Properties

Label 800.4.a.x.1.1
Level $800$
Weight $4$
Character 800.1
Self dual yes
Analytic conductor $47.202$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [800,4,Mod(1,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(47.2015280046\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.5685.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 11x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.12873\) of defining polynomial
Character \(\chi\) \(=\) 800.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.25747 q^{3} -9.15590 q^{7} +0.640965 q^{9} +O(q^{10})\) \(q-5.25747 q^{3} -9.15590 q^{7} +0.640965 q^{9} +11.8984 q^{11} -41.9826 q^{13} -75.7604 q^{17} -2.49047 q^{19} +48.1368 q^{21} +17.8566 q^{23} +138.582 q^{27} -143.538 q^{29} +88.6094 q^{31} -62.5556 q^{33} -351.059 q^{37} +220.722 q^{39} +195.922 q^{41} -366.067 q^{43} -58.5931 q^{47} -259.169 q^{49} +398.308 q^{51} +0.374917 q^{53} +13.0936 q^{57} -318.952 q^{59} +446.597 q^{61} -5.86861 q^{63} +709.343 q^{67} -93.8805 q^{69} +1137.43 q^{71} +85.3855 q^{73} -108.941 q^{77} +1249.83 q^{79} -745.895 q^{81} -926.261 q^{83} +754.648 q^{87} +973.552 q^{89} +384.389 q^{91} -465.861 q^{93} +1158.55 q^{97} +7.62648 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 5 q^{3} - 2 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 5 q^{3} - 2 q^{7} + 18 q^{9} + 31 q^{11} - 8 q^{13} + 89 q^{17} + 87 q^{19} - 70 q^{21} - 122 q^{23} + 143 q^{27} + 84 q^{29} + 294 q^{31} + 209 q^{33} + 94 q^{37} - 32 q^{39} - 345 q^{41} - 412 q^{43} + 824 q^{47} + 359 q^{49} + 1351 q^{51} - 74 q^{53} + 913 q^{57} + 1040 q^{59} + 482 q^{61} - 1532 q^{63} + 735 q^{67} - 614 q^{69} + 2048 q^{71} + 15 q^{73} - 1474 q^{77} + 998 q^{79} - 549 q^{81} - 1221 q^{83} + 2564 q^{87} + 1189 q^{89} + 4032 q^{91} - 454 q^{93} + 1010 q^{97} + 1954 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −5.25747 −1.01180 −0.505900 0.862592i \(-0.668840\pi\)
−0.505900 + 0.862592i \(0.668840\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −9.15590 −0.494372 −0.247186 0.968968i \(-0.579506\pi\)
−0.247186 + 0.968968i \(0.579506\pi\)
\(8\) 0 0
\(9\) 0.640965 0.0237394
\(10\) 0 0
\(11\) 11.8984 0.326137 0.163069 0.986615i \(-0.447861\pi\)
0.163069 + 0.986615i \(0.447861\pi\)
\(12\) 0 0
\(13\) −41.9826 −0.895684 −0.447842 0.894113i \(-0.647807\pi\)
−0.447842 + 0.894113i \(0.647807\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −75.7604 −1.08086 −0.540430 0.841389i \(-0.681738\pi\)
−0.540430 + 0.841389i \(0.681738\pi\)
\(18\) 0 0
\(19\) −2.49047 −0.0300712 −0.0150356 0.999887i \(-0.504786\pi\)
−0.0150356 + 0.999887i \(0.504786\pi\)
\(20\) 0 0
\(21\) 48.1368 0.500206
\(22\) 0 0
\(23\) 17.8566 0.161885 0.0809426 0.996719i \(-0.474207\pi\)
0.0809426 + 0.996719i \(0.474207\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 138.582 0.987781
\(28\) 0 0
\(29\) −143.538 −0.919117 −0.459558 0.888148i \(-0.651992\pi\)
−0.459558 + 0.888148i \(0.651992\pi\)
\(30\) 0 0
\(31\) 88.6094 0.513378 0.256689 0.966494i \(-0.417368\pi\)
0.256689 + 0.966494i \(0.417368\pi\)
\(32\) 0 0
\(33\) −62.5556 −0.329986
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −351.059 −1.55983 −0.779916 0.625884i \(-0.784738\pi\)
−0.779916 + 0.625884i \(0.784738\pi\)
\(38\) 0 0
\(39\) 220.722 0.906253
\(40\) 0 0
\(41\) 195.922 0.746291 0.373145 0.927773i \(-0.378279\pi\)
0.373145 + 0.927773i \(0.378279\pi\)
\(42\) 0 0
\(43\) −366.067 −1.29825 −0.649125 0.760682i \(-0.724865\pi\)
−0.649125 + 0.760682i \(0.724865\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −58.5931 −0.181844 −0.0909222 0.995858i \(-0.528981\pi\)
−0.0909222 + 0.995858i \(0.528981\pi\)
\(48\) 0 0
\(49\) −259.169 −0.755596
\(50\) 0 0
\(51\) 398.308 1.09361
\(52\) 0 0
\(53\) 0.374917 0.000971675 0 0.000485837 1.00000i \(-0.499845\pi\)
0.000485837 1.00000i \(0.499845\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 13.0936 0.0304261
\(58\) 0 0
\(59\) −318.952 −0.703796 −0.351898 0.936038i \(-0.614464\pi\)
−0.351898 + 0.936038i \(0.614464\pi\)
\(60\) 0 0
\(61\) 446.597 0.937391 0.468695 0.883360i \(-0.344724\pi\)
0.468695 + 0.883360i \(0.344724\pi\)
\(62\) 0 0
\(63\) −5.86861 −0.0117361
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 709.343 1.29343 0.646717 0.762730i \(-0.276142\pi\)
0.646717 + 0.762730i \(0.276142\pi\)
\(68\) 0 0
\(69\) −93.8805 −0.163795
\(70\) 0 0
\(71\) 1137.43 1.90124 0.950621 0.310354i \(-0.100448\pi\)
0.950621 + 0.310354i \(0.100448\pi\)
\(72\) 0 0
\(73\) 85.3855 0.136899 0.0684495 0.997655i \(-0.478195\pi\)
0.0684495 + 0.997655i \(0.478195\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −108.941 −0.161233
\(78\) 0 0
\(79\) 1249.83 1.77996 0.889981 0.455997i \(-0.150717\pi\)
0.889981 + 0.455997i \(0.150717\pi\)
\(80\) 0 0
\(81\) −745.895 −1.02318
\(82\) 0 0
\(83\) −926.261 −1.22494 −0.612472 0.790492i \(-0.709825\pi\)
−0.612472 + 0.790492i \(0.709825\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 754.648 0.929962
\(88\) 0 0
\(89\) 973.552 1.15951 0.579755 0.814791i \(-0.303148\pi\)
0.579755 + 0.814791i \(0.303148\pi\)
\(90\) 0 0
\(91\) 384.389 0.442801
\(92\) 0 0
\(93\) −465.861 −0.519436
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1158.55 1.21271 0.606355 0.795194i \(-0.292631\pi\)
0.606355 + 0.795194i \(0.292631\pi\)
\(98\) 0 0
\(99\) 7.62648 0.00774232
\(100\) 0 0
\(101\) −466.816 −0.459900 −0.229950 0.973202i \(-0.573856\pi\)
−0.229950 + 0.973202i \(0.573856\pi\)
\(102\) 0 0
\(103\) −963.139 −0.921368 −0.460684 0.887564i \(-0.652396\pi\)
−0.460684 + 0.887564i \(0.652396\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1066.94 0.963971 0.481986 0.876179i \(-0.339916\pi\)
0.481986 + 0.876179i \(0.339916\pi\)
\(108\) 0 0
\(109\) −217.521 −0.191145 −0.0955723 0.995422i \(-0.530468\pi\)
−0.0955723 + 0.995422i \(0.530468\pi\)
\(110\) 0 0
\(111\) 1845.68 1.57824
\(112\) 0 0
\(113\) 1160.71 0.966286 0.483143 0.875542i \(-0.339495\pi\)
0.483143 + 0.875542i \(0.339495\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −26.9094 −0.0212630
\(118\) 0 0
\(119\) 693.655 0.534347
\(120\) 0 0
\(121\) −1189.43 −0.893634
\(122\) 0 0
\(123\) −1030.06 −0.755097
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −34.2804 −0.0239519 −0.0119759 0.999928i \(-0.503812\pi\)
−0.0119759 + 0.999928i \(0.503812\pi\)
\(128\) 0 0
\(129\) 1924.59 1.31357
\(130\) 0 0
\(131\) 2364.02 1.57669 0.788343 0.615236i \(-0.210939\pi\)
0.788343 + 0.615236i \(0.210939\pi\)
\(132\) 0 0
\(133\) 22.8025 0.0148664
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2991.88 1.86579 0.932897 0.360144i \(-0.117272\pi\)
0.932897 + 0.360144i \(0.117272\pi\)
\(138\) 0 0
\(139\) 132.246 0.0806974 0.0403487 0.999186i \(-0.487153\pi\)
0.0403487 + 0.999186i \(0.487153\pi\)
\(140\) 0 0
\(141\) 308.052 0.183990
\(142\) 0 0
\(143\) −499.528 −0.292116
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1362.58 0.764512
\(148\) 0 0
\(149\) −788.190 −0.433363 −0.216681 0.976242i \(-0.569523\pi\)
−0.216681 + 0.976242i \(0.569523\pi\)
\(150\) 0 0
\(151\) 2200.74 1.18605 0.593025 0.805184i \(-0.297934\pi\)
0.593025 + 0.805184i \(0.297934\pi\)
\(152\) 0 0
\(153\) −48.5598 −0.0256590
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −1702.05 −0.865212 −0.432606 0.901583i \(-0.642406\pi\)
−0.432606 + 0.901583i \(0.642406\pi\)
\(158\) 0 0
\(159\) −1.97111 −0.000983140 0
\(160\) 0 0
\(161\) −163.493 −0.0800315
\(162\) 0 0
\(163\) −822.009 −0.394998 −0.197499 0.980303i \(-0.563282\pi\)
−0.197499 + 0.980303i \(0.563282\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2545.32 1.17942 0.589710 0.807615i \(-0.299242\pi\)
0.589710 + 0.807615i \(0.299242\pi\)
\(168\) 0 0
\(169\) −434.458 −0.197751
\(170\) 0 0
\(171\) −1.59631 −0.000713874 0
\(172\) 0 0
\(173\) 868.181 0.381541 0.190770 0.981635i \(-0.438901\pi\)
0.190770 + 0.981635i \(0.438901\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1676.88 0.712101
\(178\) 0 0
\(179\) 3255.65 1.35944 0.679718 0.733474i \(-0.262102\pi\)
0.679718 + 0.733474i \(0.262102\pi\)
\(180\) 0 0
\(181\) 3482.49 1.43012 0.715059 0.699065i \(-0.246400\pi\)
0.715059 + 0.699065i \(0.246400\pi\)
\(182\) 0 0
\(183\) −2347.97 −0.948452
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −901.431 −0.352509
\(188\) 0 0
\(189\) −1268.84 −0.488331
\(190\) 0 0
\(191\) −2981.37 −1.12945 −0.564724 0.825280i \(-0.691017\pi\)
−0.564724 + 0.825280i \(0.691017\pi\)
\(192\) 0 0
\(193\) 1666.53 0.621552 0.310776 0.950483i \(-0.399411\pi\)
0.310776 + 0.950483i \(0.399411\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −4355.01 −1.57503 −0.787516 0.616294i \(-0.788633\pi\)
−0.787516 + 0.616294i \(0.788633\pi\)
\(198\) 0 0
\(199\) −820.282 −0.292202 −0.146101 0.989270i \(-0.546672\pi\)
−0.146101 + 0.989270i \(0.546672\pi\)
\(200\) 0 0
\(201\) −3729.35 −1.30870
\(202\) 0 0
\(203\) 1314.22 0.454386
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 11.4455 0.00384306
\(208\) 0 0
\(209\) −29.6327 −0.00980736
\(210\) 0 0
\(211\) 4678.00 1.52629 0.763144 0.646229i \(-0.223655\pi\)
0.763144 + 0.646229i \(0.223655\pi\)
\(212\) 0 0
\(213\) −5980.00 −1.92368
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −811.299 −0.253800
\(218\) 0 0
\(219\) −448.912 −0.138514
\(220\) 0 0
\(221\) 3180.62 0.968108
\(222\) 0 0
\(223\) 2238.14 0.672094 0.336047 0.941845i \(-0.390910\pi\)
0.336047 + 0.941845i \(0.390910\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1036.89 −0.303175 −0.151588 0.988444i \(-0.548439\pi\)
−0.151588 + 0.988444i \(0.548439\pi\)
\(228\) 0 0
\(229\) −3888.44 −1.12208 −0.561038 0.827790i \(-0.689598\pi\)
−0.561038 + 0.827790i \(0.689598\pi\)
\(230\) 0 0
\(231\) 572.753 0.163136
\(232\) 0 0
\(233\) 130.042 0.0365638 0.0182819 0.999833i \(-0.494180\pi\)
0.0182819 + 0.999833i \(0.494180\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −6570.95 −1.80097
\(238\) 0 0
\(239\) −5724.03 −1.54919 −0.774595 0.632458i \(-0.782046\pi\)
−0.774595 + 0.632458i \(0.782046\pi\)
\(240\) 0 0
\(241\) 1818.75 0.486123 0.243062 0.970011i \(-0.421848\pi\)
0.243062 + 0.970011i \(0.421848\pi\)
\(242\) 0 0
\(243\) 179.812 0.0474689
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 104.557 0.0269343
\(248\) 0 0
\(249\) 4869.79 1.23940
\(250\) 0 0
\(251\) −3637.78 −0.914800 −0.457400 0.889261i \(-0.651219\pi\)
−0.457400 + 0.889261i \(0.651219\pi\)
\(252\) 0 0
\(253\) 212.466 0.0527968
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2969.73 0.720804 0.360402 0.932797i \(-0.382640\pi\)
0.360402 + 0.932797i \(0.382640\pi\)
\(258\) 0 0
\(259\) 3214.26 0.771137
\(260\) 0 0
\(261\) −92.0030 −0.0218193
\(262\) 0 0
\(263\) −3035.72 −0.711752 −0.355876 0.934533i \(-0.615817\pi\)
−0.355876 + 0.934533i \(0.615817\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −5118.42 −1.17319
\(268\) 0 0
\(269\) −6888.55 −1.56135 −0.780673 0.624939i \(-0.785124\pi\)
−0.780673 + 0.624939i \(0.785124\pi\)
\(270\) 0 0
\(271\) −6173.52 −1.38382 −0.691909 0.721985i \(-0.743230\pi\)
−0.691909 + 0.721985i \(0.743230\pi\)
\(272\) 0 0
\(273\) −2020.91 −0.448026
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −3273.30 −0.710013 −0.355007 0.934864i \(-0.615521\pi\)
−0.355007 + 0.934864i \(0.615521\pi\)
\(278\) 0 0
\(279\) 56.7955 0.0121873
\(280\) 0 0
\(281\) 458.400 0.0973161 0.0486581 0.998815i \(-0.484506\pi\)
0.0486581 + 0.998815i \(0.484506\pi\)
\(282\) 0 0
\(283\) 8064.66 1.69397 0.846987 0.531614i \(-0.178414\pi\)
0.846987 + 0.531614i \(0.178414\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1793.84 −0.368945
\(288\) 0 0
\(289\) 826.645 0.168257
\(290\) 0 0
\(291\) −6091.03 −1.22702
\(292\) 0 0
\(293\) −812.863 −0.162075 −0.0810375 0.996711i \(-0.525823\pi\)
−0.0810375 + 0.996711i \(0.525823\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1648.91 0.322152
\(298\) 0 0
\(299\) −749.667 −0.144998
\(300\) 0 0
\(301\) 3351.68 0.641819
\(302\) 0 0
\(303\) 2454.27 0.465327
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −383.469 −0.0712891 −0.0356445 0.999365i \(-0.511348\pi\)
−0.0356445 + 0.999365i \(0.511348\pi\)
\(308\) 0 0
\(309\) 5063.67 0.932240
\(310\) 0 0
\(311\) −1665.18 −0.303613 −0.151807 0.988410i \(-0.548509\pi\)
−0.151807 + 0.988410i \(0.548509\pi\)
\(312\) 0 0
\(313\) 3769.02 0.680632 0.340316 0.940311i \(-0.389466\pi\)
0.340316 + 0.940311i \(0.389466\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1859.70 0.329498 0.164749 0.986336i \(-0.447319\pi\)
0.164749 + 0.986336i \(0.447319\pi\)
\(318\) 0 0
\(319\) −1707.88 −0.299758
\(320\) 0 0
\(321\) −5609.40 −0.975346
\(322\) 0 0
\(323\) 188.679 0.0325028
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1143.61 0.193400
\(328\) 0 0
\(329\) 536.473 0.0898988
\(330\) 0 0
\(331\) −7235.06 −1.20144 −0.600718 0.799461i \(-0.705118\pi\)
−0.600718 + 0.799461i \(0.705118\pi\)
\(332\) 0 0
\(333\) −225.017 −0.0370295
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 9890.08 1.59866 0.799328 0.600895i \(-0.205189\pi\)
0.799328 + 0.600895i \(0.205189\pi\)
\(338\) 0 0
\(339\) −6102.39 −0.977688
\(340\) 0 0
\(341\) 1054.31 0.167432
\(342\) 0 0
\(343\) 5513.40 0.867918
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6269.88 −0.969984 −0.484992 0.874519i \(-0.661178\pi\)
−0.484992 + 0.874519i \(0.661178\pi\)
\(348\) 0 0
\(349\) −2436.94 −0.373772 −0.186886 0.982382i \(-0.559840\pi\)
−0.186886 + 0.982382i \(0.559840\pi\)
\(350\) 0 0
\(351\) −5818.03 −0.884739
\(352\) 0 0
\(353\) 10712.4 1.61519 0.807595 0.589738i \(-0.200769\pi\)
0.807595 + 0.589738i \(0.200769\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −3646.87 −0.540652
\(358\) 0 0
\(359\) −3540.39 −0.520487 −0.260243 0.965543i \(-0.583803\pi\)
−0.260243 + 0.965543i \(0.583803\pi\)
\(360\) 0 0
\(361\) −6852.80 −0.999096
\(362\) 0 0
\(363\) 6253.38 0.904179
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −12208.3 −1.73642 −0.868212 0.496193i \(-0.834731\pi\)
−0.868212 + 0.496193i \(0.834731\pi\)
\(368\) 0 0
\(369\) 125.579 0.0177165
\(370\) 0 0
\(371\) −3.43270 −0.000480369 0
\(372\) 0 0
\(373\) 3056.65 0.424309 0.212154 0.977236i \(-0.431952\pi\)
0.212154 + 0.977236i \(0.431952\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6026.11 0.823238
\(378\) 0 0
\(379\) −1476.91 −0.200168 −0.100084 0.994979i \(-0.531911\pi\)
−0.100084 + 0.994979i \(0.531911\pi\)
\(380\) 0 0
\(381\) 180.228 0.0242345
\(382\) 0 0
\(383\) 5567.95 0.742843 0.371422 0.928464i \(-0.378870\pi\)
0.371422 + 0.928464i \(0.378870\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −234.636 −0.0308197
\(388\) 0 0
\(389\) 5401.03 0.703966 0.351983 0.936006i \(-0.385507\pi\)
0.351983 + 0.936006i \(0.385507\pi\)
\(390\) 0 0
\(391\) −1352.82 −0.174975
\(392\) 0 0
\(393\) −12428.8 −1.59529
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 8592.52 1.08626 0.543131 0.839648i \(-0.317239\pi\)
0.543131 + 0.839648i \(0.317239\pi\)
\(398\) 0 0
\(399\) −119.884 −0.0150418
\(400\) 0 0
\(401\) −1794.62 −0.223489 −0.111745 0.993737i \(-0.535644\pi\)
−0.111745 + 0.993737i \(0.535644\pi\)
\(402\) 0 0
\(403\) −3720.06 −0.459824
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −4177.05 −0.508720
\(408\) 0 0
\(409\) 11342.3 1.37125 0.685623 0.727957i \(-0.259530\pi\)
0.685623 + 0.727957i \(0.259530\pi\)
\(410\) 0 0
\(411\) −15729.7 −1.88781
\(412\) 0 0
\(413\) 2920.29 0.347937
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −695.278 −0.0816496
\(418\) 0 0
\(419\) −8568.54 −0.999047 −0.499524 0.866300i \(-0.666492\pi\)
−0.499524 + 0.866300i \(0.666492\pi\)
\(420\) 0 0
\(421\) −7705.53 −0.892030 −0.446015 0.895026i \(-0.647157\pi\)
−0.446015 + 0.895026i \(0.647157\pi\)
\(422\) 0 0
\(423\) −37.5561 −0.00431688
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −4088.99 −0.463420
\(428\) 0 0
\(429\) 2626.25 0.295563
\(430\) 0 0
\(431\) −12316.2 −1.37645 −0.688226 0.725496i \(-0.741610\pi\)
−0.688226 + 0.725496i \(0.741610\pi\)
\(432\) 0 0
\(433\) 7105.69 0.788632 0.394316 0.918975i \(-0.370981\pi\)
0.394316 + 0.918975i \(0.370981\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −44.4714 −0.00486809
\(438\) 0 0
\(439\) 17585.8 1.91190 0.955951 0.293527i \(-0.0948292\pi\)
0.955951 + 0.293527i \(0.0948292\pi\)
\(440\) 0 0
\(441\) −166.119 −0.0179374
\(442\) 0 0
\(443\) −2154.24 −0.231041 −0.115520 0.993305i \(-0.536854\pi\)
−0.115520 + 0.993305i \(0.536854\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 4143.88 0.438476
\(448\) 0 0
\(449\) −6853.94 −0.720395 −0.360197 0.932876i \(-0.617291\pi\)
−0.360197 + 0.932876i \(0.617291\pi\)
\(450\) 0 0
\(451\) 2331.17 0.243393
\(452\) 0 0
\(453\) −11570.3 −1.20005
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1822.27 −0.186526 −0.0932629 0.995642i \(-0.529730\pi\)
−0.0932629 + 0.995642i \(0.529730\pi\)
\(458\) 0 0
\(459\) −10499.0 −1.06765
\(460\) 0 0
\(461\) 3510.72 0.354687 0.177343 0.984149i \(-0.443250\pi\)
0.177343 + 0.984149i \(0.443250\pi\)
\(462\) 0 0
\(463\) 13454.1 1.35047 0.675235 0.737603i \(-0.264042\pi\)
0.675235 + 0.737603i \(0.264042\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12549.1 1.24347 0.621736 0.783227i \(-0.286428\pi\)
0.621736 + 0.783227i \(0.286428\pi\)
\(468\) 0 0
\(469\) −6494.67 −0.639438
\(470\) 0 0
\(471\) 8948.46 0.875421
\(472\) 0 0
\(473\) −4355.63 −0.423408
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0.240308 2.30670e−5 0
\(478\) 0 0
\(479\) −6582.21 −0.627868 −0.313934 0.949445i \(-0.601647\pi\)
−0.313934 + 0.949445i \(0.601647\pi\)
\(480\) 0 0
\(481\) 14738.4 1.39712
\(482\) 0 0
\(483\) 859.561 0.0809759
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −10288.5 −0.957326 −0.478663 0.877999i \(-0.658878\pi\)
−0.478663 + 0.877999i \(0.658878\pi\)
\(488\) 0 0
\(489\) 4321.68 0.399659
\(490\) 0 0
\(491\) 3423.20 0.314637 0.157319 0.987548i \(-0.449715\pi\)
0.157319 + 0.987548i \(0.449715\pi\)
\(492\) 0 0
\(493\) 10874.5 0.993436
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −10414.2 −0.939921
\(498\) 0 0
\(499\) 8210.85 0.736609 0.368305 0.929705i \(-0.379938\pi\)
0.368305 + 0.929705i \(0.379938\pi\)
\(500\) 0 0
\(501\) −13382.0 −1.19334
\(502\) 0 0
\(503\) 7377.71 0.653988 0.326994 0.945026i \(-0.393964\pi\)
0.326994 + 0.945026i \(0.393964\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 2284.15 0.200084
\(508\) 0 0
\(509\) −456.652 −0.0397657 −0.0198829 0.999802i \(-0.506329\pi\)
−0.0198829 + 0.999802i \(0.506329\pi\)
\(510\) 0 0
\(511\) −781.781 −0.0676790
\(512\) 0 0
\(513\) −345.134 −0.0297038
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −697.166 −0.0593063
\(518\) 0 0
\(519\) −4564.43 −0.386043
\(520\) 0 0
\(521\) 3000.32 0.252297 0.126148 0.992011i \(-0.459738\pi\)
0.126148 + 0.992011i \(0.459738\pi\)
\(522\) 0 0
\(523\) −7731.71 −0.646433 −0.323216 0.946325i \(-0.604764\pi\)
−0.323216 + 0.946325i \(0.604764\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −6713.09 −0.554889
\(528\) 0 0
\(529\) −11848.1 −0.973793
\(530\) 0 0
\(531\) −204.437 −0.0167077
\(532\) 0 0
\(533\) −8225.33 −0.668440
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −17116.5 −1.37548
\(538\) 0 0
\(539\) −3083.71 −0.246428
\(540\) 0 0
\(541\) 13614.4 1.08194 0.540971 0.841041i \(-0.318057\pi\)
0.540971 + 0.841041i \(0.318057\pi\)
\(542\) 0 0
\(543\) −18309.1 −1.44699
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −3198.43 −0.250009 −0.125005 0.992156i \(-0.539895\pi\)
−0.125005 + 0.992156i \(0.539895\pi\)
\(548\) 0 0
\(549\) 286.253 0.0222531
\(550\) 0 0
\(551\) 357.478 0.0276390
\(552\) 0 0
\(553\) −11443.3 −0.879964
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −178.506 −0.0135790 −0.00678952 0.999977i \(-0.502161\pi\)
−0.00678952 + 0.999977i \(0.502161\pi\)
\(558\) 0 0
\(559\) 15368.5 1.16282
\(560\) 0 0
\(561\) 4739.24 0.356668
\(562\) 0 0
\(563\) −10610.5 −0.794280 −0.397140 0.917758i \(-0.629997\pi\)
−0.397140 + 0.917758i \(0.629997\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 6829.34 0.505830
\(568\) 0 0
\(569\) −21286.8 −1.56834 −0.784171 0.620544i \(-0.786912\pi\)
−0.784171 + 0.620544i \(0.786912\pi\)
\(570\) 0 0
\(571\) 12064.5 0.884209 0.442104 0.896964i \(-0.354232\pi\)
0.442104 + 0.896964i \(0.354232\pi\)
\(572\) 0 0
\(573\) 15674.5 1.14278
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −8292.57 −0.598309 −0.299155 0.954205i \(-0.596705\pi\)
−0.299155 + 0.954205i \(0.596705\pi\)
\(578\) 0 0
\(579\) −8761.74 −0.628887
\(580\) 0 0
\(581\) 8480.75 0.605578
\(582\) 0 0
\(583\) 4.46092 0.000316900 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 15928.8 1.12002 0.560011 0.828485i \(-0.310797\pi\)
0.560011 + 0.828485i \(0.310797\pi\)
\(588\) 0 0
\(589\) −220.679 −0.0154379
\(590\) 0 0
\(591\) 22896.3 1.59362
\(592\) 0 0
\(593\) 13330.3 0.923116 0.461558 0.887110i \(-0.347291\pi\)
0.461558 + 0.887110i \(0.347291\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 4312.61 0.295650
\(598\) 0 0
\(599\) 3311.48 0.225882 0.112941 0.993602i \(-0.463973\pi\)
0.112941 + 0.993602i \(0.463973\pi\)
\(600\) 0 0
\(601\) −6082.46 −0.412826 −0.206413 0.978465i \(-0.566179\pi\)
−0.206413 + 0.978465i \(0.566179\pi\)
\(602\) 0 0
\(603\) 454.664 0.0307054
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 6162.45 0.412069 0.206035 0.978545i \(-0.433944\pi\)
0.206035 + 0.978545i \(0.433944\pi\)
\(608\) 0 0
\(609\) −6909.48 −0.459747
\(610\) 0 0
\(611\) 2459.89 0.162875
\(612\) 0 0
\(613\) −8628.65 −0.568529 −0.284264 0.958746i \(-0.591749\pi\)
−0.284264 + 0.958746i \(0.591749\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −16309.0 −1.06414 −0.532070 0.846700i \(-0.678586\pi\)
−0.532070 + 0.846700i \(0.678586\pi\)
\(618\) 0 0
\(619\) 25185.0 1.63533 0.817665 0.575694i \(-0.195268\pi\)
0.817665 + 0.575694i \(0.195268\pi\)
\(620\) 0 0
\(621\) 2474.60 0.159907
\(622\) 0 0
\(623\) −8913.75 −0.573229
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 155.793 0.00992309
\(628\) 0 0
\(629\) 26596.4 1.68596
\(630\) 0 0
\(631\) 13579.0 0.856689 0.428344 0.903616i \(-0.359097\pi\)
0.428344 + 0.903616i \(0.359097\pi\)
\(632\) 0 0
\(633\) −24594.4 −1.54430
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 10880.6 0.676775
\(638\) 0 0
\(639\) 729.053 0.0451344
\(640\) 0 0
\(641\) 26562.1 1.63672 0.818362 0.574703i \(-0.194883\pi\)
0.818362 + 0.574703i \(0.194883\pi\)
\(642\) 0 0
\(643\) 10349.9 0.634773 0.317387 0.948296i \(-0.397195\pi\)
0.317387 + 0.948296i \(0.397195\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 30738.0 1.86775 0.933875 0.357598i \(-0.116404\pi\)
0.933875 + 0.357598i \(0.116404\pi\)
\(648\) 0 0
\(649\) −3795.02 −0.229534
\(650\) 0 0
\(651\) 4265.38 0.256795
\(652\) 0 0
\(653\) 6912.35 0.414244 0.207122 0.978315i \(-0.433590\pi\)
0.207122 + 0.978315i \(0.433590\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 54.7291 0.00324990
\(658\) 0 0
\(659\) 3793.27 0.224226 0.112113 0.993695i \(-0.464238\pi\)
0.112113 + 0.993695i \(0.464238\pi\)
\(660\) 0 0
\(661\) −30232.2 −1.77897 −0.889483 0.456968i \(-0.848935\pi\)
−0.889483 + 0.456968i \(0.848935\pi\)
\(662\) 0 0
\(663\) −16722.0 −0.979532
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −2563.11 −0.148791
\(668\) 0 0
\(669\) −11767.0 −0.680025
\(670\) 0 0
\(671\) 5313.80 0.305718
\(672\) 0 0
\(673\) −8389.60 −0.480528 −0.240264 0.970708i \(-0.577234\pi\)
−0.240264 + 0.970708i \(0.577234\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 8879.70 0.504099 0.252049 0.967714i \(-0.418895\pi\)
0.252049 + 0.967714i \(0.418895\pi\)
\(678\) 0 0
\(679\) −10607.6 −0.599530
\(680\) 0 0
\(681\) 5451.42 0.306753
\(682\) 0 0
\(683\) −12555.2 −0.703384 −0.351692 0.936116i \(-0.614394\pi\)
−0.351692 + 0.936116i \(0.614394\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 20443.3 1.13532
\(688\) 0 0
\(689\) −15.7400 −0.000870313 0
\(690\) 0 0
\(691\) 30372.7 1.67211 0.836057 0.548642i \(-0.184855\pi\)
0.836057 + 0.548642i \(0.184855\pi\)
\(692\) 0 0
\(693\) −69.8272 −0.00382759
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −14843.2 −0.806635
\(698\) 0 0
\(699\) −683.694 −0.0369953
\(700\) 0 0
\(701\) −33508.8 −1.80543 −0.902716 0.430237i \(-0.858430\pi\)
−0.902716 + 0.430237i \(0.858430\pi\)
\(702\) 0 0
\(703\) 874.303 0.0469061
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4274.12 0.227362
\(708\) 0 0
\(709\) 29743.4 1.57551 0.787756 0.615987i \(-0.211243\pi\)
0.787756 + 0.615987i \(0.211243\pi\)
\(710\) 0 0
\(711\) 801.098 0.0422553
\(712\) 0 0
\(713\) 1582.26 0.0831083
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 30093.9 1.56747
\(718\) 0 0
\(719\) 404.733 0.0209930 0.0104965 0.999945i \(-0.496659\pi\)
0.0104965 + 0.999945i \(0.496659\pi\)
\(720\) 0 0
\(721\) 8818.40 0.455499
\(722\) 0 0
\(723\) −9561.99 −0.491859
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −19096.9 −0.974230 −0.487115 0.873338i \(-0.661951\pi\)
−0.487115 + 0.873338i \(0.661951\pi\)
\(728\) 0 0
\(729\) 19193.8 0.975147
\(730\) 0 0
\(731\) 27733.4 1.40323
\(732\) 0 0
\(733\) 28856.0 1.45405 0.727026 0.686610i \(-0.240902\pi\)
0.727026 + 0.686610i \(0.240902\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8440.07 0.421837
\(738\) 0 0
\(739\) 36005.6 1.79227 0.896134 0.443784i \(-0.146364\pi\)
0.896134 + 0.443784i \(0.146364\pi\)
\(740\) 0 0
\(741\) −549.703 −0.0272522
\(742\) 0 0
\(743\) −36423.7 −1.79846 −0.899230 0.437475i \(-0.855873\pi\)
−0.899230 + 0.437475i \(0.855873\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −593.701 −0.0290795
\(748\) 0 0
\(749\) −9768.79 −0.476560
\(750\) 0 0
\(751\) 25845.4 1.25581 0.627905 0.778290i \(-0.283912\pi\)
0.627905 + 0.778290i \(0.283912\pi\)
\(752\) 0 0
\(753\) 19125.5 0.925594
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 17862.6 0.857634 0.428817 0.903391i \(-0.358931\pi\)
0.428817 + 0.903391i \(0.358931\pi\)
\(758\) 0 0
\(759\) −1117.03 −0.0534198
\(760\) 0 0
\(761\) −22733.0 −1.08288 −0.541439 0.840740i \(-0.682120\pi\)
−0.541439 + 0.840740i \(0.682120\pi\)
\(762\) 0 0
\(763\) 1991.60 0.0944965
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 13390.4 0.630379
\(768\) 0 0
\(769\) −23153.6 −1.08575 −0.542873 0.839815i \(-0.682664\pi\)
−0.542873 + 0.839815i \(0.682664\pi\)
\(770\) 0 0
\(771\) −15613.3 −0.729309
\(772\) 0 0
\(773\) −26456.0 −1.23099 −0.615494 0.788141i \(-0.711044\pi\)
−0.615494 + 0.788141i \(0.711044\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −16898.9 −0.780237
\(778\) 0 0
\(779\) −487.939 −0.0224419
\(780\) 0 0
\(781\) 13533.6 0.620066
\(782\) 0 0
\(783\) −19891.8 −0.907886
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −13415.1 −0.607619 −0.303809 0.952733i \(-0.598259\pi\)
−0.303809 + 0.952733i \(0.598259\pi\)
\(788\) 0 0
\(789\) 15960.2 0.720151
\(790\) 0 0
\(791\) −10627.3 −0.477705
\(792\) 0 0
\(793\) −18749.3 −0.839605
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 31375.6 1.39445 0.697227 0.716850i \(-0.254417\pi\)
0.697227 + 0.716850i \(0.254417\pi\)
\(798\) 0 0
\(799\) 4439.04 0.196548
\(800\) 0 0
\(801\) 624.013 0.0275261
\(802\) 0 0
\(803\) 1015.95 0.0446479
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 36216.3 1.57977
\(808\) 0 0
\(809\) 28335.1 1.23141 0.615705 0.787977i \(-0.288871\pi\)
0.615705 + 0.787977i \(0.288871\pi\)
\(810\) 0 0
\(811\) 34538.1 1.49543 0.747717 0.664018i \(-0.231150\pi\)
0.747717 + 0.664018i \(0.231150\pi\)
\(812\) 0 0
\(813\) 32457.1 1.40015
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 911.681 0.0390400
\(818\) 0 0
\(819\) 246.380 0.0105118
\(820\) 0 0
\(821\) −29834.1 −1.26823 −0.634115 0.773239i \(-0.718635\pi\)
−0.634115 + 0.773239i \(0.718635\pi\)
\(822\) 0 0
\(823\) 36246.8 1.53522 0.767608 0.640919i \(-0.221447\pi\)
0.767608 + 0.640919i \(0.221447\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 21831.3 0.917955 0.458978 0.888448i \(-0.348216\pi\)
0.458978 + 0.888448i \(0.348216\pi\)
\(828\) 0 0
\(829\) 12273.0 0.514186 0.257093 0.966387i \(-0.417235\pi\)
0.257093 + 0.966387i \(0.417235\pi\)
\(830\) 0 0
\(831\) 17209.3 0.718391
\(832\) 0 0
\(833\) 19634.8 0.816693
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 12279.6 0.507105
\(838\) 0 0
\(839\) −37441.0 −1.54065 −0.770326 0.637650i \(-0.779906\pi\)
−0.770326 + 0.637650i \(0.779906\pi\)
\(840\) 0 0
\(841\) −3785.77 −0.155224
\(842\) 0 0
\(843\) −2410.02 −0.0984644
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 10890.3 0.441788
\(848\) 0 0
\(849\) −42399.7 −1.71396
\(850\) 0 0
\(851\) −6268.72 −0.252514
\(852\) 0 0
\(853\) −1247.81 −0.0500871 −0.0250436 0.999686i \(-0.507972\pi\)
−0.0250436 + 0.999686i \(0.507972\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2927.40 0.116684 0.0583419 0.998297i \(-0.481419\pi\)
0.0583419 + 0.998297i \(0.481419\pi\)
\(858\) 0 0
\(859\) 30892.1 1.22704 0.613518 0.789681i \(-0.289754\pi\)
0.613518 + 0.789681i \(0.289754\pi\)
\(860\) 0 0
\(861\) 9431.08 0.373299
\(862\) 0 0
\(863\) −5815.96 −0.229406 −0.114703 0.993400i \(-0.536592\pi\)
−0.114703 + 0.993400i \(0.536592\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −4346.06 −0.170242
\(868\) 0 0
\(869\) 14871.0 0.580513
\(870\) 0 0
\(871\) −29780.1 −1.15851
\(872\) 0 0
\(873\) 742.589 0.0287890
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 4732.26 0.182209 0.0911044 0.995841i \(-0.470960\pi\)
0.0911044 + 0.995841i \(0.470960\pi\)
\(878\) 0 0
\(879\) 4273.60 0.163988
\(880\) 0 0
\(881\) 43833.1 1.67625 0.838124 0.545480i \(-0.183653\pi\)
0.838124 + 0.545480i \(0.183653\pi\)
\(882\) 0 0
\(883\) 37311.7 1.42201 0.711007 0.703185i \(-0.248239\pi\)
0.711007 + 0.703185i \(0.248239\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −44940.5 −1.70119 −0.850595 0.525822i \(-0.823758\pi\)
−0.850595 + 0.525822i \(0.823758\pi\)
\(888\) 0 0
\(889\) 313.867 0.0118411
\(890\) 0 0
\(891\) −8874.98 −0.333696
\(892\) 0 0
\(893\) 145.925 0.00546829
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 3941.35 0.146709
\(898\) 0 0
\(899\) −12718.8 −0.471854
\(900\) 0 0
\(901\) −28.4038 −0.00105024
\(902\) 0 0
\(903\) −17621.3 −0.649392
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 16241.6 0.594591 0.297295 0.954786i \(-0.403915\pi\)
0.297295 + 0.954786i \(0.403915\pi\)
\(908\) 0 0
\(909\) −299.213 −0.0109178
\(910\) 0 0
\(911\) −2088.21 −0.0759444 −0.0379722 0.999279i \(-0.512090\pi\)
−0.0379722 + 0.999279i \(0.512090\pi\)
\(912\) 0 0
\(913\) −11021.1 −0.399500
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −21644.8 −0.779469
\(918\) 0 0
\(919\) 40755.3 1.46289 0.731443 0.681903i \(-0.238847\pi\)
0.731443 + 0.681903i \(0.238847\pi\)
\(920\) 0 0
\(921\) 2016.08 0.0721303
\(922\) 0 0
\(923\) −47752.3 −1.70291
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −617.338 −0.0218728
\(928\) 0 0
\(929\) −24239.7 −0.856058 −0.428029 0.903765i \(-0.640792\pi\)
−0.428029 + 0.903765i \(0.640792\pi\)
\(930\) 0 0
\(931\) 645.455 0.0227217
\(932\) 0 0
\(933\) 8754.64 0.307196
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −37308.1 −1.30075 −0.650374 0.759614i \(-0.725388\pi\)
−0.650374 + 0.759614i \(0.725388\pi\)
\(938\) 0 0
\(939\) −19815.5 −0.688663
\(940\) 0 0
\(941\) 109.276 0.00378566 0.00189283 0.999998i \(-0.499397\pi\)
0.00189283 + 0.999998i \(0.499397\pi\)
\(942\) 0 0
\(943\) 3498.51 0.120813
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −13256.9 −0.454902 −0.227451 0.973790i \(-0.573039\pi\)
−0.227451 + 0.973790i \(0.573039\pi\)
\(948\) 0 0
\(949\) −3584.71 −0.122618
\(950\) 0 0
\(951\) −9777.29 −0.333386
\(952\) 0 0
\(953\) −56535.2 −1.92167 −0.960837 0.277112i \(-0.910623\pi\)
−0.960837 + 0.277112i \(0.910623\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 8979.12 0.303296
\(958\) 0 0
\(959\) −27393.4 −0.922396
\(960\) 0 0
\(961\) −21939.4 −0.736443
\(962\) 0 0
\(963\) 683.870 0.0228841
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −32548.6 −1.08241 −0.541206 0.840890i \(-0.682032\pi\)
−0.541206 + 0.840890i \(0.682032\pi\)
\(968\) 0 0
\(969\) −991.976 −0.0328863
\(970\) 0 0
\(971\) −1604.29 −0.0530216 −0.0265108 0.999649i \(-0.508440\pi\)
−0.0265108 + 0.999649i \(0.508440\pi\)
\(972\) 0 0
\(973\) −1210.83 −0.0398945
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −11269.0 −0.369014 −0.184507 0.982831i \(-0.559069\pi\)
−0.184507 + 0.982831i \(0.559069\pi\)
\(978\) 0 0
\(979\) 11583.7 0.378159
\(980\) 0 0
\(981\) −139.423 −0.00453766
\(982\) 0 0
\(983\) 48373.3 1.56955 0.784775 0.619781i \(-0.212779\pi\)
0.784775 + 0.619781i \(0.212779\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −2820.49 −0.0909596
\(988\) 0 0
\(989\) −6536.72 −0.210167
\(990\) 0 0
\(991\) −792.959 −0.0254179 −0.0127090 0.999919i \(-0.504046\pi\)
−0.0127090 + 0.999919i \(0.504046\pi\)
\(992\) 0 0
\(993\) 38038.1 1.21561
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −52187.1 −1.65775 −0.828877 0.559430i \(-0.811020\pi\)
−0.828877 + 0.559430i \(0.811020\pi\)
\(998\) 0 0
\(999\) −48650.4 −1.54077
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 800.4.a.x.1.1 yes 3
4.3 odd 2 800.4.a.u.1.3 3
5.2 odd 4 800.4.c.n.449.5 6
5.3 odd 4 800.4.c.n.449.2 6
5.4 even 2 800.4.a.v.1.3 yes 3
8.3 odd 2 1600.4.a.ct.1.1 3
8.5 even 2 1600.4.a.cq.1.3 3
20.3 even 4 800.4.c.m.449.5 6
20.7 even 4 800.4.c.m.449.2 6
20.19 odd 2 800.4.a.w.1.1 yes 3
40.19 odd 2 1600.4.a.cr.1.3 3
40.29 even 2 1600.4.a.cs.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
800.4.a.u.1.3 3 4.3 odd 2
800.4.a.v.1.3 yes 3 5.4 even 2
800.4.a.w.1.1 yes 3 20.19 odd 2
800.4.a.x.1.1 yes 3 1.1 even 1 trivial
800.4.c.m.449.2 6 20.7 even 4
800.4.c.m.449.5 6 20.3 even 4
800.4.c.n.449.2 6 5.3 odd 4
800.4.c.n.449.5 6 5.2 odd 4
1600.4.a.cq.1.3 3 8.5 even 2
1600.4.a.cr.1.3 3 40.19 odd 2
1600.4.a.cs.1.1 3 40.29 even 2
1600.4.a.ct.1.1 3 8.3 odd 2