Properties

Label 4012.2.a.j.1.13
Level $4012$
Weight $2$
Character 4012.1
Self dual yes
Analytic conductor $32.036$
Analytic rank $0$
Dimension $21$
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] = [4012,2,Mod(1,4012)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4012, 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("4012.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4012.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: \(32.0359812909\)
Analytic rank: \(0\)
Dimension: \(21\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Character \(\chi\) \(=\) 4012.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.967697 q^{3} -3.19923 q^{5} +3.75414 q^{7} -2.06356 q^{9} +O(q^{10})\) \(q+0.967697 q^{3} -3.19923 q^{5} +3.75414 q^{7} -2.06356 q^{9} +5.41425 q^{11} -0.0945861 q^{13} -3.09588 q^{15} +1.00000 q^{17} +0.489351 q^{19} +3.63287 q^{21} +6.45421 q^{23} +5.23506 q^{25} -4.89999 q^{27} -6.50307 q^{29} +4.96708 q^{31} +5.23935 q^{33} -12.0103 q^{35} -3.39405 q^{37} -0.0915307 q^{39} +8.13947 q^{41} -3.71512 q^{43} +6.60181 q^{45} -9.50432 q^{47} +7.09357 q^{49} +0.967697 q^{51} +6.30050 q^{53} -17.3214 q^{55} +0.473543 q^{57} +1.00000 q^{59} -9.32942 q^{61} -7.74690 q^{63} +0.302602 q^{65} +6.16771 q^{67} +6.24572 q^{69} +8.60812 q^{71} +6.03033 q^{73} +5.06595 q^{75} +20.3259 q^{77} -7.60416 q^{79} +1.44898 q^{81} +8.92704 q^{83} -3.19923 q^{85} -6.29300 q^{87} +6.79783 q^{89} -0.355089 q^{91} +4.80662 q^{93} -1.56554 q^{95} +5.47723 q^{97} -11.1726 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) \) 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
\(3\) 0.967697 0.558700 0.279350 0.960189i \(-0.409881\pi\)
0.279350 + 0.960189i \(0.409881\pi\)
\(4\) 0 0
\(5\) −3.19923 −1.43074 −0.715369 0.698747i \(-0.753741\pi\)
−0.715369 + 0.698747i \(0.753741\pi\)
\(6\) 0 0
\(7\) 3.75414 1.41893 0.709466 0.704740i \(-0.248936\pi\)
0.709466 + 0.704740i \(0.248936\pi\)
\(8\) 0 0
\(9\) −2.06356 −0.687854
\(10\) 0 0
\(11\) 5.41425 1.63246 0.816229 0.577728i \(-0.196061\pi\)
0.816229 + 0.577728i \(0.196061\pi\)
\(12\) 0 0
\(13\) −0.0945861 −0.0262335 −0.0131167 0.999914i \(-0.504175\pi\)
−0.0131167 + 0.999914i \(0.504175\pi\)
\(14\) 0 0
\(15\) −3.09588 −0.799353
\(16\) 0 0
\(17\) 1.00000 0.242536
\(18\) 0 0
\(19\) 0.489351 0.112265 0.0561324 0.998423i \(-0.482123\pi\)
0.0561324 + 0.998423i \(0.482123\pi\)
\(20\) 0 0
\(21\) 3.63287 0.792757
\(22\) 0 0
\(23\) 6.45421 1.34580 0.672898 0.739735i \(-0.265049\pi\)
0.672898 + 0.739735i \(0.265049\pi\)
\(24\) 0 0
\(25\) 5.23506 1.04701
\(26\) 0 0
\(27\) −4.89999 −0.943004
\(28\) 0 0
\(29\) −6.50307 −1.20759 −0.603795 0.797140i \(-0.706345\pi\)
−0.603795 + 0.797140i \(0.706345\pi\)
\(30\) 0 0
\(31\) 4.96708 0.892113 0.446057 0.895005i \(-0.352828\pi\)
0.446057 + 0.895005i \(0.352828\pi\)
\(32\) 0 0
\(33\) 5.23935 0.912055
\(34\) 0 0
\(35\) −12.0103 −2.03012
\(36\) 0 0
\(37\) −3.39405 −0.557979 −0.278990 0.960294i \(-0.589999\pi\)
−0.278990 + 0.960294i \(0.589999\pi\)
\(38\) 0 0
\(39\) −0.0915307 −0.0146566
\(40\) 0 0
\(41\) 8.13947 1.27117 0.635585 0.772031i \(-0.280759\pi\)
0.635585 + 0.772031i \(0.280759\pi\)
\(42\) 0 0
\(43\) −3.71512 −0.566551 −0.283275 0.959039i \(-0.591421\pi\)
−0.283275 + 0.959039i \(0.591421\pi\)
\(44\) 0 0
\(45\) 6.60181 0.984139
\(46\) 0 0
\(47\) −9.50432 −1.38635 −0.693174 0.720770i \(-0.743788\pi\)
−0.693174 + 0.720770i \(0.743788\pi\)
\(48\) 0 0
\(49\) 7.09357 1.01337
\(50\) 0 0
\(51\) 0.967697 0.135505
\(52\) 0 0
\(53\) 6.30050 0.865441 0.432720 0.901528i \(-0.357554\pi\)
0.432720 + 0.901528i \(0.357554\pi\)
\(54\) 0 0
\(55\) −17.3214 −2.33562
\(56\) 0 0
\(57\) 0.473543 0.0627223
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) −9.32942 −1.19451 −0.597255 0.802051i \(-0.703742\pi\)
−0.597255 + 0.802051i \(0.703742\pi\)
\(62\) 0 0
\(63\) −7.74690 −0.976018
\(64\) 0 0
\(65\) 0.302602 0.0375332
\(66\) 0 0
\(67\) 6.16771 0.753505 0.376753 0.926314i \(-0.377041\pi\)
0.376753 + 0.926314i \(0.377041\pi\)
\(68\) 0 0
\(69\) 6.24572 0.751896
\(70\) 0 0
\(71\) 8.60812 1.02160 0.510798 0.859701i \(-0.329350\pi\)
0.510798 + 0.859701i \(0.329350\pi\)
\(72\) 0 0
\(73\) 6.03033 0.705796 0.352898 0.935662i \(-0.385196\pi\)
0.352898 + 0.935662i \(0.385196\pi\)
\(74\) 0 0
\(75\) 5.06595 0.584965
\(76\) 0 0
\(77\) 20.3259 2.31635
\(78\) 0 0
\(79\) −7.60416 −0.855534 −0.427767 0.903889i \(-0.640700\pi\)
−0.427767 + 0.903889i \(0.640700\pi\)
\(80\) 0 0
\(81\) 1.44898 0.160998
\(82\) 0 0
\(83\) 8.92704 0.979870 0.489935 0.871759i \(-0.337021\pi\)
0.489935 + 0.871759i \(0.337021\pi\)
\(84\) 0 0
\(85\) −3.19923 −0.347005
\(86\) 0 0
\(87\) −6.29300 −0.674680
\(88\) 0 0
\(89\) 6.79783 0.720569 0.360284 0.932843i \(-0.382680\pi\)
0.360284 + 0.932843i \(0.382680\pi\)
\(90\) 0 0
\(91\) −0.355089 −0.0372235
\(92\) 0 0
\(93\) 4.80662 0.498424
\(94\) 0 0
\(95\) −1.56554 −0.160621
\(96\) 0 0
\(97\) 5.47723 0.556129 0.278064 0.960562i \(-0.410307\pi\)
0.278064 + 0.960562i \(0.410307\pi\)
\(98\) 0 0
\(99\) −11.1726 −1.12289
\(100\) 0 0
\(101\) 19.2743 1.91787 0.958935 0.283627i \(-0.0915378\pi\)
0.958935 + 0.283627i \(0.0915378\pi\)
\(102\) 0 0
\(103\) −4.85323 −0.478203 −0.239101 0.970995i \(-0.576853\pi\)
−0.239101 + 0.970995i \(0.576853\pi\)
\(104\) 0 0
\(105\) −11.6224 −1.13423
\(106\) 0 0
\(107\) 4.24811 0.410680 0.205340 0.978691i \(-0.434170\pi\)
0.205340 + 0.978691i \(0.434170\pi\)
\(108\) 0 0
\(109\) −11.7879 −1.12908 −0.564538 0.825407i \(-0.690946\pi\)
−0.564538 + 0.825407i \(0.690946\pi\)
\(110\) 0 0
\(111\) −3.28442 −0.311743
\(112\) 0 0
\(113\) 5.23556 0.492520 0.246260 0.969204i \(-0.420798\pi\)
0.246260 + 0.969204i \(0.420798\pi\)
\(114\) 0 0
\(115\) −20.6485 −1.92548
\(116\) 0 0
\(117\) 0.195184 0.0180448
\(118\) 0 0
\(119\) 3.75414 0.344141
\(120\) 0 0
\(121\) 18.3141 1.66492
\(122\) 0 0
\(123\) 7.87654 0.710203
\(124\) 0 0
\(125\) −0.751995 −0.0672605
\(126\) 0 0
\(127\) 13.9340 1.23644 0.618221 0.786004i \(-0.287854\pi\)
0.618221 + 0.786004i \(0.287854\pi\)
\(128\) 0 0
\(129\) −3.59511 −0.316532
\(130\) 0 0
\(131\) 2.24344 0.196010 0.0980052 0.995186i \(-0.468754\pi\)
0.0980052 + 0.995186i \(0.468754\pi\)
\(132\) 0 0
\(133\) 1.83709 0.159296
\(134\) 0 0
\(135\) 15.6762 1.34919
\(136\) 0 0
\(137\) 6.45912 0.551839 0.275920 0.961181i \(-0.411018\pi\)
0.275920 + 0.961181i \(0.411018\pi\)
\(138\) 0 0
\(139\) −11.8781 −1.00749 −0.503743 0.863853i \(-0.668044\pi\)
−0.503743 + 0.863853i \(0.668044\pi\)
\(140\) 0 0
\(141\) −9.19730 −0.774553
\(142\) 0 0
\(143\) −0.512113 −0.0428250
\(144\) 0 0
\(145\) 20.8048 1.72774
\(146\) 0 0
\(147\) 6.86443 0.566168
\(148\) 0 0
\(149\) 2.38363 0.195275 0.0976375 0.995222i \(-0.468871\pi\)
0.0976375 + 0.995222i \(0.468871\pi\)
\(150\) 0 0
\(151\) 2.52333 0.205346 0.102673 0.994715i \(-0.467260\pi\)
0.102673 + 0.994715i \(0.467260\pi\)
\(152\) 0 0
\(153\) −2.06356 −0.166829
\(154\) 0 0
\(155\) −15.8908 −1.27638
\(156\) 0 0
\(157\) 12.0785 0.963970 0.481985 0.876179i \(-0.339916\pi\)
0.481985 + 0.876179i \(0.339916\pi\)
\(158\) 0 0
\(159\) 6.09698 0.483522
\(160\) 0 0
\(161\) 24.2300 1.90959
\(162\) 0 0
\(163\) −1.98874 −0.155770 −0.0778850 0.996962i \(-0.524817\pi\)
−0.0778850 + 0.996962i \(0.524817\pi\)
\(164\) 0 0
\(165\) −16.7619 −1.30491
\(166\) 0 0
\(167\) 4.09011 0.316502 0.158251 0.987399i \(-0.449414\pi\)
0.158251 + 0.987399i \(0.449414\pi\)
\(168\) 0 0
\(169\) −12.9911 −0.999312
\(170\) 0 0
\(171\) −1.00981 −0.0772218
\(172\) 0 0
\(173\) −10.4486 −0.794389 −0.397194 0.917735i \(-0.630016\pi\)
−0.397194 + 0.917735i \(0.630016\pi\)
\(174\) 0 0
\(175\) 19.6531 1.48564
\(176\) 0 0
\(177\) 0.967697 0.0727366
\(178\) 0 0
\(179\) 7.99677 0.597707 0.298853 0.954299i \(-0.403396\pi\)
0.298853 + 0.954299i \(0.403396\pi\)
\(180\) 0 0
\(181\) 15.9418 1.18494 0.592472 0.805591i \(-0.298152\pi\)
0.592472 + 0.805591i \(0.298152\pi\)
\(182\) 0 0
\(183\) −9.02806 −0.667373
\(184\) 0 0
\(185\) 10.8583 0.798322
\(186\) 0 0
\(187\) 5.41425 0.395929
\(188\) 0 0
\(189\) −18.3953 −1.33806
\(190\) 0 0
\(191\) 10.8272 0.783429 0.391715 0.920087i \(-0.371882\pi\)
0.391715 + 0.920087i \(0.371882\pi\)
\(192\) 0 0
\(193\) −9.34664 −0.672786 −0.336393 0.941722i \(-0.609207\pi\)
−0.336393 + 0.941722i \(0.609207\pi\)
\(194\) 0 0
\(195\) 0.292827 0.0209698
\(196\) 0 0
\(197\) −17.4269 −1.24161 −0.620807 0.783964i \(-0.713195\pi\)
−0.620807 + 0.783964i \(0.713195\pi\)
\(198\) 0 0
\(199\) 7.44086 0.527469 0.263734 0.964595i \(-0.415046\pi\)
0.263734 + 0.964595i \(0.415046\pi\)
\(200\) 0 0
\(201\) 5.96847 0.420984
\(202\) 0 0
\(203\) −24.4134 −1.71349
\(204\) 0 0
\(205\) −26.0400 −1.81871
\(206\) 0 0
\(207\) −13.3187 −0.925711
\(208\) 0 0
\(209\) 2.64947 0.183268
\(210\) 0 0
\(211\) −15.0756 −1.03785 −0.518923 0.854821i \(-0.673667\pi\)
−0.518923 + 0.854821i \(0.673667\pi\)
\(212\) 0 0
\(213\) 8.33005 0.570766
\(214\) 0 0
\(215\) 11.8855 0.810585
\(216\) 0 0
\(217\) 18.6471 1.26585
\(218\) 0 0
\(219\) 5.83553 0.394329
\(220\) 0 0
\(221\) −0.0945861 −0.00636255
\(222\) 0 0
\(223\) 19.1230 1.28057 0.640286 0.768137i \(-0.278816\pi\)
0.640286 + 0.768137i \(0.278816\pi\)
\(224\) 0 0
\(225\) −10.8029 −0.720191
\(226\) 0 0
\(227\) 4.26372 0.282993 0.141496 0.989939i \(-0.454809\pi\)
0.141496 + 0.989939i \(0.454809\pi\)
\(228\) 0 0
\(229\) 26.2755 1.73634 0.868169 0.496269i \(-0.165297\pi\)
0.868169 + 0.496269i \(0.165297\pi\)
\(230\) 0 0
\(231\) 19.6693 1.29414
\(232\) 0 0
\(233\) 18.9285 1.24005 0.620025 0.784582i \(-0.287122\pi\)
0.620025 + 0.784582i \(0.287122\pi\)
\(234\) 0 0
\(235\) 30.4065 1.98350
\(236\) 0 0
\(237\) −7.35852 −0.477987
\(238\) 0 0
\(239\) 5.77782 0.373736 0.186868 0.982385i \(-0.440166\pi\)
0.186868 + 0.982385i \(0.440166\pi\)
\(240\) 0 0
\(241\) −9.15928 −0.590001 −0.295001 0.955497i \(-0.595320\pi\)
−0.295001 + 0.955497i \(0.595320\pi\)
\(242\) 0 0
\(243\) 16.1022 1.03295
\(244\) 0 0
\(245\) −22.6939 −1.44986
\(246\) 0 0
\(247\) −0.0462858 −0.00294509
\(248\) 0 0
\(249\) 8.63867 0.547453
\(250\) 0 0
\(251\) −2.28759 −0.144392 −0.0721958 0.997390i \(-0.523001\pi\)
−0.0721958 + 0.997390i \(0.523001\pi\)
\(252\) 0 0
\(253\) 34.9447 2.19695
\(254\) 0 0
\(255\) −3.09588 −0.193872
\(256\) 0 0
\(257\) 10.8830 0.678863 0.339432 0.940631i \(-0.389765\pi\)
0.339432 + 0.940631i \(0.389765\pi\)
\(258\) 0 0
\(259\) −12.7418 −0.791734
\(260\) 0 0
\(261\) 13.4195 0.830645
\(262\) 0 0
\(263\) 6.24016 0.384785 0.192393 0.981318i \(-0.438375\pi\)
0.192393 + 0.981318i \(0.438375\pi\)
\(264\) 0 0
\(265\) −20.1567 −1.23822
\(266\) 0 0
\(267\) 6.57824 0.402582
\(268\) 0 0
\(269\) 15.8170 0.964377 0.482188 0.876068i \(-0.339842\pi\)
0.482188 + 0.876068i \(0.339842\pi\)
\(270\) 0 0
\(271\) −21.1698 −1.28597 −0.642987 0.765877i \(-0.722305\pi\)
−0.642987 + 0.765877i \(0.722305\pi\)
\(272\) 0 0
\(273\) −0.343619 −0.0207968
\(274\) 0 0
\(275\) 28.3439 1.70920
\(276\) 0 0
\(277\) 30.0769 1.80715 0.903573 0.428433i \(-0.140934\pi\)
0.903573 + 0.428433i \(0.140934\pi\)
\(278\) 0 0
\(279\) −10.2499 −0.613644
\(280\) 0 0
\(281\) −13.2552 −0.790737 −0.395369 0.918523i \(-0.629383\pi\)
−0.395369 + 0.918523i \(0.629383\pi\)
\(282\) 0 0
\(283\) −9.59404 −0.570306 −0.285153 0.958482i \(-0.592044\pi\)
−0.285153 + 0.958482i \(0.592044\pi\)
\(284\) 0 0
\(285\) −1.51497 −0.0897392
\(286\) 0 0
\(287\) 30.5567 1.80370
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 5.30030 0.310709
\(292\) 0 0
\(293\) −30.0303 −1.75439 −0.877195 0.480135i \(-0.840588\pi\)
−0.877195 + 0.480135i \(0.840588\pi\)
\(294\) 0 0
\(295\) −3.19923 −0.186266
\(296\) 0 0
\(297\) −26.5298 −1.53942
\(298\) 0 0
\(299\) −0.610478 −0.0353049
\(300\) 0 0
\(301\) −13.9471 −0.803897
\(302\) 0 0
\(303\) 18.6517 1.07151
\(304\) 0 0
\(305\) 29.8469 1.70903
\(306\) 0 0
\(307\) −14.9818 −0.855054 −0.427527 0.904002i \(-0.640615\pi\)
−0.427527 + 0.904002i \(0.640615\pi\)
\(308\) 0 0
\(309\) −4.69645 −0.267172
\(310\) 0 0
\(311\) 18.3729 1.04183 0.520915 0.853609i \(-0.325591\pi\)
0.520915 + 0.853609i \(0.325591\pi\)
\(312\) 0 0
\(313\) 33.4922 1.89309 0.946544 0.322574i \(-0.104548\pi\)
0.946544 + 0.322574i \(0.104548\pi\)
\(314\) 0 0
\(315\) 24.7841 1.39643
\(316\) 0 0
\(317\) −20.0899 −1.12836 −0.564180 0.825652i \(-0.690808\pi\)
−0.564180 + 0.825652i \(0.690808\pi\)
\(318\) 0 0
\(319\) −35.2092 −1.97134
\(320\) 0 0
\(321\) 4.11088 0.229447
\(322\) 0 0
\(323\) 0.489351 0.0272282
\(324\) 0 0
\(325\) −0.495163 −0.0274667
\(326\) 0 0
\(327\) −11.4071 −0.630815
\(328\) 0 0
\(329\) −35.6806 −1.96713
\(330\) 0 0
\(331\) 3.08344 0.169481 0.0847405 0.996403i \(-0.472994\pi\)
0.0847405 + 0.996403i \(0.472994\pi\)
\(332\) 0 0
\(333\) 7.00384 0.383808
\(334\) 0 0
\(335\) −19.7319 −1.07807
\(336\) 0 0
\(337\) −27.0227 −1.47202 −0.736009 0.676971i \(-0.763292\pi\)
−0.736009 + 0.676971i \(0.763292\pi\)
\(338\) 0 0
\(339\) 5.06644 0.275171
\(340\) 0 0
\(341\) 26.8930 1.45634
\(342\) 0 0
\(343\) 0.351279 0.0189673
\(344\) 0 0
\(345\) −19.9815 −1.07577
\(346\) 0 0
\(347\) −34.8592 −1.87134 −0.935669 0.352879i \(-0.885203\pi\)
−0.935669 + 0.352879i \(0.885203\pi\)
\(348\) 0 0
\(349\) −17.1338 −0.917152 −0.458576 0.888655i \(-0.651640\pi\)
−0.458576 + 0.888655i \(0.651640\pi\)
\(350\) 0 0
\(351\) 0.463471 0.0247383
\(352\) 0 0
\(353\) 10.5647 0.562303 0.281151 0.959663i \(-0.409284\pi\)
0.281151 + 0.959663i \(0.409284\pi\)
\(354\) 0 0
\(355\) −27.5393 −1.46164
\(356\) 0 0
\(357\) 3.63287 0.192272
\(358\) 0 0
\(359\) −30.0374 −1.58531 −0.792657 0.609668i \(-0.791303\pi\)
−0.792657 + 0.609668i \(0.791303\pi\)
\(360\) 0 0
\(361\) −18.7605 −0.987397
\(362\) 0 0
\(363\) 17.7225 0.930191
\(364\) 0 0
\(365\) −19.2924 −1.00981
\(366\) 0 0
\(367\) −13.3908 −0.698992 −0.349496 0.936938i \(-0.613647\pi\)
−0.349496 + 0.936938i \(0.613647\pi\)
\(368\) 0 0
\(369\) −16.7963 −0.874380
\(370\) 0 0
\(371\) 23.6530 1.22800
\(372\) 0 0
\(373\) −0.587075 −0.0303976 −0.0151988 0.999884i \(-0.504838\pi\)
−0.0151988 + 0.999884i \(0.504838\pi\)
\(374\) 0 0
\(375\) −0.727703 −0.0375784
\(376\) 0 0
\(377\) 0.615100 0.0316792
\(378\) 0 0
\(379\) 17.3833 0.892921 0.446461 0.894803i \(-0.352684\pi\)
0.446461 + 0.894803i \(0.352684\pi\)
\(380\) 0 0
\(381\) 13.4839 0.690801
\(382\) 0 0
\(383\) −21.7571 −1.11174 −0.555868 0.831271i \(-0.687614\pi\)
−0.555868 + 0.831271i \(0.687614\pi\)
\(384\) 0 0
\(385\) −65.0270 −3.31408
\(386\) 0 0
\(387\) 7.66638 0.389704
\(388\) 0 0
\(389\) 2.95005 0.149573 0.0747867 0.997200i \(-0.476172\pi\)
0.0747867 + 0.997200i \(0.476172\pi\)
\(390\) 0 0
\(391\) 6.45421 0.326403
\(392\) 0 0
\(393\) 2.17097 0.109511
\(394\) 0 0
\(395\) 24.3274 1.22405
\(396\) 0 0
\(397\) 7.89442 0.396209 0.198105 0.980181i \(-0.436521\pi\)
0.198105 + 0.980181i \(0.436521\pi\)
\(398\) 0 0
\(399\) 1.77775 0.0889987
\(400\) 0 0
\(401\) 39.3098 1.96304 0.981519 0.191363i \(-0.0612908\pi\)
0.981519 + 0.191363i \(0.0612908\pi\)
\(402\) 0 0
\(403\) −0.469816 −0.0234032
\(404\) 0 0
\(405\) −4.63561 −0.230345
\(406\) 0 0
\(407\) −18.3763 −0.910877
\(408\) 0 0
\(409\) 15.8331 0.782895 0.391448 0.920200i \(-0.371974\pi\)
0.391448 + 0.920200i \(0.371974\pi\)
\(410\) 0 0
\(411\) 6.25047 0.308313
\(412\) 0 0
\(413\) 3.75414 0.184729
\(414\) 0 0
\(415\) −28.5596 −1.40194
\(416\) 0 0
\(417\) −11.4944 −0.562883
\(418\) 0 0
\(419\) 25.3553 1.23869 0.619344 0.785120i \(-0.287399\pi\)
0.619344 + 0.785120i \(0.287399\pi\)
\(420\) 0 0
\(421\) −31.4296 −1.53179 −0.765894 0.642967i \(-0.777703\pi\)
−0.765894 + 0.642967i \(0.777703\pi\)
\(422\) 0 0
\(423\) 19.6128 0.953605
\(424\) 0 0
\(425\) 5.23506 0.253937
\(426\) 0 0
\(427\) −35.0240 −1.69493
\(428\) 0 0
\(429\) −0.495570 −0.0239263
\(430\) 0 0
\(431\) 21.6013 1.04050 0.520249 0.854015i \(-0.325839\pi\)
0.520249 + 0.854015i \(0.325839\pi\)
\(432\) 0 0
\(433\) −5.07192 −0.243741 −0.121871 0.992546i \(-0.538889\pi\)
−0.121871 + 0.992546i \(0.538889\pi\)
\(434\) 0 0
\(435\) 20.1327 0.965291
\(436\) 0 0
\(437\) 3.15837 0.151085
\(438\) 0 0
\(439\) −14.9908 −0.715470 −0.357735 0.933823i \(-0.616451\pi\)
−0.357735 + 0.933823i \(0.616451\pi\)
\(440\) 0 0
\(441\) −14.6380 −0.697049
\(442\) 0 0
\(443\) 10.2757 0.488211 0.244106 0.969749i \(-0.421506\pi\)
0.244106 + 0.969749i \(0.421506\pi\)
\(444\) 0 0
\(445\) −21.7478 −1.03094
\(446\) 0 0
\(447\) 2.30664 0.109100
\(448\) 0 0
\(449\) −33.0309 −1.55882 −0.779411 0.626513i \(-0.784482\pi\)
−0.779411 + 0.626513i \(0.784482\pi\)
\(450\) 0 0
\(451\) 44.0691 2.07513
\(452\) 0 0
\(453\) 2.44182 0.114727
\(454\) 0 0
\(455\) 1.13601 0.0532571
\(456\) 0 0
\(457\) −6.19815 −0.289937 −0.144969 0.989436i \(-0.546308\pi\)
−0.144969 + 0.989436i \(0.546308\pi\)
\(458\) 0 0
\(459\) −4.89999 −0.228712
\(460\) 0 0
\(461\) −21.7754 −1.01418 −0.507091 0.861892i \(-0.669279\pi\)
−0.507091 + 0.861892i \(0.669279\pi\)
\(462\) 0 0
\(463\) −20.5267 −0.953956 −0.476978 0.878915i \(-0.658268\pi\)
−0.476978 + 0.878915i \(0.658268\pi\)
\(464\) 0 0
\(465\) −15.3775 −0.713114
\(466\) 0 0
\(467\) 6.23887 0.288700 0.144350 0.989527i \(-0.453891\pi\)
0.144350 + 0.989527i \(0.453891\pi\)
\(468\) 0 0
\(469\) 23.1544 1.06917
\(470\) 0 0
\(471\) 11.6883 0.538570
\(472\) 0 0
\(473\) −20.1146 −0.924870
\(474\) 0 0
\(475\) 2.56178 0.117542
\(476\) 0 0
\(477\) −13.0015 −0.595297
\(478\) 0 0
\(479\) 16.5147 0.754577 0.377289 0.926096i \(-0.376856\pi\)
0.377289 + 0.926096i \(0.376856\pi\)
\(480\) 0 0
\(481\) 0.321030 0.0146377
\(482\) 0 0
\(483\) 23.4473 1.06689
\(484\) 0 0
\(485\) −17.5229 −0.795674
\(486\) 0 0
\(487\) −28.1432 −1.27529 −0.637646 0.770330i \(-0.720092\pi\)
−0.637646 + 0.770330i \(0.720092\pi\)
\(488\) 0 0
\(489\) −1.92450 −0.0870287
\(490\) 0 0
\(491\) 42.9324 1.93751 0.968757 0.248013i \(-0.0797774\pi\)
0.968757 + 0.248013i \(0.0797774\pi\)
\(492\) 0 0
\(493\) −6.50307 −0.292883
\(494\) 0 0
\(495\) 35.7438 1.60657
\(496\) 0 0
\(497\) 32.3161 1.44957
\(498\) 0 0
\(499\) 13.9294 0.623565 0.311783 0.950153i \(-0.399074\pi\)
0.311783 + 0.950153i \(0.399074\pi\)
\(500\) 0 0
\(501\) 3.95799 0.176830
\(502\) 0 0
\(503\) −38.3172 −1.70848 −0.854240 0.519879i \(-0.825977\pi\)
−0.854240 + 0.519879i \(0.825977\pi\)
\(504\) 0 0
\(505\) −61.6630 −2.74397
\(506\) 0 0
\(507\) −12.5714 −0.558316
\(508\) 0 0
\(509\) −23.1595 −1.02653 −0.513263 0.858231i \(-0.671564\pi\)
−0.513263 + 0.858231i \(0.671564\pi\)
\(510\) 0 0
\(511\) 22.6387 1.00148
\(512\) 0 0
\(513\) −2.39782 −0.105866
\(514\) 0 0
\(515\) 15.5266 0.684183
\(516\) 0 0
\(517\) −51.4588 −2.26315
\(518\) 0 0
\(519\) −10.1110 −0.443825
\(520\) 0 0
\(521\) −34.4628 −1.50984 −0.754921 0.655815i \(-0.772325\pi\)
−0.754921 + 0.655815i \(0.772325\pi\)
\(522\) 0 0
\(523\) −14.8131 −0.647730 −0.323865 0.946103i \(-0.604982\pi\)
−0.323865 + 0.946103i \(0.604982\pi\)
\(524\) 0 0
\(525\) 19.0183 0.830026
\(526\) 0 0
\(527\) 4.96708 0.216369
\(528\) 0 0
\(529\) 18.6568 0.811166
\(530\) 0 0
\(531\) −2.06356 −0.0895510
\(532\) 0 0
\(533\) −0.769880 −0.0333472
\(534\) 0 0
\(535\) −13.5907 −0.587576
\(536\) 0 0
\(537\) 7.73845 0.333939
\(538\) 0 0
\(539\) 38.4064 1.65428
\(540\) 0 0
\(541\) −34.9944 −1.50452 −0.752262 0.658864i \(-0.771037\pi\)
−0.752262 + 0.658864i \(0.771037\pi\)
\(542\) 0 0
\(543\) 15.4268 0.662028
\(544\) 0 0
\(545\) 37.7122 1.61541
\(546\) 0 0
\(547\) 2.90475 0.124198 0.0620991 0.998070i \(-0.480221\pi\)
0.0620991 + 0.998070i \(0.480221\pi\)
\(548\) 0 0
\(549\) 19.2519 0.821649
\(550\) 0 0
\(551\) −3.18228 −0.135570
\(552\) 0 0
\(553\) −28.5471 −1.21395
\(554\) 0 0
\(555\) 10.5076 0.446022
\(556\) 0 0
\(557\) −46.7718 −1.98179 −0.990893 0.134651i \(-0.957009\pi\)
−0.990893 + 0.134651i \(0.957009\pi\)
\(558\) 0 0
\(559\) 0.351399 0.0148626
\(560\) 0 0
\(561\) 5.23935 0.221206
\(562\) 0 0
\(563\) 1.56674 0.0660303 0.0330152 0.999455i \(-0.489489\pi\)
0.0330152 + 0.999455i \(0.489489\pi\)
\(564\) 0 0
\(565\) −16.7498 −0.704668
\(566\) 0 0
\(567\) 5.43967 0.228445
\(568\) 0 0
\(569\) 34.1198 1.43037 0.715187 0.698933i \(-0.246341\pi\)
0.715187 + 0.698933i \(0.246341\pi\)
\(570\) 0 0
\(571\) 42.4531 1.77661 0.888303 0.459257i \(-0.151884\pi\)
0.888303 + 0.459257i \(0.151884\pi\)
\(572\) 0 0
\(573\) 10.4775 0.437702
\(574\) 0 0
\(575\) 33.7881 1.40906
\(576\) 0 0
\(577\) 8.93398 0.371927 0.185963 0.982557i \(-0.440459\pi\)
0.185963 + 0.982557i \(0.440459\pi\)
\(578\) 0 0
\(579\) −9.04471 −0.375885
\(580\) 0 0
\(581\) 33.5134 1.39037
\(582\) 0 0
\(583\) 34.1125 1.41280
\(584\) 0 0
\(585\) −0.624439 −0.0258174
\(586\) 0 0
\(587\) 30.4493 1.25678 0.628390 0.777899i \(-0.283714\pi\)
0.628390 + 0.777899i \(0.283714\pi\)
\(588\) 0 0
\(589\) 2.43064 0.100153
\(590\) 0 0
\(591\) −16.8639 −0.693689
\(592\) 0 0
\(593\) 24.0563 0.987872 0.493936 0.869498i \(-0.335558\pi\)
0.493936 + 0.869498i \(0.335558\pi\)
\(594\) 0 0
\(595\) −12.0103 −0.492376
\(596\) 0 0
\(597\) 7.20050 0.294697
\(598\) 0 0
\(599\) 41.8664 1.71062 0.855308 0.518120i \(-0.173368\pi\)
0.855308 + 0.518120i \(0.173368\pi\)
\(600\) 0 0
\(601\) 40.7687 1.66299 0.831495 0.555532i \(-0.187485\pi\)
0.831495 + 0.555532i \(0.187485\pi\)
\(602\) 0 0
\(603\) −12.7275 −0.518302
\(604\) 0 0
\(605\) −58.5910 −2.38206
\(606\) 0 0
\(607\) −38.9424 −1.58062 −0.790312 0.612705i \(-0.790081\pi\)
−0.790312 + 0.612705i \(0.790081\pi\)
\(608\) 0 0
\(609\) −23.6248 −0.957325
\(610\) 0 0
\(611\) 0.898977 0.0363687
\(612\) 0 0
\(613\) 13.4953 0.545068 0.272534 0.962146i \(-0.412138\pi\)
0.272534 + 0.962146i \(0.412138\pi\)
\(614\) 0 0
\(615\) −25.1988 −1.01611
\(616\) 0 0
\(617\) 13.8878 0.559101 0.279551 0.960131i \(-0.409815\pi\)
0.279551 + 0.960131i \(0.409815\pi\)
\(618\) 0 0
\(619\) 21.4746 0.863135 0.431568 0.902081i \(-0.357961\pi\)
0.431568 + 0.902081i \(0.357961\pi\)
\(620\) 0 0
\(621\) −31.6256 −1.26909
\(622\) 0 0
\(623\) 25.5200 1.02244
\(624\) 0 0
\(625\) −23.7695 −0.950779
\(626\) 0 0
\(627\) 2.56388 0.102392
\(628\) 0 0
\(629\) −3.39405 −0.135330
\(630\) 0 0
\(631\) −18.3546 −0.730685 −0.365343 0.930873i \(-0.619048\pi\)
−0.365343 + 0.930873i \(0.619048\pi\)
\(632\) 0 0
\(633\) −14.5886 −0.579844
\(634\) 0 0
\(635\) −44.5780 −1.76903
\(636\) 0 0
\(637\) −0.670953 −0.0265841
\(638\) 0 0
\(639\) −17.7634 −0.702709
\(640\) 0 0
\(641\) −37.9184 −1.49769 −0.748843 0.662747i \(-0.769390\pi\)
−0.748843 + 0.662747i \(0.769390\pi\)
\(642\) 0 0
\(643\) −2.51867 −0.0993266 −0.0496633 0.998766i \(-0.515815\pi\)
−0.0496633 + 0.998766i \(0.515815\pi\)
\(644\) 0 0
\(645\) 11.5016 0.452874
\(646\) 0 0
\(647\) 15.2313 0.598806 0.299403 0.954127i \(-0.403212\pi\)
0.299403 + 0.954127i \(0.403212\pi\)
\(648\) 0 0
\(649\) 5.41425 0.212528
\(650\) 0 0
\(651\) 18.0447 0.707229
\(652\) 0 0
\(653\) 10.2562 0.401358 0.200679 0.979657i \(-0.435685\pi\)
0.200679 + 0.979657i \(0.435685\pi\)
\(654\) 0 0
\(655\) −7.17728 −0.280439
\(656\) 0 0
\(657\) −12.4440 −0.485485
\(658\) 0 0
\(659\) −24.3376 −0.948059 −0.474029 0.880509i \(-0.657201\pi\)
−0.474029 + 0.880509i \(0.657201\pi\)
\(660\) 0 0
\(661\) 10.7069 0.416451 0.208226 0.978081i \(-0.433231\pi\)
0.208226 + 0.978081i \(0.433231\pi\)
\(662\) 0 0
\(663\) −0.0915307 −0.00355476
\(664\) 0 0
\(665\) −5.87727 −0.227911
\(666\) 0 0
\(667\) −41.9722 −1.62517
\(668\) 0 0
\(669\) 18.5053 0.715456
\(670\) 0 0
\(671\) −50.5118 −1.94999
\(672\) 0 0
\(673\) 28.7882 1.10970 0.554851 0.831950i \(-0.312775\pi\)
0.554851 + 0.831950i \(0.312775\pi\)
\(674\) 0 0
\(675\) −25.6517 −0.987336
\(676\) 0 0
\(677\) −25.6003 −0.983898 −0.491949 0.870624i \(-0.663715\pi\)
−0.491949 + 0.870624i \(0.663715\pi\)
\(678\) 0 0
\(679\) 20.5623 0.789109
\(680\) 0 0
\(681\) 4.12598 0.158108
\(682\) 0 0
\(683\) 15.6100 0.597301 0.298650 0.954363i \(-0.403464\pi\)
0.298650 + 0.954363i \(0.403464\pi\)
\(684\) 0 0
\(685\) −20.6642 −0.789538
\(686\) 0 0
\(687\) 25.4268 0.970092
\(688\) 0 0
\(689\) −0.595940 −0.0227035
\(690\) 0 0
\(691\) −33.7545 −1.28408 −0.642042 0.766670i \(-0.721912\pi\)
−0.642042 + 0.766670i \(0.721912\pi\)
\(692\) 0 0
\(693\) −41.9437 −1.59331
\(694\) 0 0
\(695\) 38.0007 1.44145
\(696\) 0 0
\(697\) 8.13947 0.308304
\(698\) 0 0
\(699\) 18.3171 0.692816
\(700\) 0 0
\(701\) −31.9310 −1.20602 −0.603010 0.797734i \(-0.706032\pi\)
−0.603010 + 0.797734i \(0.706032\pi\)
\(702\) 0 0
\(703\) −1.66088 −0.0626414
\(704\) 0 0
\(705\) 29.4243 1.10818
\(706\) 0 0
\(707\) 72.3586 2.72133
\(708\) 0 0
\(709\) 16.9943 0.638233 0.319117 0.947715i \(-0.396614\pi\)
0.319117 + 0.947715i \(0.396614\pi\)
\(710\) 0 0
\(711\) 15.6917 0.588483
\(712\) 0 0
\(713\) 32.0585 1.20060
\(714\) 0 0
\(715\) 1.63837 0.0612714
\(716\) 0 0
\(717\) 5.59118 0.208806
\(718\) 0 0
\(719\) −25.7022 −0.958531 −0.479265 0.877670i \(-0.659097\pi\)
−0.479265 + 0.877670i \(0.659097\pi\)
\(720\) 0 0
\(721\) −18.2197 −0.678537
\(722\) 0 0
\(723\) −8.86341 −0.329634
\(724\) 0 0
\(725\) −34.0439 −1.26436
\(726\) 0 0
\(727\) −40.6313 −1.50693 −0.753465 0.657488i \(-0.771619\pi\)
−0.753465 + 0.657488i \(0.771619\pi\)
\(728\) 0 0
\(729\) 11.2351 0.416114
\(730\) 0 0
\(731\) −3.71512 −0.137409
\(732\) 0 0
\(733\) −4.01414 −0.148265 −0.0741327 0.997248i \(-0.523619\pi\)
−0.0741327 + 0.997248i \(0.523619\pi\)
\(734\) 0 0
\(735\) −21.9609 −0.810039
\(736\) 0 0
\(737\) 33.3935 1.23007
\(738\) 0 0
\(739\) 3.98360 0.146539 0.0732695 0.997312i \(-0.476657\pi\)
0.0732695 + 0.997312i \(0.476657\pi\)
\(740\) 0 0
\(741\) −0.0447906 −0.00164542
\(742\) 0 0
\(743\) −3.07133 −0.112676 −0.0563381 0.998412i \(-0.517942\pi\)
−0.0563381 + 0.998412i \(0.517942\pi\)
\(744\) 0 0
\(745\) −7.62579 −0.279387
\(746\) 0 0
\(747\) −18.4215 −0.674008
\(748\) 0 0
\(749\) 15.9480 0.582727
\(750\) 0 0
\(751\) −54.1734 −1.97682 −0.988408 0.151822i \(-0.951486\pi\)
−0.988408 + 0.151822i \(0.951486\pi\)
\(752\) 0 0
\(753\) −2.21370 −0.0806716
\(754\) 0 0
\(755\) −8.07272 −0.293796
\(756\) 0 0
\(757\) −15.9569 −0.579965 −0.289982 0.957032i \(-0.593649\pi\)
−0.289982 + 0.957032i \(0.593649\pi\)
\(758\) 0 0
\(759\) 33.8159 1.22744
\(760\) 0 0
\(761\) 33.0484 1.19800 0.599002 0.800748i \(-0.295564\pi\)
0.599002 + 0.800748i \(0.295564\pi\)
\(762\) 0 0
\(763\) −44.2535 −1.60208
\(764\) 0 0
\(765\) 6.60181 0.238689
\(766\) 0 0
\(767\) −0.0945861 −0.00341531
\(768\) 0 0
\(769\) 6.46725 0.233215 0.116607 0.993178i \(-0.462798\pi\)
0.116607 + 0.993178i \(0.462798\pi\)
\(770\) 0 0
\(771\) 10.5314 0.379281
\(772\) 0 0
\(773\) 24.0822 0.866178 0.433089 0.901351i \(-0.357424\pi\)
0.433089 + 0.901351i \(0.357424\pi\)
\(774\) 0 0
\(775\) 26.0029 0.934052
\(776\) 0 0
\(777\) −12.3302 −0.442342
\(778\) 0 0
\(779\) 3.98305 0.142708
\(780\) 0 0
\(781\) 46.6065 1.66771
\(782\) 0 0
\(783\) 31.8650 1.13876
\(784\) 0 0
\(785\) −38.6419 −1.37919
\(786\) 0 0
\(787\) −24.2172 −0.863252 −0.431626 0.902053i \(-0.642060\pi\)
−0.431626 + 0.902053i \(0.642060\pi\)
\(788\) 0 0
\(789\) 6.03859 0.214979
\(790\) 0 0
\(791\) 19.6550 0.698853
\(792\) 0 0
\(793\) 0.882434 0.0313361
\(794\) 0 0
\(795\) −19.5056 −0.691793
\(796\) 0 0
\(797\) 1.93205 0.0684366 0.0342183 0.999414i \(-0.489106\pi\)
0.0342183 + 0.999414i \(0.489106\pi\)
\(798\) 0 0
\(799\) −9.50432 −0.336239
\(800\) 0 0
\(801\) −14.0278 −0.495646
\(802\) 0 0
\(803\) 32.6497 1.15218
\(804\) 0 0
\(805\) −77.5173 −2.73213
\(806\) 0 0
\(807\) 15.3060 0.538797
\(808\) 0 0
\(809\) −25.9018 −0.910660 −0.455330 0.890323i \(-0.650479\pi\)
−0.455330 + 0.890323i \(0.650479\pi\)
\(810\) 0 0
\(811\) −15.5728 −0.546836 −0.273418 0.961895i \(-0.588154\pi\)
−0.273418 + 0.961895i \(0.588154\pi\)
\(812\) 0 0
\(813\) −20.4860 −0.718474
\(814\) 0 0
\(815\) 6.36243 0.222866
\(816\) 0 0
\(817\) −1.81800 −0.0636037
\(818\) 0 0
\(819\) 0.732749 0.0256043
\(820\) 0 0
\(821\) −30.5587 −1.06651 −0.533253 0.845956i \(-0.679031\pi\)
−0.533253 + 0.845956i \(0.679031\pi\)
\(822\) 0 0
\(823\) 8.37212 0.291834 0.145917 0.989297i \(-0.453387\pi\)
0.145917 + 0.989297i \(0.453387\pi\)
\(824\) 0 0
\(825\) 27.4283 0.954931
\(826\) 0 0
\(827\) −15.5016 −0.539042 −0.269521 0.962994i \(-0.586865\pi\)
−0.269521 + 0.962994i \(0.586865\pi\)
\(828\) 0 0
\(829\) −24.0173 −0.834155 −0.417077 0.908871i \(-0.636946\pi\)
−0.417077 + 0.908871i \(0.636946\pi\)
\(830\) 0 0
\(831\) 29.1053 1.00965
\(832\) 0 0
\(833\) 7.09357 0.245778
\(834\) 0 0
\(835\) −13.0852 −0.452832
\(836\) 0 0
\(837\) −24.3386 −0.841267
\(838\) 0 0
\(839\) 29.6950 1.02519 0.512593 0.858632i \(-0.328685\pi\)
0.512593 + 0.858632i \(0.328685\pi\)
\(840\) 0 0
\(841\) 13.2899 0.458272
\(842\) 0 0
\(843\) −12.8270 −0.441785
\(844\) 0 0
\(845\) 41.5613 1.42975
\(846\) 0 0
\(847\) 68.7538 2.36241
\(848\) 0 0
\(849\) −9.28412 −0.318630
\(850\) 0 0
\(851\) −21.9059 −0.750926
\(852\) 0 0
\(853\) 17.4158 0.596307 0.298154 0.954518i \(-0.403629\pi\)
0.298154 + 0.954518i \(0.403629\pi\)
\(854\) 0 0
\(855\) 3.23060 0.110484
\(856\) 0 0
\(857\) −15.3879 −0.525641 −0.262821 0.964845i \(-0.584653\pi\)
−0.262821 + 0.964845i \(0.584653\pi\)
\(858\) 0 0
\(859\) −13.6185 −0.464656 −0.232328 0.972637i \(-0.574634\pi\)
−0.232328 + 0.972637i \(0.574634\pi\)
\(860\) 0 0
\(861\) 29.5696 1.00773
\(862\) 0 0
\(863\) 29.3267 0.998293 0.499146 0.866518i \(-0.333647\pi\)
0.499146 + 0.866518i \(0.333647\pi\)
\(864\) 0 0
\(865\) 33.4273 1.13656
\(866\) 0 0
\(867\) 0.967697 0.0328647
\(868\) 0 0
\(869\) −41.1708 −1.39662
\(870\) 0 0
\(871\) −0.583379 −0.0197671
\(872\) 0 0
\(873\) −11.3026 −0.382535
\(874\) 0 0
\(875\) −2.82309 −0.0954380
\(876\) 0 0
\(877\) 23.1989 0.783372 0.391686 0.920099i \(-0.371892\pi\)
0.391686 + 0.920099i \(0.371892\pi\)
\(878\) 0 0
\(879\) −29.0602 −0.980178
\(880\) 0 0
\(881\) 20.8370 0.702016 0.351008 0.936372i \(-0.385839\pi\)
0.351008 + 0.936372i \(0.385839\pi\)
\(882\) 0 0
\(883\) 34.4308 1.15869 0.579344 0.815083i \(-0.303309\pi\)
0.579344 + 0.815083i \(0.303309\pi\)
\(884\) 0 0
\(885\) −3.09588 −0.104067
\(886\) 0 0
\(887\) −11.0059 −0.369543 −0.184772 0.982781i \(-0.559155\pi\)
−0.184772 + 0.982781i \(0.559155\pi\)
\(888\) 0 0
\(889\) 52.3102 1.75443
\(890\) 0 0
\(891\) 7.84513 0.262822
\(892\) 0 0
\(893\) −4.65095 −0.155638
\(894\) 0 0
\(895\) −25.5835 −0.855162
\(896\) 0 0
\(897\) −0.590758 −0.0197248
\(898\) 0 0
\(899\) −32.3012 −1.07731
\(900\) 0 0
\(901\) 6.30050 0.209900
\(902\) 0 0
\(903\) −13.4966 −0.449137
\(904\) 0 0
\(905\) −51.0014 −1.69534
\(906\) 0 0
\(907\) −10.9867 −0.364809 −0.182404 0.983224i \(-0.558388\pi\)
−0.182404 + 0.983224i \(0.558388\pi\)
\(908\) 0 0
\(909\) −39.7738 −1.31921
\(910\) 0 0
\(911\) −1.52025 −0.0503682 −0.0251841 0.999683i \(-0.508017\pi\)
−0.0251841 + 0.999683i \(0.508017\pi\)
\(912\) 0 0
\(913\) 48.3332 1.59960
\(914\) 0 0
\(915\) 28.8828 0.954836
\(916\) 0 0
\(917\) 8.42219 0.278125
\(918\) 0 0
\(919\) −20.0388 −0.661018 −0.330509 0.943803i \(-0.607220\pi\)
−0.330509 + 0.943803i \(0.607220\pi\)
\(920\) 0 0
\(921\) −14.4978 −0.477719
\(922\) 0 0
\(923\) −0.814208 −0.0268000
\(924\) 0 0
\(925\) −17.7681 −0.584210
\(926\) 0 0
\(927\) 10.0149 0.328934
\(928\) 0 0
\(929\) 23.7997 0.780843 0.390421 0.920636i \(-0.372329\pi\)
0.390421 + 0.920636i \(0.372329\pi\)
\(930\) 0 0
\(931\) 3.47124 0.113765
\(932\) 0 0
\(933\) 17.7794 0.582070
\(934\) 0 0
\(935\) −17.3214 −0.566471
\(936\) 0 0
\(937\) 43.7349 1.42876 0.714378 0.699760i \(-0.246710\pi\)
0.714378 + 0.699760i \(0.246710\pi\)
\(938\) 0 0
\(939\) 32.4103 1.05767
\(940\) 0 0
\(941\) −13.9657 −0.455267 −0.227634 0.973747i \(-0.573099\pi\)
−0.227634 + 0.973747i \(0.573099\pi\)
\(942\) 0 0
\(943\) 52.5338 1.71074
\(944\) 0 0
\(945\) 58.8506 1.91441
\(946\) 0 0
\(947\) −20.4707 −0.665207 −0.332604 0.943067i \(-0.607927\pi\)
−0.332604 + 0.943067i \(0.607927\pi\)
\(948\) 0 0
\(949\) −0.570385 −0.0185155
\(950\) 0 0
\(951\) −19.4409 −0.630415
\(952\) 0 0
\(953\) 5.89960 0.191107 0.0955533 0.995424i \(-0.469538\pi\)
0.0955533 + 0.995424i \(0.469538\pi\)
\(954\) 0 0
\(955\) −34.6387 −1.12088
\(956\) 0 0
\(957\) −34.0719 −1.10139
\(958\) 0 0
\(959\) 24.2484 0.783023
\(960\) 0 0
\(961\) −6.32815 −0.204134
\(962\) 0 0
\(963\) −8.76624 −0.282488
\(964\) 0 0
\(965\) 29.9020 0.962580
\(966\) 0 0
\(967\) −18.8452 −0.606020 −0.303010 0.952987i \(-0.597992\pi\)
−0.303010 + 0.952987i \(0.597992\pi\)
\(968\) 0 0
\(969\) 0.473543 0.0152124
\(970\) 0 0
\(971\) −27.9339 −0.896440 −0.448220 0.893923i \(-0.647942\pi\)
−0.448220 + 0.893923i \(0.647942\pi\)
\(972\) 0 0
\(973\) −44.5920 −1.42955
\(974\) 0 0
\(975\) −0.479168 −0.0153457
\(976\) 0 0
\(977\) −11.0387 −0.353159 −0.176579 0.984286i \(-0.556503\pi\)
−0.176579 + 0.984286i \(0.556503\pi\)
\(978\) 0 0
\(979\) 36.8052 1.17630
\(980\) 0 0
\(981\) 24.3251 0.776640
\(982\) 0 0
\(983\) 26.7116 0.851966 0.425983 0.904731i \(-0.359928\pi\)
0.425983 + 0.904731i \(0.359928\pi\)
\(984\) 0 0
\(985\) 55.7525 1.77642
\(986\) 0 0
\(987\) −34.5280 −1.09904
\(988\) 0 0
\(989\) −23.9782 −0.762461
\(990\) 0 0
\(991\) −5.24329 −0.166559 −0.0832793 0.996526i \(-0.526539\pi\)
−0.0832793 + 0.996526i \(0.526539\pi\)
\(992\) 0 0
\(993\) 2.98383 0.0946891
\(994\) 0 0
\(995\) −23.8050 −0.754669
\(996\) 0 0
\(997\) 20.6894 0.655241 0.327620 0.944809i \(-0.393753\pi\)
0.327620 + 0.944809i \(0.393753\pi\)
\(998\) 0 0
\(999\) 16.6308 0.526177
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 4012.2.a.j.1.13 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4012.2.a.j.1.13 21 1.1 even 1 trivial