Properties

Label 8020.2.a.b.1.1
Level $8020$
Weight $2$
Character 8020.1
Self dual yes
Analytic conductor $64.040$
Analytic rank $0$
Dimension $2$
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] = [8020,2,Mod(1,8020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8020, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8020.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8020 = 2^{2} \cdot 5 \cdot 401 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8020.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: \(64.0400224211\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 8020.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.73205 q^{3} -1.00000 q^{5} -3.46410 q^{7} +4.46410 q^{9} +O(q^{10})\) \(q-2.73205 q^{3} -1.00000 q^{5} -3.46410 q^{7} +4.46410 q^{9} +3.46410 q^{11} -6.73205 q^{13} +2.73205 q^{15} +2.73205 q^{17} +6.92820 q^{19} +9.46410 q^{21} +8.19615 q^{23} +1.00000 q^{25} -4.00000 q^{27} +5.46410 q^{29} -5.46410 q^{31} -9.46410 q^{33} +3.46410 q^{35} -0.196152 q^{37} +18.3923 q^{39} -12.3923 q^{41} -2.00000 q^{43} -4.46410 q^{45} -8.92820 q^{47} +5.00000 q^{49} -7.46410 q^{51} +12.1962 q^{53} -3.46410 q^{55} -18.9282 q^{57} -8.00000 q^{59} +8.92820 q^{61} -15.4641 q^{63} +6.73205 q^{65} -8.19615 q^{67} -22.3923 q^{69} +2.53590 q^{71} -0.928203 q^{73} -2.73205 q^{75} -12.0000 q^{77} -2.46410 q^{81} -15.8564 q^{83} -2.73205 q^{85} -14.9282 q^{87} -15.8564 q^{89} +23.3205 q^{91} +14.9282 q^{93} -6.92820 q^{95} +10.7321 q^{97} +15.4641 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9} - 10 q^{13} + 2 q^{15} + 2 q^{17} + 12 q^{21} + 6 q^{23} + 2 q^{25} - 8 q^{27} + 4 q^{29} - 4 q^{31} - 12 q^{33} + 10 q^{37} + 16 q^{39} - 4 q^{41} - 4 q^{43} - 2 q^{45} - 4 q^{47} + 10 q^{49} - 8 q^{51} + 14 q^{53} - 24 q^{57} - 16 q^{59} + 4 q^{61} - 24 q^{63} + 10 q^{65} - 6 q^{67} - 24 q^{69} + 12 q^{71} + 12 q^{73} - 2 q^{75} - 24 q^{77} + 2 q^{81} - 4 q^{83} - 2 q^{85} - 16 q^{87} - 4 q^{89} + 12 q^{91} + 16 q^{93} + 18 q^{97} + 24 q^{99}+O(q^{100}) \) 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
\(3\) −2.73205 −1.57735 −0.788675 0.614810i \(-0.789233\pi\)
−0.788675 + 0.614810i \(0.789233\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.46410 −1.30931 −0.654654 0.755929i \(-0.727186\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) 4.46410 1.48803
\(10\) 0 0
\(11\) 3.46410 1.04447 0.522233 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 0 0
\(13\) −6.73205 −1.86713 −0.933567 0.358402i \(-0.883322\pi\)
−0.933567 + 0.358402i \(0.883322\pi\)
\(14\) 0 0
\(15\) 2.73205 0.705412
\(16\) 0 0
\(17\) 2.73205 0.662620 0.331310 0.943522i \(-0.392509\pi\)
0.331310 + 0.943522i \(0.392509\pi\)
\(18\) 0 0
\(19\) 6.92820 1.58944 0.794719 0.606977i \(-0.207618\pi\)
0.794719 + 0.606977i \(0.207618\pi\)
\(20\) 0 0
\(21\) 9.46410 2.06524
\(22\) 0 0
\(23\) 8.19615 1.70902 0.854508 0.519438i \(-0.173859\pi\)
0.854508 + 0.519438i \(0.173859\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) 5.46410 1.01466 0.507329 0.861752i \(-0.330633\pi\)
0.507329 + 0.861752i \(0.330633\pi\)
\(30\) 0 0
\(31\) −5.46410 −0.981382 −0.490691 0.871334i \(-0.663256\pi\)
−0.490691 + 0.871334i \(0.663256\pi\)
\(32\) 0 0
\(33\) −9.46410 −1.64749
\(34\) 0 0
\(35\) 3.46410 0.585540
\(36\) 0 0
\(37\) −0.196152 −0.0322473 −0.0161236 0.999870i \(-0.505133\pi\)
−0.0161236 + 0.999870i \(0.505133\pi\)
\(38\) 0 0
\(39\) 18.3923 2.94513
\(40\) 0 0
\(41\) −12.3923 −1.93535 −0.967676 0.252195i \(-0.918848\pi\)
−0.967676 + 0.252195i \(0.918848\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) −4.46410 −0.665469
\(46\) 0 0
\(47\) −8.92820 −1.30231 −0.651156 0.758944i \(-0.725716\pi\)
−0.651156 + 0.758944i \(0.725716\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) −7.46410 −1.04518
\(52\) 0 0
\(53\) 12.1962 1.67527 0.837635 0.546230i \(-0.183938\pi\)
0.837635 + 0.546230i \(0.183938\pi\)
\(54\) 0 0
\(55\) −3.46410 −0.467099
\(56\) 0 0
\(57\) −18.9282 −2.50710
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 8.92820 1.14314 0.571570 0.820554i \(-0.306335\pi\)
0.571570 + 0.820554i \(0.306335\pi\)
\(62\) 0 0
\(63\) −15.4641 −1.94829
\(64\) 0 0
\(65\) 6.73205 0.835008
\(66\) 0 0
\(67\) −8.19615 −1.00132 −0.500660 0.865644i \(-0.666909\pi\)
−0.500660 + 0.865644i \(0.666909\pi\)
\(68\) 0 0
\(69\) −22.3923 −2.69572
\(70\) 0 0
\(71\) 2.53590 0.300956 0.150478 0.988613i \(-0.451919\pi\)
0.150478 + 0.988613i \(0.451919\pi\)
\(72\) 0 0
\(73\) −0.928203 −0.108638 −0.0543190 0.998524i \(-0.517299\pi\)
−0.0543190 + 0.998524i \(0.517299\pi\)
\(74\) 0 0
\(75\) −2.73205 −0.315470
\(76\) 0 0
\(77\) −12.0000 −1.36753
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −2.46410 −0.273789
\(82\) 0 0
\(83\) −15.8564 −1.74047 −0.870233 0.492640i \(-0.836032\pi\)
−0.870233 + 0.492640i \(0.836032\pi\)
\(84\) 0 0
\(85\) −2.73205 −0.296333
\(86\) 0 0
\(87\) −14.9282 −1.60047
\(88\) 0 0
\(89\) −15.8564 −1.68078 −0.840388 0.541985i \(-0.817673\pi\)
−0.840388 + 0.541985i \(0.817673\pi\)
\(90\) 0 0
\(91\) 23.3205 2.44465
\(92\) 0 0
\(93\) 14.9282 1.54798
\(94\) 0 0
\(95\) −6.92820 −0.710819
\(96\) 0 0
\(97\) 10.7321 1.08967 0.544837 0.838542i \(-0.316591\pi\)
0.544837 + 0.838542i \(0.316591\pi\)
\(98\) 0 0
\(99\) 15.4641 1.55420
\(100\) 0 0
\(101\) 3.07180 0.305655 0.152828 0.988253i \(-0.451162\pi\)
0.152828 + 0.988253i \(0.451162\pi\)
\(102\) 0 0
\(103\) −10.3923 −1.02398 −0.511992 0.858990i \(-0.671092\pi\)
−0.511992 + 0.858990i \(0.671092\pi\)
\(104\) 0 0
\(105\) −9.46410 −0.923602
\(106\) 0 0
\(107\) −12.5885 −1.21697 −0.608486 0.793565i \(-0.708223\pi\)
−0.608486 + 0.793565i \(0.708223\pi\)
\(108\) 0 0
\(109\) 15.3205 1.46744 0.733719 0.679453i \(-0.237783\pi\)
0.733719 + 0.679453i \(0.237783\pi\)
\(110\) 0 0
\(111\) 0.535898 0.0508652
\(112\) 0 0
\(113\) 15.4641 1.45474 0.727370 0.686245i \(-0.240742\pi\)
0.727370 + 0.686245i \(0.240742\pi\)
\(114\) 0 0
\(115\) −8.19615 −0.764295
\(116\) 0 0
\(117\) −30.0526 −2.77836
\(118\) 0 0
\(119\) −9.46410 −0.867573
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 33.8564 3.05273
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 10.7321 0.952316 0.476158 0.879360i \(-0.342029\pi\)
0.476158 + 0.879360i \(0.342029\pi\)
\(128\) 0 0
\(129\) 5.46410 0.481087
\(130\) 0 0
\(131\) 6.53590 0.571044 0.285522 0.958372i \(-0.407833\pi\)
0.285522 + 0.958372i \(0.407833\pi\)
\(132\) 0 0
\(133\) −24.0000 −2.08106
\(134\) 0 0
\(135\) 4.00000 0.344265
\(136\) 0 0
\(137\) −9.66025 −0.825331 −0.412666 0.910883i \(-0.635402\pi\)
−0.412666 + 0.910883i \(0.635402\pi\)
\(138\) 0 0
\(139\) 10.9282 0.926918 0.463459 0.886118i \(-0.346608\pi\)
0.463459 + 0.886118i \(0.346608\pi\)
\(140\) 0 0
\(141\) 24.3923 2.05420
\(142\) 0 0
\(143\) −23.3205 −1.95016
\(144\) 0 0
\(145\) −5.46410 −0.453769
\(146\) 0 0
\(147\) −13.6603 −1.12668
\(148\) 0 0
\(149\) −3.07180 −0.251651 −0.125826 0.992052i \(-0.540158\pi\)
−0.125826 + 0.992052i \(0.540158\pi\)
\(150\) 0 0
\(151\) 15.4641 1.25845 0.629225 0.777223i \(-0.283372\pi\)
0.629225 + 0.777223i \(0.283372\pi\)
\(152\) 0 0
\(153\) 12.1962 0.986000
\(154\) 0 0
\(155\) 5.46410 0.438887
\(156\) 0 0
\(157\) 1.26795 0.101193 0.0505967 0.998719i \(-0.483888\pi\)
0.0505967 + 0.998719i \(0.483888\pi\)
\(158\) 0 0
\(159\) −33.3205 −2.64249
\(160\) 0 0
\(161\) −28.3923 −2.23763
\(162\) 0 0
\(163\) −12.1962 −0.955276 −0.477638 0.878557i \(-0.658507\pi\)
−0.477638 + 0.878557i \(0.658507\pi\)
\(164\) 0 0
\(165\) 9.46410 0.736779
\(166\) 0 0
\(167\) 12.1962 0.943767 0.471883 0.881661i \(-0.343574\pi\)
0.471883 + 0.881661i \(0.343574\pi\)
\(168\) 0 0
\(169\) 32.3205 2.48619
\(170\) 0 0
\(171\) 30.9282 2.36514
\(172\) 0 0
\(173\) −7.46410 −0.567485 −0.283743 0.958900i \(-0.591576\pi\)
−0.283743 + 0.958900i \(0.591576\pi\)
\(174\) 0 0
\(175\) −3.46410 −0.261861
\(176\) 0 0
\(177\) 21.8564 1.64283
\(178\) 0 0
\(179\) −25.3205 −1.89254 −0.946272 0.323372i \(-0.895183\pi\)
−0.946272 + 0.323372i \(0.895183\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) −24.3923 −1.80313
\(184\) 0 0
\(185\) 0.196152 0.0144214
\(186\) 0 0
\(187\) 9.46410 0.692084
\(188\) 0 0
\(189\) 13.8564 1.00791
\(190\) 0 0
\(191\) 16.3923 1.18611 0.593053 0.805164i \(-0.297923\pi\)
0.593053 + 0.805164i \(0.297923\pi\)
\(192\) 0 0
\(193\) 4.19615 0.302046 0.151023 0.988530i \(-0.451743\pi\)
0.151023 + 0.988530i \(0.451743\pi\)
\(194\) 0 0
\(195\) −18.3923 −1.31710
\(196\) 0 0
\(197\) 3.46410 0.246807 0.123404 0.992357i \(-0.460619\pi\)
0.123404 + 0.992357i \(0.460619\pi\)
\(198\) 0 0
\(199\) 16.7846 1.18983 0.594915 0.803789i \(-0.297186\pi\)
0.594915 + 0.803789i \(0.297186\pi\)
\(200\) 0 0
\(201\) 22.3923 1.57943
\(202\) 0 0
\(203\) −18.9282 −1.32850
\(204\) 0 0
\(205\) 12.3923 0.865516
\(206\) 0 0
\(207\) 36.5885 2.54307
\(208\) 0 0
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) −10.5359 −0.725321 −0.362661 0.931921i \(-0.618132\pi\)
−0.362661 + 0.931921i \(0.618132\pi\)
\(212\) 0 0
\(213\) −6.92820 −0.474713
\(214\) 0 0
\(215\) 2.00000 0.136399
\(216\) 0 0
\(217\) 18.9282 1.28493
\(218\) 0 0
\(219\) 2.53590 0.171360
\(220\) 0 0
\(221\) −18.3923 −1.23720
\(222\) 0 0
\(223\) −21.3205 −1.42773 −0.713863 0.700285i \(-0.753056\pi\)
−0.713863 + 0.700285i \(0.753056\pi\)
\(224\) 0 0
\(225\) 4.46410 0.297607
\(226\) 0 0
\(227\) −24.1962 −1.60596 −0.802978 0.596009i \(-0.796752\pi\)
−0.802978 + 0.596009i \(0.796752\pi\)
\(228\) 0 0
\(229\) 7.07180 0.467317 0.233659 0.972319i \(-0.424930\pi\)
0.233659 + 0.972319i \(0.424930\pi\)
\(230\) 0 0
\(231\) 32.7846 2.15707
\(232\) 0 0
\(233\) 15.8038 1.03534 0.517672 0.855579i \(-0.326799\pi\)
0.517672 + 0.855579i \(0.326799\pi\)
\(234\) 0 0
\(235\) 8.92820 0.582412
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −27.4641 −1.77651 −0.888253 0.459355i \(-0.848080\pi\)
−0.888253 + 0.459355i \(0.848080\pi\)
\(240\) 0 0
\(241\) 15.8564 1.02140 0.510700 0.859759i \(-0.329386\pi\)
0.510700 + 0.859759i \(0.329386\pi\)
\(242\) 0 0
\(243\) 18.7321 1.20166
\(244\) 0 0
\(245\) −5.00000 −0.319438
\(246\) 0 0
\(247\) −46.6410 −2.96770
\(248\) 0 0
\(249\) 43.3205 2.74533
\(250\) 0 0
\(251\) 27.3205 1.72446 0.862228 0.506521i \(-0.169069\pi\)
0.862228 + 0.506521i \(0.169069\pi\)
\(252\) 0 0
\(253\) 28.3923 1.78501
\(254\) 0 0
\(255\) 7.46410 0.467420
\(256\) 0 0
\(257\) 7.46410 0.465598 0.232799 0.972525i \(-0.425212\pi\)
0.232799 + 0.972525i \(0.425212\pi\)
\(258\) 0 0
\(259\) 0.679492 0.0422216
\(260\) 0 0
\(261\) 24.3923 1.50985
\(262\) 0 0
\(263\) −12.9282 −0.797187 −0.398594 0.917128i \(-0.630502\pi\)
−0.398594 + 0.917128i \(0.630502\pi\)
\(264\) 0 0
\(265\) −12.1962 −0.749204
\(266\) 0 0
\(267\) 43.3205 2.65117
\(268\) 0 0
\(269\) 3.46410 0.211210 0.105605 0.994408i \(-0.466322\pi\)
0.105605 + 0.994408i \(0.466322\pi\)
\(270\) 0 0
\(271\) 0.392305 0.0238308 0.0119154 0.999929i \(-0.496207\pi\)
0.0119154 + 0.999929i \(0.496207\pi\)
\(272\) 0 0
\(273\) −63.7128 −3.85607
\(274\) 0 0
\(275\) 3.46410 0.208893
\(276\) 0 0
\(277\) −11.8038 −0.709224 −0.354612 0.935013i \(-0.615387\pi\)
−0.354612 + 0.935013i \(0.615387\pi\)
\(278\) 0 0
\(279\) −24.3923 −1.46033
\(280\) 0 0
\(281\) 27.8564 1.66177 0.830887 0.556441i \(-0.187834\pi\)
0.830887 + 0.556441i \(0.187834\pi\)
\(282\) 0 0
\(283\) −5.66025 −0.336467 −0.168234 0.985747i \(-0.553806\pi\)
−0.168234 + 0.985747i \(0.553806\pi\)
\(284\) 0 0
\(285\) 18.9282 1.12121
\(286\) 0 0
\(287\) 42.9282 2.53397
\(288\) 0 0
\(289\) −9.53590 −0.560935
\(290\) 0 0
\(291\) −29.3205 −1.71880
\(292\) 0 0
\(293\) 4.19615 0.245142 0.122571 0.992460i \(-0.460886\pi\)
0.122571 + 0.992460i \(0.460886\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 0 0
\(297\) −13.8564 −0.804030
\(298\) 0 0
\(299\) −55.1769 −3.19096
\(300\) 0 0
\(301\) 6.92820 0.399335
\(302\) 0 0
\(303\) −8.39230 −0.482125
\(304\) 0 0
\(305\) −8.92820 −0.511227
\(306\) 0 0
\(307\) −4.92820 −0.281267 −0.140634 0.990062i \(-0.544914\pi\)
−0.140634 + 0.990062i \(0.544914\pi\)
\(308\) 0 0
\(309\) 28.3923 1.61518
\(310\) 0 0
\(311\) 6.92820 0.392862 0.196431 0.980518i \(-0.437065\pi\)
0.196431 + 0.980518i \(0.437065\pi\)
\(312\) 0 0
\(313\) −5.60770 −0.316966 −0.158483 0.987362i \(-0.550660\pi\)
−0.158483 + 0.987362i \(0.550660\pi\)
\(314\) 0 0
\(315\) 15.4641 0.871303
\(316\) 0 0
\(317\) 0.588457 0.0330511 0.0165255 0.999863i \(-0.494740\pi\)
0.0165255 + 0.999863i \(0.494740\pi\)
\(318\) 0 0
\(319\) 18.9282 1.05978
\(320\) 0 0
\(321\) 34.3923 1.91959
\(322\) 0 0
\(323\) 18.9282 1.05319
\(324\) 0 0
\(325\) −6.73205 −0.373427
\(326\) 0 0
\(327\) −41.8564 −2.31466
\(328\) 0 0
\(329\) 30.9282 1.70513
\(330\) 0 0
\(331\) −26.3923 −1.45065 −0.725326 0.688405i \(-0.758311\pi\)
−0.725326 + 0.688405i \(0.758311\pi\)
\(332\) 0 0
\(333\) −0.875644 −0.0479850
\(334\) 0 0
\(335\) 8.19615 0.447804
\(336\) 0 0
\(337\) 33.7128 1.83645 0.918227 0.396055i \(-0.129621\pi\)
0.918227 + 0.396055i \(0.129621\pi\)
\(338\) 0 0
\(339\) −42.2487 −2.29464
\(340\) 0 0
\(341\) −18.9282 −1.02502
\(342\) 0 0
\(343\) 6.92820 0.374088
\(344\) 0 0
\(345\) 22.3923 1.20556
\(346\) 0 0
\(347\) −30.0526 −1.61331 −0.806653 0.591025i \(-0.798723\pi\)
−0.806653 + 0.591025i \(0.798723\pi\)
\(348\) 0 0
\(349\) −7.46410 −0.399544 −0.199772 0.979842i \(-0.564020\pi\)
−0.199772 + 0.979842i \(0.564020\pi\)
\(350\) 0 0
\(351\) 26.9282 1.43732
\(352\) 0 0
\(353\) 1.26795 0.0674861 0.0337431 0.999431i \(-0.489257\pi\)
0.0337431 + 0.999431i \(0.489257\pi\)
\(354\) 0 0
\(355\) −2.53590 −0.134592
\(356\) 0 0
\(357\) 25.8564 1.36847
\(358\) 0 0
\(359\) 6.92820 0.365657 0.182828 0.983145i \(-0.441475\pi\)
0.182828 + 0.983145i \(0.441475\pi\)
\(360\) 0 0
\(361\) 29.0000 1.52632
\(362\) 0 0
\(363\) −2.73205 −0.143395
\(364\) 0 0
\(365\) 0.928203 0.0485844
\(366\) 0 0
\(367\) 26.7321 1.39540 0.697701 0.716389i \(-0.254206\pi\)
0.697701 + 0.716389i \(0.254206\pi\)
\(368\) 0 0
\(369\) −55.3205 −2.87987
\(370\) 0 0
\(371\) −42.2487 −2.19344
\(372\) 0 0
\(373\) −25.7128 −1.33136 −0.665679 0.746238i \(-0.731858\pi\)
−0.665679 + 0.746238i \(0.731858\pi\)
\(374\) 0 0
\(375\) 2.73205 0.141082
\(376\) 0 0
\(377\) −36.7846 −1.89450
\(378\) 0 0
\(379\) −7.46410 −0.383405 −0.191703 0.981453i \(-0.561401\pi\)
−0.191703 + 0.981453i \(0.561401\pi\)
\(380\) 0 0
\(381\) −29.3205 −1.50214
\(382\) 0 0
\(383\) 26.7846 1.36863 0.684315 0.729187i \(-0.260101\pi\)
0.684315 + 0.729187i \(0.260101\pi\)
\(384\) 0 0
\(385\) 12.0000 0.611577
\(386\) 0 0
\(387\) −8.92820 −0.453846
\(388\) 0 0
\(389\) −18.3923 −0.932527 −0.466263 0.884646i \(-0.654400\pi\)
−0.466263 + 0.884646i \(0.654400\pi\)
\(390\) 0 0
\(391\) 22.3923 1.13243
\(392\) 0 0
\(393\) −17.8564 −0.900737
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0.535898 0.0268960 0.0134480 0.999910i \(-0.495719\pi\)
0.0134480 + 0.999910i \(0.495719\pi\)
\(398\) 0 0
\(399\) 65.5692 3.28257
\(400\) 0 0
\(401\) −1.00000 −0.0499376
\(402\) 0 0
\(403\) 36.7846 1.83237
\(404\) 0 0
\(405\) 2.46410 0.122442
\(406\) 0 0
\(407\) −0.679492 −0.0336812
\(408\) 0 0
\(409\) −16.3923 −0.810547 −0.405274 0.914195i \(-0.632824\pi\)
−0.405274 + 0.914195i \(0.632824\pi\)
\(410\) 0 0
\(411\) 26.3923 1.30184
\(412\) 0 0
\(413\) 27.7128 1.36366
\(414\) 0 0
\(415\) 15.8564 0.778360
\(416\) 0 0
\(417\) −29.8564 −1.46207
\(418\) 0 0
\(419\) 1.32051 0.0645110 0.0322555 0.999480i \(-0.489731\pi\)
0.0322555 + 0.999480i \(0.489731\pi\)
\(420\) 0 0
\(421\) 13.4641 0.656200 0.328100 0.944643i \(-0.393592\pi\)
0.328100 + 0.944643i \(0.393592\pi\)
\(422\) 0 0
\(423\) −39.8564 −1.93788
\(424\) 0 0
\(425\) 2.73205 0.132524
\(426\) 0 0
\(427\) −30.9282 −1.49672
\(428\) 0 0
\(429\) 63.7128 3.07608
\(430\) 0 0
\(431\) 11.6077 0.559123 0.279562 0.960128i \(-0.409811\pi\)
0.279562 + 0.960128i \(0.409811\pi\)
\(432\) 0 0
\(433\) 5.60770 0.269489 0.134744 0.990880i \(-0.456979\pi\)
0.134744 + 0.990880i \(0.456979\pi\)
\(434\) 0 0
\(435\) 14.9282 0.715753
\(436\) 0 0
\(437\) 56.7846 2.71638
\(438\) 0 0
\(439\) 0.784610 0.0374474 0.0187237 0.999825i \(-0.494040\pi\)
0.0187237 + 0.999825i \(0.494040\pi\)
\(440\) 0 0
\(441\) 22.3205 1.06288
\(442\) 0 0
\(443\) 16.1962 0.769502 0.384751 0.923020i \(-0.374287\pi\)
0.384751 + 0.923020i \(0.374287\pi\)
\(444\) 0 0
\(445\) 15.8564 0.751666
\(446\) 0 0
\(447\) 8.39230 0.396942
\(448\) 0 0
\(449\) 37.3205 1.76126 0.880632 0.473801i \(-0.157118\pi\)
0.880632 + 0.473801i \(0.157118\pi\)
\(450\) 0 0
\(451\) −42.9282 −2.02141
\(452\) 0 0
\(453\) −42.2487 −1.98502
\(454\) 0 0
\(455\) −23.3205 −1.09328
\(456\) 0 0
\(457\) −1.32051 −0.0617708 −0.0308854 0.999523i \(-0.509833\pi\)
−0.0308854 + 0.999523i \(0.509833\pi\)
\(458\) 0 0
\(459\) −10.9282 −0.510085
\(460\) 0 0
\(461\) 10.0000 0.465746 0.232873 0.972507i \(-0.425187\pi\)
0.232873 + 0.972507i \(0.425187\pi\)
\(462\) 0 0
\(463\) −16.5885 −0.770931 −0.385465 0.922722i \(-0.625959\pi\)
−0.385465 + 0.922722i \(0.625959\pi\)
\(464\) 0 0
\(465\) −14.9282 −0.692279
\(466\) 0 0
\(467\) −3.12436 −0.144578 −0.0722890 0.997384i \(-0.523030\pi\)
−0.0722890 + 0.997384i \(0.523030\pi\)
\(468\) 0 0
\(469\) 28.3923 1.31103
\(470\) 0 0
\(471\) −3.46410 −0.159617
\(472\) 0 0
\(473\) −6.92820 −0.318559
\(474\) 0 0
\(475\) 6.92820 0.317888
\(476\) 0 0
\(477\) 54.4449 2.49286
\(478\) 0 0
\(479\) −24.7846 −1.13244 −0.566219 0.824255i \(-0.691594\pi\)
−0.566219 + 0.824255i \(0.691594\pi\)
\(480\) 0 0
\(481\) 1.32051 0.0602100
\(482\) 0 0
\(483\) 77.5692 3.52952
\(484\) 0 0
\(485\) −10.7321 −0.487317
\(486\) 0 0
\(487\) −11.8564 −0.537265 −0.268633 0.963243i \(-0.586572\pi\)
−0.268633 + 0.963243i \(0.586572\pi\)
\(488\) 0 0
\(489\) 33.3205 1.50681
\(490\) 0 0
\(491\) 10.1436 0.457774 0.228887 0.973453i \(-0.426491\pi\)
0.228887 + 0.973453i \(0.426491\pi\)
\(492\) 0 0
\(493\) 14.9282 0.672332
\(494\) 0 0
\(495\) −15.4641 −0.695060
\(496\) 0 0
\(497\) −8.78461 −0.394044
\(498\) 0 0
\(499\) 14.9282 0.668278 0.334139 0.942524i \(-0.391554\pi\)
0.334139 + 0.942524i \(0.391554\pi\)
\(500\) 0 0
\(501\) −33.3205 −1.48865
\(502\) 0 0
\(503\) 4.14359 0.184754 0.0923769 0.995724i \(-0.470554\pi\)
0.0923769 + 0.995724i \(0.470554\pi\)
\(504\) 0 0
\(505\) −3.07180 −0.136693
\(506\) 0 0
\(507\) −88.3013 −3.92160
\(508\) 0 0
\(509\) 5.60770 0.248557 0.124278 0.992247i \(-0.460338\pi\)
0.124278 + 0.992247i \(0.460338\pi\)
\(510\) 0 0
\(511\) 3.21539 0.142240
\(512\) 0 0
\(513\) −27.7128 −1.22355
\(514\) 0 0
\(515\) 10.3923 0.457940
\(516\) 0 0
\(517\) −30.9282 −1.36022
\(518\) 0 0
\(519\) 20.3923 0.895123
\(520\) 0 0
\(521\) 7.85641 0.344195 0.172098 0.985080i \(-0.444946\pi\)
0.172098 + 0.985080i \(0.444946\pi\)
\(522\) 0 0
\(523\) 28.5885 1.25009 0.625043 0.780590i \(-0.285081\pi\)
0.625043 + 0.780590i \(0.285081\pi\)
\(524\) 0 0
\(525\) 9.46410 0.413047
\(526\) 0 0
\(527\) −14.9282 −0.650283
\(528\) 0 0
\(529\) 44.1769 1.92074
\(530\) 0 0
\(531\) −35.7128 −1.54980
\(532\) 0 0
\(533\) 83.4256 3.61356
\(534\) 0 0
\(535\) 12.5885 0.544246
\(536\) 0 0
\(537\) 69.1769 2.98520
\(538\) 0 0
\(539\) 17.3205 0.746047
\(540\) 0 0
\(541\) −25.7128 −1.10548 −0.552740 0.833354i \(-0.686418\pi\)
−0.552740 + 0.833354i \(0.686418\pi\)
\(542\) 0 0
\(543\) 27.3205 1.17244
\(544\) 0 0
\(545\) −15.3205 −0.656258
\(546\) 0 0
\(547\) −22.3923 −0.957426 −0.478713 0.877971i \(-0.658897\pi\)
−0.478713 + 0.877971i \(0.658897\pi\)
\(548\) 0 0
\(549\) 39.8564 1.70103
\(550\) 0 0
\(551\) 37.8564 1.61274
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −0.535898 −0.0227476
\(556\) 0 0
\(557\) 42.0000 1.77960 0.889799 0.456354i \(-0.150845\pi\)
0.889799 + 0.456354i \(0.150845\pi\)
\(558\) 0 0
\(559\) 13.4641 0.569471
\(560\) 0 0
\(561\) −25.8564 −1.09166
\(562\) 0 0
\(563\) −18.3923 −0.775143 −0.387572 0.921840i \(-0.626686\pi\)
−0.387572 + 0.921840i \(0.626686\pi\)
\(564\) 0 0
\(565\) −15.4641 −0.650580
\(566\) 0 0
\(567\) 8.53590 0.358474
\(568\) 0 0
\(569\) −8.24871 −0.345804 −0.172902 0.984939i \(-0.555314\pi\)
−0.172902 + 0.984939i \(0.555314\pi\)
\(570\) 0 0
\(571\) 27.3205 1.14333 0.571664 0.820488i \(-0.306298\pi\)
0.571664 + 0.820488i \(0.306298\pi\)
\(572\) 0 0
\(573\) −44.7846 −1.87090
\(574\) 0 0
\(575\) 8.19615 0.341803
\(576\) 0 0
\(577\) −17.3205 −0.721062 −0.360531 0.932747i \(-0.617405\pi\)
−0.360531 + 0.932747i \(0.617405\pi\)
\(578\) 0 0
\(579\) −11.4641 −0.476432
\(580\) 0 0
\(581\) 54.9282 2.27881
\(582\) 0 0
\(583\) 42.2487 1.74976
\(584\) 0 0
\(585\) 30.0526 1.24252
\(586\) 0 0
\(587\) 35.1769 1.45191 0.725953 0.687744i \(-0.241399\pi\)
0.725953 + 0.687744i \(0.241399\pi\)
\(588\) 0 0
\(589\) −37.8564 −1.55985
\(590\) 0 0
\(591\) −9.46410 −0.389301
\(592\) 0 0
\(593\) 15.1244 0.621083 0.310541 0.950560i \(-0.399490\pi\)
0.310541 + 0.950560i \(0.399490\pi\)
\(594\) 0 0
\(595\) 9.46410 0.387990
\(596\) 0 0
\(597\) −45.8564 −1.87678
\(598\) 0 0
\(599\) −19.1769 −0.783547 −0.391774 0.920062i \(-0.628138\pi\)
−0.391774 + 0.920062i \(0.628138\pi\)
\(600\) 0 0
\(601\) 14.0000 0.571072 0.285536 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) 0 0
\(603\) −36.5885 −1.49000
\(604\) 0 0
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 45.7128 1.85543 0.927713 0.373294i \(-0.121772\pi\)
0.927713 + 0.373294i \(0.121772\pi\)
\(608\) 0 0
\(609\) 51.7128 2.09551
\(610\) 0 0
\(611\) 60.1051 2.43159
\(612\) 0 0
\(613\) 45.6603 1.84420 0.922100 0.386951i \(-0.126472\pi\)
0.922100 + 0.386951i \(0.126472\pi\)
\(614\) 0 0
\(615\) −33.8564 −1.36522
\(616\) 0 0
\(617\) 7.80385 0.314171 0.157086 0.987585i \(-0.449790\pi\)
0.157086 + 0.987585i \(0.449790\pi\)
\(618\) 0 0
\(619\) −13.3205 −0.535396 −0.267698 0.963503i \(-0.586263\pi\)
−0.267698 + 0.963503i \(0.586263\pi\)
\(620\) 0 0
\(621\) −32.7846 −1.31560
\(622\) 0 0
\(623\) 54.9282 2.20065
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −65.5692 −2.61858
\(628\) 0 0
\(629\) −0.535898 −0.0213677
\(630\) 0 0
\(631\) 36.1051 1.43732 0.718661 0.695361i \(-0.244755\pi\)
0.718661 + 0.695361i \(0.244755\pi\)
\(632\) 0 0
\(633\) 28.7846 1.14409
\(634\) 0 0
\(635\) −10.7321 −0.425888
\(636\) 0 0
\(637\) −33.6603 −1.33367
\(638\) 0 0
\(639\) 11.3205 0.447832
\(640\) 0 0
\(641\) −23.8564 −0.942271 −0.471136 0.882061i \(-0.656156\pi\)
−0.471136 + 0.882061i \(0.656156\pi\)
\(642\) 0 0
\(643\) −19.0718 −0.752118 −0.376059 0.926596i \(-0.622721\pi\)
−0.376059 + 0.926596i \(0.622721\pi\)
\(644\) 0 0
\(645\) −5.46410 −0.215149
\(646\) 0 0
\(647\) 29.6603 1.16606 0.583032 0.812449i \(-0.301866\pi\)
0.583032 + 0.812449i \(0.301866\pi\)
\(648\) 0 0
\(649\) −27.7128 −1.08782
\(650\) 0 0
\(651\) −51.7128 −2.02678
\(652\) 0 0
\(653\) 44.2487 1.73159 0.865793 0.500402i \(-0.166815\pi\)
0.865793 + 0.500402i \(0.166815\pi\)
\(654\) 0 0
\(655\) −6.53590 −0.255379
\(656\) 0 0
\(657\) −4.14359 −0.161657
\(658\) 0 0
\(659\) 6.92820 0.269884 0.134942 0.990853i \(-0.456915\pi\)
0.134942 + 0.990853i \(0.456915\pi\)
\(660\) 0 0
\(661\) 23.8564 0.927907 0.463953 0.885860i \(-0.346431\pi\)
0.463953 + 0.885860i \(0.346431\pi\)
\(662\) 0 0
\(663\) 50.2487 1.95150
\(664\) 0 0
\(665\) 24.0000 0.930680
\(666\) 0 0
\(667\) 44.7846 1.73407
\(668\) 0 0
\(669\) 58.2487 2.25203
\(670\) 0 0
\(671\) 30.9282 1.19397
\(672\) 0 0
\(673\) 38.4449 1.48194 0.740970 0.671538i \(-0.234366\pi\)
0.740970 + 0.671538i \(0.234366\pi\)
\(674\) 0 0
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) −1.60770 −0.0617887 −0.0308944 0.999523i \(-0.509836\pi\)
−0.0308944 + 0.999523i \(0.509836\pi\)
\(678\) 0 0
\(679\) −37.1769 −1.42672
\(680\) 0 0
\(681\) 66.1051 2.53315
\(682\) 0 0
\(683\) −29.6603 −1.13492 −0.567459 0.823402i \(-0.692073\pi\)
−0.567459 + 0.823402i \(0.692073\pi\)
\(684\) 0 0
\(685\) 9.66025 0.369099
\(686\) 0 0
\(687\) −19.3205 −0.737123
\(688\) 0 0
\(689\) −82.1051 −3.12796
\(690\) 0 0
\(691\) −16.0000 −0.608669 −0.304334 0.952565i \(-0.598434\pi\)
−0.304334 + 0.952565i \(0.598434\pi\)
\(692\) 0 0
\(693\) −53.5692 −2.03493
\(694\) 0 0
\(695\) −10.9282 −0.414530
\(696\) 0 0
\(697\) −33.8564 −1.28240
\(698\) 0 0
\(699\) −43.1769 −1.63310
\(700\) 0 0
\(701\) −11.0718 −0.418176 −0.209088 0.977897i \(-0.567050\pi\)
−0.209088 + 0.977897i \(0.567050\pi\)
\(702\) 0 0
\(703\) −1.35898 −0.0512550
\(704\) 0 0
\(705\) −24.3923 −0.918667
\(706\) 0 0
\(707\) −10.6410 −0.400197
\(708\) 0 0
\(709\) 1.21539 0.0456449 0.0228225 0.999740i \(-0.492735\pi\)
0.0228225 + 0.999740i \(0.492735\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −44.7846 −1.67720
\(714\) 0 0
\(715\) 23.3205 0.872138
\(716\) 0 0
\(717\) 75.0333 2.80217
\(718\) 0 0
\(719\) 0.248711 0.00927537 0.00463768 0.999989i \(-0.498524\pi\)
0.00463768 + 0.999989i \(0.498524\pi\)
\(720\) 0 0
\(721\) 36.0000 1.34071
\(722\) 0 0
\(723\) −43.3205 −1.61111
\(724\) 0 0
\(725\) 5.46410 0.202932
\(726\) 0 0
\(727\) 26.7321 0.991437 0.495718 0.868483i \(-0.334905\pi\)
0.495718 + 0.868483i \(0.334905\pi\)
\(728\) 0 0
\(729\) −43.7846 −1.62165
\(730\) 0 0
\(731\) −5.46410 −0.202097
\(732\) 0 0
\(733\) −12.5359 −0.463024 −0.231512 0.972832i \(-0.574367\pi\)
−0.231512 + 0.972832i \(0.574367\pi\)
\(734\) 0 0
\(735\) 13.6603 0.503866
\(736\) 0 0
\(737\) −28.3923 −1.04584
\(738\) 0 0
\(739\) −20.5359 −0.755425 −0.377713 0.925923i \(-0.623289\pi\)
−0.377713 + 0.925923i \(0.623289\pi\)
\(740\) 0 0
\(741\) 127.426 4.68110
\(742\) 0 0
\(743\) −36.9808 −1.35669 −0.678346 0.734742i \(-0.737303\pi\)
−0.678346 + 0.734742i \(0.737303\pi\)
\(744\) 0 0
\(745\) 3.07180 0.112542
\(746\) 0 0
\(747\) −70.7846 −2.58987
\(748\) 0 0
\(749\) 43.6077 1.59339
\(750\) 0 0
\(751\) −12.0000 −0.437886 −0.218943 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) 0 0
\(753\) −74.6410 −2.72007
\(754\) 0 0
\(755\) −15.4641 −0.562796
\(756\) 0 0
\(757\) 9.32051 0.338760 0.169380 0.985551i \(-0.445824\pi\)
0.169380 + 0.985551i \(0.445824\pi\)
\(758\) 0 0
\(759\) −77.5692 −2.81558
\(760\) 0 0
\(761\) 2.53590 0.0919262 0.0459631 0.998943i \(-0.485364\pi\)
0.0459631 + 0.998943i \(0.485364\pi\)
\(762\) 0 0
\(763\) −53.0718 −1.92133
\(764\) 0 0
\(765\) −12.1962 −0.440953
\(766\) 0 0
\(767\) 53.8564 1.94464
\(768\) 0 0
\(769\) −46.1051 −1.66259 −0.831297 0.555829i \(-0.812401\pi\)
−0.831297 + 0.555829i \(0.812401\pi\)
\(770\) 0 0
\(771\) −20.3923 −0.734411
\(772\) 0 0
\(773\) 38.7846 1.39499 0.697493 0.716592i \(-0.254299\pi\)
0.697493 + 0.716592i \(0.254299\pi\)
\(774\) 0 0
\(775\) −5.46410 −0.196276
\(776\) 0 0
\(777\) −1.85641 −0.0665982
\(778\) 0 0
\(779\) −85.8564 −3.07612
\(780\) 0 0
\(781\) 8.78461 0.314338
\(782\) 0 0
\(783\) −21.8564 −0.781084
\(784\) 0 0
\(785\) −1.26795 −0.0452550
\(786\) 0 0
\(787\) 41.2679 1.47104 0.735522 0.677501i \(-0.236937\pi\)
0.735522 + 0.677501i \(0.236937\pi\)
\(788\) 0 0
\(789\) 35.3205 1.25744
\(790\) 0 0
\(791\) −53.5692 −1.90470
\(792\) 0 0
\(793\) −60.1051 −2.13440
\(794\) 0 0
\(795\) 33.3205 1.18176
\(796\) 0 0
\(797\) −13.6077 −0.482009 −0.241005 0.970524i \(-0.577477\pi\)
−0.241005 + 0.970524i \(0.577477\pi\)
\(798\) 0 0
\(799\) −24.3923 −0.862938
\(800\) 0 0
\(801\) −70.7846 −2.50105
\(802\) 0 0
\(803\) −3.21539 −0.113469
\(804\) 0 0
\(805\) 28.3923 1.00070
\(806\) 0 0
\(807\) −9.46410 −0.333152
\(808\) 0 0
\(809\) −14.7846 −0.519799 −0.259900 0.965636i \(-0.583689\pi\)
−0.259900 + 0.965636i \(0.583689\pi\)
\(810\) 0 0
\(811\) 5.07180 0.178095 0.0890474 0.996027i \(-0.471618\pi\)
0.0890474 + 0.996027i \(0.471618\pi\)
\(812\) 0 0
\(813\) −1.07180 −0.0375896
\(814\) 0 0
\(815\) 12.1962 0.427213
\(816\) 0 0
\(817\) −13.8564 −0.484774
\(818\) 0 0
\(819\) 104.105 3.63773
\(820\) 0 0
\(821\) −35.5692 −1.24137 −0.620687 0.784058i \(-0.713146\pi\)
−0.620687 + 0.784058i \(0.713146\pi\)
\(822\) 0 0
\(823\) −42.0526 −1.46586 −0.732930 0.680304i \(-0.761848\pi\)
−0.732930 + 0.680304i \(0.761848\pi\)
\(824\) 0 0
\(825\) −9.46410 −0.329498
\(826\) 0 0
\(827\) 3.75129 0.130445 0.0652225 0.997871i \(-0.479224\pi\)
0.0652225 + 0.997871i \(0.479224\pi\)
\(828\) 0 0
\(829\) −26.1051 −0.906668 −0.453334 0.891341i \(-0.649766\pi\)
−0.453334 + 0.891341i \(0.649766\pi\)
\(830\) 0 0
\(831\) 32.2487 1.11870
\(832\) 0 0
\(833\) 13.6603 0.473300
\(834\) 0 0
\(835\) −12.1962 −0.422065
\(836\) 0 0
\(837\) 21.8564 0.755468
\(838\) 0 0
\(839\) 49.8564 1.72123 0.860617 0.509253i \(-0.170078\pi\)
0.860617 + 0.509253i \(0.170078\pi\)
\(840\) 0 0
\(841\) 0.856406 0.0295313
\(842\) 0 0
\(843\) −76.1051 −2.62120
\(844\) 0 0
\(845\) −32.3205 −1.11186
\(846\) 0 0
\(847\) −3.46410 −0.119028
\(848\) 0 0
\(849\) 15.4641 0.530727
\(850\) 0 0
\(851\) −1.60770 −0.0551111
\(852\) 0 0
\(853\) 7.07180 0.242134 0.121067 0.992644i \(-0.461368\pi\)
0.121067 + 0.992644i \(0.461368\pi\)
\(854\) 0 0
\(855\) −30.9282 −1.05772
\(856\) 0 0
\(857\) 56.2487 1.92142 0.960710 0.277555i \(-0.0895241\pi\)
0.960710 + 0.277555i \(0.0895241\pi\)
\(858\) 0 0
\(859\) −4.24871 −0.144964 −0.0724821 0.997370i \(-0.523092\pi\)
−0.0724821 + 0.997370i \(0.523092\pi\)
\(860\) 0 0
\(861\) −117.282 −3.99696
\(862\) 0 0
\(863\) 40.1962 1.36829 0.684146 0.729345i \(-0.260175\pi\)
0.684146 + 0.729345i \(0.260175\pi\)
\(864\) 0 0
\(865\) 7.46410 0.253787
\(866\) 0 0
\(867\) 26.0526 0.884791
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 55.1769 1.86960
\(872\) 0 0
\(873\) 47.9090 1.62147
\(874\) 0 0
\(875\) 3.46410 0.117108
\(876\) 0 0
\(877\) −12.4833 −0.421532 −0.210766 0.977536i \(-0.567596\pi\)
−0.210766 + 0.977536i \(0.567596\pi\)
\(878\) 0 0
\(879\) −11.4641 −0.386675
\(880\) 0 0
\(881\) 29.7128 1.00105 0.500525 0.865722i \(-0.333140\pi\)
0.500525 + 0.865722i \(0.333140\pi\)
\(882\) 0 0
\(883\) 37.7128 1.26914 0.634569 0.772867i \(-0.281178\pi\)
0.634569 + 0.772867i \(0.281178\pi\)
\(884\) 0 0
\(885\) −21.8564 −0.734695
\(886\) 0 0
\(887\) −26.4449 −0.887932 −0.443966 0.896044i \(-0.646429\pi\)
−0.443966 + 0.896044i \(0.646429\pi\)
\(888\) 0 0
\(889\) −37.1769 −1.24687
\(890\) 0 0
\(891\) −8.53590 −0.285963
\(892\) 0 0
\(893\) −61.8564 −2.06995
\(894\) 0 0
\(895\) 25.3205 0.846371
\(896\) 0 0
\(897\) 150.746 5.03327
\(898\) 0 0
\(899\) −29.8564 −0.995767
\(900\) 0 0
\(901\) 33.3205 1.11007
\(902\) 0 0
\(903\) −18.9282 −0.629891
\(904\) 0 0
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) −21.2679 −0.706191 −0.353095 0.935587i \(-0.614871\pi\)
−0.353095 + 0.935587i \(0.614871\pi\)
\(908\) 0 0
\(909\) 13.7128 0.454825
\(910\) 0 0
\(911\) −31.7128 −1.05069 −0.525346 0.850889i \(-0.676064\pi\)
−0.525346 + 0.850889i \(0.676064\pi\)
\(912\) 0 0
\(913\) −54.9282 −1.81786
\(914\) 0 0
\(915\) 24.3923 0.806385
\(916\) 0 0
\(917\) −22.6410 −0.747672
\(918\) 0 0
\(919\) −40.7846 −1.34536 −0.672680 0.739933i \(-0.734857\pi\)
−0.672680 + 0.739933i \(0.734857\pi\)
\(920\) 0 0
\(921\) 13.4641 0.443657
\(922\) 0 0
\(923\) −17.0718 −0.561925
\(924\) 0 0
\(925\) −0.196152 −0.00644945
\(926\) 0 0
\(927\) −46.3923 −1.52372
\(928\) 0 0
\(929\) −0.535898 −0.0175823 −0.00879113 0.999961i \(-0.502798\pi\)
−0.00879113 + 0.999961i \(0.502798\pi\)
\(930\) 0 0
\(931\) 34.6410 1.13531
\(932\) 0 0
\(933\) −18.9282 −0.619682
\(934\) 0 0
\(935\) −9.46410 −0.309509
\(936\) 0 0
\(937\) 41.3731 1.35160 0.675800 0.737085i \(-0.263798\pi\)
0.675800 + 0.737085i \(0.263798\pi\)
\(938\) 0 0
\(939\) 15.3205 0.499966
\(940\) 0 0
\(941\) −17.7128 −0.577421 −0.288711 0.957416i \(-0.593227\pi\)
−0.288711 + 0.957416i \(0.593227\pi\)
\(942\) 0 0
\(943\) −101.569 −3.30755
\(944\) 0 0
\(945\) −13.8564 −0.450749
\(946\) 0 0
\(947\) −48.6410 −1.58062 −0.790310 0.612707i \(-0.790081\pi\)
−0.790310 + 0.612707i \(0.790081\pi\)
\(948\) 0 0
\(949\) 6.24871 0.202842
\(950\) 0 0
\(951\) −1.60770 −0.0521331
\(952\) 0 0
\(953\) 10.3923 0.336640 0.168320 0.985732i \(-0.446166\pi\)
0.168320 + 0.985732i \(0.446166\pi\)
\(954\) 0 0
\(955\) −16.3923 −0.530443
\(956\) 0 0
\(957\) −51.7128 −1.67164
\(958\) 0 0
\(959\) 33.4641 1.08061
\(960\) 0 0
\(961\) −1.14359 −0.0368901
\(962\) 0 0
\(963\) −56.1962 −1.81090
\(964\) 0 0
\(965\) −4.19615 −0.135079
\(966\) 0 0
\(967\) 22.4449 0.721778 0.360889 0.932609i \(-0.382473\pi\)
0.360889 + 0.932609i \(0.382473\pi\)
\(968\) 0 0
\(969\) −51.7128 −1.66125
\(970\) 0 0
\(971\) −16.0000 −0.513464 −0.256732 0.966483i \(-0.582646\pi\)
−0.256732 + 0.966483i \(0.582646\pi\)
\(972\) 0 0
\(973\) −37.8564 −1.21362
\(974\) 0 0
\(975\) 18.3923 0.589025
\(976\) 0 0
\(977\) −11.4641 −0.366769 −0.183385 0.983041i \(-0.558705\pi\)
−0.183385 + 0.983041i \(0.558705\pi\)
\(978\) 0 0
\(979\) −54.9282 −1.75551
\(980\) 0 0
\(981\) 68.3923 2.18360
\(982\) 0 0
\(983\) 27.1769 0.866809 0.433404 0.901200i \(-0.357312\pi\)
0.433404 + 0.901200i \(0.357312\pi\)
\(984\) 0 0
\(985\) −3.46410 −0.110375
\(986\) 0 0
\(987\) −84.4974 −2.68958
\(988\) 0 0
\(989\) −16.3923 −0.521245
\(990\) 0 0
\(991\) −0.392305 −0.0124620 −0.00623099 0.999981i \(-0.501983\pi\)
−0.00623099 + 0.999981i \(0.501983\pi\)
\(992\) 0 0
\(993\) 72.1051 2.28819
\(994\) 0 0
\(995\) −16.7846 −0.532108
\(996\) 0 0
\(997\) 19.0718 0.604010 0.302005 0.953306i \(-0.402344\pi\)
0.302005 + 0.953306i \(0.402344\pi\)
\(998\) 0 0
\(999\) 0.784610 0.0248240
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 8020.2.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8020.2.a.b.1.1 2 1.1 even 1 trivial