Properties

Label 1792.3.g.f.127.6
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 127.6
Root \(-1.27733i\) of defining polynomial
Character \(\chi\) \(=\) 1792.127
Dual form 1792.3.g.f.127.5

$q$-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 \(-0.360000\pi\)
0.425778 + 0.904828i \(0.360000\pi\)
\(4\) 0 0
\(5\) 9.86836i 1.97367i 0.161727 + 0.986836i \(0.448294\pi\)
−0.161727 + 0.986836i \(0.551706\pi\)
\(6\) 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 \(-0.296004\pi\)
0.597896 + 0.801574i \(0.296004\pi\)
\(12\) 0 0
\(13\) 5.86836i 0.451412i 0.974195 + 0.225706i \(0.0724689\pi\)
−0.974195 + 0.225706i \(0.927531\pi\)
\(14\) 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 \(-0.364757\pi\)
0.412211 + 0.911089i \(0.364757\pi\)
\(20\) 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 \(0.171355\pi\)
−0.858567 + 0.512702i \(0.828645\pi\)
\(30\) 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 \(-0.807439\pi\)
0.822531 0.568721i \(-0.192561\pi\)
\(38\) 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 \(-0.338468\pi\)
0.485967 + 0.873977i \(0.338468\pi\)
\(44\) 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 \(0.0169787\pi\)
−0.998578 + 0.0533150i \(0.983021\pi\)
\(54\) 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 \(-0.382304\pi\)
0.361385 + 0.932417i \(0.382304\pi\)
\(60\) 0 0
\(61\) 95.9371i 1.57274i 0.617756 + 0.786370i \(0.288042\pi\)
−0.617756 + 0.786370i \(0.711958\pi\)
\(62\) 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 \(-0.674542\pi\)
−0.521271 + 0.853391i \(0.674542\pi\)
\(68\) 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 \(-0.865068\pi\)
−0.911491 + 0.411321i \(0.865068\pi\)
\(84\) 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 \(0.0665368\pi\)
−0.978232 + 0.207513i \(0.933463\pi\)
\(102\) 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 \(-0.598008\pi\)
−0.303060 + 0.952971i \(0.598008\pi\)
\(108\) 0 0
\(109\) 186.432i 1.71039i 0.518307 + 0.855195i \(0.326563\pi\)
−0.518307 + 0.855195i \(0.673437\pi\)
\(110\) 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 \(-0.428839\pi\)
0.221701 + 0.975115i \(0.428839\pi\)
\(132\) 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 \(-0.514093\pi\)
−0.0442592 + 0.999020i \(0.514093\pi\)
\(140\) 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 \(0.198184\pi\)
−0.812358 + 0.583160i \(0.801816\pi\)
\(150\) 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 \(-0.793525\pi\)
0.796894 0.604119i \(-0.206475\pi\)
\(158\) 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 \(-0.482642\pi\)
0.0545047 + 0.998514i \(0.482642\pi\)
\(164\) 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 \(-0.960071\pi\)
0.992143 0.125113i \(-0.0399293\pi\)
\(174\) 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 \(-0.678961\pi\)
−0.533068 + 0.846072i \(0.678961\pi\)
\(180\) 0 0
\(181\) − 207.153i − 1.14449i −0.820082 0.572246i \(-0.806072\pi\)
0.820082 0.572246i \(-0.193928\pi\)
\(182\) 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 \(-0.766002\pi\)
0.741746 0.670681i \(-0.233998\pi\)
\(198\) 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 \(-0.300048\pi\)
0.587664 + 0.809105i \(0.300048\pi\)
\(212\) 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 \(-0.292798\pi\)
0.605937 + 0.795513i \(0.292798\pi\)
\(228\) 0 0
\(229\) 6.29210i 0.0274764i 0.999906 + 0.0137382i \(0.00437314\pi\)
−0.999906 + 0.0137382i \(0.995627\pi\)
\(230\) 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 \(-0.507891\pi\)
−0.0247869 + 0.999693i \(0.507891\pi\)
\(252\) 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 \(0.00554701\pi\)
−0.999848 + 0.0174256i \(0.994453\pi\)
\(270\) 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 \(0.0899152\pi\)
−0.960368 + 0.278735i \(0.910085\pi\)
\(278\) 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 \(-0.223734\pi\)
0.762983 + 0.646418i \(0.223734\pi\)
\(284\) 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 \(-0.882160\pi\)
0.932253 0.361808i \(-0.117840\pi\)
\(294\) 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 \(-0.398292\pi\)
0.314117 + 0.949384i \(0.398292\pi\)
\(308\) 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 \(0.128481\pi\)
−0.919640 + 0.392763i \(0.871519\pi\)
\(318\) 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 \(-0.748128\pi\)
−0.702935 + 0.711254i \(0.748128\pi\)
\(332\) 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 \(-0.724915\pi\)
−0.649245 + 0.760580i \(0.724915\pi\)
\(348\) 0 0
\(349\) 64.7762i 0.185605i 0.995685 + 0.0928025i \(0.0295825\pi\)
−0.995685 + 0.0928025i \(0.970417\pi\)
\(350\) 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 \(0.00111263\pi\)
−0.999994 + 0.00349542i \(0.998887\pi\)
\(374\) 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 \(-0.342893\pi\)
0.473771 + 0.880648i \(0.342893\pi\)
\(380\) 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 \(-0.925568\pi\)
0.972785 0.231709i \(-0.0744318\pi\)
\(390\) 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 \(0.101208\pi\)
−0.949877 + 0.312624i \(0.898792\pi\)
\(398\) 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 \(-0.264661\pi\)
0.673800 + 0.738914i \(0.264661\pi\)
\(420\) 0 0
\(421\) 544.282i 1.29283i 0.762986 + 0.646415i \(0.223733\pi\)
−0.762986 + 0.646415i \(0.776267\pi\)
\(422\) 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 \(-0.430024\pi\)
0.218070 + 0.975933i \(0.430024\pi\)
\(444\) 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 \(-0.845488\pi\)
0.884482 0.466575i \(-0.154512\pi\)
\(462\) 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 \(-0.462294\pi\)
0.118181 + 0.992992i \(0.462294\pi\)
\(468\) 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 \(-0.308400\pi\)
0.566234 + 0.824244i \(0.308400\pi\)
\(492\) 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 \(-0.451837\pi\)
0.150730 + 0.988575i \(0.451837\pi\)
\(500\) 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 \(-0.790674\pi\)
0.791452 0.611232i \(-0.209326\pi\)
\(510\) 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 \(-0.176319\pi\)
0.850467 + 0.526028i \(0.176319\pi\)
\(524\) 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 \(-0.720774\pi\)
0.639296 0.768961i \(-0.279226\pi\)
\(542\) 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 \(-0.121822\pi\)
0.927654 + 0.373441i \(0.121822\pi\)
\(548\) 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 \(-0.860328\pi\)
0.905265 0.424847i \(-0.139672\pi\)
\(558\) 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 \(-0.528977\pi\)
−0.0909092 + 0.995859i \(0.528977\pi\)
\(564\) 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 \(-0.289478\pi\)
0.614203 + 0.789148i \(0.289478\pi\)
\(572\) 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 \(-0.526881\pi\)
−0.0843475 + 0.996436i \(0.526881\pi\)
\(588\) 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 \(-0.860025\pi\)
0.904861 0.425707i \(-0.139975\pi\)
\(614\) 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 \(-0.446945\pi\)
0.165906 + 0.986142i \(0.446945\pi\)
\(620\) 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 \(-0.343507\pi\)
0.472070 + 0.881561i \(0.343507\pi\)
\(644\) 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 \(0.225611\pi\)
−0.759159 + 0.650906i \(0.774389\pi\)
\(654\) 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 \(-0.580848\pi\)
−0.251268 + 0.967917i \(0.580848\pi\)
\(660\) 0 0
\(661\) 464.323i 0.702456i 0.936290 + 0.351228i \(0.114236\pi\)
−0.936290 + 0.351228i \(0.885764\pi\)
\(662\) 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 \(-0.836739\pi\)
0.871326 0.490705i \(-0.163261\pi\)
\(678\) 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 \(-0.645245\pi\)
−0.440631 + 0.897689i \(0.645245\pi\)
\(684\) 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 \(-0.637854\pi\)
−0.419668 + 0.907678i \(0.637854\pi\)
\(692\) 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 \(-0.788773\pi\)
0.787786 0.615949i \(-0.211227\pi\)
\(702\) 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 \(-0.933746\pi\)
0.978416 0.206645i \(-0.0662545\pi\)
\(710\) 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 \(-0.987164\pi\)
0.999187 0.0403161i \(-0.0128365\pi\)
\(734\) 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 \(-0.164351\pi\)
0.869640 + 0.493687i \(0.164351\pi\)
\(740\) 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 \(0.0941310\pi\)
−0.956592 + 0.291430i \(0.905869\pi\)
\(758\) 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 \(-0.956303\pi\)
0.990592 0.136847i \(-0.0436969\pi\)
\(774\) 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 \(-0.607862\pi\)
−0.332410 + 0.943135i \(0.607862\pi\)
\(788\) 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 \(0.102289\pi\)
−0.948810 + 0.315847i \(0.897711\pi\)
\(798\) 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 \(-0.344812\pi\)
0.468453 + 0.883489i \(0.344812\pi\)
\(812\) 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 \(-0.849074\pi\)
0.889682 0.456580i \(-0.150926\pi\)
\(822\) 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 \(-0.613073\pi\)
−0.347806 + 0.937566i \(0.613073\pi\)
\(828\) 0 0
\(829\) − 157.635i − 0.190151i −0.995470 0.0950756i \(-0.969691\pi\)
0.995470 0.0950756i \(-0.0303093\pi\)
\(830\) 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 \(0.0591834\pi\)
−0.982765 + 0.184861i \(0.940817\pi\)
\(854\) 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 \(-0.699637\pi\)
−0.586863 + 0.809687i \(0.699637\pi\)
\(860\) 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 \(0.0559627\pi\)
−0.984585 + 0.174908i \(0.944037\pi\)
\(878\) 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 \(-0.583531\pi\)
−0.259420 + 0.965765i \(0.583531\pi\)
\(884\) 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 \(-0.433594\pi\)
0.207112 + 0.978317i \(0.433594\pi\)
\(908\) 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 \(-0.956657\pi\)
0.990744 0.135747i \(-0.0433433\pi\)
\(942\) 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 \(-0.468492\pi\)
0.0988242 + 0.995105i \(0.468492\pi\)
\(948\) 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 \(-0.536770\pi\)
−0.115259 + 0.993335i \(0.536770\pi\)
\(972\) 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 \(0.114409\pi\)
−0.936099 + 0.351737i \(0.885591\pi\)
\(998\) 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.f.127.6 8
4.3 odd 2 1792.3.g.d.127.4 8
8.3 odd 2 inner 1792.3.g.f.127.5 8
8.5 even 2 1792.3.g.d.127.3 8
16.3 odd 4 224.3.d.b.127.3 8
16.5 even 4 448.3.d.e.127.3 8
16.11 odd 4 448.3.d.e.127.6 8
16.13 even 4 224.3.d.b.127.6 yes 8
48.29 odd 4 2016.3.m.c.127.2 8
48.35 even 4 2016.3.m.c.127.1 8
112.13 odd 4 1568.3.d.n.1471.3 8
112.83 even 4 1568.3.d.n.1471.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.3.d.b.127.3 8 16.3 odd 4
224.3.d.b.127.6 yes 8 16.13 even 4
448.3.d.e.127.3 8 16.5 even 4
448.3.d.e.127.6 8 16.11 odd 4
1568.3.d.n.1471.3 8 112.13 odd 4
1568.3.d.n.1471.6 8 112.83 even 4
1792.3.g.d.127.3 8 8.5 even 2
1792.3.g.d.127.4 8 4.3 odd 2
1792.3.g.f.127.5 8 8.3 odd 2 inner
1792.3.g.f.127.6 8 1.1 even 1 trivial
2016.3.m.c.127.1 8 48.35 even 4
2016.3.m.c.127.2 8 48.29 odd 4