Properties

Label 384.3.g.b.127.4
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 127.4
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 384.127
Dual form 384.3.g.b.127.8

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205i q^{3} +8.29253 q^{5} -8.55583i q^{7} -3.00000 q^{9} +O(q^{10})\) \(q-1.73205i q^{3} +8.29253 q^{5} -8.55583i q^{7} -3.00000 q^{9} -13.7980i q^{11} -17.0693 q^{13} -14.3631i q^{15} +20.3246 q^{17} +20.4440i q^{19} -14.8191 q^{21} +5.51575i q^{23} +43.7660 q^{25} +5.19615i q^{27} -41.0586 q^{29} -22.2953i q^{31} -23.8988 q^{33} -70.9495i q^{35} +11.6326 q^{37} +29.5649i q^{39} +35.9527 q^{41} -66.8648i q^{43} -24.8776 q^{45} -19.9366i q^{47} -24.2023 q^{49} -35.2032i q^{51} -17.6329 q^{53} -114.420i q^{55} +35.4100 q^{57} +62.1572i q^{59} +47.4836 q^{61} +25.6675i q^{63} -141.548 q^{65} +74.8105i q^{67} +9.55356 q^{69} -16.9150i q^{71} +101.712 q^{73} -75.8050i q^{75} -118.053 q^{77} -0.879320i q^{79} +9.00000 q^{81} +23.2131i q^{83} +168.542 q^{85} +71.1155i q^{87} -16.3089 q^{89} +146.042i q^{91} -38.6167 q^{93} +169.532i q^{95} +188.307 q^{97} +41.3939i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 16q^{5} - 24q^{9} + O(q^{10}) \) \( 8q + 16q^{5} - 24q^{9} - 48q^{13} + 16q^{17} - 8q^{25} + 80q^{29} + 16q^{37} + 80q^{41} - 48q^{45} - 88q^{49} - 176q^{53} + 96q^{57} + 272q^{61} - 160q^{65} - 16q^{73} - 320q^{77} + 72q^{81} - 32q^{85} - 240q^{89} + 192q^{93} + 400q^{97} + O(q^{100}) \)

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
\(5\) 8.29253 1.65851 0.829253 0.558874i \(-0.188766\pi\)
0.829253 + 0.558874i \(0.188766\pi\)
\(6\) 0 0
\(7\) − 8.55583i − 1.22226i −0.791529 0.611131i \(-0.790715\pi\)
0.791529 0.611131i \(-0.209285\pi\)
\(8\) 0 0
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) − 13.7980i − 1.25436i −0.778874 0.627180i \(-0.784209\pi\)
0.778874 0.627180i \(-0.215791\pi\)
\(12\) 0 0
\(13\) −17.0693 −1.31302 −0.656512 0.754316i \(-0.727969\pi\)
−0.656512 + 0.754316i \(0.727969\pi\)
\(14\) 0 0
\(15\) − 14.3631i − 0.957539i
\(16\) 0 0
\(17\) 20.3246 1.19556 0.597781 0.801659i \(-0.296049\pi\)
0.597781 + 0.801659i \(0.296049\pi\)
\(18\) 0 0
\(19\) 20.4440i 1.07600i 0.842946 + 0.537999i \(0.180819\pi\)
−0.842946 + 0.537999i \(0.819181\pi\)
\(20\) 0 0
\(21\) −14.8191 −0.705673
\(22\) 0 0
\(23\) 5.51575i 0.239815i 0.992785 + 0.119908i \(0.0382598\pi\)
−0.992785 + 0.119908i \(0.961740\pi\)
\(24\) 0 0
\(25\) 43.7660 1.75064
\(26\) 0 0
\(27\) 5.19615i 0.192450i
\(28\) 0 0
\(29\) −41.0586 −1.41581 −0.707906 0.706306i \(-0.750360\pi\)
−0.707906 + 0.706306i \(0.750360\pi\)
\(30\) 0 0
\(31\) − 22.2953i − 0.719205i −0.933106 0.359602i \(-0.882912\pi\)
0.933106 0.359602i \(-0.117088\pi\)
\(32\) 0 0
\(33\) −23.8988 −0.724205
\(34\) 0 0
\(35\) − 70.9495i − 2.02713i
\(36\) 0 0
\(37\) 11.6326 0.314396 0.157198 0.987567i \(-0.449754\pi\)
0.157198 + 0.987567i \(0.449754\pi\)
\(38\) 0 0
\(39\) 29.5649i 0.758075i
\(40\) 0 0
\(41\) 35.9527 0.876894 0.438447 0.898757i \(-0.355529\pi\)
0.438447 + 0.898757i \(0.355529\pi\)
\(42\) 0 0
\(43\) − 66.8648i − 1.55499i −0.628886 0.777497i \(-0.716489\pi\)
0.628886 0.777497i \(-0.283511\pi\)
\(44\) 0 0
\(45\) −24.8776 −0.552835
\(46\) 0 0
\(47\) − 19.9366i − 0.424182i −0.977250 0.212091i \(-0.931973\pi\)
0.977250 0.212091i \(-0.0680274\pi\)
\(48\) 0 0
\(49\) −24.2023 −0.493924
\(50\) 0 0
\(51\) − 35.2032i − 0.690259i
\(52\) 0 0
\(53\) −17.6329 −0.332697 −0.166348 0.986067i \(-0.553198\pi\)
−0.166348 + 0.986067i \(0.553198\pi\)
\(54\) 0 0
\(55\) − 114.420i − 2.08036i
\(56\) 0 0
\(57\) 35.4100 0.621227
\(58\) 0 0
\(59\) 62.1572i 1.05351i 0.850017 + 0.526756i \(0.176592\pi\)
−0.850017 + 0.526756i \(0.823408\pi\)
\(60\) 0 0
\(61\) 47.4836 0.778419 0.389210 0.921149i \(-0.372748\pi\)
0.389210 + 0.921149i \(0.372748\pi\)
\(62\) 0 0
\(63\) 25.6675i 0.407421i
\(64\) 0 0
\(65\) −141.548 −2.17766
\(66\) 0 0
\(67\) 74.8105i 1.11657i 0.829648 + 0.558287i \(0.188541\pi\)
−0.829648 + 0.558287i \(0.811459\pi\)
\(68\) 0 0
\(69\) 9.55356 0.138457
\(70\) 0 0
\(71\) − 16.9150i − 0.238240i −0.992880 0.119120i \(-0.961993\pi\)
0.992880 0.119120i \(-0.0380073\pi\)
\(72\) 0 0
\(73\) 101.712 1.39331 0.696657 0.717404i \(-0.254670\pi\)
0.696657 + 0.717404i \(0.254670\pi\)
\(74\) 0 0
\(75\) − 75.8050i − 1.01073i
\(76\) 0 0
\(77\) −118.053 −1.53316
\(78\) 0 0
\(79\) − 0.879320i − 0.0111306i −0.999985 0.00556532i \(-0.998228\pi\)
0.999985 0.00556532i \(-0.00177151\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 23.2131i 0.279676i 0.990174 + 0.139838i \(0.0446582\pi\)
−0.990174 + 0.139838i \(0.955342\pi\)
\(84\) 0 0
\(85\) 168.542 1.98285
\(86\) 0 0
\(87\) 71.1155i 0.817420i
\(88\) 0 0
\(89\) −16.3089 −0.183246 −0.0916231 0.995794i \(-0.529205\pi\)
−0.0916231 + 0.995794i \(0.529205\pi\)
\(90\) 0 0
\(91\) 146.042i 1.60486i
\(92\) 0 0
\(93\) −38.6167 −0.415233
\(94\) 0 0
\(95\) 169.532i 1.78455i
\(96\) 0 0
\(97\) 188.307 1.94131 0.970656 0.240474i \(-0.0773028\pi\)
0.970656 + 0.240474i \(0.0773028\pi\)
\(98\) 0 0
\(99\) 41.3939i 0.418120i
\(100\) 0 0
\(101\) 66.8468 0.661849 0.330925 0.943657i \(-0.392639\pi\)
0.330925 + 0.943657i \(0.392639\pi\)
\(102\) 0 0
\(103\) 166.313i 1.61469i 0.590083 + 0.807343i \(0.299095\pi\)
−0.590083 + 0.807343i \(0.700905\pi\)
\(104\) 0 0
\(105\) −122.888 −1.17036
\(106\) 0 0
\(107\) 80.1467i 0.749035i 0.927220 + 0.374517i \(0.122192\pi\)
−0.927220 + 0.374517i \(0.877808\pi\)
\(108\) 0 0
\(109\) −26.8781 −0.246589 −0.123294 0.992370i \(-0.539346\pi\)
−0.123294 + 0.992370i \(0.539346\pi\)
\(110\) 0 0
\(111\) − 20.1483i − 0.181516i
\(112\) 0 0
\(113\) −79.0958 −0.699963 −0.349981 0.936757i \(-0.613812\pi\)
−0.349981 + 0.936757i \(0.613812\pi\)
\(114\) 0 0
\(115\) 45.7395i 0.397735i
\(116\) 0 0
\(117\) 51.2079 0.437675
\(118\) 0 0
\(119\) − 173.894i − 1.46129i
\(120\) 0 0
\(121\) −69.3837 −0.573419
\(122\) 0 0
\(123\) − 62.2718i − 0.506275i
\(124\) 0 0
\(125\) 155.618 1.24494
\(126\) 0 0
\(127\) − 170.725i − 1.34429i −0.740419 0.672145i \(-0.765373\pi\)
0.740419 0.672145i \(-0.234627\pi\)
\(128\) 0 0
\(129\) −115.813 −0.897777
\(130\) 0 0
\(131\) − 178.980i − 1.36626i −0.730297 0.683129i \(-0.760619\pi\)
0.730297 0.683129i \(-0.239381\pi\)
\(132\) 0 0
\(133\) 174.915 1.31515
\(134\) 0 0
\(135\) 43.0892i 0.319180i
\(136\) 0 0
\(137\) −93.2075 −0.680347 −0.340173 0.940363i \(-0.610486\pi\)
−0.340173 + 0.940363i \(0.610486\pi\)
\(138\) 0 0
\(139\) 222.476i 1.60055i 0.599635 + 0.800274i \(0.295313\pi\)
−0.599635 + 0.800274i \(0.704687\pi\)
\(140\) 0 0
\(141\) −34.5312 −0.244902
\(142\) 0 0
\(143\) 235.522i 1.64700i
\(144\) 0 0
\(145\) −340.479 −2.34813
\(146\) 0 0
\(147\) 41.9196i 0.285167i
\(148\) 0 0
\(149\) −94.0043 −0.630901 −0.315451 0.948942i \(-0.602156\pi\)
−0.315451 + 0.948942i \(0.602156\pi\)
\(150\) 0 0
\(151\) 82.0081i 0.543100i 0.962424 + 0.271550i \(0.0875362\pi\)
−0.962424 + 0.271550i \(0.912464\pi\)
\(152\) 0 0
\(153\) −60.9737 −0.398521
\(154\) 0 0
\(155\) − 184.885i − 1.19281i
\(156\) 0 0
\(157\) 56.7180 0.361261 0.180631 0.983551i \(-0.442186\pi\)
0.180631 + 0.983551i \(0.442186\pi\)
\(158\) 0 0
\(159\) 30.5411i 0.192083i
\(160\) 0 0
\(161\) 47.1918 0.293117
\(162\) 0 0
\(163\) 189.588i 1.16312i 0.813504 + 0.581559i \(0.197557\pi\)
−0.813504 + 0.581559i \(0.802443\pi\)
\(164\) 0 0
\(165\) −198.181 −1.20110
\(166\) 0 0
\(167\) 312.777i 1.87291i 0.350783 + 0.936457i \(0.385916\pi\)
−0.350783 + 0.936457i \(0.614084\pi\)
\(168\) 0 0
\(169\) 122.361 0.724031
\(170\) 0 0
\(171\) − 61.3319i − 0.358666i
\(172\) 0 0
\(173\) 182.538 1.05513 0.527567 0.849514i \(-0.323104\pi\)
0.527567 + 0.849514i \(0.323104\pi\)
\(174\) 0 0
\(175\) − 374.455i − 2.13974i
\(176\) 0 0
\(177\) 107.659 0.608245
\(178\) 0 0
\(179\) 19.7781i 0.110492i 0.998473 + 0.0552461i \(0.0175943\pi\)
−0.998473 + 0.0552461i \(0.982406\pi\)
\(180\) 0 0
\(181\) −265.750 −1.46823 −0.734117 0.679023i \(-0.762403\pi\)
−0.734117 + 0.679023i \(0.762403\pi\)
\(182\) 0 0
\(183\) − 82.2440i − 0.449421i
\(184\) 0 0
\(185\) 96.4640 0.521427
\(186\) 0 0
\(187\) − 280.438i − 1.49967i
\(188\) 0 0
\(189\) 44.4574 0.235224
\(190\) 0 0
\(191\) 288.025i 1.50799i 0.656882 + 0.753993i \(0.271875\pi\)
−0.656882 + 0.753993i \(0.728125\pi\)
\(192\) 0 0
\(193\) 281.010 1.45601 0.728005 0.685572i \(-0.240448\pi\)
0.728005 + 0.685572i \(0.240448\pi\)
\(194\) 0 0
\(195\) 245.168i 1.25727i
\(196\) 0 0
\(197\) 33.8330 0.171741 0.0858705 0.996306i \(-0.472633\pi\)
0.0858705 + 0.996306i \(0.472633\pi\)
\(198\) 0 0
\(199\) 84.9700i 0.426985i 0.976945 + 0.213493i \(0.0684839\pi\)
−0.976945 + 0.213493i \(0.931516\pi\)
\(200\) 0 0
\(201\) 129.576 0.644654
\(202\) 0 0
\(203\) 351.290i 1.73049i
\(204\) 0 0
\(205\) 298.138 1.45433
\(206\) 0 0
\(207\) − 16.5472i − 0.0799384i
\(208\) 0 0
\(209\) 282.085 1.34969
\(210\) 0 0
\(211\) − 140.700i − 0.666826i −0.942781 0.333413i \(-0.891800\pi\)
0.942781 0.333413i \(-0.108200\pi\)
\(212\) 0 0
\(213\) −29.2977 −0.137548
\(214\) 0 0
\(215\) − 554.478i − 2.57897i
\(216\) 0 0
\(217\) −190.755 −0.879057
\(218\) 0 0
\(219\) − 176.170i − 0.804431i
\(220\) 0 0
\(221\) −346.926 −1.56980
\(222\) 0 0
\(223\) 247.529i 1.11000i 0.831851 + 0.554999i \(0.187281\pi\)
−0.831851 + 0.554999i \(0.812719\pi\)
\(224\) 0 0
\(225\) −131.298 −0.583547
\(226\) 0 0
\(227\) − 129.752i − 0.571593i −0.958290 0.285797i \(-0.907742\pi\)
0.958290 0.285797i \(-0.0922582\pi\)
\(228\) 0 0
\(229\) −294.304 −1.28517 −0.642586 0.766214i \(-0.722138\pi\)
−0.642586 + 0.766214i \(0.722138\pi\)
\(230\) 0 0
\(231\) 204.474i 0.885168i
\(232\) 0 0
\(233\) −239.948 −1.02982 −0.514911 0.857244i \(-0.672175\pi\)
−0.514911 + 0.857244i \(0.672175\pi\)
\(234\) 0 0
\(235\) − 165.325i − 0.703509i
\(236\) 0 0
\(237\) −1.52303 −0.00642628
\(238\) 0 0
\(239\) − 425.346i − 1.77969i −0.456261 0.889846i \(-0.650812\pi\)
0.456261 0.889846i \(-0.349188\pi\)
\(240\) 0 0
\(241\) 264.493 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(242\) 0 0
\(243\) − 15.5885i − 0.0641500i
\(244\) 0 0
\(245\) −200.698 −0.819176
\(246\) 0 0
\(247\) − 348.964i − 1.41281i
\(248\) 0 0
\(249\) 40.2063 0.161471
\(250\) 0 0
\(251\) 221.125i 0.880976i 0.897759 + 0.440488i \(0.145195\pi\)
−0.897759 + 0.440488i \(0.854805\pi\)
\(252\) 0 0
\(253\) 76.1061 0.300815
\(254\) 0 0
\(255\) − 291.923i − 1.14480i
\(256\) 0 0
\(257\) 87.0655 0.338776 0.169388 0.985549i \(-0.445821\pi\)
0.169388 + 0.985549i \(0.445821\pi\)
\(258\) 0 0
\(259\) − 99.5270i − 0.384274i
\(260\) 0 0
\(261\) 123.176 0.471937
\(262\) 0 0
\(263\) − 277.324i − 1.05447i −0.849721 0.527233i \(-0.823230\pi\)
0.849721 0.527233i \(-0.176770\pi\)
\(264\) 0 0
\(265\) −146.222 −0.551780
\(266\) 0 0
\(267\) 28.2479i 0.105797i
\(268\) 0 0
\(269\) 49.8005 0.185132 0.0925659 0.995707i \(-0.470493\pi\)
0.0925659 + 0.995707i \(0.470493\pi\)
\(270\) 0 0
\(271\) − 31.3145i − 0.115552i −0.998330 0.0577758i \(-0.981599\pi\)
0.998330 0.0577758i \(-0.0184009\pi\)
\(272\) 0 0
\(273\) 252.952 0.926566
\(274\) 0 0
\(275\) − 603.882i − 2.19593i
\(276\) 0 0
\(277\) 79.1431 0.285715 0.142858 0.989743i \(-0.454371\pi\)
0.142858 + 0.989743i \(0.454371\pi\)
\(278\) 0 0
\(279\) 66.8860i 0.239735i
\(280\) 0 0
\(281\) −106.190 −0.377902 −0.188951 0.981987i \(-0.560509\pi\)
−0.188951 + 0.981987i \(0.560509\pi\)
\(282\) 0 0
\(283\) − 270.251i − 0.954949i −0.878646 0.477475i \(-0.841552\pi\)
0.878646 0.477475i \(-0.158448\pi\)
\(284\) 0 0
\(285\) 293.638 1.03031
\(286\) 0 0
\(287\) − 307.605i − 1.07179i
\(288\) 0 0
\(289\) 124.088 0.429371
\(290\) 0 0
\(291\) − 326.158i − 1.12082i
\(292\) 0 0
\(293\) −24.2487 −0.0827600 −0.0413800 0.999143i \(-0.513175\pi\)
−0.0413800 + 0.999143i \(0.513175\pi\)
\(294\) 0 0
\(295\) 515.440i 1.74726i
\(296\) 0 0
\(297\) 71.6963 0.241402
\(298\) 0 0
\(299\) − 94.1500i − 0.314883i
\(300\) 0 0
\(301\) −572.084 −1.90061
\(302\) 0 0
\(303\) − 115.782i − 0.382119i
\(304\) 0 0
\(305\) 393.759 1.29101
\(306\) 0 0
\(307\) 122.865i 0.400211i 0.979774 + 0.200106i \(0.0641285\pi\)
−0.979774 + 0.200106i \(0.935871\pi\)
\(308\) 0 0
\(309\) 288.062 0.932239
\(310\) 0 0
\(311\) − 108.808i − 0.349865i −0.984580 0.174933i \(-0.944029\pi\)
0.984580 0.174933i \(-0.0559707\pi\)
\(312\) 0 0
\(313\) −52.2216 −0.166842 −0.0834211 0.996514i \(-0.526585\pi\)
−0.0834211 + 0.996514i \(0.526585\pi\)
\(314\) 0 0
\(315\) 212.848i 0.675709i
\(316\) 0 0
\(317\) −94.3251 −0.297555 −0.148778 0.988871i \(-0.547534\pi\)
−0.148778 + 0.988871i \(0.547534\pi\)
\(318\) 0 0
\(319\) 566.524i 1.77594i
\(320\) 0 0
\(321\) 138.818 0.432456
\(322\) 0 0
\(323\) 415.515i 1.28642i
\(324\) 0 0
\(325\) −747.056 −2.29863
\(326\) 0 0
\(327\) 46.5543i 0.142368i
\(328\) 0 0
\(329\) −170.574 −0.518462
\(330\) 0 0
\(331\) − 406.107i − 1.22691i −0.789730 0.613454i \(-0.789780\pi\)
0.789730 0.613454i \(-0.210220\pi\)
\(332\) 0 0
\(333\) −34.8979 −0.104799
\(334\) 0 0
\(335\) 620.368i 1.85184i
\(336\) 0 0
\(337\) −22.4140 −0.0665105 −0.0332552 0.999447i \(-0.510587\pi\)
−0.0332552 + 0.999447i \(0.510587\pi\)
\(338\) 0 0
\(339\) 136.998i 0.404124i
\(340\) 0 0
\(341\) −307.630 −0.902142
\(342\) 0 0
\(343\) − 212.165i − 0.618557i
\(344\) 0 0
\(345\) 79.2232 0.229632
\(346\) 0 0
\(347\) 458.377i 1.32097i 0.750839 + 0.660486i \(0.229650\pi\)
−0.750839 + 0.660486i \(0.770350\pi\)
\(348\) 0 0
\(349\) −282.960 −0.810774 −0.405387 0.914145i \(-0.632863\pi\)
−0.405387 + 0.914145i \(0.632863\pi\)
\(350\) 0 0
\(351\) − 88.6947i − 0.252692i
\(352\) 0 0
\(353\) −175.820 −0.498073 −0.249036 0.968494i \(-0.580114\pi\)
−0.249036 + 0.968494i \(0.580114\pi\)
\(354\) 0 0
\(355\) − 140.268i − 0.395122i
\(356\) 0 0
\(357\) −301.193 −0.843677
\(358\) 0 0
\(359\) 53.5791i 0.149246i 0.997212 + 0.0746228i \(0.0237753\pi\)
−0.997212 + 0.0746228i \(0.976225\pi\)
\(360\) 0 0
\(361\) −56.9552 −0.157771
\(362\) 0 0
\(363\) 120.176i 0.331063i
\(364\) 0 0
\(365\) 843.449 2.31082
\(366\) 0 0
\(367\) − 62.4258i − 0.170097i −0.996377 0.0850487i \(-0.972895\pi\)
0.996377 0.0850487i \(-0.0271046\pi\)
\(368\) 0 0
\(369\) −107.858 −0.292298
\(370\) 0 0
\(371\) 150.864i 0.406643i
\(372\) 0 0
\(373\) 370.552 0.993437 0.496719 0.867912i \(-0.334538\pi\)
0.496719 + 0.867912i \(0.334538\pi\)
\(374\) 0 0
\(375\) − 269.538i − 0.718768i
\(376\) 0 0
\(377\) 700.841 1.85900
\(378\) 0 0
\(379\) − 68.7286i − 0.181342i −0.995881 0.0906709i \(-0.971099\pi\)
0.995881 0.0906709i \(-0.0289012\pi\)
\(380\) 0 0
\(381\) −295.704 −0.776126
\(382\) 0 0
\(383\) 155.858i 0.406939i 0.979081 + 0.203470i \(0.0652218\pi\)
−0.979081 + 0.203470i \(0.934778\pi\)
\(384\) 0 0
\(385\) −978.958 −2.54275
\(386\) 0 0
\(387\) 200.594i 0.518332i
\(388\) 0 0
\(389\) 481.792 1.23854 0.619269 0.785179i \(-0.287429\pi\)
0.619269 + 0.785179i \(0.287429\pi\)
\(390\) 0 0
\(391\) 112.105i 0.286714i
\(392\) 0 0
\(393\) −310.002 −0.788810
\(394\) 0 0
\(395\) − 7.29179i − 0.0184602i
\(396\) 0 0
\(397\) 422.315 1.06377 0.531883 0.846818i \(-0.321485\pi\)
0.531883 + 0.846818i \(0.321485\pi\)
\(398\) 0 0
\(399\) − 302.962i − 0.759303i
\(400\) 0 0
\(401\) −641.025 −1.59857 −0.799283 0.600955i \(-0.794787\pi\)
−0.799283 + 0.600955i \(0.794787\pi\)
\(402\) 0 0
\(403\) 380.566i 0.944333i
\(404\) 0 0
\(405\) 74.6328 0.184278
\(406\) 0 0
\(407\) − 160.507i − 0.394365i
\(408\) 0 0
\(409\) −278.562 −0.681081 −0.340541 0.940230i \(-0.610610\pi\)
−0.340541 + 0.940230i \(0.610610\pi\)
\(410\) 0 0
\(411\) 161.440i 0.392798i
\(412\) 0 0
\(413\) 531.807 1.28767
\(414\) 0 0
\(415\) 192.496i 0.463845i
\(416\) 0 0
\(417\) 385.340 0.924077
\(418\) 0 0
\(419\) − 505.611i − 1.20671i −0.797473 0.603354i \(-0.793831\pi\)
0.797473 0.603354i \(-0.206169\pi\)
\(420\) 0 0
\(421\) 179.846 0.427189 0.213594 0.976922i \(-0.431483\pi\)
0.213594 + 0.976922i \(0.431483\pi\)
\(422\) 0 0
\(423\) 59.8097i 0.141394i
\(424\) 0 0
\(425\) 889.526 2.09300
\(426\) 0 0
\(427\) − 406.262i − 0.951432i
\(428\) 0 0
\(429\) 407.935 0.950898
\(430\) 0 0
\(431\) − 580.645i − 1.34720i −0.739094 0.673602i \(-0.764746\pi\)
0.739094 0.673602i \(-0.235254\pi\)
\(432\) 0 0
\(433\) −425.621 −0.982959 −0.491479 0.870889i \(-0.663544\pi\)
−0.491479 + 0.870889i \(0.663544\pi\)
\(434\) 0 0
\(435\) 589.727i 1.35570i
\(436\) 0 0
\(437\) −112.764 −0.258041
\(438\) 0 0
\(439\) 129.991i 0.296107i 0.988979 + 0.148053i \(0.0473008\pi\)
−0.988979 + 0.148053i \(0.952699\pi\)
\(440\) 0 0
\(441\) 72.6069 0.164641
\(442\) 0 0
\(443\) 205.564i 0.464027i 0.972713 + 0.232014i \(0.0745314\pi\)
−0.972713 + 0.232014i \(0.925469\pi\)
\(444\) 0 0
\(445\) −135.242 −0.303915
\(446\) 0 0
\(447\) 162.820i 0.364251i
\(448\) 0 0
\(449\) 415.190 0.924698 0.462349 0.886698i \(-0.347007\pi\)
0.462349 + 0.886698i \(0.347007\pi\)
\(450\) 0 0
\(451\) − 496.073i − 1.09994i
\(452\) 0 0
\(453\) 142.042 0.313559
\(454\) 0 0
\(455\) 1211.06i 2.66167i
\(456\) 0 0
\(457\) −598.927 −1.31056 −0.655281 0.755385i \(-0.727450\pi\)
−0.655281 + 0.755385i \(0.727450\pi\)
\(458\) 0 0
\(459\) 105.610i 0.230086i
\(460\) 0 0
\(461\) −376.804 −0.817361 −0.408681 0.912677i \(-0.634011\pi\)
−0.408681 + 0.912677i \(0.634011\pi\)
\(462\) 0 0
\(463\) 218.577i 0.472088i 0.971742 + 0.236044i \(0.0758510\pi\)
−0.971742 + 0.236044i \(0.924149\pi\)
\(464\) 0 0
\(465\) −320.230 −0.688666
\(466\) 0 0
\(467\) 443.608i 0.949911i 0.880010 + 0.474955i \(0.157536\pi\)
−0.880010 + 0.474955i \(0.842464\pi\)
\(468\) 0 0
\(469\) 640.066 1.36475
\(470\) 0 0
\(471\) − 98.2385i − 0.208574i
\(472\) 0 0
\(473\) −922.597 −1.95052
\(474\) 0 0
\(475\) 894.751i 1.88369i
\(476\) 0 0
\(477\) 52.8988 0.110899
\(478\) 0 0
\(479\) 222.133i 0.463743i 0.972746 + 0.231871i \(0.0744849\pi\)
−0.972746 + 0.231871i \(0.925515\pi\)
\(480\) 0 0
\(481\) −198.561 −0.412809
\(482\) 0 0
\(483\) − 81.7387i − 0.169231i
\(484\) 0 0
\(485\) 1561.54 3.21968
\(486\) 0 0
\(487\) 403.795i 0.829148i 0.910016 + 0.414574i \(0.136069\pi\)
−0.910016 + 0.414574i \(0.863931\pi\)
\(488\) 0 0
\(489\) 328.376 0.671526
\(490\) 0 0
\(491\) − 808.347i − 1.64633i −0.567803 0.823164i \(-0.692207\pi\)
0.567803 0.823164i \(-0.307793\pi\)
\(492\) 0 0
\(493\) −834.498 −1.69269
\(494\) 0 0
\(495\) 343.260i 0.693454i
\(496\) 0 0
\(497\) −144.722 −0.291192
\(498\) 0 0
\(499\) 132.172i 0.264874i 0.991191 + 0.132437i \(0.0422802\pi\)
−0.991191 + 0.132437i \(0.957720\pi\)
\(500\) 0 0
\(501\) 541.745 1.08133
\(502\) 0 0
\(503\) 92.8360i 0.184565i 0.995733 + 0.0922823i \(0.0294162\pi\)
−0.995733 + 0.0922823i \(0.970584\pi\)
\(504\) 0 0
\(505\) 554.329 1.09768
\(506\) 0 0
\(507\) − 211.936i − 0.418020i
\(508\) 0 0
\(509\) −927.417 −1.82204 −0.911018 0.412366i \(-0.864703\pi\)
−0.911018 + 0.412366i \(0.864703\pi\)
\(510\) 0 0
\(511\) − 870.231i − 1.70300i
\(512\) 0 0
\(513\) −106.230 −0.207076
\(514\) 0 0
\(515\) 1379.15i 2.67796i
\(516\) 0 0
\(517\) −275.084 −0.532077
\(518\) 0 0
\(519\) − 316.165i − 0.609182i
\(520\) 0 0
\(521\) 527.145 1.01179 0.505897 0.862594i \(-0.331162\pi\)
0.505897 + 0.862594i \(0.331162\pi\)
\(522\) 0 0
\(523\) 972.249i 1.85898i 0.368843 + 0.929492i \(0.379754\pi\)
−0.368843 + 0.929492i \(0.620246\pi\)
\(524\) 0 0
\(525\) −648.575 −1.23538
\(526\) 0 0
\(527\) − 453.143i − 0.859854i
\(528\) 0 0
\(529\) 498.577 0.942489
\(530\) 0 0
\(531\) − 186.472i − 0.351171i
\(532\) 0 0
\(533\) −613.687 −1.15138
\(534\) 0 0
\(535\) 664.619i 1.24228i
\(536\) 0 0
\(537\) 34.2567 0.0637927
\(538\) 0 0
\(539\) 333.942i 0.619559i
\(540\) 0 0
\(541\) −4.66591 −0.00862461 −0.00431231 0.999991i \(-0.501373\pi\)
−0.00431231 + 0.999991i \(0.501373\pi\)
\(542\) 0 0
\(543\) 460.293i 0.847685i
\(544\) 0 0
\(545\) −222.888 −0.408968
\(546\) 0 0
\(547\) − 386.796i − 0.707123i −0.935411 0.353561i \(-0.884971\pi\)
0.935411 0.353561i \(-0.115029\pi\)
\(548\) 0 0
\(549\) −142.451 −0.259473
\(550\) 0 0
\(551\) − 839.399i − 1.52341i
\(552\) 0 0
\(553\) −7.52332 −0.0136046
\(554\) 0 0
\(555\) − 167.081i − 0.301046i
\(556\) 0 0
\(557\) −219.630 −0.394308 −0.197154 0.980373i \(-0.563170\pi\)
−0.197154 + 0.980373i \(0.563170\pi\)
\(558\) 0 0
\(559\) 1141.34i 2.04175i
\(560\) 0 0
\(561\) −485.732 −0.865833
\(562\) 0 0
\(563\) − 266.807i − 0.473902i −0.971522 0.236951i \(-0.923852\pi\)
0.971522 0.236951i \(-0.0761480\pi\)
\(564\) 0 0
\(565\) −655.904 −1.16089
\(566\) 0 0
\(567\) − 77.0025i − 0.135807i
\(568\) 0 0
\(569\) −972.046 −1.70834 −0.854170 0.519993i \(-0.825934\pi\)
−0.854170 + 0.519993i \(0.825934\pi\)
\(570\) 0 0
\(571\) − 340.194i − 0.595786i −0.954599 0.297893i \(-0.903716\pi\)
0.954599 0.297893i \(-0.0962838\pi\)
\(572\) 0 0
\(573\) 498.875 0.870637
\(574\) 0 0
\(575\) 241.402i 0.419830i
\(576\) 0 0
\(577\) 65.0584 0.112753 0.0563764 0.998410i \(-0.482045\pi\)
0.0563764 + 0.998410i \(0.482045\pi\)
\(578\) 0 0
\(579\) − 486.723i − 0.840628i
\(580\) 0 0
\(581\) 198.608 0.341838
\(582\) 0 0
\(583\) 243.299i 0.417322i
\(584\) 0 0
\(585\) 424.643 0.725886
\(586\) 0 0
\(587\) − 162.144i − 0.276226i −0.990417 0.138113i \(-0.955896\pi\)
0.990417 0.138113i \(-0.0441036\pi\)
\(588\) 0 0
\(589\) 455.805 0.773862
\(590\) 0 0
\(591\) − 58.6004i − 0.0991547i
\(592\) 0 0
\(593\) −793.227 −1.33765 −0.668825 0.743420i \(-0.733203\pi\)
−0.668825 + 0.743420i \(0.733203\pi\)
\(594\) 0 0
\(595\) − 1442.02i − 2.42356i
\(596\) 0 0
\(597\) 147.172 0.246520
\(598\) 0 0
\(599\) − 432.194i − 0.721525i −0.932658 0.360763i \(-0.882516\pi\)
0.932658 0.360763i \(-0.117484\pi\)
\(600\) 0 0
\(601\) 936.310 1.55792 0.778960 0.627074i \(-0.215748\pi\)
0.778960 + 0.627074i \(0.215748\pi\)
\(602\) 0 0
\(603\) − 224.431i − 0.372191i
\(604\) 0 0
\(605\) −575.366 −0.951018
\(606\) 0 0
\(607\) − 223.972i − 0.368982i −0.982834 0.184491i \(-0.940936\pi\)
0.982834 0.184491i \(-0.0590637\pi\)
\(608\) 0 0
\(609\) 608.453 0.999101
\(610\) 0 0
\(611\) 340.304i 0.556962i
\(612\) 0 0
\(613\) 1010.40 1.64828 0.824141 0.566385i \(-0.191658\pi\)
0.824141 + 0.566385i \(0.191658\pi\)
\(614\) 0 0
\(615\) − 516.391i − 0.839660i
\(616\) 0 0
\(617\) −400.206 −0.648633 −0.324316 0.945949i \(-0.605134\pi\)
−0.324316 + 0.945949i \(0.605134\pi\)
\(618\) 0 0
\(619\) − 417.349i − 0.674231i −0.941463 0.337116i \(-0.890549\pi\)
0.941463 0.337116i \(-0.109451\pi\)
\(620\) 0 0
\(621\) −28.6607 −0.0461525
\(622\) 0 0
\(623\) 139.536i 0.223975i
\(624\) 0 0
\(625\) 196.315 0.314104
\(626\) 0 0
\(627\) − 488.585i − 0.779243i
\(628\) 0 0
\(629\) 236.428 0.375880
\(630\) 0 0
\(631\) − 718.033i − 1.13793i −0.822362 0.568965i \(-0.807344\pi\)
0.822362 0.568965i \(-0.192656\pi\)
\(632\) 0 0
\(633\) −243.700 −0.384992
\(634\) 0 0
\(635\) − 1415.74i − 2.22951i
\(636\) 0 0
\(637\) 413.116 0.648534
\(638\) 0 0
\(639\) 50.7451i 0.0794133i
\(640\) 0 0
\(641\) 139.926 0.218293 0.109146 0.994026i \(-0.465188\pi\)
0.109146 + 0.994026i \(0.465188\pi\)
\(642\) 0 0
\(643\) 777.990i 1.20994i 0.796249 + 0.604969i \(0.206814\pi\)
−0.796249 + 0.604969i \(0.793186\pi\)
\(644\) 0 0
\(645\) −960.384 −1.48897
\(646\) 0 0
\(647\) − 673.336i − 1.04070i −0.853952 0.520352i \(-0.825801\pi\)
0.853952 0.520352i \(-0.174199\pi\)
\(648\) 0 0
\(649\) 857.642 1.32148
\(650\) 0 0
\(651\) 330.398i 0.507524i
\(652\) 0 0
\(653\) −223.410 −0.342128 −0.171064 0.985260i \(-0.554721\pi\)
−0.171064 + 0.985260i \(0.554721\pi\)
\(654\) 0 0
\(655\) − 1484.20i − 2.26595i
\(656\) 0 0
\(657\) −305.136 −0.464438
\(658\) 0 0
\(659\) − 166.367i − 0.252453i −0.992001 0.126227i \(-0.959713\pi\)
0.992001 0.126227i \(-0.0402866\pi\)
\(660\) 0 0
\(661\) 655.773 0.992092 0.496046 0.868296i \(-0.334785\pi\)
0.496046 + 0.868296i \(0.334785\pi\)
\(662\) 0 0
\(663\) 600.894i 0.906326i
\(664\) 0 0
\(665\) 1450.49 2.18119
\(666\) 0 0
\(667\) − 226.469i − 0.339533i
\(668\) 0 0
\(669\) 428.734 0.640857
\(670\) 0 0
\(671\) − 655.176i − 0.976418i
\(672\) 0 0
\(673\) −863.477 −1.28303 −0.641513 0.767112i \(-0.721693\pi\)
−0.641513 + 0.767112i \(0.721693\pi\)
\(674\) 0 0
\(675\) 227.415i 0.336911i
\(676\) 0 0
\(677\) −831.520 −1.22824 −0.614121 0.789212i \(-0.710489\pi\)
−0.614121 + 0.789212i \(0.710489\pi\)
\(678\) 0 0
\(679\) − 1611.13i − 2.37279i
\(680\) 0 0
\(681\) −224.737 −0.330010
\(682\) 0 0
\(683\) 308.273i 0.451351i 0.974202 + 0.225676i \(0.0724590\pi\)
−0.974202 + 0.225676i \(0.927541\pi\)
\(684\) 0 0
\(685\) −772.926 −1.12836
\(686\) 0 0
\(687\) 509.750i 0.741994i
\(688\) 0 0
\(689\) 300.982 0.436839
\(690\) 0 0
\(691\) 666.811i 0.964994i 0.875898 + 0.482497i \(0.160270\pi\)
−0.875898 + 0.482497i \(0.839730\pi\)
\(692\) 0 0
\(693\) 354.159 0.511052
\(694\) 0 0
\(695\) 1844.89i 2.65452i
\(696\) 0 0
\(697\) 730.722 1.04838
\(698\) 0 0
\(699\) 415.603i 0.594568i
\(700\) 0 0
\(701\) −242.205 −0.345514 −0.172757 0.984964i \(-0.555268\pi\)
−0.172757 + 0.984964i \(0.555268\pi\)
\(702\) 0 0
\(703\) 237.817i 0.338289i
\(704\) 0 0
\(705\) −286.351 −0.406171
\(706\) 0 0
\(707\) − 571.930i − 0.808953i
\(708\) 0 0
\(709\) −1380.24 −1.94674 −0.973371 0.229237i \(-0.926377\pi\)
−0.973371 + 0.229237i \(0.926377\pi\)
\(710\) 0 0
\(711\) 2.63796i 0.00371021i
\(712\) 0 0
\(713\) 122.976 0.172476
\(714\) 0 0
\(715\) 1953.07i 2.73157i
\(716\) 0 0
\(717\) −736.722 −1.02751
\(718\) 0 0
\(719\) − 968.365i − 1.34682i −0.739268 0.673411i \(-0.764828\pi\)
0.739268 0.673411i \(-0.235172\pi\)
\(720\) 0 0
\(721\) 1422.94 1.97357
\(722\) 0 0
\(723\) − 458.115i − 0.633631i
\(724\) 0 0
\(725\) −1796.97 −2.47858
\(726\) 0 0
\(727\) − 1017.52i − 1.39962i −0.714329 0.699810i \(-0.753268\pi\)
0.714329 0.699810i \(-0.246732\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) 0 0
\(731\) − 1359.00i − 1.85909i
\(732\) 0 0
\(733\) 1026.22 1.40003 0.700016 0.714127i \(-0.253176\pi\)
0.700016 + 0.714127i \(0.253176\pi\)
\(734\) 0 0
\(735\) 347.619i 0.472952i
\(736\) 0 0
\(737\) 1032.23 1.40059
\(738\) 0 0
\(739\) − 923.444i − 1.24959i −0.780791 0.624793i \(-0.785183\pi\)
0.780791 0.624793i \(-0.214817\pi\)
\(740\) 0 0
\(741\) −604.424 −0.815686
\(742\) 0 0
\(743\) 1267.12i 1.70542i 0.522387 + 0.852708i \(0.325042\pi\)
−0.522387 + 0.852708i \(0.674958\pi\)
\(744\) 0 0
\(745\) −779.533 −1.04635
\(746\) 0 0
\(747\) − 69.6394i − 0.0932254i
\(748\) 0 0
\(749\) 685.722 0.915517
\(750\) 0 0
\(751\) 791.442i 1.05385i 0.849912 + 0.526925i \(0.176655\pi\)
−0.849912 + 0.526925i \(0.823345\pi\)
\(752\) 0 0
\(753\) 383.000 0.508632
\(754\) 0 0
\(755\) 680.054i 0.900734i
\(756\) 0 0
\(757\) 333.990 0.441202 0.220601 0.975364i \(-0.429198\pi\)
0.220601 + 0.975364i \(0.429198\pi\)
\(758\) 0 0
\(759\) − 131.820i − 0.173675i
\(760\) 0 0
\(761\) 617.122 0.810936 0.405468 0.914109i \(-0.367109\pi\)
0.405468 + 0.914109i \(0.367109\pi\)
\(762\) 0 0
\(763\) 229.965i 0.301396i
\(764\) 0 0
\(765\) −505.626 −0.660949
\(766\) 0 0
\(767\) − 1060.98i − 1.38329i
\(768\) 0 0
\(769\) −657.604 −0.855142 −0.427571 0.903982i \(-0.640631\pi\)
−0.427571 + 0.903982i \(0.640631\pi\)
\(770\) 0 0
\(771\) − 150.802i − 0.195593i
\(772\) 0 0
\(773\) 1314.59 1.70064 0.850319 0.526268i \(-0.176409\pi\)
0.850319 + 0.526268i \(0.176409\pi\)
\(774\) 0 0
\(775\) − 975.779i − 1.25907i
\(776\) 0 0
\(777\) −172.386 −0.221861
\(778\) 0 0
\(779\) 735.015i 0.943536i
\(780\) 0 0
\(781\) −233.393 −0.298839
\(782\) 0 0
\(783\) − 213.347i − 0.272473i
\(784\) 0 0
\(785\) 470.336 0.599154
\(786\) 0 0
\(787\) − 301.018i − 0.382488i −0.981542 0.191244i \(-0.938748\pi\)
0.981542 0.191244i \(-0.0612522\pi\)
\(788\) 0 0
\(789\) −480.340 −0.608796
\(790\) 0 0
\(791\) 676.731i 0.855538i
\(792\) 0 0
\(793\) −810.512 −1.02208
\(794\) 0 0
\(795\) 253.263i 0.318570i
\(796\) 0 0
\(797\) 476.193 0.597481 0.298741 0.954334i \(-0.403433\pi\)
0.298741 + 0.954334i \(0.403433\pi\)
\(798\) 0 0
\(799\) − 405.202i − 0.507137i
\(800\) 0 0
\(801\) 48.9267 0.0610821
\(802\) 0 0
\(803\) − 1403.42i − 1.74772i
\(804\) 0 0
\(805\) 391.340 0.486136
\(806\) 0 0
\(807\) − 86.2570i − 0.106886i
\(808\) 0 0
\(809\) −692.874 −0.856457 −0.428228 0.903671i \(-0.640862\pi\)
−0.428228 + 0.903671i \(0.640862\pi\)
\(810\) 0 0
\(811\) − 785.074i − 0.968032i −0.875059 0.484016i \(-0.839178\pi\)
0.875059 0.484016i \(-0.160822\pi\)
\(812\) 0 0
\(813\) −54.2383 −0.0667138
\(814\) 0 0
\(815\) 1572.17i 1.92904i
\(816\) 0 0
\(817\) 1366.98 1.67317
\(818\) 0 0
\(819\) − 438.126i − 0.534953i
\(820\) 0 0
\(821\) −527.681 −0.642729 −0.321365 0.946956i \(-0.604141\pi\)
−0.321365 + 0.946956i \(0.604141\pi\)
\(822\) 0 0
\(823\) 228.552i 0.277706i 0.990313 + 0.138853i \(0.0443416\pi\)
−0.990313 + 0.138853i \(0.955658\pi\)
\(824\) 0 0
\(825\) −1045.95 −1.26782
\(826\) 0 0
\(827\) − 608.409i − 0.735683i −0.929889 0.367841i \(-0.880097\pi\)
0.929889 0.367841i \(-0.119903\pi\)
\(828\) 0 0
\(829\) 151.637 0.182915 0.0914577 0.995809i \(-0.470847\pi\)
0.0914577 + 0.995809i \(0.470847\pi\)
\(830\) 0 0
\(831\) − 137.080i − 0.164958i
\(832\) 0 0
\(833\) −491.901 −0.590518
\(834\) 0 0
\(835\) 2593.71i 3.10624i
\(836\) 0 0
\(837\) 115.850 0.138411
\(838\) 0 0
\(839\) 653.351i 0.778725i 0.921085 + 0.389363i \(0.127305\pi\)
−0.921085 + 0.389363i \(0.872695\pi\)
\(840\) 0 0
\(841\) 844.805 1.00452
\(842\) 0 0
\(843\) 183.927i 0.218182i
\(844\) 0 0
\(845\) 1014.68 1.20081
\(846\) 0 0
\(847\) 593.635i 0.700868i
\(848\) 0 0
\(849\) −468.088 −0.551340
\(850\) 0 0
\(851\) 64.1628i 0.0753969i
\(852\) 0 0
\(853\) 1372.25 1.60873 0.804367 0.594132i \(-0.202505\pi\)
0.804367 + 0.594132i \(0.202505\pi\)
\(854\) 0 0
\(855\) − 508.596i − 0.594849i
\(856\) 0 0
\(857\) −1453.61 −1.69617 −0.848083 0.529863i \(-0.822243\pi\)
−0.848083 + 0.529863i \(0.822243\pi\)
\(858\) 0 0
\(859\) − 962.466i − 1.12045i −0.828341 0.560225i \(-0.810715\pi\)
0.828341 0.560225i \(-0.189285\pi\)
\(860\) 0 0
\(861\) −532.787 −0.618801
\(862\) 0 0
\(863\) − 695.890i − 0.806362i −0.915120 0.403181i \(-0.867905\pi\)
0.915120 0.403181i \(-0.132095\pi\)
\(864\) 0 0
\(865\) 1513.70 1.74994
\(866\) 0 0
\(867\) − 214.927i − 0.247897i
\(868\) 0 0
\(869\) −12.1328 −0.0139618
\(870\) 0 0
\(871\) − 1276.96i − 1.46609i
\(872\) 0 0
\(873\) −564.922 −0.647104
\(874\) 0 0
\(875\) − 1331.44i − 1.52165i
\(876\) 0 0
\(877\) 338.191 0.385622 0.192811 0.981236i \(-0.438240\pi\)
0.192811 + 0.981236i \(0.438240\pi\)
\(878\) 0 0
\(879\) 42.0000i 0.0477815i
\(880\) 0 0
\(881\) −459.985 −0.522117 −0.261058 0.965323i \(-0.584072\pi\)
−0.261058 + 0.965323i \(0.584072\pi\)
\(882\) 0 0
\(883\) 1208.31i 1.36842i 0.729286 + 0.684209i \(0.239853\pi\)
−0.729286 + 0.684209i \(0.760147\pi\)
\(884\) 0 0
\(885\) 892.769 1.00878
\(886\) 0 0
\(887\) 1500.73i 1.69192i 0.533250 + 0.845958i \(0.320971\pi\)
−0.533250 + 0.845958i \(0.679029\pi\)
\(888\) 0 0
\(889\) −1460.69 −1.64307
\(890\) 0 0
\(891\) − 124.182i − 0.139373i
\(892\) 0 0
\(893\) 407.582 0.456419
\(894\) 0 0
\(895\) 164.010i 0.183252i
\(896\) 0 0
\(897\) −163.073 −0.181798
\(898\) 0 0
\(899\) 915.415i 1.01826i
\(900\) 0 0
\(901\) −358.382 −0.397760
\(902\) 0 0
\(903\) 990.878i 1.09732i
\(904\) 0 0
\(905\) −2203.74 −2.43507
\(906\) 0 0
\(907\) − 665.128i − 0.733328i −0.930353 0.366664i \(-0.880500\pi\)
0.930353 0.366664i \(-0.119500\pi\)
\(908\) 0 0
\(909\) −200.540 −0.220616
\(910\) 0 0
\(911\) − 345.687i − 0.379459i −0.981836 0.189729i \(-0.939239\pi\)
0.981836 0.189729i \(-0.0607611\pi\)
\(912\) 0 0
\(913\) 320.294 0.350815
\(914\) 0 0
\(915\) − 682.010i − 0.745367i
\(916\) 0 0
\(917\) −1531.32 −1.66993
\(918\) 0 0
\(919\) 1102.20i 1.19934i 0.800246 + 0.599671i \(0.204702\pi\)
−0.800246 + 0.599671i \(0.795298\pi\)
\(920\) 0 0
\(921\) 212.808 0.231062
\(922\) 0 0
\(923\) 288.728i 0.312815i
\(924\) 0 0
\(925\) 509.115 0.550394
\(926\) 0 0
\(927\) − 498.938i − 0.538228i
\(928\) 0 0
\(929\) −1256.03 −1.35202 −0.676010 0.736893i \(-0.736292\pi\)
−0.676010 + 0.736893i \(0.736292\pi\)
\(930\) 0 0
\(931\) − 494.790i − 0.531461i
\(932\) 0 0
\(933\) −188.461 −0.201995
\(934\) 0 0
\(935\) − 2325.54i − 2.48720i
\(936\) 0 0
\(937\) −822.214 −0.877496 −0.438748 0.898610i \(-0.644578\pi\)
−0.438748 + 0.898610i \(0.644578\pi\)
\(938\) 0 0
\(939\) 90.4505i 0.0963264i
\(940\) 0 0
\(941\) −1185.72 −1.26006 −0.630031 0.776570i \(-0.716958\pi\)
−0.630031 + 0.776570i \(0.716958\pi\)
\(942\) 0 0
\(943\) 198.306i 0.210293i
\(944\) 0 0
\(945\) 368.664 0.390121
\(946\) 0 0
\(947\) 1335.34i 1.41007i 0.709170 + 0.705037i \(0.249070\pi\)
−0.709170 + 0.705037i \(0.750930\pi\)
\(948\) 0 0
\(949\) −1736.15 −1.82945
\(950\) 0 0
\(951\) 163.376i 0.171794i
\(952\) 0 0
\(953\) 663.428 0.696147 0.348073 0.937467i \(-0.386836\pi\)
0.348073 + 0.937467i \(0.386836\pi\)
\(954\) 0 0
\(955\) 2388.46i 2.50100i
\(956\) 0 0
\(957\) 981.249 1.02534
\(958\) 0 0
\(959\) 797.468i 0.831562i
\(960\) 0 0
\(961\) 463.918 0.482745
\(962\) 0 0
\(963\) − 240.440i − 0.249678i
\(964\) 0 0
\(965\) 2330.28 2.41480
\(966\) 0 0
\(967\) 155.996i 0.161320i 0.996742 + 0.0806600i \(0.0257028\pi\)
−0.996742 + 0.0806600i \(0.974297\pi\)
\(968\) 0 0
\(969\) 719.692 0.742716
\(970\) 0 0
\(971\) 1280.67i 1.31892i 0.751740 + 0.659460i \(0.229215\pi\)
−0.751740 + 0.659460i \(0.770785\pi\)
\(972\) 0 0
\(973\) 1903.47 1.95629
\(974\) 0 0
\(975\) 1293.94i 1.32712i
\(976\) 0 0
\(977\) −217.470 −0.222590 −0.111295 0.993787i \(-0.535500\pi\)
−0.111295 + 0.993787i \(0.535500\pi\)
\(978\) 0 0
\(979\) 225.030i 0.229857i
\(980\) 0 0
\(981\) 80.6344 0.0821962
\(982\) 0 0
\(983\) 1027.63i 1.04540i 0.852515 + 0.522702i \(0.175076\pi\)
−0.852515 + 0.522702i \(0.824924\pi\)
\(984\) 0 0
\(985\) 280.561 0.284833
\(986\) 0 0
\(987\) 295.443i 0.299334i
\(988\) 0 0
\(989\) 368.809 0.372911
\(990\) 0 0
\(991\) 797.792i 0.805038i 0.915412 + 0.402519i \(0.131865\pi\)
−0.915412 + 0.402519i \(0.868135\pi\)
\(992\) 0 0
\(993\) −703.397 −0.708356
\(994\) 0 0
\(995\) 704.616i 0.708157i
\(996\) 0 0
\(997\) −1237.58 −1.24131 −0.620654 0.784085i \(-0.713133\pi\)
−0.620654 + 0.784085i \(0.713133\pi\)
\(998\) 0 0
\(999\) 60.4450i 0.0605055i
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.127.4 yes 8
3.2 odd 2 1152.3.g.c.127.1 8
4.3 odd 2 inner 384.3.g.b.127.8 yes 8
8.3 odd 2 384.3.g.a.127.1 8
8.5 even 2 384.3.g.a.127.5 yes 8
12.11 even 2 1152.3.g.c.127.2 8
16.3 odd 4 768.3.b.f.127.1 8
16.5 even 4 768.3.b.f.127.4 8
16.11 odd 4 768.3.b.e.127.8 8
16.13 even 4 768.3.b.e.127.5 8
24.5 odd 2 1152.3.g.f.127.7 8
24.11 even 2 1152.3.g.f.127.8 8
48.5 odd 4 2304.3.b.q.127.1 8
48.11 even 4 2304.3.b.t.127.1 8
48.29 odd 4 2304.3.b.t.127.8 8
48.35 even 4 2304.3.b.q.127.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.3.g.a.127.1 8 8.3 odd 2
384.3.g.a.127.5 yes 8 8.5 even 2
384.3.g.b.127.4 yes 8 1.1 even 1 trivial
384.3.g.b.127.8 yes 8 4.3 odd 2 inner
768.3.b.e.127.5 8 16.13 even 4
768.3.b.e.127.8 8 16.11 odd 4
768.3.b.f.127.1 8 16.3 odd 4
768.3.b.f.127.4 8 16.5 even 4
1152.3.g.c.127.1 8 3.2 odd 2
1152.3.g.c.127.2 8 12.11 even 2
1152.3.g.f.127.7 8 24.5 odd 2
1152.3.g.f.127.8 8 24.11 even 2
2304.3.b.q.127.1 8 48.5 odd 4
2304.3.b.q.127.8 8 48.35 even 4
2304.3.b.t.127.1 8 48.11 even 4
2304.3.b.t.127.8 8 48.29 odd 4