Properties

Label 800.4.a.v.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 \(3.59535\) 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)
 
\(f(q)\) \(=\) \(q-8.19070 q^{3} +21.7061 q^{7} +40.0875 q^{9} +O(q^{10})\) \(q-8.19070 q^{3} +21.7061 q^{7} +40.0875 q^{9} +37.8968 q^{11} +52.7369 q^{13} -99.9636 q^{17} +116.747 q^{19} -177.788 q^{21} +37.7320 q^{23} -107.196 q^{27} +218.664 q^{29} -67.3808 q^{31} -310.401 q^{33} -362.591 q^{37} -431.952 q^{39} -291.803 q^{41} +183.160 q^{43} -443.479 q^{47} +128.155 q^{49} +818.772 q^{51} -416.349 q^{53} -956.240 q^{57} +828.632 q^{59} +442.983 q^{61} +870.143 q^{63} +570.007 q^{67} -309.051 q^{69} +341.295 q^{71} +506.312 q^{73} +822.591 q^{77} +426.757 q^{79} -204.356 q^{81} -982.438 q^{83} -1791.01 q^{87} +926.879 q^{89} +1144.71 q^{91} +551.895 q^{93} +1889.63 q^{97} +1519.19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 5 q^{3} + 2 q^{7} + 18 q^{9}+O(q^{10}) \) 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
\(3\) −8.19070 −1.57630 −0.788150 0.615483i \(-0.788961\pi\)
−0.788150 + 0.615483i \(0.788961\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 21.7061 1.17202 0.586010 0.810304i \(-0.300698\pi\)
0.586010 + 0.810304i \(0.300698\pi\)
\(8\) 0 0
\(9\) 40.0875 1.48472
\(10\) 0 0
\(11\) 37.8968 1.03876 0.519378 0.854545i \(-0.326164\pi\)
0.519378 + 0.854545i \(0.326164\pi\)
\(12\) 0 0
\(13\) 52.7369 1.12512 0.562561 0.826756i \(-0.309816\pi\)
0.562561 + 0.826756i \(0.309816\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −99.9636 −1.42616 −0.713081 0.701082i \(-0.752701\pi\)
−0.713081 + 0.701082i \(0.752701\pi\)
\(18\) 0 0
\(19\) 116.747 1.40966 0.704832 0.709374i \(-0.251022\pi\)
0.704832 + 0.709374i \(0.251022\pi\)
\(20\) 0 0
\(21\) −177.788 −1.84745
\(22\) 0 0
\(23\) 37.7320 0.342072 0.171036 0.985265i \(-0.445289\pi\)
0.171036 + 0.985265i \(0.445289\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −107.196 −0.764067
\(28\) 0 0
\(29\) 218.664 1.40017 0.700085 0.714060i \(-0.253146\pi\)
0.700085 + 0.714060i \(0.253146\pi\)
\(30\) 0 0
\(31\) −67.3808 −0.390385 −0.195193 0.980765i \(-0.562533\pi\)
−0.195193 + 0.980765i \(0.562533\pi\)
\(32\) 0 0
\(33\) −310.401 −1.63739
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −362.591 −1.61107 −0.805536 0.592547i \(-0.798123\pi\)
−0.805536 + 0.592547i \(0.798123\pi\)
\(38\) 0 0
\(39\) −431.952 −1.77353
\(40\) 0 0
\(41\) −291.803 −1.11151 −0.555757 0.831345i \(-0.687571\pi\)
−0.555757 + 0.831345i \(0.687571\pi\)
\(42\) 0 0
\(43\) 183.160 0.649571 0.324786 0.945788i \(-0.394708\pi\)
0.324786 + 0.945788i \(0.394708\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −443.479 −1.37634 −0.688170 0.725549i \(-0.741586\pi\)
−0.688170 + 0.725549i \(0.741586\pi\)
\(48\) 0 0
\(49\) 128.155 0.373629
\(50\) 0 0
\(51\) 818.772 2.24806
\(52\) 0 0
\(53\) −416.349 −1.07905 −0.539527 0.841968i \(-0.681397\pi\)
−0.539527 + 0.841968i \(0.681397\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −956.240 −2.22205
\(58\) 0 0
\(59\) 828.632 1.82845 0.914226 0.405204i \(-0.132799\pi\)
0.914226 + 0.405204i \(0.132799\pi\)
\(60\) 0 0
\(61\) 442.983 0.929807 0.464903 0.885361i \(-0.346089\pi\)
0.464903 + 0.885361i \(0.346089\pi\)
\(62\) 0 0
\(63\) 870.143 1.74012
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 570.007 1.03936 0.519682 0.854360i \(-0.326050\pi\)
0.519682 + 0.854360i \(0.326050\pi\)
\(68\) 0 0
\(69\) −309.051 −0.539208
\(70\) 0 0
\(71\) 341.295 0.570483 0.285241 0.958456i \(-0.407926\pi\)
0.285241 + 0.958456i \(0.407926\pi\)
\(72\) 0 0
\(73\) 506.312 0.811772 0.405886 0.913924i \(-0.366963\pi\)
0.405886 + 0.913924i \(0.366963\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 822.591 1.21744
\(78\) 0 0
\(79\) 426.757 0.607771 0.303886 0.952708i \(-0.401716\pi\)
0.303886 + 0.952708i \(0.401716\pi\)
\(80\) 0 0
\(81\) −204.356 −0.280323
\(82\) 0 0
\(83\) −982.438 −1.29924 −0.649618 0.760261i \(-0.725071\pi\)
−0.649618 + 0.760261i \(0.725071\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1791.01 −2.20709
\(88\) 0 0
\(89\) 926.879 1.10392 0.551961 0.833870i \(-0.313880\pi\)
0.551961 + 0.833870i \(0.313880\pi\)
\(90\) 0 0
\(91\) 1144.71 1.31866
\(92\) 0 0
\(93\) 551.895 0.615364
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1889.63 1.97797 0.988984 0.148024i \(-0.0472912\pi\)
0.988984 + 0.148024i \(0.0472912\pi\)
\(98\) 0 0
\(99\) 1519.19 1.54226
\(100\) 0 0
\(101\) 96.2748 0.0948486 0.0474243 0.998875i \(-0.484899\pi\)
0.0474243 + 0.998875i \(0.484899\pi\)
\(102\) 0 0
\(103\) 1701.85 1.62804 0.814019 0.580838i \(-0.197275\pi\)
0.814019 + 0.580838i \(0.197275\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −746.877 −0.674798 −0.337399 0.941362i \(-0.609547\pi\)
−0.337399 + 0.941362i \(0.609547\pi\)
\(108\) 0 0
\(109\) −2150.44 −1.88968 −0.944840 0.327532i \(-0.893783\pi\)
−0.944840 + 0.327532i \(0.893783\pi\)
\(110\) 0 0
\(111\) 2969.88 2.53953
\(112\) 0 0
\(113\) 661.058 0.550328 0.275164 0.961397i \(-0.411268\pi\)
0.275164 + 0.961397i \(0.411268\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2114.09 1.67049
\(118\) 0 0
\(119\) −2169.82 −1.67149
\(120\) 0 0
\(121\) 105.167 0.0790134
\(122\) 0 0
\(123\) 2390.07 1.75208
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −989.619 −0.691452 −0.345726 0.938335i \(-0.612367\pi\)
−0.345726 + 0.938335i \(0.612367\pi\)
\(128\) 0 0
\(129\) −1500.20 −1.02392
\(130\) 0 0
\(131\) −6.42246 −0.00428346 −0.00214173 0.999998i \(-0.500682\pi\)
−0.00214173 + 0.999998i \(0.500682\pi\)
\(132\) 0 0
\(133\) 2534.12 1.65215
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1033.75 −0.644668 −0.322334 0.946626i \(-0.604467\pi\)
−0.322334 + 0.946626i \(0.604467\pi\)
\(138\) 0 0
\(139\) −56.8412 −0.0346850 −0.0173425 0.999850i \(-0.505521\pi\)
−0.0173425 + 0.999850i \(0.505521\pi\)
\(140\) 0 0
\(141\) 3632.40 2.16953
\(142\) 0 0
\(143\) 1998.56 1.16873
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −1049.68 −0.588951
\(148\) 0 0
\(149\) 542.238 0.298133 0.149067 0.988827i \(-0.452373\pi\)
0.149067 + 0.988827i \(0.452373\pi\)
\(150\) 0 0
\(151\) −390.225 −0.210305 −0.105152 0.994456i \(-0.533533\pi\)
−0.105152 + 0.994456i \(0.533533\pi\)
\(152\) 0 0
\(153\) −4007.29 −2.11745
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3674.37 1.86782 0.933908 0.357514i \(-0.116376\pi\)
0.933908 + 0.357514i \(0.116376\pi\)
\(158\) 0 0
\(159\) 3410.19 1.70091
\(160\) 0 0
\(161\) 819.014 0.400915
\(162\) 0 0
\(163\) −1219.38 −0.585944 −0.292972 0.956121i \(-0.594644\pi\)
−0.292972 + 0.956121i \(0.594644\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 610.547 0.282907 0.141454 0.989945i \(-0.454822\pi\)
0.141454 + 0.989945i \(0.454822\pi\)
\(168\) 0 0
\(169\) 584.181 0.265899
\(170\) 0 0
\(171\) 4680.10 2.09296
\(172\) 0 0
\(173\) −50.4300 −0.0221626 −0.0110813 0.999939i \(-0.503527\pi\)
−0.0110813 + 0.999939i \(0.503527\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −6787.07 −2.88219
\(178\) 0 0
\(179\) 373.064 0.155777 0.0778885 0.996962i \(-0.475182\pi\)
0.0778885 + 0.996962i \(0.475182\pi\)
\(180\) 0 0
\(181\) −232.883 −0.0956358 −0.0478179 0.998856i \(-0.515227\pi\)
−0.0478179 + 0.998856i \(0.515227\pi\)
\(182\) 0 0
\(183\) −3628.34 −1.46565
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −3788.30 −1.48143
\(188\) 0 0
\(189\) −2326.80 −0.895501
\(190\) 0 0
\(191\) 2769.18 1.04906 0.524531 0.851391i \(-0.324241\pi\)
0.524531 + 0.851391i \(0.324241\pi\)
\(192\) 0 0
\(193\) −307.412 −0.114653 −0.0573264 0.998355i \(-0.518258\pi\)
−0.0573264 + 0.998355i \(0.518258\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 317.010 0.114650 0.0573249 0.998356i \(-0.481743\pi\)
0.0573249 + 0.998356i \(0.481743\pi\)
\(198\) 0 0
\(199\) 3917.06 1.39534 0.697671 0.716418i \(-0.254220\pi\)
0.697671 + 0.716418i \(0.254220\pi\)
\(200\) 0 0
\(201\) −4668.75 −1.63835
\(202\) 0 0
\(203\) 4746.35 1.64103
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1512.58 0.507882
\(208\) 0 0
\(209\) 4424.34 1.46430
\(210\) 0 0
\(211\) −2054.34 −0.670267 −0.335133 0.942171i \(-0.608781\pi\)
−0.335133 + 0.942171i \(0.608781\pi\)
\(212\) 0 0
\(213\) −2795.44 −0.899252
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −1462.57 −0.457539
\(218\) 0 0
\(219\) −4147.05 −1.27960
\(220\) 0 0
\(221\) −5271.77 −1.60461
\(222\) 0 0
\(223\) −3087.92 −0.927275 −0.463638 0.886025i \(-0.653456\pi\)
−0.463638 + 0.886025i \(0.653456\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3572.18 1.04447 0.522233 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(228\) 0 0
\(229\) 1799.26 0.519207 0.259603 0.965715i \(-0.416408\pi\)
0.259603 + 0.965715i \(0.416408\pi\)
\(230\) 0 0
\(231\) −6737.60 −1.91905
\(232\) 0 0
\(233\) 3900.64 1.09674 0.548368 0.836237i \(-0.315249\pi\)
0.548368 + 0.836237i \(0.315249\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −3495.44 −0.958030
\(238\) 0 0
\(239\) 60.6726 0.0164209 0.00821043 0.999966i \(-0.497387\pi\)
0.00821043 + 0.999966i \(0.497387\pi\)
\(240\) 0 0
\(241\) −6160.51 −1.64661 −0.823305 0.567599i \(-0.807872\pi\)
−0.823305 + 0.567599i \(0.807872\pi\)
\(242\) 0 0
\(243\) 4568.10 1.20594
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 6156.88 1.58604
\(248\) 0 0
\(249\) 8046.85 2.04798
\(250\) 0 0
\(251\) −880.907 −0.221523 −0.110762 0.993847i \(-0.535329\pi\)
−0.110762 + 0.993847i \(0.535329\pi\)
\(252\) 0 0
\(253\) 1429.92 0.355329
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5845.84 −1.41889 −0.709443 0.704763i \(-0.751053\pi\)
−0.709443 + 0.704763i \(0.751053\pi\)
\(258\) 0 0
\(259\) −7870.44 −1.88821
\(260\) 0 0
\(261\) 8765.70 2.07886
\(262\) 0 0
\(263\) −1994.10 −0.467535 −0.233767 0.972293i \(-0.575105\pi\)
−0.233767 + 0.972293i \(0.575105\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −7591.78 −1.74011
\(268\) 0 0
\(269\) −426.120 −0.0965836 −0.0482918 0.998833i \(-0.515378\pi\)
−0.0482918 + 0.998833i \(0.515378\pi\)
\(270\) 0 0
\(271\) 7527.32 1.68728 0.843638 0.536912i \(-0.180409\pi\)
0.843638 + 0.536912i \(0.180409\pi\)
\(272\) 0 0
\(273\) −9375.99 −2.07861
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2409.09 0.522557 0.261279 0.965263i \(-0.415856\pi\)
0.261279 + 0.965263i \(0.415856\pi\)
\(278\) 0 0
\(279\) −2701.13 −0.579613
\(280\) 0 0
\(281\) −5487.57 −1.16499 −0.582493 0.812836i \(-0.697923\pi\)
−0.582493 + 0.812836i \(0.697923\pi\)
\(282\) 0 0
\(283\) −4999.15 −1.05006 −0.525032 0.851082i \(-0.675947\pi\)
−0.525032 + 0.851082i \(0.675947\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6333.91 −1.30271
\(288\) 0 0
\(289\) 5079.73 1.03394
\(290\) 0 0
\(291\) −15477.4 −3.11787
\(292\) 0 0
\(293\) 8992.96 1.79309 0.896543 0.442956i \(-0.146070\pi\)
0.896543 + 0.442956i \(0.146070\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −4062.37 −0.793679
\(298\) 0 0
\(299\) 1989.87 0.384873
\(300\) 0 0
\(301\) 3975.68 0.761310
\(302\) 0 0
\(303\) −788.558 −0.149510
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 5584.49 1.03819 0.519094 0.854717i \(-0.326270\pi\)
0.519094 + 0.854717i \(0.326270\pi\)
\(308\) 0 0
\(309\) −13939.3 −2.56628
\(310\) 0 0
\(311\) −6218.04 −1.13374 −0.566869 0.823808i \(-0.691845\pi\)
−0.566869 + 0.823808i \(0.691845\pi\)
\(312\) 0 0
\(313\) 5046.00 0.911236 0.455618 0.890175i \(-0.349418\pi\)
0.455618 + 0.890175i \(0.349418\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5370.16 0.951476 0.475738 0.879587i \(-0.342181\pi\)
0.475738 + 0.879587i \(0.342181\pi\)
\(318\) 0 0
\(319\) 8286.67 1.45443
\(320\) 0 0
\(321\) 6117.44 1.06368
\(322\) 0 0
\(323\) −11670.5 −2.01041
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 17613.6 2.97870
\(328\) 0 0
\(329\) −9626.19 −1.61310
\(330\) 0 0
\(331\) 9.92458 0.00164805 0.000824024 1.00000i \(-0.499738\pi\)
0.000824024 1.00000i \(0.499738\pi\)
\(332\) 0 0
\(333\) −14535.4 −2.39199
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1684.61 −0.272304 −0.136152 0.990688i \(-0.543474\pi\)
−0.136152 + 0.990688i \(0.543474\pi\)
\(338\) 0 0
\(339\) −5414.52 −0.867482
\(340\) 0 0
\(341\) −2553.51 −0.405515
\(342\) 0 0
\(343\) −4663.45 −0.734119
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 8690.93 1.34453 0.672267 0.740308i \(-0.265320\pi\)
0.672267 + 0.740308i \(0.265320\pi\)
\(348\) 0 0
\(349\) 6861.60 1.05242 0.526208 0.850356i \(-0.323613\pi\)
0.526208 + 0.850356i \(0.323613\pi\)
\(350\) 0 0
\(351\) −5653.16 −0.859668
\(352\) 0 0
\(353\) −5010.05 −0.755405 −0.377703 0.925927i \(-0.623286\pi\)
−0.377703 + 0.925927i \(0.623286\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 17772.3 2.63477
\(358\) 0 0
\(359\) −2871.64 −0.422171 −0.211085 0.977468i \(-0.567700\pi\)
−0.211085 + 0.977468i \(0.567700\pi\)
\(360\) 0 0
\(361\) 6770.88 0.987152
\(362\) 0 0
\(363\) −861.390 −0.124549
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −2096.96 −0.298257 −0.149129 0.988818i \(-0.547647\pi\)
−0.149129 + 0.988818i \(0.547647\pi\)
\(368\) 0 0
\(369\) −11697.7 −1.65029
\(370\) 0 0
\(371\) −9037.30 −1.26467
\(372\) 0 0
\(373\) 9530.96 1.32304 0.661521 0.749927i \(-0.269911\pi\)
0.661521 + 0.749927i \(0.269911\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 11531.7 1.57536
\(378\) 0 0
\(379\) 9603.41 1.30157 0.650784 0.759263i \(-0.274440\pi\)
0.650784 + 0.759263i \(0.274440\pi\)
\(380\) 0 0
\(381\) 8105.66 1.08994
\(382\) 0 0
\(383\) 7688.07 1.02570 0.512849 0.858479i \(-0.328590\pi\)
0.512849 + 0.858479i \(0.328590\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 7342.41 0.964433
\(388\) 0 0
\(389\) 3579.00 0.466485 0.233242 0.972419i \(-0.425066\pi\)
0.233242 + 0.972419i \(0.425066\pi\)
\(390\) 0 0
\(391\) −3771.83 −0.487850
\(392\) 0 0
\(393\) 52.6044 0.00675202
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 1262.66 0.159624 0.0798122 0.996810i \(-0.474568\pi\)
0.0798122 + 0.996810i \(0.474568\pi\)
\(398\) 0 0
\(399\) −20756.2 −2.60429
\(400\) 0 0
\(401\) 1738.14 0.216455 0.108228 0.994126i \(-0.465482\pi\)
0.108228 + 0.994126i \(0.465482\pi\)
\(402\) 0 0
\(403\) −3553.45 −0.439231
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −13741.1 −1.67351
\(408\) 0 0
\(409\) 2168.41 0.262154 0.131077 0.991372i \(-0.458156\pi\)
0.131077 + 0.991372i \(0.458156\pi\)
\(410\) 0 0
\(411\) 8467.16 1.01619
\(412\) 0 0
\(413\) 17986.4 2.14298
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 465.569 0.0546739
\(418\) 0 0
\(419\) −7505.84 −0.875141 −0.437571 0.899184i \(-0.644161\pi\)
−0.437571 + 0.899184i \(0.644161\pi\)
\(420\) 0 0
\(421\) −11097.3 −1.28468 −0.642341 0.766419i \(-0.722037\pi\)
−0.642341 + 0.766419i \(0.722037\pi\)
\(422\) 0 0
\(423\) −17777.9 −2.04348
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 9615.44 1.08975
\(428\) 0 0
\(429\) −16369.6 −1.84226
\(430\) 0 0
\(431\) −8935.68 −0.998646 −0.499323 0.866416i \(-0.666418\pi\)
−0.499323 + 0.866416i \(0.666418\pi\)
\(432\) 0 0
\(433\) 6453.45 0.716243 0.358122 0.933675i \(-0.383417\pi\)
0.358122 + 0.933675i \(0.383417\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4405.10 0.482207
\(438\) 0 0
\(439\) −5254.88 −0.571303 −0.285651 0.958334i \(-0.592210\pi\)
−0.285651 + 0.958334i \(0.592210\pi\)
\(440\) 0 0
\(441\) 5137.40 0.554735
\(442\) 0 0
\(443\) −242.163 −0.0259718 −0.0129859 0.999916i \(-0.504134\pi\)
−0.0129859 + 0.999916i \(0.504134\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −4441.30 −0.469947
\(448\) 0 0
\(449\) 10154.7 1.06733 0.533666 0.845696i \(-0.320814\pi\)
0.533666 + 0.845696i \(0.320814\pi\)
\(450\) 0 0
\(451\) −11058.4 −1.15459
\(452\) 0 0
\(453\) 3196.21 0.331504
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −967.379 −0.0990198 −0.0495099 0.998774i \(-0.515766\pi\)
−0.0495099 + 0.998774i \(0.515766\pi\)
\(458\) 0 0
\(459\) 10715.7 1.08968
\(460\) 0 0
\(461\) 17022.8 1.71980 0.859902 0.510460i \(-0.170525\pi\)
0.859902 + 0.510460i \(0.170525\pi\)
\(462\) 0 0
\(463\) 13976.3 1.40288 0.701440 0.712728i \(-0.252541\pi\)
0.701440 + 0.712728i \(0.252541\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −4454.27 −0.441369 −0.220684 0.975345i \(-0.570829\pi\)
−0.220684 + 0.975345i \(0.570829\pi\)
\(468\) 0 0
\(469\) 12372.6 1.21816
\(470\) 0 0
\(471\) −30095.7 −2.94424
\(472\) 0 0
\(473\) 6941.16 0.674746
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −16690.4 −1.60210
\(478\) 0 0
\(479\) −9617.44 −0.917394 −0.458697 0.888593i \(-0.651684\pi\)
−0.458697 + 0.888593i \(0.651684\pi\)
\(480\) 0 0
\(481\) −19121.9 −1.81265
\(482\) 0 0
\(483\) −6708.29 −0.631963
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 8346.22 0.776599 0.388299 0.921533i \(-0.373063\pi\)
0.388299 + 0.921533i \(0.373063\pi\)
\(488\) 0 0
\(489\) 9987.53 0.923623
\(490\) 0 0
\(491\) 11867.9 1.09081 0.545406 0.838172i \(-0.316375\pi\)
0.545406 + 0.838172i \(0.316375\pi\)
\(492\) 0 0
\(493\) −21858.5 −1.99687
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7408.19 0.668617
\(498\) 0 0
\(499\) 17696.1 1.58755 0.793773 0.608214i \(-0.208114\pi\)
0.793773 + 0.608214i \(0.208114\pi\)
\(500\) 0 0
\(501\) −5000.80 −0.445947
\(502\) 0 0
\(503\) 9911.58 0.878599 0.439300 0.898341i \(-0.355227\pi\)
0.439300 + 0.898341i \(0.355227\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −4784.85 −0.419137
\(508\) 0 0
\(509\) −3999.53 −0.348283 −0.174141 0.984721i \(-0.555715\pi\)
−0.174141 + 0.984721i \(0.555715\pi\)
\(510\) 0 0
\(511\) 10990.1 0.951413
\(512\) 0 0
\(513\) −12514.8 −1.07708
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −16806.4 −1.42968
\(518\) 0 0
\(519\) 413.057 0.0349348
\(520\) 0 0
\(521\) −11301.1 −0.950306 −0.475153 0.879903i \(-0.657607\pi\)
−0.475153 + 0.879903i \(0.657607\pi\)
\(522\) 0 0
\(523\) −3377.88 −0.282417 −0.141209 0.989980i \(-0.545099\pi\)
−0.141209 + 0.989980i \(0.545099\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6735.62 0.556752
\(528\) 0 0
\(529\) −10743.3 −0.882987
\(530\) 0 0
\(531\) 33217.8 2.71474
\(532\) 0 0
\(533\) −15388.8 −1.25059
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −3055.65 −0.245551
\(538\) 0 0
\(539\) 4856.65 0.388109
\(540\) 0 0
\(541\) −19694.0 −1.56508 −0.782542 0.622598i \(-0.786077\pi\)
−0.782542 + 0.622598i \(0.786077\pi\)
\(542\) 0 0
\(543\) 1907.48 0.150751
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 11353.7 0.887474 0.443737 0.896157i \(-0.353652\pi\)
0.443737 + 0.896157i \(0.353652\pi\)
\(548\) 0 0
\(549\) 17758.1 1.38050
\(550\) 0 0
\(551\) 25528.4 1.97377
\(552\) 0 0
\(553\) 9263.23 0.712320
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −13675.1 −1.04027 −0.520137 0.854083i \(-0.674119\pi\)
−0.520137 + 0.854083i \(0.674119\pi\)
\(558\) 0 0
\(559\) 9659.27 0.730847
\(560\) 0 0
\(561\) 31028.8 2.33518
\(562\) 0 0
\(563\) −8690.01 −0.650515 −0.325258 0.945625i \(-0.605451\pi\)
−0.325258 + 0.945625i \(0.605451\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −4435.76 −0.328544
\(568\) 0 0
\(569\) 10774.4 0.793827 0.396914 0.917856i \(-0.370081\pi\)
0.396914 + 0.917856i \(0.370081\pi\)
\(570\) 0 0
\(571\) −11320.2 −0.829657 −0.414828 0.909900i \(-0.636158\pi\)
−0.414828 + 0.909900i \(0.636158\pi\)
\(572\) 0 0
\(573\) −22681.5 −1.65364
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −13525.2 −0.975842 −0.487921 0.872888i \(-0.662245\pi\)
−0.487921 + 0.872888i \(0.662245\pi\)
\(578\) 0 0
\(579\) 2517.92 0.180727
\(580\) 0 0
\(581\) −21324.9 −1.52273
\(582\) 0 0
\(583\) −15778.3 −1.12087
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −3492.61 −0.245580 −0.122790 0.992433i \(-0.539184\pi\)
−0.122790 + 0.992433i \(0.539184\pi\)
\(588\) 0 0
\(589\) −7866.50 −0.550312
\(590\) 0 0
\(591\) −2596.53 −0.180722
\(592\) 0 0
\(593\) −8431.49 −0.583878 −0.291939 0.956437i \(-0.594300\pi\)
−0.291939 + 0.956437i \(0.594300\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −32083.5 −2.19948
\(598\) 0 0
\(599\) −16783.6 −1.14484 −0.572420 0.819960i \(-0.693995\pi\)
−0.572420 + 0.819960i \(0.693995\pi\)
\(600\) 0 0
\(601\) −10152.2 −0.689044 −0.344522 0.938778i \(-0.611959\pi\)
−0.344522 + 0.938778i \(0.611959\pi\)
\(602\) 0 0
\(603\) 22850.1 1.54317
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −14130.3 −0.944859 −0.472430 0.881368i \(-0.656623\pi\)
−0.472430 + 0.881368i \(0.656623\pi\)
\(608\) 0 0
\(609\) −38875.9 −2.58675
\(610\) 0 0
\(611\) −23387.7 −1.54855
\(612\) 0 0
\(613\) 5458.31 0.359639 0.179820 0.983700i \(-0.442449\pi\)
0.179820 + 0.983700i \(0.442449\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −4490.56 −0.293004 −0.146502 0.989210i \(-0.546801\pi\)
−0.146502 + 0.989210i \(0.546801\pi\)
\(618\) 0 0
\(619\) 7386.94 0.479655 0.239827 0.970816i \(-0.422909\pi\)
0.239827 + 0.970816i \(0.422909\pi\)
\(620\) 0 0
\(621\) −4044.70 −0.261366
\(622\) 0 0
\(623\) 20118.9 1.29382
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −36238.4 −2.30817
\(628\) 0 0
\(629\) 36246.0 2.29765
\(630\) 0 0
\(631\) 26268.5 1.65726 0.828631 0.559796i \(-0.189120\pi\)
0.828631 + 0.559796i \(0.189120\pi\)
\(632\) 0 0
\(633\) 16826.4 1.05654
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 6758.48 0.420378
\(638\) 0 0
\(639\) 13681.7 0.847008
\(640\) 0 0
\(641\) 21533.8 1.32688 0.663442 0.748228i \(-0.269095\pi\)
0.663442 + 0.748228i \(0.269095\pi\)
\(642\) 0 0
\(643\) 8517.72 0.522405 0.261202 0.965284i \(-0.415881\pi\)
0.261202 + 0.965284i \(0.415881\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −17887.7 −1.08692 −0.543459 0.839435i \(-0.682886\pi\)
−0.543459 + 0.839435i \(0.682886\pi\)
\(648\) 0 0
\(649\) 31402.5 1.89932
\(650\) 0 0
\(651\) 11979.5 0.721219
\(652\) 0 0
\(653\) 11322.5 0.678537 0.339269 0.940690i \(-0.389820\pi\)
0.339269 + 0.940690i \(0.389820\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 20296.8 1.20526
\(658\) 0 0
\(659\) −23067.9 −1.36358 −0.681788 0.731550i \(-0.738797\pi\)
−0.681788 + 0.731550i \(0.738797\pi\)
\(660\) 0 0
\(661\) 16492.0 0.970445 0.485222 0.874391i \(-0.338739\pi\)
0.485222 + 0.874391i \(0.338739\pi\)
\(662\) 0 0
\(663\) 43179.5 2.52934
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 8250.63 0.478959
\(668\) 0 0
\(669\) 25292.2 1.46166
\(670\) 0 0
\(671\) 16787.6 0.965842
\(672\) 0 0
\(673\) −6314.44 −0.361670 −0.180835 0.983513i \(-0.557880\pi\)
−0.180835 + 0.983513i \(0.557880\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 12486.8 0.708872 0.354436 0.935080i \(-0.384673\pi\)
0.354436 + 0.935080i \(0.384673\pi\)
\(678\) 0 0
\(679\) 41016.5 2.31822
\(680\) 0 0
\(681\) −29258.6 −1.64639
\(682\) 0 0
\(683\) 9593.65 0.537468 0.268734 0.963214i \(-0.413395\pi\)
0.268734 + 0.963214i \(0.413395\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −14737.2 −0.818425
\(688\) 0 0
\(689\) −21956.9 −1.21407
\(690\) 0 0
\(691\) 31333.7 1.72502 0.862510 0.506040i \(-0.168891\pi\)
0.862510 + 0.506040i \(0.168891\pi\)
\(692\) 0 0
\(693\) 32975.6 1.80756
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 29169.7 1.58520
\(698\) 0 0
\(699\) −31949.0 −1.72879
\(700\) 0 0
\(701\) 4337.52 0.233703 0.116851 0.993149i \(-0.462720\pi\)
0.116851 + 0.993149i \(0.462720\pi\)
\(702\) 0 0
\(703\) −42331.5 −2.27107
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2089.75 0.111164
\(708\) 0 0
\(709\) −26437.5 −1.40040 −0.700198 0.713948i \(-0.746905\pi\)
−0.700198 + 0.713948i \(0.746905\pi\)
\(710\) 0 0
\(711\) 17107.6 0.902371
\(712\) 0 0
\(713\) −2542.41 −0.133540
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −496.951 −0.0258842
\(718\) 0 0
\(719\) 26143.2 1.35602 0.678009 0.735053i \(-0.262843\pi\)
0.678009 + 0.735053i \(0.262843\pi\)
\(720\) 0 0
\(721\) 36940.5 1.90809
\(722\) 0 0
\(723\) 50458.8 2.59555
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −1355.22 −0.0691366 −0.0345683 0.999402i \(-0.511006\pi\)
−0.0345683 + 0.999402i \(0.511006\pi\)
\(728\) 0 0
\(729\) −31898.3 −1.62060
\(730\) 0 0
\(731\) −18309.3 −0.926394
\(732\) 0 0
\(733\) 632.549 0.0318741 0.0159371 0.999873i \(-0.494927\pi\)
0.0159371 + 0.999873i \(0.494927\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 21601.4 1.07965
\(738\) 0 0
\(739\) 10611.4 0.528209 0.264104 0.964494i \(-0.414924\pi\)
0.264104 + 0.964494i \(0.414924\pi\)
\(740\) 0 0
\(741\) −50429.1 −2.50008
\(742\) 0 0
\(743\) 5293.51 0.261373 0.130686 0.991424i \(-0.458282\pi\)
0.130686 + 0.991424i \(0.458282\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −39383.5 −1.92900
\(748\) 0 0
\(749\) −16211.8 −0.790876
\(750\) 0 0
\(751\) 19063.6 0.926287 0.463143 0.886283i \(-0.346721\pi\)
0.463143 + 0.886283i \(0.346721\pi\)
\(752\) 0 0
\(753\) 7215.24 0.349187
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −12698.5 −0.609687 −0.304844 0.952402i \(-0.598604\pi\)
−0.304844 + 0.952402i \(0.598604\pi\)
\(758\) 0 0
\(759\) −11712.0 −0.560106
\(760\) 0 0
\(761\) 19828.4 0.944517 0.472259 0.881460i \(-0.343439\pi\)
0.472259 + 0.881460i \(0.343439\pi\)
\(762\) 0 0
\(763\) −46677.7 −2.21474
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 43699.5 2.05723
\(768\) 0 0
\(769\) −2141.74 −0.100433 −0.0502166 0.998738i \(-0.515991\pi\)
−0.0502166 + 0.998738i \(0.515991\pi\)
\(770\) 0 0
\(771\) 47881.5 2.23659
\(772\) 0 0
\(773\) 9705.16 0.451579 0.225789 0.974176i \(-0.427504\pi\)
0.225789 + 0.974176i \(0.427504\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 64464.4 2.97638
\(778\) 0 0
\(779\) −34067.2 −1.56686
\(780\) 0 0
\(781\) 12934.0 0.592592
\(782\) 0 0
\(783\) −23439.8 −1.06982
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 24334.7 1.10221 0.551104 0.834437i \(-0.314207\pi\)
0.551104 + 0.834437i \(0.314207\pi\)
\(788\) 0 0
\(789\) 16333.1 0.736975
\(790\) 0 0
\(791\) 14349.0 0.644995
\(792\) 0 0
\(793\) 23361.6 1.04615
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −5913.01 −0.262797 −0.131399 0.991330i \(-0.541947\pi\)
−0.131399 + 0.991330i \(0.541947\pi\)
\(798\) 0 0
\(799\) 44331.7 1.96288
\(800\) 0 0
\(801\) 37156.3 1.63902
\(802\) 0 0
\(803\) 19187.6 0.843233
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 3490.22 0.152245
\(808\) 0 0
\(809\) −38547.6 −1.67523 −0.837614 0.546262i \(-0.816050\pi\)
−0.837614 + 0.546262i \(0.816050\pi\)
\(810\) 0 0
\(811\) −11741.9 −0.508400 −0.254200 0.967152i \(-0.581812\pi\)
−0.254200 + 0.967152i \(0.581812\pi\)
\(812\) 0 0
\(813\) −61654.0 −2.65965
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 21383.3 0.915677
\(818\) 0 0
\(819\) 45888.6 1.95785
\(820\) 0 0
\(821\) −8066.49 −0.342902 −0.171451 0.985193i \(-0.554845\pi\)
−0.171451 + 0.985193i \(0.554845\pi\)
\(822\) 0 0
\(823\) 34620.2 1.46633 0.733163 0.680053i \(-0.238043\pi\)
0.733163 + 0.680053i \(0.238043\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −37537.2 −1.57835 −0.789176 0.614167i \(-0.789492\pi\)
−0.789176 + 0.614167i \(0.789492\pi\)
\(828\) 0 0
\(829\) −32392.5 −1.35710 −0.678551 0.734554i \(-0.737392\pi\)
−0.678551 + 0.734554i \(0.737392\pi\)
\(830\) 0 0
\(831\) −19732.1 −0.823707
\(832\) 0 0
\(833\) −12810.8 −0.532855
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 7222.92 0.298280
\(838\) 0 0
\(839\) 46962.7 1.93246 0.966229 0.257684i \(-0.0829594\pi\)
0.966229 + 0.257684i \(0.0829594\pi\)
\(840\) 0 0
\(841\) 23425.0 0.960475
\(842\) 0 0
\(843\) 44947.0 1.83637
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 2282.76 0.0926052
\(848\) 0 0
\(849\) 40946.5 1.65522
\(850\) 0 0
\(851\) −13681.3 −0.551103
\(852\) 0 0
\(853\) −3879.81 −0.155735 −0.0778676 0.996964i \(-0.524811\pi\)
−0.0778676 + 0.996964i \(0.524811\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −37194.7 −1.48255 −0.741276 0.671201i \(-0.765779\pi\)
−0.741276 + 0.671201i \(0.765779\pi\)
\(858\) 0 0
\(859\) 7966.82 0.316443 0.158221 0.987404i \(-0.449424\pi\)
0.158221 + 0.987404i \(0.449424\pi\)
\(860\) 0 0
\(861\) 51879.2 2.05347
\(862\) 0 0
\(863\) −21821.7 −0.860742 −0.430371 0.902652i \(-0.641617\pi\)
−0.430371 + 0.902652i \(0.641617\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −41606.5 −1.62979
\(868\) 0 0
\(869\) 16172.7 0.631326
\(870\) 0 0
\(871\) 30060.4 1.16941
\(872\) 0 0
\(873\) 75750.5 2.93673
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −9394.86 −0.361735 −0.180868 0.983507i \(-0.557891\pi\)
−0.180868 + 0.983507i \(0.557891\pi\)
\(878\) 0 0
\(879\) −73658.6 −2.82644
\(880\) 0 0
\(881\) −27007.4 −1.03280 −0.516402 0.856346i \(-0.672729\pi\)
−0.516402 + 0.856346i \(0.672729\pi\)
\(882\) 0 0
\(883\) 1376.56 0.0524633 0.0262317 0.999656i \(-0.491649\pi\)
0.0262317 + 0.999656i \(0.491649\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −49523.6 −1.87468 −0.937339 0.348420i \(-0.886718\pi\)
−0.937339 + 0.348420i \(0.886718\pi\)
\(888\) 0 0
\(889\) −21480.8 −0.810395
\(890\) 0 0
\(891\) −7744.42 −0.291187
\(892\) 0 0
\(893\) −51774.8 −1.94018
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −16298.4 −0.606675
\(898\) 0 0
\(899\) −14733.8 −0.546605
\(900\) 0 0
\(901\) 41619.7 1.53891
\(902\) 0 0
\(903\) −32563.6 −1.20005
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 27592.7 1.01015 0.505073 0.863077i \(-0.331466\pi\)
0.505073 + 0.863077i \(0.331466\pi\)
\(908\) 0 0
\(909\) 3859.42 0.140824
\(910\) 0 0
\(911\) −34374.2 −1.25013 −0.625065 0.780572i \(-0.714928\pi\)
−0.625065 + 0.780572i \(0.714928\pi\)
\(912\) 0 0
\(913\) −37231.2 −1.34959
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −139.407 −0.00502030
\(918\) 0 0
\(919\) −49860.3 −1.78970 −0.894852 0.446362i \(-0.852719\pi\)
−0.894852 + 0.446362i \(0.852719\pi\)
\(920\) 0 0
\(921\) −45740.8 −1.63649
\(922\) 0 0
\(923\) 17998.9 0.641863
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 68222.8 2.41718
\(928\) 0 0
\(929\) −33348.6 −1.17775 −0.588876 0.808223i \(-0.700429\pi\)
−0.588876 + 0.808223i \(0.700429\pi\)
\(930\) 0 0
\(931\) 14961.7 0.526691
\(932\) 0 0
\(933\) 50930.0 1.78711
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 20404.8 0.711415 0.355707 0.934597i \(-0.384240\pi\)
0.355707 + 0.934597i \(0.384240\pi\)
\(938\) 0 0
\(939\) −41330.3 −1.43638
\(940\) 0 0
\(941\) −47495.4 −1.64538 −0.822692 0.568488i \(-0.807529\pi\)
−0.822692 + 0.568488i \(0.807529\pi\)
\(942\) 0 0
\(943\) −11010.3 −0.380218
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −38728.7 −1.32895 −0.664473 0.747312i \(-0.731344\pi\)
−0.664473 + 0.747312i \(0.731344\pi\)
\(948\) 0 0
\(949\) 26701.3 0.913343
\(950\) 0 0
\(951\) −43985.3 −1.49981
\(952\) 0 0
\(953\) −14200.9 −0.482699 −0.241349 0.970438i \(-0.577590\pi\)
−0.241349 + 0.970438i \(0.577590\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −67873.6 −2.29262
\(958\) 0 0
\(959\) −22438.7 −0.755563
\(960\) 0 0
\(961\) −25250.8 −0.847599
\(962\) 0 0
\(963\) −29940.4 −1.00189
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 38637.2 1.28489 0.642445 0.766332i \(-0.277920\pi\)
0.642445 + 0.766332i \(0.277920\pi\)
\(968\) 0 0
\(969\) 95589.2 3.16901
\(970\) 0 0
\(971\) −18828.7 −0.622288 −0.311144 0.950363i \(-0.600712\pi\)
−0.311144 + 0.950363i \(0.600712\pi\)
\(972\) 0 0
\(973\) −1233.80 −0.0406514
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −59016.1 −1.93254 −0.966271 0.257527i \(-0.917092\pi\)
−0.966271 + 0.257527i \(0.917092\pi\)
\(978\) 0 0
\(979\) 35125.7 1.14670
\(980\) 0 0
\(981\) −86205.9 −2.80565
\(982\) 0 0
\(983\) 42329.5 1.37345 0.686724 0.726918i \(-0.259048\pi\)
0.686724 + 0.726918i \(0.259048\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 78845.2 2.54273
\(988\) 0 0
\(989\) 6910.97 0.222200
\(990\) 0 0
\(991\) −6895.28 −0.221025 −0.110513 0.993875i \(-0.535249\pi\)
−0.110513 + 0.993875i \(0.535249\pi\)
\(992\) 0 0
\(993\) −81.2892 −0.00259782
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 34160.9 1.08514 0.542570 0.840010i \(-0.317451\pi\)
0.542570 + 0.840010i \(0.317451\pi\)
\(998\) 0 0
\(999\) 38868.2 1.23097
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.v.1.1 yes 3
4.3 odd 2 800.4.a.w.1.3 yes 3
5.2 odd 4 800.4.c.n.449.6 6
5.3 odd 4 800.4.c.n.449.1 6
5.4 even 2 800.4.a.x.1.3 yes 3
8.3 odd 2 1600.4.a.cr.1.1 3
8.5 even 2 1600.4.a.cs.1.3 3
20.3 even 4 800.4.c.m.449.6 6
20.7 even 4 800.4.c.m.449.1 6
20.19 odd 2 800.4.a.u.1.1 3
40.19 odd 2 1600.4.a.ct.1.3 3
40.29 even 2 1600.4.a.cq.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
800.4.a.u.1.1 3 20.19 odd 2
800.4.a.v.1.1 yes 3 1.1 even 1 trivial
800.4.a.w.1.3 yes 3 4.3 odd 2
800.4.a.x.1.3 yes 3 5.4 even 2
800.4.c.m.449.1 6 20.7 even 4
800.4.c.m.449.6 6 20.3 even 4
800.4.c.n.449.1 6 5.3 odd 4
800.4.c.n.449.6 6 5.2 odd 4
1600.4.a.cq.1.1 3 40.29 even 2
1600.4.a.cr.1.1 3 8.3 odd 2
1600.4.a.cs.1.3 3 8.5 even 2
1600.4.a.ct.1.3 3 40.19 odd 2