Properties

Label 882.3.b.i.197.3
Level $882$
Weight $3$
Character 882.197
Analytic conductor $24.033$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 882.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.0327593166\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 197.3
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 882.197
Dual form 882.3.b.i.197.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421i q^{2} -2.00000 q^{4} -2.00000i q^{5} -2.82843i q^{8} +O(q^{10})\) \(q+1.41421i q^{2} -2.00000 q^{4} -2.00000i q^{5} -2.82843i q^{8} +2.82843 q^{10} -2.82843i q^{11} -12.7279 q^{13} +4.00000 q^{16} +22.0000i q^{17} +5.65685 q^{19} +4.00000i q^{20} +4.00000 q^{22} +2.82843i q^{23} +21.0000 q^{25} -18.0000i q^{26} -35.3553i q^{29} +33.9411 q^{31} +5.65685i q^{32} -31.1127 q^{34} +64.0000 q^{37} +8.00000i q^{38} -5.65685 q^{40} -20.0000i q^{41} +44.0000 q^{43} +5.65685i q^{44} -4.00000 q^{46} -68.0000i q^{47} +29.6985i q^{50} +25.4558 q^{52} +18.3848i q^{53} -5.65685 q^{55} +50.0000 q^{58} +100.000i q^{59} -52.3259 q^{61} +48.0000i q^{62} -8.00000 q^{64} +25.4558i q^{65} +120.000 q^{67} -44.0000i q^{68} +8.48528i q^{71} +74.9533 q^{73} +90.5097i q^{74} -11.3137 q^{76} +92.0000 q^{79} -8.00000i q^{80} +28.2843 q^{82} -112.000i q^{83} +44.0000 q^{85} +62.2254i q^{86} -8.00000 q^{88} +20.0000i q^{89} -5.65685i q^{92} +96.1665 q^{94} -11.3137i q^{95} +26.8701 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} + O(q^{10}) \) \( 4q - 8q^{4} + 16q^{16} + 16q^{22} + 84q^{25} + 256q^{37} + 176q^{43} - 16q^{46} + 200q^{58} - 32q^{64} + 480q^{67} + 368q^{79} + 176q^{85} - 32q^{88} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 0.707107i
\(3\) 0 0
\(4\) −2.00000 −0.500000
\(5\) − 2.00000i − 0.400000i −0.979796 0.200000i \(-0.935906\pi\)
0.979796 0.200000i \(-0.0640942\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) − 2.82843i − 0.353553i
\(9\) 0 0
\(10\) 2.82843 0.282843
\(11\) − 2.82843i − 0.257130i −0.991701 0.128565i \(-0.958963\pi\)
0.991701 0.128565i \(-0.0410371\pi\)
\(12\) 0 0
\(13\) −12.7279 −0.979071 −0.489535 0.871983i \(-0.662834\pi\)
−0.489535 + 0.871983i \(0.662834\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 22.0000i 1.29412i 0.762440 + 0.647059i \(0.224001\pi\)
−0.762440 + 0.647059i \(0.775999\pi\)
\(18\) 0 0
\(19\) 5.65685 0.297729 0.148865 0.988858i \(-0.452438\pi\)
0.148865 + 0.988858i \(0.452438\pi\)
\(20\) 4.00000i 0.200000i
\(21\) 0 0
\(22\) 4.00000 0.181818
\(23\) 2.82843i 0.122975i 0.998108 + 0.0614875i \(0.0195844\pi\)
−0.998108 + 0.0614875i \(0.980416\pi\)
\(24\) 0 0
\(25\) 21.0000 0.840000
\(26\) − 18.0000i − 0.692308i
\(27\) 0 0
\(28\) 0 0
\(29\) − 35.3553i − 1.21915i −0.792729 0.609575i \(-0.791340\pi\)
0.792729 0.609575i \(-0.208660\pi\)
\(30\) 0 0
\(31\) 33.9411 1.09488 0.547438 0.836847i \(-0.315603\pi\)
0.547438 + 0.836847i \(0.315603\pi\)
\(32\) 5.65685i 0.176777i
\(33\) 0 0
\(34\) −31.1127 −0.915079
\(35\) 0 0
\(36\) 0 0
\(37\) 64.0000 1.72973 0.864865 0.502005i \(-0.167404\pi\)
0.864865 + 0.502005i \(0.167404\pi\)
\(38\) 8.00000i 0.210526i
\(39\) 0 0
\(40\) −5.65685 −0.141421
\(41\) − 20.0000i − 0.487805i −0.969800 0.243902i \(-0.921572\pi\)
0.969800 0.243902i \(-0.0784277\pi\)
\(42\) 0 0
\(43\) 44.0000 1.02326 0.511628 0.859207i \(-0.329043\pi\)
0.511628 + 0.859207i \(0.329043\pi\)
\(44\) 5.65685i 0.128565i
\(45\) 0 0
\(46\) −4.00000 −0.0869565
\(47\) − 68.0000i − 1.44681i −0.690425 0.723404i \(-0.742576\pi\)
0.690425 0.723404i \(-0.257424\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 29.6985i 0.593970i
\(51\) 0 0
\(52\) 25.4558 0.489535
\(53\) 18.3848i 0.346883i 0.984844 + 0.173441i \(0.0554887\pi\)
−0.984844 + 0.173441i \(0.944511\pi\)
\(54\) 0 0
\(55\) −5.65685 −0.102852
\(56\) 0 0
\(57\) 0 0
\(58\) 50.0000 0.862069
\(59\) 100.000i 1.69492i 0.530863 + 0.847458i \(0.321868\pi\)
−0.530863 + 0.847458i \(0.678132\pi\)
\(60\) 0 0
\(61\) −52.3259 −0.857802 −0.428901 0.903352i \(-0.641099\pi\)
−0.428901 + 0.903352i \(0.641099\pi\)
\(62\) 48.0000i 0.774194i
\(63\) 0 0
\(64\) −8.00000 −0.125000
\(65\) 25.4558i 0.391628i
\(66\) 0 0
\(67\) 120.000 1.79104 0.895522 0.445016i \(-0.146802\pi\)
0.895522 + 0.445016i \(0.146802\pi\)
\(68\) − 44.0000i − 0.647059i
\(69\) 0 0
\(70\) 0 0
\(71\) 8.48528i 0.119511i 0.998213 + 0.0597555i \(0.0190321\pi\)
−0.998213 + 0.0597555i \(0.980968\pi\)
\(72\) 0 0
\(73\) 74.9533 1.02676 0.513379 0.858162i \(-0.328394\pi\)
0.513379 + 0.858162i \(0.328394\pi\)
\(74\) 90.5097i 1.22310i
\(75\) 0 0
\(76\) −11.3137 −0.148865
\(77\) 0 0
\(78\) 0 0
\(79\) 92.0000 1.16456 0.582278 0.812989i \(-0.302161\pi\)
0.582278 + 0.812989i \(0.302161\pi\)
\(80\) − 8.00000i − 0.100000i
\(81\) 0 0
\(82\) 28.2843 0.344930
\(83\) − 112.000i − 1.34940i −0.738093 0.674699i \(-0.764274\pi\)
0.738093 0.674699i \(-0.235726\pi\)
\(84\) 0 0
\(85\) 44.0000 0.517647
\(86\) 62.2254i 0.723551i
\(87\) 0 0
\(88\) −8.00000 −0.0909091
\(89\) 20.0000i 0.224719i 0.993668 + 0.112360i \(0.0358408\pi\)
−0.993668 + 0.112360i \(0.964159\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 5.65685i − 0.0614875i
\(93\) 0 0
\(94\) 96.1665 1.02305
\(95\) − 11.3137i − 0.119092i
\(96\) 0 0
\(97\) 26.8701 0.277011 0.138505 0.990362i \(-0.455770\pi\)
0.138505 + 0.990362i \(0.455770\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −42.0000 −0.420000
\(101\) 84.0000i 0.831683i 0.909437 + 0.415842i \(0.136513\pi\)
−0.909437 + 0.415842i \(0.863487\pi\)
\(102\) 0 0
\(103\) −158.392 −1.53779 −0.768893 0.639378i \(-0.779192\pi\)
−0.768893 + 0.639378i \(0.779192\pi\)
\(104\) 36.0000i 0.346154i
\(105\) 0 0
\(106\) −26.0000 −0.245283
\(107\) 138.593i 1.29526i 0.761955 + 0.647631i \(0.224240\pi\)
−0.761955 + 0.647631i \(0.775760\pi\)
\(108\) 0 0
\(109\) 70.0000 0.642202 0.321101 0.947045i \(-0.395947\pi\)
0.321101 + 0.947045i \(0.395947\pi\)
\(110\) − 8.00000i − 0.0727273i
\(111\) 0 0
\(112\) 0 0
\(113\) 21.2132i 0.187727i 0.995585 + 0.0938637i \(0.0299218\pi\)
−0.995585 + 0.0938637i \(0.970078\pi\)
\(114\) 0 0
\(115\) 5.65685 0.0491900
\(116\) 70.7107i 0.609575i
\(117\) 0 0
\(118\) −141.421 −1.19849
\(119\) 0 0
\(120\) 0 0
\(121\) 113.000 0.933884
\(122\) − 74.0000i − 0.606557i
\(123\) 0 0
\(124\) −67.8823 −0.547438
\(125\) − 92.0000i − 0.736000i
\(126\) 0 0
\(127\) −20.0000 −0.157480 −0.0787402 0.996895i \(-0.525090\pi\)
−0.0787402 + 0.996895i \(0.525090\pi\)
\(128\) − 11.3137i − 0.0883883i
\(129\) 0 0
\(130\) −36.0000 −0.276923
\(131\) 144.000i 1.09924i 0.835416 + 0.549618i \(0.185227\pi\)
−0.835416 + 0.549618i \(0.814773\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 169.706i 1.26646i
\(135\) 0 0
\(136\) 62.2254 0.457540
\(137\) − 199.404i − 1.45550i −0.685840 0.727752i \(-0.740565\pi\)
0.685840 0.727752i \(-0.259435\pi\)
\(138\) 0 0
\(139\) 5.65685 0.0406968 0.0203484 0.999793i \(-0.493522\pi\)
0.0203484 + 0.999793i \(0.493522\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −12.0000 −0.0845070
\(143\) 36.0000i 0.251748i
\(144\) 0 0
\(145\) −70.7107 −0.487660
\(146\) 106.000i 0.726027i
\(147\) 0 0
\(148\) −128.000 −0.864865
\(149\) 18.3848i 0.123388i 0.998095 + 0.0616939i \(0.0196503\pi\)
−0.998095 + 0.0616939i \(0.980350\pi\)
\(150\) 0 0
\(151\) −64.0000 −0.423841 −0.211921 0.977287i \(-0.567972\pi\)
−0.211921 + 0.977287i \(0.567972\pi\)
\(152\) − 16.0000i − 0.105263i
\(153\) 0 0
\(154\) 0 0
\(155\) − 67.8823i − 0.437950i
\(156\) 0 0
\(157\) 162.635 1.03589 0.517944 0.855414i \(-0.326697\pi\)
0.517944 + 0.855414i \(0.326697\pi\)
\(158\) 130.108i 0.823466i
\(159\) 0 0
\(160\) 11.3137 0.0707107
\(161\) 0 0
\(162\) 0 0
\(163\) −224.000 −1.37423 −0.687117 0.726547i \(-0.741124\pi\)
−0.687117 + 0.726547i \(0.741124\pi\)
\(164\) 40.0000i 0.243902i
\(165\) 0 0
\(166\) 158.392 0.954168
\(167\) − 292.000i − 1.74850i −0.485473 0.874251i \(-0.661353\pi\)
0.485473 0.874251i \(-0.338647\pi\)
\(168\) 0 0
\(169\) −7.00000 −0.0414201
\(170\) 62.2254i 0.366032i
\(171\) 0 0
\(172\) −88.0000 −0.511628
\(173\) 156.000i 0.901734i 0.892591 + 0.450867i \(0.148885\pi\)
−0.892591 + 0.450867i \(0.851115\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) − 11.3137i − 0.0642824i
\(177\) 0 0
\(178\) −28.2843 −0.158900
\(179\) 93.3381i 0.521442i 0.965414 + 0.260721i \(0.0839602\pi\)
−0.965414 + 0.260721i \(0.916040\pi\)
\(180\) 0 0
\(181\) 190.919 1.05480 0.527400 0.849617i \(-0.323167\pi\)
0.527400 + 0.849617i \(0.323167\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 8.00000 0.0434783
\(185\) − 128.000i − 0.691892i
\(186\) 0 0
\(187\) 62.2254 0.332756
\(188\) 136.000i 0.723404i
\(189\) 0 0
\(190\) 16.0000 0.0842105
\(191\) − 132.936i − 0.696000i −0.937494 0.348000i \(-0.886861\pi\)
0.937494 0.348000i \(-0.113139\pi\)
\(192\) 0 0
\(193\) 122.000 0.632124 0.316062 0.948738i \(-0.397639\pi\)
0.316062 + 0.948738i \(0.397639\pi\)
\(194\) 38.0000i 0.195876i
\(195\) 0 0
\(196\) 0 0
\(197\) 7.07107i 0.0358937i 0.999839 + 0.0179469i \(0.00571297\pi\)
−0.999839 + 0.0179469i \(0.994287\pi\)
\(198\) 0 0
\(199\) −373.352 −1.87614 −0.938071 0.346442i \(-0.887390\pi\)
−0.938071 + 0.346442i \(0.887390\pi\)
\(200\) − 59.3970i − 0.296985i
\(201\) 0 0
\(202\) −118.794 −0.588089
\(203\) 0 0
\(204\) 0 0
\(205\) −40.0000 −0.195122
\(206\) − 224.000i − 1.08738i
\(207\) 0 0
\(208\) −50.9117 −0.244768
\(209\) − 16.0000i − 0.0765550i
\(210\) 0 0
\(211\) −12.0000 −0.0568720 −0.0284360 0.999596i \(-0.509053\pi\)
−0.0284360 + 0.999596i \(0.509053\pi\)
\(212\) − 36.7696i − 0.173441i
\(213\) 0 0
\(214\) −196.000 −0.915888
\(215\) − 88.0000i − 0.409302i
\(216\) 0 0
\(217\) 0 0
\(218\) 98.9949i 0.454105i
\(219\) 0 0
\(220\) 11.3137 0.0514259
\(221\) − 280.014i − 1.26703i
\(222\) 0 0
\(223\) 203.647 0.913214 0.456607 0.889668i \(-0.349065\pi\)
0.456607 + 0.889668i \(0.349065\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −30.0000 −0.132743
\(227\) 204.000i 0.898678i 0.893361 + 0.449339i \(0.148341\pi\)
−0.893361 + 0.449339i \(0.851659\pi\)
\(228\) 0 0
\(229\) 46.6690 0.203795 0.101897 0.994795i \(-0.467509\pi\)
0.101897 + 0.994795i \(0.467509\pi\)
\(230\) 8.00000i 0.0347826i
\(231\) 0 0
\(232\) −100.000 −0.431034
\(233\) − 309.713i − 1.32924i −0.747182 0.664620i \(-0.768594\pi\)
0.747182 0.664620i \(-0.231406\pi\)
\(234\) 0 0
\(235\) −136.000 −0.578723
\(236\) − 200.000i − 0.847458i
\(237\) 0 0
\(238\) 0 0
\(239\) − 42.4264i − 0.177516i −0.996053 0.0887582i \(-0.971710\pi\)
0.996053 0.0887582i \(-0.0282898\pi\)
\(240\) 0 0
\(241\) −377.595 −1.56678 −0.783392 0.621528i \(-0.786512\pi\)
−0.783392 + 0.621528i \(0.786512\pi\)
\(242\) 159.806i 0.660356i
\(243\) 0 0
\(244\) 104.652 0.428901
\(245\) 0 0
\(246\) 0 0
\(247\) −72.0000 −0.291498
\(248\) − 96.0000i − 0.387097i
\(249\) 0 0
\(250\) 130.108 0.520431
\(251\) 332.000i 1.32271i 0.750073 + 0.661355i \(0.230018\pi\)
−0.750073 + 0.661355i \(0.769982\pi\)
\(252\) 0 0
\(253\) 8.00000 0.0316206
\(254\) − 28.2843i − 0.111355i
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) 180.000i 0.700389i 0.936677 + 0.350195i \(0.113885\pi\)
−0.936677 + 0.350195i \(0.886115\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) − 50.9117i − 0.195814i
\(261\) 0 0
\(262\) −203.647 −0.777278
\(263\) − 161.220i − 0.613005i −0.951870 0.306503i \(-0.900841\pi\)
0.951870 0.306503i \(-0.0991588\pi\)
\(264\) 0 0
\(265\) 36.7696 0.138753
\(266\) 0 0
\(267\) 0 0
\(268\) −240.000 −0.895522
\(269\) − 526.000i − 1.95539i −0.210029 0.977695i \(-0.567356\pi\)
0.210029 0.977695i \(-0.432644\pi\)
\(270\) 0 0
\(271\) 429.921 1.58642 0.793212 0.608946i \(-0.208407\pi\)
0.793212 + 0.608946i \(0.208407\pi\)
\(272\) 88.0000i 0.323529i
\(273\) 0 0
\(274\) 282.000 1.02920
\(275\) − 59.3970i − 0.215989i
\(276\) 0 0
\(277\) 64.0000 0.231047 0.115523 0.993305i \(-0.463145\pi\)
0.115523 + 0.993305i \(0.463145\pi\)
\(278\) 8.00000i 0.0287770i
\(279\) 0 0
\(280\) 0 0
\(281\) − 448.306i − 1.59539i −0.603058 0.797697i \(-0.706051\pi\)
0.603058 0.797697i \(-0.293949\pi\)
\(282\) 0 0
\(283\) −124.451 −0.439755 −0.219878 0.975527i \(-0.570566\pi\)
−0.219878 + 0.975527i \(0.570566\pi\)
\(284\) − 16.9706i − 0.0597555i
\(285\) 0 0
\(286\) −50.9117 −0.178013
\(287\) 0 0
\(288\) 0 0
\(289\) −195.000 −0.674740
\(290\) − 100.000i − 0.344828i
\(291\) 0 0
\(292\) −149.907 −0.513379
\(293\) 284.000i 0.969283i 0.874713 + 0.484642i \(0.161050\pi\)
−0.874713 + 0.484642i \(0.838950\pi\)
\(294\) 0 0
\(295\) 200.000 0.677966
\(296\) − 181.019i − 0.611552i
\(297\) 0 0
\(298\) −26.0000 −0.0872483
\(299\) − 36.0000i − 0.120401i
\(300\) 0 0
\(301\) 0 0
\(302\) − 90.5097i − 0.299701i
\(303\) 0 0
\(304\) 22.6274 0.0744323
\(305\) 104.652i 0.343121i
\(306\) 0 0
\(307\) 282.843 0.921312 0.460656 0.887579i \(-0.347614\pi\)
0.460656 + 0.887579i \(0.347614\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 96.0000 0.309677
\(311\) 188.000i 0.604502i 0.953228 + 0.302251i \(0.0977380\pi\)
−0.953228 + 0.302251i \(0.902262\pi\)
\(312\) 0 0
\(313\) 216.375 0.691293 0.345646 0.938365i \(-0.387660\pi\)
0.345646 + 0.938365i \(0.387660\pi\)
\(314\) 230.000i 0.732484i
\(315\) 0 0
\(316\) −184.000 −0.582278
\(317\) − 456.791i − 1.44098i −0.693465 0.720491i \(-0.743917\pi\)
0.693465 0.720491i \(-0.256083\pi\)
\(318\) 0 0
\(319\) −100.000 −0.313480
\(320\) 16.0000i 0.0500000i
\(321\) 0 0
\(322\) 0 0
\(323\) 124.451i 0.385297i
\(324\) 0 0
\(325\) −267.286 −0.822420
\(326\) − 316.784i − 0.971730i
\(327\) 0 0
\(328\) −56.5685 −0.172465
\(329\) 0 0
\(330\) 0 0
\(331\) −500.000 −1.51057 −0.755287 0.655394i \(-0.772503\pi\)
−0.755287 + 0.655394i \(0.772503\pi\)
\(332\) 224.000i 0.674699i
\(333\) 0 0
\(334\) 412.950 1.23638
\(335\) − 240.000i − 0.716418i
\(336\) 0 0
\(337\) 86.0000 0.255193 0.127596 0.991826i \(-0.459274\pi\)
0.127596 + 0.991826i \(0.459274\pi\)
\(338\) − 9.89949i − 0.0292884i
\(339\) 0 0
\(340\) −88.0000 −0.258824
\(341\) − 96.0000i − 0.281525i
\(342\) 0 0
\(343\) 0 0
\(344\) − 124.451i − 0.361776i
\(345\) 0 0
\(346\) −220.617 −0.637622
\(347\) − 246.073i − 0.709145i −0.935029 0.354572i \(-0.884626\pi\)
0.935029 0.354572i \(-0.115374\pi\)
\(348\) 0 0
\(349\) −445.477 −1.27644 −0.638220 0.769854i \(-0.720329\pi\)
−0.638220 + 0.769854i \(0.720329\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 16.0000 0.0454545
\(353\) − 218.000i − 0.617564i −0.951133 0.308782i \(-0.900079\pi\)
0.951133 0.308782i \(-0.0999213\pi\)
\(354\) 0 0
\(355\) 16.9706 0.0478044
\(356\) − 40.0000i − 0.112360i
\(357\) 0 0
\(358\) −132.000 −0.368715
\(359\) 370.524i 1.03210i 0.856558 + 0.516050i \(0.172598\pi\)
−0.856558 + 0.516050i \(0.827402\pi\)
\(360\) 0 0
\(361\) −329.000 −0.911357
\(362\) 270.000i 0.745856i
\(363\) 0 0
\(364\) 0 0
\(365\) − 149.907i − 0.410703i
\(366\) 0 0
\(367\) −446.891 −1.21769 −0.608844 0.793290i \(-0.708366\pi\)
−0.608844 + 0.793290i \(0.708366\pi\)
\(368\) 11.3137i 0.0307438i
\(369\) 0 0
\(370\) 181.019 0.489241
\(371\) 0 0
\(372\) 0 0
\(373\) 286.000 0.766756 0.383378 0.923592i \(-0.374761\pi\)
0.383378 + 0.923592i \(0.374761\pi\)
\(374\) 88.0000i 0.235294i
\(375\) 0 0
\(376\) −192.333 −0.511524
\(377\) 450.000i 1.19363i
\(378\) 0 0
\(379\) −188.000 −0.496042 −0.248021 0.968755i \(-0.579780\pi\)
−0.248021 + 0.968755i \(0.579780\pi\)
\(380\) 22.6274i 0.0595458i
\(381\) 0 0
\(382\) 188.000 0.492147
\(383\) − 352.000i − 0.919060i −0.888162 0.459530i \(-0.848018\pi\)
0.888162 0.459530i \(-0.151982\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 172.534i 0.446979i
\(387\) 0 0
\(388\) −53.7401 −0.138505
\(389\) − 205.061i − 0.527149i −0.964639 0.263575i \(-0.915099\pi\)
0.964639 0.263575i \(-0.0849015\pi\)
\(390\) 0 0
\(391\) −62.2254 −0.159144
\(392\) 0 0
\(393\) 0 0
\(394\) −10.0000 −0.0253807
\(395\) − 184.000i − 0.465823i
\(396\) 0 0
\(397\) −244.659 −0.616269 −0.308135 0.951343i \(-0.599705\pi\)
−0.308135 + 0.951343i \(0.599705\pi\)
\(398\) − 528.000i − 1.32663i
\(399\) 0 0
\(400\) 84.0000 0.210000
\(401\) − 445.477i − 1.11092i −0.831545 0.555458i \(-0.812543\pi\)
0.831545 0.555458i \(-0.187457\pi\)
\(402\) 0 0
\(403\) −432.000 −1.07196
\(404\) − 168.000i − 0.415842i
\(405\) 0 0
\(406\) 0 0
\(407\) − 181.019i − 0.444765i
\(408\) 0 0
\(409\) 750.947 1.83606 0.918029 0.396514i \(-0.129780\pi\)
0.918029 + 0.396514i \(0.129780\pi\)
\(410\) − 56.5685i − 0.137972i
\(411\) 0 0
\(412\) 316.784 0.768893
\(413\) 0 0
\(414\) 0 0
\(415\) −224.000 −0.539759
\(416\) − 72.0000i − 0.173077i
\(417\) 0 0
\(418\) 22.6274 0.0541326
\(419\) − 236.000i − 0.563246i −0.959525 0.281623i \(-0.909127\pi\)
0.959525 0.281623i \(-0.0908727\pi\)
\(420\) 0 0
\(421\) −96.0000 −0.228029 −0.114014 0.993479i \(-0.536371\pi\)
−0.114014 + 0.993479i \(0.536371\pi\)
\(422\) − 16.9706i − 0.0402146i
\(423\) 0 0
\(424\) 52.0000 0.122642
\(425\) 462.000i 1.08706i
\(426\) 0 0
\(427\) 0 0
\(428\) − 277.186i − 0.647631i
\(429\) 0 0
\(430\) 124.451 0.289420
\(431\) 857.013i 1.98843i 0.107407 + 0.994215i \(0.465745\pi\)
−0.107407 + 0.994215i \(0.534255\pi\)
\(432\) 0 0
\(433\) −156.978 −0.362535 −0.181268 0.983434i \(-0.558020\pi\)
−0.181268 + 0.983434i \(0.558020\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −140.000 −0.321101
\(437\) 16.0000i 0.0366133i
\(438\) 0 0
\(439\) 644.881 1.46898 0.734489 0.678621i \(-0.237422\pi\)
0.734489 + 0.678621i \(0.237422\pi\)
\(440\) 16.0000i 0.0363636i
\(441\) 0 0
\(442\) 396.000 0.895928
\(443\) 579.828i 1.30887i 0.756120 + 0.654433i \(0.227093\pi\)
−0.756120 + 0.654433i \(0.772907\pi\)
\(444\) 0 0
\(445\) 40.0000 0.0898876
\(446\) 288.000i 0.645740i
\(447\) 0 0
\(448\) 0 0
\(449\) 869.741i 1.93706i 0.248891 + 0.968532i \(0.419934\pi\)
−0.248891 + 0.968532i \(0.580066\pi\)
\(450\) 0 0
\(451\) −56.5685 −0.125429
\(452\) − 42.4264i − 0.0938637i
\(453\) 0 0
\(454\) −288.500 −0.635462
\(455\) 0 0
\(456\) 0 0
\(457\) 384.000 0.840263 0.420131 0.907463i \(-0.361984\pi\)
0.420131 + 0.907463i \(0.361984\pi\)
\(458\) 66.0000i 0.144105i
\(459\) 0 0
\(460\) −11.3137 −0.0245950
\(461\) − 268.000i − 0.581345i −0.956823 0.290672i \(-0.906121\pi\)
0.956823 0.290672i \(-0.0938790\pi\)
\(462\) 0 0
\(463\) −716.000 −1.54644 −0.773218 0.634140i \(-0.781354\pi\)
−0.773218 + 0.634140i \(0.781354\pi\)
\(464\) − 141.421i − 0.304787i
\(465\) 0 0
\(466\) 438.000 0.939914
\(467\) − 308.000i − 0.659529i −0.944063 0.329764i \(-0.893031\pi\)
0.944063 0.329764i \(-0.106969\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) − 192.333i − 0.409219i
\(471\) 0 0
\(472\) 282.843 0.599243
\(473\) − 124.451i − 0.263109i
\(474\) 0 0
\(475\) 118.794 0.250093
\(476\) 0 0
\(477\) 0 0
\(478\) 60.0000 0.125523
\(479\) 388.000i 0.810021i 0.914312 + 0.405010i \(0.132732\pi\)
−0.914312 + 0.405010i \(0.867268\pi\)
\(480\) 0 0
\(481\) −814.587 −1.69353
\(482\) − 534.000i − 1.10788i
\(483\) 0 0
\(484\) −226.000 −0.466942
\(485\) − 53.7401i − 0.110804i
\(486\) 0 0
\(487\) 120.000 0.246407 0.123203 0.992381i \(-0.460683\pi\)
0.123203 + 0.992381i \(0.460683\pi\)
\(488\) 148.000i 0.303279i
\(489\) 0 0
\(490\) 0 0
\(491\) 619.426i 1.26156i 0.775962 + 0.630780i \(0.217265\pi\)
−0.775962 + 0.630780i \(0.782735\pi\)
\(492\) 0 0
\(493\) 777.817 1.57772
\(494\) − 101.823i − 0.206120i
\(495\) 0 0
\(496\) 135.765 0.273719
\(497\) 0 0
\(498\) 0 0
\(499\) 476.000 0.953908 0.476954 0.878928i \(-0.341741\pi\)
0.476954 + 0.878928i \(0.341741\pi\)
\(500\) 184.000i 0.368000i
\(501\) 0 0
\(502\) −469.519 −0.935297
\(503\) 968.000i 1.92445i 0.272250 + 0.962227i \(0.412232\pi\)
−0.272250 + 0.962227i \(0.587768\pi\)
\(504\) 0 0
\(505\) 168.000 0.332673
\(506\) 11.3137i 0.0223591i
\(507\) 0 0
\(508\) 40.0000 0.0787402
\(509\) 254.000i 0.499018i 0.968373 + 0.249509i \(0.0802692\pi\)
−0.968373 + 0.249509i \(0.919731\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 22.6274i 0.0441942i
\(513\) 0 0
\(514\) −254.558 −0.495250
\(515\) 316.784i 0.615114i
\(516\) 0 0
\(517\) −192.333 −0.372017
\(518\) 0 0
\(519\) 0 0
\(520\) 72.0000 0.138462
\(521\) − 806.000i − 1.54702i −0.633781 0.773512i \(-0.718498\pi\)
0.633781 0.773512i \(-0.281502\pi\)
\(522\) 0 0
\(523\) −610.940 −1.16815 −0.584073 0.811701i \(-0.698542\pi\)
−0.584073 + 0.811701i \(0.698542\pi\)
\(524\) − 288.000i − 0.549618i
\(525\) 0 0
\(526\) 228.000 0.433460
\(527\) 746.705i 1.41690i
\(528\) 0 0
\(529\) 521.000 0.984877
\(530\) 52.0000i 0.0981132i
\(531\) 0 0
\(532\) 0 0
\(533\) 254.558i 0.477596i
\(534\) 0 0
\(535\) 277.186 0.518104
\(536\) − 339.411i − 0.633230i
\(537\) 0 0
\(538\) 743.876 1.38267
\(539\) 0 0
\(540\) 0 0
\(541\) −80.0000 −0.147874 −0.0739372 0.997263i \(-0.523556\pi\)
−0.0739372 + 0.997263i \(0.523556\pi\)
\(542\) 608.000i 1.12177i
\(543\) 0 0
\(544\) −124.451 −0.228770
\(545\) − 140.000i − 0.256881i
\(546\) 0 0
\(547\) 256.000 0.468007 0.234004 0.972236i \(-0.424817\pi\)
0.234004 + 0.972236i \(0.424817\pi\)
\(548\) 398.808i 0.727752i
\(549\) 0 0
\(550\) 84.0000 0.152727
\(551\) − 200.000i − 0.362976i
\(552\) 0 0
\(553\) 0 0
\(554\) 90.5097i 0.163375i
\(555\) 0 0
\(556\) −11.3137 −0.0203484
\(557\) 784.889i 1.40914i 0.709637 + 0.704568i \(0.248859\pi\)
−0.709637 + 0.704568i \(0.751141\pi\)
\(558\) 0 0
\(559\) −560.029 −1.00184
\(560\) 0 0
\(561\) 0 0
\(562\) 634.000 1.12811
\(563\) 604.000i 1.07282i 0.843956 + 0.536412i \(0.180221\pi\)
−0.843956 + 0.536412i \(0.819779\pi\)
\(564\) 0 0
\(565\) 42.4264 0.0750910
\(566\) − 176.000i − 0.310954i
\(567\) 0 0
\(568\) 24.0000 0.0422535
\(569\) 434.164i 0.763029i 0.924363 + 0.381515i \(0.124597\pi\)
−0.924363 + 0.381515i \(0.875403\pi\)
\(570\) 0 0
\(571\) −248.000 −0.434326 −0.217163 0.976135i \(-0.569680\pi\)
−0.217163 + 0.976135i \(0.569680\pi\)
\(572\) − 72.0000i − 0.125874i
\(573\) 0 0
\(574\) 0 0
\(575\) 59.3970i 0.103299i
\(576\) 0 0
\(577\) 767.918 1.33088 0.665440 0.746451i \(-0.268244\pi\)
0.665440 + 0.746451i \(0.268244\pi\)
\(578\) − 275.772i − 0.477114i
\(579\) 0 0
\(580\) 141.421 0.243830
\(581\) 0 0
\(582\) 0 0
\(583\) 52.0000 0.0891938
\(584\) − 212.000i − 0.363014i
\(585\) 0 0
\(586\) −401.637 −0.685387
\(587\) − 1140.00i − 1.94208i −0.238920 0.971039i \(-0.576793\pi\)
0.238920 0.971039i \(-0.423207\pi\)
\(588\) 0 0
\(589\) 192.000 0.325976
\(590\) 282.843i 0.479394i
\(591\) 0 0
\(592\) 256.000 0.432432
\(593\) − 682.000i − 1.15008i −0.818124 0.575042i \(-0.804986\pi\)
0.818124 0.575042i \(-0.195014\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) − 36.7696i − 0.0616939i
\(597\) 0 0
\(598\) 50.9117 0.0851366
\(599\) 1055.00i 1.76127i 0.473791 + 0.880637i \(0.342885\pi\)
−0.473791 + 0.880637i \(0.657115\pi\)
\(600\) 0 0
\(601\) −555.786 −0.924769 −0.462384 0.886680i \(-0.653006\pi\)
−0.462384 + 0.886680i \(0.653006\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 128.000 0.211921
\(605\) − 226.000i − 0.373554i
\(606\) 0 0
\(607\) −424.264 −0.698952 −0.349476 0.936945i \(-0.613640\pi\)
−0.349476 + 0.936945i \(0.613640\pi\)
\(608\) 32.0000i 0.0526316i
\(609\) 0 0
\(610\) −148.000 −0.242623
\(611\) 865.499i 1.41653i
\(612\) 0 0
\(613\) 312.000 0.508972 0.254486 0.967076i \(-0.418094\pi\)
0.254486 + 0.967076i \(0.418094\pi\)
\(614\) 400.000i 0.651466i
\(615\) 0 0
\(616\) 0 0
\(617\) − 490.732i − 0.795352i −0.917526 0.397676i \(-0.869817\pi\)
0.917526 0.397676i \(-0.130183\pi\)
\(618\) 0 0
\(619\) 277.186 0.447796 0.223898 0.974613i \(-0.428122\pi\)
0.223898 + 0.974613i \(0.428122\pi\)
\(620\) 135.765i 0.218975i
\(621\) 0 0
\(622\) −265.872 −0.427447
\(623\) 0 0
\(624\) 0 0
\(625\) 341.000 0.545600
\(626\) 306.000i 0.488818i
\(627\) 0 0
\(628\) −325.269 −0.517944
\(629\) 1408.00i 2.23847i
\(630\) 0 0
\(631\) −316.000 −0.500792 −0.250396 0.968143i \(-0.580561\pi\)
−0.250396 + 0.968143i \(0.580561\pi\)
\(632\) − 260.215i − 0.411733i
\(633\) 0 0
\(634\) 646.000 1.01893
\(635\) 40.0000i 0.0629921i
\(636\) 0 0
\(637\) 0 0
\(638\) − 141.421i − 0.221664i
\(639\) 0 0
\(640\) −22.6274 −0.0353553
\(641\) − 434.164i − 0.677322i −0.940908 0.338661i \(-0.890026\pi\)
0.940908 0.338661i \(-0.109974\pi\)
\(642\) 0 0
\(643\) 158.392 0.246333 0.123166 0.992386i \(-0.460695\pi\)
0.123166 + 0.992386i \(0.460695\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −176.000 −0.272446
\(647\) 876.000i 1.35394i 0.736010 + 0.676971i \(0.236708\pi\)
−0.736010 + 0.676971i \(0.763292\pi\)
\(648\) 0 0
\(649\) 282.843 0.435813
\(650\) − 378.000i − 0.581538i
\(651\) 0 0
\(652\) 448.000 0.687117
\(653\) − 292.742i − 0.448304i −0.974554 0.224152i \(-0.928039\pi\)
0.974554 0.224152i \(-0.0719611\pi\)
\(654\) 0 0
\(655\) 288.000 0.439695
\(656\) − 80.0000i − 0.121951i
\(657\) 0 0
\(658\) 0 0
\(659\) 907.925i 1.37773i 0.724889 + 0.688866i \(0.241891\pi\)
−0.724889 + 0.688866i \(0.758109\pi\)
\(660\) 0 0
\(661\) 784.889 1.18743 0.593713 0.804677i \(-0.297661\pi\)
0.593713 + 0.804677i \(0.297661\pi\)
\(662\) − 707.107i − 1.06814i
\(663\) 0 0
\(664\) −316.784 −0.477084
\(665\) 0 0
\(666\) 0 0
\(667\) 100.000 0.149925
\(668\) 584.000i 0.874251i
\(669\) 0 0
\(670\) 339.411 0.506584
\(671\) 148.000i 0.220566i
\(672\) 0 0
\(673\) −264.000 −0.392273 −0.196137 0.980577i \(-0.562840\pi\)
−0.196137 + 0.980577i \(0.562840\pi\)
\(674\) 121.622i 0.180449i
\(675\) 0 0
\(676\) 14.0000 0.0207101
\(677\) 300.000i 0.443131i 0.975145 + 0.221566i \(0.0711167\pi\)
−0.975145 + 0.221566i \(0.928883\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) − 124.451i − 0.183016i
\(681\) 0 0
\(682\) 135.765 0.199068
\(683\) 1162.48i 1.70203i 0.525145 + 0.851013i \(0.324011\pi\)
−0.525145 + 0.851013i \(0.675989\pi\)
\(684\) 0 0
\(685\) −398.808 −0.582202
\(686\) 0 0
\(687\) 0 0
\(688\) 176.000 0.255814
\(689\) − 234.000i − 0.339623i
\(690\) 0 0
\(691\) 509.117 0.736783 0.368391 0.929671i \(-0.379909\pi\)
0.368391 + 0.929671i \(0.379909\pi\)
\(692\) − 312.000i − 0.450867i
\(693\) 0 0
\(694\) 348.000 0.501441
\(695\) − 11.3137i − 0.0162787i
\(696\) 0 0
\(697\) 440.000 0.631277
\(698\) − 630.000i − 0.902579i
\(699\) 0 0
\(700\) 0 0
\(701\) 182.434i 0.260248i 0.991498 + 0.130124i \(0.0415375\pi\)
−0.991498 + 0.130124i \(0.958463\pi\)
\(702\) 0 0
\(703\) 362.039 0.514991
\(704\) 22.6274i 0.0321412i
\(705\) 0 0
\(706\) 308.299 0.436684
\(707\) 0 0
\(708\) 0 0
\(709\) 1318.00 1.85896 0.929478 0.368877i \(-0.120258\pi\)
0.929478 + 0.368877i \(0.120258\pi\)
\(710\) 24.0000i 0.0338028i
\(711\) 0 0
\(712\) 56.5685 0.0794502
\(713\) 96.0000i 0.134642i
\(714\) 0 0
\(715\) 72.0000 0.100699
\(716\) − 186.676i − 0.260721i
\(717\) 0 0
\(718\) −524.000 −0.729805
\(719\) − 1016.00i − 1.41307i −0.707676 0.706537i \(-0.750256\pi\)
0.707676 0.706537i \(-0.249744\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 465.276i − 0.644427i
\(723\) 0 0
\(724\) −381.838 −0.527400
\(725\) − 742.462i − 1.02409i
\(726\) 0 0
\(727\) −384.666 −0.529114 −0.264557 0.964370i \(-0.585226\pi\)
−0.264557 + 0.964370i \(0.585226\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 212.000 0.290411
\(731\) 968.000i 1.32421i
\(732\) 0 0
\(733\) −555.786 −0.758235 −0.379117 0.925349i \(-0.623772\pi\)
−0.379117 + 0.925349i \(0.623772\pi\)
\(734\) − 632.000i − 0.861035i
\(735\) 0 0
\(736\) −16.0000 −0.0217391
\(737\) − 339.411i − 0.460531i
\(738\) 0 0
\(739\) −1200.00 −1.62382 −0.811908 0.583785i \(-0.801571\pi\)
−0.811908 + 0.583785i \(0.801571\pi\)
\(740\) 256.000i 0.345946i
\(741\) 0 0
\(742\) 0 0
\(743\) − 1309.56i − 1.76253i −0.472620 0.881266i \(-0.656692\pi\)
0.472620 0.881266i \(-0.343308\pi\)
\(744\) 0 0
\(745\) 36.7696 0.0493551
\(746\) 404.465i 0.542178i
\(747\) 0 0
\(748\) −124.451 −0.166378
\(749\) 0 0
\(750\) 0 0
\(751\) −568.000 −0.756325 −0.378162 0.925739i \(-0.623444\pi\)
−0.378162 + 0.925739i \(0.623444\pi\)
\(752\) − 272.000i − 0.361702i
\(753\) 0 0
\(754\) −636.396 −0.844027
\(755\) 128.000i 0.169536i
\(756\) 0 0
\(757\) −358.000 −0.472919 −0.236460 0.971641i \(-0.575987\pi\)
−0.236460 + 0.971641i \(0.575987\pi\)
\(758\) − 265.872i − 0.350755i
\(759\) 0 0
\(760\) −32.0000 −0.0421053
\(761\) − 508.000i − 0.667543i −0.942654 0.333771i \(-0.891679\pi\)
0.942654 0.333771i \(-0.108321\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 265.872i 0.348000i
\(765\) 0 0
\(766\) 497.803 0.649874
\(767\) − 1272.79i − 1.65944i
\(768\) 0 0
\(769\) −861.256 −1.11997 −0.559984 0.828503i \(-0.689193\pi\)
−0.559984 + 0.828503i \(0.689193\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −244.000 −0.316062
\(773\) − 892.000i − 1.15395i −0.816763 0.576973i \(-0.804234\pi\)
0.816763 0.576973i \(-0.195766\pi\)
\(774\) 0 0
\(775\) 712.764 0.919695
\(776\) − 76.0000i − 0.0979381i
\(777\) 0 0
\(778\) 290.000 0.372751
\(779\) − 113.137i − 0.145234i
\(780\) 0 0
\(781\) 24.0000 0.0307298
\(782\) − 88.0000i − 0.112532i
\(783\) 0 0
\(784\) 0 0
\(785\) − 325.269i − 0.414356i
\(786\) 0 0
\(787\) −429.921 −0.546278 −0.273139 0.961975i \(-0.588062\pi\)
−0.273139 + 0.961975i \(0.588062\pi\)
\(788\) − 14.1421i − 0.0179469i
\(789\) 0 0
\(790\) 260.215 0.329386
\(791\) 0 0
\(792\) 0 0
\(793\) 666.000 0.839849
\(794\) − 346.000i − 0.435768i
\(795\) 0 0
\(796\) 746.705 0.938071
\(797\) − 540.000i − 0.677541i −0.940869 0.338770i \(-0.889989\pi\)
0.940869 0.338770i \(-0.110011\pi\)
\(798\) 0 0
\(799\) 1496.00 1.87234
\(800\) 118.794i 0.148492i
\(801\) 0 0
\(802\) 630.000 0.785536
\(803\) − 212.000i − 0.264010i
\(804\) 0 0
\(805\) 0 0
\(806\) − 610.940i − 0.757990i
\(807\) 0 0
\(808\) 237.588 0.294044
\(809\) 872.570i 1.07858i 0.842121 + 0.539289i \(0.181307\pi\)
−0.842121 + 0.539289i \(0.818693\pi\)
\(810\) 0 0
\(811\) 492.146 0.606839 0.303419 0.952857i \(-0.401872\pi\)
0.303419 + 0.952857i \(0.401872\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 256.000 0.314496
\(815\) 448.000i 0.549693i
\(816\) 0 0
\(817\) 248.902 0.304653
\(818\) 1062.00i 1.29829i
\(819\) 0 0
\(820\) 80.0000 0.0975610
\(821\) − 750.947i − 0.914674i −0.889293 0.457337i \(-0.848803\pi\)
0.889293 0.457337i \(-0.151197\pi\)
\(822\) 0 0
\(823\) −116.000 −0.140948 −0.0704739 0.997514i \(-0.522451\pi\)
−0.0704739 + 0.997514i \(0.522451\pi\)
\(824\) 448.000i 0.543689i
\(825\) 0 0
\(826\) 0 0
\(827\) 873.984i 1.05681i 0.848992 + 0.528406i \(0.177210\pi\)
−0.848992 + 0.528406i \(0.822790\pi\)
\(828\) 0 0
\(829\) −272.943 −0.329244 −0.164622 0.986357i \(-0.552640\pi\)
−0.164622 + 0.986357i \(0.552640\pi\)
\(830\) − 316.784i − 0.381667i
\(831\) 0 0
\(832\) 101.823 0.122384
\(833\) 0 0
\(834\) 0 0
\(835\) −584.000 −0.699401
\(836\) 32.0000i 0.0382775i
\(837\) 0 0
\(838\) 333.754 0.398275
\(839\) − 188.000i − 0.224076i −0.993704 0.112038i \(-0.964262\pi\)
0.993704 0.112038i \(-0.0357379\pi\)
\(840\) 0 0
\(841\) −409.000 −0.486326
\(842\) − 135.765i − 0.161241i
\(843\) 0 0
\(844\) 24.0000 0.0284360
\(845\) 14.0000i 0.0165680i
\(846\) 0 0
\(847\) 0 0
\(848\) 73.5391i 0.0867206i
\(849\) 0 0
\(850\) −653.367 −0.768667
\(851\) 181.019i 0.212714i
\(852\) 0 0
\(853\) −63.6396 −0.0746068 −0.0373034 0.999304i \(-0.511877\pi\)
−0.0373034 + 0.999304i \(0.511877\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 392.000 0.457944
\(857\) 260.000i 0.303384i 0.988428 + 0.151692i \(0.0484722\pi\)
−0.988428 + 0.151692i \(0.951528\pi\)
\(858\) 0 0
\(859\) 791.960 0.921955 0.460978 0.887412i \(-0.347499\pi\)
0.460978 + 0.887412i \(0.347499\pi\)
\(860\) 176.000i 0.204651i
\(861\) 0 0
\(862\) −1212.00 −1.40603
\(863\) − 421.436i − 0.488338i −0.969733 0.244169i \(-0.921485\pi\)
0.969733 0.244169i \(-0.0785152\pi\)
\(864\) 0 0
\(865\) 312.000 0.360694
\(866\) − 222.000i − 0.256351i
\(867\) 0 0
\(868\) 0 0
\(869\) − 260.215i − 0.299442i
\(870\) 0 0
\(871\) −1527.35 −1.75356
\(872\) − 197.990i − 0.227053i
\(873\) 0 0
\(874\) −22.6274 −0.0258895
\(875\) 0 0
\(876\) 0 0
\(877\) −952.000 −1.08552 −0.542759 0.839888i \(-0.682620\pi\)
−0.542759 + 0.839888i \(0.682620\pi\)
\(878\) 912.000i 1.03872i
\(879\) 0 0
\(880\) −22.6274 −0.0257130
\(881\) 1158.00i 1.31442i 0.753710 + 0.657208i \(0.228263\pi\)
−0.753710 + 0.657208i \(0.771737\pi\)
\(882\) 0 0
\(883\) −1036.00 −1.17327 −0.586636 0.809850i \(-0.699548\pi\)
−0.586636 + 0.809850i \(0.699548\pi\)
\(884\) 560.029i 0.633516i
\(885\) 0 0
\(886\) −820.000 −0.925508
\(887\) − 460.000i − 0.518602i −0.965797 0.259301i \(-0.916508\pi\)
0.965797 0.259301i \(-0.0834922\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 56.5685i 0.0635602i
\(891\) 0 0
\(892\) −407.294 −0.456607
\(893\) − 384.666i − 0.430757i
\(894\) 0 0
\(895\) 186.676 0.208577
\(896\) 0 0
\(897\) 0 0
\(898\) −1230.00 −1.36971
\(899\) − 1200.00i − 1.33482i
\(900\) 0 0
\(901\) −404.465 −0.448907
\(902\) − 80.0000i − 0.0886918i
\(903\) 0 0
\(904\) 60.0000 0.0663717
\(905\) − 381.838i − 0.421920i
\(906\) 0 0
\(907\) 220.000 0.242558 0.121279 0.992618i \(-0.461300\pi\)
0.121279 + 0.992618i \(0.461300\pi\)
\(908\) − 408.000i − 0.449339i
\(909\) 0 0
\(910\) 0 0
\(911\) − 1049.35i − 1.15186i −0.817498 0.575931i \(-0.804640\pi\)
0.817498 0.575931i \(-0.195360\pi\)
\(912\) 0 0
\(913\) −316.784 −0.346970
\(914\) 543.058i 0.594155i
\(915\) 0 0
\(916\) −93.3381 −0.101897
\(917\) 0 0
\(918\) 0 0
\(919\) −320.000 −0.348205 −0.174102 0.984728i \(-0.555702\pi\)
−0.174102 + 0.984728i \(0.555702\pi\)
\(920\) − 16.0000i − 0.0173913i
\(921\) 0 0
\(922\) 379.009 0.411073
\(923\) − 108.000i − 0.117010i
\(924\) 0 0
\(925\) 1344.00 1.45297
\(926\) − 1012.58i − 1.09350i
\(927\) 0 0
\(928\) 200.000 0.215517
\(929\) 1756.00i 1.89020i 0.326775 + 0.945102i \(0.394038\pi\)
−0.326775 + 0.945102i \(0.605962\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 619.426i 0.664620i
\(933\) 0 0
\(934\) 435.578 0.466357
\(935\) − 124.451i − 0.133102i
\(936\) 0 0
\(937\) 1740.90 1.85795 0.928974 0.370145i \(-0.120692\pi\)
0.928974 + 0.370145i \(0.120692\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 272.000 0.289362
\(941\) − 626.000i − 0.665250i −0.943059 0.332625i \(-0.892066\pi\)
0.943059 0.332625i \(-0.107934\pi\)
\(942\) 0 0
\(943\) 56.5685 0.0599878
\(944\) 400.000i 0.423729i
\(945\) 0 0
\(946\) 176.000 0.186047
\(947\) 789.131i 0.833296i 0.909068 + 0.416648i \(0.136795\pi\)
−0.909068 + 0.416648i \(0.863205\pi\)
\(948\) 0 0
\(949\) −954.000 −1.00527
\(950\) 168.000i 0.176842i
\(951\) 0 0
\(952\) 0 0
\(953\) − 193.747i − 0.203302i −0.994820 0.101651i \(-0.967587\pi\)
0.994820 0.101651i \(-0.0324126\pi\)
\(954\) 0 0
\(955\) −265.872 −0.278400
\(956\) 84.8528i 0.0887582i
\(957\) 0 0
\(958\) −548.715 −0.572771
\(959\) 0 0
\(960\) 0 0
\(961\) 191.000 0.198751
\(962\) − 1152.00i − 1.19751i
\(963\) 0 0
\(964\) 755.190 0.783392
\(965\) − 244.000i − 0.252850i
\(966\) 0 0
\(967\) 132.000 0.136505 0.0682523 0.997668i \(-0.478258\pi\)
0.0682523 + 0.997668i \(0.478258\pi\)
\(968\) − 319.612i − 0.330178i
\(969\) 0 0
\(970\) 76.0000 0.0783505
\(971\) 160.000i 0.164779i 0.996600 + 0.0823893i \(0.0262551\pi\)
−0.996600 + 0.0823893i \(0.973745\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 169.706i 0.174236i
\(975\) 0 0
\(976\) −209.304 −0.214450
\(977\) − 1497.65i − 1.53291i −0.642299 0.766455i \(-0.722019\pi\)
0.642299 0.766455i \(-0.277981\pi\)
\(978\) 0 0
\(979\) 56.5685 0.0577820
\(980\) 0 0
\(981\) 0 0
\(982\) −876.000 −0.892057
\(983\) − 328.000i − 0.333672i −0.985985 0.166836i \(-0.946645\pi\)
0.985985 0.166836i \(-0.0533551\pi\)
\(984\) 0 0
\(985\) 14.1421 0.0143575
\(986\) 1100.00i 1.11562i
\(987\) 0 0
\(988\) 144.000 0.145749
\(989\) 124.451i 0.125835i
\(990\) 0 0
\(991\) −16.0000 −0.0161453 −0.00807265 0.999967i \(-0.502570\pi\)
−0.00807265 + 0.999967i \(0.502570\pi\)
\(992\) 192.000i 0.193548i
\(993\) 0 0
\(994\) 0 0
\(995\) 746.705i 0.750457i
\(996\) 0 0
\(997\) −875.398 −0.878032 −0.439016 0.898479i \(-0.644673\pi\)
−0.439016 + 0.898479i \(0.644673\pi\)
\(998\) 673.166i 0.674515i
\(999\) 0 0
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 882.3.b.i.197.3 yes 4
3.2 odd 2 inner 882.3.b.i.197.2 yes 4
7.2 even 3 882.3.s.f.557.3 8
7.3 odd 6 882.3.s.f.863.1 8
7.4 even 3 882.3.s.f.863.2 8
7.5 odd 6 882.3.s.f.557.4 8
7.6 odd 2 inner 882.3.b.i.197.4 yes 4
21.2 odd 6 882.3.s.f.557.2 8
21.5 even 6 882.3.s.f.557.1 8
21.11 odd 6 882.3.s.f.863.3 8
21.17 even 6 882.3.s.f.863.4 8
21.20 even 2 inner 882.3.b.i.197.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
882.3.b.i.197.1 4 21.20 even 2 inner
882.3.b.i.197.2 yes 4 3.2 odd 2 inner
882.3.b.i.197.3 yes 4 1.1 even 1 trivial
882.3.b.i.197.4 yes 4 7.6 odd 2 inner
882.3.s.f.557.1 8 21.5 even 6
882.3.s.f.557.2 8 21.2 odd 6
882.3.s.f.557.3 8 7.2 even 3
882.3.s.f.557.4 8 7.5 odd 6
882.3.s.f.863.1 8 7.3 odd 6
882.3.s.f.863.2 8 7.4 even 3
882.3.s.f.863.3 8 21.11 odd 6
882.3.s.f.863.4 8 21.17 even 6