Properties

Label 384.3.g.b.127.1
Level $384$
Weight $3$
Character 384.127
Analytic conductor $10.463$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 384.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.4632421514\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.1
Root \(0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 384.127
Dual form 384.3.g.b.127.5

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205i q^{3} -4.29253 q^{5} -2.75787i q^{7} -3.00000 q^{9} +O(q^{10})\) \(q-1.73205i q^{3} -4.29253 q^{5} -2.75787i q^{7} -3.00000 q^{9} +13.7980i q^{11} -14.5266 q^{13} +7.43488i q^{15} +22.8673 q^{17} +16.0399i q^{19} -4.77678 q^{21} +17.1117i q^{23} -6.57420 q^{25} +5.19615i q^{27} +21.8667 q^{29} +38.6944i q^{31} +23.8988 q^{33} +11.8383i q^{35} -66.4204 q^{37} +25.1608i q^{39} +23.2392 q^{41} +47.9230i q^{43} +12.8776 q^{45} +14.8512i q^{47} +41.3941 q^{49} -39.6073i q^{51} -65.5589 q^{53} -59.2281i q^{55} +27.7819 q^{57} -65.8428i q^{59} +40.1123 q^{61} +8.27362i q^{63} +62.3559 q^{65} +74.8105i q^{67} +29.6383 q^{69} +122.681i q^{71} -144.904 q^{73} +11.3868i q^{75} +38.0530 q^{77} -128.657i q^{79} +9.00000 q^{81} +22.0417i q^{83} -98.1584 q^{85} -37.8743i q^{87} -122.075 q^{89} +40.0626i q^{91} +67.0208 q^{93} -68.8516i q^{95} -88.3072 q^{97} -41.3939i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 16q^{5} - 24q^{9} + O(q^{10}) \) \( 8q + 16q^{5} - 24q^{9} - 48q^{13} + 16q^{17} - 8q^{25} + 80q^{29} + 16q^{37} + 80q^{41} - 48q^{45} - 88q^{49} - 176q^{53} + 96q^{57} + 272q^{61} - 160q^{65} - 16q^{73} - 320q^{77} + 72q^{81} - 32q^{85} - 240q^{89} + 192q^{93} + 400q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) − 1.73205i − 0.577350i
\(4\) 0 0
\(5\) −4.29253 −0.858506 −0.429253 0.903184i \(-0.641223\pi\)
−0.429253 + 0.903184i \(0.641223\pi\)
\(6\) 0 0
\(7\) − 2.75787i − 0.393982i −0.980405 0.196991i \(-0.936883\pi\)
0.980405 0.196991i \(-0.0631170\pi\)
\(8\) 0 0
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) 13.7980i 1.25436i 0.778874 + 0.627180i \(0.215791\pi\)
−0.778874 + 0.627180i \(0.784209\pi\)
\(12\) 0 0
\(13\) −14.5266 −1.11743 −0.558716 0.829359i \(-0.688706\pi\)
−0.558716 + 0.829359i \(0.688706\pi\)
\(14\) 0 0
\(15\) 7.43488i 0.495659i
\(16\) 0 0
\(17\) 22.8673 1.34513 0.672567 0.740037i \(-0.265192\pi\)
0.672567 + 0.740037i \(0.265192\pi\)
\(18\) 0 0
\(19\) 16.0399i 0.844204i 0.906548 + 0.422102i \(0.138708\pi\)
−0.906548 + 0.422102i \(0.861292\pi\)
\(20\) 0 0
\(21\) −4.77678 −0.227466
\(22\) 0 0
\(23\) 17.1117i 0.743986i 0.928236 + 0.371993i \(0.121325\pi\)
−0.928236 + 0.371993i \(0.878675\pi\)
\(24\) 0 0
\(25\) −6.57420 −0.262968
\(26\) 0 0
\(27\) 5.19615i 0.192450i
\(28\) 0 0
\(29\) 21.8667 0.754025 0.377013 0.926208i \(-0.376951\pi\)
0.377013 + 0.926208i \(0.376951\pi\)
\(30\) 0 0
\(31\) 38.6944i 1.24821i 0.781341 + 0.624104i \(0.214536\pi\)
−0.781341 + 0.624104i \(0.785464\pi\)
\(32\) 0 0
\(33\) 23.8988 0.724205
\(34\) 0 0
\(35\) 11.8383i 0.338236i
\(36\) 0 0
\(37\) −66.4204 −1.79515 −0.897573 0.440866i \(-0.854671\pi\)
−0.897573 + 0.440866i \(0.854671\pi\)
\(38\) 0 0
\(39\) 25.1608i 0.645149i
\(40\) 0 0
\(41\) 23.2392 0.566809 0.283405 0.959000i \(-0.408536\pi\)
0.283405 + 0.959000i \(0.408536\pi\)
\(42\) 0 0
\(43\) 47.9230i 1.11449i 0.830349 + 0.557244i \(0.188141\pi\)
−0.830349 + 0.557244i \(0.811859\pi\)
\(44\) 0 0
\(45\) 12.8776 0.286169
\(46\) 0 0
\(47\) 14.8512i 0.315983i 0.987441 + 0.157991i \(0.0505018\pi\)
−0.987441 + 0.157991i \(0.949498\pi\)
\(48\) 0 0
\(49\) 41.3941 0.844778
\(50\) 0 0
\(51\) − 39.6073i − 0.776613i
\(52\) 0 0
\(53\) −65.5589 −1.23696 −0.618480 0.785800i \(-0.712251\pi\)
−0.618480 + 0.785800i \(0.712251\pi\)
\(54\) 0 0
\(55\) − 59.2281i − 1.07688i
\(56\) 0 0
\(57\) 27.7819 0.487401
\(58\) 0 0
\(59\) − 65.8428i − 1.11598i −0.829848 0.557990i \(-0.811573\pi\)
0.829848 0.557990i \(-0.188427\pi\)
\(60\) 0 0
\(61\) 40.1123 0.657579 0.328790 0.944403i \(-0.393359\pi\)
0.328790 + 0.944403i \(0.393359\pi\)
\(62\) 0 0
\(63\) 8.27362i 0.131327i
\(64\) 0 0
\(65\) 62.3559 0.959321
\(66\) 0 0
\(67\) 74.8105i 1.11657i 0.829648 + 0.558287i \(0.188541\pi\)
−0.829648 + 0.558287i \(0.811459\pi\)
\(68\) 0 0
\(69\) 29.6383 0.429540
\(70\) 0 0
\(71\) 122.681i 1.72790i 0.503578 + 0.863950i \(0.332017\pi\)
−0.503578 + 0.863950i \(0.667983\pi\)
\(72\) 0 0
\(73\) −144.904 −1.98498 −0.992492 0.122312i \(-0.960969\pi\)
−0.992492 + 0.122312i \(0.960969\pi\)
\(74\) 0 0
\(75\) 11.3868i 0.151825i
\(76\) 0 0
\(77\) 38.0530 0.494195
\(78\) 0 0
\(79\) − 128.657i − 1.62857i −0.580467 0.814284i \(-0.697130\pi\)
0.580467 0.814284i \(-0.302870\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 22.0417i 0.265563i 0.991145 + 0.132781i \(0.0423908\pi\)
−0.991145 + 0.132781i \(0.957609\pi\)
\(84\) 0 0
\(85\) −98.1584 −1.15480
\(86\) 0 0
\(87\) − 37.8743i − 0.435337i
\(88\) 0 0
\(89\) −122.075 −1.37163 −0.685813 0.727778i \(-0.740553\pi\)
−0.685813 + 0.727778i \(0.740553\pi\)
\(90\) 0 0
\(91\) 40.0626i 0.440248i
\(92\) 0 0
\(93\) 67.0208 0.720653
\(94\) 0 0
\(95\) − 68.8516i − 0.724754i
\(96\) 0 0
\(97\) −88.3072 −0.910384 −0.455192 0.890393i \(-0.650429\pi\)
−0.455192 + 0.890393i \(0.650429\pi\)
\(98\) 0 0
\(99\) − 41.3939i − 0.418120i
\(100\) 0 0
\(101\) 104.345 1.03312 0.516560 0.856251i \(-0.327212\pi\)
0.516560 + 0.856251i \(0.327212\pi\)
\(102\) 0 0
\(103\) − 69.0609i − 0.670494i −0.942130 0.335247i \(-0.891180\pi\)
0.942130 0.335247i \(-0.108820\pi\)
\(104\) 0 0
\(105\) 20.5045 0.195281
\(106\) 0 0
\(107\) − 149.429i − 1.39653i −0.715839 0.698265i \(-0.753956\pi\)
0.715839 0.698265i \(-0.246044\pi\)
\(108\) 0 0
\(109\) −54.3341 −0.498478 −0.249239 0.968442i \(-0.580180\pi\)
−0.249239 + 0.968442i \(0.580180\pi\)
\(110\) 0 0
\(111\) 115.044i 1.03643i
\(112\) 0 0
\(113\) 6.32846 0.0560041 0.0280021 0.999608i \(-0.491086\pi\)
0.0280021 + 0.999608i \(0.491086\pi\)
\(114\) 0 0
\(115\) − 73.4523i − 0.638716i
\(116\) 0 0
\(117\) 43.5798 0.372477
\(118\) 0 0
\(119\) − 63.0651i − 0.529958i
\(120\) 0 0
\(121\) −69.3837 −0.573419
\(122\) 0 0
\(123\) − 40.2514i − 0.327247i
\(124\) 0 0
\(125\) 135.533 1.08427
\(126\) 0 0
\(127\) − 16.5228i − 0.130101i −0.997882 0.0650504i \(-0.979279\pi\)
0.997882 0.0650504i \(-0.0207208\pi\)
\(128\) 0 0
\(129\) 83.0050 0.643450
\(130\) 0 0
\(131\) − 50.9799i − 0.389159i −0.980887 0.194580i \(-0.937666\pi\)
0.980887 0.194580i \(-0.0623343\pi\)
\(132\) 0 0
\(133\) 44.2360 0.332601
\(134\) 0 0
\(135\) − 22.3046i − 0.165220i
\(136\) 0 0
\(137\) 10.0157 0.0731070 0.0365535 0.999332i \(-0.488362\pi\)
0.0365535 + 0.999332i \(0.488362\pi\)
\(138\) 0 0
\(139\) 65.7088i 0.472725i 0.971665 + 0.236363i \(0.0759553\pi\)
−0.971665 + 0.236363i \(0.924045\pi\)
\(140\) 0 0
\(141\) 25.7230 0.182433
\(142\) 0 0
\(143\) − 200.438i − 1.40166i
\(144\) 0 0
\(145\) −93.8635 −0.647335
\(146\) 0 0
\(147\) − 71.6967i − 0.487733i
\(148\) 0 0
\(149\) 94.7716 0.636051 0.318026 0.948082i \(-0.396980\pi\)
0.318026 + 0.948082i \(0.396980\pi\)
\(150\) 0 0
\(151\) − 269.769i − 1.78655i −0.449508 0.893276i \(-0.648401\pi\)
0.449508 0.893276i \(-0.351599\pi\)
\(152\) 0 0
\(153\) −68.6018 −0.448378
\(154\) 0 0
\(155\) − 166.097i − 1.07159i
\(156\) 0 0
\(157\) −31.5058 −0.200674 −0.100337 0.994954i \(-0.531992\pi\)
−0.100337 + 0.994954i \(0.531992\pi\)
\(158\) 0 0
\(159\) 113.551i 0.714159i
\(160\) 0 0
\(161\) 47.1918 0.293117
\(162\) 0 0
\(163\) − 201.159i − 1.23410i −0.786923 0.617051i \(-0.788327\pi\)
0.786923 0.617051i \(-0.211673\pi\)
\(164\) 0 0
\(165\) −102.586 −0.621734
\(166\) 0 0
\(167\) 266.393i 1.59517i 0.603208 + 0.797584i \(0.293889\pi\)
−0.603208 + 0.797584i \(0.706111\pi\)
\(168\) 0 0
\(169\) 42.0224 0.248653
\(170\) 0 0
\(171\) − 48.1196i − 0.281401i
\(172\) 0 0
\(173\) 43.8456 0.253443 0.126721 0.991938i \(-0.459555\pi\)
0.126721 + 0.991938i \(0.459555\pi\)
\(174\) 0 0
\(175\) 18.1308i 0.103605i
\(176\) 0 0
\(177\) −114.043 −0.644311
\(178\) 0 0
\(179\) 275.778i 1.54066i 0.637646 + 0.770330i \(0.279908\pi\)
−0.637646 + 0.770330i \(0.720092\pi\)
\(180\) 0 0
\(181\) 79.0033 0.436482 0.218241 0.975895i \(-0.429968\pi\)
0.218241 + 0.975895i \(0.429968\pi\)
\(182\) 0 0
\(183\) − 69.4766i − 0.379654i
\(184\) 0 0
\(185\) 285.111 1.54114
\(186\) 0 0
\(187\) 315.522i 1.68728i
\(188\) 0 0
\(189\) 14.3303 0.0758219
\(190\) 0 0
\(191\) 142.409i 0.745598i 0.927912 + 0.372799i \(0.121602\pi\)
−0.927912 + 0.372799i \(0.878398\pi\)
\(192\) 0 0
\(193\) −277.818 −1.43947 −0.719736 0.694248i \(-0.755737\pi\)
−0.719736 + 0.694248i \(0.755737\pi\)
\(194\) 0 0
\(195\) − 108.004i − 0.553864i
\(196\) 0 0
\(197\) −3.40848 −0.0173019 −0.00865095 0.999963i \(-0.502754\pi\)
−0.00865095 + 0.999963i \(0.502754\pi\)
\(198\) 0 0
\(199\) 314.323i 1.57951i 0.613421 + 0.789756i \(0.289793\pi\)
−0.613421 + 0.789756i \(0.710207\pi\)
\(200\) 0 0
\(201\) 129.576 0.644654
\(202\) 0 0
\(203\) − 60.3057i − 0.297072i
\(204\) 0 0
\(205\) −99.7548 −0.486609
\(206\) 0 0
\(207\) − 51.3350i − 0.247995i
\(208\) 0 0
\(209\) −221.317 −1.05894
\(210\) 0 0
\(211\) − 1.54933i − 0.00734281i −0.999993 0.00367140i \(-0.998831\pi\)
0.999993 0.00367140i \(-0.00116865\pi\)
\(212\) 0 0
\(213\) 212.490 0.997603
\(214\) 0 0
\(215\) − 205.711i − 0.956794i
\(216\) 0 0
\(217\) 106.714 0.491772
\(218\) 0 0
\(219\) 250.981i 1.14603i
\(220\) 0 0
\(221\) −332.184 −1.50309
\(222\) 0 0
\(223\) 148.964i 0.668001i 0.942573 + 0.334000i \(0.108399\pi\)
−0.942573 + 0.334000i \(0.891601\pi\)
\(224\) 0 0
\(225\) 19.7226 0.0876560
\(226\) 0 0
\(227\) 237.803i 1.04759i 0.851844 + 0.523796i \(0.175485\pi\)
−0.851844 + 0.523796i \(0.824515\pi\)
\(228\) 0 0
\(229\) −70.0590 −0.305935 −0.152967 0.988231i \(-0.548883\pi\)
−0.152967 + 0.988231i \(0.548883\pi\)
\(230\) 0 0
\(231\) − 65.9098i − 0.285324i
\(232\) 0 0
\(233\) 7.18107 0.0308200 0.0154100 0.999881i \(-0.495095\pi\)
0.0154100 + 0.999881i \(0.495095\pi\)
\(234\) 0 0
\(235\) − 63.7491i − 0.271273i
\(236\) 0 0
\(237\) −222.840 −0.940254
\(238\) 0 0
\(239\) − 216.620i − 0.906359i −0.891419 0.453180i \(-0.850290\pi\)
0.891419 0.453180i \(-0.149710\pi\)
\(240\) 0 0
\(241\) 385.001 1.59751 0.798757 0.601653i \(-0.205491\pi\)
0.798757 + 0.601653i \(0.205491\pi\)
\(242\) 0 0
\(243\) − 15.5885i − 0.0641500i
\(244\) 0 0
\(245\) −177.685 −0.725247
\(246\) 0 0
\(247\) − 233.005i − 0.943340i
\(248\) 0 0
\(249\) 38.1774 0.153323
\(250\) 0 0
\(251\) 85.4882i 0.340590i 0.985393 + 0.170295i \(0.0544721\pi\)
−0.985393 + 0.170295i \(0.945528\pi\)
\(252\) 0 0
\(253\) −236.106 −0.933226
\(254\) 0 0
\(255\) 170.015i 0.666727i
\(256\) 0 0
\(257\) −54.2981 −0.211277 −0.105638 0.994405i \(-0.533689\pi\)
−0.105638 + 0.994405i \(0.533689\pi\)
\(258\) 0 0
\(259\) 183.179i 0.707255i
\(260\) 0 0
\(261\) −65.6002 −0.251342
\(262\) 0 0
\(263\) 501.827i 1.90809i 0.299670 + 0.954043i \(0.403123\pi\)
−0.299670 + 0.954043i \(0.596877\pi\)
\(264\) 0 0
\(265\) 281.413 1.06194
\(266\) 0 0
\(267\) 211.440i 0.791909i
\(268\) 0 0
\(269\) −284.911 −1.05915 −0.529574 0.848264i \(-0.677648\pi\)
−0.529574 + 0.848264i \(0.677648\pi\)
\(270\) 0 0
\(271\) 241.190i 0.889998i 0.895531 + 0.444999i \(0.146796\pi\)
−0.895531 + 0.444999i \(0.853204\pi\)
\(272\) 0 0
\(273\) 69.3904 0.254177
\(274\) 0 0
\(275\) − 90.7105i − 0.329856i
\(276\) 0 0
\(277\) 242.877 0.876813 0.438407 0.898777i \(-0.355543\pi\)
0.438407 + 0.898777i \(0.355543\pi\)
\(278\) 0 0
\(279\) − 116.083i − 0.416069i
\(280\) 0 0
\(281\) 105.341 0.374880 0.187440 0.982276i \(-0.439981\pi\)
0.187440 + 0.982276i \(0.439981\pi\)
\(282\) 0 0
\(283\) − 223.867i − 0.791049i −0.918455 0.395525i \(-0.870563\pi\)
0.918455 0.395525i \(-0.129437\pi\)
\(284\) 0 0
\(285\) −119.254 −0.418437
\(286\) 0 0
\(287\) − 64.0907i − 0.223313i
\(288\) 0 0
\(289\) 233.912 0.809384
\(290\) 0 0
\(291\) 152.953i 0.525610i
\(292\) 0 0
\(293\) 421.057 1.43705 0.718527 0.695499i \(-0.244817\pi\)
0.718527 + 0.695499i \(0.244817\pi\)
\(294\) 0 0
\(295\) 282.632i 0.958075i
\(296\) 0 0
\(297\) −71.6963 −0.241402
\(298\) 0 0
\(299\) − 248.575i − 0.831353i
\(300\) 0 0
\(301\) 132.166 0.439088
\(302\) 0 0
\(303\) − 180.731i − 0.596472i
\(304\) 0 0
\(305\) −172.183 −0.564536
\(306\) 0 0
\(307\) 122.865i 0.400211i 0.979774 + 0.200106i \(0.0641285\pi\)
−0.979774 + 0.200106i \(0.935871\pi\)
\(308\) 0 0
\(309\) −119.617 −0.387110
\(310\) 0 0
\(311\) 437.535i 1.40686i 0.710762 + 0.703432i \(0.248350\pi\)
−0.710762 + 0.703432i \(0.751650\pi\)
\(312\) 0 0
\(313\) 375.413 1.19940 0.599702 0.800223i \(-0.295286\pi\)
0.599702 + 0.800223i \(0.295286\pi\)
\(314\) 0 0
\(315\) − 35.5148i − 0.112745i
\(316\) 0 0
\(317\) −388.867 −1.22671 −0.613355 0.789808i \(-0.710180\pi\)
−0.613355 + 0.789808i \(0.710180\pi\)
\(318\) 0 0
\(319\) 301.716i 0.945819i
\(320\) 0 0
\(321\) −258.818 −0.806287
\(322\) 0 0
\(323\) 366.788i 1.13557i
\(324\) 0 0
\(325\) 95.5008 0.293849
\(326\) 0 0
\(327\) 94.1094i 0.287796i
\(328\) 0 0
\(329\) 40.9577 0.124491
\(330\) 0 0
\(331\) 15.4690i 0.0467341i 0.999727 + 0.0233670i \(0.00743864\pi\)
−0.999727 + 0.0233670i \(0.992561\pi\)
\(332\) 0 0
\(333\) 199.261 0.598382
\(334\) 0 0
\(335\) − 321.126i − 0.958585i
\(336\) 0 0
\(337\) −88.0105 −0.261159 −0.130579 0.991438i \(-0.541684\pi\)
−0.130579 + 0.991438i \(0.541684\pi\)
\(338\) 0 0
\(339\) − 10.9612i − 0.0323340i
\(340\) 0 0
\(341\) −533.904 −1.56570
\(342\) 0 0
\(343\) − 249.296i − 0.726810i
\(344\) 0 0
\(345\) −127.223 −0.368763
\(346\) 0 0
\(347\) 183.589i 0.529076i 0.964375 + 0.264538i \(0.0852194\pi\)
−0.964375 + 0.264538i \(0.914781\pi\)
\(348\) 0 0
\(349\) 242.556 0.695003 0.347501 0.937679i \(-0.387030\pi\)
0.347501 + 0.937679i \(0.387030\pi\)
\(350\) 0 0
\(351\) − 75.4825i − 0.215050i
\(352\) 0 0
\(353\) 10.2850 0.0291360 0.0145680 0.999894i \(-0.495363\pi\)
0.0145680 + 0.999894i \(0.495363\pi\)
\(354\) 0 0
\(355\) − 526.611i − 1.48341i
\(356\) 0 0
\(357\) −109.232 −0.305972
\(358\) 0 0
\(359\) − 137.976i − 0.384334i −0.981362 0.192167i \(-0.938448\pi\)
0.981362 0.192167i \(-0.0615515\pi\)
\(360\) 0 0
\(361\) 103.723 0.287320
\(362\) 0 0
\(363\) 120.176i 0.331063i
\(364\) 0 0
\(365\) 622.004 1.70412
\(366\) 0 0
\(367\) 400.180i 1.09041i 0.838303 + 0.545205i \(0.183548\pi\)
−0.838303 + 0.545205i \(0.816452\pi\)
\(368\) 0 0
\(369\) −69.7175 −0.188936
\(370\) 0 0
\(371\) 180.803i 0.487340i
\(372\) 0 0
\(373\) 644.881 1.72890 0.864451 0.502717i \(-0.167666\pi\)
0.864451 + 0.502717i \(0.167666\pi\)
\(374\) 0 0
\(375\) − 234.750i − 0.626001i
\(376\) 0 0
\(377\) −317.649 −0.842571
\(378\) 0 0
\(379\) − 485.900i − 1.28206i −0.767517 0.641029i \(-0.778508\pi\)
0.767517 0.641029i \(-0.221492\pi\)
\(380\) 0 0
\(381\) −28.6183 −0.0751137
\(382\) 0 0
\(383\) − 168.828i − 0.440804i −0.975409 0.220402i \(-0.929263\pi\)
0.975409 0.220402i \(-0.0707370\pi\)
\(384\) 0 0
\(385\) −163.344 −0.424270
\(386\) 0 0
\(387\) − 143.769i − 0.371496i
\(388\) 0 0
\(389\) 192.592 0.495095 0.247548 0.968876i \(-0.420375\pi\)
0.247548 + 0.968876i \(0.420375\pi\)
\(390\) 0 0
\(391\) 391.297i 1.00076i
\(392\) 0 0
\(393\) −88.2998 −0.224681
\(394\) 0 0
\(395\) 552.263i 1.39813i
\(396\) 0 0
\(397\) 62.0483 0.156293 0.0781465 0.996942i \(-0.475100\pi\)
0.0781465 + 0.996942i \(0.475100\pi\)
\(398\) 0 0
\(399\) − 76.6189i − 0.192027i
\(400\) 0 0
\(401\) 353.066 0.880464 0.440232 0.897884i \(-0.354896\pi\)
0.440232 + 0.897884i \(0.354896\pi\)
\(402\) 0 0
\(403\) − 562.099i − 1.39479i
\(404\) 0 0
\(405\) −38.6328 −0.0953895
\(406\) 0 0
\(407\) − 916.466i − 2.25176i
\(408\) 0 0
\(409\) −62.9725 −0.153967 −0.0769835 0.997032i \(-0.524529\pi\)
−0.0769835 + 0.997032i \(0.524529\pi\)
\(410\) 0 0
\(411\) − 17.3476i − 0.0422084i
\(412\) 0 0
\(413\) −181.586 −0.439676
\(414\) 0 0
\(415\) − 94.6146i − 0.227987i
\(416\) 0 0
\(417\) 113.811 0.272928
\(418\) 0 0
\(419\) − 398.741i − 0.951650i −0.879540 0.475825i \(-0.842150\pi\)
0.879540 0.475825i \(-0.157850\pi\)
\(420\) 0 0
\(421\) −533.059 −1.26617 −0.633086 0.774081i \(-0.718212\pi\)
−0.633086 + 0.774081i \(0.718212\pi\)
\(422\) 0 0
\(423\) − 44.5535i − 0.105328i
\(424\) 0 0
\(425\) −150.334 −0.353727
\(426\) 0 0
\(427\) − 110.625i − 0.259075i
\(428\) 0 0
\(429\) −347.168 −0.809250
\(430\) 0 0
\(431\) − 284.282i − 0.659587i −0.944053 0.329793i \(-0.893021\pi\)
0.944053 0.329793i \(-0.106979\pi\)
\(432\) 0 0
\(433\) −304.599 −0.703462 −0.351731 0.936101i \(-0.614407\pi\)
−0.351731 + 0.936101i \(0.614407\pi\)
\(434\) 0 0
\(435\) 162.576i 0.373739i
\(436\) 0 0
\(437\) −274.469 −0.628075
\(438\) 0 0
\(439\) − 204.615i − 0.466094i −0.972465 0.233047i \(-0.925130\pi\)
0.972465 0.233047i \(-0.0748696\pi\)
\(440\) 0 0
\(441\) −124.182 −0.281593
\(442\) 0 0
\(443\) − 625.828i − 1.41270i −0.707861 0.706352i \(-0.750340\pi\)
0.707861 0.706352i \(-0.249660\pi\)
\(444\) 0 0
\(445\) 524.009 1.17755
\(446\) 0 0
\(447\) − 164.149i − 0.367224i
\(448\) 0 0
\(449\) −557.532 −1.24172 −0.620860 0.783921i \(-0.713217\pi\)
−0.620860 + 0.783921i \(0.713217\pi\)
\(450\) 0 0
\(451\) 320.653i 0.710983i
\(452\) 0 0
\(453\) −467.254 −1.03147
\(454\) 0 0
\(455\) − 171.970i − 0.377955i
\(456\) 0 0
\(457\) −312.142 −0.683024 −0.341512 0.939877i \(-0.610939\pi\)
−0.341512 + 0.939877i \(0.610939\pi\)
\(458\) 0 0
\(459\) 118.822i 0.258871i
\(460\) 0 0
\(461\) 603.187 1.30843 0.654216 0.756308i \(-0.272999\pi\)
0.654216 + 0.756308i \(0.272999\pi\)
\(462\) 0 0
\(463\) − 707.866i − 1.52887i −0.644702 0.764434i \(-0.723019\pi\)
0.644702 0.764434i \(-0.276981\pi\)
\(464\) 0 0
\(465\) −287.689 −0.618685
\(466\) 0 0
\(467\) − 195.220i − 0.418031i −0.977912 0.209015i \(-0.932974\pi\)
0.977912 0.209015i \(-0.0670259\pi\)
\(468\) 0 0
\(469\) 206.318 0.439910
\(470\) 0 0
\(471\) 54.5696i 0.115859i
\(472\) 0 0
\(473\) −661.239 −1.39797
\(474\) 0 0
\(475\) − 105.449i − 0.221998i
\(476\) 0 0
\(477\) 196.677 0.412320
\(478\) 0 0
\(479\) 233.729i 0.487952i 0.969781 + 0.243976i \(0.0784517\pi\)
−0.969781 + 0.243976i \(0.921548\pi\)
\(480\) 0 0
\(481\) 964.863 2.00595
\(482\) 0 0
\(483\) − 81.7387i − 0.169231i
\(484\) 0 0
\(485\) 379.061 0.781569
\(486\) 0 0
\(487\) 224.058i 0.460078i 0.973181 + 0.230039i \(0.0738854\pi\)
−0.973181 + 0.230039i \(0.926115\pi\)
\(488\) 0 0
\(489\) −348.417 −0.712510
\(490\) 0 0
\(491\) − 194.772i − 0.396684i −0.980133 0.198342i \(-0.936444\pi\)
0.980133 0.198342i \(-0.0635556\pi\)
\(492\) 0 0
\(493\) 500.032 1.01426
\(494\) 0 0
\(495\) 177.684i 0.358958i
\(496\) 0 0
\(497\) 338.339 0.680762
\(498\) 0 0
\(499\) − 99.7462i − 0.199892i −0.994993 0.0999461i \(-0.968133\pi\)
0.994993 0.0999461i \(-0.0318670\pi\)
\(500\) 0 0
\(501\) 461.406 0.920970
\(502\) 0 0
\(503\) − 772.172i − 1.53513i −0.640969 0.767567i \(-0.721467\pi\)
0.640969 0.767567i \(-0.278533\pi\)
\(504\) 0 0
\(505\) −447.904 −0.886939
\(506\) 0 0
\(507\) − 72.7849i − 0.143560i
\(508\) 0 0
\(509\) 292.306 0.574276 0.287138 0.957889i \(-0.407296\pi\)
0.287138 + 0.957889i \(0.407296\pi\)
\(510\) 0 0
\(511\) 399.627i 0.782048i
\(512\) 0 0
\(513\) −83.3456 −0.162467
\(514\) 0 0
\(515\) 296.446i 0.575623i
\(516\) 0 0
\(517\) −204.916 −0.396356
\(518\) 0 0
\(519\) − 75.9428i − 0.146325i
\(520\) 0 0
\(521\) −412.035 −0.790853 −0.395427 0.918498i \(-0.629403\pi\)
−0.395427 + 0.918498i \(0.629403\pi\)
\(522\) 0 0
\(523\) 124.693i 0.238420i 0.992869 + 0.119210i \(0.0380361\pi\)
−0.992869 + 0.119210i \(0.961964\pi\)
\(524\) 0 0
\(525\) 31.4035 0.0598162
\(526\) 0 0
\(527\) 884.836i 1.67901i
\(528\) 0 0
\(529\) 236.191 0.446486
\(530\) 0 0
\(531\) 197.528i 0.371993i
\(532\) 0 0
\(533\) −337.586 −0.633370
\(534\) 0 0
\(535\) 641.427i 1.19893i
\(536\) 0 0
\(537\) 477.662 0.889500
\(538\) 0 0
\(539\) 571.154i 1.05966i
\(540\) 0 0
\(541\) 572.988 1.05913 0.529564 0.848270i \(-0.322356\pi\)
0.529564 + 0.848270i \(0.322356\pi\)
\(542\) 0 0
\(543\) − 136.838i − 0.252003i
\(544\) 0 0
\(545\) 233.231 0.427946
\(546\) 0 0
\(547\) − 682.433i − 1.24759i −0.781587 0.623796i \(-0.785590\pi\)
0.781587 0.623796i \(-0.214410\pi\)
\(548\) 0 0
\(549\) −120.337 −0.219193
\(550\) 0 0
\(551\) 350.739i 0.636551i
\(552\) 0 0
\(553\) −354.820 −0.641627
\(554\) 0 0
\(555\) − 493.828i − 0.889779i
\(556\) 0 0
\(557\) −856.125 −1.53703 −0.768515 0.639832i \(-0.779004\pi\)
−0.768515 + 0.639832i \(0.779004\pi\)
\(558\) 0 0
\(559\) − 696.158i − 1.24536i
\(560\) 0 0
\(561\) 546.499 0.974152
\(562\) 0 0
\(563\) − 680.745i − 1.20914i −0.796552 0.604570i \(-0.793345\pi\)
0.796552 0.604570i \(-0.206655\pi\)
\(564\) 0 0
\(565\) −27.1651 −0.0480798
\(566\) 0 0
\(567\) − 24.8209i − 0.0437758i
\(568\) 0 0
\(569\) 343.401 0.603516 0.301758 0.953384i \(-0.402426\pi\)
0.301758 + 0.953384i \(0.402426\pi\)
\(570\) 0 0
\(571\) 329.137i 0.576422i 0.957567 + 0.288211i \(0.0930604\pi\)
−0.957567 + 0.288211i \(0.906940\pi\)
\(572\) 0 0
\(573\) 246.660 0.430471
\(574\) 0 0
\(575\) − 112.495i − 0.195644i
\(576\) 0 0
\(577\) 734.738 1.27338 0.636688 0.771122i \(-0.280304\pi\)
0.636688 + 0.771122i \(0.280304\pi\)
\(578\) 0 0
\(579\) 481.195i 0.831079i
\(580\) 0 0
\(581\) 60.7883 0.104627
\(582\) 0 0
\(583\) − 904.579i − 1.55159i
\(584\) 0 0
\(585\) −187.068 −0.319774
\(586\) 0 0
\(587\) 433.815i 0.739037i 0.929223 + 0.369519i \(0.120477\pi\)
−0.929223 + 0.369519i \(0.879523\pi\)
\(588\) 0 0
\(589\) −620.654 −1.05374
\(590\) 0 0
\(591\) 5.90365i 0.00998926i
\(592\) 0 0
\(593\) −425.075 −0.716822 −0.358411 0.933564i \(-0.616681\pi\)
−0.358411 + 0.933564i \(0.616681\pi\)
\(594\) 0 0
\(595\) 270.709i 0.454972i
\(596\) 0 0
\(597\) 544.424 0.911932
\(598\) 0 0
\(599\) 1048.73i 1.75081i 0.483394 + 0.875403i \(0.339404\pi\)
−0.483394 + 0.875403i \(0.660596\pi\)
\(600\) 0 0
\(601\) −478.816 −0.796699 −0.398349 0.917234i \(-0.630417\pi\)
−0.398349 + 0.917234i \(0.630417\pi\)
\(602\) 0 0
\(603\) − 224.431i − 0.372191i
\(604\) 0 0
\(605\) 297.831 0.492283
\(606\) 0 0
\(607\) − 18.7009i − 0.0308088i −0.999881 0.0154044i \(-0.995096\pi\)
0.999881 0.0154044i \(-0.00490356\pi\)
\(608\) 0 0
\(609\) −104.453 −0.171515
\(610\) 0 0
\(611\) − 215.737i − 0.353089i
\(612\) 0 0
\(613\) −314.964 −0.513807 −0.256904 0.966437i \(-0.582702\pi\)
−0.256904 + 0.966437i \(0.582702\pi\)
\(614\) 0 0
\(615\) 172.780i 0.280944i
\(616\) 0 0
\(617\) 786.590 1.27486 0.637431 0.770507i \(-0.279997\pi\)
0.637431 + 0.770507i \(0.279997\pi\)
\(618\) 0 0
\(619\) − 505.431i − 0.816528i −0.912864 0.408264i \(-0.866134\pi\)
0.912864 0.408264i \(-0.133866\pi\)
\(620\) 0 0
\(621\) −88.9148 −0.143180
\(622\) 0 0
\(623\) 336.667i 0.540396i
\(624\) 0 0
\(625\) −417.425 −0.667880
\(626\) 0 0
\(627\) 383.333i 0.611377i
\(628\) 0 0
\(629\) −1518.85 −2.41471
\(630\) 0 0
\(631\) 682.834i 1.08215i 0.840976 + 0.541073i \(0.181982\pi\)
−0.840976 + 0.541073i \(0.818018\pi\)
\(632\) 0 0
\(633\) −2.68352 −0.00423937
\(634\) 0 0
\(635\) 70.9246i 0.111692i
\(636\) 0 0
\(637\) −601.316 −0.943982
\(638\) 0 0
\(639\) − 368.043i − 0.575967i
\(640\) 0 0
\(641\) 816.719 1.27413 0.637066 0.770809i \(-0.280148\pi\)
0.637066 + 0.770809i \(0.280148\pi\)
\(642\) 0 0
\(643\) 1164.05i 1.81034i 0.425045 + 0.905172i \(0.360258\pi\)
−0.425045 + 0.905172i \(0.639742\pi\)
\(644\) 0 0
\(645\) −356.301 −0.552405
\(646\) 0 0
\(647\) − 471.519i − 0.728778i −0.931247 0.364389i \(-0.881278\pi\)
0.931247 0.364389i \(-0.118722\pi\)
\(648\) 0 0
\(649\) 908.496 1.39984
\(650\) 0 0
\(651\) − 184.835i − 0.283924i
\(652\) 0 0
\(653\) 1187.25 1.81814 0.909071 0.416642i \(-0.136793\pi\)
0.909071 + 0.416642i \(0.136793\pi\)
\(654\) 0 0
\(655\) 218.833i 0.334096i
\(656\) 0 0
\(657\) 434.711 0.661661
\(658\) 0 0
\(659\) − 245.640i − 0.372747i −0.982479 0.186373i \(-0.940327\pi\)
0.982479 0.186373i \(-0.0596734\pi\)
\(660\) 0 0
\(661\) −1244.70 −1.88305 −0.941527 0.336936i \(-0.890609\pi\)
−0.941527 + 0.336936i \(0.890609\pi\)
\(662\) 0 0
\(663\) 575.359i 0.867812i
\(664\) 0 0
\(665\) −189.884 −0.285540
\(666\) 0 0
\(667\) 374.176i 0.560984i
\(668\) 0 0
\(669\) 258.013 0.385670
\(670\) 0 0
\(671\) 553.468i 0.824841i
\(672\) 0 0
\(673\) −852.278 −1.26639 −0.633193 0.773994i \(-0.718256\pi\)
−0.633193 + 0.773994i \(0.718256\pi\)
\(674\) 0 0
\(675\) − 34.1605i − 0.0506082i
\(676\) 0 0
\(677\) 555.765 0.820923 0.410461 0.911878i \(-0.365368\pi\)
0.410461 + 0.911878i \(0.365368\pi\)
\(678\) 0 0
\(679\) 243.540i 0.358675i
\(680\) 0 0
\(681\) 411.888 0.604828
\(682\) 0 0
\(683\) − 335.241i − 0.490836i −0.969417 0.245418i \(-0.921075\pi\)
0.969417 0.245418i \(-0.0789253\pi\)
\(684\) 0 0
\(685\) −42.9925 −0.0627628
\(686\) 0 0
\(687\) 121.346i 0.176631i
\(688\) 0 0
\(689\) 952.349 1.38222
\(690\) 0 0
\(691\) 1274.20i 1.84400i 0.387192 + 0.921999i \(0.373445\pi\)
−0.387192 + 0.921999i \(0.626555\pi\)
\(692\) 0 0
\(693\) −114.159 −0.164732
\(694\) 0 0
\(695\) − 282.057i − 0.405837i
\(696\) 0 0
\(697\) 531.416 0.762434
\(698\) 0 0
\(699\) − 12.4380i − 0.0177940i
\(700\) 0 0
\(701\) −676.056 −0.964416 −0.482208 0.876057i \(-0.660165\pi\)
−0.482208 + 0.876057i \(0.660165\pi\)
\(702\) 0 0
\(703\) − 1065.37i − 1.51547i
\(704\) 0 0
\(705\) −110.417 −0.156619
\(706\) 0 0
\(707\) − 287.771i − 0.407031i
\(708\) 0 0
\(709\) 62.3418 0.0879292 0.0439646 0.999033i \(-0.486001\pi\)
0.0439646 + 0.999033i \(0.486001\pi\)
\(710\) 0 0
\(711\) 385.971i 0.542856i
\(712\) 0 0
\(713\) −662.127 −0.928649
\(714\) 0 0
\(715\) 860.384i 1.20333i
\(716\) 0 0
\(717\) −375.197 −0.523287
\(718\) 0 0
\(719\) 322.341i 0.448318i 0.974553 + 0.224159i \(0.0719636\pi\)
−0.974553 + 0.224159i \(0.928036\pi\)
\(720\) 0 0
\(721\) −190.461 −0.264163
\(722\) 0 0
\(723\) − 666.841i − 0.922326i
\(724\) 0 0
\(725\) −143.756 −0.198284
\(726\) 0 0
\(727\) 368.070i 0.506287i 0.967429 + 0.253143i \(0.0814644\pi\)
−0.967429 + 0.253143i \(0.918536\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) 0 0
\(731\) 1095.87i 1.49913i
\(732\) 0 0
\(733\) −430.587 −0.587431 −0.293715 0.955893i \(-0.594892\pi\)
−0.293715 + 0.955893i \(0.594892\pi\)
\(734\) 0 0
\(735\) 307.760i 0.418721i
\(736\) 0 0
\(737\) −1032.23 −1.40059
\(738\) 0 0
\(739\) 818.050i 1.10697i 0.832859 + 0.553485i \(0.186702\pi\)
−0.832859 + 0.553485i \(0.813298\pi\)
\(740\) 0 0
\(741\) −403.576 −0.544638
\(742\) 0 0
\(743\) − 685.438i − 0.922528i −0.887263 0.461264i \(-0.847396\pi\)
0.887263 0.461264i \(-0.152604\pi\)
\(744\) 0 0
\(745\) −406.810 −0.546053
\(746\) 0 0
\(747\) − 66.1251i − 0.0885209i
\(748\) 0 0
\(749\) −412.106 −0.550208
\(750\) 0 0
\(751\) − 104.336i − 0.138929i −0.997584 0.0694647i \(-0.977871\pi\)
0.997584 0.0694647i \(-0.0221291\pi\)
\(752\) 0 0
\(753\) 148.070 0.196640
\(754\) 0 0
\(755\) 1157.99i 1.53377i
\(756\) 0 0
\(757\) 539.949 0.713274 0.356637 0.934243i \(-0.383923\pi\)
0.356637 + 0.934243i \(0.383923\pi\)
\(758\) 0 0
\(759\) 408.948i 0.538798i
\(760\) 0 0
\(761\) 1192.21 1.56663 0.783317 0.621623i \(-0.213526\pi\)
0.783317 + 0.621623i \(0.213526\pi\)
\(762\) 0 0
\(763\) 149.847i 0.196391i
\(764\) 0 0
\(765\) 294.475 0.384935
\(766\) 0 0
\(767\) 956.473i 1.24703i
\(768\) 0 0
\(769\) −321.506 −0.418083 −0.209042 0.977907i \(-0.567034\pi\)
−0.209042 + 0.977907i \(0.567034\pi\)
\(770\) 0 0
\(771\) 94.0471i 0.121981i
\(772\) 0 0
\(773\) 1453.29 1.88006 0.940029 0.341094i \(-0.110798\pi\)
0.940029 + 0.341094i \(0.110798\pi\)
\(774\) 0 0
\(775\) − 254.385i − 0.328239i
\(776\) 0 0
\(777\) 317.276 0.408334
\(778\) 0 0
\(779\) 372.753i 0.478502i
\(780\) 0 0
\(781\) −1692.75 −2.16741
\(782\) 0 0
\(783\) 113.623i 0.145112i
\(784\) 0 0
\(785\) 135.240 0.172280
\(786\) 0 0
\(787\) 231.223i 0.293802i 0.989151 + 0.146901i \(0.0469299\pi\)
−0.989151 + 0.146901i \(0.953070\pi\)
\(788\) 0 0
\(789\) 869.189 1.10163
\(790\) 0 0
\(791\) − 17.4531i − 0.0220646i
\(792\) 0 0
\(793\) −582.696 −0.734800
\(794\) 0 0
\(795\) − 487.422i − 0.613110i
\(796\) 0 0
\(797\) −51.5068 −0.0646259 −0.0323129 0.999478i \(-0.510287\pi\)
−0.0323129 + 0.999478i \(0.510287\pi\)
\(798\) 0 0
\(799\) 339.606i 0.425039i
\(800\) 0 0
\(801\) 366.224 0.457209
\(802\) 0 0
\(803\) − 1999.38i − 2.48988i
\(804\) 0 0
\(805\) −202.572 −0.251643
\(806\) 0 0
\(807\) 493.480i 0.611499i
\(808\) 0 0
\(809\) 580.914 0.718065 0.359032 0.933325i \(-0.383107\pi\)
0.359032 + 0.933325i \(0.383107\pi\)
\(810\) 0 0
\(811\) − 1351.98i − 1.66706i −0.552475 0.833529i \(-0.686317\pi\)
0.552475 0.833529i \(-0.313683\pi\)
\(812\) 0 0
\(813\) 417.753 0.513841
\(814\) 0 0
\(815\) 863.480i 1.05948i
\(816\) 0 0
\(817\) −768.678 −0.940855
\(818\) 0 0
\(819\) − 120.188i − 0.146749i
\(820\) 0 0
\(821\) −179.511 −0.218650 −0.109325 0.994006i \(-0.534869\pi\)
−0.109325 + 0.994006i \(0.534869\pi\)
\(822\) 0 0
\(823\) − 665.446i − 0.808561i −0.914635 0.404281i \(-0.867522\pi\)
0.914635 0.404281i \(-0.132478\pi\)
\(824\) 0 0
\(825\) −157.115 −0.190443
\(826\) 0 0
\(827\) 901.166i 1.08968i 0.838540 + 0.544840i \(0.183410\pi\)
−0.838540 + 0.544840i \(0.816590\pi\)
\(828\) 0 0
\(829\) 183.151 0.220930 0.110465 0.993880i \(-0.464766\pi\)
0.110465 + 0.993880i \(0.464766\pi\)
\(830\) 0 0
\(831\) − 420.676i − 0.506228i
\(832\) 0 0
\(833\) 946.571 1.13634
\(834\) 0 0
\(835\) − 1143.50i − 1.36946i
\(836\) 0 0
\(837\) −201.062 −0.240218
\(838\) 0 0
\(839\) − 846.082i − 1.00844i −0.863575 0.504220i \(-0.831780\pi\)
0.863575 0.504220i \(-0.168220\pi\)
\(840\) 0 0
\(841\) −362.846 −0.431446
\(842\) 0 0
\(843\) − 182.457i − 0.216437i
\(844\) 0 0
\(845\) −180.382 −0.213470
\(846\) 0 0
\(847\) 191.351i 0.225917i
\(848\) 0 0
\(849\) −387.749 −0.456713
\(850\) 0 0
\(851\) − 1136.56i − 1.33556i
\(852\) 0 0
\(853\) 540.635 0.633804 0.316902 0.948458i \(-0.397357\pi\)
0.316902 + 0.948458i \(0.397357\pi\)
\(854\) 0 0
\(855\) 206.555i 0.241585i
\(856\) 0 0
\(857\) 1333.41 1.55590 0.777952 0.628323i \(-0.216258\pi\)
0.777952 + 0.628323i \(0.216258\pi\)
\(858\) 0 0
\(859\) − 439.034i − 0.511099i −0.966796 0.255549i \(-0.917744\pi\)
0.966796 0.255549i \(-0.0822563\pi\)
\(860\) 0 0
\(861\) −111.008 −0.128930
\(862\) 0 0
\(863\) − 1164.41i − 1.34926i −0.738155 0.674631i \(-0.764303\pi\)
0.738155 0.674631i \(-0.235697\pi\)
\(864\) 0 0
\(865\) −188.208 −0.217582
\(866\) 0 0
\(867\) − 405.147i − 0.467298i
\(868\) 0 0
\(869\) 1775.20 2.04281
\(870\) 0 0
\(871\) − 1086.74i − 1.24769i
\(872\) 0 0
\(873\) 264.922 0.303461
\(874\) 0 0
\(875\) − 373.783i − 0.427181i
\(876\) 0 0
\(877\) −404.456 −0.461181 −0.230591 0.973051i \(-0.574066\pi\)
−0.230591 + 0.973051i \(0.574066\pi\)
\(878\) 0 0
\(879\) − 729.292i − 0.829684i
\(880\) 0 0
\(881\) −227.084 −0.257757 −0.128879 0.991660i \(-0.541138\pi\)
−0.128879 + 0.991660i \(0.541138\pi\)
\(882\) 0 0
\(883\) 673.730i 0.763001i 0.924369 + 0.381501i \(0.124593\pi\)
−0.924369 + 0.381501i \(0.875407\pi\)
\(884\) 0 0
\(885\) 489.533 0.553145
\(886\) 0 0
\(887\) 1370.95i 1.54560i 0.634648 + 0.772801i \(0.281145\pi\)
−0.634648 + 0.772801i \(0.718855\pi\)
\(888\) 0 0
\(889\) −45.5678 −0.0512574
\(890\) 0 0
\(891\) 124.182i 0.139373i
\(892\) 0 0
\(893\) −238.211 −0.266754
\(894\) 0 0
\(895\) − 1183.79i − 1.32267i
\(896\) 0 0
\(897\) −430.544 −0.479982
\(898\) 0 0
\(899\) 846.121i 0.941180i
\(900\) 0 0
\(901\) −1499.15 −1.66388
\(902\) 0 0
\(903\) − 228.917i − 0.253508i
\(904\) 0 0
\(905\) −339.124 −0.374723
\(906\) 0 0
\(907\) 1469.46i 1.62013i 0.586342 + 0.810063i \(0.300567\pi\)
−0.586342 + 0.810063i \(0.699433\pi\)
\(908\) 0 0
\(909\) −313.035 −0.344373
\(910\) 0 0
\(911\) 871.203i 0.956315i 0.878274 + 0.478157i \(0.158695\pi\)
−0.878274 + 0.478157i \(0.841305\pi\)
\(912\) 0 0
\(913\) −304.131 −0.333111
\(914\) 0 0
\(915\) 298.230i 0.325935i
\(916\) 0 0
\(917\) −140.596 −0.153322
\(918\) 0 0
\(919\) 256.361i 0.278957i 0.990225 + 0.139478i \(0.0445426\pi\)
−0.990225 + 0.139478i \(0.955457\pi\)
\(920\) 0 0
\(921\) 212.808 0.231062
\(922\) 0 0
\(923\) − 1782.14i − 1.93081i
\(924\) 0 0
\(925\) 436.661 0.472066
\(926\) 0 0
\(927\) 207.183i 0.223498i
\(928\) 0 0
\(929\) −1399.93 −1.50692 −0.753462 0.657491i \(-0.771618\pi\)
−0.753462 + 0.657491i \(0.771618\pi\)
\(930\) 0 0
\(931\) 663.956i 0.713165i
\(932\) 0 0
\(933\) 757.832 0.812253
\(934\) 0 0
\(935\) − 1354.39i − 1.44854i
\(936\) 0 0
\(937\) 365.610 0.390192 0.195096 0.980784i \(-0.437498\pi\)
0.195096 + 0.980784i \(0.437498\pi\)
\(938\) 0 0
\(939\) − 650.235i − 0.692476i
\(940\) 0 0
\(941\) 407.498 0.433048 0.216524 0.976277i \(-0.430528\pi\)
0.216524 + 0.976277i \(0.430528\pi\)
\(942\) 0 0
\(943\) 397.661i 0.421698i
\(944\) 0 0
\(945\) −61.5134 −0.0650935
\(946\) 0 0
\(947\) − 1150.64i − 1.21503i −0.794307 0.607516i \(-0.792166\pi\)
0.794307 0.607516i \(-0.207834\pi\)
\(948\) 0 0
\(949\) 2104.96 2.21808
\(950\) 0 0
\(951\) 673.537i 0.708241i
\(952\) 0 0
\(953\) −1341.53 −1.40769 −0.703844 0.710355i \(-0.748534\pi\)
−0.703844 + 0.710355i \(0.748534\pi\)
\(954\) 0 0
\(955\) − 611.295i − 0.640100i
\(956\) 0 0
\(957\) 522.588 0.546069
\(958\) 0 0
\(959\) − 27.6219i − 0.0288029i
\(960\) 0 0
\(961\) −536.260 −0.558023
\(962\) 0 0
\(963\) 448.286i 0.465510i
\(964\) 0 0
\(965\) 1192.54 1.23579
\(966\) 0 0
\(967\) − 1227.25i − 1.26914i −0.772867 0.634568i \(-0.781178\pi\)
0.772867 0.634568i \(-0.218822\pi\)
\(968\) 0 0
\(969\) 635.295 0.655620
\(970\) 0 0
\(971\) 1864.31i 1.91999i 0.280020 + 0.959994i \(0.409659\pi\)
−0.280020 + 0.959994i \(0.590341\pi\)
\(972\) 0 0
\(973\) 181.217 0.186245
\(974\) 0 0
\(975\) − 165.412i − 0.169654i
\(976\) 0 0
\(977\) −1209.56 −1.23803 −0.619017 0.785378i \(-0.712469\pi\)
−0.619017 + 0.785378i \(0.712469\pi\)
\(978\) 0 0
\(979\) − 1684.38i − 1.72051i
\(980\) 0 0
\(981\) 163.002 0.166159
\(982\) 0 0
\(983\) 1018.26i 1.03587i 0.855420 + 0.517935i \(0.173299\pi\)
−0.855420 + 0.517935i \(0.826701\pi\)
\(984\) 0 0
\(985\) 14.6310 0.0148538
\(986\) 0 0
\(987\) − 70.9408i − 0.0718752i
\(988\) 0 0
\(989\) −820.042 −0.829163
\(990\) 0 0
\(991\) − 1627.52i − 1.64230i −0.570712 0.821150i \(-0.693333\pi\)
0.570712 0.821150i \(-0.306667\pi\)
\(992\) 0 0
\(993\) 26.7931 0.0269819
\(994\) 0 0
\(995\) − 1349.24i − 1.35602i
\(996\) 0 0
\(997\) −540.192 −0.541817 −0.270909 0.962605i \(-0.587324\pi\)
−0.270909 + 0.962605i \(0.587324\pi\)
\(998\) 0 0
\(999\) − 345.131i − 0.345476i
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 384.3.g.b.127.1 yes 8
3.2 odd 2 1152.3.g.c.127.7 8
4.3 odd 2 inner 384.3.g.b.127.5 yes 8
8.3 odd 2 384.3.g.a.127.4 8
8.5 even 2 384.3.g.a.127.8 yes 8
12.11 even 2 1152.3.g.c.127.8 8
16.3 odd 4 768.3.b.e.127.4 8
16.5 even 4 768.3.b.e.127.1 8
16.11 odd 4 768.3.b.f.127.5 8
16.13 even 4 768.3.b.f.127.8 8
24.5 odd 2 1152.3.g.f.127.1 8
24.11 even 2 1152.3.g.f.127.2 8
48.5 odd 4 2304.3.b.t.127.7 8
48.11 even 4 2304.3.b.q.127.7 8
48.29 odd 4 2304.3.b.q.127.2 8
48.35 even 4 2304.3.b.t.127.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.3.g.a.127.4 8 8.3 odd 2
384.3.g.a.127.8 yes 8 8.5 even 2
384.3.g.b.127.1 yes 8 1.1 even 1 trivial
384.3.g.b.127.5 yes 8 4.3 odd 2 inner
768.3.b.e.127.1 8 16.5 even 4
768.3.b.e.127.4 8 16.3 odd 4
768.3.b.f.127.5 8 16.11 odd 4
768.3.b.f.127.8 8 16.13 even 4
1152.3.g.c.127.7 8 3.2 odd 2
1152.3.g.c.127.8 8 12.11 even 2
1152.3.g.f.127.1 8 24.5 odd 2
1152.3.g.f.127.2 8 24.11 even 2
2304.3.b.q.127.2 8 48.29 odd 4
2304.3.b.q.127.7 8 48.11 even 4
2304.3.b.t.127.2 8 48.35 even 4
2304.3.b.t.127.7 8 48.5 odd 4