Properties

Label 4016.2.a.c.1.4
Level $4016$
Weight $2$
Character 4016.1
Self dual yes
Analytic conductor $32.068$
Analytic rank $1$
Dimension $4$
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] = [4016,2,Mod(1,4016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4016, 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("4016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4016 = 2^{4} \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4016.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: \(32.0679214517\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 3x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 251)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.35567\) of defining polynomial
Character \(\chi\) \(=\) 4016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.19353 q^{3} +0.455887 q^{5} -2.54920 q^{7} +1.81156 q^{9} +O(q^{10})\) \(q+2.19353 q^{3} +0.455887 q^{5} -2.54920 q^{7} +1.81156 q^{9} +0.262360 q^{11} -5.81156 q^{13} +1.00000 q^{15} +2.25546 q^{17} +7.64252 q^{19} -5.59174 q^{21} -2.29374 q^{23} -4.79217 q^{25} -2.60687 q^{27} -7.68079 q^{29} -6.67390 q^{31} +0.575493 q^{33} -1.16215 q^{35} -4.67997 q^{37} -12.7478 q^{39} -6.97880 q^{41} +9.67816 q^{43} +0.825867 q^{45} -0.331192 q^{47} -0.501572 q^{49} +4.94742 q^{51} +4.11646 q^{53} +0.119606 q^{55} +16.7641 q^{57} -3.70312 q^{59} -7.97371 q^{61} -4.61803 q^{63} -2.64941 q^{65} -3.81074 q^{67} -5.03138 q^{69} +10.4735 q^{71} +8.70131 q^{73} -10.5117 q^{75} -0.668808 q^{77} +8.03513 q^{79} -11.1529 q^{81} -6.34188 q^{83} +1.02824 q^{85} -16.8480 q^{87} -7.06388 q^{89} +14.8148 q^{91} -14.6394 q^{93} +3.48412 q^{95} +1.60980 q^{97} +0.475281 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 3 q^{5} + 3 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 3 q^{5} + 3 q^{7} - 4 q^{9} + 3 q^{11} - 12 q^{13} + 4 q^{15} + q^{17} + 9 q^{19} - 7 q^{21} - 4 q^{23} - 7 q^{25} - q^{27} - 12 q^{29} + 2 q^{31} - 5 q^{35} - 13 q^{37} - 13 q^{39} + q^{41} + 5 q^{43} + 11 q^{45} - 12 q^{47} - 9 q^{49} - 2 q^{51} + 5 q^{53} + 3 q^{55} + 16 q^{57} - 6 q^{59} - 21 q^{61} - 14 q^{63} + q^{65} - 17 q^{67} - 13 q^{69} + 10 q^{71} - 2 q^{73} - 13 q^{75} + 8 q^{77} + 21 q^{79} - 8 q^{81} + q^{83} - 17 q^{85} - 31 q^{87} + 5 q^{89} + 2 q^{91} - 23 q^{93} - 12 q^{95} + 6 q^{97} - 2 q^{99}+O(q^{100}) \) 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
\(3\) 2.19353 1.26643 0.633217 0.773975i \(-0.281734\pi\)
0.633217 + 0.773975i \(0.281734\pi\)
\(4\) 0 0
\(5\) 0.455887 0.203879 0.101939 0.994791i \(-0.467495\pi\)
0.101939 + 0.994791i \(0.467495\pi\)
\(6\) 0 0
\(7\) −2.54920 −0.963508 −0.481754 0.876307i \(-0.660000\pi\)
−0.481754 + 0.876307i \(0.660000\pi\)
\(8\) 0 0
\(9\) 1.81156 0.603854
\(10\) 0 0
\(11\) 0.262360 0.0791044 0.0395522 0.999218i \(-0.487407\pi\)
0.0395522 + 0.999218i \(0.487407\pi\)
\(12\) 0 0
\(13\) −5.81156 −1.61184 −0.805919 0.592026i \(-0.798328\pi\)
−0.805919 + 0.592026i \(0.798328\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 2.25546 0.547030 0.273515 0.961868i \(-0.411814\pi\)
0.273515 + 0.961868i \(0.411814\pi\)
\(18\) 0 0
\(19\) 7.64252 1.75331 0.876657 0.481116i \(-0.159769\pi\)
0.876657 + 0.481116i \(0.159769\pi\)
\(20\) 0 0
\(21\) −5.59174 −1.22022
\(22\) 0 0
\(23\) −2.29374 −0.478278 −0.239139 0.970985i \(-0.576865\pi\)
−0.239139 + 0.970985i \(0.576865\pi\)
\(24\) 0 0
\(25\) −4.79217 −0.958433
\(26\) 0 0
\(27\) −2.60687 −0.501693
\(28\) 0 0
\(29\) −7.68079 −1.42629 −0.713144 0.701018i \(-0.752729\pi\)
−0.713144 + 0.701018i \(0.752729\pi\)
\(30\) 0 0
\(31\) −6.67390 −1.19867 −0.599334 0.800499i \(-0.704568\pi\)
−0.599334 + 0.800499i \(0.704568\pi\)
\(32\) 0 0
\(33\) 0.575493 0.100180
\(34\) 0 0
\(35\) −1.16215 −0.196439
\(36\) 0 0
\(37\) −4.67997 −0.769382 −0.384691 0.923045i \(-0.625692\pi\)
−0.384691 + 0.923045i \(0.625692\pi\)
\(38\) 0 0
\(39\) −12.7478 −2.04128
\(40\) 0 0
\(41\) −6.97880 −1.08990 −0.544952 0.838467i \(-0.683452\pi\)
−0.544952 + 0.838467i \(0.683452\pi\)
\(42\) 0 0
\(43\) 9.67816 1.47591 0.737953 0.674852i \(-0.235793\pi\)
0.737953 + 0.674852i \(0.235793\pi\)
\(44\) 0 0
\(45\) 0.825867 0.123113
\(46\) 0 0
\(47\) −0.331192 −0.0483094 −0.0241547 0.999708i \(-0.507689\pi\)
−0.0241547 + 0.999708i \(0.507689\pi\)
\(48\) 0 0
\(49\) −0.501572 −0.0716532
\(50\) 0 0
\(51\) 4.94742 0.692777
\(52\) 0 0
\(53\) 4.11646 0.565439 0.282720 0.959203i \(-0.408763\pi\)
0.282720 + 0.959203i \(0.408763\pi\)
\(54\) 0 0
\(55\) 0.119606 0.0161277
\(56\) 0 0
\(57\) 16.7641 2.22045
\(58\) 0 0
\(59\) −3.70312 −0.482105 −0.241052 0.970512i \(-0.577493\pi\)
−0.241052 + 0.970512i \(0.577493\pi\)
\(60\) 0 0
\(61\) −7.97371 −1.02093 −0.510464 0.859899i \(-0.670526\pi\)
−0.510464 + 0.859899i \(0.670526\pi\)
\(62\) 0 0
\(63\) −4.61803 −0.581818
\(64\) 0 0
\(65\) −2.64941 −0.328619
\(66\) 0 0
\(67\) −3.81074 −0.465556 −0.232778 0.972530i \(-0.574781\pi\)
−0.232778 + 0.972530i \(0.574781\pi\)
\(68\) 0 0
\(69\) −5.03138 −0.605707
\(70\) 0 0
\(71\) 10.4735 1.24297 0.621486 0.783425i \(-0.286529\pi\)
0.621486 + 0.783425i \(0.286529\pi\)
\(72\) 0 0
\(73\) 8.70131 1.01841 0.509205 0.860645i \(-0.329939\pi\)
0.509205 + 0.860645i \(0.329939\pi\)
\(74\) 0 0
\(75\) −10.5117 −1.21379
\(76\) 0 0
\(77\) −0.668808 −0.0762177
\(78\) 0 0
\(79\) 8.03513 0.904023 0.452012 0.892012i \(-0.350707\pi\)
0.452012 + 0.892012i \(0.350707\pi\)
\(80\) 0 0
\(81\) −11.1529 −1.23921
\(82\) 0 0
\(83\) −6.34188 −0.696112 −0.348056 0.937474i \(-0.613158\pi\)
−0.348056 + 0.937474i \(0.613158\pi\)
\(84\) 0 0
\(85\) 1.02824 0.111528
\(86\) 0 0
\(87\) −16.8480 −1.80630
\(88\) 0 0
\(89\) −7.06388 −0.748770 −0.374385 0.927273i \(-0.622146\pi\)
−0.374385 + 0.927273i \(0.622146\pi\)
\(90\) 0 0
\(91\) 14.8148 1.55302
\(92\) 0 0
\(93\) −14.6394 −1.51803
\(94\) 0 0
\(95\) 3.48412 0.357463
\(96\) 0 0
\(97\) 1.60980 0.163451 0.0817253 0.996655i \(-0.473957\pi\)
0.0817253 + 0.996655i \(0.473957\pi\)
\(98\) 0 0
\(99\) 0.475281 0.0477675
\(100\) 0 0
\(101\) 17.2290 1.71435 0.857175 0.515025i \(-0.172217\pi\)
0.857175 + 0.515025i \(0.172217\pi\)
\(102\) 0 0
\(103\) −3.70019 −0.364590 −0.182295 0.983244i \(-0.558353\pi\)
−0.182295 + 0.983244i \(0.558353\pi\)
\(104\) 0 0
\(105\) −2.54920 −0.248777
\(106\) 0 0
\(107\) −8.57054 −0.828545 −0.414273 0.910153i \(-0.635964\pi\)
−0.414273 + 0.910153i \(0.635964\pi\)
\(108\) 0 0
\(109\) −13.4835 −1.29149 −0.645743 0.763555i \(-0.723452\pi\)
−0.645743 + 0.763555i \(0.723452\pi\)
\(110\) 0 0
\(111\) −10.2656 −0.974371
\(112\) 0 0
\(113\) −8.21111 −0.772436 −0.386218 0.922407i \(-0.626219\pi\)
−0.386218 + 0.922407i \(0.626219\pi\)
\(114\) 0 0
\(115\) −1.04569 −0.0975107
\(116\) 0 0
\(117\) −10.5280 −0.973314
\(118\) 0 0
\(119\) −5.74963 −0.527067
\(120\) 0 0
\(121\) −10.9312 −0.993742
\(122\) 0 0
\(123\) −15.3082 −1.38029
\(124\) 0 0
\(125\) −4.46412 −0.399283
\(126\) 0 0
\(127\) −11.5249 −1.02267 −0.511333 0.859383i \(-0.670848\pi\)
−0.511333 + 0.859383i \(0.670848\pi\)
\(128\) 0 0
\(129\) 21.2293 1.86914
\(130\) 0 0
\(131\) −8.43407 −0.736889 −0.368444 0.929650i \(-0.620109\pi\)
−0.368444 + 0.929650i \(0.620109\pi\)
\(132\) 0 0
\(133\) −19.4823 −1.68933
\(134\) 0 0
\(135\) −1.18844 −0.102285
\(136\) 0 0
\(137\) −13.1227 −1.12115 −0.560573 0.828105i \(-0.689419\pi\)
−0.560573 + 0.828105i \(0.689419\pi\)
\(138\) 0 0
\(139\) 3.24430 0.275178 0.137589 0.990489i \(-0.456065\pi\)
0.137589 + 0.990489i \(0.456065\pi\)
\(140\) 0 0
\(141\) −0.726479 −0.0611806
\(142\) 0 0
\(143\) −1.52472 −0.127503
\(144\) 0 0
\(145\) −3.50157 −0.290790
\(146\) 0 0
\(147\) −1.10021 −0.0907440
\(148\) 0 0
\(149\) 19.6846 1.61262 0.806312 0.591491i \(-0.201460\pi\)
0.806312 + 0.591491i \(0.201460\pi\)
\(150\) 0 0
\(151\) 5.55985 0.452454 0.226227 0.974075i \(-0.427361\pi\)
0.226227 + 0.974075i \(0.427361\pi\)
\(152\) 0 0
\(153\) 4.08591 0.330326
\(154\) 0 0
\(155\) −3.04254 −0.244383
\(156\) 0 0
\(157\) 20.1782 1.61040 0.805199 0.593004i \(-0.202058\pi\)
0.805199 + 0.593004i \(0.202058\pi\)
\(158\) 0 0
\(159\) 9.02957 0.716091
\(160\) 0 0
\(161\) 5.84720 0.460824
\(162\) 0 0
\(163\) 10.7329 0.840666 0.420333 0.907370i \(-0.361913\pi\)
0.420333 + 0.907370i \(0.361913\pi\)
\(164\) 0 0
\(165\) 0.262360 0.0204247
\(166\) 0 0
\(167\) −0.902422 −0.0698315 −0.0349158 0.999390i \(-0.511116\pi\)
−0.0349158 + 0.999390i \(0.511116\pi\)
\(168\) 0 0
\(169\) 20.7742 1.59802
\(170\) 0 0
\(171\) 13.8449 1.05874
\(172\) 0 0
\(173\) −18.8342 −1.43194 −0.715970 0.698131i \(-0.754015\pi\)
−0.715970 + 0.698131i \(0.754015\pi\)
\(174\) 0 0
\(175\) 12.2162 0.923458
\(176\) 0 0
\(177\) −8.12288 −0.610553
\(178\) 0 0
\(179\) −3.62493 −0.270940 −0.135470 0.990781i \(-0.543254\pi\)
−0.135470 + 0.990781i \(0.543254\pi\)
\(180\) 0 0
\(181\) 12.9275 0.960897 0.480449 0.877023i \(-0.340474\pi\)
0.480449 + 0.877023i \(0.340474\pi\)
\(182\) 0 0
\(183\) −17.4905 −1.29294
\(184\) 0 0
\(185\) −2.13354 −0.156861
\(186\) 0 0
\(187\) 0.591742 0.0432725
\(188\) 0 0
\(189\) 6.64544 0.483385
\(190\) 0 0
\(191\) −18.1631 −1.31424 −0.657118 0.753788i \(-0.728225\pi\)
−0.657118 + 0.753788i \(0.728225\pi\)
\(192\) 0 0
\(193\) −6.01737 −0.433140 −0.216570 0.976267i \(-0.569487\pi\)
−0.216570 + 0.976267i \(0.569487\pi\)
\(194\) 0 0
\(195\) −5.81156 −0.416175
\(196\) 0 0
\(197\) −4.69571 −0.334555 −0.167278 0.985910i \(-0.553498\pi\)
−0.167278 + 0.985910i \(0.553498\pi\)
\(198\) 0 0
\(199\) −20.3162 −1.44018 −0.720088 0.693882i \(-0.755899\pi\)
−0.720088 + 0.693882i \(0.755899\pi\)
\(200\) 0 0
\(201\) −8.35895 −0.589595
\(202\) 0 0
\(203\) 19.5799 1.37424
\(204\) 0 0
\(205\) −3.18154 −0.222208
\(206\) 0 0
\(207\) −4.15525 −0.288810
\(208\) 0 0
\(209\) 2.00509 0.138695
\(210\) 0 0
\(211\) 6.99017 0.481223 0.240612 0.970621i \(-0.422652\pi\)
0.240612 + 0.970621i \(0.422652\pi\)
\(212\) 0 0
\(213\) 22.9738 1.57414
\(214\) 0 0
\(215\) 4.41214 0.300906
\(216\) 0 0
\(217\) 17.0131 1.15492
\(218\) 0 0
\(219\) 19.0866 1.28975
\(220\) 0 0
\(221\) −13.1078 −0.881723
\(222\) 0 0
\(223\) 2.89566 0.193908 0.0969538 0.995289i \(-0.469090\pi\)
0.0969538 + 0.995289i \(0.469090\pi\)
\(224\) 0 0
\(225\) −8.68130 −0.578754
\(226\) 0 0
\(227\) 6.83372 0.453570 0.226785 0.973945i \(-0.427179\pi\)
0.226785 + 0.973945i \(0.427179\pi\)
\(228\) 0 0
\(229\) 21.2121 1.40173 0.700867 0.713292i \(-0.252797\pi\)
0.700867 + 0.713292i \(0.252797\pi\)
\(230\) 0 0
\(231\) −1.46705 −0.0965247
\(232\) 0 0
\(233\) 30.0079 1.96588 0.982941 0.183920i \(-0.0588788\pi\)
0.982941 + 0.183920i \(0.0588788\pi\)
\(234\) 0 0
\(235\) −0.150986 −0.00984925
\(236\) 0 0
\(237\) 17.6253 1.14489
\(238\) 0 0
\(239\) −10.6721 −0.690320 −0.345160 0.938544i \(-0.612175\pi\)
−0.345160 + 0.938544i \(0.612175\pi\)
\(240\) 0 0
\(241\) −11.7527 −0.757057 −0.378529 0.925590i \(-0.623570\pi\)
−0.378529 + 0.925590i \(0.623570\pi\)
\(242\) 0 0
\(243\) −16.6436 −1.06769
\(244\) 0 0
\(245\) −0.228660 −0.0146086
\(246\) 0 0
\(247\) −44.4149 −2.82606
\(248\) 0 0
\(249\) −13.9111 −0.881579
\(250\) 0 0
\(251\) 1.00000 0.0631194
\(252\) 0 0
\(253\) −0.601785 −0.0378339
\(254\) 0 0
\(255\) 2.25546 0.141242
\(256\) 0 0
\(257\) −17.9181 −1.11770 −0.558851 0.829268i \(-0.688757\pi\)
−0.558851 + 0.829268i \(0.688757\pi\)
\(258\) 0 0
\(259\) 11.9302 0.741305
\(260\) 0 0
\(261\) −13.9142 −0.861269
\(262\) 0 0
\(263\) 14.8907 0.918200 0.459100 0.888385i \(-0.348172\pi\)
0.459100 + 0.888385i \(0.348172\pi\)
\(264\) 0 0
\(265\) 1.87664 0.115281
\(266\) 0 0
\(267\) −15.4948 −0.948267
\(268\) 0 0
\(269\) 1.40922 0.0859214 0.0429607 0.999077i \(-0.486321\pi\)
0.0429607 + 0.999077i \(0.486321\pi\)
\(270\) 0 0
\(271\) −0.918888 −0.0558185 −0.0279092 0.999610i \(-0.508885\pi\)
−0.0279092 + 0.999610i \(0.508885\pi\)
\(272\) 0 0
\(273\) 32.4968 1.96679
\(274\) 0 0
\(275\) −1.25727 −0.0758163
\(276\) 0 0
\(277\) 22.4782 1.35058 0.675292 0.737551i \(-0.264018\pi\)
0.675292 + 0.737551i \(0.264018\pi\)
\(278\) 0 0
\(279\) −12.0902 −0.723820
\(280\) 0 0
\(281\) −18.3905 −1.09709 −0.548544 0.836122i \(-0.684818\pi\)
−0.548544 + 0.836122i \(0.684818\pi\)
\(282\) 0 0
\(283\) 13.1472 0.781517 0.390758 0.920493i \(-0.372213\pi\)
0.390758 + 0.920493i \(0.372213\pi\)
\(284\) 0 0
\(285\) 7.64252 0.452704
\(286\) 0 0
\(287\) 17.7904 1.05013
\(288\) 0 0
\(289\) −11.9129 −0.700758
\(290\) 0 0
\(291\) 3.53114 0.206999
\(292\) 0 0
\(293\) −21.5099 −1.25662 −0.628312 0.777961i \(-0.716254\pi\)
−0.628312 + 0.777961i \(0.716254\pi\)
\(294\) 0 0
\(295\) −1.68820 −0.0982909
\(296\) 0 0
\(297\) −0.683938 −0.0396861
\(298\) 0 0
\(299\) 13.3302 0.770906
\(300\) 0 0
\(301\) −24.6716 −1.42205
\(302\) 0 0
\(303\) 37.7923 2.17111
\(304\) 0 0
\(305\) −3.63511 −0.208146
\(306\) 0 0
\(307\) −5.66639 −0.323398 −0.161699 0.986840i \(-0.551697\pi\)
−0.161699 + 0.986840i \(0.551697\pi\)
\(308\) 0 0
\(309\) −8.11646 −0.461729
\(310\) 0 0
\(311\) 26.3965 1.49681 0.748405 0.663242i \(-0.230820\pi\)
0.748405 + 0.663242i \(0.230820\pi\)
\(312\) 0 0
\(313\) −15.4172 −0.871434 −0.435717 0.900084i \(-0.643505\pi\)
−0.435717 + 0.900084i \(0.643505\pi\)
\(314\) 0 0
\(315\) −2.10530 −0.118620
\(316\) 0 0
\(317\) −19.2318 −1.08017 −0.540084 0.841611i \(-0.681607\pi\)
−0.540084 + 0.841611i \(0.681607\pi\)
\(318\) 0 0
\(319\) −2.01513 −0.112826
\(320\) 0 0
\(321\) −18.7997 −1.04930
\(322\) 0 0
\(323\) 17.2374 0.959115
\(324\) 0 0
\(325\) 27.8500 1.54484
\(326\) 0 0
\(327\) −29.5765 −1.63558
\(328\) 0 0
\(329\) 0.844276 0.0465464
\(330\) 0 0
\(331\) 7.61743 0.418692 0.209346 0.977842i \(-0.432867\pi\)
0.209346 + 0.977842i \(0.432867\pi\)
\(332\) 0 0
\(333\) −8.47805 −0.464594
\(334\) 0 0
\(335\) −1.73726 −0.0949169
\(336\) 0 0
\(337\) 2.32541 0.126673 0.0633367 0.997992i \(-0.479826\pi\)
0.0633367 + 0.997992i \(0.479826\pi\)
\(338\) 0 0
\(339\) −18.0113 −0.978239
\(340\) 0 0
\(341\) −1.75096 −0.0948199
\(342\) 0 0
\(343\) 19.1230 1.03255
\(344\) 0 0
\(345\) −2.29374 −0.123491
\(346\) 0 0
\(347\) −15.7701 −0.846585 −0.423293 0.905993i \(-0.639126\pi\)
−0.423293 + 0.905993i \(0.639126\pi\)
\(348\) 0 0
\(349\) −31.5020 −1.68626 −0.843132 0.537706i \(-0.819291\pi\)
−0.843132 + 0.537706i \(0.819291\pi\)
\(350\) 0 0
\(351\) 15.1500 0.808647
\(352\) 0 0
\(353\) 37.4767 1.99468 0.997341 0.0728782i \(-0.0232184\pi\)
0.997341 + 0.0728782i \(0.0232184\pi\)
\(354\) 0 0
\(355\) 4.77472 0.253416
\(356\) 0 0
\(357\) −12.6120 −0.667496
\(358\) 0 0
\(359\) 22.9690 1.21226 0.606128 0.795367i \(-0.292722\pi\)
0.606128 + 0.795367i \(0.292722\pi\)
\(360\) 0 0
\(361\) 39.4081 2.07411
\(362\) 0 0
\(363\) −23.9778 −1.25851
\(364\) 0 0
\(365\) 3.96681 0.207632
\(366\) 0 0
\(367\) 18.9369 0.988499 0.494250 0.869320i \(-0.335443\pi\)
0.494250 + 0.869320i \(0.335443\pi\)
\(368\) 0 0
\(369\) −12.6425 −0.658143
\(370\) 0 0
\(371\) −10.4937 −0.544805
\(372\) 0 0
\(373\) 5.83341 0.302042 0.151021 0.988531i \(-0.451744\pi\)
0.151021 + 0.988531i \(0.451744\pi\)
\(374\) 0 0
\(375\) −9.79217 −0.505665
\(376\) 0 0
\(377\) 44.6374 2.29894
\(378\) 0 0
\(379\) −30.7528 −1.57967 −0.789833 0.613322i \(-0.789833\pi\)
−0.789833 + 0.613322i \(0.789833\pi\)
\(380\) 0 0
\(381\) −25.2801 −1.29514
\(382\) 0 0
\(383\) −10.9560 −0.559824 −0.279912 0.960026i \(-0.590305\pi\)
−0.279912 + 0.960026i \(0.590305\pi\)
\(384\) 0 0
\(385\) −0.304901 −0.0155392
\(386\) 0 0
\(387\) 17.5326 0.891231
\(388\) 0 0
\(389\) 0.331033 0.0167840 0.00839201 0.999965i \(-0.497329\pi\)
0.00839201 + 0.999965i \(0.497329\pi\)
\(390\) 0 0
\(391\) −5.17344 −0.261632
\(392\) 0 0
\(393\) −18.5004 −0.933220
\(394\) 0 0
\(395\) 3.66311 0.184311
\(396\) 0 0
\(397\) 17.3473 0.870635 0.435318 0.900277i \(-0.356636\pi\)
0.435318 + 0.900277i \(0.356636\pi\)
\(398\) 0 0
\(399\) −42.7350 −2.13942
\(400\) 0 0
\(401\) −2.14177 −0.106955 −0.0534774 0.998569i \(-0.517031\pi\)
−0.0534774 + 0.998569i \(0.517031\pi\)
\(402\) 0 0
\(403\) 38.7858 1.93206
\(404\) 0 0
\(405\) −5.08447 −0.252650
\(406\) 0 0
\(407\) −1.22784 −0.0608615
\(408\) 0 0
\(409\) −21.4823 −1.06223 −0.531117 0.847299i \(-0.678227\pi\)
−0.531117 + 0.847299i \(0.678227\pi\)
\(410\) 0 0
\(411\) −28.7849 −1.41986
\(412\) 0 0
\(413\) 9.43999 0.464511
\(414\) 0 0
\(415\) −2.89118 −0.141922
\(416\) 0 0
\(417\) 7.11646 0.348495
\(418\) 0 0
\(419\) 11.5083 0.562217 0.281108 0.959676i \(-0.409298\pi\)
0.281108 + 0.959676i \(0.409298\pi\)
\(420\) 0 0
\(421\) −15.1653 −0.739111 −0.369556 0.929209i \(-0.620490\pi\)
−0.369556 + 0.929209i \(0.620490\pi\)
\(422\) 0 0
\(423\) −0.599975 −0.0291718
\(424\) 0 0
\(425\) −10.8086 −0.524292
\(426\) 0 0
\(427\) 20.3266 0.983673
\(428\) 0 0
\(429\) −3.34451 −0.161475
\(430\) 0 0
\(431\) −31.0776 −1.49696 −0.748478 0.663159i \(-0.769215\pi\)
−0.748478 + 0.663159i \(0.769215\pi\)
\(432\) 0 0
\(433\) 37.4376 1.79914 0.899568 0.436781i \(-0.143882\pi\)
0.899568 + 0.436781i \(0.143882\pi\)
\(434\) 0 0
\(435\) −7.68079 −0.368266
\(436\) 0 0
\(437\) −17.5299 −0.838571
\(438\) 0 0
\(439\) 4.32227 0.206291 0.103145 0.994666i \(-0.467109\pi\)
0.103145 + 0.994666i \(0.467109\pi\)
\(440\) 0 0
\(441\) −0.908629 −0.0432680
\(442\) 0 0
\(443\) 37.6334 1.78802 0.894008 0.448050i \(-0.147881\pi\)
0.894008 + 0.448050i \(0.147881\pi\)
\(444\) 0 0
\(445\) −3.22033 −0.152658
\(446\) 0 0
\(447\) 43.1787 2.04228
\(448\) 0 0
\(449\) 30.7504 1.45120 0.725600 0.688117i \(-0.241562\pi\)
0.725600 + 0.688117i \(0.241562\pi\)
\(450\) 0 0
\(451\) −1.83095 −0.0862163
\(452\) 0 0
\(453\) 12.1957 0.573003
\(454\) 0 0
\(455\) 6.75389 0.316627
\(456\) 0 0
\(457\) 6.13474 0.286971 0.143485 0.989652i \(-0.454169\pi\)
0.143485 + 0.989652i \(0.454169\pi\)
\(458\) 0 0
\(459\) −5.87970 −0.274441
\(460\) 0 0
\(461\) −9.48434 −0.441730 −0.220865 0.975304i \(-0.570888\pi\)
−0.220865 + 0.975304i \(0.570888\pi\)
\(462\) 0 0
\(463\) −1.49963 −0.0696937 −0.0348468 0.999393i \(-0.511094\pi\)
−0.0348468 + 0.999393i \(0.511094\pi\)
\(464\) 0 0
\(465\) −6.67390 −0.309495
\(466\) 0 0
\(467\) 21.6702 1.00278 0.501389 0.865222i \(-0.332823\pi\)
0.501389 + 0.865222i \(0.332823\pi\)
\(468\) 0 0
\(469\) 9.71433 0.448566
\(470\) 0 0
\(471\) 44.2615 2.03946
\(472\) 0 0
\(473\) 2.53916 0.116751
\(474\) 0 0
\(475\) −36.6242 −1.68043
\(476\) 0 0
\(477\) 7.45722 0.341443
\(478\) 0 0
\(479\) −4.96540 −0.226875 −0.113437 0.993545i \(-0.536186\pi\)
−0.113437 + 0.993545i \(0.536186\pi\)
\(480\) 0 0
\(481\) 27.1979 1.24012
\(482\) 0 0
\(483\) 12.8260 0.583603
\(484\) 0 0
\(485\) 0.733887 0.0333241
\(486\) 0 0
\(487\) 33.1715 1.50314 0.751571 0.659652i \(-0.229296\pi\)
0.751571 + 0.659652i \(0.229296\pi\)
\(488\) 0 0
\(489\) 23.5429 1.06465
\(490\) 0 0
\(491\) −28.8677 −1.30278 −0.651390 0.758743i \(-0.725814\pi\)
−0.651390 + 0.758743i \(0.725814\pi\)
\(492\) 0 0
\(493\) −17.3237 −0.780222
\(494\) 0 0
\(495\) 0.216674 0.00973878
\(496\) 0 0
\(497\) −26.6990 −1.19761
\(498\) 0 0
\(499\) 42.8314 1.91740 0.958699 0.284422i \(-0.0918016\pi\)
0.958699 + 0.284422i \(0.0918016\pi\)
\(500\) 0 0
\(501\) −1.97949 −0.0884370
\(502\) 0 0
\(503\) −17.6267 −0.785935 −0.392968 0.919552i \(-0.628552\pi\)
−0.392968 + 0.919552i \(0.628552\pi\)
\(504\) 0 0
\(505\) 7.85448 0.349520
\(506\) 0 0
\(507\) 45.5689 2.02378
\(508\) 0 0
\(509\) 29.4363 1.30474 0.652371 0.757900i \(-0.273775\pi\)
0.652371 + 0.757900i \(0.273775\pi\)
\(510\) 0 0
\(511\) −22.1814 −0.981247
\(512\) 0 0
\(513\) −19.9231 −0.879625
\(514\) 0 0
\(515\) −1.68687 −0.0743322
\(516\) 0 0
\(517\) −0.0868915 −0.00382148
\(518\) 0 0
\(519\) −41.3134 −1.81346
\(520\) 0 0
\(521\) 40.8139 1.78809 0.894045 0.447976i \(-0.147855\pi\)
0.894045 + 0.447976i \(0.147855\pi\)
\(522\) 0 0
\(523\) −2.92026 −0.127694 −0.0638471 0.997960i \(-0.520337\pi\)
−0.0638471 + 0.997960i \(0.520337\pi\)
\(524\) 0 0
\(525\) 26.7966 1.16950
\(526\) 0 0
\(527\) −15.0527 −0.655707
\(528\) 0 0
\(529\) −17.7388 −0.771250
\(530\) 0 0
\(531\) −6.70842 −0.291121
\(532\) 0 0
\(533\) 40.5577 1.75675
\(534\) 0 0
\(535\) −3.90720 −0.168923
\(536\) 0 0
\(537\) −7.95139 −0.343128
\(538\) 0 0
\(539\) −0.131592 −0.00566808
\(540\) 0 0
\(541\) −28.4385 −1.22267 −0.611333 0.791373i \(-0.709366\pi\)
−0.611333 + 0.791373i \(0.709366\pi\)
\(542\) 0 0
\(543\) 28.3569 1.21691
\(544\) 0 0
\(545\) −6.14696 −0.263307
\(546\) 0 0
\(547\) −1.23749 −0.0529111 −0.0264555 0.999650i \(-0.508422\pi\)
−0.0264555 + 0.999650i \(0.508422\pi\)
\(548\) 0 0
\(549\) −14.4449 −0.616492
\(550\) 0 0
\(551\) −58.7006 −2.50073
\(552\) 0 0
\(553\) −20.4832 −0.871033
\(554\) 0 0
\(555\) −4.67997 −0.198654
\(556\) 0 0
\(557\) 29.1991 1.23721 0.618603 0.785704i \(-0.287699\pi\)
0.618603 + 0.785704i \(0.287699\pi\)
\(558\) 0 0
\(559\) −56.2452 −2.37892
\(560\) 0 0
\(561\) 1.29800 0.0548017
\(562\) 0 0
\(563\) 2.13098 0.0898103 0.0449051 0.998991i \(-0.485701\pi\)
0.0449051 + 0.998991i \(0.485701\pi\)
\(564\) 0 0
\(565\) −3.74334 −0.157483
\(566\) 0 0
\(567\) 28.4311 1.19399
\(568\) 0 0
\(569\) −9.35256 −0.392080 −0.196040 0.980596i \(-0.562808\pi\)
−0.196040 + 0.980596i \(0.562808\pi\)
\(570\) 0 0
\(571\) −27.3673 −1.14529 −0.572644 0.819804i \(-0.694082\pi\)
−0.572644 + 0.819804i \(0.694082\pi\)
\(572\) 0 0
\(573\) −39.8413 −1.66439
\(574\) 0 0
\(575\) 10.9920 0.458397
\(576\) 0 0
\(577\) −5.00690 −0.208440 −0.104220 0.994554i \(-0.533235\pi\)
−0.104220 + 0.994554i \(0.533235\pi\)
\(578\) 0 0
\(579\) −13.1993 −0.548542
\(580\) 0 0
\(581\) 16.1667 0.670709
\(582\) 0 0
\(583\) 1.07999 0.0447288
\(584\) 0 0
\(585\) −4.79958 −0.198438
\(586\) 0 0
\(587\) −36.7558 −1.51707 −0.758537 0.651630i \(-0.774086\pi\)
−0.758537 + 0.651630i \(0.774086\pi\)
\(588\) 0 0
\(589\) −51.0054 −2.10164
\(590\) 0 0
\(591\) −10.3002 −0.423692
\(592\) 0 0
\(593\) 10.5502 0.433244 0.216622 0.976256i \(-0.430496\pi\)
0.216622 + 0.976256i \(0.430496\pi\)
\(594\) 0 0
\(595\) −2.62118 −0.107458
\(596\) 0 0
\(597\) −44.5641 −1.82389
\(598\) 0 0
\(599\) 43.6049 1.78165 0.890823 0.454350i \(-0.150129\pi\)
0.890823 + 0.454350i \(0.150129\pi\)
\(600\) 0 0
\(601\) −33.1345 −1.35158 −0.675792 0.737093i \(-0.736198\pi\)
−0.675792 + 0.737093i \(0.736198\pi\)
\(602\) 0 0
\(603\) −6.90338 −0.281127
\(604\) 0 0
\(605\) −4.98337 −0.202603
\(606\) 0 0
\(607\) −26.1597 −1.06179 −0.530896 0.847437i \(-0.678144\pi\)
−0.530896 + 0.847437i \(0.678144\pi\)
\(608\) 0 0
\(609\) 42.9490 1.74038
\(610\) 0 0
\(611\) 1.92474 0.0778668
\(612\) 0 0
\(613\) −10.7113 −0.432624 −0.216312 0.976324i \(-0.569403\pi\)
−0.216312 + 0.976324i \(0.569403\pi\)
\(614\) 0 0
\(615\) −6.97880 −0.281412
\(616\) 0 0
\(617\) 29.2739 1.17852 0.589261 0.807943i \(-0.299419\pi\)
0.589261 + 0.807943i \(0.299419\pi\)
\(618\) 0 0
\(619\) −20.3879 −0.819457 −0.409729 0.912207i \(-0.634377\pi\)
−0.409729 + 0.912207i \(0.634377\pi\)
\(620\) 0 0
\(621\) 5.97949 0.239949
\(622\) 0 0
\(623\) 18.0072 0.721445
\(624\) 0 0
\(625\) 21.9257 0.877028
\(626\) 0 0
\(627\) 4.39822 0.175648
\(628\) 0 0
\(629\) −10.5555 −0.420875
\(630\) 0 0
\(631\) 22.9356 0.913050 0.456525 0.889710i \(-0.349094\pi\)
0.456525 + 0.889710i \(0.349094\pi\)
\(632\) 0 0
\(633\) 15.3331 0.609437
\(634\) 0 0
\(635\) −5.25403 −0.208500
\(636\) 0 0
\(637\) 2.91492 0.115493
\(638\) 0 0
\(639\) 18.9733 0.750573
\(640\) 0 0
\(641\) −6.48994 −0.256337 −0.128169 0.991752i \(-0.540910\pi\)
−0.128169 + 0.991752i \(0.540910\pi\)
\(642\) 0 0
\(643\) −40.9651 −1.61550 −0.807752 0.589522i \(-0.799316\pi\)
−0.807752 + 0.589522i \(0.799316\pi\)
\(644\) 0 0
\(645\) 9.67816 0.381077
\(646\) 0 0
\(647\) 19.5401 0.768202 0.384101 0.923291i \(-0.374511\pi\)
0.384101 + 0.923291i \(0.374511\pi\)
\(648\) 0 0
\(649\) −0.971548 −0.0381366
\(650\) 0 0
\(651\) 37.3187 1.46264
\(652\) 0 0
\(653\) −25.6102 −1.00220 −0.501102 0.865388i \(-0.667072\pi\)
−0.501102 + 0.865388i \(0.667072\pi\)
\(654\) 0 0
\(655\) −3.84498 −0.150236
\(656\) 0 0
\(657\) 15.7629 0.614971
\(658\) 0 0
\(659\) −23.8852 −0.930436 −0.465218 0.885196i \(-0.654024\pi\)
−0.465218 + 0.885196i \(0.654024\pi\)
\(660\) 0 0
\(661\) −16.9682 −0.659987 −0.329993 0.943983i \(-0.607047\pi\)
−0.329993 + 0.943983i \(0.607047\pi\)
\(662\) 0 0
\(663\) −28.7522 −1.11664
\(664\) 0 0
\(665\) −8.88173 −0.344419
\(666\) 0 0
\(667\) 17.6177 0.682162
\(668\) 0 0
\(669\) 6.35170 0.245571
\(670\) 0 0
\(671\) −2.09198 −0.0807600
\(672\) 0 0
\(673\) −29.7852 −1.14813 −0.574067 0.818808i \(-0.694635\pi\)
−0.574067 + 0.818808i \(0.694635\pi\)
\(674\) 0 0
\(675\) 12.4926 0.480839
\(676\) 0 0
\(677\) −31.3504 −1.20489 −0.602447 0.798159i \(-0.705808\pi\)
−0.602447 + 0.798159i \(0.705808\pi\)
\(678\) 0 0
\(679\) −4.10371 −0.157486
\(680\) 0 0
\(681\) 14.9900 0.574416
\(682\) 0 0
\(683\) 4.52663 0.173207 0.0866033 0.996243i \(-0.472399\pi\)
0.0866033 + 0.996243i \(0.472399\pi\)
\(684\) 0 0
\(685\) −5.98245 −0.228578
\(686\) 0 0
\(687\) 46.5293 1.77520
\(688\) 0 0
\(689\) −23.9231 −0.911396
\(690\) 0 0
\(691\) −31.9329 −1.21478 −0.607392 0.794402i \(-0.707784\pi\)
−0.607392 + 0.794402i \(0.707784\pi\)
\(692\) 0 0
\(693\) −1.21159 −0.0460243
\(694\) 0 0
\(695\) 1.47903 0.0561030
\(696\) 0 0
\(697\) −15.7404 −0.596210
\(698\) 0 0
\(699\) 65.8231 2.48966
\(700\) 0 0
\(701\) −24.9368 −0.941849 −0.470924 0.882174i \(-0.656080\pi\)
−0.470924 + 0.882174i \(0.656080\pi\)
\(702\) 0 0
\(703\) −35.7667 −1.34897
\(704\) 0 0
\(705\) −0.331192 −0.0124734
\(706\) 0 0
\(707\) −43.9202 −1.65179
\(708\) 0 0
\(709\) −7.35240 −0.276125 −0.138063 0.990424i \(-0.544087\pi\)
−0.138063 + 0.990424i \(0.544087\pi\)
\(710\) 0 0
\(711\) 14.5561 0.545898
\(712\) 0 0
\(713\) 15.3082 0.573296
\(714\) 0 0
\(715\) −0.695099 −0.0259952
\(716\) 0 0
\(717\) −23.4095 −0.874244
\(718\) 0 0
\(719\) 44.4992 1.65954 0.829771 0.558105i \(-0.188471\pi\)
0.829771 + 0.558105i \(0.188471\pi\)
\(720\) 0 0
\(721\) 9.43252 0.351286
\(722\) 0 0
\(723\) −25.7798 −0.958763
\(724\) 0 0
\(725\) 36.8076 1.36700
\(726\) 0 0
\(727\) 5.68657 0.210903 0.105452 0.994424i \(-0.466371\pi\)
0.105452 + 0.994424i \(0.466371\pi\)
\(728\) 0 0
\(729\) −3.04947 −0.112943
\(730\) 0 0
\(731\) 21.8287 0.807364
\(732\) 0 0
\(733\) −34.2952 −1.26672 −0.633361 0.773856i \(-0.718325\pi\)
−0.633361 + 0.773856i \(0.718325\pi\)
\(734\) 0 0
\(735\) −0.501572 −0.0185008
\(736\) 0 0
\(737\) −0.999784 −0.0368275
\(738\) 0 0
\(739\) 5.56934 0.204871 0.102436 0.994740i \(-0.467336\pi\)
0.102436 + 0.994740i \(0.467336\pi\)
\(740\) 0 0
\(741\) −97.4254 −3.57901
\(742\) 0 0
\(743\) 19.0644 0.699407 0.349703 0.936860i \(-0.386282\pi\)
0.349703 + 0.936860i \(0.386282\pi\)
\(744\) 0 0
\(745\) 8.97394 0.328780
\(746\) 0 0
\(747\) −11.4887 −0.420350
\(748\) 0 0
\(749\) 21.8480 0.798310
\(750\) 0 0
\(751\) 41.6818 1.52099 0.760496 0.649343i \(-0.224956\pi\)
0.760496 + 0.649343i \(0.224956\pi\)
\(752\) 0 0
\(753\) 2.19353 0.0799366
\(754\) 0 0
\(755\) 2.53466 0.0922458
\(756\) 0 0
\(757\) −54.3880 −1.97677 −0.988383 0.151984i \(-0.951434\pi\)
−0.988383 + 0.151984i \(0.951434\pi\)
\(758\) 0 0
\(759\) −1.32003 −0.0479141
\(760\) 0 0
\(761\) 3.55028 0.128698 0.0643489 0.997927i \(-0.479503\pi\)
0.0643489 + 0.997927i \(0.479503\pi\)
\(762\) 0 0
\(763\) 34.3722 1.24436
\(764\) 0 0
\(765\) 1.86271 0.0673465
\(766\) 0 0
\(767\) 21.5209 0.777074
\(768\) 0 0
\(769\) −32.3633 −1.16705 −0.583526 0.812094i \(-0.698327\pi\)
−0.583526 + 0.812094i \(0.698327\pi\)
\(770\) 0 0
\(771\) −39.3039 −1.41549
\(772\) 0 0
\(773\) 21.4219 0.770492 0.385246 0.922814i \(-0.374117\pi\)
0.385246 + 0.922814i \(0.374117\pi\)
\(774\) 0 0
\(775\) 31.9824 1.14884
\(776\) 0 0
\(777\) 26.1692 0.938814
\(778\) 0 0
\(779\) −53.3356 −1.91094
\(780\) 0 0
\(781\) 2.74782 0.0983246
\(782\) 0 0
\(783\) 20.0229 0.715558
\(784\) 0 0
\(785\) 9.19899 0.328326
\(786\) 0 0
\(787\) −0.0246158 −0.000877457 0 −0.000438729 1.00000i \(-0.500140\pi\)
−0.000438729 1.00000i \(0.500140\pi\)
\(788\) 0 0
\(789\) 32.6632 1.16284
\(790\) 0 0
\(791\) 20.9318 0.744248
\(792\) 0 0
\(793\) 46.3397 1.64557
\(794\) 0 0
\(795\) 4.11646 0.145996
\(796\) 0 0
\(797\) −1.27168 −0.0450451 −0.0225225 0.999746i \(-0.507170\pi\)
−0.0225225 + 0.999746i \(0.507170\pi\)
\(798\) 0 0
\(799\) −0.746992 −0.0264267
\(800\) 0 0
\(801\) −12.7966 −0.452147
\(802\) 0 0
\(803\) 2.28287 0.0805608
\(804\) 0 0
\(805\) 2.66566 0.0939523
\(806\) 0 0
\(807\) 3.09115 0.108814
\(808\) 0 0
\(809\) −15.4312 −0.542533 −0.271266 0.962504i \(-0.587442\pi\)
−0.271266 + 0.962504i \(0.587442\pi\)
\(810\) 0 0
\(811\) 38.0794 1.33715 0.668575 0.743645i \(-0.266905\pi\)
0.668575 + 0.743645i \(0.266905\pi\)
\(812\) 0 0
\(813\) −2.01561 −0.0706904
\(814\) 0 0
\(815\) 4.89299 0.171394
\(816\) 0 0
\(817\) 73.9655 2.58772
\(818\) 0 0
\(819\) 26.8380 0.937795
\(820\) 0 0
\(821\) 30.9233 1.07923 0.539615 0.841912i \(-0.318570\pi\)
0.539615 + 0.841912i \(0.318570\pi\)
\(822\) 0 0
\(823\) −37.9040 −1.32125 −0.660626 0.750716i \(-0.729709\pi\)
−0.660626 + 0.750716i \(0.729709\pi\)
\(824\) 0 0
\(825\) −2.75786 −0.0960163
\(826\) 0 0
\(827\) 49.6417 1.72621 0.863106 0.505022i \(-0.168516\pi\)
0.863106 + 0.505022i \(0.168516\pi\)
\(828\) 0 0
\(829\) −38.8020 −1.34765 −0.673825 0.738891i \(-0.735350\pi\)
−0.673825 + 0.738891i \(0.735350\pi\)
\(830\) 0 0
\(831\) 49.3065 1.71042
\(832\) 0 0
\(833\) −1.13128 −0.0391964
\(834\) 0 0
\(835\) −0.411402 −0.0142372
\(836\) 0 0
\(837\) 17.3980 0.601363
\(838\) 0 0
\(839\) −41.6653 −1.43845 −0.719223 0.694780i \(-0.755502\pi\)
−0.719223 + 0.694780i \(0.755502\pi\)
\(840\) 0 0
\(841\) 29.9946 1.03430
\(842\) 0 0
\(843\) −40.3402 −1.38939
\(844\) 0 0
\(845\) 9.47070 0.325802
\(846\) 0 0
\(847\) 27.8657 0.957478
\(848\) 0 0
\(849\) 28.8386 0.989739
\(850\) 0 0
\(851\) 10.7346 0.367978
\(852\) 0 0
\(853\) 28.6962 0.982540 0.491270 0.871007i \(-0.336533\pi\)
0.491270 + 0.871007i \(0.336533\pi\)
\(854\) 0 0
\(855\) 6.31170 0.215856
\(856\) 0 0
\(857\) 20.3022 0.693510 0.346755 0.937956i \(-0.387284\pi\)
0.346755 + 0.937956i \(0.387284\pi\)
\(858\) 0 0
\(859\) 12.7410 0.434717 0.217359 0.976092i \(-0.430256\pi\)
0.217359 + 0.976092i \(0.430256\pi\)
\(860\) 0 0
\(861\) 39.0236 1.32992
\(862\) 0 0
\(863\) −39.4251 −1.34204 −0.671022 0.741437i \(-0.734145\pi\)
−0.671022 + 0.741437i \(0.734145\pi\)
\(864\) 0 0
\(865\) −8.58628 −0.291942
\(866\) 0 0
\(867\) −26.1313 −0.887464
\(868\) 0 0
\(869\) 2.10810 0.0715122
\(870\) 0 0
\(871\) 22.1463 0.750400
\(872\) 0 0
\(873\) 2.91625 0.0987002
\(874\) 0 0
\(875\) 11.3799 0.384712
\(876\) 0 0
\(877\) 18.3896 0.620974 0.310487 0.950578i \(-0.399508\pi\)
0.310487 + 0.950578i \(0.399508\pi\)
\(878\) 0 0
\(879\) −47.1826 −1.59143
\(880\) 0 0
\(881\) −8.24414 −0.277752 −0.138876 0.990310i \(-0.544349\pi\)
−0.138876 + 0.990310i \(0.544349\pi\)
\(882\) 0 0
\(883\) −19.3808 −0.652217 −0.326109 0.945332i \(-0.605738\pi\)
−0.326109 + 0.945332i \(0.605738\pi\)
\(884\) 0 0
\(885\) −3.70312 −0.124479
\(886\) 0 0
\(887\) 9.71734 0.326276 0.163138 0.986603i \(-0.447838\pi\)
0.163138 + 0.986603i \(0.447838\pi\)
\(888\) 0 0
\(889\) 29.3792 0.985346
\(890\) 0 0
\(891\) −2.92608 −0.0980273
\(892\) 0 0
\(893\) −2.53114 −0.0847015
\(894\) 0 0
\(895\) −1.65256 −0.0552389
\(896\) 0 0
\(897\) 29.2402 0.976301
\(898\) 0 0
\(899\) 51.2608 1.70964
\(900\) 0 0
\(901\) 9.28452 0.309312
\(902\) 0 0
\(903\) −54.1178 −1.80093
\(904\) 0 0
\(905\) 5.89350 0.195907
\(906\) 0 0
\(907\) 12.1518 0.403492 0.201746 0.979438i \(-0.435338\pi\)
0.201746 + 0.979438i \(0.435338\pi\)
\(908\) 0 0
\(909\) 31.2114 1.03522
\(910\) 0 0
\(911\) 26.0279 0.862343 0.431172 0.902270i \(-0.358100\pi\)
0.431172 + 0.902270i \(0.358100\pi\)
\(912\) 0 0
\(913\) −1.66385 −0.0550655
\(914\) 0 0
\(915\) −7.97371 −0.263603
\(916\) 0 0
\(917\) 21.5002 0.709998
\(918\) 0 0
\(919\) 7.54348 0.248836 0.124418 0.992230i \(-0.460294\pi\)
0.124418 + 0.992230i \(0.460294\pi\)
\(920\) 0 0
\(921\) −12.4294 −0.409562
\(922\) 0 0
\(923\) −60.8672 −2.00347
\(924\) 0 0
\(925\) 22.4272 0.737402
\(926\) 0 0
\(927\) −6.70312 −0.220159
\(928\) 0 0
\(929\) 58.9637 1.93454 0.967268 0.253758i \(-0.0816667\pi\)
0.967268 + 0.253758i \(0.0816667\pi\)
\(930\) 0 0
\(931\) −3.83327 −0.125630
\(932\) 0 0
\(933\) 57.9015 1.89561
\(934\) 0 0
\(935\) 0.269767 0.00882234
\(936\) 0 0
\(937\) −10.7067 −0.349774 −0.174887 0.984589i \(-0.555956\pi\)
−0.174887 + 0.984589i \(0.555956\pi\)
\(938\) 0 0
\(939\) −33.8181 −1.10361
\(940\) 0 0
\(941\) 15.4127 0.502439 0.251220 0.967930i \(-0.419168\pi\)
0.251220 + 0.967930i \(0.419168\pi\)
\(942\) 0 0
\(943\) 16.0075 0.521277
\(944\) 0 0
\(945\) 3.02957 0.0985519
\(946\) 0 0
\(947\) 24.5351 0.797285 0.398643 0.917106i \(-0.369481\pi\)
0.398643 + 0.917106i \(0.369481\pi\)
\(948\) 0 0
\(949\) −50.5682 −1.64151
\(950\) 0 0
\(951\) −42.1856 −1.36796
\(952\) 0 0
\(953\) −24.3819 −0.789807 −0.394903 0.918723i \(-0.629222\pi\)
−0.394903 + 0.918723i \(0.629222\pi\)
\(954\) 0 0
\(955\) −8.28032 −0.267945
\(956\) 0 0
\(957\) −4.42024 −0.142886
\(958\) 0 0
\(959\) 33.4523 1.08023
\(960\) 0 0
\(961\) 13.5409 0.436803
\(962\) 0 0
\(963\) −15.5261 −0.500320
\(964\) 0 0
\(965\) −2.74324 −0.0883080
\(966\) 0 0
\(967\) 37.8858 1.21832 0.609162 0.793046i \(-0.291506\pi\)
0.609162 + 0.793046i \(0.291506\pi\)
\(968\) 0 0
\(969\) 37.8107 1.21466
\(970\) 0 0
\(971\) −9.04262 −0.290192 −0.145096 0.989418i \(-0.546349\pi\)
−0.145096 + 0.989418i \(0.546349\pi\)
\(972\) 0 0
\(973\) −8.27038 −0.265136
\(974\) 0 0
\(975\) 61.0897 1.95644
\(976\) 0 0
\(977\) 16.3756 0.523902 0.261951 0.965081i \(-0.415634\pi\)
0.261951 + 0.965081i \(0.415634\pi\)
\(978\) 0 0
\(979\) −1.85328 −0.0592310
\(980\) 0 0
\(981\) −24.4262 −0.779869
\(982\) 0 0
\(983\) 23.6092 0.753018 0.376509 0.926413i \(-0.377124\pi\)
0.376509 + 0.926413i \(0.377124\pi\)
\(984\) 0 0
\(985\) −2.14071 −0.0682087
\(986\) 0 0
\(987\) 1.85194 0.0589480
\(988\) 0 0
\(989\) −22.1992 −0.705893
\(990\) 0 0
\(991\) −51.1253 −1.62405 −0.812024 0.583625i \(-0.801634\pi\)
−0.812024 + 0.583625i \(0.801634\pi\)
\(992\) 0 0
\(993\) 16.7090 0.530245
\(994\) 0 0
\(995\) −9.26188 −0.293621
\(996\) 0 0
\(997\) −30.0524 −0.951768 −0.475884 0.879508i \(-0.657872\pi\)
−0.475884 + 0.879508i \(0.657872\pi\)
\(998\) 0 0
\(999\) 12.2001 0.385994
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 4016.2.a.c.1.4 4
4.3 odd 2 251.2.a.a.1.1 4
12.11 even 2 2259.2.a.f.1.3 4
20.19 odd 2 6275.2.a.c.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
251.2.a.a.1.1 4 4.3 odd 2
2259.2.a.f.1.3 4 12.11 even 2
4016.2.a.c.1.4 4 1.1 even 1 trivial
6275.2.a.c.1.4 4 20.19 odd 2