Properties

Label 4840.2.a.bg.1.2
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
Analytic rank $0$
Dimension $8$
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] = [4840,2,Mod(1,4840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.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: \(38.6475945783\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 21x^{6} + 15x^{5} + 140x^{4} - 60x^{3} - 295x^{2} + 50x + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.77190\) of defining polynomial
Character \(\chi\) \(=\) 4840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.77190 q^{3} +1.00000 q^{5} -4.55506 q^{7} +4.68342 q^{9} +O(q^{10})\) \(q-2.77190 q^{3} +1.00000 q^{5} -4.55506 q^{7} +4.68342 q^{9} +2.90026 q^{13} -2.77190 q^{15} -7.08802 q^{17} -7.14347 q^{19} +12.6262 q^{21} -1.86699 q^{23} +1.00000 q^{25} -4.66626 q^{27} -1.01070 q^{29} +4.41034 q^{31} -4.55506 q^{35} -5.39094 q^{37} -8.03922 q^{39} -4.48790 q^{41} -6.73002 q^{43} +4.68342 q^{45} +7.85017 q^{47} +13.7486 q^{49} +19.6473 q^{51} +0.714458 q^{53} +19.8010 q^{57} -3.80330 q^{59} +2.11595 q^{61} -21.3332 q^{63} +2.90026 q^{65} -12.8384 q^{67} +5.17511 q^{69} +4.21095 q^{71} +1.36576 q^{73} -2.77190 q^{75} -12.2375 q^{79} -1.11585 q^{81} -16.3351 q^{83} -7.08802 q^{85} +2.80156 q^{87} -11.8165 q^{89} -13.2108 q^{91} -12.2250 q^{93} -7.14347 q^{95} -10.8544 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{3} + 8 q^{5} - 6 q^{7} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{3} + 8 q^{5} - 6 q^{7} + 19 q^{9} + 12 q^{13} + q^{15} + 2 q^{17} - 6 q^{19} + 6 q^{21} + 10 q^{23} + 8 q^{25} + 13 q^{27} + 8 q^{29} + 19 q^{31} - 6 q^{35} + 12 q^{37} - 21 q^{39} - 3 q^{41} - 8 q^{43} + 19 q^{45} + 10 q^{47} + 22 q^{49} - 7 q^{51} + 28 q^{53} + 25 q^{57} + 25 q^{59} + 10 q^{61} - 64 q^{63} + 12 q^{65} - 2 q^{67} + 18 q^{69} + 25 q^{71} + 38 q^{73} + q^{75} - 38 q^{79} + 32 q^{81} - 28 q^{83} + 2 q^{85} + 15 q^{87} + 12 q^{89} + 8 q^{91} - 15 q^{93} - 6 q^{95} + 21 q^{97}+O(q^{100}) \) 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
\(3\) −2.77190 −1.60036 −0.800178 0.599763i \(-0.795262\pi\)
−0.800178 + 0.599763i \(0.795262\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −4.55506 −1.72165 −0.860825 0.508901i \(-0.830052\pi\)
−0.860825 + 0.508901i \(0.830052\pi\)
\(8\) 0 0
\(9\) 4.68342 1.56114
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 2.90026 0.804387 0.402193 0.915555i \(-0.368248\pi\)
0.402193 + 0.915555i \(0.368248\pi\)
\(14\) 0 0
\(15\) −2.77190 −0.715701
\(16\) 0 0
\(17\) −7.08802 −1.71910 −0.859548 0.511054i \(-0.829255\pi\)
−0.859548 + 0.511054i \(0.829255\pi\)
\(18\) 0 0
\(19\) −7.14347 −1.63882 −0.819412 0.573205i \(-0.805700\pi\)
−0.819412 + 0.573205i \(0.805700\pi\)
\(20\) 0 0
\(21\) 12.6262 2.75525
\(22\) 0 0
\(23\) −1.86699 −0.389295 −0.194647 0.980873i \(-0.562356\pi\)
−0.194647 + 0.980873i \(0.562356\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −4.66626 −0.898023
\(28\) 0 0
\(29\) −1.01070 −0.187682 −0.0938412 0.995587i \(-0.529915\pi\)
−0.0938412 + 0.995587i \(0.529915\pi\)
\(30\) 0 0
\(31\) 4.41034 0.792121 0.396061 0.918224i \(-0.370377\pi\)
0.396061 + 0.918224i \(0.370377\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.55506 −0.769945
\(36\) 0 0
\(37\) −5.39094 −0.886265 −0.443132 0.896456i \(-0.646133\pi\)
−0.443132 + 0.896456i \(0.646133\pi\)
\(38\) 0 0
\(39\) −8.03922 −1.28730
\(40\) 0 0
\(41\) −4.48790 −0.700892 −0.350446 0.936583i \(-0.613970\pi\)
−0.350446 + 0.936583i \(0.613970\pi\)
\(42\) 0 0
\(43\) −6.73002 −1.02632 −0.513159 0.858293i \(-0.671525\pi\)
−0.513159 + 0.858293i \(0.671525\pi\)
\(44\) 0 0
\(45\) 4.68342 0.698163
\(46\) 0 0
\(47\) 7.85017 1.14507 0.572533 0.819882i \(-0.305961\pi\)
0.572533 + 0.819882i \(0.305961\pi\)
\(48\) 0 0
\(49\) 13.7486 1.96408
\(50\) 0 0
\(51\) 19.6473 2.75117
\(52\) 0 0
\(53\) 0.714458 0.0981383 0.0490692 0.998795i \(-0.484375\pi\)
0.0490692 + 0.998795i \(0.484375\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 19.8010 2.62270
\(58\) 0 0
\(59\) −3.80330 −0.495147 −0.247574 0.968869i \(-0.579633\pi\)
−0.247574 + 0.968869i \(0.579633\pi\)
\(60\) 0 0
\(61\) 2.11595 0.270920 0.135460 0.990783i \(-0.456749\pi\)
0.135460 + 0.990783i \(0.456749\pi\)
\(62\) 0 0
\(63\) −21.3332 −2.68774
\(64\) 0 0
\(65\) 2.90026 0.359733
\(66\) 0 0
\(67\) −12.8384 −1.56847 −0.784233 0.620467i \(-0.786943\pi\)
−0.784233 + 0.620467i \(0.786943\pi\)
\(68\) 0 0
\(69\) 5.17511 0.623010
\(70\) 0 0
\(71\) 4.21095 0.499747 0.249874 0.968278i \(-0.419611\pi\)
0.249874 + 0.968278i \(0.419611\pi\)
\(72\) 0 0
\(73\) 1.36576 0.159850 0.0799249 0.996801i \(-0.474532\pi\)
0.0799249 + 0.996801i \(0.474532\pi\)
\(74\) 0 0
\(75\) −2.77190 −0.320071
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −12.2375 −1.37682 −0.688411 0.725320i \(-0.741692\pi\)
−0.688411 + 0.725320i \(0.741692\pi\)
\(80\) 0 0
\(81\) −1.11585 −0.123983
\(82\) 0 0
\(83\) −16.3351 −1.79301 −0.896503 0.443037i \(-0.853901\pi\)
−0.896503 + 0.443037i \(0.853901\pi\)
\(84\) 0 0
\(85\) −7.08802 −0.768803
\(86\) 0 0
\(87\) 2.80156 0.300359
\(88\) 0 0
\(89\) −11.8165 −1.25255 −0.626273 0.779604i \(-0.715420\pi\)
−0.626273 + 0.779604i \(0.715420\pi\)
\(90\) 0 0
\(91\) −13.2108 −1.38487
\(92\) 0 0
\(93\) −12.2250 −1.26768
\(94\) 0 0
\(95\) −7.14347 −0.732904
\(96\) 0 0
\(97\) −10.8544 −1.10210 −0.551048 0.834473i \(-0.685772\pi\)
−0.551048 + 0.834473i \(0.685772\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 12.6665 1.26037 0.630184 0.776446i \(-0.282979\pi\)
0.630184 + 0.776446i \(0.282979\pi\)
\(102\) 0 0
\(103\) −6.18162 −0.609093 −0.304547 0.952497i \(-0.598505\pi\)
−0.304547 + 0.952497i \(0.598505\pi\)
\(104\) 0 0
\(105\) 12.6262 1.23219
\(106\) 0 0
\(107\) 1.00774 0.0974223 0.0487111 0.998813i \(-0.484489\pi\)
0.0487111 + 0.998813i \(0.484489\pi\)
\(108\) 0 0
\(109\) 6.60399 0.632548 0.316274 0.948668i \(-0.397568\pi\)
0.316274 + 0.948668i \(0.397568\pi\)
\(110\) 0 0
\(111\) 14.9431 1.41834
\(112\) 0 0
\(113\) −10.2611 −0.965279 −0.482640 0.875819i \(-0.660322\pi\)
−0.482640 + 0.875819i \(0.660322\pi\)
\(114\) 0 0
\(115\) −1.86699 −0.174098
\(116\) 0 0
\(117\) 13.5831 1.25576
\(118\) 0 0
\(119\) 32.2863 2.95968
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 12.4400 1.12168
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 17.7211 1.57249 0.786245 0.617914i \(-0.212022\pi\)
0.786245 + 0.617914i \(0.212022\pi\)
\(128\) 0 0
\(129\) 18.6549 1.64247
\(130\) 0 0
\(131\) 14.3999 1.25812 0.629061 0.777356i \(-0.283440\pi\)
0.629061 + 0.777356i \(0.283440\pi\)
\(132\) 0 0
\(133\) 32.5389 2.82148
\(134\) 0 0
\(135\) −4.66626 −0.401608
\(136\) 0 0
\(137\) 4.46970 0.381872 0.190936 0.981603i \(-0.438848\pi\)
0.190936 + 0.981603i \(0.438848\pi\)
\(138\) 0 0
\(139\) 4.84523 0.410967 0.205483 0.978661i \(-0.434123\pi\)
0.205483 + 0.978661i \(0.434123\pi\)
\(140\) 0 0
\(141\) −21.7599 −1.83251
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −1.01070 −0.0839341
\(146\) 0 0
\(147\) −38.1096 −3.14323
\(148\) 0 0
\(149\) −10.7263 −0.878734 −0.439367 0.898308i \(-0.644797\pi\)
−0.439367 + 0.898308i \(0.644797\pi\)
\(150\) 0 0
\(151\) −0.201814 −0.0164234 −0.00821172 0.999966i \(-0.502614\pi\)
−0.00821172 + 0.999966i \(0.502614\pi\)
\(152\) 0 0
\(153\) −33.1961 −2.68375
\(154\) 0 0
\(155\) 4.41034 0.354247
\(156\) 0 0
\(157\) −2.92198 −0.233199 −0.116600 0.993179i \(-0.537199\pi\)
−0.116600 + 0.993179i \(0.537199\pi\)
\(158\) 0 0
\(159\) −1.98040 −0.157056
\(160\) 0 0
\(161\) 8.50425 0.670229
\(162\) 0 0
\(163\) 8.48158 0.664329 0.332164 0.943222i \(-0.392221\pi\)
0.332164 + 0.943222i \(0.392221\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −10.0642 −0.778790 −0.389395 0.921071i \(-0.627316\pi\)
−0.389395 + 0.921071i \(0.627316\pi\)
\(168\) 0 0
\(169\) −4.58851 −0.352962
\(170\) 0 0
\(171\) −33.4558 −2.55843
\(172\) 0 0
\(173\) 5.77262 0.438884 0.219442 0.975626i \(-0.429576\pi\)
0.219442 + 0.975626i \(0.429576\pi\)
\(174\) 0 0
\(175\) −4.55506 −0.344330
\(176\) 0 0
\(177\) 10.5424 0.792412
\(178\) 0 0
\(179\) 16.8671 1.26071 0.630353 0.776309i \(-0.282910\pi\)
0.630353 + 0.776309i \(0.282910\pi\)
\(180\) 0 0
\(181\) 15.1054 1.12277 0.561387 0.827553i \(-0.310268\pi\)
0.561387 + 0.827553i \(0.310268\pi\)
\(182\) 0 0
\(183\) −5.86520 −0.433568
\(184\) 0 0
\(185\) −5.39094 −0.396350
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 21.2551 1.54608
\(190\) 0 0
\(191\) 10.4338 0.754966 0.377483 0.926016i \(-0.376790\pi\)
0.377483 + 0.926016i \(0.376790\pi\)
\(192\) 0 0
\(193\) 13.9250 1.00234 0.501172 0.865348i \(-0.332902\pi\)
0.501172 + 0.865348i \(0.332902\pi\)
\(194\) 0 0
\(195\) −8.03922 −0.575700
\(196\) 0 0
\(197\) −2.25234 −0.160473 −0.0802363 0.996776i \(-0.525567\pi\)
−0.0802363 + 0.996776i \(0.525567\pi\)
\(198\) 0 0
\(199\) 22.1246 1.56837 0.784184 0.620528i \(-0.213082\pi\)
0.784184 + 0.620528i \(0.213082\pi\)
\(200\) 0 0
\(201\) 35.5869 2.51010
\(202\) 0 0
\(203\) 4.60380 0.323123
\(204\) 0 0
\(205\) −4.48790 −0.313448
\(206\) 0 0
\(207\) −8.74390 −0.607743
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 7.77119 0.534991 0.267495 0.963559i \(-0.413804\pi\)
0.267495 + 0.963559i \(0.413804\pi\)
\(212\) 0 0
\(213\) −11.6723 −0.799774
\(214\) 0 0
\(215\) −6.73002 −0.458984
\(216\) 0 0
\(217\) −20.0894 −1.36376
\(218\) 0 0
\(219\) −3.78574 −0.255817
\(220\) 0 0
\(221\) −20.5571 −1.38282
\(222\) 0 0
\(223\) −7.92084 −0.530419 −0.265209 0.964191i \(-0.585441\pi\)
−0.265209 + 0.964191i \(0.585441\pi\)
\(224\) 0 0
\(225\) 4.68342 0.312228
\(226\) 0 0
\(227\) −20.2672 −1.34518 −0.672590 0.740015i \(-0.734818\pi\)
−0.672590 + 0.740015i \(0.734818\pi\)
\(228\) 0 0
\(229\) 9.88050 0.652922 0.326461 0.945211i \(-0.394144\pi\)
0.326461 + 0.945211i \(0.394144\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.79865 0.641931 0.320966 0.947091i \(-0.395993\pi\)
0.320966 + 0.947091i \(0.395993\pi\)
\(234\) 0 0
\(235\) 7.85017 0.512089
\(236\) 0 0
\(237\) 33.9210 2.20341
\(238\) 0 0
\(239\) −11.0149 −0.712493 −0.356247 0.934392i \(-0.615944\pi\)
−0.356247 + 0.934392i \(0.615944\pi\)
\(240\) 0 0
\(241\) −13.3108 −0.857421 −0.428711 0.903442i \(-0.641032\pi\)
−0.428711 + 0.903442i \(0.641032\pi\)
\(242\) 0 0
\(243\) 17.0918 1.09644
\(244\) 0 0
\(245\) 13.7486 0.878363
\(246\) 0 0
\(247\) −20.7179 −1.31825
\(248\) 0 0
\(249\) 45.2791 2.86945
\(250\) 0 0
\(251\) −14.5916 −0.921011 −0.460506 0.887657i \(-0.652332\pi\)
−0.460506 + 0.887657i \(0.652332\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 19.6473 1.23036
\(256\) 0 0
\(257\) 0.965528 0.0602280 0.0301140 0.999546i \(-0.490413\pi\)
0.0301140 + 0.999546i \(0.490413\pi\)
\(258\) 0 0
\(259\) 24.5560 1.52584
\(260\) 0 0
\(261\) −4.73353 −0.292998
\(262\) 0 0
\(263\) 20.7753 1.28106 0.640529 0.767934i \(-0.278715\pi\)
0.640529 + 0.767934i \(0.278715\pi\)
\(264\) 0 0
\(265\) 0.714458 0.0438888
\(266\) 0 0
\(267\) 32.7541 2.00452
\(268\) 0 0
\(269\) −3.91387 −0.238633 −0.119317 0.992856i \(-0.538070\pi\)
−0.119317 + 0.992856i \(0.538070\pi\)
\(270\) 0 0
\(271\) −14.2782 −0.867338 −0.433669 0.901072i \(-0.642781\pi\)
−0.433669 + 0.901072i \(0.642781\pi\)
\(272\) 0 0
\(273\) 36.6191 2.21629
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 3.76298 0.226095 0.113048 0.993590i \(-0.463939\pi\)
0.113048 + 0.993590i \(0.463939\pi\)
\(278\) 0 0
\(279\) 20.6555 1.23661
\(280\) 0 0
\(281\) −10.3189 −0.615576 −0.307788 0.951455i \(-0.599589\pi\)
−0.307788 + 0.951455i \(0.599589\pi\)
\(282\) 0 0
\(283\) 30.4565 1.81045 0.905224 0.424934i \(-0.139703\pi\)
0.905224 + 0.424934i \(0.139703\pi\)
\(284\) 0 0
\(285\) 19.8010 1.17291
\(286\) 0 0
\(287\) 20.4426 1.20669
\(288\) 0 0
\(289\) 33.2400 1.95529
\(290\) 0 0
\(291\) 30.0873 1.76375
\(292\) 0 0
\(293\) −7.90548 −0.461843 −0.230922 0.972972i \(-0.574174\pi\)
−0.230922 + 0.972972i \(0.574174\pi\)
\(294\) 0 0
\(295\) −3.80330 −0.221437
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −5.41475 −0.313143
\(300\) 0 0
\(301\) 30.6556 1.76696
\(302\) 0 0
\(303\) −35.1104 −2.01704
\(304\) 0 0
\(305\) 2.11595 0.121159
\(306\) 0 0
\(307\) 14.2585 0.813777 0.406889 0.913478i \(-0.366614\pi\)
0.406889 + 0.913478i \(0.366614\pi\)
\(308\) 0 0
\(309\) 17.1348 0.974766
\(310\) 0 0
\(311\) 17.2118 0.975991 0.487995 0.872846i \(-0.337728\pi\)
0.487995 + 0.872846i \(0.337728\pi\)
\(312\) 0 0
\(313\) −7.96400 −0.450152 −0.225076 0.974341i \(-0.572263\pi\)
−0.225076 + 0.974341i \(0.572263\pi\)
\(314\) 0 0
\(315\) −21.3332 −1.20199
\(316\) 0 0
\(317\) −13.7307 −0.771191 −0.385595 0.922668i \(-0.626004\pi\)
−0.385595 + 0.922668i \(0.626004\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −2.79336 −0.155910
\(322\) 0 0
\(323\) 50.6330 2.81730
\(324\) 0 0
\(325\) 2.90026 0.160877
\(326\) 0 0
\(327\) −18.3056 −1.01230
\(328\) 0 0
\(329\) −35.7580 −1.97140
\(330\) 0 0
\(331\) 10.6971 0.587967 0.293983 0.955811i \(-0.405019\pi\)
0.293983 + 0.955811i \(0.405019\pi\)
\(332\) 0 0
\(333\) −25.2480 −1.38358
\(334\) 0 0
\(335\) −12.8384 −0.701439
\(336\) 0 0
\(337\) 11.4797 0.625338 0.312669 0.949862i \(-0.398777\pi\)
0.312669 + 0.949862i \(0.398777\pi\)
\(338\) 0 0
\(339\) 28.4426 1.54479
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −30.7401 −1.65981
\(344\) 0 0
\(345\) 5.17511 0.278618
\(346\) 0 0
\(347\) −0.423181 −0.0227176 −0.0113588 0.999935i \(-0.503616\pi\)
−0.0113588 + 0.999935i \(0.503616\pi\)
\(348\) 0 0
\(349\) 0.847568 0.0453693 0.0226846 0.999743i \(-0.492779\pi\)
0.0226846 + 0.999743i \(0.492779\pi\)
\(350\) 0 0
\(351\) −13.5334 −0.722357
\(352\) 0 0
\(353\) 9.79855 0.521524 0.260762 0.965403i \(-0.416026\pi\)
0.260762 + 0.965403i \(0.416026\pi\)
\(354\) 0 0
\(355\) 4.21095 0.223494
\(356\) 0 0
\(357\) −89.4944 −4.73655
\(358\) 0 0
\(359\) 7.60668 0.401465 0.200733 0.979646i \(-0.435668\pi\)
0.200733 + 0.979646i \(0.435668\pi\)
\(360\) 0 0
\(361\) 32.0291 1.68574
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.36576 0.0714870
\(366\) 0 0
\(367\) −1.86780 −0.0974981 −0.0487491 0.998811i \(-0.515523\pi\)
−0.0487491 + 0.998811i \(0.515523\pi\)
\(368\) 0 0
\(369\) −21.0187 −1.09419
\(370\) 0 0
\(371\) −3.25440 −0.168960
\(372\) 0 0
\(373\) 20.0214 1.03667 0.518335 0.855178i \(-0.326552\pi\)
0.518335 + 0.855178i \(0.326552\pi\)
\(374\) 0 0
\(375\) −2.77190 −0.143140
\(376\) 0 0
\(377\) −2.93129 −0.150969
\(378\) 0 0
\(379\) 24.0052 1.23307 0.616533 0.787329i \(-0.288537\pi\)
0.616533 + 0.787329i \(0.288537\pi\)
\(380\) 0 0
\(381\) −49.1210 −2.51655
\(382\) 0 0
\(383\) 19.6767 1.00543 0.502716 0.864452i \(-0.332334\pi\)
0.502716 + 0.864452i \(0.332334\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −31.5195 −1.60223
\(388\) 0 0
\(389\) 25.0051 1.26781 0.633904 0.773412i \(-0.281451\pi\)
0.633904 + 0.773412i \(0.281451\pi\)
\(390\) 0 0
\(391\) 13.2333 0.669235
\(392\) 0 0
\(393\) −39.9149 −2.01344
\(394\) 0 0
\(395\) −12.2375 −0.615734
\(396\) 0 0
\(397\) −19.3120 −0.969240 −0.484620 0.874725i \(-0.661042\pi\)
−0.484620 + 0.874725i \(0.661042\pi\)
\(398\) 0 0
\(399\) −90.1946 −4.51538
\(400\) 0 0
\(401\) 23.2469 1.16090 0.580448 0.814297i \(-0.302877\pi\)
0.580448 + 0.814297i \(0.302877\pi\)
\(402\) 0 0
\(403\) 12.7911 0.637172
\(404\) 0 0
\(405\) −1.11585 −0.0554470
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −5.92254 −0.292851 −0.146425 0.989222i \(-0.546777\pi\)
−0.146425 + 0.989222i \(0.546777\pi\)
\(410\) 0 0
\(411\) −12.3895 −0.611131
\(412\) 0 0
\(413\) 17.3242 0.852470
\(414\) 0 0
\(415\) −16.3351 −0.801857
\(416\) 0 0
\(417\) −13.4305 −0.657693
\(418\) 0 0
\(419\) −1.11739 −0.0545881 −0.0272940 0.999627i \(-0.508689\pi\)
−0.0272940 + 0.999627i \(0.508689\pi\)
\(420\) 0 0
\(421\) 17.2344 0.839956 0.419978 0.907534i \(-0.362038\pi\)
0.419978 + 0.907534i \(0.362038\pi\)
\(422\) 0 0
\(423\) 36.7656 1.78761
\(424\) 0 0
\(425\) −7.08802 −0.343819
\(426\) 0 0
\(427\) −9.63828 −0.466429
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 7.74895 0.373254 0.186627 0.982431i \(-0.440244\pi\)
0.186627 + 0.982431i \(0.440244\pi\)
\(432\) 0 0
\(433\) 3.93864 0.189279 0.0946395 0.995512i \(-0.469830\pi\)
0.0946395 + 0.995512i \(0.469830\pi\)
\(434\) 0 0
\(435\) 2.80156 0.134324
\(436\) 0 0
\(437\) 13.3368 0.637985
\(438\) 0 0
\(439\) −34.7505 −1.65855 −0.829275 0.558841i \(-0.811246\pi\)
−0.829275 + 0.558841i \(0.811246\pi\)
\(440\) 0 0
\(441\) 64.3903 3.06620
\(442\) 0 0
\(443\) 24.3227 1.15560 0.577802 0.816177i \(-0.303910\pi\)
0.577802 + 0.816177i \(0.303910\pi\)
\(444\) 0 0
\(445\) −11.8165 −0.560155
\(446\) 0 0
\(447\) 29.7322 1.40629
\(448\) 0 0
\(449\) −18.9235 −0.893057 −0.446528 0.894770i \(-0.647340\pi\)
−0.446528 + 0.894770i \(0.647340\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0.559409 0.0262833
\(454\) 0 0
\(455\) −13.2108 −0.619334
\(456\) 0 0
\(457\) −16.2210 −0.758787 −0.379393 0.925235i \(-0.623867\pi\)
−0.379393 + 0.925235i \(0.623867\pi\)
\(458\) 0 0
\(459\) 33.0745 1.54379
\(460\) 0 0
\(461\) −11.3851 −0.530258 −0.265129 0.964213i \(-0.585415\pi\)
−0.265129 + 0.964213i \(0.585415\pi\)
\(462\) 0 0
\(463\) −36.4523 −1.69408 −0.847040 0.531529i \(-0.821618\pi\)
−0.847040 + 0.531529i \(0.821618\pi\)
\(464\) 0 0
\(465\) −12.2250 −0.566922
\(466\) 0 0
\(467\) 28.0455 1.29779 0.648895 0.760878i \(-0.275232\pi\)
0.648895 + 0.760878i \(0.275232\pi\)
\(468\) 0 0
\(469\) 58.4799 2.70035
\(470\) 0 0
\(471\) 8.09943 0.373202
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −7.14347 −0.327765
\(476\) 0 0
\(477\) 3.34610 0.153208
\(478\) 0 0
\(479\) −36.3433 −1.66057 −0.830284 0.557340i \(-0.811822\pi\)
−0.830284 + 0.557340i \(0.811822\pi\)
\(480\) 0 0
\(481\) −15.6351 −0.712899
\(482\) 0 0
\(483\) −23.5729 −1.07261
\(484\) 0 0
\(485\) −10.8544 −0.492873
\(486\) 0 0
\(487\) 41.8132 1.89474 0.947368 0.320146i \(-0.103732\pi\)
0.947368 + 0.320146i \(0.103732\pi\)
\(488\) 0 0
\(489\) −23.5101 −1.06316
\(490\) 0 0
\(491\) 19.3555 0.873501 0.436751 0.899583i \(-0.356129\pi\)
0.436751 + 0.899583i \(0.356129\pi\)
\(492\) 0 0
\(493\) 7.16386 0.322644
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −19.1811 −0.860390
\(498\) 0 0
\(499\) 13.3514 0.597691 0.298846 0.954301i \(-0.403398\pi\)
0.298846 + 0.954301i \(0.403398\pi\)
\(500\) 0 0
\(501\) 27.8969 1.24634
\(502\) 0 0
\(503\) 35.8888 1.60020 0.800100 0.599866i \(-0.204780\pi\)
0.800100 + 0.599866i \(0.204780\pi\)
\(504\) 0 0
\(505\) 12.6665 0.563654
\(506\) 0 0
\(507\) 12.7189 0.564865
\(508\) 0 0
\(509\) 17.9153 0.794080 0.397040 0.917801i \(-0.370037\pi\)
0.397040 + 0.917801i \(0.370037\pi\)
\(510\) 0 0
\(511\) −6.22111 −0.275206
\(512\) 0 0
\(513\) 33.3333 1.47170
\(514\) 0 0
\(515\) −6.18162 −0.272395
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −16.0011 −0.702370
\(520\) 0 0
\(521\) 7.25941 0.318040 0.159020 0.987275i \(-0.449167\pi\)
0.159020 + 0.987275i \(0.449167\pi\)
\(522\) 0 0
\(523\) 34.0176 1.48748 0.743742 0.668467i \(-0.233049\pi\)
0.743742 + 0.668467i \(0.233049\pi\)
\(524\) 0 0
\(525\) 12.6262 0.551051
\(526\) 0 0
\(527\) −31.2606 −1.36173
\(528\) 0 0
\(529\) −19.5143 −0.848450
\(530\) 0 0
\(531\) −17.8124 −0.772994
\(532\) 0 0
\(533\) −13.0161 −0.563788
\(534\) 0 0
\(535\) 1.00774 0.0435686
\(536\) 0 0
\(537\) −46.7539 −2.01758
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 19.0875 0.820637 0.410319 0.911942i \(-0.365417\pi\)
0.410319 + 0.911942i \(0.365417\pi\)
\(542\) 0 0
\(543\) −41.8706 −1.79684
\(544\) 0 0
\(545\) 6.60399 0.282884
\(546\) 0 0
\(547\) −23.3090 −0.996621 −0.498311 0.866999i \(-0.666046\pi\)
−0.498311 + 0.866999i \(0.666046\pi\)
\(548\) 0 0
\(549\) 9.90988 0.422944
\(550\) 0 0
\(551\) 7.21991 0.307578
\(552\) 0 0
\(553\) 55.7424 2.37041
\(554\) 0 0
\(555\) 14.9431 0.634300
\(556\) 0 0
\(557\) −28.3838 −1.20266 −0.601329 0.799001i \(-0.705362\pi\)
−0.601329 + 0.799001i \(0.705362\pi\)
\(558\) 0 0
\(559\) −19.5188 −0.825557
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −8.13215 −0.342729 −0.171365 0.985208i \(-0.554818\pi\)
−0.171365 + 0.985208i \(0.554818\pi\)
\(564\) 0 0
\(565\) −10.2611 −0.431686
\(566\) 0 0
\(567\) 5.08276 0.213456
\(568\) 0 0
\(569\) 16.3414 0.685067 0.342534 0.939506i \(-0.388715\pi\)
0.342534 + 0.939506i \(0.388715\pi\)
\(570\) 0 0
\(571\) 33.1999 1.38937 0.694687 0.719312i \(-0.255543\pi\)
0.694687 + 0.719312i \(0.255543\pi\)
\(572\) 0 0
\(573\) −28.9215 −1.20822
\(574\) 0 0
\(575\) −1.86699 −0.0778589
\(576\) 0 0
\(577\) −45.0285 −1.87456 −0.937281 0.348576i \(-0.886665\pi\)
−0.937281 + 0.348576i \(0.886665\pi\)
\(578\) 0 0
\(579\) −38.5987 −1.60411
\(580\) 0 0
\(581\) 74.4072 3.08693
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 13.5831 0.561593
\(586\) 0 0
\(587\) −44.6981 −1.84489 −0.922444 0.386131i \(-0.873811\pi\)
−0.922444 + 0.386131i \(0.873811\pi\)
\(588\) 0 0
\(589\) −31.5052 −1.29815
\(590\) 0 0
\(591\) 6.24326 0.256813
\(592\) 0 0
\(593\) −18.7395 −0.769538 −0.384769 0.923013i \(-0.625719\pi\)
−0.384769 + 0.923013i \(0.625719\pi\)
\(594\) 0 0
\(595\) 32.2863 1.32361
\(596\) 0 0
\(597\) −61.3270 −2.50995
\(598\) 0 0
\(599\) −19.5241 −0.797735 −0.398867 0.917009i \(-0.630597\pi\)
−0.398867 + 0.917009i \(0.630597\pi\)
\(600\) 0 0
\(601\) −31.3146 −1.27735 −0.638674 0.769477i \(-0.720517\pi\)
−0.638674 + 0.769477i \(0.720517\pi\)
\(602\) 0 0
\(603\) −60.1278 −2.44859
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −22.0334 −0.894309 −0.447154 0.894457i \(-0.647563\pi\)
−0.447154 + 0.894457i \(0.647563\pi\)
\(608\) 0 0
\(609\) −12.7613 −0.517112
\(610\) 0 0
\(611\) 22.7675 0.921075
\(612\) 0 0
\(613\) 37.8937 1.53051 0.765256 0.643726i \(-0.222612\pi\)
0.765256 + 0.643726i \(0.222612\pi\)
\(614\) 0 0
\(615\) 12.4400 0.501629
\(616\) 0 0
\(617\) −34.0287 −1.36995 −0.684973 0.728569i \(-0.740186\pi\)
−0.684973 + 0.728569i \(0.740186\pi\)
\(618\) 0 0
\(619\) −26.4162 −1.06176 −0.530878 0.847448i \(-0.678138\pi\)
−0.530878 + 0.847448i \(0.678138\pi\)
\(620\) 0 0
\(621\) 8.71187 0.349595
\(622\) 0 0
\(623\) 53.8248 2.15645
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 38.2111 1.52357
\(630\) 0 0
\(631\) −38.5953 −1.53645 −0.768226 0.640178i \(-0.778861\pi\)
−0.768226 + 0.640178i \(0.778861\pi\)
\(632\) 0 0
\(633\) −21.5410 −0.856176
\(634\) 0 0
\(635\) 17.7211 0.703239
\(636\) 0 0
\(637\) 39.8744 1.57988
\(638\) 0 0
\(639\) 19.7216 0.780175
\(640\) 0 0
\(641\) −18.2250 −0.719843 −0.359921 0.932983i \(-0.617196\pi\)
−0.359921 + 0.932983i \(0.617196\pi\)
\(642\) 0 0
\(643\) 10.5456 0.415879 0.207940 0.978142i \(-0.433324\pi\)
0.207940 + 0.978142i \(0.433324\pi\)
\(644\) 0 0
\(645\) 18.6549 0.734537
\(646\) 0 0
\(647\) 16.9165 0.665057 0.332529 0.943093i \(-0.392098\pi\)
0.332529 + 0.943093i \(0.392098\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 55.6857 2.18249
\(652\) 0 0
\(653\) 47.9383 1.87597 0.937986 0.346673i \(-0.112689\pi\)
0.937986 + 0.346673i \(0.112689\pi\)
\(654\) 0 0
\(655\) 14.3999 0.562649
\(656\) 0 0
\(657\) 6.39641 0.249548
\(658\) 0 0
\(659\) −5.96793 −0.232477 −0.116239 0.993221i \(-0.537084\pi\)
−0.116239 + 0.993221i \(0.537084\pi\)
\(660\) 0 0
\(661\) −22.5701 −0.877875 −0.438937 0.898518i \(-0.644645\pi\)
−0.438937 + 0.898518i \(0.644645\pi\)
\(662\) 0 0
\(663\) 56.9821 2.21300
\(664\) 0 0
\(665\) 32.5389 1.26181
\(666\) 0 0
\(667\) 1.88697 0.0730637
\(668\) 0 0
\(669\) 21.9558 0.848858
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 6.69924 0.258237 0.129118 0.991629i \(-0.458785\pi\)
0.129118 + 0.991629i \(0.458785\pi\)
\(674\) 0 0
\(675\) −4.66626 −0.179605
\(676\) 0 0
\(677\) 41.5199 1.59574 0.797870 0.602830i \(-0.205960\pi\)
0.797870 + 0.602830i \(0.205960\pi\)
\(678\) 0 0
\(679\) 49.4424 1.89743
\(680\) 0 0
\(681\) 56.1786 2.15277
\(682\) 0 0
\(683\) −13.4576 −0.514940 −0.257470 0.966286i \(-0.582889\pi\)
−0.257470 + 0.966286i \(0.582889\pi\)
\(684\) 0 0
\(685\) 4.46970 0.170778
\(686\) 0 0
\(687\) −27.3877 −1.04491
\(688\) 0 0
\(689\) 2.07211 0.0789411
\(690\) 0 0
\(691\) −29.4815 −1.12153 −0.560765 0.827975i \(-0.689493\pi\)
−0.560765 + 0.827975i \(0.689493\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4.84523 0.183790
\(696\) 0 0
\(697\) 31.8103 1.20490
\(698\) 0 0
\(699\) −27.1609 −1.02732
\(700\) 0 0
\(701\) −9.03034 −0.341071 −0.170536 0.985352i \(-0.554550\pi\)
−0.170536 + 0.985352i \(0.554550\pi\)
\(702\) 0 0
\(703\) 38.5100 1.45243
\(704\) 0 0
\(705\) −21.7599 −0.819524
\(706\) 0 0
\(707\) −57.6969 −2.16991
\(708\) 0 0
\(709\) 49.0834 1.84336 0.921682 0.387945i \(-0.126815\pi\)
0.921682 + 0.387945i \(0.126815\pi\)
\(710\) 0 0
\(711\) −57.3132 −2.14941
\(712\) 0 0
\(713\) −8.23407 −0.308368
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 30.5321 1.14024
\(718\) 0 0
\(719\) −48.7090 −1.81654 −0.908269 0.418387i \(-0.862596\pi\)
−0.908269 + 0.418387i \(0.862596\pi\)
\(720\) 0 0
\(721\) 28.1576 1.04865
\(722\) 0 0
\(723\) 36.8961 1.37218
\(724\) 0 0
\(725\) −1.01070 −0.0375365
\(726\) 0 0
\(727\) −50.6964 −1.88023 −0.940113 0.340864i \(-0.889281\pi\)
−0.940113 + 0.340864i \(0.889281\pi\)
\(728\) 0 0
\(729\) −44.0292 −1.63071
\(730\) 0 0
\(731\) 47.7025 1.76434
\(732\) 0 0
\(733\) −47.1585 −1.74184 −0.870920 0.491425i \(-0.836476\pi\)
−0.870920 + 0.491425i \(0.836476\pi\)
\(734\) 0 0
\(735\) −38.1096 −1.40569
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 18.1569 0.667912 0.333956 0.942589i \(-0.391616\pi\)
0.333956 + 0.942589i \(0.391616\pi\)
\(740\) 0 0
\(741\) 57.4279 2.10967
\(742\) 0 0
\(743\) −3.73625 −0.137070 −0.0685349 0.997649i \(-0.521832\pi\)
−0.0685349 + 0.997649i \(0.521832\pi\)
\(744\) 0 0
\(745\) −10.7263 −0.392982
\(746\) 0 0
\(747\) −76.5040 −2.79913
\(748\) 0 0
\(749\) −4.59033 −0.167727
\(750\) 0 0
\(751\) −27.8430 −1.01600 −0.508002 0.861356i \(-0.669616\pi\)
−0.508002 + 0.861356i \(0.669616\pi\)
\(752\) 0 0
\(753\) 40.4463 1.47395
\(754\) 0 0
\(755\) −0.201814 −0.00734478
\(756\) 0 0
\(757\) 45.0313 1.63669 0.818346 0.574726i \(-0.194891\pi\)
0.818346 + 0.574726i \(0.194891\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −18.7068 −0.678120 −0.339060 0.940765i \(-0.610109\pi\)
−0.339060 + 0.940765i \(0.610109\pi\)
\(762\) 0 0
\(763\) −30.0816 −1.08903
\(764\) 0 0
\(765\) −33.1961 −1.20021
\(766\) 0 0
\(767\) −11.0305 −0.398290
\(768\) 0 0
\(769\) −13.5701 −0.489351 −0.244675 0.969605i \(-0.578681\pi\)
−0.244675 + 0.969605i \(0.578681\pi\)
\(770\) 0 0
\(771\) −2.67635 −0.0963862
\(772\) 0 0
\(773\) −3.27790 −0.117898 −0.0589489 0.998261i \(-0.518775\pi\)
−0.0589489 + 0.998261i \(0.518775\pi\)
\(774\) 0 0
\(775\) 4.41034 0.158424
\(776\) 0 0
\(777\) −68.0668 −2.44188
\(778\) 0 0
\(779\) 32.0592 1.14864
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 4.71619 0.168543
\(784\) 0 0
\(785\) −2.92198 −0.104290
\(786\) 0 0
\(787\) 2.54160 0.0905981 0.0452991 0.998973i \(-0.485576\pi\)
0.0452991 + 0.998973i \(0.485576\pi\)
\(788\) 0 0
\(789\) −57.5870 −2.05015
\(790\) 0 0
\(791\) 46.7397 1.66187
\(792\) 0 0
\(793\) 6.13680 0.217924
\(794\) 0 0
\(795\) −1.98040 −0.0702377
\(796\) 0 0
\(797\) 26.2914 0.931290 0.465645 0.884972i \(-0.345822\pi\)
0.465645 + 0.884972i \(0.345822\pi\)
\(798\) 0 0
\(799\) −55.6422 −1.96848
\(800\) 0 0
\(801\) −55.3416 −1.95540
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 8.50425 0.299736
\(806\) 0 0
\(807\) 10.8489 0.381898
\(808\) 0 0
\(809\) −7.94284 −0.279255 −0.139628 0.990204i \(-0.544591\pi\)
−0.139628 + 0.990204i \(0.544591\pi\)
\(810\) 0 0
\(811\) 19.4167 0.681811 0.340906 0.940098i \(-0.389266\pi\)
0.340906 + 0.940098i \(0.389266\pi\)
\(812\) 0 0
\(813\) 39.5777 1.38805
\(814\) 0 0
\(815\) 8.48158 0.297097
\(816\) 0 0
\(817\) 48.0757 1.68196
\(818\) 0 0
\(819\) −61.8719 −2.16198
\(820\) 0 0
\(821\) −13.0819 −0.456561 −0.228281 0.973595i \(-0.573310\pi\)
−0.228281 + 0.973595i \(0.573310\pi\)
\(822\) 0 0
\(823\) 31.4670 1.09687 0.548435 0.836193i \(-0.315224\pi\)
0.548435 + 0.836193i \(0.315224\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −6.47448 −0.225140 −0.112570 0.993644i \(-0.535908\pi\)
−0.112570 + 0.993644i \(0.535908\pi\)
\(828\) 0 0
\(829\) −50.0421 −1.73804 −0.869018 0.494781i \(-0.835248\pi\)
−0.869018 + 0.494781i \(0.835248\pi\)
\(830\) 0 0
\(831\) −10.4306 −0.361833
\(832\) 0 0
\(833\) −97.4500 −3.37644
\(834\) 0 0
\(835\) −10.0642 −0.348286
\(836\) 0 0
\(837\) −20.5798 −0.711343
\(838\) 0 0
\(839\) 10.0716 0.347709 0.173855 0.984771i \(-0.444378\pi\)
0.173855 + 0.984771i \(0.444378\pi\)
\(840\) 0 0
\(841\) −27.9785 −0.964775
\(842\) 0 0
\(843\) 28.6030 0.985141
\(844\) 0 0
\(845\) −4.58851 −0.157850
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −84.4222 −2.89736
\(850\) 0 0
\(851\) 10.0648 0.345018
\(852\) 0 0
\(853\) −24.2805 −0.831348 −0.415674 0.909514i \(-0.636454\pi\)
−0.415674 + 0.909514i \(0.636454\pi\)
\(854\) 0 0
\(855\) −33.4558 −1.14417
\(856\) 0 0
\(857\) 23.8353 0.814200 0.407100 0.913384i \(-0.366540\pi\)
0.407100 + 0.913384i \(0.366540\pi\)
\(858\) 0 0
\(859\) 27.1693 0.927004 0.463502 0.886096i \(-0.346593\pi\)
0.463502 + 0.886096i \(0.346593\pi\)
\(860\) 0 0
\(861\) −56.6649 −1.93114
\(862\) 0 0
\(863\) 33.5003 1.14036 0.570182 0.821518i \(-0.306873\pi\)
0.570182 + 0.821518i \(0.306873\pi\)
\(864\) 0 0
\(865\) 5.77262 0.196275
\(866\) 0 0
\(867\) −92.1378 −3.12917
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −37.2348 −1.26165
\(872\) 0 0
\(873\) −50.8357 −1.72053
\(874\) 0 0
\(875\) −4.55506 −0.153989
\(876\) 0 0
\(877\) −4.10027 −0.138456 −0.0692281 0.997601i \(-0.522054\pi\)
−0.0692281 + 0.997601i \(0.522054\pi\)
\(878\) 0 0
\(879\) 21.9132 0.739113
\(880\) 0 0
\(881\) 34.8285 1.17340 0.586701 0.809804i \(-0.300426\pi\)
0.586701 + 0.809804i \(0.300426\pi\)
\(882\) 0 0
\(883\) −26.6934 −0.898304 −0.449152 0.893455i \(-0.648274\pi\)
−0.449152 + 0.893455i \(0.648274\pi\)
\(884\) 0 0
\(885\) 10.5424 0.354377
\(886\) 0 0
\(887\) 51.4083 1.72612 0.863060 0.505101i \(-0.168545\pi\)
0.863060 + 0.505101i \(0.168545\pi\)
\(888\) 0 0
\(889\) −80.7205 −2.70728
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −56.0775 −1.87656
\(894\) 0 0
\(895\) 16.8671 0.563805
\(896\) 0 0
\(897\) 15.0091 0.501141
\(898\) 0 0
\(899\) −4.45754 −0.148667
\(900\) 0 0
\(901\) −5.06409 −0.168709
\(902\) 0 0
\(903\) −84.9743 −2.82777
\(904\) 0 0
\(905\) 15.1054 0.502120
\(906\) 0 0
\(907\) −6.75322 −0.224237 −0.112118 0.993695i \(-0.535764\pi\)
−0.112118 + 0.993695i \(0.535764\pi\)
\(908\) 0 0
\(909\) 59.3227 1.96761
\(910\) 0 0
\(911\) −13.0900 −0.433692 −0.216846 0.976206i \(-0.569577\pi\)
−0.216846 + 0.976206i \(0.569577\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −5.86520 −0.193898
\(916\) 0 0
\(917\) −65.5922 −2.16605
\(918\) 0 0
\(919\) 0.533564 0.0176007 0.00880033 0.999961i \(-0.497199\pi\)
0.00880033 + 0.999961i \(0.497199\pi\)
\(920\) 0 0
\(921\) −39.5232 −1.30233
\(922\) 0 0
\(923\) 12.2128 0.401990
\(924\) 0 0
\(925\) −5.39094 −0.177253
\(926\) 0 0
\(927\) −28.9511 −0.950879
\(928\) 0 0
\(929\) −9.92150 −0.325514 −0.162757 0.986666i \(-0.552039\pi\)
−0.162757 + 0.986666i \(0.552039\pi\)
\(930\) 0 0
\(931\) −98.2124 −3.21878
\(932\) 0 0
\(933\) −47.7093 −1.56193
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −3.57002 −0.116627 −0.0583137 0.998298i \(-0.518572\pi\)
−0.0583137 + 0.998298i \(0.518572\pi\)
\(938\) 0 0
\(939\) 22.0754 0.720403
\(940\) 0 0
\(941\) −48.8240 −1.59162 −0.795809 0.605548i \(-0.792954\pi\)
−0.795809 + 0.605548i \(0.792954\pi\)
\(942\) 0 0
\(943\) 8.37887 0.272853
\(944\) 0 0
\(945\) 21.2551 0.691428
\(946\) 0 0
\(947\) 15.1463 0.492190 0.246095 0.969246i \(-0.420852\pi\)
0.246095 + 0.969246i \(0.420852\pi\)
\(948\) 0 0
\(949\) 3.96105 0.128581
\(950\) 0 0
\(951\) 38.0600 1.23418
\(952\) 0 0
\(953\) 25.6087 0.829547 0.414773 0.909925i \(-0.363861\pi\)
0.414773 + 0.909925i \(0.363861\pi\)
\(954\) 0 0
\(955\) 10.4338 0.337631
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −20.3597 −0.657450
\(960\) 0 0
\(961\) −11.5489 −0.372544
\(962\) 0 0
\(963\) 4.71969 0.152090
\(964\) 0 0
\(965\) 13.9250 0.448262
\(966\) 0 0
\(967\) 19.9187 0.640544 0.320272 0.947326i \(-0.396226\pi\)
0.320272 + 0.947326i \(0.396226\pi\)
\(968\) 0 0
\(969\) −140.350 −4.50868
\(970\) 0 0
\(971\) −14.3646 −0.460980 −0.230490 0.973075i \(-0.574033\pi\)
−0.230490 + 0.973075i \(0.574033\pi\)
\(972\) 0 0
\(973\) −22.0703 −0.707541
\(974\) 0 0
\(975\) −8.03922 −0.257461
\(976\) 0 0
\(977\) 48.4441 1.54987 0.774933 0.632044i \(-0.217784\pi\)
0.774933 + 0.632044i \(0.217784\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 30.9293 0.987496
\(982\) 0 0
\(983\) 10.0332 0.320010 0.160005 0.987116i \(-0.448849\pi\)
0.160005 + 0.987116i \(0.448849\pi\)
\(984\) 0 0
\(985\) −2.25234 −0.0717655
\(986\) 0 0
\(987\) 99.1175 3.15495
\(988\) 0 0
\(989\) 12.5649 0.399540
\(990\) 0 0
\(991\) −4.44920 −0.141333 −0.0706667 0.997500i \(-0.522513\pi\)
−0.0706667 + 0.997500i \(0.522513\pi\)
\(992\) 0 0
\(993\) −29.6513 −0.940956
\(994\) 0 0
\(995\) 22.1246 0.701396
\(996\) 0 0
\(997\) 42.4383 1.34404 0.672018 0.740535i \(-0.265428\pi\)
0.672018 + 0.740535i \(0.265428\pi\)
\(998\) 0 0
\(999\) 25.1555 0.795886
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 4840.2.a.bg.1.2 8
4.3 odd 2 9680.2.a.df.1.7 8
11.5 even 5 440.2.y.d.201.1 yes 16
11.9 even 5 440.2.y.d.81.1 16
11.10 odd 2 4840.2.a.bh.1.2 8
44.27 odd 10 880.2.bo.k.641.4 16
44.31 odd 10 880.2.bo.k.81.4 16
44.43 even 2 9680.2.a.de.1.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.d.81.1 16 11.9 even 5
440.2.y.d.201.1 yes 16 11.5 even 5
880.2.bo.k.81.4 16 44.31 odd 10
880.2.bo.k.641.4 16 44.27 odd 10
4840.2.a.bg.1.2 8 1.1 even 1 trivial
4840.2.a.bh.1.2 8 11.10 odd 2
9680.2.a.de.1.7 8 44.43 even 2
9680.2.a.df.1.7 8 4.3 odd 2