Properties

Label 224.3.d.b.127.8
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
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 127.8
Root \(2.92812i\) of \(x^{8} + 14 x^{6} + 53 x^{4} + 56 x^{2} + 4\)
Character \(\chi\) \(=\) 224.127
Dual form 224.3.d.b.127.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.85623i q^{3} -5.78167 q^{5} -2.64575i q^{7} -25.2955 q^{9} +O(q^{10})\) \(q+5.85623i q^{3} -5.78167 q^{5} -2.64575i q^{7} -25.2955 q^{9} -3.01966i q^{11} +9.78167 q^{13} -33.8588i q^{15} -11.6027 q^{17} +25.5687i q^{19} +15.4941 q^{21} -26.1571i q^{23} +8.42771 q^{25} -95.4301i q^{27} +1.56334 q^{29} +12.0107i q^{31} +17.6839 q^{33} +15.2969i q^{35} -70.6721 q^{37} +57.2837i q^{39} -49.8775 q^{41} +73.2730i q^{43} +146.250 q^{45} +44.2248i q^{47} -7.00000 q^{49} -67.9479i q^{51} -54.2355 q^{53} +17.4587i q^{55} -149.736 q^{57} -12.4706i q^{59} +35.6770 q^{61} +66.9255i q^{63} -56.5544 q^{65} +24.4891i q^{67} +153.182 q^{69} -11.0480i q^{71} +74.3713 q^{73} +49.3547i q^{75} -7.98928 q^{77} +22.8912i q^{79} +331.202 q^{81} +48.1934i q^{83} +67.0828 q^{85} +9.15529i q^{87} +67.4801 q^{89} -25.8799i q^{91} -70.3376 q^{93} -147.830i q^{95} -7.75949 q^{97} +76.3838i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 40q^{9} + O(q^{10}) \) \( 8q - 40q^{9} + 32q^{13} - 16q^{17} + 104q^{25} - 80q^{29} - 176q^{37} + 144q^{41} + 256q^{45} - 56q^{49} + 48q^{53} - 400q^{57} - 192q^{61} - 304q^{65} + 576q^{69} + 272q^{73} - 112q^{77} + 504q^{81} - 160q^{85} - 80q^{89} - 608q^{93} + 528q^{97} + O(q^{100}) \)

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
\(3\) 5.85623i 1.95208i 0.217596 + 0.976039i \(0.430179\pi\)
−0.217596 + 0.976039i \(0.569821\pi\)
\(4\) 0 0
\(5\) −5.78167 −1.15633 −0.578167 0.815918i \(-0.696232\pi\)
−0.578167 + 0.815918i \(0.696232\pi\)
\(6\) 0 0
\(7\) − 2.64575i − 0.377964i
\(8\) 0 0
\(9\) −25.2955 −2.81061
\(10\) 0 0
\(11\) − 3.01966i − 0.274515i −0.990535 0.137257i \(-0.956171\pi\)
0.990535 0.137257i \(-0.0438288\pi\)
\(12\) 0 0
\(13\) 9.78167 0.752436 0.376218 0.926531i \(-0.377224\pi\)
0.376218 + 0.926531i \(0.377224\pi\)
\(14\) 0 0
\(15\) − 33.8588i − 2.25725i
\(16\) 0 0
\(17\) −11.6027 −0.682510 −0.341255 0.939971i \(-0.610852\pi\)
−0.341255 + 0.939971i \(0.610852\pi\)
\(18\) 0 0
\(19\) 25.5687i 1.34572i 0.739769 + 0.672861i \(0.234935\pi\)
−0.739769 + 0.672861i \(0.765065\pi\)
\(20\) 0 0
\(21\) 15.4941 0.737816
\(22\) 0 0
\(23\) − 26.1571i − 1.13726i −0.822592 0.568632i \(-0.807473\pi\)
0.822592 0.568632i \(-0.192527\pi\)
\(24\) 0 0
\(25\) 8.42771 0.337109
\(26\) 0 0
\(27\) − 95.4301i − 3.53445i
\(28\) 0 0
\(29\) 1.56334 0.0539083 0.0269542 0.999637i \(-0.491419\pi\)
0.0269542 + 0.999637i \(0.491419\pi\)
\(30\) 0 0
\(31\) 12.0107i 0.387443i 0.981057 + 0.193721i \(0.0620557\pi\)
−0.981057 + 0.193721i \(0.937944\pi\)
\(32\) 0 0
\(33\) 17.6839 0.535875
\(34\) 0 0
\(35\) 15.2969i 0.437053i
\(36\) 0 0
\(37\) −70.6721 −1.91006 −0.955029 0.296513i \(-0.904176\pi\)
−0.955029 + 0.296513i \(0.904176\pi\)
\(38\) 0 0
\(39\) 57.2837i 1.46881i
\(40\) 0 0
\(41\) −49.8775 −1.21652 −0.608262 0.793736i \(-0.708133\pi\)
−0.608262 + 0.793736i \(0.708133\pi\)
\(42\) 0 0
\(43\) 73.2730i 1.70402i 0.523522 + 0.852012i \(0.324618\pi\)
−0.523522 + 0.852012i \(0.675382\pi\)
\(44\) 0 0
\(45\) 146.250 3.25000
\(46\) 0 0
\(47\) 44.2248i 0.940952i 0.882413 + 0.470476i \(0.155918\pi\)
−0.882413 + 0.470476i \(0.844082\pi\)
\(48\) 0 0
\(49\) −7.00000 −0.142857
\(50\) 0 0
\(51\) − 67.9479i − 1.33231i
\(52\) 0 0
\(53\) −54.2355 −1.02331 −0.511655 0.859191i \(-0.670968\pi\)
−0.511655 + 0.859191i \(0.670968\pi\)
\(54\) 0 0
\(55\) 17.4587i 0.317431i
\(56\) 0 0
\(57\) −149.736 −2.62695
\(58\) 0 0
\(59\) − 12.4706i − 0.211367i −0.994400 0.105683i \(-0.966297\pi\)
0.994400 0.105683i \(-0.0337030\pi\)
\(60\) 0 0
\(61\) 35.6770 0.584869 0.292435 0.956285i \(-0.405535\pi\)
0.292435 + 0.956285i \(0.405535\pi\)
\(62\) 0 0
\(63\) 66.9255i 1.06231i
\(64\) 0 0
\(65\) −56.5544 −0.870068
\(66\) 0 0
\(67\) 24.4891i 0.365509i 0.983159 + 0.182754i \(0.0585013\pi\)
−0.983159 + 0.182754i \(0.941499\pi\)
\(68\) 0 0
\(69\) 153.182 2.22003
\(70\) 0 0
\(71\) − 11.0480i − 0.155606i −0.996969 0.0778030i \(-0.975209\pi\)
0.996969 0.0778030i \(-0.0247905\pi\)
\(72\) 0 0
\(73\) 74.3713 1.01879 0.509393 0.860534i \(-0.329870\pi\)
0.509393 + 0.860534i \(0.329870\pi\)
\(74\) 0 0
\(75\) 49.3547i 0.658062i
\(76\) 0 0
\(77\) −7.98928 −0.103757
\(78\) 0 0
\(79\) 22.8912i 0.289762i 0.989449 + 0.144881i \(0.0462800\pi\)
−0.989449 + 0.144881i \(0.953720\pi\)
\(80\) 0 0
\(81\) 331.202 4.08891
\(82\) 0 0
\(83\) 48.1934i 0.580644i 0.956929 + 0.290322i \(0.0937623\pi\)
−0.956929 + 0.290322i \(0.906238\pi\)
\(84\) 0 0
\(85\) 67.0828 0.789210
\(86\) 0 0
\(87\) 9.15529i 0.105233i
\(88\) 0 0
\(89\) 67.4801 0.758204 0.379102 0.925355i \(-0.376233\pi\)
0.379102 + 0.925355i \(0.376233\pi\)
\(90\) 0 0
\(91\) − 25.8799i − 0.284394i
\(92\) 0 0
\(93\) −70.3376 −0.756318
\(94\) 0 0
\(95\) − 147.830i − 1.55610i
\(96\) 0 0
\(97\) −7.75949 −0.0799947 −0.0399974 0.999200i \(-0.512735\pi\)
−0.0399974 + 0.999200i \(0.512735\pi\)
\(98\) 0 0
\(99\) 76.3838i 0.771554i
\(100\) 0 0
\(101\) −102.524 −1.01509 −0.507546 0.861624i \(-0.669447\pi\)
−0.507546 + 0.861624i \(0.669447\pi\)
\(102\) 0 0
\(103\) − 14.6874i − 0.142596i −0.997455 0.0712980i \(-0.977286\pi\)
0.997455 0.0712980i \(-0.0227141\pi\)
\(104\) 0 0
\(105\) −89.5820 −0.853162
\(106\) 0 0
\(107\) 70.0212i 0.654404i 0.944954 + 0.327202i \(0.106106\pi\)
−0.944954 + 0.327202i \(0.893894\pi\)
\(108\) 0 0
\(109\) −65.9204 −0.604774 −0.302387 0.953185i \(-0.597783\pi\)
−0.302387 + 0.953185i \(0.597783\pi\)
\(110\) 0 0
\(111\) − 413.873i − 3.72858i
\(112\) 0 0
\(113\) 83.2184 0.736446 0.368223 0.929737i \(-0.379966\pi\)
0.368223 + 0.929737i \(0.379966\pi\)
\(114\) 0 0
\(115\) 151.232i 1.31506i
\(116\) 0 0
\(117\) −247.432 −2.11480
\(118\) 0 0
\(119\) 30.6978i 0.257965i
\(120\) 0 0
\(121\) 111.882 0.924642
\(122\) 0 0
\(123\) − 292.094i − 2.37475i
\(124\) 0 0
\(125\) 95.8155 0.766524
\(126\) 0 0
\(127\) 102.292i 0.805452i 0.915321 + 0.402726i \(0.131937\pi\)
−0.915321 + 0.402726i \(0.868063\pi\)
\(128\) 0 0
\(129\) −429.104 −3.32639
\(130\) 0 0
\(131\) 89.1759i 0.680732i 0.940293 + 0.340366i \(0.110551\pi\)
−0.940293 + 0.340366i \(0.889449\pi\)
\(132\) 0 0
\(133\) 67.6484 0.508635
\(134\) 0 0
\(135\) 551.745i 4.08700i
\(136\) 0 0
\(137\) −11.3800 −0.0830660 −0.0415330 0.999137i \(-0.513224\pi\)
−0.0415330 + 0.999137i \(0.513224\pi\)
\(138\) 0 0
\(139\) 121.256i 0.872343i 0.899863 + 0.436172i \(0.143666\pi\)
−0.899863 + 0.436172i \(0.856334\pi\)
\(140\) 0 0
\(141\) −258.991 −1.83681
\(142\) 0 0
\(143\) − 29.5374i − 0.206555i
\(144\) 0 0
\(145\) −9.03872 −0.0623360
\(146\) 0 0
\(147\) − 40.9936i − 0.278868i
\(148\) 0 0
\(149\) −63.5731 −0.426665 −0.213333 0.976980i \(-0.568432\pi\)
−0.213333 + 0.976980i \(0.568432\pi\)
\(150\) 0 0
\(151\) − 122.414i − 0.810689i −0.914164 0.405344i \(-0.867152\pi\)
0.914164 0.405344i \(-0.132848\pi\)
\(152\) 0 0
\(153\) 293.495 1.91827
\(154\) 0 0
\(155\) − 69.4420i − 0.448013i
\(156\) 0 0
\(157\) 39.6070 0.252274 0.126137 0.992013i \(-0.459742\pi\)
0.126137 + 0.992013i \(0.459742\pi\)
\(158\) 0 0
\(159\) − 317.616i − 1.99758i
\(160\) 0 0
\(161\) −69.2051 −0.429845
\(162\) 0 0
\(163\) − 32.9017i − 0.201851i −0.994894 0.100925i \(-0.967820\pi\)
0.994894 0.100925i \(-0.0321803\pi\)
\(164\) 0 0
\(165\) −102.242 −0.619650
\(166\) 0 0
\(167\) 98.0405i 0.587069i 0.955949 + 0.293534i \(0.0948315\pi\)
−0.955949 + 0.293534i \(0.905168\pi\)
\(168\) 0 0
\(169\) −73.3189 −0.433840
\(170\) 0 0
\(171\) − 646.772i − 3.78229i
\(172\) 0 0
\(173\) −336.891 −1.94734 −0.973672 0.227951i \(-0.926797\pi\)
−0.973672 + 0.227951i \(0.926797\pi\)
\(174\) 0 0
\(175\) − 22.2976i − 0.127415i
\(176\) 0 0
\(177\) 73.0309 0.412604
\(178\) 0 0
\(179\) − 51.4537i − 0.287451i −0.989618 0.143725i \(-0.954092\pi\)
0.989618 0.143725i \(-0.0459082\pi\)
\(180\) 0 0
\(181\) 281.807 1.55695 0.778473 0.627678i \(-0.215994\pi\)
0.778473 + 0.627678i \(0.215994\pi\)
\(182\) 0 0
\(183\) 208.933i 1.14171i
\(184\) 0 0
\(185\) 408.603 2.20866
\(186\) 0 0
\(187\) 35.0362i 0.187359i
\(188\) 0 0
\(189\) −252.484 −1.33590
\(190\) 0 0
\(191\) − 165.004i − 0.863897i −0.901898 0.431949i \(-0.857826\pi\)
0.901898 0.431949i \(-0.142174\pi\)
\(192\) 0 0
\(193\) −64.6933 −0.335199 −0.167599 0.985855i \(-0.553601\pi\)
−0.167599 + 0.985855i \(0.553601\pi\)
\(194\) 0 0
\(195\) − 331.196i − 1.69844i
\(196\) 0 0
\(197\) 189.712 0.963003 0.481501 0.876445i \(-0.340092\pi\)
0.481501 + 0.876445i \(0.340092\pi\)
\(198\) 0 0
\(199\) − 205.590i − 1.03312i −0.856252 0.516559i \(-0.827213\pi\)
0.856252 0.516559i \(-0.172787\pi\)
\(200\) 0 0
\(201\) −143.414 −0.713502
\(202\) 0 0
\(203\) − 4.13621i − 0.0203754i
\(204\) 0 0
\(205\) 288.375 1.40671
\(206\) 0 0
\(207\) 661.655i 3.19640i
\(208\) 0 0
\(209\) 77.2089 0.369421
\(210\) 0 0
\(211\) 220.725i 1.04609i 0.852305 + 0.523045i \(0.175204\pi\)
−0.852305 + 0.523045i \(0.824796\pi\)
\(212\) 0 0
\(213\) 64.6998 0.303755
\(214\) 0 0
\(215\) − 423.641i − 1.97042i
\(216\) 0 0
\(217\) 31.7774 0.146440
\(218\) 0 0
\(219\) 435.536i 1.98875i
\(220\) 0 0
\(221\) −113.493 −0.513545
\(222\) 0 0
\(223\) 163.489i 0.733134i 0.930392 + 0.366567i \(0.119467\pi\)
−0.930392 + 0.366567i \(0.880533\pi\)
\(224\) 0 0
\(225\) −213.183 −0.947480
\(226\) 0 0
\(227\) − 221.227i − 0.974569i −0.873243 0.487285i \(-0.837987\pi\)
0.873243 0.487285i \(-0.162013\pi\)
\(228\) 0 0
\(229\) 233.563 1.01993 0.509964 0.860196i \(-0.329659\pi\)
0.509964 + 0.860196i \(0.329659\pi\)
\(230\) 0 0
\(231\) − 46.7871i − 0.202542i
\(232\) 0 0
\(233\) −13.2573 −0.0568983 −0.0284491 0.999595i \(-0.509057\pi\)
−0.0284491 + 0.999595i \(0.509057\pi\)
\(234\) 0 0
\(235\) − 255.693i − 1.08806i
\(236\) 0 0
\(237\) −134.056 −0.565638
\(238\) 0 0
\(239\) − 353.231i − 1.47796i −0.673730 0.738978i \(-0.735309\pi\)
0.673730 0.738978i \(-0.264691\pi\)
\(240\) 0 0
\(241\) 181.261 0.752121 0.376060 0.926595i \(-0.377279\pi\)
0.376060 + 0.926595i \(0.377279\pi\)
\(242\) 0 0
\(243\) 1080.72i 4.44742i
\(244\) 0 0
\(245\) 40.4717 0.165191
\(246\) 0 0
\(247\) 250.105i 1.01257i
\(248\) 0 0
\(249\) −282.232 −1.13346
\(250\) 0 0
\(251\) 98.4522i 0.392240i 0.980580 + 0.196120i \(0.0628342\pi\)
−0.980580 + 0.196120i \(0.937166\pi\)
\(252\) 0 0
\(253\) −78.9856 −0.312196
\(254\) 0 0
\(255\) 392.853i 1.54060i
\(256\) 0 0
\(257\) −279.527 −1.08765 −0.543827 0.839197i \(-0.683025\pi\)
−0.543827 + 0.839197i \(0.683025\pi\)
\(258\) 0 0
\(259\) 186.981i 0.721934i
\(260\) 0 0
\(261\) −39.5455 −0.151515
\(262\) 0 0
\(263\) − 45.3085i − 0.172276i −0.996283 0.0861378i \(-0.972547\pi\)
0.996283 0.0861378i \(-0.0274525\pi\)
\(264\) 0 0
\(265\) 313.572 1.18329
\(266\) 0 0
\(267\) 395.179i 1.48007i
\(268\) 0 0
\(269\) −220.227 −0.818686 −0.409343 0.912380i \(-0.634242\pi\)
−0.409343 + 0.912380i \(0.634242\pi\)
\(270\) 0 0
\(271\) 54.8463i 0.202385i 0.994867 + 0.101193i \(0.0322658\pi\)
−0.994867 + 0.101193i \(0.967734\pi\)
\(272\) 0 0
\(273\) 151.559 0.555160
\(274\) 0 0
\(275\) − 25.4489i − 0.0925413i
\(276\) 0 0
\(277\) 320.078 1.15552 0.577758 0.816208i \(-0.303928\pi\)
0.577758 + 0.816208i \(0.303928\pi\)
\(278\) 0 0
\(279\) − 303.817i − 1.08895i
\(280\) 0 0
\(281\) 526.445 1.87347 0.936734 0.350041i \(-0.113832\pi\)
0.936734 + 0.350041i \(0.113832\pi\)
\(282\) 0 0
\(283\) 331.932i 1.17290i 0.809984 + 0.586452i \(0.199476\pi\)
−0.809984 + 0.586452i \(0.800524\pi\)
\(284\) 0 0
\(285\) 865.726 3.03763
\(286\) 0 0
\(287\) 131.963i 0.459803i
\(288\) 0 0
\(289\) −154.378 −0.534180
\(290\) 0 0
\(291\) − 45.4414i − 0.156156i
\(292\) 0 0
\(293\) −240.715 −0.821554 −0.410777 0.911736i \(-0.634742\pi\)
−0.410777 + 0.911736i \(0.634742\pi\)
\(294\) 0 0
\(295\) 72.1011i 0.244410i
\(296\) 0 0
\(297\) −288.167 −0.970259
\(298\) 0 0
\(299\) − 255.860i − 0.855718i
\(300\) 0 0
\(301\) 193.862 0.644061
\(302\) 0 0
\(303\) − 600.407i − 1.98154i
\(304\) 0 0
\(305\) −206.273 −0.676304
\(306\) 0 0
\(307\) − 221.862i − 0.722678i −0.932435 0.361339i \(-0.882320\pi\)
0.932435 0.361339i \(-0.117680\pi\)
\(308\) 0 0
\(309\) 86.0128 0.278359
\(310\) 0 0
\(311\) 476.048i 1.53070i 0.643614 + 0.765350i \(0.277434\pi\)
−0.643614 + 0.765350i \(0.722566\pi\)
\(312\) 0 0
\(313\) −495.170 −1.58201 −0.791006 0.611808i \(-0.790442\pi\)
−0.791006 + 0.611808i \(0.790442\pi\)
\(314\) 0 0
\(315\) − 386.941i − 1.22839i
\(316\) 0 0
\(317\) −493.029 −1.55530 −0.777649 0.628699i \(-0.783588\pi\)
−0.777649 + 0.628699i \(0.783588\pi\)
\(318\) 0 0
\(319\) − 4.72077i − 0.0147986i
\(320\) 0 0
\(321\) −410.060 −1.27745
\(322\) 0 0
\(323\) − 296.665i − 0.918468i
\(324\) 0 0
\(325\) 82.4371 0.253653
\(326\) 0 0
\(327\) − 386.045i − 1.18057i
\(328\) 0 0
\(329\) 117.008 0.355647
\(330\) 0 0
\(331\) − 586.929i − 1.77320i −0.462539 0.886599i \(-0.653061\pi\)
0.462539 0.886599i \(-0.346939\pi\)
\(332\) 0 0
\(333\) 1787.69 5.36842
\(334\) 0 0
\(335\) − 141.588i − 0.422650i
\(336\) 0 0
\(337\) 22.2636 0.0660639 0.0330320 0.999454i \(-0.489484\pi\)
0.0330320 + 0.999454i \(0.489484\pi\)
\(338\) 0 0
\(339\) 487.347i 1.43760i
\(340\) 0 0
\(341\) 36.2683 0.106359
\(342\) 0 0
\(343\) 18.5203i 0.0539949i
\(344\) 0 0
\(345\) −885.647 −2.56709
\(346\) 0 0
\(347\) − 264.797i − 0.763103i −0.924348 0.381552i \(-0.875390\pi\)
0.924348 0.381552i \(-0.124610\pi\)
\(348\) 0 0
\(349\) 101.730 0.291489 0.145745 0.989322i \(-0.453442\pi\)
0.145745 + 0.989322i \(0.453442\pi\)
\(350\) 0 0
\(351\) − 933.466i − 2.65945i
\(352\) 0 0
\(353\) 472.399 1.33824 0.669121 0.743154i \(-0.266671\pi\)
0.669121 + 0.743154i \(0.266671\pi\)
\(354\) 0 0
\(355\) 63.8760i 0.179933i
\(356\) 0 0
\(357\) −179.773 −0.503567
\(358\) 0 0
\(359\) 473.453i 1.31881i 0.751788 + 0.659405i \(0.229192\pi\)
−0.751788 + 0.659405i \(0.770808\pi\)
\(360\) 0 0
\(361\) −292.758 −0.810965
\(362\) 0 0
\(363\) 655.205i 1.80497i
\(364\) 0 0
\(365\) −429.991 −1.17806
\(366\) 0 0
\(367\) 467.459i 1.27373i 0.770975 + 0.636866i \(0.219769\pi\)
−0.770975 + 0.636866i \(0.780231\pi\)
\(368\) 0 0
\(369\) 1261.67 3.41917
\(370\) 0 0
\(371\) 143.494i 0.386775i
\(372\) 0 0
\(373\) 534.491 1.43295 0.716475 0.697613i \(-0.245754\pi\)
0.716475 + 0.697613i \(0.245754\pi\)
\(374\) 0 0
\(375\) 561.118i 1.49631i
\(376\) 0 0
\(377\) 15.2921 0.0405626
\(378\) 0 0
\(379\) − 204.355i − 0.539196i −0.962973 0.269598i \(-0.913109\pi\)
0.962973 0.269598i \(-0.0868908\pi\)
\(380\) 0 0
\(381\) −599.048 −1.57231
\(382\) 0 0
\(383\) 622.590i 1.62556i 0.582570 + 0.812781i \(0.302047\pi\)
−0.582570 + 0.812781i \(0.697953\pi\)
\(384\) 0 0
\(385\) 46.1914 0.119978
\(386\) 0 0
\(387\) − 1853.48i − 4.78934i
\(388\) 0 0
\(389\) −510.568 −1.31251 −0.656257 0.754537i \(-0.727861\pi\)
−0.656257 + 0.754537i \(0.727861\pi\)
\(390\) 0 0
\(391\) 303.492i 0.776194i
\(392\) 0 0
\(393\) −522.235 −1.32884
\(394\) 0 0
\(395\) − 132.349i − 0.335062i
\(396\) 0 0
\(397\) −591.459 −1.48982 −0.744910 0.667165i \(-0.767508\pi\)
−0.744910 + 0.667165i \(0.767508\pi\)
\(398\) 0 0
\(399\) 396.165i 0.992895i
\(400\) 0 0
\(401\) 29.0538 0.0724534 0.0362267 0.999344i \(-0.488466\pi\)
0.0362267 + 0.999344i \(0.488466\pi\)
\(402\) 0 0
\(403\) 117.485i 0.291526i
\(404\) 0 0
\(405\) −1914.90 −4.72815
\(406\) 0 0
\(407\) 213.406i 0.524339i
\(408\) 0 0
\(409\) 120.749 0.295230 0.147615 0.989045i \(-0.452840\pi\)
0.147615 + 0.989045i \(0.452840\pi\)
\(410\) 0 0
\(411\) − 66.6442i − 0.162151i
\(412\) 0 0
\(413\) −32.9942 −0.0798891
\(414\) 0 0
\(415\) − 278.638i − 0.671418i
\(416\) 0 0
\(417\) −710.102 −1.70288
\(418\) 0 0
\(419\) 379.985i 0.906885i 0.891286 + 0.453443i \(0.149804\pi\)
−0.891286 + 0.453443i \(0.850196\pi\)
\(420\) 0 0
\(421\) −536.480 −1.27430 −0.637149 0.770740i \(-0.719887\pi\)
−0.637149 + 0.770740i \(0.719887\pi\)
\(422\) 0 0
\(423\) − 1118.69i − 2.64465i
\(424\) 0 0
\(425\) −97.7840 −0.230080
\(426\) 0 0
\(427\) − 94.3926i − 0.221060i
\(428\) 0 0
\(429\) 172.978 0.403211
\(430\) 0 0
\(431\) − 384.935i − 0.893120i −0.894754 0.446560i \(-0.852649\pi\)
0.894754 0.446560i \(-0.147351\pi\)
\(432\) 0 0
\(433\) −667.115 −1.54068 −0.770340 0.637633i \(-0.779914\pi\)
−0.770340 + 0.637633i \(0.779914\pi\)
\(434\) 0 0
\(435\) − 52.9329i − 0.121685i
\(436\) 0 0
\(437\) 668.802 1.53044
\(438\) 0 0
\(439\) − 521.733i − 1.18846i −0.804296 0.594229i \(-0.797457\pi\)
0.804296 0.594229i \(-0.202543\pi\)
\(440\) 0 0
\(441\) 177.068 0.401515
\(442\) 0 0
\(443\) 204.156i 0.460848i 0.973090 + 0.230424i \(0.0740114\pi\)
−0.973090 + 0.230424i \(0.925989\pi\)
\(444\) 0 0
\(445\) −390.148 −0.876737
\(446\) 0 0
\(447\) − 372.299i − 0.832884i
\(448\) 0 0
\(449\) 513.424 1.14348 0.571742 0.820434i \(-0.306268\pi\)
0.571742 + 0.820434i \(0.306268\pi\)
\(450\) 0 0
\(451\) 150.613i 0.333954i
\(452\) 0 0
\(453\) 716.885 1.58253
\(454\) 0 0
\(455\) 149.629i 0.328855i
\(456\) 0 0
\(457\) −134.029 −0.293281 −0.146641 0.989190i \(-0.546846\pi\)
−0.146641 + 0.989190i \(0.546846\pi\)
\(458\) 0 0
\(459\) 1107.24i 2.41230i
\(460\) 0 0
\(461\) 8.06740 0.0174998 0.00874990 0.999962i \(-0.497215\pi\)
0.00874990 + 0.999962i \(0.497215\pi\)
\(462\) 0 0
\(463\) 837.707i 1.80930i 0.426152 + 0.904652i \(0.359869\pi\)
−0.426152 + 0.904652i \(0.640131\pi\)
\(464\) 0 0
\(465\) 406.669 0.874556
\(466\) 0 0
\(467\) − 495.780i − 1.06163i −0.847489 0.530814i \(-0.821886\pi\)
0.847489 0.530814i \(-0.178114\pi\)
\(468\) 0 0
\(469\) 64.7920 0.138149
\(470\) 0 0
\(471\) 231.948i 0.492458i
\(472\) 0 0
\(473\) 221.260 0.467780
\(474\) 0 0
\(475\) 215.486i 0.453654i
\(476\) 0 0
\(477\) 1371.91 2.87613
\(478\) 0 0
\(479\) 225.993i 0.471802i 0.971777 + 0.235901i \(0.0758042\pi\)
−0.971777 + 0.235901i \(0.924196\pi\)
\(480\) 0 0
\(481\) −691.292 −1.43720
\(482\) 0 0
\(483\) − 405.281i − 0.839091i
\(484\) 0 0
\(485\) 44.8628 0.0925006
\(486\) 0 0
\(487\) − 728.173i − 1.49522i −0.664137 0.747611i \(-0.731201\pi\)
0.664137 0.747611i \(-0.268799\pi\)
\(488\) 0 0
\(489\) 192.680 0.394028
\(490\) 0 0
\(491\) 724.908i 1.47639i 0.674587 + 0.738196i \(0.264322\pi\)
−0.674587 + 0.738196i \(0.735678\pi\)
\(492\) 0 0
\(493\) −18.1389 −0.0367930
\(494\) 0 0
\(495\) − 441.626i − 0.892174i
\(496\) 0 0
\(497\) −29.2303 −0.0588135
\(498\) 0 0
\(499\) − 691.520i − 1.38581i −0.721028 0.692906i \(-0.756330\pi\)
0.721028 0.692906i \(-0.243670\pi\)
\(500\) 0 0
\(501\) −574.148 −1.14600
\(502\) 0 0
\(503\) 518.951i 1.03171i 0.856675 + 0.515856i \(0.172526\pi\)
−0.856675 + 0.515856i \(0.827474\pi\)
\(504\) 0 0
\(505\) 592.762 1.17379
\(506\) 0 0
\(507\) − 429.373i − 0.846889i
\(508\) 0 0
\(509\) 823.772 1.61841 0.809206 0.587525i \(-0.199898\pi\)
0.809206 + 0.587525i \(0.199898\pi\)
\(510\) 0 0
\(511\) − 196.768i − 0.385065i
\(512\) 0 0
\(513\) 2440.02 4.75638
\(514\) 0 0
\(515\) 84.9177i 0.164889i
\(516\) 0 0
\(517\) 133.544 0.258305
\(518\) 0 0
\(519\) − 1972.91i − 3.80137i
\(520\) 0 0
\(521\) −533.963 −1.02488 −0.512440 0.858723i \(-0.671258\pi\)
−0.512440 + 0.858723i \(0.671258\pi\)
\(522\) 0 0
\(523\) 709.660i 1.35690i 0.734646 + 0.678451i \(0.237348\pi\)
−0.734646 + 0.678451i \(0.762652\pi\)
\(524\) 0 0
\(525\) 130.580 0.248724
\(526\) 0 0
\(527\) − 139.356i − 0.264433i
\(528\) 0 0
\(529\) −155.192 −0.293369
\(530\) 0 0
\(531\) 315.450i 0.594069i
\(532\) 0 0
\(533\) −487.885 −0.915356
\(534\) 0 0
\(535\) − 404.839i − 0.756709i
\(536\) 0 0
\(537\) 301.325 0.561126
\(538\) 0 0
\(539\) 21.1376i 0.0392164i
\(540\) 0 0
\(541\) −666.282 −1.23157 −0.615787 0.787913i \(-0.711162\pi\)
−0.615787 + 0.787913i \(0.711162\pi\)
\(542\) 0 0
\(543\) 1650.33i 3.03928i
\(544\) 0 0
\(545\) 381.130 0.699321
\(546\) 0 0
\(547\) − 65.3904i − 0.119544i −0.998212 0.0597718i \(-0.980963\pi\)
0.998212 0.0597718i \(-0.0190373\pi\)
\(548\) 0 0
\(549\) −902.467 −1.64384
\(550\) 0 0
\(551\) 39.9726i 0.0725456i
\(552\) 0 0
\(553\) 60.5644 0.109520
\(554\) 0 0
\(555\) 2392.87i 4.31149i
\(556\) 0 0
\(557\) −671.474 −1.20552 −0.602759 0.797923i \(-0.705932\pi\)
−0.602759 + 0.797923i \(0.705932\pi\)
\(558\) 0 0
\(559\) 716.733i 1.28217i
\(560\) 0 0
\(561\) −205.180 −0.365740
\(562\) 0 0
\(563\) 629.362i 1.11787i 0.829211 + 0.558936i \(0.188790\pi\)
−0.829211 + 0.558936i \(0.811210\pi\)
\(564\) 0 0
\(565\) −481.142 −0.851578
\(566\) 0 0
\(567\) − 876.277i − 1.54546i
\(568\) 0 0
\(569\) −49.5333 −0.0870533 −0.0435267 0.999052i \(-0.513859\pi\)
−0.0435267 + 0.999052i \(0.513859\pi\)
\(570\) 0 0
\(571\) − 988.807i − 1.73171i −0.500294 0.865856i \(-0.666775\pi\)
0.500294 0.865856i \(-0.333225\pi\)
\(572\) 0 0
\(573\) 966.304 1.68639
\(574\) 0 0
\(575\) − 220.444i − 0.383381i
\(576\) 0 0
\(577\) −63.1355 −0.109420 −0.0547101 0.998502i \(-0.517423\pi\)
−0.0547101 + 0.998502i \(0.517423\pi\)
\(578\) 0 0
\(579\) − 378.859i − 0.654334i
\(580\) 0 0
\(581\) 127.508 0.219463
\(582\) 0 0
\(583\) 163.773i 0.280914i
\(584\) 0 0
\(585\) 1430.57 2.44542
\(586\) 0 0
\(587\) 9.88169i 0.0168342i 0.999965 + 0.00841711i \(0.00267928\pi\)
−0.999965 + 0.00841711i \(0.997321\pi\)
\(588\) 0 0
\(589\) −307.098 −0.521390
\(590\) 0 0
\(591\) 1111.00i 1.87986i
\(592\) 0 0
\(593\) −147.865 −0.249351 −0.124676 0.992198i \(-0.539789\pi\)
−0.124676 + 0.992198i \(0.539789\pi\)
\(594\) 0 0
\(595\) − 177.484i − 0.298293i
\(596\) 0 0
\(597\) 1203.99 2.01673
\(598\) 0 0
\(599\) 464.007i 0.774636i 0.921946 + 0.387318i \(0.126599\pi\)
−0.921946 + 0.387318i \(0.873401\pi\)
\(600\) 0 0
\(601\) 215.359 0.358334 0.179167 0.983819i \(-0.442660\pi\)
0.179167 + 0.983819i \(0.442660\pi\)
\(602\) 0 0
\(603\) − 619.463i − 1.02730i
\(604\) 0 0
\(605\) −646.863 −1.06919
\(606\) 0 0
\(607\) 310.053i 0.510796i 0.966836 + 0.255398i \(0.0822065\pi\)
−0.966836 + 0.255398i \(0.917794\pi\)
\(608\) 0 0
\(609\) 24.2226 0.0397744
\(610\) 0 0
\(611\) 432.592i 0.708007i
\(612\) 0 0
\(613\) −791.288 −1.29084 −0.645422 0.763826i \(-0.723319\pi\)
−0.645422 + 0.763826i \(0.723319\pi\)
\(614\) 0 0
\(615\) 1688.79i 2.74600i
\(616\) 0 0
\(617\) 1139.17 1.84631 0.923153 0.384433i \(-0.125603\pi\)
0.923153 + 0.384433i \(0.125603\pi\)
\(618\) 0 0
\(619\) − 419.673i − 0.677985i −0.940789 0.338992i \(-0.889914\pi\)
0.940789 0.338992i \(-0.110086\pi\)
\(620\) 0 0
\(621\) −2496.17 −4.01960
\(622\) 0 0
\(623\) − 178.536i − 0.286574i
\(624\) 0 0
\(625\) −764.666 −1.22347
\(626\) 0 0
\(627\) 452.153i 0.721138i
\(628\) 0 0
\(629\) 819.985 1.30363
\(630\) 0 0
\(631\) 506.397i 0.802530i 0.915962 + 0.401265i \(0.131429\pi\)
−0.915962 + 0.401265i \(0.868571\pi\)
\(632\) 0 0
\(633\) −1292.62 −2.04205
\(634\) 0 0
\(635\) − 591.421i − 0.931372i
\(636\) 0 0
\(637\) −68.4717 −0.107491
\(638\) 0 0
\(639\) 279.465i 0.437347i
\(640\) 0 0
\(641\) −501.692 −0.782671 −0.391335 0.920248i \(-0.627987\pi\)
−0.391335 + 0.920248i \(0.627987\pi\)
\(642\) 0 0
\(643\) − 691.217i − 1.07499i −0.843268 0.537493i \(-0.819371\pi\)
0.843268 0.537493i \(-0.180629\pi\)
\(644\) 0 0
\(645\) 2480.94 3.84642
\(646\) 0 0
\(647\) 711.553i 1.09977i 0.835239 + 0.549886i \(0.185329\pi\)
−0.835239 + 0.549886i \(0.814671\pi\)
\(648\) 0 0
\(649\) −37.6571 −0.0580233
\(650\) 0 0
\(651\) 186.096i 0.285861i
\(652\) 0 0
\(653\) 455.205 0.697099 0.348549 0.937290i \(-0.386674\pi\)
0.348549 + 0.937290i \(0.386674\pi\)
\(654\) 0 0
\(655\) − 515.586i − 0.787154i
\(656\) 0 0
\(657\) −1881.26 −2.86341
\(658\) 0 0
\(659\) − 156.616i − 0.237656i −0.992915 0.118828i \(-0.962086\pi\)
0.992915 0.118828i \(-0.0379138\pi\)
\(660\) 0 0
\(661\) 241.002 0.364602 0.182301 0.983243i \(-0.441646\pi\)
0.182301 + 0.983243i \(0.441646\pi\)
\(662\) 0 0
\(663\) − 664.644i − 1.00248i
\(664\) 0 0
\(665\) −391.121 −0.588152
\(666\) 0 0
\(667\) − 40.8924i − 0.0613080i
\(668\) 0 0
\(669\) −957.429 −1.43114
\(670\) 0 0
\(671\) − 107.733i − 0.160555i
\(672\) 0 0
\(673\) 623.074 0.925816 0.462908 0.886406i \(-0.346806\pi\)
0.462908 + 0.886406i \(0.346806\pi\)
\(674\) 0 0
\(675\) − 804.258i − 1.19149i
\(676\) 0 0
\(677\) 343.461 0.507328 0.253664 0.967292i \(-0.418364\pi\)
0.253664 + 0.967292i \(0.418364\pi\)
\(678\) 0 0
\(679\) 20.5297i 0.0302352i
\(680\) 0 0
\(681\) 1295.56 1.90243
\(682\) 0 0
\(683\) − 909.578i − 1.33174i −0.746068 0.665870i \(-0.768061\pi\)
0.746068 0.665870i \(-0.231939\pi\)
\(684\) 0 0
\(685\) 65.7957 0.0960521
\(686\) 0 0
\(687\) 1367.80i 1.99098i
\(688\) 0 0
\(689\) −530.514 −0.769976
\(690\) 0 0
\(691\) 325.753i 0.471423i 0.971823 + 0.235711i \(0.0757420\pi\)
−0.971823 + 0.235711i \(0.924258\pi\)
\(692\) 0 0
\(693\) 202.093 0.291620
\(694\) 0 0
\(695\) − 701.061i − 1.00872i
\(696\) 0 0
\(697\) 578.712 0.830290
\(698\) 0 0
\(699\) − 77.6379i − 0.111070i
\(700\) 0 0
\(701\) −414.315 −0.591034 −0.295517 0.955338i \(-0.595492\pi\)
−0.295517 + 0.955338i \(0.595492\pi\)
\(702\) 0 0
\(703\) − 1806.99i − 2.57040i
\(704\) 0 0
\(705\) 1497.40 2.12397
\(706\) 0 0
\(707\) 271.254i 0.383669i
\(708\) 0 0
\(709\) 1146.48 1.61703 0.808517 0.588472i \(-0.200270\pi\)
0.808517 + 0.588472i \(0.200270\pi\)
\(710\) 0 0
\(711\) − 579.044i − 0.814408i
\(712\) 0 0
\(713\) 314.165 0.440624
\(714\) 0 0
\(715\) 170.775i 0.238847i
\(716\) 0 0
\(717\) 2068.60 2.88508
\(718\) 0 0
\(719\) − 387.316i − 0.538687i −0.963044 0.269344i \(-0.913193\pi\)
0.963044 0.269344i \(-0.0868067\pi\)
\(720\) 0 0
\(721\) −38.8592 −0.0538963
\(722\) 0 0
\(723\) 1061.51i 1.46820i
\(724\) 0 0
\(725\) 13.1754 0.0181730
\(726\) 0 0
\(727\) − 620.563i − 0.853594i −0.904347 0.426797i \(-0.859642\pi\)
0.904347 0.426797i \(-0.140358\pi\)
\(728\) 0 0
\(729\) −3348.15 −4.59280
\(730\) 0 0
\(731\) − 850.163i − 1.16301i
\(732\) 0 0
\(733\) 96.8796 0.132169 0.0660843 0.997814i \(-0.478949\pi\)
0.0660843 + 0.997814i \(0.478949\pi\)
\(734\) 0 0
\(735\) 237.012i 0.322465i
\(736\) 0 0
\(737\) 73.9488 0.100338
\(738\) 0 0
\(739\) − 782.913i − 1.05942i −0.848178 0.529711i \(-0.822300\pi\)
0.848178 0.529711i \(-0.177700\pi\)
\(740\) 0 0
\(741\) −1464.67 −1.97661
\(742\) 0 0
\(743\) 48.0641i 0.0646892i 0.999477 + 0.0323446i \(0.0102974\pi\)
−0.999477 + 0.0323446i \(0.989703\pi\)
\(744\) 0 0
\(745\) 367.559 0.493368
\(746\) 0 0
\(747\) − 1219.08i − 1.63196i
\(748\) 0 0
\(749\) 185.259 0.247341
\(750\) 0 0
\(751\) − 284.489i − 0.378814i −0.981899 0.189407i \(-0.939343\pi\)
0.981899 0.189407i \(-0.0606565\pi\)
\(752\) 0 0
\(753\) −576.559 −0.765683
\(754\) 0 0
\(755\) 707.757i 0.937427i
\(756\) 0 0
\(757\) −797.514 −1.05352 −0.526759 0.850014i \(-0.676593\pi\)
−0.526759 + 0.850014i \(0.676593\pi\)
\(758\) 0 0
\(759\) − 462.558i − 0.609431i
\(760\) 0 0
\(761\) 1004.15 1.31951 0.659756 0.751480i \(-0.270660\pi\)
0.659756 + 0.751480i \(0.270660\pi\)
\(762\) 0 0
\(763\) 174.409i 0.228583i
\(764\) 0 0
\(765\) −1696.89 −2.21816
\(766\) 0 0
\(767\) − 121.984i − 0.159040i
\(768\) 0 0
\(769\) −959.769 −1.24807 −0.624037 0.781395i \(-0.714509\pi\)
−0.624037 + 0.781395i \(0.714509\pi\)
\(770\) 0 0
\(771\) − 1636.98i − 2.12319i
\(772\) 0 0
\(773\) −94.4740 −0.122217 −0.0611086 0.998131i \(-0.519464\pi\)
−0.0611086 + 0.998131i \(0.519464\pi\)
\(774\) 0 0
\(775\) 101.223i 0.130610i
\(776\) 0 0
\(777\) −1095.00 −1.40927
\(778\) 0 0
\(779\) − 1275.30i − 1.63710i
\(780\) 0 0
\(781\) −33.3613 −0.0427162
\(782\) 0 0
\(783\) − 149.190i − 0.190536i
\(784\) 0 0
\(785\) −228.994 −0.291713
\(786\) 0 0
\(787\) − 108.864i − 0.138328i −0.997605 0.0691642i \(-0.977967\pi\)
0.997605 0.0691642i \(-0.0220332\pi\)
\(788\) 0 0
\(789\) 265.337 0.336295
\(790\) 0 0
\(791\) − 220.175i − 0.278351i
\(792\) 0 0
\(793\) 348.981 0.440077
\(794\) 0 0
\(795\) 1836.35i 2.30987i
\(796\) 0 0
\(797\) 887.200 1.11317 0.556587 0.830789i \(-0.312111\pi\)
0.556587 + 0.830789i \(0.312111\pi\)
\(798\) 0 0
\(799\) − 513.125i − 0.642209i
\(800\) 0 0
\(801\) −1706.94 −2.13101
\(802\) 0 0
\(803\) − 224.576i − 0.279672i
\(804\) 0 0
\(805\) 400.121 0.497045
\(806\) 0 0
\(807\) − 1289.70i − 1.59814i
\(808\) 0 0
\(809\) 485.529 0.600160 0.300080 0.953914i \(-0.402987\pi\)
0.300080 + 0.953914i \(0.402987\pi\)
\(810\) 0 0
\(811\) 1118.90i 1.37966i 0.723972 + 0.689830i \(0.242315\pi\)
−0.723972 + 0.689830i \(0.757685\pi\)
\(812\) 0 0
\(813\) −321.193 −0.395071
\(814\) 0 0
\(815\) 190.227i 0.233407i
\(816\) 0 0
\(817\) −1873.50 −2.29314
\(818\) 0 0
\(819\) 654.643i 0.799320i
\(820\) 0 0
\(821\) −486.539 −0.592617 −0.296308 0.955092i \(-0.595756\pi\)
−0.296308 + 0.955092i \(0.595756\pi\)
\(822\) 0 0
\(823\) − 1049.45i − 1.27515i −0.770389 0.637575i \(-0.779938\pi\)
0.770389 0.637575i \(-0.220062\pi\)
\(824\) 0 0
\(825\) 149.035 0.180648
\(826\) 0 0
\(827\) 1472.00i 1.77993i 0.456031 + 0.889964i \(0.349271\pi\)
−0.456031 + 0.889964i \(0.650729\pi\)
\(828\) 0 0
\(829\) 1049.47 1.26595 0.632973 0.774174i \(-0.281834\pi\)
0.632973 + 0.774174i \(0.281834\pi\)
\(830\) 0 0
\(831\) 1874.45i 2.25566i
\(832\) 0 0
\(833\) 81.2187 0.0975014
\(834\) 0 0
\(835\) − 566.838i − 0.678848i
\(836\) 0 0
\(837\) 1146.18 1.36940
\(838\) 0 0
\(839\) − 1456.31i − 1.73576i −0.496771 0.867882i \(-0.665481\pi\)
0.496771 0.867882i \(-0.334519\pi\)
\(840\) 0 0
\(841\) −838.556 −0.997094
\(842\) 0 0
\(843\) 3082.98i 3.65716i
\(844\) 0 0
\(845\) 423.906 0.501664
\(846\) 0 0
\(847\) − 296.011i − 0.349482i
\(848\) 0 0
\(849\) −1943.87 −2.28960
\(850\) 0 0
\(851\) 1848.58i 2.17224i
\(852\) 0 0
\(853\) 121.003 0.141855 0.0709277 0.997481i \(-0.477404\pi\)
0.0709277 + 0.997481i \(0.477404\pi\)
\(854\) 0 0
\(855\) 3739.42i 4.37360i
\(856\) 0 0
\(857\) 1405.84 1.64041 0.820207 0.572066i \(-0.193858\pi\)
0.820207 + 0.572066i \(0.193858\pi\)
\(858\) 0 0
\(859\) 991.383i 1.15411i 0.816704 + 0.577057i \(0.195799\pi\)
−0.816704 + 0.577057i \(0.804201\pi\)
\(860\) 0 0
\(861\) −772.808 −0.897571
\(862\) 0 0
\(863\) 284.139i 0.329245i 0.986357 + 0.164623i \(0.0526406\pi\)
−0.986357 + 0.164623i \(0.947359\pi\)
\(864\) 0 0
\(865\) 1947.79 2.25178
\(866\) 0 0
\(867\) − 904.074i − 1.04276i
\(868\) 0 0
\(869\) 69.1238 0.0795440
\(870\) 0 0
\(871\) 239.544i 0.275022i
\(872\) 0 0
\(873\) 196.280 0.224834
\(874\) 0 0
\(875\) − 253.504i − 0.289719i
\(876\) 0 0
\(877\) −255.174 −0.290963 −0.145481 0.989361i \(-0.546473\pi\)
−0.145481 + 0.989361i \(0.546473\pi\)
\(878\) 0 0
\(879\) − 1409.68i − 1.60374i
\(880\) 0 0
\(881\) −421.803 −0.478778 −0.239389 0.970924i \(-0.576947\pi\)
−0.239389 + 0.970924i \(0.576947\pi\)
\(882\) 0 0
\(883\) − 1069.35i − 1.21105i −0.795827 0.605524i \(-0.792964\pi\)
0.795827 0.605524i \(-0.207036\pi\)
\(884\) 0 0
\(885\) −422.241 −0.477108
\(886\) 0 0
\(887\) − 619.581i − 0.698513i −0.937027 0.349256i \(-0.886434\pi\)
0.937027 0.349256i \(-0.113566\pi\)
\(888\) 0 0
\(889\) 270.640 0.304432
\(890\) 0 0
\(891\) − 1000.12i − 1.12247i
\(892\) 0 0
\(893\) −1130.77 −1.26626
\(894\) 0 0
\(895\) 297.488i 0.332389i
\(896\) 0 0
\(897\) 1498.37 1.67043
\(898\) 0 0
\(899\) 18.7769i 0.0208864i
\(900\) 0 0
\(901\) 629.276 0.698420
\(902\) 0 0
\(903\) 1135.30i 1.25726i
\(904\) 0 0
\(905\) −1629.32 −1.80035
\(906\) 0 0
\(907\) 302.666i 0.333700i 0.985982 + 0.166850i \(0.0533595\pi\)
−0.985982 + 0.166850i \(0.946640\pi\)
\(908\) 0 0
\(909\) 2593.40 2.85303
\(910\) 0 0
\(911\) − 1004.97i − 1.10315i −0.834127 0.551573i \(-0.814028\pi\)
0.834127 0.551573i \(-0.185972\pi\)
\(912\) 0 0
\(913\) 145.528 0.159395
\(914\) 0 0
\(915\) − 1207.98i − 1.32020i
\(916\) 0 0
\(917\) 235.937 0.257293
\(918\) 0 0
\(919\) − 73.2026i − 0.0796546i −0.999207 0.0398273i \(-0.987319\pi\)
0.999207 0.0398273i \(-0.0126808\pi\)
\(920\) 0 0
\(921\) 1299.28 1.41072
\(922\) 0 0
\(923\) − 108.068i − 0.117084i
\(924\) 0 0
\(925\) −595.605 −0.643897
\(926\) 0 0
\(927\) 371.525i 0.400782i
\(928\) 0 0
\(929\) 225.883 0.243146 0.121573 0.992582i \(-0.461206\pi\)
0.121573 + 0.992582i \(0.461206\pi\)
\(930\) 0 0
\(931\) − 178.981i − 0.192246i
\(932\) 0 0
\(933\) −2787.85 −2.98805
\(934\) 0 0
\(935\) − 202.568i − 0.216650i
\(936\) 0 0
\(937\) −1168.51 −1.24708 −0.623540 0.781792i \(-0.714306\pi\)
−0.623540 + 0.781792i \(0.714306\pi\)
\(938\) 0 0
\(939\) − 2899.83i − 3.08821i
\(940\) 0 0
\(941\) −119.216 −0.126690 −0.0633452 0.997992i \(-0.520177\pi\)
−0.0633452 + 0.997992i \(0.520177\pi\)
\(942\) 0 0
\(943\) 1304.65i 1.38351i
\(944\) 0 0
\(945\) 1459.78 1.54474
\(946\) 0 0
\(947\) 875.062i 0.924036i 0.886871 + 0.462018i \(0.152874\pi\)
−0.886871 + 0.462018i \(0.847126\pi\)
\(948\) 0 0
\(949\) 727.476 0.766571
\(950\) 0 0
\(951\) − 2887.30i − 3.03606i
\(952\) 0 0
\(953\) −468.167 −0.491256 −0.245628 0.969364i \(-0.578994\pi\)
−0.245628 + 0.969364i \(0.578994\pi\)
\(954\) 0 0
\(955\) 954.001i 0.998954i
\(956\) 0 0
\(957\) 27.6459 0.0288881
\(958\) 0 0
\(959\) 30.1088i 0.0313960i
\(960\) 0 0
\(961\) 816.743 0.849888
\(962\) 0 0
\(963\) − 1771.22i − 1.83927i
\(964\) 0 0
\(965\) 374.036 0.387602
\(966\) 0 0
\(967\) − 1139.67i − 1.17856i −0.807930 0.589279i \(-0.799412\pi\)
0.807930 0.589279i \(-0.200588\pi\)
\(968\) 0 0
\(969\) 1737.34 1.79292
\(970\) 0 0
\(971\) − 237.061i − 0.244141i −0.992521 0.122071i \(-0.961047\pi\)
0.992521 0.122071i \(-0.0389534\pi\)
\(972\) 0 0
\(973\) 320.812 0.329715
\(974\) 0 0
\(975\) 482.771i 0.495150i
\(976\) 0 0
\(977\) 422.059 0.431994 0.215997 0.976394i \(-0.430700\pi\)
0.215997 + 0.976394i \(0.430700\pi\)
\(978\) 0 0
\(979\) − 203.767i − 0.208138i
\(980\) 0 0
\(981\) 1667.49 1.69978
\(982\) 0 0
\(983\) 1614.06i 1.64197i 0.570946 + 0.820987i \(0.306576\pi\)
−0.570946 + 0.820987i \(0.693424\pi\)
\(984\) 0 0
\(985\) −1096.85 −1.11355
\(986\) 0 0
\(987\) 685.224i 0.694250i
\(988\) 0 0
\(989\) 1916.61 1.93792
\(990\) 0 0
\(991\) − 226.772i − 0.228832i −0.993433 0.114416i \(-0.963500\pi\)
0.993433 0.114416i \(-0.0364996\pi\)
\(992\) 0 0
\(993\) 3437.19 3.46142
\(994\) 0 0
\(995\) 1188.66i 1.19463i
\(996\) 0 0
\(997\) −1555.56 −1.56024 −0.780120 0.625629i \(-0.784842\pi\)
−0.780120 + 0.625629i \(0.784842\pi\)
\(998\) 0 0
\(999\) 6744.25i 6.75100i
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.127.8 yes 8
3.2 odd 2 2016.3.m.c.127.7 8
4.3 odd 2 inner 224.3.d.b.127.1 8
7.6 odd 2 1568.3.d.n.1471.1 8
8.3 odd 2 448.3.d.e.127.8 8
8.5 even 2 448.3.d.e.127.1 8
12.11 even 2 2016.3.m.c.127.8 8
16.3 odd 4 1792.3.g.f.127.8 8
16.5 even 4 1792.3.g.f.127.7 8
16.11 odd 4 1792.3.g.d.127.1 8
16.13 even 4 1792.3.g.d.127.2 8
28.27 even 2 1568.3.d.n.1471.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.3.d.b.127.1 8 4.3 odd 2 inner
224.3.d.b.127.8 yes 8 1.1 even 1 trivial
448.3.d.e.127.1 8 8.5 even 2
448.3.d.e.127.8 8 8.3 odd 2
1568.3.d.n.1471.1 8 7.6 odd 2
1568.3.d.n.1471.8 8 28.27 even 2
1792.3.g.d.127.1 8 16.11 odd 4
1792.3.g.d.127.2 8 16.13 even 4
1792.3.g.f.127.7 8 16.5 even 4
1792.3.g.f.127.8 8 16.3 odd 4
2016.3.m.c.127.7 8 3.2 odd 2
2016.3.m.c.127.8 8 12.11 even 2