^{21} +16.4817i q^{23} -72.3844 q^{25} +29.3114 q^{27} -29.7367i q^{29} +54.8014i q^{31} +33.6034 q^{33} -26.1092 q^{35} +42.0853i q^{37} +14.9917i q^{39} -0.773275 q^{41} -41.7931 q^{43} +24.4110i q^{45} -58.4528i q^{47} -7.00000 q^{49} +1.45795 q^{51} -5.65139i q^{53} +129.805i q^{55} +40.0163 q^{57} -42.6434 q^{59} -95.9371i q^{61} +6.54471i q^{63} -57.9110 q^{65} +69.8503 q^{67} -42.1053i q^{69} -92.0882i q^{71} -9.97539 q^{73} +184.918 q^{75} +34.8014i q^{77} +20.1780i q^{79} -52.6180 q^{81} +151.307 q^{83} +5.63187i q^{85} +75.9674i q^{87} -5.79743 q^{89} -15.5262 q^{91} -140.000i q^{93} +154.578i q^{95} +103.696 q^{97} +32.5379 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{3} + 40q^{9} + O(q^{10}) \) \( 8q - 8q^{3} + 40q^{9} - 32q^{11} - 16q^{17} - 88q^{19} - 104q^{25} - 176q^{27} - 56q^{35} - 144q^{41} + 224q^{43} - 56q^{49} - 16q^{51} + 400q^{57} - 232q^{59} - 304q^{65} + 368q^{67} - 272q^{73} + 664q^{75} + 504q^{81} + 424q^{83} + 80q^{89} - 56q^{91} + 528q^{97} - 544q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) −2.55467 −0.851556 −0.425778 0.904828i \(-0.640000\pi\)
−0.425778 + 0.904828i \(0.640000\pi\)
\(4\) 0 0
\(5\) − 9.86836i − 1.97367i −0.161727 0.986836i \(-0.551706\pi\)
0.161727 0.986836i \(-0.448294\pi\)
\(6\) 0 0
\(7\) − 2.64575i − 0.377964i
\(8\) 0 0
\(9\) −2.47367 −0.274852
\(10\) 0 0
\(11\) −13.1537 −1.19579 −0.597896 0.801574i \(-0.703996\pi\)
−0.597896 + 0.801574i \(0.703996\pi\)
\(12\) 0 0
\(13\) − 5.86836i − 0.451412i −0.974195 0.225706i \(-0.927531\pi\)
0.974195 0.225706i \(-0.0724689\pi\)
\(14\) 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
\(19\) −15.6640 −0.824421 −0.412211 0.911089i \(-0.635243\pi\)
−0.412211 + 0.911089i \(0.635243\pi\)
\(20\) 0 0
\(21\) 6.75902i 0.321858i
\(22\) 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
\(27\) 29.3114 1.08561
\(28\) 0 0
\(29\) − 29.7367i − 1.02540i −0.858567 0.512702i \(-0.828645\pi\)
0.858567 0.512702i \(-0.171355\pi\)
\(30\) 0 0
\(31\) 54.8014i 1.76779i 0.467687 + 0.883894i \(0.345087\pi\)
−0.467687 + 0.883894i \(0.654913\pi\)
\(32\) 0 0
\(33\) 33.6034 1.01828
\(34\) 0 0
\(35\) −26.1092 −0.745978
\(36\) 0 0
\(37\) 42.0853i 1.13744i 0.822531 + 0.568721i \(0.192561\pi\)
−0.822531 + 0.568721i \(0.807439\pi\)
\(38\) 0 0
\(39\) 14.9917i 0.384403i
\(40\) 0 0
\(41\) −0.773275 −0.0188604 −0.00943019 0.999956i \(-0.503002\pi\)
−0.00943019 + 0.999956i \(0.503002\pi\)
\(42\) 0 0
\(43\) −41.7931 −0.971933 −0.485967 0.873977i \(-0.661532\pi\)
−0.485967 + 0.873977i \(0.661532\pi\)
\(44\) 0 0
\(45\) 24.4110i 0.542467i
\(46\) 0 0
\(47\) − 58.4528i − 1.24368i −0.783145 0.621839i \(-0.786386\pi\)
0.783145 0.621839i \(-0.213614\pi\)
\(48\) 0 0
\(49\) −7.00000 −0.142857
\(50\) 0 0
\(51\) 1.45795 0.0285873
\(52\) 0 0
\(53\) − 5.65139i − 0.106630i −0.998578 0.0533150i \(-0.983021\pi\)
0.998578 0.0533150i \(-0.0169787\pi\)
\(54\) 0 0
\(55\) 129.805i 2.36010i
\(56\) 0 0
\(57\) 40.0163 0.702041
\(58\) 0 0
\(59\) −42.6434 −0.722770 −0.361385 0.932417i \(-0.617696\pi\)
−0.361385 + 0.932417i \(0.617696\pi\)
\(60\) 0 0
\(61\) − 95.9371i − 1.57274i −0.617756 0.786370i \(-0.711958\pi\)
0.617756 0.786370i \(-0.288042\pi\)
\(62\) 0 0
\(63\) 6.54471i 0.103884i
\(64\) 0 0
\(65\) −57.9110 −0.890939
\(66\) 0 0
\(67\) 69.8503 1.04254 0.521271 0.853391i \(-0.325458\pi\)
0.521271 + 0.853391i \(0.325458\pi\)
\(68\) 0 0
\(69\) − 42.1053i − 0.610222i
\(70\) 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 \(-0.521765\pi\)
−0.0683246 + 0.997663i \(0.521765\pi\)
\(74\) 0 0
\(75\) 184.918 2.46558
\(76\) 0 0
\(77\) 34.8014i 0.451967i
\(78\) 0 0
\(79\) 20.1780i 0.255417i 0.991812 + 0.127709i \(0.0407622\pi\)
−0.991812 + 0.127709i \(0.959238\pi\)
\(80\) 0 0
\(81\) −52.6180 −0.649604
\(82\) 0 0
\(83\) 151.307 1.82298 0.911491 0.411321i \(-0.134932\pi\)
0.911491 + 0.411321i \(0.134932\pi\)
\(84\) 0 0
\(85\) 5.63187i 0.0662573i
\(86\) 0 0
\(87\) 75.9674i 0.873189i
\(88\) 0 0
\(89\) −5.79743 −0.0651396 −0.0325698 0.999469i \(-0.510369\pi\)
−0.0325698 + 0.999469i \(0.510369\pi\)
\(90\) 0 0
\(91\) −15.5262 −0.170618
\(92\) 0 0
\(93\) − 140.000i − 1.50537i
\(94\) 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
\(99\) 32.5379 0.328666
\(100\) 0 0
\(101\) − 41.9176i − 0.415025i −0.978232 0.207513i \(-0.933463\pi\)
0.978232 0.207513i \(-0.0665368\pi\)
\(102\) 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
\(107\) 64.8549 0.606121 0.303060 0.952971i \(-0.401992\pi\)
0.303060 + 0.952971i \(0.401992\pi\)
\(108\) 0 0
\(109\) − 186.432i − 1.71039i −0.518307 0.855195i \(-0.673437\pi\)
0.518307 0.855195i \(-0.326563\pi\)
\(110\) 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
\(115\) 162.647 1.41433
\(116\) 0 0
\(117\) 14.5164i 0.124071i
\(118\) 0 0
\(119\) 1.50993i 0.0126885i
\(120\) 0 0
\(121\) 52.0200 0.429917
\(122\) 0 0
\(123\) 1.97546 0.0160607
\(124\) 0 0
\(125\) 467.606i 3.74085i
\(126\) 0 0
\(127\) − 93.3874i − 0.735334i −0.929958 0.367667i \(-0.880157\pi\)
0.929958 0.367667i \(-0.119843\pi\)
\(128\) 0 0
\(129\) 106.768 0.827656
\(130\) 0 0
\(131\) −58.0857 −0.443403 −0.221701 0.975115i \(-0.571161\pi\)
−0.221701 + 0.975115i \(0.571161\pi\)
\(132\) 0 0
\(133\) 41.4431i 0.311602i
\(134\) 0 0
\(135\) − 289.256i − 2.14263i
\(136\) 0 0
\(137\) −176.420 −1.28774 −0.643870 0.765135i \(-0.722672\pi\)
−0.643870 + 0.765135i \(0.722672\pi\)
\(138\) 0 0
\(139\) 12.3041 0.0885185 0.0442592 0.999020i \(-0.485907\pi\)
0.0442592 + 0.999020i \(0.485907\pi\)
\(140\) 0 0
\(141\) 149.328i 1.05906i
\(142\) 0 0
\(143\) 77.1906i 0.539795i
\(144\) 0 0
\(145\) −293.452 −2.02381
\(146\) 0 0
\(147\) 17.8827 0.121651
\(148\) 0 0
\(149\) − 173.782i − 1.16632i −0.812358 0.583160i \(-0.801816\pi\)
0.812358 0.583160i \(-0.198184\pi\)
\(150\) 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
\(155\) 540.800 3.48903
\(156\) 0 0
\(157\) 189.693i 1.20824i 0.796894 + 0.604119i \(0.206475\pi\)
−0.796894 + 0.604119i \(0.793525\pi\)
\(158\) 0 0
\(159\) 14.4374i 0.0908014i
\(160\) 0 0
\(161\) 43.6065 0.270848
\(162\) 0 0
\(163\) −17.7685 −0.109009 −0.0545047 0.998514i \(-0.517358\pi\)
−0.0545047 + 0.998514i \(0.517358\pi\)
\(164\) 0 0
\(165\) − 331.610i − 2.00976i
\(166\) 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
\(171\) 38.7475 0.226594
\(172\) 0 0
\(173\) 43.2891i 0.250226i 0.992143 + 0.125113i \(0.0399293\pi\)
−0.992143 + 0.125113i \(0.960071\pi\)
\(174\) 0 0
\(175\) 191.511i 1.09435i
\(176\) 0 0
\(177\) 108.940 0.615479
\(178\) 0 0
\(179\) 190.838 1.06614 0.533068 0.846072i \(-0.321039\pi\)
0.533068 + 0.846072i \(0.321039\pi\)
\(180\) 0 0
\(181\) 207.153i 1.14449i 0.820082 + 0.572246i \(0.193928\pi\)
−0.820082 + 0.572246i \(0.806072\pi\)
\(182\) 0 0
\(183\) 245.087i 1.33928i
\(184\) 0 0
\(185\) 415.313 2.24493
\(186\) 0 0
\(187\) 7.50683 0.0401434
\(188\) 0 0
\(189\) − 77.5507i − 0.410321i
\(190\) 0 0
\(191\) 315.489i 1.65178i 0.563833 + 0.825889i \(0.309326\pi\)
−0.563833 + 0.825889i \(0.690674\pi\)
\(192\) 0 0
\(193\) −30.9403 −0.160312 −0.0801561 0.996782i \(-0.525542\pi\)
−0.0801561 + 0.996782i \(0.525542\pi\)
\(194\) 0 0
\(195\) 147.943 0.758684
\(196\) 0 0
\(197\) 264.248i 1.34136i 0.741746 + 0.670681i \(0.233998\pi\)
−0.741746 + 0.670681i \(0.766002\pi\)
\(198\) 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
\(203\) −78.6759 −0.387566
\(204\) 0 0
\(205\) 7.63095i 0.0372242i
\(206\) 0 0
\(207\) − 40.7703i − 0.196958i
\(208\) 0 0
\(209\) 206.040 0.985836
\(210\) 0 0
\(211\) −247.994 −1.17533 −0.587664 0.809105i \(-0.699952\pi\)
−0.587664 + 0.809105i \(0.699952\pi\)
\(212\) 0 0
\(213\) 235.255i 1.10448i
\(214\) 0 0
\(215\) 412.430i 1.91828i
\(216\) 0 0
\(217\) 144.991 0.668161
\(218\) 0 0
\(219\) 25.4838 0.116364
\(220\) 0 0
\(221\) 3.34907i 0.0151542i
\(222\) 0 0
\(223\) 56.7999i 0.254708i 0.991857 + 0.127354i \(0.0406484\pi\)
−0.991857 + 0.127354i \(0.959352\pi\)
\(224\) 0 0
\(225\) 179.055 0.795800
\(226\) 0 0
\(227\) −275.095 −1.21187 −0.605937 0.795513i \(-0.707202\pi\)
−0.605937 + 0.795513i \(0.707202\pi\)
\(228\) 0 0
\(229\) − 6.29210i − 0.0274764i −0.999906 0.0137382i \(-0.995627\pi\)
0.999906 0.0137382i \(-0.00437314\pi\)
\(230\) 0 0
\(231\) − 88.9061i − 0.384875i
\(232\) 0 0
\(233\) 142.049 0.609653 0.304827 0.952408i \(-0.401401\pi\)
0.304827 + 0.952408i \(0.401401\pi\)
\(234\) 0 0
\(235\) −576.833 −2.45461
\(236\) 0 0
\(237\) − 51.5480i − 0.217502i
\(238\) 0 0
\(239\) 51.3954i 0.215043i 0.994203 + 0.107522i \(0.0342915\pi\)
−0.994203 + 0.107522i \(0.965708\pi\)
\(240\) 0 0
\(241\) 93.7048 0.388817 0.194408 0.980921i \(-0.437721\pi\)
0.194408 + 0.980921i \(0.437721\pi\)
\(242\) 0 0
\(243\) −129.381 −0.532433
\(244\) 0 0
\(245\) 69.0785i 0.281953i
\(246\) 0 0
\(247\) 91.9220i 0.372154i
\(248\) 0 0
\(249\) −386.540 −1.55237
\(250\) 0 0
\(251\) 12.4430 0.0495737 0.0247869 0.999693i \(-0.492109\pi\)
0.0247869 + 0.999693i \(0.492109\pi\)
\(252\) 0 0
\(253\) − 216.796i − 0.856900i
\(254\) 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
\(259\) 111.347 0.429912
\(260\) 0 0
\(261\) 73.5587i 0.281834i
\(262\) 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
\(267\) 14.8105 0.0554700
\(268\) 0 0
\(269\) − 9.37496i − 0.0348512i −0.999848 0.0174256i \(-0.994453\pi\)
0.999848 0.0174256i \(-0.00554701\pi\)
\(270\) 0 0
\(271\) 203.067i 0.749324i 0.927161 + 0.374662i \(0.122241\pi\)
−0.927161 + 0.374662i \(0.877759\pi\)
\(272\) 0 0
\(273\) 39.6643 0.145291
\(274\) 0 0
\(275\) 952.124 3.46227
\(276\) 0 0
\(277\) − 154.419i − 0.557471i −0.960368 0.278735i \(-0.910085\pi\)
0.960368 0.278735i \(-0.0899152\pi\)
\(278\) 0 0
\(279\) − 135.561i − 0.485880i
\(280\) 0 0
\(281\) 217.495 0.774005 0.387002 0.922079i \(-0.373511\pi\)
0.387002 + 0.922079i \(0.373511\pi\)
\(282\) 0 0
\(283\) −431.849 −1.52597 −0.762983 0.646418i \(-0.776266\pi\)
−0.762983 + 0.646418i \(0.776266\pi\)
\(284\) 0 0
\(285\) − 394.896i − 1.38560i
\(286\) 0 0
\(287\) 2.04589i 0.00712855i
\(288\) 0 0
\(289\) −288.674 −0.998873
\(290\) 0 0
\(291\) −264.908 −0.910336
\(292\) 0 0
\(293\) 212.019i 0.723616i 0.932253 + 0.361808i \(0.117840\pi\)
−0.932253 + 0.361808i \(0.882160\pi\)
\(294\) 0 0
\(295\) 420.820i 1.42651i
\(296\) 0 0
\(297\) −385.554 −1.29816
\(298\) 0 0
\(299\) 96.7206 0.323480
\(300\) 0 0
\(301\) 110.574i 0.367356i
\(302\) 0 0
\(303\) 107.086i 0.353418i
\(304\) 0 0
\(305\) −946.741 −3.10407
\(306\) 0 0
\(307\) −192.868 −0.628234 −0.314117 0.949384i \(-0.601708\pi\)
−0.314117 + 0.949384i \(0.601708\pi\)
\(308\) 0 0
\(309\) − 47.8689i − 0.154915i
\(310\) 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 \(-0.376688\pi\)
0.377779 + 0.925896i \(0.376688\pi\)
\(314\) 0 0
\(315\) 64.5855 0.205033
\(316\) 0 0
\(317\) − 249.012i − 0.785526i −0.919640 0.392763i \(-0.871519\pi\)
0.919640 0.392763i \(-0.128481\pi\)
\(318\) 0 0
\(319\) 391.148i 1.22617i
\(320\) 0 0
\(321\) −165.683 −0.516146
\(322\) 0 0
\(323\) 8.93945 0.0276763
\(324\) 0 0
\(325\) 424.778i 1.30701i
\(326\) 0 0
\(327\) 476.273i 1.45649i
\(328\) 0 0
\(329\) −154.652 −0.470066
\(330\) 0 0
\(331\) 465.343 1.40587 0.702935 0.711254i \(-0.251872\pi\)
0.702935 + 0.711254i \(0.251872\pi\)
\(332\) 0 0
\(333\) − 104.105i − 0.312628i
\(334\) 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
\(339\) 216.516 0.638690
\(340\) 0 0
\(341\) − 720.842i − 2.11391i
\(342\) 0 0
\(343\) 18.5203i 0.0539949i
\(344\) 0 0
\(345\) −415.510 −1.20438
\(346\) 0 0
\(347\) 450.576 1.29849 0.649245 0.760580i \(-0.275085\pi\)
0.649245 + 0.760580i \(0.275085\pi\)
\(348\) 0 0
\(349\) − 64.7762i − 0.185605i −0.995685 0.0928025i \(-0.970417\pi\)
0.995685 0.0928025i \(-0.0295825\pi\)
\(350\) 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
\(355\) −908.760 −2.55989
\(356\) 0 0
\(357\) − 3.85737i − 0.0108050i
\(358\) 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
\(363\) −132.894 −0.366099
\(364\) 0 0
\(365\) 98.4407i 0.269701i
\(366\) 0 0
\(367\) 430.196i 1.17220i 0.810240 + 0.586098i \(0.199337\pi\)
−0.810240 + 0.586098i \(0.800663\pi\)
\(368\) 0 0
\(369\) 1.91283 0.00518381
\(370\) 0 0
\(371\) −14.9522 −0.0403023
\(372\) 0 0
\(373\) − 2.60758i − 0.00699083i −0.999994 0.00349542i \(-0.998887\pi\)
0.999994 0.00349542i \(-0.00111263\pi\)
\(374\) 0 0
\(375\) − 1194.58i − 3.18555i
\(376\) 0 0
\(377\) −174.506 −0.462880
\(378\) 0 0
\(379\) −359.118 −0.947541 −0.473771 0.880648i \(-0.657107\pi\)
−0.473771 + 0.880648i \(0.657107\pi\)
\(380\) 0 0
\(381\) 238.574i 0.626178i
\(382\) 0 0
\(383\) − 470.758i − 1.22913i −0.788865 0.614567i \(-0.789331\pi\)
0.788865 0.614567i \(-0.210669\pi\)
\(384\) 0 0
\(385\) 343.433 0.892034
\(386\) 0 0
\(387\) 103.382 0.267138
\(388\) 0 0
\(389\) 180.270i 0.463418i 0.972785 + 0.231709i \(0.0744318\pi\)
−0.972785 + 0.231709i \(0.925568\pi\)
\(390\) 0 0
\(391\) − 9.40612i − 0.0240566i
\(392\) 0 0
\(393\) 148.390 0.377582
\(394\) 0 0
\(395\) 199.123 0.504110
\(396\) 0 0
\(397\) − 248.223i − 0.625248i −0.949877 0.312624i \(-0.898792\pi\)
0.949877 0.312624i \(-0.101208\pi\)
\(398\) 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
\(403\) 321.594 0.798001
\(404\) 0 0
\(405\) 519.253i 1.28211i
\(406\) 0 0
\(407\) − 553.578i − 1.36014i
\(408\) 0 0
\(409\) −479.494 −1.17236 −0.586179 0.810182i \(-0.699368\pi\)
−0.586179 + 0.810182i \(0.699368\pi\)
\(410\) 0 0
\(411\) 450.695 1.09658
\(412\) 0 0
\(413\) 112.824i 0.273181i
\(414\) 0 0
\(415\) − 1493.16i − 3.59797i
\(416\) 0 0
\(417\) −31.4328 −0.0753785
\(418\) 0 0
\(419\) −564.644 −1.34760 −0.673800 0.738914i \(-0.735339\pi\)
−0.673800 + 0.738914i \(0.735339\pi\)
\(420\) 0 0
\(421\) − 544.282i − 1.29283i −0.762986 0.646415i \(-0.776267\pi\)
0.762986 0.646415i \(-0.223733\pi\)
\(422\) 0 0
\(423\) 144.593i 0.341827i
\(424\) 0 0
\(425\) 41.3098 0.0971996
\(426\) 0 0
\(427\) −253.826 −0.594440
\(428\) 0 0
\(429\) − 197.196i − 0.459665i
\(430\) 0 0
\(431\) 90.3539i 0.209638i 0.994491 + 0.104819i \(0.0334263\pi\)
−0.994491 + 0.104819i \(0.966574\pi\)
\(432\) 0 0
\(433\) −131.383 −0.303425 −0.151713 0.988425i \(-0.548479\pi\)
−0.151713 + 0.988425i \(0.548479\pi\)
\(434\) 0 0
\(435\) 749.674 1.72339
\(436\) 0 0
\(437\) − 258.170i − 0.590777i
\(438\) 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
\(443\) −193.210 −0.436139 −0.218070 0.975933i \(-0.569976\pi\)
−0.218070 + 0.975933i \(0.569976\pi\)
\(444\) 0 0
\(445\) 57.2111i 0.128564i
\(446\) 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
\(451\) 10.1714 0.0225531
\(452\) 0 0
\(453\) − 353.651i − 0.780686i
\(454\) 0 0
\(455\) 153.218i 0.336743i
\(456\) 0 0
\(457\) 426.401 0.933043 0.466521 0.884510i \(-0.345507\pi\)
0.466521 + 0.884510i \(0.345507\pi\)
\(458\) 0 0
\(459\) −16.7280 −0.0364445
\(460\) 0 0
\(461\) 430.182i 0.933150i 0.884482 + 0.466575i \(0.154512\pi\)
−0.884482 + 0.466575i \(0.845488\pi\)
\(462\) 0 0
\(463\) 183.454i 0.396230i 0.980179 + 0.198115i \(0.0634819\pi\)
−0.980179 + 0.198115i \(0.936518\pi\)
\(464\) 0 0
\(465\) −1381.56 −2.97111
\(466\) 0 0
\(467\) −110.381 −0.236362 −0.118181 0.992992i \(-0.537706\pi\)
−0.118181 + 0.992992i \(0.537706\pi\)
\(468\) 0 0
\(469\) − 184.807i − 0.394044i
\(470\) 0 0
\(471\) − 484.604i − 1.02888i
\(472\) 0 0
\(473\) 549.735 1.16223
\(474\) 0 0
\(475\) 1133.83 2.38701
\(476\) 0 0
\(477\) 13.9797i 0.0293075i
\(478\) 0 0
\(479\) 515.593i 1.07639i 0.842819 + 0.538197i \(0.180894\pi\)
−0.842819 + 0.538197i \(0.819106\pi\)
\(480\) 0 0
\(481\) 246.972 0.513455
\(482\) 0 0
\(483\) −111.400 −0.230642
\(484\) 0 0
\(485\) − 1023.30i − 2.10991i
\(486\) 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
\(491\) −556.042 −1.13247 −0.566234 0.824244i \(-0.691600\pi\)
−0.566234 + 0.824244i \(0.691600\pi\)
\(492\) 0 0
\(493\) 16.9708i 0.0344234i
\(494\) 0 0
\(495\) − 321.096i − 0.648678i
\(496\) 0 0
\(497\) −243.643 −0.490227
\(498\) 0 0
\(499\) −150.429 −0.301461 −0.150730 0.988575i \(-0.548163\pi\)
−0.150730 + 0.988575i \(0.548163\pi\)
\(500\) 0 0
\(501\) − 0.227601i 0 0.000454293i
\(502\) 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
\(507\) −343.762 −0.678032
\(508\) 0 0
\(509\) 622.234i 1.22246i 0.791452 + 0.611232i \(0.209326\pi\)
−0.791452 + 0.611232i \(0.790674\pi\)
\(510\) 0 0
\(511\) 26.3924i 0.0516485i
\(512\) 0 0
\(513\) −459.134 −0.894999
\(514\) 0 0
\(515\) 184.911 0.359051
\(516\) 0 0
\(517\) 768.871i 1.48718i
\(518\) 0 0
\(519\) − 110.589i − 0.213081i
\(520\) 0 0
\(521\) 644.142 1.23636 0.618179 0.786037i \(-0.287871\pi\)
0.618179 + 0.786037i \(0.287871\pi\)
\(522\) 0 0
\(523\) −889.589 −1.70093 −0.850467 0.526028i \(-0.823681\pi\)
−0.850467 + 0.526028i \(0.823681\pi\)
\(524\) 0 0
\(525\) − 489.248i − 0.931900i
\(526\) 0 0
\(527\) − 31.2752i − 0.0593457i
\(528\) 0 0
\(529\) 257.353 0.486490
\(530\) 0 0
\(531\) 105.486 0.198655
\(532\) 0 0
\(533\) 4.53785i 0.00851380i
\(534\) 0 0
\(535\) − 640.012i − 1.19628i
\(536\) 0 0
\(537\) −487.529 −0.907875
\(538\) 0 0
\(539\) 92.0759 0.170827
\(540\) 0 0
\(541\) 832.015i 1.53792i 0.639296 + 0.768961i \(0.279226\pi\)
−0.639296 + 0.768961i \(0.720774\pi\)
\(542\) 0 0
\(543\) − 529.207i − 0.974599i
\(544\) 0 0
\(545\) −1839.78 −3.37575
\(546\) 0 0
\(547\) −1014.85 −1.85531 −0.927654 0.373441i \(-0.878178\pi\)
−0.927654 + 0.373441i \(0.878178\pi\)
\(548\) 0 0
\(549\) 237.316i 0.432270i
\(550\) 0 0
\(551\) 465.796i 0.845365i
\(552\) 0 0
\(553\) 53.3859 0.0965387
\(554\) 0 0
\(555\) −1060.99 −1.91169
\(556\) 0 0
\(557\) 473.280i 0.849694i 0.905265 + 0.424847i \(0.139672\pi\)
−0.905265 + 0.424847i \(0.860328\pi\)
\(558\) 0 0
\(559\) 245.257i 0.438742i
\(560\) 0 0
\(561\) −19.1775 −0.0341844
\(562\) 0 0
\(563\) 102.364 0.181818 0.0909092 0.995859i \(-0.471023\pi\)
0.0909092 + 0.995859i \(0.471023\pi\)
\(564\) 0 0
\(565\) 836.373i 1.48031i
\(566\) 0 0
\(567\) 139.214i 0.245527i
\(568\) 0 0
\(569\) 68.4059 0.120221 0.0601106 0.998192i \(-0.480855\pi\)
0.0601106 + 0.998192i \(0.480855\pi\)
\(570\) 0 0
\(571\) −701.419 −1.22841 −0.614203 0.789148i \(-0.710522\pi\)
−0.614203 + 0.789148i \(0.710522\pi\)
\(572\) 0 0
\(573\) − 805.971i − 1.40658i
\(574\) 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
\(579\) 79.0421 0.136515
\(580\) 0 0
\(581\) − 400.322i − 0.689022i
\(582\) 0 0
\(583\) 74.3367i 0.127507i
\(584\) 0 0
\(585\) 143.253 0.244876
\(586\) 0 0
\(587\) 99.0239 0.168695 0.0843475 0.996436i \(-0.473119\pi\)
0.0843475 + 0.996436i \(0.473119\pi\)
\(588\) 0 0
\(589\) − 858.410i − 1.45740i
\(590\) 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
\(595\) 14.9005 0.0250429
\(596\) 0 0
\(597\) − 335.481i − 0.561945i
\(598\) 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 \(-0.484629\pi\)
0.0482722 + 0.998834i \(0.484629\pi\)
\(602\) 0 0
\(603\) −172.786 −0.286545
\(604\) 0 0
\(605\) − 513.352i − 0.848515i
\(606\) 0 0
\(607\) − 67.3822i − 0.111009i −0.998458 0.0555043i \(-0.982323\pi\)
0.998458 0.0555043i \(-0.0176767\pi\)
\(608\) 0 0
\(609\) 200.991 0.330034
\(610\) 0 0
\(611\) −343.022 −0.561411
\(612\) 0 0
\(613\) 521.917i 0.851415i 0.904861 + 0.425707i \(0.139975\pi\)
−0.904861 + 0.425707i \(0.860025\pi\)
\(614\) 0 0
\(615\) − 19.4946i − 0.0316985i
\(616\) 0 0
\(617\) −165.257 −0.267840 −0.133920 0.990992i \(-0.542757\pi\)
−0.133920 + 0.990992i \(0.542757\pi\)
\(618\) 0 0
\(619\) −205.392 −0.331812 −0.165906 0.986142i \(-0.553055\pi\)
−0.165906 + 0.986142i \(0.553055\pi\)
\(620\) 0 0
\(621\) 483.103i 0.777943i
\(622\) 0 0
\(623\) 15.3385i 0.0246205i
\(624\) 0 0
\(625\) 2804.90 4.48783
\(626\) 0 0
\(627\) −526.363 −0.839495
\(628\) 0 0
\(629\) − 24.0181i − 0.0381846i
\(630\) 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
\(635\) −921.580 −1.45131
\(636\) 0 0
\(637\) 41.0785i 0.0644874i
\(638\) 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
\(643\) −607.082 −0.944140 −0.472070 0.881561i \(-0.656493\pi\)
−0.472070 + 0.881561i \(0.656493\pi\)
\(644\) 0 0
\(645\) − 1053.62i − 1.63352i
\(646\) 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
\(651\) −370.404 −0.568977
\(652\) 0 0
\(653\) − 850.083i − 1.30181i −0.759159 0.650906i \(-0.774389\pi\)
0.759159 0.650906i \(-0.225611\pi\)
\(654\) 0 0
\(655\) 573.211i 0.875131i
\(656\) 0 0
\(657\) 24.6758 0.0375583
\(658\) 0 0
\(659\) 331.172 0.502537 0.251268 0.967917i \(-0.419152\pi\)
0.251268 + 0.967917i \(0.419152\pi\)
\(660\) 0 0
\(661\) − 464.323i − 0.702456i −0.936290 0.351228i \(-0.885764\pi\)
0.936290 0.351228i \(-0.114236\pi\)
\(662\) 0 0
\(663\) − 8.55577i − 0.0129046i
\(664\) 0 0
\(665\) 408.975 0.615000
\(666\) 0 0
\(667\) 490.112 0.734801
\(668\) 0 0
\(669\) − 145.105i − 0.216898i
\(670\) 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
\(675\) −2121.69 −3.14325
\(676\) 0 0
\(677\) 664.414i 0.981410i 0.871326 + 0.490705i \(0.163261\pi\)
−0.871326 + 0.490705i \(0.836739\pi\)
\(678\) 0 0
\(679\) − 274.353i − 0.404054i
\(680\) 0 0
\(681\) 702.778 1.03198
\(682\) 0 0
\(683\) 601.901 0.881261 0.440631 0.897689i \(-0.354755\pi\)
0.440631 + 0.897689i \(0.354755\pi\)
\(684\) 0 0
\(685\) 1740.98i 2.54157i
\(686\) 0 0
\(687\) 16.0742i 0.0233977i
\(688\) 0 0
\(689\) −33.1643 −0.0481340
\(690\) 0 0
\(691\) 579.981 0.839336 0.419668 0.907678i \(-0.362146\pi\)
0.419668 + 0.907678i \(0.362146\pi\)
\(692\) 0 0
\(693\) − 86.0872i − 0.124224i
\(694\) 0 0
\(695\) − 121.421i − 0.174706i
\(696\) 0 0
\(697\) 0.441308 0.000633154 0
\(698\) 0 0
\(699\) −362.889 −0.519154
\(700\) 0 0
\(701\) 863.561i 1.23190i 0.787786 + 0.615949i \(0.211227\pi\)
−0.787786 + 0.615949i \(0.788773\pi\)
\(702\) 0 0
\(703\) − 659.225i − 0.937731i
\(704\) 0 0
\(705\) 1473.62 2.09024
\(706\) 0 0
\(707\) −110.903 −0.156865
\(708\) 0 0
\(709\) 293.022i 0.413290i 0.978416 + 0.206645i \(0.0662545\pi\)
−0.978416 + 0.206645i \(0.933746\pi\)
\(710\) 0 0
\(711\) − 49.9136i − 0.0702019i
\(712\) 0 0
\(713\) −903.222 −1.26679
\(714\) 0 0
\(715\) 761.744 1.06538
\(716\) 0 0
\(717\) − 131.298i − 0.183122i
\(718\) 0 0
\(719\) 381.821i 0.531044i 0.964105 + 0.265522i \(0.0855443\pi\)
−0.964105 + 0.265522i \(0.914456\pi\)
\(720\) 0 0
\(721\) 49.5756 0.0687594
\(722\) 0 0
\(723\) −239.385 −0.331099
\(724\) 0 0
\(725\) 2152.48i 2.96893i
\(726\) 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
\(731\) 23.8514 0.0326284
\(732\) 0 0
\(733\) 59.1034i 0.0806322i 0.999187 + 0.0403161i \(0.0128365\pi\)
−0.999187 + 0.0403161i \(0.987164\pi\)
\(734\) 0 0
\(735\) − 176.473i − 0.240099i
\(736\) 0 0
\(737\) −918.790 −1.24666
\(738\) 0 0
\(739\) −1285.33 −1.73928 −0.869640 0.493687i \(-0.835649\pi\)
−0.869640 + 0.493687i \(0.835649\pi\)
\(740\) 0 0
\(741\) − 234.830i − 0.316910i
\(742\) 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
\(747\) −374.284 −0.501050
\(748\) 0 0
\(749\) − 171.590i − 0.229092i
\(750\) 0 0
\(751\) − 190.150i − 0.253195i −0.991954 0.126598i \(-0.959594\pi\)
0.991954 0.126598i \(-0.0404057\pi\)
\(752\) 0 0
\(753\) −31.7878 −0.0422148
\(754\) 0 0
\(755\) 1366.11 1.80941
\(756\) 0 0
\(757\) − 441.225i − 0.582859i −0.956592 0.291430i \(-0.905869\pi\)
0.956592 0.291430i \(-0.0941310\pi\)
\(758\) 0 0
\(759\) 553.841i 0.729698i
\(760\) 0 0
\(761\) 1044.28 1.37225 0.686123 0.727486i \(-0.259311\pi\)
0.686123 + 0.727486i \(0.259311\pi\)
\(762\) 0 0
\(763\) −493.254 −0.646466
\(764\) 0 0
\(765\) − 13.9314i − 0.0182110i
\(766\) 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
\(771\) −433.998 −0.562903
\(772\) 0 0
\(773\) 211.566i 0.273694i 0.990592 + 0.136847i \(0.0436969\pi\)
−0.990592 + 0.136847i \(0.956303\pi\)
\(774\) 0 0
\(775\) − 3966.77i − 5.11841i
\(776\) 0 0
\(777\) −284.455 −0.366095
\(778\) 0 0
\(779\) 12.1126 0.0155489
\(780\) 0 0
\(781\) 1211.30i 1.55096i
\(782\) 0 0
\(783\) − 871.625i − 1.11319i
\(784\) 0 0
\(785\) 1871.96 2.38466
\(786\) 0 0
\(787\) 523.213 0.664820 0.332410 0.943135i \(-0.392138\pi\)
0.332410 + 0.943135i \(0.392138\pi\)
\(788\) 0 0
\(789\) − 303.396i − 0.384533i
\(790\) 0 0
\(791\) 224.235i 0.283483i
\(792\) 0 0
\(793\) −562.993 −0.709953
\(794\) 0 0
\(795\) 142.474 0.179212
\(796\) 0 0
\(797\) − 503.460i − 0.631693i −0.948810 0.315847i \(-0.897711\pi\)
0.948810 0.315847i \(-0.102289\pi\)
\(798\) 0 0
\(799\) 33.3590i 0.0417510i
\(800\) 0 0
\(801\) 14.3409 0.0179038
\(802\) 0 0
\(803\) 131.213 0.163404
\(804\) 0 0
\(805\) − 430.325i − 0.534565i
\(806\) 0 0
\(807\) 23.9499i 0.0296777i
\(808\) 0 0
\(809\) −100.517 −0.124248 −0.0621242 0.998068i \(-0.519788\pi\)
−0.0621242 + 0.998068i \(0.519788\pi\)
\(810\) 0 0
\(811\) −759.830 −0.936905 −0.468453 0.883489i \(-0.655188\pi\)
−0.468453 + 0.883489i \(0.655188\pi\)
\(812\) 0 0
\(813\) − 518.768i − 0.638091i
\(814\) 0 0
\(815\) 175.346i 0.215149i
\(816\) 0 0
\(817\) 654.648 0.801283
\(818\) 0 0
\(819\) 38.4067 0.0468946
\(820\) 0 0
\(821\) 749.705i 0.913160i 0.889682 + 0.456580i \(0.150926\pi\)
−0.889682 + 0.456580i \(0.849074\pi\)
\(822\) 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
\(827\) 575.272 0.695613 0.347806 0.937566i \(-0.386927\pi\)
0.347806 + 0.937566i \(0.386927\pi\)
\(828\) 0 0
\(829\) 157.635i 0.190151i 0.995470 + 0.0950756i \(0.0303093\pi\)
−0.995470 + 0.0950756i \(0.969691\pi\)
\(830\) 0 0
\(831\) 394.490i 0.474718i
\(832\) 0 0
\(833\) 3.99490 0.00479580
\(834\) 0 0
\(835\) 0.879193 0.00105293
\(836\) 0 0
\(837\) 1606.31i 1.91913i
\(838\) 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
\(843\) −555.628 −0.659108
\(844\) 0 0
\(845\) − 1327.91i − 1.57149i
\(846\) 0 0
\(847\) − 137.632i − 0.162493i
\(848\) 0 0
\(849\) 1103.23 1.29945
\(850\) 0 0
\(851\) −693.638 −0.815086
\(852\) 0 0
\(853\) − 315.373i − 0.369722i −0.982765 0.184861i \(-0.940817\pi\)
0.982765 0.184861i \(-0.0591834\pi\)
\(854\) 0 0
\(855\) − 382.375i − 0.447222i
\(856\) 0 0
\(857\) −1222.53 −1.42652 −0.713260 0.700900i \(-0.752782\pi\)
−0.713260 + 0.700900i \(0.752782\pi\)
\(858\) 0 0
\(859\) 1008.23 1.17373 0.586863 0.809687i \(-0.300363\pi\)
0.586863 + 0.809687i \(0.300363\pi\)
\(860\) 0 0
\(861\) − 5.22658i − 0.00607036i
\(862\) 0 0
\(863\) 1140.31i 1.32133i 0.750679 + 0.660667i \(0.229726\pi\)
−0.750679 + 0.660667i \(0.770274\pi\)
\(864\) 0 0
\(865\) 427.192 0.493864
\(866\) 0 0
\(867\) 737.467 0.850597
\(868\) 0 0
\(869\) − 265.415i − 0.305426i
\(870\) 0 0
\(871\) − 409.906i − 0.470616i
\(872\) 0 0
\(873\) −256.508 −0.293824
\(874\) 0 0
\(875\) 1237.17 1.41391
\(876\) 0 0
\(877\) − 306.788i − 0.349816i −0.984585 0.174908i \(-0.944037\pi\)
0.984585 0.174908i \(-0.0559627\pi\)
\(878\) 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
\(883\) 458.136 0.518841 0.259420 0.965765i \(-0.416469\pi\)
0.259420 + 0.965765i \(0.416469\pi\)
\(884\) 0 0
\(885\) − 1075.06i − 1.21475i
\(886\) 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
\(891\) 692.121 0.776791
\(892\) 0 0
\(893\) 915.605i 1.02531i
\(894\) 0 0
\(895\) − 1883.26i − 2.10420i
\(896\) 0 0
\(897\) −247.089 −0.275462
\(898\) 0 0
\(899\) 1629.61 1.81270
\(900\) 0 0
\(901\) 3.22525i 0.00357963i
\(902\) 0 0
\(903\) − 282.481i − 0.312825i
\(904\) 0 0
\(905\) 2044.26 2.25885
\(906\) 0 0
\(907\) −375.701 −0.414224 −0.207112 0.978317i \(-0.566406\pi\)
−0.207112 + 0.978317i \(0.566406\pi\)
\(908\) 0 0
\(909\) 103.690i 0.114071i
\(910\) 0 0
\(911\) − 41.9614i − 0.0460608i −0.999735 0.0230304i \(-0.992669\pi\)
0.999735 0.0230304i \(-0.00733145\pi\)
\(912\) 0 0
\(913\) −1990.25 −2.17991
\(914\) 0 0
\(915\) 2418.61 2.64329
\(916\) 0 0
\(917\) 153.680i 0.167590i
\(918\) 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
\(923\) −540.407 −0.585489
\(924\) 0 0
\(925\) − 3046.32i − 3.29332i
\(926\) 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
\(931\) 109.648 0.117774
\(932\) 0 0
\(933\) 689.285i 0.738784i
\(934\) 0 0
\(935\) − 74.0800i − 0.0792300i
\(936\) 0 0
\(937\) 183.028 0.195334 0.0976668 0.995219i \(-0.468862\pi\)
0.0976668 + 0.995219i \(0.468862\pi\)
\(938\) 0 0
\(939\) −604.152 −0.643400
\(940\) 0 0
\(941\) 255.475i 0.271493i 0.990744 + 0.135747i \(0.0433433\pi\)
−0.990744 + 0.135747i \(0.956657\pi\)
\(942\) 0 0
\(943\) − 12.7449i − 0.0135153i
\(944\) 0 0
\(945\) −765.298 −0.809839
\(946\) 0 0
\(947\) −187.173 −0.197648 −0.0988242 0.995105i \(-0.531508\pi\)
−0.0988242 + 0.995105i \(0.531508\pi\)
\(948\) 0 0
\(949\) 58.5391i 0.0616851i
\(950\) 0 0
\(951\) 636.142i 0.668920i
\(952\) 0 0
\(953\) −362.350 −0.380220 −0.190110 0.981763i \(-0.560885\pi\)
−0.190110 + 0.981763i \(0.560885\pi\)
\(954\) 0 0
\(955\) 3113.36 3.26007
\(956\) 0 0
\(957\) − 999.253i − 1.04415i
\(958\) 0 0
\(959\) 466.764i 0.486720i
\(960\) 0 0
\(961\) −2042.20 −2.12508
\(962\) 0 0
\(963\) −160.430 −0.166594
\(964\) 0 0
\(965\) 305.330i 0.316404i
\(966\) 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
\(971\) 223.833 0.230518 0.115259 0.993335i \(-0.463230\pi\)
0.115259 + 0.993335i \(0.463230\pi\)
\(972\) 0 0
\(973\) − 32.5535i − 0.0334568i
\(974\) 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
\(979\) 76.2576 0.0778934
\(980\) 0 0
\(981\) 461.172i 0.470104i
\(982\) 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
\(987\) 395.084 0.400287
\(988\) 0 0
\(989\) − 688.823i − 0.696484i
\(990\) 0 0
\(991\) − 1607.53i − 1.62213i −0.584959 0.811063i \(-0.698889\pi\)
0.584959 0.811063i \(-0.301111\pi\)
\(992\) 0 0
\(993\) −1188.80 −1.19718
\(994\) 0 0
\(995\) 1295.92 1.30243
\(996\) 0 0
\(997\) − 701.364i − 0.703474i −0.936099 0.351737i \(-0.885591\pi\)
0.936099 0.351737i \(-0.114409\pi\)
\(998\) 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 1792.3.g.d.127.3 8
4.3 odd 2 1792.3.g.f.127.5 8
8.3 odd 2 inner 1792.3.g.d.127.4 8
8.5 even 2 1792.3.g.f.127.6 8
16.3 odd 4 448.3.d.e.127.6 8
16.5 even 4 224.3.d.b.127.6 yes 8
16.11 odd 4 224.3.d.b.127.3 8
16.13 even 4 448.3.d.e.127.3 8
48.5 odd 4 2016.3.m.c.127.2 8
48.11 even 4 2016.3.m.c.127.1 8
112.27 even 4 1568.3.d.n.1471.6 8
112.69 odd 4 1568.3.d.n.1471.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.3.d.b.127.3 8 16.11 odd 4
224.3.d.b.127.6 yes 8 16.5 even 4
448.3.d.e.127.3 8 16.13 even 4
448.3.d.e.127.6 8 16.3 odd 4
1568.3.d.n.1471.3 8 112.69 odd 4
1568.3.d.n.1471.6 8 112.27 even 4
1792.3.g.d.127.3 8 1.1 even 1 trivial
1792.3.g.d.127.4 8 8.3 odd 2 inner
1792.3.g.f.127.5 8 4.3 odd 2
1792.3.g.f.127.6 8 8.5 even 2
2016.3.m.c.127.1 8 48.11 even 4
2016.3.m.c.127.2 8 48.5 odd 4