Properties

Label 8002.2.a.f.1.8
Level $8002$
Weight $2$
Character 8002.1
Self dual yes
Analytic conductor $63.896$
Analytic rank $1$
Dimension $89$
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] = [8002,2,Mod(1,8002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8002, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8002.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8002 = 2 \cdot 4001 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8002.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: \(63.8962916974\)
Analytic rank: \(1\)
Dimension: \(89\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Character \(\chi\) \(=\) 8002.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.96590 q^{3} +1.00000 q^{4} -3.55325 q^{5} +2.96590 q^{6} +2.76315 q^{7} -1.00000 q^{8} +5.79657 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.96590 q^{3} +1.00000 q^{4} -3.55325 q^{5} +2.96590 q^{6} +2.76315 q^{7} -1.00000 q^{8} +5.79657 q^{9} +3.55325 q^{10} +0.525847 q^{11} -2.96590 q^{12} +1.74773 q^{13} -2.76315 q^{14} +10.5386 q^{15} +1.00000 q^{16} +0.404331 q^{17} -5.79657 q^{18} +6.67379 q^{19} -3.55325 q^{20} -8.19522 q^{21} -0.525847 q^{22} -9.22100 q^{23} +2.96590 q^{24} +7.62560 q^{25} -1.74773 q^{26} -8.29436 q^{27} +2.76315 q^{28} +8.91123 q^{29} -10.5386 q^{30} +0.930175 q^{31} -1.00000 q^{32} -1.55961 q^{33} -0.404331 q^{34} -9.81816 q^{35} +5.79657 q^{36} -9.29553 q^{37} -6.67379 q^{38} -5.18358 q^{39} +3.55325 q^{40} -1.05461 q^{41} +8.19522 q^{42} -0.266653 q^{43} +0.525847 q^{44} -20.5967 q^{45} +9.22100 q^{46} -11.6624 q^{47} -2.96590 q^{48} +0.634983 q^{49} -7.62560 q^{50} -1.19921 q^{51} +1.74773 q^{52} -11.3924 q^{53} +8.29436 q^{54} -1.86847 q^{55} -2.76315 q^{56} -19.7938 q^{57} -8.91123 q^{58} +2.58507 q^{59} +10.5386 q^{60} +2.18120 q^{61} -0.930175 q^{62} +16.0168 q^{63} +1.00000 q^{64} -6.21011 q^{65} +1.55961 q^{66} +10.0583 q^{67} +0.404331 q^{68} +27.3486 q^{69} +9.81816 q^{70} +12.2881 q^{71} -5.79657 q^{72} +11.1510 q^{73} +9.29553 q^{74} -22.6168 q^{75} +6.67379 q^{76} +1.45299 q^{77} +5.18358 q^{78} -11.9811 q^{79} -3.55325 q^{80} +7.21053 q^{81} +1.05461 q^{82} -6.06295 q^{83} -8.19522 q^{84} -1.43669 q^{85} +0.266653 q^{86} -26.4298 q^{87} -0.525847 q^{88} -16.8855 q^{89} +20.5967 q^{90} +4.82922 q^{91} -9.22100 q^{92} -2.75881 q^{93} +11.6624 q^{94} -23.7137 q^{95} +2.96590 q^{96} -7.78289 q^{97} -0.634983 q^{98} +3.04811 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9} + 18 q^{10} - 26 q^{11} - 12 q^{12} + 2 q^{13} + 27 q^{14} - 21 q^{15} + 89 q^{16} - 60 q^{17} - 95 q^{18} + q^{19} - 18 q^{20} - 6 q^{21} + 26 q^{22} - 45 q^{23} + 12 q^{24} + 107 q^{25} - 2 q^{26} - 45 q^{27} - 27 q^{28} - 18 q^{29} + 21 q^{30} - 38 q^{31} - 89 q^{32} - 29 q^{33} + 60 q^{34} - 47 q^{35} + 95 q^{36} - 15 q^{37} - q^{38} - 38 q^{39} + 18 q^{40} - 50 q^{41} + 6 q^{42} - 15 q^{43} - 26 q^{44} - 35 q^{45} + 45 q^{46} - 121 q^{47} - 12 q^{48} + 132 q^{49} - 107 q^{50} + 6 q^{51} + 2 q^{52} - 46 q^{53} + 45 q^{54} - 37 q^{55} + 27 q^{56} - 42 q^{57} + 18 q^{58} - 34 q^{59} - 21 q^{60} + 41 q^{61} + 38 q^{62} - 131 q^{63} + 89 q^{64} - 57 q^{65} + 29 q^{66} - 11 q^{67} - 60 q^{68} + 15 q^{69} + 47 q^{70} - 66 q^{71} - 95 q^{72} - 47 q^{73} + 15 q^{74} - 46 q^{75} + q^{76} - 106 q^{77} + 38 q^{78} - 51 q^{79} - 18 q^{80} + 113 q^{81} + 50 q^{82} - 141 q^{83} - 6 q^{84} - 7 q^{85} + 15 q^{86} - 110 q^{87} + 26 q^{88} - 30 q^{89} + 35 q^{90} + 37 q^{91} - 45 q^{92} - 44 q^{93} + 121 q^{94} - 98 q^{95} + 12 q^{96} + 3 q^{97} - 132 q^{98} - 71 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −2.96590 −1.71236 −0.856182 0.516674i \(-0.827170\pi\)
−0.856182 + 0.516674i \(0.827170\pi\)
\(4\) 1.00000 0.500000
\(5\) −3.55325 −1.58906 −0.794531 0.607223i \(-0.792283\pi\)
−0.794531 + 0.607223i \(0.792283\pi\)
\(6\) 2.96590 1.21082
\(7\) 2.76315 1.04437 0.522186 0.852832i \(-0.325117\pi\)
0.522186 + 0.852832i \(0.325117\pi\)
\(8\) −1.00000 −0.353553
\(9\) 5.79657 1.93219
\(10\) 3.55325 1.12364
\(11\) 0.525847 0.158549 0.0792744 0.996853i \(-0.474740\pi\)
0.0792744 + 0.996853i \(0.474740\pi\)
\(12\) −2.96590 −0.856182
\(13\) 1.74773 0.484732 0.242366 0.970185i \(-0.422077\pi\)
0.242366 + 0.970185i \(0.422077\pi\)
\(14\) −2.76315 −0.738482
\(15\) 10.5386 2.72105
\(16\) 1.00000 0.250000
\(17\) 0.404331 0.0980647 0.0490323 0.998797i \(-0.484386\pi\)
0.0490323 + 0.998797i \(0.484386\pi\)
\(18\) −5.79657 −1.36627
\(19\) 6.67379 1.53107 0.765537 0.643392i \(-0.222474\pi\)
0.765537 + 0.643392i \(0.222474\pi\)
\(20\) −3.55325 −0.794531
\(21\) −8.19522 −1.78834
\(22\) −0.525847 −0.112111
\(23\) −9.22100 −1.92271 −0.961356 0.275309i \(-0.911220\pi\)
−0.961356 + 0.275309i \(0.911220\pi\)
\(24\) 2.96590 0.605412
\(25\) 7.62560 1.52512
\(26\) −1.74773 −0.342757
\(27\) −8.29436 −1.59625
\(28\) 2.76315 0.522186
\(29\) 8.91123 1.65477 0.827387 0.561632i \(-0.189826\pi\)
0.827387 + 0.561632i \(0.189826\pi\)
\(30\) −10.5386 −1.92408
\(31\) 0.930175 0.167064 0.0835322 0.996505i \(-0.473380\pi\)
0.0835322 + 0.996505i \(0.473380\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.55961 −0.271493
\(34\) −0.404331 −0.0693422
\(35\) −9.81816 −1.65957
\(36\) 5.79657 0.966095
\(37\) −9.29553 −1.52818 −0.764088 0.645112i \(-0.776811\pi\)
−0.764088 + 0.645112i \(0.776811\pi\)
\(38\) −6.67379 −1.08263
\(39\) −5.18358 −0.830037
\(40\) 3.55325 0.561818
\(41\) −1.05461 −0.164703 −0.0823513 0.996603i \(-0.526243\pi\)
−0.0823513 + 0.996603i \(0.526243\pi\)
\(42\) 8.19522 1.26455
\(43\) −0.266653 −0.0406641 −0.0203321 0.999793i \(-0.506472\pi\)
−0.0203321 + 0.999793i \(0.506472\pi\)
\(44\) 0.525847 0.0792744
\(45\) −20.5967 −3.07037
\(46\) 9.22100 1.35956
\(47\) −11.6624 −1.70114 −0.850571 0.525861i \(-0.823743\pi\)
−0.850571 + 0.525861i \(0.823743\pi\)
\(48\) −2.96590 −0.428091
\(49\) 0.634983 0.0907118
\(50\) −7.62560 −1.07842
\(51\) −1.19921 −0.167922
\(52\) 1.74773 0.242366
\(53\) −11.3924 −1.56486 −0.782430 0.622739i \(-0.786020\pi\)
−0.782430 + 0.622739i \(0.786020\pi\)
\(54\) 8.29436 1.12872
\(55\) −1.86847 −0.251944
\(56\) −2.76315 −0.369241
\(57\) −19.7938 −2.62176
\(58\) −8.91123 −1.17010
\(59\) 2.58507 0.336548 0.168274 0.985740i \(-0.446181\pi\)
0.168274 + 0.985740i \(0.446181\pi\)
\(60\) 10.5386 1.36053
\(61\) 2.18120 0.279274 0.139637 0.990203i \(-0.455406\pi\)
0.139637 + 0.990203i \(0.455406\pi\)
\(62\) −0.930175 −0.118132
\(63\) 16.0168 2.01792
\(64\) 1.00000 0.125000
\(65\) −6.21011 −0.770269
\(66\) 1.55961 0.191975
\(67\) 10.0583 1.22882 0.614411 0.788986i \(-0.289394\pi\)
0.614411 + 0.788986i \(0.289394\pi\)
\(68\) 0.404331 0.0490323
\(69\) 27.3486 3.29238
\(70\) 9.81816 1.17349
\(71\) 12.2881 1.45832 0.729162 0.684341i \(-0.239910\pi\)
0.729162 + 0.684341i \(0.239910\pi\)
\(72\) −5.79657 −0.683133
\(73\) 11.1510 1.30513 0.652564 0.757734i \(-0.273693\pi\)
0.652564 + 0.757734i \(0.273693\pi\)
\(74\) 9.29553 1.08058
\(75\) −22.6168 −2.61156
\(76\) 6.67379 0.765537
\(77\) 1.45299 0.165584
\(78\) 5.18358 0.586925
\(79\) −11.9811 −1.34797 −0.673987 0.738743i \(-0.735420\pi\)
−0.673987 + 0.738743i \(0.735420\pi\)
\(80\) −3.55325 −0.397266
\(81\) 7.21053 0.801170
\(82\) 1.05461 0.116462
\(83\) −6.06295 −0.665495 −0.332748 0.943016i \(-0.607976\pi\)
−0.332748 + 0.943016i \(0.607976\pi\)
\(84\) −8.19522 −0.894172
\(85\) −1.43669 −0.155831
\(86\) 0.266653 0.0287539
\(87\) −26.4298 −2.83358
\(88\) −0.525847 −0.0560555
\(89\) −16.8855 −1.78986 −0.894931 0.446205i \(-0.852775\pi\)
−0.894931 + 0.446205i \(0.852775\pi\)
\(90\) 20.5967 2.17108
\(91\) 4.82922 0.506240
\(92\) −9.22100 −0.961356
\(93\) −2.75881 −0.286075
\(94\) 11.6624 1.20289
\(95\) −23.7137 −2.43297
\(96\) 2.96590 0.302706
\(97\) −7.78289 −0.790233 −0.395117 0.918631i \(-0.629296\pi\)
−0.395117 + 0.918631i \(0.629296\pi\)
\(98\) −0.634983 −0.0641430
\(99\) 3.04811 0.306347
\(100\) 7.62560 0.762560
\(101\) 16.5848 1.65025 0.825125 0.564950i \(-0.191104\pi\)
0.825125 + 0.564950i \(0.191104\pi\)
\(102\) 1.19921 0.118739
\(103\) 9.35729 0.922001 0.461001 0.887400i \(-0.347491\pi\)
0.461001 + 0.887400i \(0.347491\pi\)
\(104\) −1.74773 −0.171379
\(105\) 29.1197 2.84179
\(106\) 11.3924 1.10652
\(107\) −12.2662 −1.18582 −0.592912 0.805268i \(-0.702022\pi\)
−0.592912 + 0.805268i \(0.702022\pi\)
\(108\) −8.29436 −0.798125
\(109\) 10.5900 1.01433 0.507167 0.861848i \(-0.330693\pi\)
0.507167 + 0.861848i \(0.330693\pi\)
\(110\) 1.86847 0.178151
\(111\) 27.5696 2.61679
\(112\) 2.76315 0.261093
\(113\) 17.3588 1.63298 0.816488 0.577362i \(-0.195918\pi\)
0.816488 + 0.577362i \(0.195918\pi\)
\(114\) 19.7938 1.85386
\(115\) 32.7645 3.05531
\(116\) 8.91123 0.827387
\(117\) 10.1308 0.936594
\(118\) −2.58507 −0.237975
\(119\) 1.11723 0.102416
\(120\) −10.5386 −0.962038
\(121\) −10.7235 −0.974862
\(122\) −2.18120 −0.197477
\(123\) 3.12787 0.282031
\(124\) 0.930175 0.0835322
\(125\) −9.32942 −0.834448
\(126\) −16.0168 −1.42689
\(127\) −10.8304 −0.961046 −0.480523 0.876982i \(-0.659553\pi\)
−0.480523 + 0.876982i \(0.659553\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0.790865 0.0696318
\(130\) 6.21011 0.544663
\(131\) −4.60183 −0.402064 −0.201032 0.979585i \(-0.564429\pi\)
−0.201032 + 0.979585i \(0.564429\pi\)
\(132\) −1.55961 −0.135747
\(133\) 18.4407 1.59901
\(134\) −10.0583 −0.868909
\(135\) 29.4719 2.53654
\(136\) −0.404331 −0.0346711
\(137\) 6.63810 0.567131 0.283566 0.958953i \(-0.408483\pi\)
0.283566 + 0.958953i \(0.408483\pi\)
\(138\) −27.3486 −2.32807
\(139\) −8.64286 −0.733078 −0.366539 0.930403i \(-0.619457\pi\)
−0.366539 + 0.930403i \(0.619457\pi\)
\(140\) −9.81816 −0.829786
\(141\) 34.5896 2.91297
\(142\) −12.2881 −1.03119
\(143\) 0.919036 0.0768537
\(144\) 5.79657 0.483048
\(145\) −31.6639 −2.62954
\(146\) −11.1510 −0.922865
\(147\) −1.88330 −0.155332
\(148\) −9.29553 −0.764088
\(149\) 8.43034 0.690640 0.345320 0.938485i \(-0.387770\pi\)
0.345320 + 0.938485i \(0.387770\pi\)
\(150\) 22.6168 1.84665
\(151\) 6.86075 0.558320 0.279160 0.960245i \(-0.409944\pi\)
0.279160 + 0.960245i \(0.409944\pi\)
\(152\) −6.67379 −0.541316
\(153\) 2.34373 0.189480
\(154\) −1.45299 −0.117086
\(155\) −3.30514 −0.265476
\(156\) −5.18358 −0.415019
\(157\) −3.40902 −0.272070 −0.136035 0.990704i \(-0.543436\pi\)
−0.136035 + 0.990704i \(0.543436\pi\)
\(158\) 11.9811 0.953162
\(159\) 33.7886 2.67961
\(160\) 3.55325 0.280909
\(161\) −25.4790 −2.00802
\(162\) −7.21053 −0.566513
\(163\) 8.29841 0.649982 0.324991 0.945717i \(-0.394639\pi\)
0.324991 + 0.945717i \(0.394639\pi\)
\(164\) −1.05461 −0.0823513
\(165\) 5.54169 0.431420
\(166\) 6.06295 0.470576
\(167\) −8.46894 −0.655346 −0.327673 0.944791i \(-0.606264\pi\)
−0.327673 + 0.944791i \(0.606264\pi\)
\(168\) 8.19522 0.632275
\(169\) −9.94546 −0.765035
\(170\) 1.43669 0.110189
\(171\) 38.6851 2.95833
\(172\) −0.266653 −0.0203321
\(173\) −17.5748 −1.33619 −0.668094 0.744077i \(-0.732889\pi\)
−0.668094 + 0.744077i \(0.732889\pi\)
\(174\) 26.4298 2.00364
\(175\) 21.0707 1.59279
\(176\) 0.525847 0.0396372
\(177\) −7.66707 −0.576293
\(178\) 16.8855 1.26562
\(179\) 16.6868 1.24723 0.623614 0.781732i \(-0.285664\pi\)
0.623614 + 0.781732i \(0.285664\pi\)
\(180\) −20.5967 −1.53519
\(181\) −21.7269 −1.61495 −0.807473 0.589904i \(-0.799166\pi\)
−0.807473 + 0.589904i \(0.799166\pi\)
\(182\) −4.82922 −0.357966
\(183\) −6.46923 −0.478219
\(184\) 9.22100 0.679781
\(185\) 33.0294 2.42837
\(186\) 2.75881 0.202286
\(187\) 0.212616 0.0155480
\(188\) −11.6624 −0.850571
\(189\) −22.9185 −1.66708
\(190\) 23.7137 1.72037
\(191\) 13.3276 0.964350 0.482175 0.876075i \(-0.339847\pi\)
0.482175 + 0.876075i \(0.339847\pi\)
\(192\) −2.96590 −0.214046
\(193\) −6.42123 −0.462210 −0.231105 0.972929i \(-0.574234\pi\)
−0.231105 + 0.972929i \(0.574234\pi\)
\(194\) 7.78289 0.558779
\(195\) 18.4186 1.31898
\(196\) 0.634983 0.0453559
\(197\) 12.8234 0.913630 0.456815 0.889562i \(-0.348990\pi\)
0.456815 + 0.889562i \(0.348990\pi\)
\(198\) −3.04811 −0.216620
\(199\) 16.3445 1.15863 0.579315 0.815104i \(-0.303320\pi\)
0.579315 + 0.815104i \(0.303320\pi\)
\(200\) −7.62560 −0.539211
\(201\) −29.8321 −2.10419
\(202\) −16.5848 −1.16690
\(203\) 24.6231 1.72820
\(204\) −1.19921 −0.0839612
\(205\) 3.74730 0.261723
\(206\) −9.35729 −0.651953
\(207\) −53.4502 −3.71505
\(208\) 1.74773 0.121183
\(209\) 3.50940 0.242750
\(210\) −29.1197 −2.00945
\(211\) 9.74762 0.671054 0.335527 0.942031i \(-0.391086\pi\)
0.335527 + 0.942031i \(0.391086\pi\)
\(212\) −11.3924 −0.782430
\(213\) −36.4452 −2.49718
\(214\) 12.2662 0.838504
\(215\) 0.947484 0.0646178
\(216\) 8.29436 0.564360
\(217\) 2.57021 0.174477
\(218\) −10.5900 −0.717242
\(219\) −33.0728 −2.23485
\(220\) −1.86847 −0.125972
\(221\) 0.706660 0.0475351
\(222\) −27.5696 −1.85035
\(223\) 23.4967 1.57345 0.786726 0.617302i \(-0.211774\pi\)
0.786726 + 0.617302i \(0.211774\pi\)
\(224\) −2.76315 −0.184621
\(225\) 44.2023 2.94682
\(226\) −17.3588 −1.15469
\(227\) −15.5615 −1.03285 −0.516427 0.856331i \(-0.672738\pi\)
−0.516427 + 0.856331i \(0.672738\pi\)
\(228\) −19.7938 −1.31088
\(229\) 14.5342 0.960446 0.480223 0.877147i \(-0.340556\pi\)
0.480223 + 0.877147i \(0.340556\pi\)
\(230\) −32.7645 −2.16043
\(231\) −4.30943 −0.283540
\(232\) −8.91123 −0.585051
\(233\) −9.69239 −0.634970 −0.317485 0.948263i \(-0.602838\pi\)
−0.317485 + 0.948263i \(0.602838\pi\)
\(234\) −10.1308 −0.662272
\(235\) 41.4396 2.70322
\(236\) 2.58507 0.168274
\(237\) 35.5347 2.30822
\(238\) −1.11723 −0.0724190
\(239\) −20.6560 −1.33613 −0.668064 0.744104i \(-0.732877\pi\)
−0.668064 + 0.744104i \(0.732877\pi\)
\(240\) 10.5386 0.680263
\(241\) −19.0386 −1.22638 −0.613192 0.789934i \(-0.710115\pi\)
−0.613192 + 0.789934i \(0.710115\pi\)
\(242\) 10.7235 0.689332
\(243\) 3.49734 0.224355
\(244\) 2.18120 0.139637
\(245\) −2.25625 −0.144147
\(246\) −3.12787 −0.199426
\(247\) 11.6640 0.742160
\(248\) −0.930175 −0.0590662
\(249\) 17.9821 1.13957
\(250\) 9.32942 0.590044
\(251\) 20.4574 1.29126 0.645629 0.763651i \(-0.276595\pi\)
0.645629 + 0.763651i \(0.276595\pi\)
\(252\) 16.0168 1.00896
\(253\) −4.84884 −0.304844
\(254\) 10.8304 0.679562
\(255\) 4.26108 0.266839
\(256\) 1.00000 0.0625000
\(257\) 30.5506 1.90570 0.952848 0.303447i \(-0.0981377\pi\)
0.952848 + 0.303447i \(0.0981377\pi\)
\(258\) −0.790865 −0.0492371
\(259\) −25.6849 −1.59598
\(260\) −6.21011 −0.385135
\(261\) 51.6546 3.19734
\(262\) 4.60183 0.284302
\(263\) 13.1452 0.810571 0.405285 0.914190i \(-0.367172\pi\)
0.405285 + 0.914190i \(0.367172\pi\)
\(264\) 1.55961 0.0959874
\(265\) 40.4799 2.48666
\(266\) −18.4407 −1.13067
\(267\) 50.0808 3.06489
\(268\) 10.0583 0.614411
\(269\) 2.18874 0.133450 0.0667248 0.997771i \(-0.478745\pi\)
0.0667248 + 0.997771i \(0.478745\pi\)
\(270\) −29.4719 −1.79361
\(271\) 3.13707 0.190563 0.0952817 0.995450i \(-0.469625\pi\)
0.0952817 + 0.995450i \(0.469625\pi\)
\(272\) 0.404331 0.0245162
\(273\) −14.3230 −0.866867
\(274\) −6.63810 −0.401022
\(275\) 4.00990 0.241806
\(276\) 27.3486 1.64619
\(277\) 21.6799 1.30262 0.651309 0.758812i \(-0.274220\pi\)
0.651309 + 0.758812i \(0.274220\pi\)
\(278\) 8.64286 0.518364
\(279\) 5.39182 0.322800
\(280\) 9.81816 0.586747
\(281\) 4.47708 0.267080 0.133540 0.991043i \(-0.457365\pi\)
0.133540 + 0.991043i \(0.457365\pi\)
\(282\) −34.5896 −2.05978
\(283\) −30.2162 −1.79616 −0.898082 0.439827i \(-0.855040\pi\)
−0.898082 + 0.439827i \(0.855040\pi\)
\(284\) 12.2881 0.729162
\(285\) 70.3324 4.16613
\(286\) −0.919036 −0.0543438
\(287\) −2.91404 −0.172011
\(288\) −5.79657 −0.341566
\(289\) −16.8365 −0.990383
\(290\) 31.6639 1.85937
\(291\) 23.0833 1.35317
\(292\) 11.1510 0.652564
\(293\) −23.6619 −1.38234 −0.691171 0.722691i \(-0.742905\pi\)
−0.691171 + 0.722691i \(0.742905\pi\)
\(294\) 1.88330 0.109836
\(295\) −9.18542 −0.534796
\(296\) 9.29553 0.540292
\(297\) −4.36156 −0.253084
\(298\) −8.43034 −0.488356
\(299\) −16.1158 −0.931999
\(300\) −22.6168 −1.30578
\(301\) −0.736800 −0.0424685
\(302\) −6.86075 −0.394792
\(303\) −49.1889 −2.82583
\(304\) 6.67379 0.382768
\(305\) −7.75036 −0.443784
\(306\) −2.34373 −0.133982
\(307\) 21.4101 1.22194 0.610968 0.791655i \(-0.290780\pi\)
0.610968 + 0.791655i \(0.290780\pi\)
\(308\) 1.45299 0.0827920
\(309\) −27.7528 −1.57880
\(310\) 3.30514 0.187720
\(311\) −28.4872 −1.61536 −0.807680 0.589622i \(-0.799277\pi\)
−0.807680 + 0.589622i \(0.799277\pi\)
\(312\) 5.18358 0.293462
\(313\) −15.6149 −0.882605 −0.441303 0.897358i \(-0.645483\pi\)
−0.441303 + 0.897358i \(0.645483\pi\)
\(314\) 3.40902 0.192382
\(315\) −56.9117 −3.20661
\(316\) −11.9811 −0.673987
\(317\) 19.5055 1.09554 0.547769 0.836629i \(-0.315477\pi\)
0.547769 + 0.836629i \(0.315477\pi\)
\(318\) −33.7886 −1.89477
\(319\) 4.68595 0.262363
\(320\) −3.55325 −0.198633
\(321\) 36.3805 2.03056
\(322\) 25.4790 1.41989
\(323\) 2.69842 0.150144
\(324\) 7.21053 0.400585
\(325\) 13.3275 0.739274
\(326\) −8.29841 −0.459607
\(327\) −31.4087 −1.73691
\(328\) 1.05461 0.0582311
\(329\) −32.2250 −1.77662
\(330\) −5.54169 −0.305060
\(331\) 23.6814 1.30165 0.650823 0.759229i \(-0.274424\pi\)
0.650823 + 0.759229i \(0.274424\pi\)
\(332\) −6.06295 −0.332748
\(333\) −53.8822 −2.95273
\(334\) 8.46894 0.463400
\(335\) −35.7398 −1.95268
\(336\) −8.19522 −0.447086
\(337\) 11.9124 0.648910 0.324455 0.945901i \(-0.394819\pi\)
0.324455 + 0.945901i \(0.394819\pi\)
\(338\) 9.94546 0.540962
\(339\) −51.4844 −2.79625
\(340\) −1.43669 −0.0779155
\(341\) 0.489130 0.0264879
\(342\) −38.6851 −2.09185
\(343\) −17.5875 −0.949635
\(344\) 0.266653 0.0143769
\(345\) −97.1764 −5.23180
\(346\) 17.5748 0.944827
\(347\) 11.1351 0.597762 0.298881 0.954290i \(-0.403387\pi\)
0.298881 + 0.954290i \(0.403387\pi\)
\(348\) −26.4298 −1.41679
\(349\) −29.6691 −1.58815 −0.794075 0.607819i \(-0.792044\pi\)
−0.794075 + 0.607819i \(0.792044\pi\)
\(350\) −21.0707 −1.12627
\(351\) −14.4963 −0.773753
\(352\) −0.525847 −0.0280277
\(353\) 33.4239 1.77897 0.889487 0.456961i \(-0.151062\pi\)
0.889487 + 0.456961i \(0.151062\pi\)
\(354\) 7.66707 0.407500
\(355\) −43.6625 −2.31737
\(356\) −16.8855 −0.894931
\(357\) −3.31358 −0.175373
\(358\) −16.6868 −0.881924
\(359\) −12.3282 −0.650659 −0.325329 0.945601i \(-0.605475\pi\)
−0.325329 + 0.945601i \(0.605475\pi\)
\(360\) 20.5967 1.08554
\(361\) 25.5395 1.34419
\(362\) 21.7269 1.14194
\(363\) 31.8048 1.66932
\(364\) 4.82922 0.253120
\(365\) −39.6224 −2.07393
\(366\) 6.46923 0.338152
\(367\) −12.9043 −0.673598 −0.336799 0.941577i \(-0.609344\pi\)
−0.336799 + 0.941577i \(0.609344\pi\)
\(368\) −9.22100 −0.480678
\(369\) −6.11313 −0.318237
\(370\) −33.0294 −1.71712
\(371\) −31.4787 −1.63430
\(372\) −2.75881 −0.143037
\(373\) 21.1322 1.09418 0.547092 0.837072i \(-0.315735\pi\)
0.547092 + 0.837072i \(0.315735\pi\)
\(374\) −0.212616 −0.0109941
\(375\) 27.6701 1.42888
\(376\) 11.6624 0.601444
\(377\) 15.5744 0.802122
\(378\) 22.9185 1.17880
\(379\) −23.5472 −1.20954 −0.604770 0.796400i \(-0.706735\pi\)
−0.604770 + 0.796400i \(0.706735\pi\)
\(380\) −23.7137 −1.21649
\(381\) 32.1220 1.64566
\(382\) −13.3276 −0.681898
\(383\) −31.1585 −1.59212 −0.796062 0.605215i \(-0.793087\pi\)
−0.796062 + 0.605215i \(0.793087\pi\)
\(384\) 2.96590 0.151353
\(385\) −5.16285 −0.263123
\(386\) 6.42123 0.326832
\(387\) −1.54567 −0.0785708
\(388\) −7.78289 −0.395117
\(389\) −15.5165 −0.786720 −0.393360 0.919385i \(-0.628687\pi\)
−0.393360 + 0.919385i \(0.628687\pi\)
\(390\) −18.4186 −0.932661
\(391\) −3.72834 −0.188550
\(392\) −0.634983 −0.0320715
\(393\) 13.6486 0.688480
\(394\) −12.8234 −0.646034
\(395\) 42.5717 2.14202
\(396\) 3.04811 0.153173
\(397\) 8.60718 0.431982 0.215991 0.976395i \(-0.430702\pi\)
0.215991 + 0.976395i \(0.430702\pi\)
\(398\) −16.3445 −0.819275
\(399\) −54.6932 −2.73809
\(400\) 7.62560 0.381280
\(401\) −6.41977 −0.320588 −0.160294 0.987069i \(-0.551244\pi\)
−0.160294 + 0.987069i \(0.551244\pi\)
\(402\) 29.8321 1.48789
\(403\) 1.62569 0.0809814
\(404\) 16.5848 0.825125
\(405\) −25.6208 −1.27311
\(406\) −24.6231 −1.22202
\(407\) −4.88803 −0.242291
\(408\) 1.19921 0.0593696
\(409\) −26.7924 −1.32480 −0.662399 0.749151i \(-0.730462\pi\)
−0.662399 + 0.749151i \(0.730462\pi\)
\(410\) −3.74730 −0.185066
\(411\) −19.6879 −0.971135
\(412\) 9.35729 0.461001
\(413\) 7.14294 0.351481
\(414\) 53.4502 2.62693
\(415\) 21.5432 1.05751
\(416\) −1.74773 −0.0856893
\(417\) 25.6339 1.25530
\(418\) −3.50940 −0.171650
\(419\) 2.70657 0.132225 0.0661123 0.997812i \(-0.478940\pi\)
0.0661123 + 0.997812i \(0.478940\pi\)
\(420\) 29.1197 1.42090
\(421\) −33.3581 −1.62577 −0.812886 0.582423i \(-0.802105\pi\)
−0.812886 + 0.582423i \(0.802105\pi\)
\(422\) −9.74762 −0.474507
\(423\) −67.6022 −3.28693
\(424\) 11.3924 0.553262
\(425\) 3.08327 0.149560
\(426\) 36.4452 1.76577
\(427\) 6.02698 0.291666
\(428\) −12.2662 −0.592912
\(429\) −2.72577 −0.131601
\(430\) −0.947484 −0.0456917
\(431\) 4.33168 0.208650 0.104325 0.994543i \(-0.466732\pi\)
0.104325 + 0.994543i \(0.466732\pi\)
\(432\) −8.29436 −0.399062
\(433\) 15.5145 0.745578 0.372789 0.927916i \(-0.378402\pi\)
0.372789 + 0.927916i \(0.378402\pi\)
\(434\) −2.57021 −0.123374
\(435\) 93.9119 4.50273
\(436\) 10.5900 0.507167
\(437\) −61.5391 −2.94381
\(438\) 33.0728 1.58028
\(439\) −11.2187 −0.535440 −0.267720 0.963497i \(-0.586270\pi\)
−0.267720 + 0.963497i \(0.586270\pi\)
\(440\) 1.86847 0.0890757
\(441\) 3.68072 0.175273
\(442\) −0.706660 −0.0336124
\(443\) 8.86735 0.421300 0.210650 0.977562i \(-0.432442\pi\)
0.210650 + 0.977562i \(0.432442\pi\)
\(444\) 27.5696 1.30840
\(445\) 59.9985 2.84420
\(446\) −23.4967 −1.11260
\(447\) −25.0035 −1.18263
\(448\) 2.76315 0.130546
\(449\) −14.5144 −0.684976 −0.342488 0.939522i \(-0.611270\pi\)
−0.342488 + 0.939522i \(0.611270\pi\)
\(450\) −44.2023 −2.08372
\(451\) −0.554564 −0.0261134
\(452\) 17.3588 0.816488
\(453\) −20.3483 −0.956047
\(454\) 15.5615 0.730338
\(455\) −17.1594 −0.804447
\(456\) 19.7938 0.926930
\(457\) 25.0221 1.17048 0.585242 0.810859i \(-0.301000\pi\)
0.585242 + 0.810859i \(0.301000\pi\)
\(458\) −14.5342 −0.679138
\(459\) −3.35367 −0.156536
\(460\) 32.7645 1.52765
\(461\) 19.7791 0.921203 0.460602 0.887607i \(-0.347634\pi\)
0.460602 + 0.887607i \(0.347634\pi\)
\(462\) 4.30943 0.200493
\(463\) −29.3979 −1.36623 −0.683117 0.730309i \(-0.739376\pi\)
−0.683117 + 0.730309i \(0.739376\pi\)
\(464\) 8.91123 0.413694
\(465\) 9.80273 0.454591
\(466\) 9.69239 0.448991
\(467\) −7.21932 −0.334070 −0.167035 0.985951i \(-0.553419\pi\)
−0.167035 + 0.985951i \(0.553419\pi\)
\(468\) 10.1308 0.468297
\(469\) 27.7927 1.28335
\(470\) −41.4396 −1.91147
\(471\) 10.1108 0.465883
\(472\) −2.58507 −0.118988
\(473\) −0.140218 −0.00644725
\(474\) −35.5347 −1.63216
\(475\) 50.8917 2.33507
\(476\) 1.11723 0.0512080
\(477\) −66.0366 −3.02361
\(478\) 20.6560 0.944785
\(479\) −40.0520 −1.83002 −0.915011 0.403428i \(-0.867819\pi\)
−0.915011 + 0.403428i \(0.867819\pi\)
\(480\) −10.5386 −0.481019
\(481\) −16.2460 −0.740756
\(482\) 19.0386 0.867185
\(483\) 75.5681 3.43847
\(484\) −10.7235 −0.487431
\(485\) 27.6546 1.25573
\(486\) −3.49734 −0.158643
\(487\) −12.1659 −0.551288 −0.275644 0.961260i \(-0.588891\pi\)
−0.275644 + 0.961260i \(0.588891\pi\)
\(488\) −2.18120 −0.0987384
\(489\) −24.6123 −1.11301
\(490\) 2.25625 0.101927
\(491\) −8.63426 −0.389659 −0.194829 0.980837i \(-0.562415\pi\)
−0.194829 + 0.980837i \(0.562415\pi\)
\(492\) 3.12787 0.141015
\(493\) 3.60309 0.162275
\(494\) −11.6640 −0.524786
\(495\) −10.8307 −0.486804
\(496\) 0.930175 0.0417661
\(497\) 33.9537 1.52303
\(498\) −17.9821 −0.805798
\(499\) 17.1640 0.768367 0.384184 0.923257i \(-0.374483\pi\)
0.384184 + 0.923257i \(0.374483\pi\)
\(500\) −9.32942 −0.417224
\(501\) 25.1180 1.12219
\(502\) −20.4574 −0.913057
\(503\) −8.05769 −0.359274 −0.179637 0.983733i \(-0.557492\pi\)
−0.179637 + 0.983733i \(0.557492\pi\)
\(504\) −16.0168 −0.713444
\(505\) −58.9300 −2.62235
\(506\) 4.84884 0.215557
\(507\) 29.4972 1.31002
\(508\) −10.8304 −0.480523
\(509\) 41.6844 1.84763 0.923814 0.382841i \(-0.125054\pi\)
0.923814 + 0.382841i \(0.125054\pi\)
\(510\) −4.26108 −0.188684
\(511\) 30.8119 1.36304
\(512\) −1.00000 −0.0441942
\(513\) −55.3548 −2.44398
\(514\) −30.5506 −1.34753
\(515\) −33.2488 −1.46512
\(516\) 0.790865 0.0348159
\(517\) −6.13266 −0.269714
\(518\) 25.6849 1.12853
\(519\) 52.1251 2.28804
\(520\) 6.21011 0.272331
\(521\) −10.7712 −0.471894 −0.235947 0.971766i \(-0.575819\pi\)
−0.235947 + 0.971766i \(0.575819\pi\)
\(522\) −51.6546 −2.26086
\(523\) −37.5747 −1.64303 −0.821513 0.570189i \(-0.806870\pi\)
−0.821513 + 0.570189i \(0.806870\pi\)
\(524\) −4.60183 −0.201032
\(525\) −62.4935 −2.72744
\(526\) −13.1452 −0.573160
\(527\) 0.376099 0.0163831
\(528\) −1.55961 −0.0678734
\(529\) 62.0268 2.69682
\(530\) −40.4799 −1.75833
\(531\) 14.9846 0.650275
\(532\) 18.4407 0.799505
\(533\) −1.84317 −0.0798365
\(534\) −50.0808 −2.16721
\(535\) 43.5851 1.88435
\(536\) −10.0583 −0.434454
\(537\) −49.4914 −2.13571
\(538\) −2.18874 −0.0943631
\(539\) 0.333904 0.0143823
\(540\) 29.4719 1.26827
\(541\) −14.1269 −0.607363 −0.303681 0.952774i \(-0.598216\pi\)
−0.303681 + 0.952774i \(0.598216\pi\)
\(542\) −3.13707 −0.134749
\(543\) 64.4398 2.76538
\(544\) −0.404331 −0.0173356
\(545\) −37.6288 −1.61184
\(546\) 14.3230 0.612968
\(547\) −32.6020 −1.39396 −0.696980 0.717091i \(-0.745473\pi\)
−0.696980 + 0.717091i \(0.745473\pi\)
\(548\) 6.63810 0.283566
\(549\) 12.6435 0.539611
\(550\) −4.00990 −0.170983
\(551\) 59.4717 2.53358
\(552\) −27.3486 −1.16403
\(553\) −33.1054 −1.40779
\(554\) −21.6799 −0.921090
\(555\) −97.9619 −4.15825
\(556\) −8.64286 −0.366539
\(557\) 12.9021 0.546681 0.273340 0.961917i \(-0.411871\pi\)
0.273340 + 0.961917i \(0.411871\pi\)
\(558\) −5.39182 −0.228254
\(559\) −0.466035 −0.0197112
\(560\) −9.81816 −0.414893
\(561\) −0.630599 −0.0266239
\(562\) −4.47708 −0.188854
\(563\) 2.86995 0.120954 0.0604771 0.998170i \(-0.480738\pi\)
0.0604771 + 0.998170i \(0.480738\pi\)
\(564\) 34.5896 1.45649
\(565\) −61.6801 −2.59490
\(566\) 30.2162 1.27008
\(567\) 19.9238 0.836719
\(568\) −12.2881 −0.515595
\(569\) −14.0169 −0.587620 −0.293810 0.955864i \(-0.594923\pi\)
−0.293810 + 0.955864i \(0.594923\pi\)
\(570\) −70.3324 −2.94590
\(571\) 6.16938 0.258181 0.129090 0.991633i \(-0.458794\pi\)
0.129090 + 0.991633i \(0.458794\pi\)
\(572\) 0.919036 0.0384268
\(573\) −39.5283 −1.65132
\(574\) 2.91404 0.121630
\(575\) −70.3157 −2.93237
\(576\) 5.79657 0.241524
\(577\) 3.97321 0.165407 0.0827034 0.996574i \(-0.473645\pi\)
0.0827034 + 0.996574i \(0.473645\pi\)
\(578\) 16.8365 0.700307
\(579\) 19.0447 0.791472
\(580\) −31.6639 −1.31477
\(581\) −16.7528 −0.695024
\(582\) −23.0833 −0.956834
\(583\) −5.99064 −0.248107
\(584\) −11.1510 −0.461432
\(585\) −35.9973 −1.48831
\(586\) 23.6619 0.977464
\(587\) 7.12695 0.294161 0.147080 0.989125i \(-0.453012\pi\)
0.147080 + 0.989125i \(0.453012\pi\)
\(588\) −1.88330 −0.0776659
\(589\) 6.20779 0.255788
\(590\) 9.18542 0.378158
\(591\) −38.0330 −1.56447
\(592\) −9.29553 −0.382044
\(593\) −7.00548 −0.287681 −0.143840 0.989601i \(-0.545945\pi\)
−0.143840 + 0.989601i \(0.545945\pi\)
\(594\) 4.36156 0.178957
\(595\) −3.96979 −0.162745
\(596\) 8.43034 0.345320
\(597\) −48.4761 −1.98400
\(598\) 16.1158 0.659023
\(599\) −27.9284 −1.14113 −0.570563 0.821254i \(-0.693275\pi\)
−0.570563 + 0.821254i \(0.693275\pi\)
\(600\) 22.6168 0.923326
\(601\) −26.2857 −1.07222 −0.536108 0.844149i \(-0.680106\pi\)
−0.536108 + 0.844149i \(0.680106\pi\)
\(602\) 0.736800 0.0300297
\(603\) 58.3039 2.37432
\(604\) 6.86075 0.279160
\(605\) 38.1032 1.54912
\(606\) 49.1889 1.99816
\(607\) −25.9106 −1.05168 −0.525839 0.850584i \(-0.676248\pi\)
−0.525839 + 0.850584i \(0.676248\pi\)
\(608\) −6.67379 −0.270658
\(609\) −73.0295 −2.95931
\(610\) 7.75036 0.313803
\(611\) −20.3827 −0.824597
\(612\) 2.34373 0.0947398
\(613\) −0.653109 −0.0263788 −0.0131894 0.999913i \(-0.504198\pi\)
−0.0131894 + 0.999913i \(0.504198\pi\)
\(614\) −21.4101 −0.864039
\(615\) −11.1141 −0.448164
\(616\) −1.45299 −0.0585428
\(617\) 11.2932 0.454649 0.227324 0.973819i \(-0.427002\pi\)
0.227324 + 0.973819i \(0.427002\pi\)
\(618\) 27.7528 1.11638
\(619\) 39.7936 1.59944 0.799720 0.600373i \(-0.204981\pi\)
0.799720 + 0.600373i \(0.204981\pi\)
\(620\) −3.30514 −0.132738
\(621\) 76.4823 3.06913
\(622\) 28.4872 1.14223
\(623\) −46.6572 −1.86928
\(624\) −5.18358 −0.207509
\(625\) −4.97823 −0.199129
\(626\) 15.6149 0.624096
\(627\) −10.4085 −0.415676
\(628\) −3.40902 −0.136035
\(629\) −3.75847 −0.149860
\(630\) 56.9117 2.26741
\(631\) −7.64944 −0.304520 −0.152260 0.988341i \(-0.548655\pi\)
−0.152260 + 0.988341i \(0.548655\pi\)
\(632\) 11.9811 0.476581
\(633\) −28.9105 −1.14909
\(634\) −19.5055 −0.774663
\(635\) 38.4833 1.52716
\(636\) 33.7886 1.33981
\(637\) 1.10978 0.0439709
\(638\) −4.68595 −0.185518
\(639\) 71.2286 2.81776
\(640\) 3.55325 0.140455
\(641\) 24.3399 0.961369 0.480684 0.876894i \(-0.340388\pi\)
0.480684 + 0.876894i \(0.340388\pi\)
\(642\) −36.3805 −1.43582
\(643\) 5.13312 0.202431 0.101215 0.994865i \(-0.467727\pi\)
0.101215 + 0.994865i \(0.467727\pi\)
\(644\) −25.4790 −1.00401
\(645\) −2.81014 −0.110649
\(646\) −2.69842 −0.106168
\(647\) −1.12682 −0.0442997 −0.0221499 0.999755i \(-0.507051\pi\)
−0.0221499 + 0.999755i \(0.507051\pi\)
\(648\) −7.21053 −0.283256
\(649\) 1.35935 0.0533593
\(650\) −13.3275 −0.522746
\(651\) −7.62299 −0.298768
\(652\) 8.29841 0.324991
\(653\) 28.9590 1.13325 0.566626 0.823975i \(-0.308248\pi\)
0.566626 + 0.823975i \(0.308248\pi\)
\(654\) 31.4087 1.22818
\(655\) 16.3515 0.638904
\(656\) −1.05461 −0.0411756
\(657\) 64.6377 2.52176
\(658\) 32.2250 1.25626
\(659\) 10.4674 0.407750 0.203875 0.978997i \(-0.434646\pi\)
0.203875 + 0.978997i \(0.434646\pi\)
\(660\) 5.54169 0.215710
\(661\) −12.4685 −0.484967 −0.242484 0.970156i \(-0.577962\pi\)
−0.242484 + 0.970156i \(0.577962\pi\)
\(662\) −23.6814 −0.920403
\(663\) −2.09588 −0.0813973
\(664\) 6.06295 0.235288
\(665\) −65.5244 −2.54093
\(666\) 53.8822 2.08789
\(667\) −82.1705 −3.18165
\(668\) −8.46894 −0.327673
\(669\) −69.6888 −2.69432
\(670\) 35.7398 1.38075
\(671\) 1.14698 0.0442786
\(672\) 8.19522 0.316138
\(673\) −2.34563 −0.0904175 −0.0452087 0.998978i \(-0.514395\pi\)
−0.0452087 + 0.998978i \(0.514395\pi\)
\(674\) −11.9124 −0.458849
\(675\) −63.2495 −2.43447
\(676\) −9.94546 −0.382518
\(677\) 16.0140 0.615469 0.307735 0.951472i \(-0.400429\pi\)
0.307735 + 0.951472i \(0.400429\pi\)
\(678\) 51.4844 1.97725
\(679\) −21.5053 −0.825297
\(680\) 1.43669 0.0550946
\(681\) 46.1539 1.76862
\(682\) −0.489130 −0.0187297
\(683\) 0.738062 0.0282412 0.0141206 0.999900i \(-0.495505\pi\)
0.0141206 + 0.999900i \(0.495505\pi\)
\(684\) 38.6851 1.47916
\(685\) −23.5868 −0.901207
\(686\) 17.5875 0.671493
\(687\) −43.1069 −1.64463
\(688\) −0.266653 −0.0101660
\(689\) −19.9107 −0.758537
\(690\) 97.1764 3.69944
\(691\) 42.8442 1.62987 0.814935 0.579552i \(-0.196773\pi\)
0.814935 + 0.579552i \(0.196773\pi\)
\(692\) −17.5748 −0.668094
\(693\) 8.42238 0.319940
\(694\) −11.1351 −0.422681
\(695\) 30.7103 1.16491
\(696\) 26.4298 1.00182
\(697\) −0.426412 −0.0161515
\(698\) 29.6691 1.12299
\(699\) 28.7467 1.08730
\(700\) 21.0707 0.796396
\(701\) −30.9266 −1.16808 −0.584041 0.811724i \(-0.698529\pi\)
−0.584041 + 0.811724i \(0.698529\pi\)
\(702\) 14.4963 0.547126
\(703\) −62.0365 −2.33975
\(704\) 0.525847 0.0198186
\(705\) −122.906 −4.62890
\(706\) −33.4239 −1.25792
\(707\) 45.8263 1.72347
\(708\) −7.66707 −0.288146
\(709\) 29.3904 1.10378 0.551889 0.833917i \(-0.313907\pi\)
0.551889 + 0.833917i \(0.313907\pi\)
\(710\) 43.6625 1.63863
\(711\) −69.4491 −2.60454
\(712\) 16.8855 0.632812
\(713\) −8.57714 −0.321216
\(714\) 3.31358 0.124008
\(715\) −3.26557 −0.122125
\(716\) 16.6868 0.623614
\(717\) 61.2637 2.28794
\(718\) 12.3282 0.460085
\(719\) −46.8953 −1.74890 −0.874450 0.485115i \(-0.838778\pi\)
−0.874450 + 0.485115i \(0.838778\pi\)
\(720\) −20.5967 −0.767593
\(721\) 25.8556 0.962912
\(722\) −25.5395 −0.950483
\(723\) 56.4667 2.10002
\(724\) −21.7269 −0.807473
\(725\) 67.9535 2.52373
\(726\) −31.8048 −1.18039
\(727\) −39.6718 −1.47134 −0.735672 0.677338i \(-0.763134\pi\)
−0.735672 + 0.677338i \(0.763134\pi\)
\(728\) −4.82922 −0.178983
\(729\) −32.0044 −1.18535
\(730\) 39.6224 1.46649
\(731\) −0.107816 −0.00398771
\(732\) −6.46923 −0.239110
\(733\) 20.3814 0.752804 0.376402 0.926456i \(-0.377161\pi\)
0.376402 + 0.926456i \(0.377161\pi\)
\(734\) 12.9043 0.476306
\(735\) 6.69183 0.246832
\(736\) 9.22100 0.339891
\(737\) 5.28915 0.194828
\(738\) 6.11313 0.225027
\(739\) −17.1026 −0.629130 −0.314565 0.949236i \(-0.601859\pi\)
−0.314565 + 0.949236i \(0.601859\pi\)
\(740\) 33.0294 1.21418
\(741\) −34.5942 −1.27085
\(742\) 31.4787 1.15562
\(743\) 2.64080 0.0968817 0.0484409 0.998826i \(-0.484575\pi\)
0.0484409 + 0.998826i \(0.484575\pi\)
\(744\) 2.75881 0.101143
\(745\) −29.9551 −1.09747
\(746\) −21.1322 −0.773706
\(747\) −35.1443 −1.28586
\(748\) 0.212616 0.00777402
\(749\) −33.8935 −1.23844
\(750\) −27.6701 −1.01037
\(751\) −5.57353 −0.203381 −0.101690 0.994816i \(-0.532425\pi\)
−0.101690 + 0.994816i \(0.532425\pi\)
\(752\) −11.6624 −0.425285
\(753\) −60.6745 −2.21110
\(754\) −15.5744 −0.567186
\(755\) −24.3780 −0.887205
\(756\) −22.9185 −0.833539
\(757\) −1.11091 −0.0403768 −0.0201884 0.999796i \(-0.506427\pi\)
−0.0201884 + 0.999796i \(0.506427\pi\)
\(758\) 23.5472 0.855274
\(759\) 14.3812 0.522003
\(760\) 23.7137 0.860185
\(761\) 40.3452 1.46251 0.731256 0.682103i \(-0.238935\pi\)
0.731256 + 0.682103i \(0.238935\pi\)
\(762\) −32.1220 −1.16366
\(763\) 29.2616 1.05934
\(764\) 13.3276 0.482175
\(765\) −8.32788 −0.301095
\(766\) 31.1585 1.12580
\(767\) 4.51800 0.163135
\(768\) −2.96590 −0.107023
\(769\) 22.7954 0.822025 0.411012 0.911630i \(-0.365175\pi\)
0.411012 + 0.911630i \(0.365175\pi\)
\(770\) 5.16285 0.186056
\(771\) −90.6102 −3.26325
\(772\) −6.42123 −0.231105
\(773\) −8.20615 −0.295155 −0.147577 0.989051i \(-0.547148\pi\)
−0.147577 + 0.989051i \(0.547148\pi\)
\(774\) 1.54567 0.0555580
\(775\) 7.09314 0.254793
\(776\) 7.78289 0.279390
\(777\) 76.1790 2.73291
\(778\) 15.5165 0.556295
\(779\) −7.03826 −0.252172
\(780\) 18.4186 0.659491
\(781\) 6.46164 0.231216
\(782\) 3.72834 0.133325
\(783\) −73.9130 −2.64143
\(784\) 0.634983 0.0226780
\(785\) 12.1131 0.432336
\(786\) −13.6486 −0.486829
\(787\) 10.6059 0.378058 0.189029 0.981971i \(-0.439466\pi\)
0.189029 + 0.981971i \(0.439466\pi\)
\(788\) 12.8234 0.456815
\(789\) −38.9875 −1.38799
\(790\) −42.5717 −1.51463
\(791\) 47.9649 1.70543
\(792\) −3.04811 −0.108310
\(793\) 3.81214 0.135373
\(794\) −8.60718 −0.305457
\(795\) −120.059 −4.25807
\(796\) 16.3445 0.579315
\(797\) 13.4124 0.475092 0.237546 0.971376i \(-0.423657\pi\)
0.237546 + 0.971376i \(0.423657\pi\)
\(798\) 54.6932 1.93612
\(799\) −4.71549 −0.166822
\(800\) −7.62560 −0.269606
\(801\) −97.8781 −3.45835
\(802\) 6.41977 0.226690
\(803\) 5.86373 0.206927
\(804\) −29.8321 −1.05210
\(805\) 90.5332 3.19088
\(806\) −1.62569 −0.0572625
\(807\) −6.49157 −0.228514
\(808\) −16.5848 −0.583452
\(809\) −22.0775 −0.776203 −0.388101 0.921617i \(-0.626869\pi\)
−0.388101 + 0.921617i \(0.626869\pi\)
\(810\) 25.6208 0.900225
\(811\) −13.7393 −0.482453 −0.241227 0.970469i \(-0.577550\pi\)
−0.241227 + 0.970469i \(0.577550\pi\)
\(812\) 24.6231 0.864100
\(813\) −9.30424 −0.326314
\(814\) 4.88803 0.171325
\(815\) −29.4864 −1.03286
\(816\) −1.19921 −0.0419806
\(817\) −1.77958 −0.0622598
\(818\) 26.7924 0.936774
\(819\) 27.9929 0.978152
\(820\) 3.74730 0.130861
\(821\) −39.2727 −1.37063 −0.685314 0.728248i \(-0.740335\pi\)
−0.685314 + 0.728248i \(0.740335\pi\)
\(822\) 19.6879 0.686696
\(823\) −2.25543 −0.0786193 −0.0393096 0.999227i \(-0.512516\pi\)
−0.0393096 + 0.999227i \(0.512516\pi\)
\(824\) −9.35729 −0.325977
\(825\) −11.8930 −0.414060
\(826\) −7.14294 −0.248535
\(827\) −24.5139 −0.852433 −0.426216 0.904621i \(-0.640154\pi\)
−0.426216 + 0.904621i \(0.640154\pi\)
\(828\) −53.4502 −1.85752
\(829\) −13.4813 −0.468225 −0.234112 0.972210i \(-0.575218\pi\)
−0.234112 + 0.972210i \(0.575218\pi\)
\(830\) −21.5432 −0.747775
\(831\) −64.3004 −2.23056
\(832\) 1.74773 0.0605915
\(833\) 0.256743 0.00889563
\(834\) −25.6339 −0.887628
\(835\) 30.0923 1.04139
\(836\) 3.50940 0.121375
\(837\) −7.71520 −0.266676
\(838\) −2.70657 −0.0934969
\(839\) −10.7520 −0.371200 −0.185600 0.982625i \(-0.559423\pi\)
−0.185600 + 0.982625i \(0.559423\pi\)
\(840\) −29.1197 −1.00472
\(841\) 50.4101 1.73828
\(842\) 33.3581 1.14959
\(843\) −13.2786 −0.457339
\(844\) 9.74762 0.335527
\(845\) 35.3387 1.21569
\(846\) 67.6022 2.32421
\(847\) −29.6306 −1.01812
\(848\) −11.3924 −0.391215
\(849\) 89.6182 3.07569
\(850\) −3.08327 −0.105755
\(851\) 85.7141 2.93824
\(852\) −36.4452 −1.24859
\(853\) −37.8022 −1.29432 −0.647161 0.762354i \(-0.724044\pi\)
−0.647161 + 0.762354i \(0.724044\pi\)
\(854\) −6.02698 −0.206239
\(855\) −137.458 −4.70097
\(856\) 12.2662 0.419252
\(857\) 29.9124 1.02179 0.510895 0.859643i \(-0.329314\pi\)
0.510895 + 0.859643i \(0.329314\pi\)
\(858\) 2.72577 0.0930563
\(859\) 9.07654 0.309687 0.154844 0.987939i \(-0.450513\pi\)
0.154844 + 0.987939i \(0.450513\pi\)
\(860\) 0.947484 0.0323089
\(861\) 8.64277 0.294545
\(862\) −4.33168 −0.147538
\(863\) −29.2026 −0.994068 −0.497034 0.867731i \(-0.665578\pi\)
−0.497034 + 0.867731i \(0.665578\pi\)
\(864\) 8.29436 0.282180
\(865\) 62.4477 2.12329
\(866\) −15.5145 −0.527203
\(867\) 49.9355 1.69590
\(868\) 2.57021 0.0872386
\(869\) −6.30021 −0.213720
\(870\) −93.9119 −3.18391
\(871\) 17.5792 0.595649
\(872\) −10.5900 −0.358621
\(873\) −45.1141 −1.52688
\(874\) 61.5391 2.08159
\(875\) −25.7786 −0.871474
\(876\) −33.0728 −1.11743
\(877\) 23.3461 0.788342 0.394171 0.919037i \(-0.371032\pi\)
0.394171 + 0.919037i \(0.371032\pi\)
\(878\) 11.2187 0.378613
\(879\) 70.1788 2.36707
\(880\) −1.86847 −0.0629860
\(881\) −48.0477 −1.61877 −0.809385 0.587279i \(-0.800199\pi\)
−0.809385 + 0.587279i \(0.800199\pi\)
\(882\) −3.68072 −0.123936
\(883\) −14.7204 −0.495382 −0.247691 0.968839i \(-0.579672\pi\)
−0.247691 + 0.968839i \(0.579672\pi\)
\(884\) 0.706660 0.0237675
\(885\) 27.2430 0.915765
\(886\) −8.86735 −0.297904
\(887\) 42.0926 1.41333 0.706665 0.707548i \(-0.250199\pi\)
0.706665 + 0.707548i \(0.250199\pi\)
\(888\) −27.5696 −0.925176
\(889\) −29.9261 −1.00369
\(890\) −59.9985 −2.01115
\(891\) 3.79164 0.127025
\(892\) 23.4967 0.786726
\(893\) −77.8327 −2.60457
\(894\) 25.0035 0.836244
\(895\) −59.2923 −1.98192
\(896\) −2.76315 −0.0923103
\(897\) 47.7978 1.59592
\(898\) 14.5144 0.484351
\(899\) 8.28900 0.276454
\(900\) 44.2023 1.47341
\(901\) −4.60628 −0.153458
\(902\) 0.554564 0.0184650
\(903\) 2.18528 0.0727215
\(904\) −17.3588 −0.577345
\(905\) 77.2011 2.56625
\(906\) 20.3483 0.676027
\(907\) 12.6681 0.420636 0.210318 0.977633i \(-0.432550\pi\)
0.210318 + 0.977633i \(0.432550\pi\)
\(908\) −15.5615 −0.516427
\(909\) 96.1351 3.18860
\(910\) 17.1594 0.568830
\(911\) 18.8793 0.625498 0.312749 0.949836i \(-0.398750\pi\)
0.312749 + 0.949836i \(0.398750\pi\)
\(912\) −19.7938 −0.655439
\(913\) −3.18819 −0.105514
\(914\) −25.0221 −0.827657
\(915\) 22.9868 0.759920
\(916\) 14.5342 0.480223
\(917\) −12.7155 −0.419904
\(918\) 3.35367 0.110687
\(919\) 8.08733 0.266777 0.133388 0.991064i \(-0.457414\pi\)
0.133388 + 0.991064i \(0.457414\pi\)
\(920\) −32.7645 −1.08021
\(921\) −63.5001 −2.09240
\(922\) −19.7791 −0.651389
\(923\) 21.4761 0.706896
\(924\) −4.30943 −0.141770
\(925\) −70.8840 −2.33065
\(926\) 29.3979 0.966074
\(927\) 54.2402 1.78148
\(928\) −8.91123 −0.292526
\(929\) −26.8763 −0.881784 −0.440892 0.897560i \(-0.645338\pi\)
−0.440892 + 0.897560i \(0.645338\pi\)
\(930\) −9.80273 −0.321444
\(931\) 4.23775 0.138887
\(932\) −9.69239 −0.317485
\(933\) 84.4901 2.76608
\(934\) 7.21932 0.236223
\(935\) −0.755479 −0.0247068
\(936\) −10.1308 −0.331136
\(937\) −3.67287 −0.119987 −0.0599937 0.998199i \(-0.519108\pi\)
−0.0599937 + 0.998199i \(0.519108\pi\)
\(938\) −27.7927 −0.907464
\(939\) 46.3122 1.51134
\(940\) 41.4396 1.35161
\(941\) 14.7441 0.480643 0.240321 0.970693i \(-0.422747\pi\)
0.240321 + 0.970693i \(0.422747\pi\)
\(942\) −10.1108 −0.329429
\(943\) 9.72457 0.316675
\(944\) 2.58507 0.0841370
\(945\) 81.4353 2.64909
\(946\) 0.140218 0.00455890
\(947\) −29.9323 −0.972670 −0.486335 0.873772i \(-0.661667\pi\)
−0.486335 + 0.873772i \(0.661667\pi\)
\(948\) 35.5347 1.15411
\(949\) 19.4889 0.632637
\(950\) −50.8917 −1.65114
\(951\) −57.8514 −1.87596
\(952\) −1.11723 −0.0362095
\(953\) 35.0333 1.13484 0.567420 0.823429i \(-0.307942\pi\)
0.567420 + 0.823429i \(0.307942\pi\)
\(954\) 66.0366 2.13801
\(955\) −47.3563 −1.53241
\(956\) −20.6560 −0.668064
\(957\) −13.8981 −0.449260
\(958\) 40.0520 1.29402
\(959\) 18.3420 0.592296
\(960\) 10.5386 0.340132
\(961\) −30.1348 −0.972090
\(962\) 16.2460 0.523793
\(963\) −71.1022 −2.29124
\(964\) −19.0386 −0.613192
\(965\) 22.8163 0.734481
\(966\) −75.5681 −2.43137
\(967\) 22.8316 0.734214 0.367107 0.930179i \(-0.380348\pi\)
0.367107 + 0.930179i \(0.380348\pi\)
\(968\) 10.7235 0.344666
\(969\) −8.00326 −0.257102
\(970\) −27.6546 −0.887935
\(971\) −57.0807 −1.83181 −0.915904 0.401397i \(-0.868525\pi\)
−0.915904 + 0.401397i \(0.868525\pi\)
\(972\) 3.49734 0.112177
\(973\) −23.8815 −0.765605
\(974\) 12.1659 0.389820
\(975\) −39.5279 −1.26591
\(976\) 2.18120 0.0698186
\(977\) −3.91047 −0.125107 −0.0625535 0.998042i \(-0.519924\pi\)
−0.0625535 + 0.998042i \(0.519924\pi\)
\(978\) 24.6123 0.787014
\(979\) −8.87920 −0.283781
\(980\) −2.25625 −0.0720734
\(981\) 61.3854 1.95989
\(982\) 8.63426 0.275530
\(983\) −41.5646 −1.32571 −0.662853 0.748749i \(-0.730655\pi\)
−0.662853 + 0.748749i \(0.730655\pi\)
\(984\) −3.12787 −0.0997129
\(985\) −45.5648 −1.45182
\(986\) −3.60309 −0.114746
\(987\) 95.5763 3.04223
\(988\) 11.6640 0.371080
\(989\) 2.45880 0.0781854
\(990\) 10.8307 0.344222
\(991\) 27.5585 0.875424 0.437712 0.899115i \(-0.355789\pi\)
0.437712 + 0.899115i \(0.355789\pi\)
\(992\) −0.930175 −0.0295331
\(993\) −70.2366 −2.22889
\(994\) −33.9537 −1.07695
\(995\) −58.0761 −1.84113
\(996\) 17.9821 0.569785
\(997\) −41.2516 −1.30645 −0.653226 0.757163i \(-0.726585\pi\)
−0.653226 + 0.757163i \(0.726585\pi\)
\(998\) −17.1640 −0.543318
\(999\) 77.1005 2.43935
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 8002.2.a.f.1.8 89
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8002.2.a.f.1.8 89 1.1 even 1 trivial