Properties

Label 8020.2.a.c.1.21
Level $8020$
Weight $2$
Character 8020.1
Self dual yes
Analytic conductor $64.040$
Analytic rank $1$
Dimension $28$
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: \(1\)
Dimension: \(28\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.21
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)
 
\(f(q)\) \(=\) \(q+1.78695 q^{3} -1.00000 q^{5} -3.35040 q^{7} +0.193201 q^{9} +O(q^{10})\) \(q+1.78695 q^{3} -1.00000 q^{5} -3.35040 q^{7} +0.193201 q^{9} +3.67045 q^{11} -2.18268 q^{13} -1.78695 q^{15} +2.43273 q^{17} +0.922582 q^{19} -5.98700 q^{21} -0.388193 q^{23} +1.00000 q^{25} -5.01562 q^{27} -3.57159 q^{29} +4.21772 q^{31} +6.55892 q^{33} +3.35040 q^{35} -7.05508 q^{37} -3.90034 q^{39} +7.95489 q^{41} +8.97665 q^{43} -0.193201 q^{45} +0.141117 q^{47} +4.22517 q^{49} +4.34717 q^{51} +13.4249 q^{53} -3.67045 q^{55} +1.64861 q^{57} -8.54689 q^{59} -5.52029 q^{61} -0.647300 q^{63} +2.18268 q^{65} -1.45721 q^{67} -0.693682 q^{69} -0.463799 q^{71} -6.01284 q^{73} +1.78695 q^{75} -12.2975 q^{77} -16.8055 q^{79} -9.54228 q^{81} -13.6170 q^{83} -2.43273 q^{85} -6.38227 q^{87} -3.56062 q^{89} +7.31284 q^{91} +7.53687 q^{93} -0.922582 q^{95} -4.44507 q^{97} +0.709134 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) \) 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
\(3\) 1.78695 1.03170 0.515849 0.856680i \(-0.327477\pi\)
0.515849 + 0.856680i \(0.327477\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.35040 −1.26633 −0.633166 0.774016i \(-0.718245\pi\)
−0.633166 + 0.774016i \(0.718245\pi\)
\(8\) 0 0
\(9\) 0.193201 0.0644003
\(10\) 0 0
\(11\) 3.67045 1.10668 0.553341 0.832955i \(-0.313353\pi\)
0.553341 + 0.832955i \(0.313353\pi\)
\(12\) 0 0
\(13\) −2.18268 −0.605366 −0.302683 0.953091i \(-0.597882\pi\)
−0.302683 + 0.953091i \(0.597882\pi\)
\(14\) 0 0
\(15\) −1.78695 −0.461389
\(16\) 0 0
\(17\) 2.43273 0.590023 0.295011 0.955494i \(-0.404677\pi\)
0.295011 + 0.955494i \(0.404677\pi\)
\(18\) 0 0
\(19\) 0.922582 0.211655 0.105827 0.994385i \(-0.466251\pi\)
0.105827 + 0.994385i \(0.466251\pi\)
\(20\) 0 0
\(21\) −5.98700 −1.30647
\(22\) 0 0
\(23\) −0.388193 −0.0809438 −0.0404719 0.999181i \(-0.512886\pi\)
−0.0404719 + 0.999181i \(0.512886\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.01562 −0.965256
\(28\) 0 0
\(29\) −3.57159 −0.663228 −0.331614 0.943415i \(-0.607593\pi\)
−0.331614 + 0.943415i \(0.607593\pi\)
\(30\) 0 0
\(31\) 4.21772 0.757525 0.378762 0.925494i \(-0.376350\pi\)
0.378762 + 0.925494i \(0.376350\pi\)
\(32\) 0 0
\(33\) 6.55892 1.14176
\(34\) 0 0
\(35\) 3.35040 0.566321
\(36\) 0 0
\(37\) −7.05508 −1.15985 −0.579924 0.814671i \(-0.696918\pi\)
−0.579924 + 0.814671i \(0.696918\pi\)
\(38\) 0 0
\(39\) −3.90034 −0.624555
\(40\) 0 0
\(41\) 7.95489 1.24234 0.621172 0.783674i \(-0.286657\pi\)
0.621172 + 0.783674i \(0.286657\pi\)
\(42\) 0 0
\(43\) 8.97665 1.36893 0.684463 0.729047i \(-0.260037\pi\)
0.684463 + 0.729047i \(0.260037\pi\)
\(44\) 0 0
\(45\) −0.193201 −0.0288007
\(46\) 0 0
\(47\) 0.141117 0.0205840 0.0102920 0.999947i \(-0.496724\pi\)
0.0102920 + 0.999947i \(0.496724\pi\)
\(48\) 0 0
\(49\) 4.22517 0.603595
\(50\) 0 0
\(51\) 4.34717 0.608725
\(52\) 0 0
\(53\) 13.4249 1.84405 0.922025 0.387130i \(-0.126533\pi\)
0.922025 + 0.387130i \(0.126533\pi\)
\(54\) 0 0
\(55\) −3.67045 −0.494923
\(56\) 0 0
\(57\) 1.64861 0.218364
\(58\) 0 0
\(59\) −8.54689 −1.11271 −0.556355 0.830944i \(-0.687801\pi\)
−0.556355 + 0.830944i \(0.687801\pi\)
\(60\) 0 0
\(61\) −5.52029 −0.706801 −0.353400 0.935472i \(-0.614975\pi\)
−0.353400 + 0.935472i \(0.614975\pi\)
\(62\) 0 0
\(63\) −0.647300 −0.0815521
\(64\) 0 0
\(65\) 2.18268 0.270728
\(66\) 0 0
\(67\) −1.45721 −0.178027 −0.0890133 0.996030i \(-0.528371\pi\)
−0.0890133 + 0.996030i \(0.528371\pi\)
\(68\) 0 0
\(69\) −0.693682 −0.0835095
\(70\) 0 0
\(71\) −0.463799 −0.0550428 −0.0275214 0.999621i \(-0.508761\pi\)
−0.0275214 + 0.999621i \(0.508761\pi\)
\(72\) 0 0
\(73\) −6.01284 −0.703750 −0.351875 0.936047i \(-0.614456\pi\)
−0.351875 + 0.936047i \(0.614456\pi\)
\(74\) 0 0
\(75\) 1.78695 0.206340
\(76\) 0 0
\(77\) −12.2975 −1.40143
\(78\) 0 0
\(79\) −16.8055 −1.89076 −0.945382 0.325963i \(-0.894311\pi\)
−0.945382 + 0.325963i \(0.894311\pi\)
\(80\) 0 0
\(81\) −9.54228 −1.06025
\(82\) 0 0
\(83\) −13.6170 −1.49466 −0.747331 0.664452i \(-0.768665\pi\)
−0.747331 + 0.664452i \(0.768665\pi\)
\(84\) 0 0
\(85\) −2.43273 −0.263866
\(86\) 0 0
\(87\) −6.38227 −0.684251
\(88\) 0 0
\(89\) −3.56062 −0.377425 −0.188713 0.982032i \(-0.560431\pi\)
−0.188713 + 0.982032i \(0.560431\pi\)
\(90\) 0 0
\(91\) 7.31284 0.766594
\(92\) 0 0
\(93\) 7.53687 0.781537
\(94\) 0 0
\(95\) −0.922582 −0.0946549
\(96\) 0 0
\(97\) −4.44507 −0.451328 −0.225664 0.974205i \(-0.572455\pi\)
−0.225664 + 0.974205i \(0.572455\pi\)
\(98\) 0 0
\(99\) 0.709134 0.0712706
\(100\) 0 0
\(101\) −5.90925 −0.587992 −0.293996 0.955807i \(-0.594985\pi\)
−0.293996 + 0.955807i \(0.594985\pi\)
\(102\) 0 0
\(103\) −14.2801 −1.40706 −0.703532 0.710664i \(-0.748395\pi\)
−0.703532 + 0.710664i \(0.748395\pi\)
\(104\) 0 0
\(105\) 5.98700 0.584272
\(106\) 0 0
\(107\) 10.2869 0.994468 0.497234 0.867616i \(-0.334349\pi\)
0.497234 + 0.867616i \(0.334349\pi\)
\(108\) 0 0
\(109\) −11.1935 −1.07214 −0.536071 0.844173i \(-0.680092\pi\)
−0.536071 + 0.844173i \(0.680092\pi\)
\(110\) 0 0
\(111\) −12.6071 −1.19661
\(112\) 0 0
\(113\) 1.94884 0.183332 0.0916659 0.995790i \(-0.470781\pi\)
0.0916659 + 0.995790i \(0.470781\pi\)
\(114\) 0 0
\(115\) 0.388193 0.0361992
\(116\) 0 0
\(117\) −0.421696 −0.0389858
\(118\) 0 0
\(119\) −8.15060 −0.747164
\(120\) 0 0
\(121\) 2.47218 0.224744
\(122\) 0 0
\(123\) 14.2150 1.28172
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −2.92151 −0.259242 −0.129621 0.991564i \(-0.541376\pi\)
−0.129621 + 0.991564i \(0.541376\pi\)
\(128\) 0 0
\(129\) 16.0409 1.41232
\(130\) 0 0
\(131\) −2.29337 −0.200373 −0.100187 0.994969i \(-0.531944\pi\)
−0.100187 + 0.994969i \(0.531944\pi\)
\(132\) 0 0
\(133\) −3.09102 −0.268025
\(134\) 0 0
\(135\) 5.01562 0.431676
\(136\) 0 0
\(137\) −4.09705 −0.350035 −0.175017 0.984565i \(-0.555998\pi\)
−0.175017 + 0.984565i \(0.555998\pi\)
\(138\) 0 0
\(139\) −13.7915 −1.16978 −0.584891 0.811112i \(-0.698863\pi\)
−0.584891 + 0.811112i \(0.698863\pi\)
\(140\) 0 0
\(141\) 0.252169 0.0212364
\(142\) 0 0
\(143\) −8.01141 −0.669948
\(144\) 0 0
\(145\) 3.57159 0.296605
\(146\) 0 0
\(147\) 7.55018 0.622728
\(148\) 0 0
\(149\) −0.864421 −0.0708161 −0.0354081 0.999373i \(-0.511273\pi\)
−0.0354081 + 0.999373i \(0.511273\pi\)
\(150\) 0 0
\(151\) −15.7319 −1.28025 −0.640123 0.768273i \(-0.721116\pi\)
−0.640123 + 0.768273i \(0.721116\pi\)
\(152\) 0 0
\(153\) 0.470005 0.0379976
\(154\) 0 0
\(155\) −4.21772 −0.338775
\(156\) 0 0
\(157\) 18.6334 1.48711 0.743554 0.668675i \(-0.233138\pi\)
0.743554 + 0.668675i \(0.233138\pi\)
\(158\) 0 0
\(159\) 23.9896 1.90250
\(160\) 0 0
\(161\) 1.30060 0.102502
\(162\) 0 0
\(163\) 5.97284 0.467829 0.233914 0.972257i \(-0.424846\pi\)
0.233914 + 0.972257i \(0.424846\pi\)
\(164\) 0 0
\(165\) −6.55892 −0.510611
\(166\) 0 0
\(167\) 22.8344 1.76698 0.883490 0.468450i \(-0.155187\pi\)
0.883490 + 0.468450i \(0.155187\pi\)
\(168\) 0 0
\(169\) −8.23591 −0.633532
\(170\) 0 0
\(171\) 0.178244 0.0136306
\(172\) 0 0
\(173\) −14.7784 −1.12358 −0.561789 0.827280i \(-0.689887\pi\)
−0.561789 + 0.827280i \(0.689887\pi\)
\(174\) 0 0
\(175\) −3.35040 −0.253266
\(176\) 0 0
\(177\) −15.2729 −1.14798
\(178\) 0 0
\(179\) −2.50945 −0.187565 −0.0937825 0.995593i \(-0.529896\pi\)
−0.0937825 + 0.995593i \(0.529896\pi\)
\(180\) 0 0
\(181\) 10.6277 0.789948 0.394974 0.918692i \(-0.370754\pi\)
0.394974 + 0.918692i \(0.370754\pi\)
\(182\) 0 0
\(183\) −9.86450 −0.729205
\(184\) 0 0
\(185\) 7.05508 0.518700
\(186\) 0 0
\(187\) 8.92919 0.652967
\(188\) 0 0
\(189\) 16.8043 1.22233
\(190\) 0 0
\(191\) −0.925727 −0.0669833 −0.0334916 0.999439i \(-0.510663\pi\)
−0.0334916 + 0.999439i \(0.510663\pi\)
\(192\) 0 0
\(193\) 2.29130 0.164931 0.0824656 0.996594i \(-0.473721\pi\)
0.0824656 + 0.996594i \(0.473721\pi\)
\(194\) 0 0
\(195\) 3.90034 0.279309
\(196\) 0 0
\(197\) −23.9368 −1.70542 −0.852712 0.522382i \(-0.825044\pi\)
−0.852712 + 0.522382i \(0.825044\pi\)
\(198\) 0 0
\(199\) −22.4764 −1.59331 −0.796655 0.604434i \(-0.793399\pi\)
−0.796655 + 0.604434i \(0.793399\pi\)
\(200\) 0 0
\(201\) −2.60397 −0.183670
\(202\) 0 0
\(203\) 11.9663 0.839867
\(204\) 0 0
\(205\) −7.95489 −0.555593
\(206\) 0 0
\(207\) −0.0749992 −0.00521281
\(208\) 0 0
\(209\) 3.38629 0.234235
\(210\) 0 0
\(211\) −25.0752 −1.72625 −0.863125 0.504990i \(-0.831496\pi\)
−0.863125 + 0.504990i \(0.831496\pi\)
\(212\) 0 0
\(213\) −0.828787 −0.0567876
\(214\) 0 0
\(215\) −8.97665 −0.612203
\(216\) 0 0
\(217\) −14.1310 −0.959277
\(218\) 0 0
\(219\) −10.7447 −0.726057
\(220\) 0 0
\(221\) −5.30986 −0.357180
\(222\) 0 0
\(223\) −1.30732 −0.0875446 −0.0437723 0.999042i \(-0.513938\pi\)
−0.0437723 + 0.999042i \(0.513938\pi\)
\(224\) 0 0
\(225\) 0.193201 0.0128801
\(226\) 0 0
\(227\) 16.0007 1.06200 0.531001 0.847371i \(-0.321816\pi\)
0.531001 + 0.847371i \(0.321816\pi\)
\(228\) 0 0
\(229\) 0.775853 0.0512698 0.0256349 0.999671i \(-0.491839\pi\)
0.0256349 + 0.999671i \(0.491839\pi\)
\(230\) 0 0
\(231\) −21.9750 −1.44585
\(232\) 0 0
\(233\) 19.5965 1.28381 0.641904 0.766785i \(-0.278145\pi\)
0.641904 + 0.766785i \(0.278145\pi\)
\(234\) 0 0
\(235\) −0.141117 −0.00920543
\(236\) 0 0
\(237\) −30.0306 −1.95070
\(238\) 0 0
\(239\) −13.7558 −0.889791 −0.444895 0.895583i \(-0.646759\pi\)
−0.444895 + 0.895583i \(0.646759\pi\)
\(240\) 0 0
\(241\) 0.269687 0.0173721 0.00868604 0.999962i \(-0.497235\pi\)
0.00868604 + 0.999962i \(0.497235\pi\)
\(242\) 0 0
\(243\) −2.00475 −0.128604
\(244\) 0 0
\(245\) −4.22517 −0.269936
\(246\) 0 0
\(247\) −2.01370 −0.128129
\(248\) 0 0
\(249\) −24.3330 −1.54204
\(250\) 0 0
\(251\) −7.93704 −0.500982 −0.250491 0.968119i \(-0.580592\pi\)
−0.250491 + 0.968119i \(0.580592\pi\)
\(252\) 0 0
\(253\) −1.42484 −0.0895790
\(254\) 0 0
\(255\) −4.34717 −0.272230
\(256\) 0 0
\(257\) −7.58454 −0.473111 −0.236555 0.971618i \(-0.576018\pi\)
−0.236555 + 0.971618i \(0.576018\pi\)
\(258\) 0 0
\(259\) 23.6373 1.46875
\(260\) 0 0
\(261\) −0.690035 −0.0427121
\(262\) 0 0
\(263\) −13.9321 −0.859088 −0.429544 0.903046i \(-0.641326\pi\)
−0.429544 + 0.903046i \(0.641326\pi\)
\(264\) 0 0
\(265\) −13.4249 −0.824685
\(266\) 0 0
\(267\) −6.36266 −0.389389
\(268\) 0 0
\(269\) −12.7172 −0.775382 −0.387691 0.921789i \(-0.626727\pi\)
−0.387691 + 0.921789i \(0.626727\pi\)
\(270\) 0 0
\(271\) −0.586141 −0.0356055 −0.0178028 0.999842i \(-0.505667\pi\)
−0.0178028 + 0.999842i \(0.505667\pi\)
\(272\) 0 0
\(273\) 13.0677 0.790894
\(274\) 0 0
\(275\) 3.67045 0.221336
\(276\) 0 0
\(277\) 20.2552 1.21702 0.608509 0.793547i \(-0.291768\pi\)
0.608509 + 0.793547i \(0.291768\pi\)
\(278\) 0 0
\(279\) 0.814867 0.0487848
\(280\) 0 0
\(281\) 31.1365 1.85745 0.928725 0.370769i \(-0.120906\pi\)
0.928725 + 0.370769i \(0.120906\pi\)
\(282\) 0 0
\(283\) 15.7551 0.936543 0.468272 0.883585i \(-0.344877\pi\)
0.468272 + 0.883585i \(0.344877\pi\)
\(284\) 0 0
\(285\) −1.64861 −0.0976553
\(286\) 0 0
\(287\) −26.6520 −1.57322
\(288\) 0 0
\(289\) −11.0818 −0.651873
\(290\) 0 0
\(291\) −7.94313 −0.465634
\(292\) 0 0
\(293\) −19.8065 −1.15711 −0.578554 0.815644i \(-0.696383\pi\)
−0.578554 + 0.815644i \(0.696383\pi\)
\(294\) 0 0
\(295\) 8.54689 0.497619
\(296\) 0 0
\(297\) −18.4096 −1.06823
\(298\) 0 0
\(299\) 0.847300 0.0490006
\(300\) 0 0
\(301\) −30.0754 −1.73352
\(302\) 0 0
\(303\) −10.5595 −0.606630
\(304\) 0 0
\(305\) 5.52029 0.316091
\(306\) 0 0
\(307\) 21.8761 1.24853 0.624266 0.781212i \(-0.285398\pi\)
0.624266 + 0.781212i \(0.285398\pi\)
\(308\) 0 0
\(309\) −25.5179 −1.45166
\(310\) 0 0
\(311\) 15.4538 0.876306 0.438153 0.898900i \(-0.355633\pi\)
0.438153 + 0.898900i \(0.355633\pi\)
\(312\) 0 0
\(313\) −14.4191 −0.815016 −0.407508 0.913202i \(-0.633602\pi\)
−0.407508 + 0.913202i \(0.633602\pi\)
\(314\) 0 0
\(315\) 0.647300 0.0364712
\(316\) 0 0
\(317\) 32.0743 1.80147 0.900737 0.434366i \(-0.143027\pi\)
0.900737 + 0.434366i \(0.143027\pi\)
\(318\) 0 0
\(319\) −13.1093 −0.733982
\(320\) 0 0
\(321\) 18.3821 1.02599
\(322\) 0 0
\(323\) 2.24439 0.124881
\(324\) 0 0
\(325\) −2.18268 −0.121073
\(326\) 0 0
\(327\) −20.0022 −1.10613
\(328\) 0 0
\(329\) −0.472797 −0.0260661
\(330\) 0 0
\(331\) −3.40996 −0.187428 −0.0937142 0.995599i \(-0.529874\pi\)
−0.0937142 + 0.995599i \(0.529874\pi\)
\(332\) 0 0
\(333\) −1.36305 −0.0746945
\(334\) 0 0
\(335\) 1.45721 0.0796159
\(336\) 0 0
\(337\) −14.5511 −0.792650 −0.396325 0.918110i \(-0.629715\pi\)
−0.396325 + 0.918110i \(0.629715\pi\)
\(338\) 0 0
\(339\) 3.48249 0.189143
\(340\) 0 0
\(341\) 15.4809 0.838339
\(342\) 0 0
\(343\) 9.29679 0.501980
\(344\) 0 0
\(345\) 0.693682 0.0373466
\(346\) 0 0
\(347\) −12.9684 −0.696179 −0.348089 0.937461i \(-0.613169\pi\)
−0.348089 + 0.937461i \(0.613169\pi\)
\(348\) 0 0
\(349\) −3.67490 −0.196713 −0.0983563 0.995151i \(-0.531358\pi\)
−0.0983563 + 0.995151i \(0.531358\pi\)
\(350\) 0 0
\(351\) 10.9475 0.584333
\(352\) 0 0
\(353\) −3.31330 −0.176349 −0.0881747 0.996105i \(-0.528103\pi\)
−0.0881747 + 0.996105i \(0.528103\pi\)
\(354\) 0 0
\(355\) 0.463799 0.0246159
\(356\) 0 0
\(357\) −14.5647 −0.770848
\(358\) 0 0
\(359\) −7.94756 −0.419456 −0.209728 0.977760i \(-0.567258\pi\)
−0.209728 + 0.977760i \(0.567258\pi\)
\(360\) 0 0
\(361\) −18.1488 −0.955202
\(362\) 0 0
\(363\) 4.41768 0.231868
\(364\) 0 0
\(365\) 6.01284 0.314727
\(366\) 0 0
\(367\) −13.4816 −0.703731 −0.351866 0.936050i \(-0.614453\pi\)
−0.351866 + 0.936050i \(0.614453\pi\)
\(368\) 0 0
\(369\) 1.53689 0.0800074
\(370\) 0 0
\(371\) −44.9787 −2.33518
\(372\) 0 0
\(373\) 2.24708 0.116349 0.0581746 0.998306i \(-0.481472\pi\)
0.0581746 + 0.998306i \(0.481472\pi\)
\(374\) 0 0
\(375\) −1.78695 −0.0922779
\(376\) 0 0
\(377\) 7.79564 0.401496
\(378\) 0 0
\(379\) 13.1523 0.675586 0.337793 0.941220i \(-0.390320\pi\)
0.337793 + 0.941220i \(0.390320\pi\)
\(380\) 0 0
\(381\) −5.22060 −0.267459
\(382\) 0 0
\(383\) 34.6831 1.77222 0.886111 0.463473i \(-0.153397\pi\)
0.886111 + 0.463473i \(0.153397\pi\)
\(384\) 0 0
\(385\) 12.2975 0.626737
\(386\) 0 0
\(387\) 1.73430 0.0881593
\(388\) 0 0
\(389\) −16.8872 −0.856214 −0.428107 0.903728i \(-0.640819\pi\)
−0.428107 + 0.903728i \(0.640819\pi\)
\(390\) 0 0
\(391\) −0.944367 −0.0477587
\(392\) 0 0
\(393\) −4.09815 −0.206724
\(394\) 0 0
\(395\) 16.8055 0.845576
\(396\) 0 0
\(397\) −15.1796 −0.761841 −0.380921 0.924608i \(-0.624393\pi\)
−0.380921 + 0.924608i \(0.624393\pi\)
\(398\) 0 0
\(399\) −5.52350 −0.276521
\(400\) 0 0
\(401\) 1.00000 0.0499376
\(402\) 0 0
\(403\) −9.20593 −0.458580
\(404\) 0 0
\(405\) 9.54228 0.474160
\(406\) 0 0
\(407\) −25.8953 −1.28358
\(408\) 0 0
\(409\) −7.58297 −0.374954 −0.187477 0.982269i \(-0.560031\pi\)
−0.187477 + 0.982269i \(0.560031\pi\)
\(410\) 0 0
\(411\) −7.32124 −0.361130
\(412\) 0 0
\(413\) 28.6355 1.40906
\(414\) 0 0
\(415\) 13.6170 0.668433
\(416\) 0 0
\(417\) −24.6448 −1.20686
\(418\) 0 0
\(419\) 24.3497 1.18956 0.594781 0.803888i \(-0.297239\pi\)
0.594781 + 0.803888i \(0.297239\pi\)
\(420\) 0 0
\(421\) −32.2232 −1.57046 −0.785231 0.619203i \(-0.787456\pi\)
−0.785231 + 0.619203i \(0.787456\pi\)
\(422\) 0 0
\(423\) 0.0272639 0.00132561
\(424\) 0 0
\(425\) 2.43273 0.118005
\(426\) 0 0
\(427\) 18.4952 0.895044
\(428\) 0 0
\(429\) −14.3160 −0.691183
\(430\) 0 0
\(431\) −7.85869 −0.378540 −0.189270 0.981925i \(-0.560612\pi\)
−0.189270 + 0.981925i \(0.560612\pi\)
\(432\) 0 0
\(433\) 32.3280 1.55358 0.776792 0.629757i \(-0.216846\pi\)
0.776792 + 0.629757i \(0.216846\pi\)
\(434\) 0 0
\(435\) 6.38227 0.306006
\(436\) 0 0
\(437\) −0.358140 −0.0171321
\(438\) 0 0
\(439\) 5.65347 0.269825 0.134913 0.990857i \(-0.456925\pi\)
0.134913 + 0.990857i \(0.456925\pi\)
\(440\) 0 0
\(441\) 0.816306 0.0388717
\(442\) 0 0
\(443\) −28.9347 −1.37473 −0.687364 0.726313i \(-0.741232\pi\)
−0.687364 + 0.726313i \(0.741232\pi\)
\(444\) 0 0
\(445\) 3.56062 0.168790
\(446\) 0 0
\(447\) −1.54468 −0.0730609
\(448\) 0 0
\(449\) −27.1416 −1.28089 −0.640445 0.768004i \(-0.721250\pi\)
−0.640445 + 0.768004i \(0.721250\pi\)
\(450\) 0 0
\(451\) 29.1980 1.37488
\(452\) 0 0
\(453\) −28.1122 −1.32083
\(454\) 0 0
\(455\) −7.31284 −0.342831
\(456\) 0 0
\(457\) −9.87447 −0.461908 −0.230954 0.972965i \(-0.574185\pi\)
−0.230954 + 0.972965i \(0.574185\pi\)
\(458\) 0 0
\(459\) −12.2016 −0.569523
\(460\) 0 0
\(461\) 15.6815 0.730361 0.365181 0.930937i \(-0.381007\pi\)
0.365181 + 0.930937i \(0.381007\pi\)
\(462\) 0 0
\(463\) 16.6644 0.774459 0.387229 0.921983i \(-0.373432\pi\)
0.387229 + 0.921983i \(0.373432\pi\)
\(464\) 0 0
\(465\) −7.53687 −0.349514
\(466\) 0 0
\(467\) −7.82136 −0.361929 −0.180965 0.983490i \(-0.557922\pi\)
−0.180965 + 0.983490i \(0.557922\pi\)
\(468\) 0 0
\(469\) 4.88223 0.225441
\(470\) 0 0
\(471\) 33.2970 1.53425
\(472\) 0 0
\(473\) 32.9483 1.51497
\(474\) 0 0
\(475\) 0.922582 0.0423310
\(476\) 0 0
\(477\) 2.59370 0.118757
\(478\) 0 0
\(479\) −10.2431 −0.468021 −0.234010 0.972234i \(-0.575185\pi\)
−0.234010 + 0.972234i \(0.575185\pi\)
\(480\) 0 0
\(481\) 15.3990 0.702132
\(482\) 0 0
\(483\) 2.32411 0.105751
\(484\) 0 0
\(485\) 4.44507 0.201840
\(486\) 0 0
\(487\) −33.8658 −1.53460 −0.767302 0.641286i \(-0.778401\pi\)
−0.767302 + 0.641286i \(0.778401\pi\)
\(488\) 0 0
\(489\) 10.6732 0.482658
\(490\) 0 0
\(491\) 32.9507 1.48704 0.743521 0.668712i \(-0.233154\pi\)
0.743521 + 0.668712i \(0.233154\pi\)
\(492\) 0 0
\(493\) −8.68871 −0.391320
\(494\) 0 0
\(495\) −0.709134 −0.0318732
\(496\) 0 0
\(497\) 1.55391 0.0697025
\(498\) 0 0
\(499\) −14.9132 −0.667608 −0.333804 0.942643i \(-0.608332\pi\)
−0.333804 + 0.942643i \(0.608332\pi\)
\(500\) 0 0
\(501\) 40.8040 1.82299
\(502\) 0 0
\(503\) −30.5064 −1.36021 −0.680106 0.733114i \(-0.738066\pi\)
−0.680106 + 0.733114i \(0.738066\pi\)
\(504\) 0 0
\(505\) 5.90925 0.262958
\(506\) 0 0
\(507\) −14.7172 −0.653613
\(508\) 0 0
\(509\) −3.33146 −0.147664 −0.0738321 0.997271i \(-0.523523\pi\)
−0.0738321 + 0.997271i \(0.523523\pi\)
\(510\) 0 0
\(511\) 20.1454 0.891181
\(512\) 0 0
\(513\) −4.62732 −0.204301
\(514\) 0 0
\(515\) 14.2801 0.629258
\(516\) 0 0
\(517\) 0.517961 0.0227799
\(518\) 0 0
\(519\) −26.4082 −1.15919
\(520\) 0 0
\(521\) 23.6262 1.03508 0.517541 0.855658i \(-0.326847\pi\)
0.517541 + 0.855658i \(0.326847\pi\)
\(522\) 0 0
\(523\) 12.9782 0.567498 0.283749 0.958899i \(-0.408422\pi\)
0.283749 + 0.958899i \(0.408422\pi\)
\(524\) 0 0
\(525\) −5.98700 −0.261294
\(526\) 0 0
\(527\) 10.2606 0.446957
\(528\) 0 0
\(529\) −22.8493 −0.993448
\(530\) 0 0
\(531\) −1.65127 −0.0716589
\(532\) 0 0
\(533\) −17.3630 −0.752073
\(534\) 0 0
\(535\) −10.2869 −0.444740
\(536\) 0 0
\(537\) −4.48427 −0.193510
\(538\) 0 0
\(539\) 15.5083 0.667988
\(540\) 0 0
\(541\) 13.7194 0.589841 0.294921 0.955522i \(-0.404707\pi\)
0.294921 + 0.955522i \(0.404707\pi\)
\(542\) 0 0
\(543\) 18.9911 0.814988
\(544\) 0 0
\(545\) 11.1935 0.479476
\(546\) 0 0
\(547\) −16.4791 −0.704597 −0.352299 0.935888i \(-0.614600\pi\)
−0.352299 + 0.935888i \(0.614600\pi\)
\(548\) 0 0
\(549\) −1.06653 −0.0455182
\(550\) 0 0
\(551\) −3.29509 −0.140375
\(552\) 0 0
\(553\) 56.3051 2.39433
\(554\) 0 0
\(555\) 12.6071 0.535141
\(556\) 0 0
\(557\) −33.9333 −1.43780 −0.718901 0.695113i \(-0.755354\pi\)
−0.718901 + 0.695113i \(0.755354\pi\)
\(558\) 0 0
\(559\) −19.5932 −0.828702
\(560\) 0 0
\(561\) 15.9560 0.673665
\(562\) 0 0
\(563\) 3.00240 0.126536 0.0632680 0.997997i \(-0.479848\pi\)
0.0632680 + 0.997997i \(0.479848\pi\)
\(564\) 0 0
\(565\) −1.94884 −0.0819885
\(566\) 0 0
\(567\) 31.9704 1.34263
\(568\) 0 0
\(569\) 2.42927 0.101840 0.0509201 0.998703i \(-0.483785\pi\)
0.0509201 + 0.998703i \(0.483785\pi\)
\(570\) 0 0
\(571\) −35.2844 −1.47661 −0.738303 0.674469i \(-0.764373\pi\)
−0.738303 + 0.674469i \(0.764373\pi\)
\(572\) 0 0
\(573\) −1.65423 −0.0691065
\(574\) 0 0
\(575\) −0.388193 −0.0161888
\(576\) 0 0
\(577\) 5.24045 0.218163 0.109081 0.994033i \(-0.465209\pi\)
0.109081 + 0.994033i \(0.465209\pi\)
\(578\) 0 0
\(579\) 4.09444 0.170159
\(580\) 0 0
\(581\) 45.6224 1.89274
\(582\) 0 0
\(583\) 49.2754 2.04078
\(584\) 0 0
\(585\) 0.421696 0.0174350
\(586\) 0 0
\(587\) −10.2392 −0.422616 −0.211308 0.977419i \(-0.567772\pi\)
−0.211308 + 0.977419i \(0.567772\pi\)
\(588\) 0 0
\(589\) 3.89119 0.160334
\(590\) 0 0
\(591\) −42.7739 −1.75948
\(592\) 0 0
\(593\) 40.5841 1.66659 0.833295 0.552829i \(-0.186452\pi\)
0.833295 + 0.552829i \(0.186452\pi\)
\(594\) 0 0
\(595\) 8.15060 0.334142
\(596\) 0 0
\(597\) −40.1643 −1.64381
\(598\) 0 0
\(599\) 17.0021 0.694687 0.347344 0.937738i \(-0.387084\pi\)
0.347344 + 0.937738i \(0.387084\pi\)
\(600\) 0 0
\(601\) −31.7608 −1.29555 −0.647774 0.761832i \(-0.724300\pi\)
−0.647774 + 0.761832i \(0.724300\pi\)
\(602\) 0 0
\(603\) −0.281534 −0.0114650
\(604\) 0 0
\(605\) −2.47218 −0.100509
\(606\) 0 0
\(607\) −26.2822 −1.06676 −0.533382 0.845875i \(-0.679079\pi\)
−0.533382 + 0.845875i \(0.679079\pi\)
\(608\) 0 0
\(609\) 21.3831 0.866489
\(610\) 0 0
\(611\) −0.308012 −0.0124608
\(612\) 0 0
\(613\) 48.7765 1.97007 0.985033 0.172367i \(-0.0551415\pi\)
0.985033 + 0.172367i \(0.0551415\pi\)
\(614\) 0 0
\(615\) −14.2150 −0.573204
\(616\) 0 0
\(617\) 40.6957 1.63835 0.819173 0.573546i \(-0.194433\pi\)
0.819173 + 0.573546i \(0.194433\pi\)
\(618\) 0 0
\(619\) 22.3689 0.899082 0.449541 0.893260i \(-0.351588\pi\)
0.449541 + 0.893260i \(0.351588\pi\)
\(620\) 0 0
\(621\) 1.94703 0.0781315
\(622\) 0 0
\(623\) 11.9295 0.477945
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 6.05114 0.241659
\(628\) 0 0
\(629\) −17.1631 −0.684336
\(630\) 0 0
\(631\) 7.40314 0.294714 0.147357 0.989083i \(-0.452923\pi\)
0.147357 + 0.989083i \(0.452923\pi\)
\(632\) 0 0
\(633\) −44.8083 −1.78097
\(634\) 0 0
\(635\) 2.92151 0.115936
\(636\) 0 0
\(637\) −9.22218 −0.365396
\(638\) 0 0
\(639\) −0.0896064 −0.00354478
\(640\) 0 0
\(641\) 4.92817 0.194651 0.0973255 0.995253i \(-0.468971\pi\)
0.0973255 + 0.995253i \(0.468971\pi\)
\(642\) 0 0
\(643\) −13.4930 −0.532113 −0.266057 0.963957i \(-0.585721\pi\)
−0.266057 + 0.963957i \(0.585721\pi\)
\(644\) 0 0
\(645\) −16.0409 −0.631608
\(646\) 0 0
\(647\) 43.0533 1.69260 0.846301 0.532706i \(-0.178825\pi\)
0.846301 + 0.532706i \(0.178825\pi\)
\(648\) 0 0
\(649\) −31.3709 −1.23142
\(650\) 0 0
\(651\) −25.2515 −0.989684
\(652\) 0 0
\(653\) −7.13210 −0.279101 −0.139550 0.990215i \(-0.544566\pi\)
−0.139550 + 0.990215i \(0.544566\pi\)
\(654\) 0 0
\(655\) 2.29337 0.0896096
\(656\) 0 0
\(657\) −1.16169 −0.0453217
\(658\) 0 0
\(659\) 1.68310 0.0655642 0.0327821 0.999463i \(-0.489563\pi\)
0.0327821 + 0.999463i \(0.489563\pi\)
\(660\) 0 0
\(661\) −10.7950 −0.419878 −0.209939 0.977714i \(-0.567327\pi\)
−0.209939 + 0.977714i \(0.567327\pi\)
\(662\) 0 0
\(663\) −9.48847 −0.368502
\(664\) 0 0
\(665\) 3.09102 0.119865
\(666\) 0 0
\(667\) 1.38647 0.0536842
\(668\) 0 0
\(669\) −2.33612 −0.0903196
\(670\) 0 0
\(671\) −20.2619 −0.782204
\(672\) 0 0
\(673\) 35.5476 1.37026 0.685129 0.728422i \(-0.259746\pi\)
0.685129 + 0.728422i \(0.259746\pi\)
\(674\) 0 0
\(675\) −5.01562 −0.193051
\(676\) 0 0
\(677\) −40.1798 −1.54423 −0.772117 0.635480i \(-0.780802\pi\)
−0.772117 + 0.635480i \(0.780802\pi\)
\(678\) 0 0
\(679\) 14.8927 0.571531
\(680\) 0 0
\(681\) 28.5924 1.09566
\(682\) 0 0
\(683\) 23.7900 0.910298 0.455149 0.890415i \(-0.349586\pi\)
0.455149 + 0.890415i \(0.349586\pi\)
\(684\) 0 0
\(685\) 4.09705 0.156540
\(686\) 0 0
\(687\) 1.38641 0.0528950
\(688\) 0 0
\(689\) −29.3022 −1.11633
\(690\) 0 0
\(691\) −12.2862 −0.467388 −0.233694 0.972310i \(-0.575081\pi\)
−0.233694 + 0.972310i \(0.575081\pi\)
\(692\) 0 0
\(693\) −2.37588 −0.0902523
\(694\) 0 0
\(695\) 13.7915 0.523142
\(696\) 0 0
\(697\) 19.3521 0.733012
\(698\) 0 0
\(699\) 35.0180 1.32450
\(700\) 0 0
\(701\) −0.649217 −0.0245206 −0.0122603 0.999925i \(-0.503903\pi\)
−0.0122603 + 0.999925i \(0.503903\pi\)
\(702\) 0 0
\(703\) −6.50889 −0.245487
\(704\) 0 0
\(705\) −0.252169 −0.00949723
\(706\) 0 0
\(707\) 19.7983 0.744593
\(708\) 0 0
\(709\) 14.0672 0.528303 0.264152 0.964481i \(-0.414908\pi\)
0.264152 + 0.964481i \(0.414908\pi\)
\(710\) 0 0
\(711\) −3.24684 −0.121766
\(712\) 0 0
\(713\) −1.63729 −0.0613169
\(714\) 0 0
\(715\) 8.01141 0.299610
\(716\) 0 0
\(717\) −24.5810 −0.917995
\(718\) 0 0
\(719\) 30.0719 1.12149 0.560747 0.827987i \(-0.310514\pi\)
0.560747 + 0.827987i \(0.310514\pi\)
\(720\) 0 0
\(721\) 47.8441 1.78181
\(722\) 0 0
\(723\) 0.481919 0.0179227
\(724\) 0 0
\(725\) −3.57159 −0.132646
\(726\) 0 0
\(727\) 7.13470 0.264611 0.132306 0.991209i \(-0.457762\pi\)
0.132306 + 0.991209i \(0.457762\pi\)
\(728\) 0 0
\(729\) 25.0444 0.927572
\(730\) 0 0
\(731\) 21.8377 0.807698
\(732\) 0 0
\(733\) −6.14338 −0.226911 −0.113455 0.993543i \(-0.536192\pi\)
−0.113455 + 0.993543i \(0.536192\pi\)
\(734\) 0 0
\(735\) −7.55018 −0.278492
\(736\) 0 0
\(737\) −5.34861 −0.197019
\(738\) 0 0
\(739\) −21.9056 −0.805811 −0.402906 0.915242i \(-0.632000\pi\)
−0.402906 + 0.915242i \(0.632000\pi\)
\(740\) 0 0
\(741\) −3.59839 −0.132190
\(742\) 0 0
\(743\) −11.3938 −0.417999 −0.209000 0.977916i \(-0.567021\pi\)
−0.209000 + 0.977916i \(0.567021\pi\)
\(744\) 0 0
\(745\) 0.864421 0.0316699
\(746\) 0 0
\(747\) −2.63082 −0.0962567
\(748\) 0 0
\(749\) −34.4651 −1.25933
\(750\) 0 0
\(751\) 15.5295 0.566679 0.283339 0.959020i \(-0.408558\pi\)
0.283339 + 0.959020i \(0.408558\pi\)
\(752\) 0 0
\(753\) −14.1831 −0.516862
\(754\) 0 0
\(755\) 15.7319 0.572543
\(756\) 0 0
\(757\) 22.3814 0.813466 0.406733 0.913547i \(-0.366668\pi\)
0.406733 + 0.913547i \(0.366668\pi\)
\(758\) 0 0
\(759\) −2.54612 −0.0924185
\(760\) 0 0
\(761\) 9.29721 0.337024 0.168512 0.985700i \(-0.446104\pi\)
0.168512 + 0.985700i \(0.446104\pi\)
\(762\) 0 0
\(763\) 37.5026 1.35769
\(764\) 0 0
\(765\) −0.470005 −0.0169931
\(766\) 0 0
\(767\) 18.6551 0.673598
\(768\) 0 0
\(769\) 30.4922 1.09958 0.549789 0.835304i \(-0.314708\pi\)
0.549789 + 0.835304i \(0.314708\pi\)
\(770\) 0 0
\(771\) −13.5532 −0.488107
\(772\) 0 0
\(773\) −37.7976 −1.35948 −0.679742 0.733451i \(-0.737908\pi\)
−0.679742 + 0.733451i \(0.737908\pi\)
\(774\) 0 0
\(775\) 4.21772 0.151505
\(776\) 0 0
\(777\) 42.2388 1.51531
\(778\) 0 0
\(779\) 7.33904 0.262948
\(780\) 0 0
\(781\) −1.70235 −0.0609149
\(782\) 0 0
\(783\) 17.9137 0.640185
\(784\) 0 0
\(785\) −18.6334 −0.665055
\(786\) 0 0
\(787\) 15.0189 0.535366 0.267683 0.963507i \(-0.413742\pi\)
0.267683 + 0.963507i \(0.413742\pi\)
\(788\) 0 0
\(789\) −24.8959 −0.886319
\(790\) 0 0
\(791\) −6.52940 −0.232159
\(792\) 0 0
\(793\) 12.0490 0.427873
\(794\) 0 0
\(795\) −23.9896 −0.850825
\(796\) 0 0
\(797\) −24.9916 −0.885249 −0.442624 0.896707i \(-0.645953\pi\)
−0.442624 + 0.896707i \(0.645953\pi\)
\(798\) 0 0
\(799\) 0.343298 0.0121450
\(800\) 0 0
\(801\) −0.687915 −0.0243063
\(802\) 0 0
\(803\) −22.0698 −0.778827
\(804\) 0 0
\(805\) −1.30060 −0.0458401
\(806\) 0 0
\(807\) −22.7251 −0.799960
\(808\) 0 0
\(809\) −16.4758 −0.579258 −0.289629 0.957139i \(-0.593532\pi\)
−0.289629 + 0.957139i \(0.593532\pi\)
\(810\) 0 0
\(811\) −25.5489 −0.897142 −0.448571 0.893747i \(-0.648067\pi\)
−0.448571 + 0.893747i \(0.648067\pi\)
\(812\) 0 0
\(813\) −1.04741 −0.0367342
\(814\) 0 0
\(815\) −5.97284 −0.209219
\(816\) 0 0
\(817\) 8.28170 0.289740
\(818\) 0 0
\(819\) 1.41285 0.0493689
\(820\) 0 0
\(821\) 29.7596 1.03862 0.519308 0.854587i \(-0.326190\pi\)
0.519308 + 0.854587i \(0.326190\pi\)
\(822\) 0 0
\(823\) −30.1579 −1.05124 −0.525619 0.850720i \(-0.676166\pi\)
−0.525619 + 0.850720i \(0.676166\pi\)
\(824\) 0 0
\(825\) 6.55892 0.228352
\(826\) 0 0
\(827\) 0.0353484 0.00122918 0.000614592 1.00000i \(-0.499804\pi\)
0.000614592 1.00000i \(0.499804\pi\)
\(828\) 0 0
\(829\) 35.9544 1.24875 0.624374 0.781125i \(-0.285354\pi\)
0.624374 + 0.781125i \(0.285354\pi\)
\(830\) 0 0
\(831\) 36.1951 1.25559
\(832\) 0 0
\(833\) 10.2787 0.356135
\(834\) 0 0
\(835\) −22.8344 −0.790218
\(836\) 0 0
\(837\) −21.1545 −0.731205
\(838\) 0 0
\(839\) 5.74075 0.198193 0.0990964 0.995078i \(-0.468405\pi\)
0.0990964 + 0.995078i \(0.468405\pi\)
\(840\) 0 0
\(841\) −16.2437 −0.560128
\(842\) 0 0
\(843\) 55.6395 1.91633
\(844\) 0 0
\(845\) 8.23591 0.283324
\(846\) 0 0
\(847\) −8.28280 −0.284600
\(848\) 0 0
\(849\) 28.1536 0.966230
\(850\) 0 0
\(851\) 2.73873 0.0938824
\(852\) 0 0
\(853\) −32.8946 −1.12629 −0.563145 0.826358i \(-0.690409\pi\)
−0.563145 + 0.826358i \(0.690409\pi\)
\(854\) 0 0
\(855\) −0.178244 −0.00609581
\(856\) 0 0
\(857\) −6.92868 −0.236679 −0.118340 0.992973i \(-0.537757\pi\)
−0.118340 + 0.992973i \(0.537757\pi\)
\(858\) 0 0
\(859\) 36.4826 1.24477 0.622385 0.782711i \(-0.286164\pi\)
0.622385 + 0.782711i \(0.286164\pi\)
\(860\) 0 0
\(861\) −47.6259 −1.62309
\(862\) 0 0
\(863\) 27.6369 0.940772 0.470386 0.882461i \(-0.344115\pi\)
0.470386 + 0.882461i \(0.344115\pi\)
\(864\) 0 0
\(865\) 14.7784 0.502480
\(866\) 0 0
\(867\) −19.8027 −0.672536
\(868\) 0 0
\(869\) −61.6836 −2.09247
\(870\) 0 0
\(871\) 3.18062 0.107771
\(872\) 0 0
\(873\) −0.858791 −0.0290657
\(874\) 0 0
\(875\) 3.35040 0.113264
\(876\) 0 0
\(877\) −15.0675 −0.508794 −0.254397 0.967100i \(-0.581877\pi\)
−0.254397 + 0.967100i \(0.581877\pi\)
\(878\) 0 0
\(879\) −35.3933 −1.19379
\(880\) 0 0
\(881\) 9.22760 0.310886 0.155443 0.987845i \(-0.450320\pi\)
0.155443 + 0.987845i \(0.450320\pi\)
\(882\) 0 0
\(883\) −40.7426 −1.37110 −0.685549 0.728026i \(-0.740438\pi\)
−0.685549 + 0.728026i \(0.740438\pi\)
\(884\) 0 0
\(885\) 15.2729 0.513393
\(886\) 0 0
\(887\) 29.9799 1.00663 0.503313 0.864104i \(-0.332114\pi\)
0.503313 + 0.864104i \(0.332114\pi\)
\(888\) 0 0
\(889\) 9.78821 0.328286
\(890\) 0 0
\(891\) −35.0244 −1.17336
\(892\) 0 0
\(893\) 0.130192 0.00435670
\(894\) 0 0
\(895\) 2.50945 0.0838816
\(896\) 0 0
\(897\) 1.51409 0.0505538
\(898\) 0 0
\(899\) −15.0640 −0.502412
\(900\) 0 0
\(901\) 32.6591 1.08803
\(902\) 0 0
\(903\) −53.7433 −1.78846
\(904\) 0 0
\(905\) −10.6277 −0.353276
\(906\) 0 0
\(907\) 1.07576 0.0357200 0.0178600 0.999840i \(-0.494315\pi\)
0.0178600 + 0.999840i \(0.494315\pi\)
\(908\) 0 0
\(909\) −1.14167 −0.0378669
\(910\) 0 0
\(911\) −0.536627 −0.0177792 −0.00888962 0.999960i \(-0.502830\pi\)
−0.00888962 + 0.999960i \(0.502830\pi\)
\(912\) 0 0
\(913\) −49.9806 −1.65412
\(914\) 0 0
\(915\) 9.86450 0.326110
\(916\) 0 0
\(917\) 7.68372 0.253739
\(918\) 0 0
\(919\) 37.3306 1.23142 0.615711 0.787972i \(-0.288869\pi\)
0.615711 + 0.787972i \(0.288869\pi\)
\(920\) 0 0
\(921\) 39.0915 1.28811
\(922\) 0 0
\(923\) 1.01232 0.0333211
\(924\) 0 0
\(925\) −7.05508 −0.231969
\(926\) 0 0
\(927\) −2.75894 −0.0906154
\(928\) 0 0
\(929\) 32.9873 1.08228 0.541139 0.840933i \(-0.317993\pi\)
0.541139 + 0.840933i \(0.317993\pi\)
\(930\) 0 0
\(931\) 3.89806 0.127754
\(932\) 0 0
\(933\) 27.6152 0.904083
\(934\) 0 0
\(935\) −8.92919 −0.292016
\(936\) 0 0
\(937\) −48.4870 −1.58400 −0.792000 0.610521i \(-0.790960\pi\)
−0.792000 + 0.610521i \(0.790960\pi\)
\(938\) 0 0
\(939\) −25.7663 −0.840850
\(940\) 0 0
\(941\) −21.2876 −0.693956 −0.346978 0.937873i \(-0.612792\pi\)
−0.346978 + 0.937873i \(0.612792\pi\)
\(942\) 0 0
\(943\) −3.08803 −0.100560
\(944\) 0 0
\(945\) −16.8043 −0.546644
\(946\) 0 0
\(947\) 21.3715 0.694481 0.347241 0.937776i \(-0.387119\pi\)
0.347241 + 0.937776i \(0.387119\pi\)
\(948\) 0 0
\(949\) 13.1241 0.426026
\(950\) 0 0
\(951\) 57.3153 1.85858
\(952\) 0 0
\(953\) −52.9098 −1.71391 −0.856957 0.515387i \(-0.827648\pi\)
−0.856957 + 0.515387i \(0.827648\pi\)
\(954\) 0 0
\(955\) 0.925727 0.0299558
\(956\) 0 0
\(957\) −23.4258 −0.757248
\(958\) 0 0
\(959\) 13.7268 0.443260
\(960\) 0 0
\(961\) −13.2108 −0.426156
\(962\) 0 0
\(963\) 1.98743 0.0640441
\(964\) 0 0
\(965\) −2.29130 −0.0737594
\(966\) 0 0
\(967\) 3.65921 0.117672 0.0588362 0.998268i \(-0.481261\pi\)
0.0588362 + 0.998268i \(0.481261\pi\)
\(968\) 0 0
\(969\) 4.01062 0.128840
\(970\) 0 0
\(971\) −8.03188 −0.257755 −0.128878 0.991661i \(-0.541137\pi\)
−0.128878 + 0.991661i \(0.541137\pi\)
\(972\) 0 0
\(973\) 46.2071 1.48133
\(974\) 0 0
\(975\) −3.90034 −0.124911
\(976\) 0 0
\(977\) −21.0646 −0.673916 −0.336958 0.941520i \(-0.609398\pi\)
−0.336958 + 0.941520i \(0.609398\pi\)
\(978\) 0 0
\(979\) −13.0691 −0.417689
\(980\) 0 0
\(981\) −2.16259 −0.0690463
\(982\) 0 0
\(983\) −27.1729 −0.866680 −0.433340 0.901231i \(-0.642665\pi\)
−0.433340 + 0.901231i \(0.642665\pi\)
\(984\) 0 0
\(985\) 23.9368 0.762688
\(986\) 0 0
\(987\) −0.844866 −0.0268924
\(988\) 0 0
\(989\) −3.48467 −0.110806
\(990\) 0 0
\(991\) 1.70561 0.0541805 0.0270902 0.999633i \(-0.491376\pi\)
0.0270902 + 0.999633i \(0.491376\pi\)
\(992\) 0 0
\(993\) −6.09344 −0.193369
\(994\) 0 0
\(995\) 22.4764 0.712550
\(996\) 0 0
\(997\) 58.1166 1.84057 0.920285 0.391248i \(-0.127956\pi\)
0.920285 + 0.391248i \(0.127956\pi\)
\(998\) 0 0
\(999\) 35.3856 1.11955
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.c.1.21 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8020.2.a.c.1.21 28 1.1 even 1 trivial