Properties

Label 384.3.g.b.127.6
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 127.6
Root \(-0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 384.127
Dual form 384.3.g.b.127.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.73205i q^{3} +1.36433 q^{5} +1.24213i q^{7} -3.00000 q^{9} +O(q^{10})\) \(q+1.73205i q^{3} +1.36433 q^{5} +1.24213i q^{7} -3.00000 q^{9} +5.79796i q^{11} +16.3830 q^{13} +2.36308i q^{15} -5.01086 q^{17} +26.1835i q^{19} -2.15142 q^{21} +25.1117i q^{23} -23.1386 q^{25} -5.19615i q^{27} +32.7743 q^{29} -1.01836i q^{31} -10.0424 q^{33} +1.69466i q^{35} +14.9948 q^{37} +28.3762i q^{39} -72.5212 q^{41} +33.4922i q^{43} -4.09298 q^{45} +66.5640i q^{47} +47.4571 q^{49} -8.67906i q^{51} -54.6513 q^{53} +7.91030i q^{55} -45.3511 q^{57} +20.5880i q^{59} +111.026 q^{61} -3.72638i q^{63} +22.3518 q^{65} +60.9540i q^{67} -43.4947 q^{69} -80.4576i q^{71} +30.0525 q^{73} -40.0773i q^{75} -7.20179 q^{77} -80.9441i q^{79} +9.00000 q^{81} -113.958i q^{83} -6.83644 q^{85} +56.7667i q^{87} -21.0637 q^{89} +20.3498i q^{91} +1.76386 q^{93} +35.7228i q^{95} +160.594 q^{97} -17.3939i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 16q^{5} - 24q^{9} + O(q^{10}) \) \( 8q + 16q^{5} - 24q^{9} - 48q^{13} + 16q^{17} - 8q^{25} + 80q^{29} + 16q^{37} + 80q^{41} - 48q^{45} - 88q^{49} - 176q^{53} + 96q^{57} + 272q^{61} - 160q^{65} - 16q^{73} - 320q^{77} + 72q^{81} - 32q^{85} - 240q^{89} + 192q^{93} + 400q^{97} + O(q^{100}) \)

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
\(5\) 1.36433 0.272865 0.136433 0.990649i \(-0.456436\pi\)
0.136433 + 0.990649i \(0.456436\pi\)
\(6\) 0 0
\(7\) 1.24213i 0.177446i 0.996056 + 0.0887232i \(0.0282787\pi\)
−0.996056 + 0.0887232i \(0.971721\pi\)
\(8\) 0 0
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) 5.79796i 0.527087i 0.964647 + 0.263544i \(0.0848913\pi\)
−0.964647 + 0.263544i \(0.915109\pi\)
\(12\) 0 0
\(13\) 16.3830 1.26023 0.630116 0.776501i \(-0.283007\pi\)
0.630116 + 0.776501i \(0.283007\pi\)
\(14\) 0 0
\(15\) 2.36308i 0.157539i
\(16\) 0 0
\(17\) −5.01086 −0.294756 −0.147378 0.989080i \(-0.547083\pi\)
−0.147378 + 0.989080i \(0.547083\pi\)
\(18\) 0 0
\(19\) 26.1835i 1.37808i 0.724725 + 0.689039i \(0.241967\pi\)
−0.724725 + 0.689039i \(0.758033\pi\)
\(20\) 0 0
\(21\) −2.15142 −0.102449
\(22\) 0 0
\(23\) 25.1117i 1.09181i 0.837847 + 0.545906i \(0.183814\pi\)
−0.837847 + 0.545906i \(0.816186\pi\)
\(24\) 0 0
\(25\) −23.1386 −0.925545
\(26\) 0 0
\(27\) − 5.19615i − 0.192450i
\(28\) 0 0
\(29\) 32.7743 1.13015 0.565074 0.825040i \(-0.308848\pi\)
0.565074 + 0.825040i \(0.308848\pi\)
\(30\) 0 0
\(31\) − 1.01836i − 0.0328504i −0.999865 0.0164252i \(-0.994771\pi\)
0.999865 0.0164252i \(-0.00522854\pi\)
\(32\) 0 0
\(33\) −10.0424 −0.304314
\(34\) 0 0
\(35\) 1.69466i 0.0484189i
\(36\) 0 0
\(37\) 14.9948 0.405264 0.202632 0.979255i \(-0.435050\pi\)
0.202632 + 0.979255i \(0.435050\pi\)
\(38\) 0 0
\(39\) 28.3762i 0.727595i
\(40\) 0 0
\(41\) −72.5212 −1.76881 −0.884405 0.466720i \(-0.845435\pi\)
−0.884405 + 0.466720i \(0.845435\pi\)
\(42\) 0 0
\(43\) 33.4922i 0.778888i 0.921050 + 0.389444i \(0.127333\pi\)
−0.921050 + 0.389444i \(0.872667\pi\)
\(44\) 0 0
\(45\) −4.09298 −0.0909550
\(46\) 0 0
\(47\) 66.5640i 1.41626i 0.706085 + 0.708128i \(0.250460\pi\)
−0.706085 + 0.708128i \(0.749540\pi\)
\(48\) 0 0
\(49\) 47.4571 0.968513
\(50\) 0 0
\(51\) − 8.67906i − 0.170178i
\(52\) 0 0
\(53\) −54.6513 −1.03116 −0.515579 0.856842i \(-0.672423\pi\)
−0.515579 + 0.856842i \(0.672423\pi\)
\(54\) 0 0
\(55\) 7.91030i 0.143824i
\(56\) 0 0
\(57\) −45.3511 −0.795633
\(58\) 0 0
\(59\) 20.5880i 0.348949i 0.984662 + 0.174474i \(0.0558226\pi\)
−0.984662 + 0.174474i \(0.944177\pi\)
\(60\) 0 0
\(61\) 111.026 1.82010 0.910050 0.414499i \(-0.136043\pi\)
0.910050 + 0.414499i \(0.136043\pi\)
\(62\) 0 0
\(63\) − 3.72638i − 0.0591488i
\(64\) 0 0
\(65\) 22.3518 0.343873
\(66\) 0 0
\(67\) 60.9540i 0.909762i 0.890552 + 0.454881i \(0.150318\pi\)
−0.890552 + 0.454881i \(0.849682\pi\)
\(68\) 0 0
\(69\) −43.4947 −0.630358
\(70\) 0 0
\(71\) − 80.4576i − 1.13320i −0.823991 0.566602i \(-0.808258\pi\)
0.823991 0.566602i \(-0.191742\pi\)
\(72\) 0 0
\(73\) 30.0525 0.411679 0.205839 0.978586i \(-0.434008\pi\)
0.205839 + 0.978586i \(0.434008\pi\)
\(74\) 0 0
\(75\) − 40.0773i − 0.534363i
\(76\) 0 0
\(77\) −7.20179 −0.0935298
\(78\) 0 0
\(79\) − 80.9441i − 1.02461i −0.858804 0.512304i \(-0.828792\pi\)
0.858804 0.512304i \(-0.171208\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) − 113.958i − 1.37299i −0.727134 0.686496i \(-0.759148\pi\)
0.727134 0.686496i \(-0.240852\pi\)
\(84\) 0 0
\(85\) −6.83644 −0.0804288
\(86\) 0 0
\(87\) 56.7667i 0.652491i
\(88\) 0 0
\(89\) −21.0637 −0.236671 −0.118335 0.992974i \(-0.537756\pi\)
−0.118335 + 0.992974i \(0.537756\pi\)
\(90\) 0 0
\(91\) 20.3498i 0.223624i
\(92\) 0 0
\(93\) 1.76386 0.0189662
\(94\) 0 0
\(95\) 35.7228i 0.376029i
\(96\) 0 0
\(97\) 160.594 1.65561 0.827806 0.561014i \(-0.189589\pi\)
0.827806 + 0.561014i \(0.189589\pi\)
\(98\) 0 0
\(99\) − 17.3939i − 0.175696i
\(100\) 0 0
\(101\) −76.2681 −0.755130 −0.377565 0.925983i \(-0.623239\pi\)
−0.377565 + 0.925983i \(0.623239\pi\)
\(102\) 0 0
\(103\) − 182.763i − 1.77440i −0.461383 0.887201i \(-0.652647\pi\)
0.461383 0.887201i \(-0.347353\pi\)
\(104\) 0 0
\(105\) −2.93524 −0.0279547
\(106\) 0 0
\(107\) 31.8533i 0.297694i 0.988860 + 0.148847i \(0.0475562\pi\)
−0.988860 + 0.148847i \(0.952444\pi\)
\(108\) 0 0
\(109\) 11.3289 0.103935 0.0519676 0.998649i \(-0.483451\pi\)
0.0519676 + 0.998649i \(0.483451\pi\)
\(110\) 0 0
\(111\) 25.9717i 0.233979i
\(112\) 0 0
\(113\) 49.9587 0.442113 0.221056 0.975261i \(-0.429050\pi\)
0.221056 + 0.975261i \(0.429050\pi\)
\(114\) 0 0
\(115\) 34.2605i 0.297917i
\(116\) 0 0
\(117\) −49.1490 −0.420077
\(118\) 0 0
\(119\) − 6.22412i − 0.0523035i
\(120\) 0 0
\(121\) 87.3837 0.722179
\(122\) 0 0
\(123\) − 125.610i − 1.02122i
\(124\) 0 0
\(125\) −65.6767 −0.525414
\(126\) 0 0
\(127\) − 208.236i − 1.63965i −0.572614 0.819825i \(-0.694071\pi\)
0.572614 0.819825i \(-0.305929\pi\)
\(128\) 0 0
\(129\) −58.0102 −0.449691
\(130\) 0 0
\(131\) − 220.549i − 1.68358i −0.539804 0.841791i \(-0.681502\pi\)
0.539804 0.841791i \(-0.318498\pi\)
\(132\) 0 0
\(133\) −32.5231 −0.244535
\(134\) 0 0
\(135\) − 7.08924i − 0.0525129i
\(136\) 0 0
\(137\) 15.2664 0.111433 0.0557167 0.998447i \(-0.482256\pi\)
0.0557167 + 0.998447i \(0.482256\pi\)
\(138\) 0 0
\(139\) − 86.7117i − 0.623825i −0.950111 0.311912i \(-0.899030\pi\)
0.950111 0.311912i \(-0.100970\pi\)
\(140\) 0 0
\(141\) −115.292 −0.817675
\(142\) 0 0
\(143\) 94.9881i 0.664252i
\(144\) 0 0
\(145\) 44.7148 0.308378
\(146\) 0 0
\(147\) 82.1982i 0.559171i
\(148\) 0 0
\(149\) −146.849 −0.985561 −0.492780 0.870154i \(-0.664019\pi\)
−0.492780 + 0.870154i \(0.664019\pi\)
\(150\) 0 0
\(151\) 195.933i 1.29757i 0.760972 + 0.648785i \(0.224723\pi\)
−0.760972 + 0.648785i \(0.775277\pi\)
\(152\) 0 0
\(153\) 15.0326 0.0982522
\(154\) 0 0
\(155\) − 1.38938i − 0.00896374i
\(156\) 0 0
\(157\) 4.65454 0.0296468 0.0148234 0.999890i \(-0.495281\pi\)
0.0148234 + 0.999890i \(0.495281\pi\)
\(158\) 0 0
\(159\) − 94.6589i − 0.595339i
\(160\) 0 0
\(161\) −31.1918 −0.193738
\(162\) 0 0
\(163\) 59.5489i 0.365331i 0.983175 + 0.182665i \(0.0584725\pi\)
−0.983175 + 0.182665i \(0.941528\pi\)
\(164\) 0 0
\(165\) −13.7010 −0.0830367
\(166\) 0 0
\(167\) − 209.012i − 1.25157i −0.779996 0.625785i \(-0.784779\pi\)
0.779996 0.625785i \(-0.215221\pi\)
\(168\) 0 0
\(169\) 99.4032 0.588185
\(170\) 0 0
\(171\) − 78.5504i − 0.459359i
\(172\) 0 0
\(173\) −96.7635 −0.559326 −0.279663 0.960098i \(-0.590223\pi\)
−0.279663 + 0.960098i \(0.590223\pi\)
\(174\) 0 0
\(175\) − 28.7411i − 0.164235i
\(176\) 0 0
\(177\) −35.6594 −0.201466
\(178\) 0 0
\(179\) − 49.5039i − 0.276558i −0.990393 0.138279i \(-0.955843\pi\)
0.990393 0.138279i \(-0.0441571\pi\)
\(180\) 0 0
\(181\) 141.417 0.781310 0.390655 0.920537i \(-0.372248\pi\)
0.390655 + 0.920537i \(0.372248\pi\)
\(182\) 0 0
\(183\) 192.303i 1.05084i
\(184\) 0 0
\(185\) 20.4578 0.110582
\(186\) 0 0
\(187\) − 29.0528i − 0.155362i
\(188\) 0 0
\(189\) 6.45427 0.0341496
\(190\) 0 0
\(191\) 116.994i 0.612533i 0.951946 + 0.306267i \(0.0990799\pi\)
−0.951946 + 0.306267i \(0.900920\pi\)
\(192\) 0 0
\(193\) −90.7357 −0.470133 −0.235067 0.971979i \(-0.575531\pi\)
−0.235067 + 0.971979i \(0.575531\pi\)
\(194\) 0 0
\(195\) 38.7144i 0.198535i
\(196\) 0 0
\(197\) 380.039 1.92913 0.964566 0.263840i \(-0.0849891\pi\)
0.964566 + 0.263840i \(0.0849891\pi\)
\(198\) 0 0
\(199\) − 77.6563i − 0.390233i −0.980780 0.195116i \(-0.937492\pi\)
0.980780 0.195116i \(-0.0625084\pi\)
\(200\) 0 0
\(201\) −105.576 −0.525251
\(202\) 0 0
\(203\) 40.7098i 0.200541i
\(204\) 0 0
\(205\) −98.9425 −0.482646
\(206\) 0 0
\(207\) − 75.3350i − 0.363937i
\(208\) 0 0
\(209\) −151.811 −0.726367
\(210\) 0 0
\(211\) 191.446i 0.907325i 0.891174 + 0.453662i \(0.149883\pi\)
−0.891174 + 0.453662i \(0.850117\pi\)
\(212\) 0 0
\(213\) 139.357 0.654256
\(214\) 0 0
\(215\) 45.6943i 0.212531i
\(216\) 0 0
\(217\) 1.26493 0.00582919
\(218\) 0 0
\(219\) 52.0525i 0.237683i
\(220\) 0 0
\(221\) −82.0930 −0.371462
\(222\) 0 0
\(223\) − 168.451i − 0.755387i −0.925931 0.377693i \(-0.876717\pi\)
0.925931 0.377693i \(-0.123283\pi\)
\(224\) 0 0
\(225\) 69.4158 0.308515
\(226\) 0 0
\(227\) 113.516i 0.500071i 0.968237 + 0.250036i \(0.0804424\pi\)
−0.968237 + 0.250036i \(0.919558\pi\)
\(228\) 0 0
\(229\) 117.618 0.513615 0.256808 0.966463i \(-0.417329\pi\)
0.256808 + 0.966463i \(0.417329\pi\)
\(230\) 0 0
\(231\) − 12.4739i − 0.0539994i
\(232\) 0 0
\(233\) 277.085 1.18921 0.594604 0.804019i \(-0.297309\pi\)
0.594604 + 0.804019i \(0.297309\pi\)
\(234\) 0 0
\(235\) 90.8150i 0.386447i
\(236\) 0 0
\(237\) 140.199 0.591558
\(238\) 0 0
\(239\) 343.072i 1.43545i 0.696327 + 0.717724i \(0.254816\pi\)
−0.696327 + 0.717724i \(0.745184\pi\)
\(240\) 0 0
\(241\) −328.140 −1.36157 −0.680787 0.732481i \(-0.738362\pi\)
−0.680787 + 0.732481i \(0.738362\pi\)
\(242\) 0 0
\(243\) 15.5885i 0.0641500i
\(244\) 0 0
\(245\) 64.7470 0.264273
\(246\) 0 0
\(247\) 428.964i 1.73670i
\(248\) 0 0
\(249\) 197.382 0.792697
\(250\) 0 0
\(251\) 452.914i 1.80444i 0.431279 + 0.902219i \(0.358063\pi\)
−0.431279 + 0.902219i \(0.641937\pi\)
\(252\) 0 0
\(253\) −145.596 −0.575480
\(254\) 0 0
\(255\) − 11.8411i − 0.0464356i
\(256\) 0 0
\(257\) −346.830 −1.34953 −0.674767 0.738031i \(-0.735756\pi\)
−0.674767 + 0.738031i \(0.735756\pi\)
\(258\) 0 0
\(259\) 18.6254i 0.0719127i
\(260\) 0 0
\(261\) −98.3229 −0.376716
\(262\) 0 0
\(263\) − 402.440i − 1.53019i −0.643917 0.765095i \(-0.722692\pi\)
0.643917 0.765095i \(-0.277308\pi\)
\(264\) 0 0
\(265\) −74.5622 −0.281367
\(266\) 0 0
\(267\) − 36.4834i − 0.136642i
\(268\) 0 0
\(269\) 321.562 1.19540 0.597699 0.801721i \(-0.296082\pi\)
0.597699 + 0.801721i \(0.296082\pi\)
\(270\) 0 0
\(271\) 456.902i 1.68599i 0.537924 + 0.842993i \(0.319209\pi\)
−0.537924 + 0.842993i \(0.680791\pi\)
\(272\) 0 0
\(273\) −35.2468 −0.129109
\(274\) 0 0
\(275\) − 134.157i − 0.487843i
\(276\) 0 0
\(277\) 329.543 1.18969 0.594843 0.803842i \(-0.297214\pi\)
0.594843 + 0.803842i \(0.297214\pi\)
\(278\) 0 0
\(279\) 3.05509i 0.0109501i
\(280\) 0 0
\(281\) −175.064 −0.623005 −0.311503 0.950245i \(-0.600832\pi\)
−0.311503 + 0.950245i \(0.600832\pi\)
\(282\) 0 0
\(283\) − 150.298i − 0.531087i −0.964099 0.265544i \(-0.914449\pi\)
0.964099 0.265544i \(-0.0855515\pi\)
\(284\) 0 0
\(285\) −61.8736 −0.217101
\(286\) 0 0
\(287\) − 90.0804i − 0.313869i
\(288\) 0 0
\(289\) −263.891 −0.913119
\(290\) 0 0
\(291\) 278.158i 0.955868i
\(292\) 0 0
\(293\) 160.435 0.547561 0.273781 0.961792i \(-0.411726\pi\)
0.273781 + 0.961792i \(0.411726\pi\)
\(294\) 0 0
\(295\) 28.0887i 0.0952159i
\(296\) 0 0
\(297\) 30.1271 0.101438
\(298\) 0 0
\(299\) 411.405i 1.37594i
\(300\) 0 0
\(301\) −41.6015 −0.138211
\(302\) 0 0
\(303\) − 132.100i − 0.435974i
\(304\) 0 0
\(305\) 151.476 0.496642
\(306\) 0 0
\(307\) − 168.120i − 0.547621i −0.961784 0.273811i \(-0.911716\pi\)
0.961784 0.273811i \(-0.0882841\pi\)
\(308\) 0 0
\(309\) 316.555 1.02445
\(310\) 0 0
\(311\) 470.376i 1.51246i 0.654305 + 0.756231i \(0.272961\pi\)
−0.654305 + 0.756231i \(0.727039\pi\)
\(312\) 0 0
\(313\) 19.4378 0.0621016 0.0310508 0.999518i \(-0.490115\pi\)
0.0310508 + 0.999518i \(0.490115\pi\)
\(314\) 0 0
\(315\) − 5.08399i − 0.0161396i
\(316\) 0 0
\(317\) −242.195 −0.764021 −0.382011 0.924158i \(-0.624768\pi\)
−0.382011 + 0.924158i \(0.624768\pi\)
\(318\) 0 0
\(319\) 190.024i 0.595686i
\(320\) 0 0
\(321\) −55.1715 −0.171874
\(322\) 0 0
\(323\) − 131.202i − 0.406197i
\(324\) 0 0
\(325\) −379.080 −1.16640
\(326\) 0 0
\(327\) 19.6223i 0.0600070i
\(328\) 0 0
\(329\) −82.6808 −0.251309
\(330\) 0 0
\(331\) − 440.951i − 1.33218i −0.745872 0.666090i \(-0.767967\pi\)
0.745872 0.666090i \(-0.232033\pi\)
\(332\) 0 0
\(333\) −44.9843 −0.135088
\(334\) 0 0
\(335\) 83.1612i 0.248242i
\(336\) 0 0
\(337\) −250.841 −0.744335 −0.372167 0.928166i \(-0.621385\pi\)
−0.372167 + 0.928166i \(0.621385\pi\)
\(338\) 0 0
\(339\) 86.5310i 0.255254i
\(340\) 0 0
\(341\) 5.90443 0.0173150
\(342\) 0 0
\(343\) 119.812i 0.349306i
\(344\) 0 0
\(345\) −59.3409 −0.172003
\(346\) 0 0
\(347\) − 16.1029i − 0.0464060i −0.999731 0.0232030i \(-0.992614\pi\)
0.999731 0.0232030i \(-0.00738641\pi\)
\(348\) 0 0
\(349\) −274.843 −0.787516 −0.393758 0.919214i \(-0.628825\pi\)
−0.393758 + 0.919214i \(0.628825\pi\)
\(350\) 0 0
\(351\) − 85.1287i − 0.242532i
\(352\) 0 0
\(353\) 165.428 0.468634 0.234317 0.972160i \(-0.424715\pi\)
0.234317 + 0.972160i \(0.424715\pi\)
\(354\) 0 0
\(355\) − 109.770i − 0.309212i
\(356\) 0 0
\(357\) 10.7805 0.0301974
\(358\) 0 0
\(359\) − 688.519i − 1.91788i −0.283607 0.958941i \(-0.591531\pi\)
0.283607 0.958941i \(-0.408469\pi\)
\(360\) 0 0
\(361\) −324.574 −0.899096
\(362\) 0 0
\(363\) 151.353i 0.416950i
\(364\) 0 0
\(365\) 41.0015 0.112333
\(366\) 0 0
\(367\) 102.170i 0.278393i 0.990265 + 0.139196i \(0.0444519\pi\)
−0.990265 + 0.139196i \(0.955548\pi\)
\(368\) 0 0
\(369\) 217.564 0.589603
\(370\) 0 0
\(371\) − 67.8838i − 0.182975i
\(372\) 0 0
\(373\) −294.317 −0.789052 −0.394526 0.918885i \(-0.629091\pi\)
−0.394526 + 0.918885i \(0.629091\pi\)
\(374\) 0 0
\(375\) − 113.755i − 0.303348i
\(376\) 0 0
\(377\) 536.942 1.42425
\(378\) 0 0
\(379\) − 81.1923i − 0.214228i −0.994247 0.107114i \(-0.965839\pi\)
0.994247 0.107114i \(-0.0341610\pi\)
\(380\) 0 0
\(381\) 360.675 0.946653
\(382\) 0 0
\(383\) − 198.838i − 0.519160i −0.965722 0.259580i \(-0.916416\pi\)
0.965722 0.259580i \(-0.0835841\pi\)
\(384\) 0 0
\(385\) −9.82559 −0.0255210
\(386\) 0 0
\(387\) − 100.477i − 0.259629i
\(388\) 0 0
\(389\) 368.767 0.947987 0.473993 0.880528i \(-0.342812\pi\)
0.473993 + 0.880528i \(0.342812\pi\)
\(390\) 0 0
\(391\) − 125.831i − 0.321819i
\(392\) 0 0
\(393\) 382.002 0.972016
\(394\) 0 0
\(395\) − 110.434i − 0.279580i
\(396\) 0 0
\(397\) −114.315 −0.287947 −0.143973 0.989582i \(-0.545988\pi\)
−0.143973 + 0.989582i \(0.545988\pi\)
\(398\) 0 0
\(399\) − 56.3317i − 0.141182i
\(400\) 0 0
\(401\) 39.9083 0.0995218 0.0497609 0.998761i \(-0.484154\pi\)
0.0497609 + 0.998761i \(0.484154\pi\)
\(402\) 0 0
\(403\) − 16.6839i − 0.0413992i
\(404\) 0 0
\(405\) 12.2789 0.0303183
\(406\) 0 0
\(407\) 86.9391i 0.213610i
\(408\) 0 0
\(409\) −269.868 −0.659825 −0.329912 0.944012i \(-0.607019\pi\)
−0.329912 + 0.944012i \(0.607019\pi\)
\(410\) 0 0
\(411\) 26.4421i 0.0643361i
\(412\) 0 0
\(413\) −25.5728 −0.0619197
\(414\) 0 0
\(415\) − 155.476i − 0.374641i
\(416\) 0 0
\(417\) 150.189 0.360165
\(418\) 0 0
\(419\) 20.3559i 0.0485821i 0.999705 + 0.0242910i \(0.00773283\pi\)
−0.999705 + 0.0242910i \(0.992267\pi\)
\(420\) 0 0
\(421\) −557.905 −1.32519 −0.662595 0.748978i \(-0.730545\pi\)
−0.662595 + 0.748978i \(0.730545\pi\)
\(422\) 0 0
\(423\) − 199.692i − 0.472085i
\(424\) 0 0
\(425\) 115.944 0.272810
\(426\) 0 0
\(427\) 137.908i 0.322970i
\(428\) 0 0
\(429\) −164.524 −0.383506
\(430\) 0 0
\(431\) − 376.569i − 0.873710i −0.899532 0.436855i \(-0.856092\pi\)
0.899532 0.436855i \(-0.143908\pi\)
\(432\) 0 0
\(433\) 602.876 1.39232 0.696162 0.717885i \(-0.254890\pi\)
0.696162 + 0.717885i \(0.254890\pi\)
\(434\) 0 0
\(435\) 77.4483i 0.178042i
\(436\) 0 0
\(437\) −657.510 −1.50460
\(438\) 0 0
\(439\) 381.087i 0.868080i 0.900894 + 0.434040i \(0.142912\pi\)
−0.900894 + 0.434040i \(0.857088\pi\)
\(440\) 0 0
\(441\) −142.371 −0.322838
\(442\) 0 0
\(443\) − 599.838i − 1.35404i −0.735966 0.677018i \(-0.763272\pi\)
0.735966 0.677018i \(-0.236728\pi\)
\(444\) 0 0
\(445\) −28.7377 −0.0645791
\(446\) 0 0
\(447\) − 254.349i − 0.569014i
\(448\) 0 0
\(449\) 814.240 1.81345 0.906726 0.421720i \(-0.138573\pi\)
0.906726 + 0.421720i \(0.138573\pi\)
\(450\) 0 0
\(451\) − 420.475i − 0.932317i
\(452\) 0 0
\(453\) −339.366 −0.749152
\(454\) 0 0
\(455\) 27.7637i 0.0610191i
\(456\) 0 0
\(457\) 111.281 0.243502 0.121751 0.992561i \(-0.461149\pi\)
0.121751 + 0.992561i \(0.461149\pi\)
\(458\) 0 0
\(459\) 26.0372i 0.0567259i
\(460\) 0 0
\(461\) 507.833 1.10159 0.550795 0.834641i \(-0.314325\pi\)
0.550795 + 0.834641i \(0.314325\pi\)
\(462\) 0 0
\(463\) − 397.302i − 0.858103i −0.903280 0.429052i \(-0.858848\pi\)
0.903280 0.429052i \(-0.141152\pi\)
\(464\) 0 0
\(465\) 2.40648 0.00517522
\(466\) 0 0
\(467\) 830.195i 1.77772i 0.458179 + 0.888860i \(0.348502\pi\)
−0.458179 + 0.888860i \(0.651498\pi\)
\(468\) 0 0
\(469\) −75.7126 −0.161434
\(470\) 0 0
\(471\) 8.06190i 0.0171166i
\(472\) 0 0
\(473\) −194.186 −0.410542
\(474\) 0 0
\(475\) − 605.849i − 1.27547i
\(476\) 0 0
\(477\) 163.954 0.343719
\(478\) 0 0
\(479\) − 146.251i − 0.305325i −0.988278 0.152662i \(-0.951215\pi\)
0.988278 0.152662i \(-0.0487847\pi\)
\(480\) 0 0
\(481\) 245.660 0.510727
\(482\) 0 0
\(483\) − 54.0258i − 0.111855i
\(484\) 0 0
\(485\) 219.103 0.451759
\(486\) 0 0
\(487\) − 177.070i − 0.363593i −0.983336 0.181797i \(-0.941809\pi\)
0.983336 0.181797i \(-0.0581913\pi\)
\(488\) 0 0
\(489\) −103.142 −0.210924
\(490\) 0 0
\(491\) − 94.9463i − 0.193373i −0.995315 0.0966866i \(-0.969176\pi\)
0.995315 0.0966866i \(-0.0308245\pi\)
\(492\) 0 0
\(493\) −164.227 −0.333118
\(494\) 0 0
\(495\) − 23.7309i − 0.0479412i
\(496\) 0 0
\(497\) 99.9384 0.201083
\(498\) 0 0
\(499\) − 744.720i − 1.49243i −0.665707 0.746213i \(-0.731870\pi\)
0.665707 0.746213i \(-0.268130\pi\)
\(500\) 0 0
\(501\) 362.020 0.722594
\(502\) 0 0
\(503\) − 578.757i − 1.15061i −0.817939 0.575305i \(-0.804883\pi\)
0.817939 0.575305i \(-0.195117\pi\)
\(504\) 0 0
\(505\) −104.055 −0.206049
\(506\) 0 0
\(507\) 172.171i 0.339589i
\(508\) 0 0
\(509\) −323.101 −0.634777 −0.317388 0.948296i \(-0.602806\pi\)
−0.317388 + 0.948296i \(0.602806\pi\)
\(510\) 0 0
\(511\) 37.3290i 0.0730509i
\(512\) 0 0
\(513\) 136.053 0.265211
\(514\) 0 0
\(515\) − 249.349i − 0.484172i
\(516\) 0 0
\(517\) −385.935 −0.746490
\(518\) 0 0
\(519\) − 167.599i − 0.322927i
\(520\) 0 0
\(521\) −582.929 −1.11887 −0.559433 0.828875i \(-0.688981\pi\)
−0.559433 + 0.828875i \(0.688981\pi\)
\(522\) 0 0
\(523\) − 227.111i − 0.434247i −0.976144 0.217124i \(-0.930332\pi\)
0.976144 0.217124i \(-0.0696675\pi\)
\(524\) 0 0
\(525\) 49.7810 0.0948209
\(526\) 0 0
\(527\) 5.10288i 0.00968288i
\(528\) 0 0
\(529\) −101.596 −0.192053
\(530\) 0 0
\(531\) − 61.7639i − 0.116316i
\(532\) 0 0
\(533\) −1188.12 −2.22911
\(534\) 0 0
\(535\) 43.4582i 0.0812303i
\(536\) 0 0
\(537\) 85.7433 0.159671
\(538\) 0 0
\(539\) 275.154i 0.510491i
\(540\) 0 0
\(541\) 551.391 1.01921 0.509603 0.860409i \(-0.329792\pi\)
0.509603 + 0.860409i \(0.329792\pi\)
\(542\) 0 0
\(543\) 244.942i 0.451090i
\(544\) 0 0
\(545\) 15.4564 0.0283603
\(546\) 0 0
\(547\) 745.659i 1.36318i 0.731735 + 0.681590i \(0.238711\pi\)
−0.731735 + 0.681590i \(0.761289\pi\)
\(548\) 0 0
\(549\) −333.078 −0.606700
\(550\) 0 0
\(551\) 858.144i 1.55743i
\(552\) 0 0
\(553\) 100.543 0.181813
\(554\) 0 0
\(555\) 35.4339i 0.0638448i
\(556\) 0 0
\(557\) 755.207 1.35585 0.677924 0.735132i \(-0.262880\pi\)
0.677924 + 0.735132i \(0.262880\pi\)
\(558\) 0 0
\(559\) 548.703i 0.981580i
\(560\) 0 0
\(561\) 50.3209 0.0896985
\(562\) 0 0
\(563\) − 699.309i − 1.24211i −0.783766 0.621056i \(-0.786704\pi\)
0.783766 0.621056i \(-0.213296\pi\)
\(564\) 0 0
\(565\) 68.1600 0.120637
\(566\) 0 0
\(567\) 11.1791i 0.0197163i
\(568\) 0 0
\(569\) 35.8709 0.0630419 0.0315210 0.999503i \(-0.489965\pi\)
0.0315210 + 0.999503i \(0.489965\pi\)
\(570\) 0 0
\(571\) 828.429i 1.45084i 0.688307 + 0.725420i \(0.258354\pi\)
−0.688307 + 0.725420i \(0.741646\pi\)
\(572\) 0 0
\(573\) −202.639 −0.353646
\(574\) 0 0
\(575\) − 581.049i − 1.01052i
\(576\) 0 0
\(577\) −471.333 −0.816867 −0.408434 0.912788i \(-0.633925\pi\)
−0.408434 + 0.912788i \(0.633925\pi\)
\(578\) 0 0
\(579\) − 157.159i − 0.271432i
\(580\) 0 0
\(581\) 141.550 0.243632
\(582\) 0 0
\(583\) − 316.866i − 0.543510i
\(584\) 0 0
\(585\) −67.0553 −0.114624
\(586\) 0 0
\(587\) − 645.149i − 1.09906i −0.835473 0.549531i \(-0.814807\pi\)
0.835473 0.549531i \(-0.185193\pi\)
\(588\) 0 0
\(589\) 26.6643 0.0452704
\(590\) 0 0
\(591\) 658.247i 1.11379i
\(592\) 0 0
\(593\) 203.619 0.343370 0.171685 0.985152i \(-0.445079\pi\)
0.171685 + 0.985152i \(0.445079\pi\)
\(594\) 0 0
\(595\) − 8.49172i − 0.0142718i
\(596\) 0 0
\(597\) 134.505 0.225301
\(598\) 0 0
\(599\) 603.605i 1.00769i 0.863795 + 0.503844i \(0.168081\pi\)
−0.863795 + 0.503844i \(0.831919\pi\)
\(600\) 0 0
\(601\) −626.271 −1.04205 −0.521024 0.853542i \(-0.674450\pi\)
−0.521024 + 0.853542i \(0.674450\pi\)
\(602\) 0 0
\(603\) − 182.862i − 0.303254i
\(604\) 0 0
\(605\) 119.220 0.197057
\(606\) 0 0
\(607\) 421.012i 0.693595i 0.937940 + 0.346797i \(0.112731\pi\)
−0.937940 + 0.346797i \(0.887269\pi\)
\(608\) 0 0
\(609\) −70.5114 −0.115782
\(610\) 0 0
\(611\) 1090.52i 1.78481i
\(612\) 0 0
\(613\) 12.9743 0.0211652 0.0105826 0.999944i \(-0.496631\pi\)
0.0105826 + 0.999944i \(0.496631\pi\)
\(614\) 0 0
\(615\) − 171.373i − 0.278656i
\(616\) 0 0
\(617\) −423.164 −0.685842 −0.342921 0.939364i \(-0.611416\pi\)
−0.342921 + 0.939364i \(0.611416\pi\)
\(618\) 0 0
\(619\) 625.820i 1.01102i 0.862821 + 0.505509i \(0.168695\pi\)
−0.862821 + 0.505509i \(0.831305\pi\)
\(620\) 0 0
\(621\) 130.484 0.210119
\(622\) 0 0
\(623\) − 26.1637i − 0.0419963i
\(624\) 0 0
\(625\) 488.861 0.782178
\(626\) 0 0
\(627\) − 262.944i − 0.419368i
\(628\) 0 0
\(629\) −75.1367 −0.119454
\(630\) 0 0
\(631\) − 690.848i − 1.09485i −0.836856 0.547423i \(-0.815609\pi\)
0.836856 0.547423i \(-0.184391\pi\)
\(632\) 0 0
\(633\) −331.593 −0.523844
\(634\) 0 0
\(635\) − 284.101i − 0.447403i
\(636\) 0 0
\(637\) 777.491 1.22055
\(638\) 0 0
\(639\) 241.373i 0.377735i
\(640\) 0 0
\(641\) −369.160 −0.575913 −0.287957 0.957643i \(-0.592976\pi\)
−0.287957 + 0.957643i \(0.592976\pi\)
\(642\) 0 0
\(643\) − 666.030i − 1.03582i −0.855436 0.517909i \(-0.826711\pi\)
0.855436 0.517909i \(-0.173289\pi\)
\(644\) 0 0
\(645\) −79.1448 −0.122705
\(646\) 0 0
\(647\) 651.886i 1.00755i 0.863835 + 0.503776i \(0.168056\pi\)
−0.863835 + 0.503776i \(0.831944\pi\)
\(648\) 0 0
\(649\) −119.368 −0.183926
\(650\) 0 0
\(651\) 2.19093i 0.00336549i
\(652\) 0 0
\(653\) −903.324 −1.38334 −0.691672 0.722212i \(-0.743126\pi\)
−0.691672 + 0.722212i \(0.743126\pi\)
\(654\) 0 0
\(655\) − 300.901i − 0.459391i
\(656\) 0 0
\(657\) −90.1576 −0.137226
\(658\) 0 0
\(659\) 643.621i 0.976664i 0.872658 + 0.488332i \(0.162394\pi\)
−0.872658 + 0.488332i \(0.837606\pi\)
\(660\) 0 0
\(661\) 860.187 1.30134 0.650671 0.759360i \(-0.274488\pi\)
0.650671 + 0.759360i \(0.274488\pi\)
\(662\) 0 0
\(663\) − 142.189i − 0.214463i
\(664\) 0 0
\(665\) −44.3722 −0.0667250
\(666\) 0 0
\(667\) 823.017i 1.23391i
\(668\) 0 0
\(669\) 291.766 0.436123
\(670\) 0 0
\(671\) 643.725i 0.959351i
\(672\) 0 0
\(673\) 866.535 1.28757 0.643785 0.765206i \(-0.277363\pi\)
0.643785 + 0.765206i \(0.277363\pi\)
\(674\) 0 0
\(675\) 120.232i 0.178121i
\(676\) 0 0
\(677\) 1307.26 1.93095 0.965477 0.260489i \(-0.0838838\pi\)
0.965477 + 0.260489i \(0.0838838\pi\)
\(678\) 0 0
\(679\) 199.478i 0.293783i
\(680\) 0 0
\(681\) −196.616 −0.288716
\(682\) 0 0
\(683\) 783.569i 1.14725i 0.819120 + 0.573623i \(0.194462\pi\)
−0.819120 + 0.573623i \(0.805538\pi\)
\(684\) 0 0
\(685\) 20.8283 0.0304063
\(686\) 0 0
\(687\) 203.720i 0.296536i
\(688\) 0 0
\(689\) −895.354 −1.29950
\(690\) 0 0
\(691\) 1014.95i 1.46882i 0.678708 + 0.734408i \(0.262540\pi\)
−0.678708 + 0.734408i \(0.737460\pi\)
\(692\) 0 0
\(693\) 21.6054 0.0311766
\(694\) 0 0
\(695\) − 118.303i − 0.170220i
\(696\) 0 0
\(697\) 363.394 0.521368
\(698\) 0 0
\(699\) 479.926i 0.686590i
\(700\) 0 0
\(701\) 957.527 1.36595 0.682973 0.730444i \(-0.260687\pi\)
0.682973 + 0.730444i \(0.260687\pi\)
\(702\) 0 0
\(703\) 392.615i 0.558485i
\(704\) 0 0
\(705\) −157.296 −0.223115
\(706\) 0 0
\(707\) − 94.7346i − 0.133995i
\(708\) 0 0
\(709\) 65.7503 0.0927366 0.0463683 0.998924i \(-0.485235\pi\)
0.0463683 + 0.998924i \(0.485235\pi\)
\(710\) 0 0
\(711\) 242.832i 0.341536i
\(712\) 0 0
\(713\) 25.5728 0.0358665
\(714\) 0 0
\(715\) 129.595i 0.181251i
\(716\) 0 0
\(717\) −594.219 −0.828757
\(718\) 0 0
\(719\) − 573.085i − 0.797058i −0.917156 0.398529i \(-0.869521\pi\)
0.917156 0.398529i \(-0.130479\pi\)
\(720\) 0 0
\(721\) 227.015 0.314861
\(722\) 0 0
\(723\) − 568.354i − 0.786106i
\(724\) 0 0
\(725\) −758.352 −1.04600
\(726\) 0 0
\(727\) − 249.632i − 0.343373i −0.985152 0.171686i \(-0.945078\pi\)
0.985152 0.171686i \(-0.0549216\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) 0 0
\(731\) − 167.825i − 0.229582i
\(732\) 0 0
\(733\) 662.187 0.903393 0.451696 0.892172i \(-0.350819\pi\)
0.451696 + 0.892172i \(0.350819\pi\)
\(734\) 0 0
\(735\) 112.145i 0.152578i
\(736\) 0 0
\(737\) −353.409 −0.479524
\(738\) 0 0
\(739\) 98.7372i 0.133609i 0.997766 + 0.0668046i \(0.0212804\pi\)
−0.997766 + 0.0668046i \(0.978720\pi\)
\(740\) 0 0
\(741\) −742.988 −1.00268
\(742\) 0 0
\(743\) 906.520i 1.22008i 0.792370 + 0.610041i \(0.208847\pi\)
−0.792370 + 0.610041i \(0.791153\pi\)
\(744\) 0 0
\(745\) −200.349 −0.268925
\(746\) 0 0
\(747\) 341.875i 0.457664i
\(748\) 0 0
\(749\) −39.5657 −0.0528248
\(750\) 0 0
\(751\) − 286.284i − 0.381204i −0.981667 0.190602i \(-0.938956\pi\)
0.981667 0.190602i \(-0.0610440\pi\)
\(752\) 0 0
\(753\) −784.470 −1.04179
\(754\) 0 0
\(755\) 267.316i 0.354062i
\(756\) 0 0
\(757\) 1162.42 1.53556 0.767781 0.640712i \(-0.221361\pi\)
0.767781 + 0.640712i \(0.221361\pi\)
\(758\) 0 0
\(759\) − 252.180i − 0.332253i
\(760\) 0 0
\(761\) −994.905 −1.30737 −0.653683 0.756769i \(-0.726777\pi\)
−0.653683 + 0.756769i \(0.726777\pi\)
\(762\) 0 0
\(763\) 14.0720i 0.0184429i
\(764\) 0 0
\(765\) 20.5093 0.0268096
\(766\) 0 0
\(767\) 337.293i 0.439756i
\(768\) 0 0
\(769\) −614.473 −0.799055 −0.399527 0.916721i \(-0.630826\pi\)
−0.399527 + 0.916721i \(0.630826\pi\)
\(770\) 0 0
\(771\) − 600.727i − 0.779153i
\(772\) 0 0
\(773\) 927.633 1.20004 0.600021 0.799984i \(-0.295159\pi\)
0.600021 + 0.799984i \(0.295159\pi\)
\(774\) 0 0
\(775\) 23.5635i 0.0304045i
\(776\) 0 0
\(777\) −32.2601 −0.0415188
\(778\) 0 0
\(779\) − 1898.86i − 2.43756i
\(780\) 0 0
\(781\) 466.490 0.597298
\(782\) 0 0
\(783\) − 170.300i − 0.217497i
\(784\) 0 0
\(785\) 6.35031 0.00808957
\(786\) 0 0
\(787\) 781.920i 0.993545i 0.867881 + 0.496772i \(0.165482\pi\)
−0.867881 + 0.496772i \(0.834518\pi\)
\(788\) 0 0
\(789\) 697.047 0.883456
\(790\) 0 0
\(791\) 62.0550i 0.0784513i
\(792\) 0 0
\(793\) 1818.94 2.29375
\(794\) 0 0
\(795\) − 129.146i − 0.162447i
\(796\) 0 0
\(797\) −1160.22 −1.45574 −0.727868 0.685718i \(-0.759489\pi\)
−0.727868 + 0.685718i \(0.759489\pi\)
\(798\) 0 0
\(799\) − 333.543i − 0.417450i
\(800\) 0 0
\(801\) 63.1910 0.0788902
\(802\) 0 0
\(803\) 174.243i 0.216991i
\(804\) 0 0
\(805\) −42.5558 −0.0528644
\(806\) 0 0
\(807\) 556.962i 0.690163i
\(808\) 0 0
\(809\) 1512.26 1.86930 0.934651 0.355568i \(-0.115712\pi\)
0.934651 + 0.355568i \(0.115712\pi\)
\(810\) 0 0
\(811\) 1586.92i 1.95674i 0.206858 + 0.978371i \(0.433676\pi\)
−0.206858 + 0.978371i \(0.566324\pi\)
\(812\) 0 0
\(813\) −791.378 −0.973405
\(814\) 0 0
\(815\) 81.2441i 0.0996860i
\(816\) 0 0
\(817\) −876.942 −1.07337
\(818\) 0 0
\(819\) − 61.0493i − 0.0745412i
\(820\) 0 0
\(821\) −1118.96 −1.36292 −0.681459 0.731857i \(-0.738654\pi\)
−0.681459 + 0.731857i \(0.738654\pi\)
\(822\) 0 0
\(823\) − 1628.26i − 1.97844i −0.146438 0.989220i \(-0.546781\pi\)
0.146438 0.989220i \(-0.453219\pi\)
\(824\) 0 0
\(825\) 232.366 0.281656
\(826\) 0 0
\(827\) − 421.552i − 0.509736i −0.966976 0.254868i \(-0.917968\pi\)
0.966976 0.254868i \(-0.0820321\pi\)
\(828\) 0 0
\(829\) −475.263 −0.573297 −0.286649 0.958036i \(-0.592541\pi\)
−0.286649 + 0.958036i \(0.592541\pi\)
\(830\) 0 0
\(831\) 570.786i 0.686866i
\(832\) 0 0
\(833\) −237.801 −0.285475
\(834\) 0 0
\(835\) − 285.161i − 0.341510i
\(836\) 0 0
\(837\) −5.29157 −0.00632207
\(838\) 0 0
\(839\) 653.590i 0.779010i 0.921024 + 0.389505i \(0.127354\pi\)
−0.921024 + 0.389505i \(0.872646\pi\)
\(840\) 0 0
\(841\) 233.154 0.277234
\(842\) 0 0
\(843\) − 303.221i − 0.359692i
\(844\) 0 0
\(845\) 135.618 0.160495
\(846\) 0 0
\(847\) 108.541i 0.128148i
\(848\) 0 0
\(849\) 260.323 0.306623
\(850\) 0 0
\(851\) 376.544i 0.442472i
\(852\) 0 0
\(853\) 140.493 0.164705 0.0823523 0.996603i \(-0.473757\pi\)
0.0823523 + 0.996603i \(0.473757\pi\)
\(854\) 0 0
\(855\) − 107.168i − 0.125343i
\(856\) 0 0
\(857\) −562.796 −0.656704 −0.328352 0.944555i \(-0.606493\pi\)
−0.328352 + 0.944555i \(0.606493\pi\)
\(858\) 0 0
\(859\) − 228.316i − 0.265792i −0.991130 0.132896i \(-0.957572\pi\)
0.991130 0.132896i \(-0.0424277\pi\)
\(860\) 0 0
\(861\) 156.024 0.181212
\(862\) 0 0
\(863\) − 892.187i − 1.03382i −0.856040 0.516910i \(-0.827082\pi\)
0.856040 0.516910i \(-0.172918\pi\)
\(864\) 0 0
\(865\) −132.017 −0.152621
\(866\) 0 0
\(867\) − 457.073i − 0.527189i
\(868\) 0 0
\(869\) 469.310 0.540058
\(870\) 0 0
\(871\) 998.611i 1.14651i
\(872\) 0 0
\(873\) −481.783 −0.551871
\(874\) 0 0
\(875\) − 81.5787i − 0.0932328i
\(876\) 0 0
\(877\) −1406.66 −1.60395 −0.801974 0.597359i \(-0.796217\pi\)
−0.801974 + 0.597359i \(0.796217\pi\)
\(878\) 0 0
\(879\) 277.882i 0.316135i
\(880\) 0 0
\(881\) 943.043 1.07042 0.535212 0.844718i \(-0.320232\pi\)
0.535212 + 0.844718i \(0.320232\pi\)
\(882\) 0 0
\(883\) − 1146.63i − 1.29856i −0.760549 0.649280i \(-0.775070\pi\)
0.760549 0.649280i \(-0.224930\pi\)
\(884\) 0 0
\(885\) −48.6511 −0.0549729
\(886\) 0 0
\(887\) 894.171i 1.00808i 0.863679 + 0.504042i \(0.168154\pi\)
−0.863679 + 0.504042i \(0.831846\pi\)
\(888\) 0 0
\(889\) 258.655 0.290950
\(890\) 0 0
\(891\) 52.1816i 0.0585652i
\(892\) 0 0
\(893\) −1742.88 −1.95171
\(894\) 0 0
\(895\) − 67.5395i − 0.0754631i
\(896\) 0 0
\(897\) −712.574 −0.794397
\(898\) 0 0
\(899\) − 33.3761i − 0.0371258i
\(900\) 0 0
\(901\) 273.850 0.303940
\(902\) 0 0
\(903\) − 72.0559i − 0.0797961i
\(904\) 0 0
\(905\) 192.939 0.213192
\(906\) 0 0
\(907\) − 1358.60i − 1.49790i −0.662626 0.748950i \(-0.730558\pi\)
0.662626 0.748950i \(-0.269442\pi\)
\(908\) 0 0
\(909\) 228.804 0.251710
\(910\) 0 0
\(911\) − 804.510i − 0.883106i −0.897235 0.441553i \(-0.854428\pi\)
0.897235 0.441553i \(-0.145572\pi\)
\(912\) 0 0
\(913\) 660.726 0.723686
\(914\) 0 0
\(915\) 262.364i 0.286736i
\(916\) 0 0
\(917\) 273.950 0.298745
\(918\) 0 0
\(919\) − 1704.73i − 1.85498i −0.373849 0.927490i \(-0.621962\pi\)
0.373849 0.927490i \(-0.378038\pi\)
\(920\) 0 0
\(921\) 291.192 0.316169
\(922\) 0 0
\(923\) − 1318.14i − 1.42810i
\(924\) 0 0
\(925\) −346.958 −0.375090
\(926\) 0 0
\(927\) 548.290i 0.591467i
\(928\) 0 0
\(929\) −1351.05 −1.45431 −0.727154 0.686475i \(-0.759157\pi\)
−0.727154 + 0.686475i \(0.759157\pi\)
\(930\) 0 0
\(931\) 1242.59i 1.33469i
\(932\) 0 0
\(933\) −814.715 −0.873220
\(934\) 0 0
\(935\) − 39.6374i − 0.0423930i
\(936\) 0 0
\(937\) 672.646 0.717872 0.358936 0.933362i \(-0.383140\pi\)
0.358936 + 0.933362i \(0.383140\pi\)
\(938\) 0 0
\(939\) 33.6672i 0.0358544i
\(940\) 0 0
\(941\) 528.671 0.561818 0.280909 0.959734i \(-0.409364\pi\)
0.280909 + 0.959734i \(0.409364\pi\)
\(942\) 0 0
\(943\) − 1821.13i − 1.93121i
\(944\) 0 0
\(945\) 8.80573 0.00931823
\(946\) 0 0
\(947\) − 661.066i − 0.698063i −0.937111 0.349032i \(-0.886511\pi\)
0.937111 0.349032i \(-0.113489\pi\)
\(948\) 0 0
\(949\) 492.351 0.518811
\(950\) 0 0
\(951\) − 419.494i − 0.441108i
\(952\) 0 0
\(953\) 1545.41 1.62163 0.810815 0.585303i \(-0.199024\pi\)
0.810815 + 0.585303i \(0.199024\pi\)
\(954\) 0 0
\(955\) 159.618i 0.167139i
\(956\) 0 0
\(957\) −329.131 −0.343920
\(958\) 0 0
\(959\) 18.9627i 0.0197735i
\(960\) 0 0
\(961\) 959.963 0.998921
\(962\) 0 0
\(963\) − 95.5598i − 0.0992313i
\(964\) 0 0
\(965\) −123.793 −0.128283
\(966\) 0 0
\(967\) 161.279i 0.166782i 0.996517 + 0.0833912i \(0.0265751\pi\)
−0.996517 + 0.0833912i \(0.973425\pi\)
\(968\) 0 0
\(969\) 227.248 0.234518
\(970\) 0 0
\(971\) − 4.20412i − 0.00432969i −0.999998 0.00216484i \(-0.999311\pi\)
0.999998 0.00216484i \(-0.000689091\pi\)
\(972\) 0 0
\(973\) 107.707 0.110696
\(974\) 0 0
\(975\) − 656.586i − 0.673422i
\(976\) 0 0
\(977\) −1348.19 −1.37993 −0.689967 0.723841i \(-0.742375\pi\)
−0.689967 + 0.723841i \(0.742375\pi\)
\(978\) 0 0
\(979\) − 122.126i − 0.124746i
\(980\) 0 0
\(981\) −33.9868 −0.0346451
\(982\) 0 0
\(983\) − 984.262i − 1.00128i −0.865655 0.500642i \(-0.833097\pi\)
0.865655 0.500642i \(-0.166903\pi\)
\(984\) 0 0
\(985\) 518.497 0.526393
\(986\) 0 0
\(987\) − 143.207i − 0.145094i
\(988\) 0 0
\(989\) −841.045 −0.850399
\(990\) 0 0
\(991\) 1013.87i 1.02308i 0.859259 + 0.511541i \(0.170925\pi\)
−0.859259 + 0.511541i \(0.829075\pi\)
\(992\) 0 0
\(993\) 763.750 0.769134
\(994\) 0 0
\(995\) − 105.948i − 0.106481i
\(996\) 0 0
\(997\) 311.310 0.312246 0.156123 0.987738i \(-0.450100\pi\)
0.156123 + 0.987738i \(0.450100\pi\)
\(998\) 0 0
\(999\) − 77.9151i − 0.0779931i
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.127.6 yes 8
3.2 odd 2 1152.3.g.c.127.6 8
4.3 odd 2 inner 384.3.g.b.127.2 yes 8
8.3 odd 2 384.3.g.a.127.7 yes 8
8.5 even 2 384.3.g.a.127.3 8
12.11 even 2 1152.3.g.c.127.5 8
16.3 odd 4 768.3.b.f.127.6 8
16.5 even 4 768.3.b.f.127.7 8
16.11 odd 4 768.3.b.e.127.3 8
16.13 even 4 768.3.b.e.127.2 8
24.5 odd 2 1152.3.g.f.127.4 8
24.11 even 2 1152.3.g.f.127.3 8
48.5 odd 4 2304.3.b.q.127.4 8
48.11 even 4 2304.3.b.t.127.4 8
48.29 odd 4 2304.3.b.t.127.5 8
48.35 even 4 2304.3.b.q.127.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.3.g.a.127.3 8 8.5 even 2
384.3.g.a.127.7 yes 8 8.3 odd 2
384.3.g.b.127.2 yes 8 4.3 odd 2 inner
384.3.g.b.127.6 yes 8 1.1 even 1 trivial
768.3.b.e.127.2 8 16.13 even 4
768.3.b.e.127.3 8 16.11 odd 4
768.3.b.f.127.6 8 16.3 odd 4
768.3.b.f.127.7 8 16.5 even 4
1152.3.g.c.127.5 8 12.11 even 2
1152.3.g.c.127.6 8 3.2 odd 2
1152.3.g.f.127.3 8 24.11 even 2
1152.3.g.f.127.4 8 24.5 odd 2
2304.3.b.q.127.4 8 48.5 odd 4
2304.3.b.q.127.5 8 48.35 even 4
2304.3.b.t.127.4 8 48.11 even 4
2304.3.b.t.127.5 8 48.29 odd 4