Properties

Label 4031.2.a.c.1.43
Level $4031$
Weight $2$
Character 4031.1
Self dual yes
Analytic conductor $32.188$
Analytic rank $1$
Dimension $61$
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] = [4031,2,Mod(1,4031)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4031, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4031.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4031 = 29 \cdot 139 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4031.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: \(32.1876970548\)
Analytic rank: \(1\)
Dimension: \(61\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.43
Character \(\chi\) \(=\) 4031.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.19557 q^{2} -0.378969 q^{3} -0.570603 q^{4} +0.0302315 q^{5} -0.453085 q^{6} +3.94410 q^{7} -3.07335 q^{8} -2.85638 q^{9} +O(q^{10})\) \(q+1.19557 q^{2} -0.378969 q^{3} -0.570603 q^{4} +0.0302315 q^{5} -0.453085 q^{6} +3.94410 q^{7} -3.07335 q^{8} -2.85638 q^{9} +0.0361440 q^{10} +1.87601 q^{11} +0.216240 q^{12} +0.677227 q^{13} +4.71547 q^{14} -0.0114568 q^{15} -2.53321 q^{16} -4.16638 q^{17} -3.41502 q^{18} -3.45637 q^{19} -0.0172502 q^{20} -1.49469 q^{21} +2.24291 q^{22} +6.14590 q^{23} +1.16470 q^{24} -4.99909 q^{25} +0.809675 q^{26} +2.21938 q^{27} -2.25052 q^{28} -1.00000 q^{29} -0.0136974 q^{30} -3.10222 q^{31} +3.11805 q^{32} -0.710949 q^{33} -4.98122 q^{34} +0.119236 q^{35} +1.62986 q^{36} -4.04246 q^{37} -4.13235 q^{38} -0.256648 q^{39} -0.0929118 q^{40} -6.63153 q^{41} -1.78701 q^{42} -3.15900 q^{43} -1.07046 q^{44} -0.0863527 q^{45} +7.34788 q^{46} +1.79205 q^{47} +0.960006 q^{48} +8.55594 q^{49} -5.97678 q^{50} +1.57893 q^{51} -0.386428 q^{52} +5.84228 q^{53} +2.65344 q^{54} +0.0567146 q^{55} -12.1216 q^{56} +1.30986 q^{57} -1.19557 q^{58} -7.81490 q^{59} +0.00653727 q^{60} +6.45746 q^{61} -3.70893 q^{62} -11.2659 q^{63} +8.79428 q^{64} +0.0204736 q^{65} -0.849993 q^{66} -13.0351 q^{67} +2.37735 q^{68} -2.32910 q^{69} +0.142555 q^{70} +1.57776 q^{71} +8.77865 q^{72} +0.772201 q^{73} -4.83306 q^{74} +1.89450 q^{75} +1.97222 q^{76} +7.39918 q^{77} -0.306841 q^{78} -7.96706 q^{79} -0.0765826 q^{80} +7.72807 q^{81} -7.92848 q^{82} -10.1333 q^{83} +0.852874 q^{84} -0.125956 q^{85} -3.77682 q^{86} +0.378969 q^{87} -5.76563 q^{88} -7.68535 q^{89} -0.103241 q^{90} +2.67105 q^{91} -3.50687 q^{92} +1.17564 q^{93} +2.14253 q^{94} -0.104491 q^{95} -1.18164 q^{96} +4.70310 q^{97} +10.2293 q^{98} -5.35861 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.19557 0.845399 0.422699 0.906270i \(-0.361083\pi\)
0.422699 + 0.906270i \(0.361083\pi\)
\(3\) −0.378969 −0.218798 −0.109399 0.993998i \(-0.534893\pi\)
−0.109399 + 0.993998i \(0.534893\pi\)
\(4\) −0.570603 −0.285301
\(5\) 0.0302315 0.0135199 0.00675996 0.999977i \(-0.497848\pi\)
0.00675996 + 0.999977i \(0.497848\pi\)
\(6\) −0.453085 −0.184971
\(7\) 3.94410 1.49073 0.745365 0.666656i \(-0.232275\pi\)
0.745365 + 0.666656i \(0.232275\pi\)
\(8\) −3.07335 −1.08659
\(9\) −2.85638 −0.952128
\(10\) 0.0361440 0.0114297
\(11\) 1.87601 0.565639 0.282819 0.959173i \(-0.408730\pi\)
0.282819 + 0.959173i \(0.408730\pi\)
\(12\) 0.216240 0.0624232
\(13\) 0.677227 0.187829 0.0939145 0.995580i \(-0.470062\pi\)
0.0939145 + 0.995580i \(0.470062\pi\)
\(14\) 4.71547 1.26026
\(15\) −0.0114568 −0.00295813
\(16\) −2.53321 −0.633302
\(17\) −4.16638 −1.01050 −0.505248 0.862974i \(-0.668599\pi\)
−0.505248 + 0.862974i \(0.668599\pi\)
\(18\) −3.41502 −0.804927
\(19\) −3.45637 −0.792946 −0.396473 0.918046i \(-0.629766\pi\)
−0.396473 + 0.918046i \(0.629766\pi\)
\(20\) −0.0172502 −0.00385725
\(21\) −1.49469 −0.326168
\(22\) 2.24291 0.478190
\(23\) 6.14590 1.28151 0.640755 0.767746i \(-0.278622\pi\)
0.640755 + 0.767746i \(0.278622\pi\)
\(24\) 1.16470 0.237744
\(25\) −4.99909 −0.999817
\(26\) 0.809675 0.158790
\(27\) 2.21938 0.427121
\(28\) −2.25052 −0.425307
\(29\) −1.00000 −0.185695
\(30\) −0.0136974 −0.00250080
\(31\) −3.10222 −0.557174 −0.278587 0.960411i \(-0.589866\pi\)
−0.278587 + 0.960411i \(0.589866\pi\)
\(32\) 3.11805 0.551199
\(33\) −0.710949 −0.123760
\(34\) −4.98122 −0.854272
\(35\) 0.119236 0.0201546
\(36\) 1.62986 0.271643
\(37\) −4.04246 −0.664576 −0.332288 0.943178i \(-0.607821\pi\)
−0.332288 + 0.943178i \(0.607821\pi\)
\(38\) −4.13235 −0.670356
\(39\) −0.256648 −0.0410965
\(40\) −0.0929118 −0.0146906
\(41\) −6.63153 −1.03567 −0.517835 0.855480i \(-0.673262\pi\)
−0.517835 + 0.855480i \(0.673262\pi\)
\(42\) −1.78701 −0.275742
\(43\) −3.15900 −0.481743 −0.240872 0.970557i \(-0.577433\pi\)
−0.240872 + 0.970557i \(0.577433\pi\)
\(44\) −1.07046 −0.161378
\(45\) −0.0863527 −0.0128727
\(46\) 7.34788 1.08339
\(47\) 1.79205 0.261397 0.130698 0.991422i \(-0.458278\pi\)
0.130698 + 0.991422i \(0.458278\pi\)
\(48\) 0.960006 0.138565
\(49\) 8.55594 1.22228
\(50\) −5.97678 −0.845244
\(51\) 1.57893 0.221094
\(52\) −0.386428 −0.0535879
\(53\) 5.84228 0.802499 0.401249 0.915969i \(-0.368576\pi\)
0.401249 + 0.915969i \(0.368576\pi\)
\(54\) 2.65344 0.361087
\(55\) 0.0567146 0.00764739
\(56\) −12.1216 −1.61982
\(57\) 1.30986 0.173495
\(58\) −1.19557 −0.156987
\(59\) −7.81490 −1.01741 −0.508707 0.860940i \(-0.669876\pi\)
−0.508707 + 0.860940i \(0.669876\pi\)
\(60\) 0.00653727 0.000843957 0
\(61\) 6.45746 0.826793 0.413397 0.910551i \(-0.364342\pi\)
0.413397 + 0.910551i \(0.364342\pi\)
\(62\) −3.70893 −0.471034
\(63\) −11.2659 −1.41937
\(64\) 8.79428 1.09929
\(65\) 0.0204736 0.00253943
\(66\) −0.849993 −0.104627
\(67\) −13.0351 −1.59249 −0.796243 0.604977i \(-0.793182\pi\)
−0.796243 + 0.604977i \(0.793182\pi\)
\(68\) 2.37735 0.288296
\(69\) −2.32910 −0.280391
\(70\) 0.142555 0.0170386
\(71\) 1.57776 0.187246 0.0936231 0.995608i \(-0.470155\pi\)
0.0936231 + 0.995608i \(0.470155\pi\)
\(72\) 8.77865 1.03457
\(73\) 0.772201 0.0903793 0.0451897 0.998978i \(-0.485611\pi\)
0.0451897 + 0.998978i \(0.485611\pi\)
\(74\) −4.83306 −0.561832
\(75\) 1.89450 0.218758
\(76\) 1.97222 0.226229
\(77\) 7.39918 0.843215
\(78\) −0.306841 −0.0347430
\(79\) −7.96706 −0.896364 −0.448182 0.893942i \(-0.647928\pi\)
−0.448182 + 0.893942i \(0.647928\pi\)
\(80\) −0.0765826 −0.00856219
\(81\) 7.72807 0.858675
\(82\) −7.92848 −0.875555
\(83\) −10.1333 −1.11228 −0.556140 0.831089i \(-0.687718\pi\)
−0.556140 + 0.831089i \(0.687718\pi\)
\(84\) 0.852874 0.0930562
\(85\) −0.125956 −0.0136618
\(86\) −3.77682 −0.407265
\(87\) 0.378969 0.0406297
\(88\) −5.76563 −0.614619
\(89\) −7.68535 −0.814646 −0.407323 0.913284i \(-0.633538\pi\)
−0.407323 + 0.913284i \(0.633538\pi\)
\(90\) −0.103241 −0.0108826
\(91\) 2.67105 0.280002
\(92\) −3.50687 −0.365616
\(93\) 1.17564 0.121908
\(94\) 2.14253 0.220985
\(95\) −0.104491 −0.0107206
\(96\) −1.18164 −0.120601
\(97\) 4.70310 0.477527 0.238763 0.971078i \(-0.423258\pi\)
0.238763 + 0.971078i \(0.423258\pi\)
\(98\) 10.2293 1.03331
\(99\) −5.35861 −0.538560
\(100\) 2.85249 0.285249
\(101\) −10.3987 −1.03471 −0.517356 0.855770i \(-0.673084\pi\)
−0.517356 + 0.855770i \(0.673084\pi\)
\(102\) 1.88773 0.186913
\(103\) 12.0298 1.18533 0.592666 0.805448i \(-0.298075\pi\)
0.592666 + 0.805448i \(0.298075\pi\)
\(104\) −2.08135 −0.204093
\(105\) −0.0451867 −0.00440977
\(106\) 6.98488 0.678431
\(107\) −18.8308 −1.82045 −0.910223 0.414119i \(-0.864090\pi\)
−0.910223 + 0.414119i \(0.864090\pi\)
\(108\) −1.26639 −0.121858
\(109\) −0.644546 −0.0617363 −0.0308681 0.999523i \(-0.509827\pi\)
−0.0308681 + 0.999523i \(0.509827\pi\)
\(110\) 0.0678065 0.00646510
\(111\) 1.53197 0.145408
\(112\) −9.99123 −0.944082
\(113\) 11.6268 1.09376 0.546880 0.837211i \(-0.315815\pi\)
0.546880 + 0.837211i \(0.315815\pi\)
\(114\) 1.56603 0.146672
\(115\) 0.185800 0.0173259
\(116\) 0.570603 0.0529791
\(117\) −1.93442 −0.178837
\(118\) −9.34329 −0.860120
\(119\) −16.4326 −1.50638
\(120\) 0.0352106 0.00321428
\(121\) −7.48058 −0.680053
\(122\) 7.72037 0.698970
\(123\) 2.51314 0.226602
\(124\) 1.77013 0.158963
\(125\) −0.302287 −0.0270374
\(126\) −13.4692 −1.19993
\(127\) −4.49114 −0.398524 −0.199262 0.979946i \(-0.563855\pi\)
−0.199262 + 0.979946i \(0.563855\pi\)
\(128\) 4.27810 0.378135
\(129\) 1.19716 0.105404
\(130\) 0.0244777 0.00214683
\(131\) 8.42874 0.736423 0.368211 0.929742i \(-0.379970\pi\)
0.368211 + 0.929742i \(0.379970\pi\)
\(132\) 0.405670 0.0353090
\(133\) −13.6323 −1.18207
\(134\) −15.5844 −1.34629
\(135\) 0.0670953 0.00577464
\(136\) 12.8047 1.09800
\(137\) −21.2023 −1.81143 −0.905716 0.423885i \(-0.860666\pi\)
−0.905716 + 0.423885i \(0.860666\pi\)
\(138\) −2.78462 −0.237042
\(139\) −1.00000 −0.0848189
\(140\) −0.0680364 −0.00575012
\(141\) −0.679130 −0.0571930
\(142\) 1.88633 0.158298
\(143\) 1.27049 0.106243
\(144\) 7.23581 0.602984
\(145\) −0.0302315 −0.00251059
\(146\) 0.923224 0.0764066
\(147\) −3.24243 −0.267431
\(148\) 2.30664 0.189605
\(149\) 4.97969 0.407952 0.203976 0.978976i \(-0.434614\pi\)
0.203976 + 0.978976i \(0.434614\pi\)
\(150\) 2.26501 0.184937
\(151\) −18.4816 −1.50401 −0.752004 0.659158i \(-0.770913\pi\)
−0.752004 + 0.659158i \(0.770913\pi\)
\(152\) 10.6226 0.861609
\(153\) 11.9008 0.962122
\(154\) 8.84627 0.712853
\(155\) −0.0937845 −0.00753295
\(156\) 0.146444 0.0117249
\(157\) 5.86492 0.468072 0.234036 0.972228i \(-0.424807\pi\)
0.234036 + 0.972228i \(0.424807\pi\)
\(158\) −9.52521 −0.757785
\(159\) −2.21404 −0.175585
\(160\) 0.0942634 0.00745217
\(161\) 24.2401 1.91039
\(162\) 9.23948 0.725922
\(163\) 9.25562 0.724956 0.362478 0.931992i \(-0.381931\pi\)
0.362478 + 0.931992i \(0.381931\pi\)
\(164\) 3.78397 0.295478
\(165\) −0.0214930 −0.00167323
\(166\) −12.1152 −0.940319
\(167\) −15.6357 −1.20993 −0.604965 0.796252i \(-0.706813\pi\)
−0.604965 + 0.796252i \(0.706813\pi\)
\(168\) 4.59370 0.354412
\(169\) −12.5414 −0.964720
\(170\) −0.150590 −0.0115497
\(171\) 9.87272 0.754986
\(172\) 1.80254 0.137442
\(173\) −21.0242 −1.59844 −0.799221 0.601038i \(-0.794754\pi\)
−0.799221 + 0.601038i \(0.794754\pi\)
\(174\) 0.453085 0.0343483
\(175\) −19.7169 −1.49046
\(176\) −4.75233 −0.358220
\(177\) 2.96160 0.222608
\(178\) −9.18841 −0.688700
\(179\) 1.91109 0.142841 0.0714207 0.997446i \(-0.477247\pi\)
0.0714207 + 0.997446i \(0.477247\pi\)
\(180\) 0.0492731 0.00367260
\(181\) 20.3001 1.50890 0.754448 0.656359i \(-0.227904\pi\)
0.754448 + 0.656359i \(0.227904\pi\)
\(182\) 3.19344 0.236714
\(183\) −2.44717 −0.180900
\(184\) −18.8885 −1.39248
\(185\) −0.122210 −0.00898502
\(186\) 1.40557 0.103061
\(187\) −7.81619 −0.571576
\(188\) −1.02255 −0.0745769
\(189\) 8.75348 0.636722
\(190\) −0.124927 −0.00906316
\(191\) −13.1332 −0.950285 −0.475143 0.879909i \(-0.657604\pi\)
−0.475143 + 0.879909i \(0.657604\pi\)
\(192\) −3.33276 −0.240521
\(193\) 15.8725 1.14253 0.571264 0.820766i \(-0.306453\pi\)
0.571264 + 0.820766i \(0.306453\pi\)
\(194\) 5.62290 0.403701
\(195\) −0.00775884 −0.000555622 0
\(196\) −4.88204 −0.348717
\(197\) −19.2357 −1.37049 −0.685245 0.728313i \(-0.740305\pi\)
−0.685245 + 0.728313i \(0.740305\pi\)
\(198\) −6.40661 −0.455298
\(199\) 1.62911 0.115485 0.0577423 0.998332i \(-0.481610\pi\)
0.0577423 + 0.998332i \(0.481610\pi\)
\(200\) 15.3639 1.08639
\(201\) 4.93988 0.348432
\(202\) −12.4324 −0.874744
\(203\) −3.94410 −0.276822
\(204\) −0.900941 −0.0630785
\(205\) −0.200481 −0.0140022
\(206\) 14.3825 1.00208
\(207\) −17.5551 −1.22016
\(208\) −1.71556 −0.118952
\(209\) −6.48420 −0.448521
\(210\) −0.0540240 −0.00372801
\(211\) 11.7265 0.807285 0.403642 0.914917i \(-0.367744\pi\)
0.403642 + 0.914917i \(0.367744\pi\)
\(212\) −3.33362 −0.228954
\(213\) −0.597923 −0.0409690
\(214\) −22.5137 −1.53900
\(215\) −0.0955013 −0.00651313
\(216\) −6.82094 −0.464106
\(217\) −12.2355 −0.830597
\(218\) −0.770602 −0.0521918
\(219\) −0.292640 −0.0197748
\(220\) −0.0323615 −0.00218181
\(221\) −2.82159 −0.189801
\(222\) 1.83158 0.122927
\(223\) −22.6519 −1.51688 −0.758441 0.651742i \(-0.774039\pi\)
−0.758441 + 0.651742i \(0.774039\pi\)
\(224\) 12.2979 0.821690
\(225\) 14.2793 0.951954
\(226\) 13.9007 0.924663
\(227\) 10.7530 0.713705 0.356852 0.934161i \(-0.383850\pi\)
0.356852 + 0.934161i \(0.383850\pi\)
\(228\) −0.747407 −0.0494983
\(229\) 7.38393 0.487944 0.243972 0.969782i \(-0.421550\pi\)
0.243972 + 0.969782i \(0.421550\pi\)
\(230\) 0.222137 0.0146473
\(231\) −2.80406 −0.184493
\(232\) 3.07335 0.201775
\(233\) 15.9135 1.04253 0.521265 0.853395i \(-0.325460\pi\)
0.521265 + 0.853395i \(0.325460\pi\)
\(234\) −2.31274 −0.151189
\(235\) 0.0541762 0.00353407
\(236\) 4.45920 0.290269
\(237\) 3.01926 0.196122
\(238\) −19.6464 −1.27349
\(239\) −24.1341 −1.56111 −0.780553 0.625089i \(-0.785063\pi\)
−0.780553 + 0.625089i \(0.785063\pi\)
\(240\) 0.0290224 0.00187339
\(241\) −18.1709 −1.17049 −0.585247 0.810855i \(-0.699002\pi\)
−0.585247 + 0.810855i \(0.699002\pi\)
\(242\) −8.94359 −0.574916
\(243\) −9.58685 −0.614997
\(244\) −3.68464 −0.235885
\(245\) 0.258659 0.0165251
\(246\) 3.00465 0.191569
\(247\) −2.34075 −0.148938
\(248\) 9.53418 0.605421
\(249\) 3.84022 0.243364
\(250\) −0.361407 −0.0228574
\(251\) −9.09251 −0.573914 −0.286957 0.957943i \(-0.592644\pi\)
−0.286957 + 0.957943i \(0.592644\pi\)
\(252\) 6.42833 0.404947
\(253\) 11.5298 0.724872
\(254\) −5.36949 −0.336912
\(255\) 0.0477333 0.00298918
\(256\) −12.4738 −0.779611
\(257\) 20.6452 1.28781 0.643907 0.765104i \(-0.277312\pi\)
0.643907 + 0.765104i \(0.277312\pi\)
\(258\) 1.43130 0.0891086
\(259\) −15.9439 −0.990704
\(260\) −0.0116823 −0.000724504 0
\(261\) 2.85638 0.176806
\(262\) 10.0772 0.622571
\(263\) −8.88540 −0.547898 −0.273949 0.961744i \(-0.588330\pi\)
−0.273949 + 0.961744i \(0.588330\pi\)
\(264\) 2.18499 0.134477
\(265\) 0.176621 0.0108497
\(266\) −16.2984 −0.999320
\(267\) 2.91251 0.178243
\(268\) 7.43784 0.454338
\(269\) 9.39241 0.572665 0.286333 0.958130i \(-0.407564\pi\)
0.286333 + 0.958130i \(0.407564\pi\)
\(270\) 0.0802174 0.00488187
\(271\) −12.0112 −0.729628 −0.364814 0.931080i \(-0.618867\pi\)
−0.364814 + 0.931080i \(0.618867\pi\)
\(272\) 10.5543 0.639949
\(273\) −1.01225 −0.0612639
\(274\) −25.3489 −1.53138
\(275\) −9.37834 −0.565535
\(276\) 1.32899 0.0799960
\(277\) −15.5181 −0.932390 −0.466195 0.884682i \(-0.654376\pi\)
−0.466195 + 0.884682i \(0.654376\pi\)
\(278\) −1.19557 −0.0717058
\(279\) 8.86111 0.530501
\(280\) −0.366454 −0.0218998
\(281\) 20.3300 1.21279 0.606395 0.795164i \(-0.292615\pi\)
0.606395 + 0.795164i \(0.292615\pi\)
\(282\) −0.811950 −0.0483509
\(283\) 8.62730 0.512840 0.256420 0.966565i \(-0.417457\pi\)
0.256420 + 0.966565i \(0.417457\pi\)
\(284\) −0.900277 −0.0534216
\(285\) 0.0395989 0.00234564
\(286\) 1.51896 0.0898180
\(287\) −26.1554 −1.54391
\(288\) −8.90636 −0.524812
\(289\) 0.358757 0.0211034
\(290\) −0.0361440 −0.00212245
\(291\) −1.78232 −0.104482
\(292\) −0.440620 −0.0257853
\(293\) 15.3786 0.898427 0.449213 0.893424i \(-0.351704\pi\)
0.449213 + 0.893424i \(0.351704\pi\)
\(294\) −3.87657 −0.226086
\(295\) −0.236256 −0.0137554
\(296\) 12.4239 0.722123
\(297\) 4.16359 0.241596
\(298\) 5.95359 0.344882
\(299\) 4.16217 0.240705
\(300\) −1.08100 −0.0624118
\(301\) −12.4594 −0.718150
\(302\) −22.0961 −1.27149
\(303\) 3.94079 0.226392
\(304\) 8.75571 0.502174
\(305\) 0.195219 0.0111782
\(306\) 14.2283 0.813376
\(307\) −3.69798 −0.211055 −0.105527 0.994416i \(-0.533653\pi\)
−0.105527 + 0.994416i \(0.533653\pi\)
\(308\) −4.22199 −0.240570
\(309\) −4.55892 −0.259348
\(310\) −0.112126 −0.00636835
\(311\) 25.1148 1.42413 0.712064 0.702114i \(-0.247760\pi\)
0.712064 + 0.702114i \(0.247760\pi\)
\(312\) 0.788767 0.0446552
\(313\) 31.9991 1.80870 0.904348 0.426797i \(-0.140358\pi\)
0.904348 + 0.426797i \(0.140358\pi\)
\(314\) 7.01195 0.395707
\(315\) −0.340584 −0.0191897
\(316\) 4.54603 0.255734
\(317\) 32.4190 1.82083 0.910417 0.413691i \(-0.135761\pi\)
0.910417 + 0.413691i \(0.135761\pi\)
\(318\) −2.64705 −0.148439
\(319\) −1.87601 −0.105037
\(320\) 0.265864 0.0148623
\(321\) 7.13629 0.398309
\(322\) 28.9808 1.61504
\(323\) 14.4006 0.801269
\(324\) −4.40966 −0.244981
\(325\) −3.38552 −0.187795
\(326\) 11.0658 0.612877
\(327\) 0.244262 0.0135077
\(328\) 20.3810 1.12535
\(329\) 7.06802 0.389672
\(330\) −0.0256965 −0.00141455
\(331\) 29.3420 1.61278 0.806392 0.591381i \(-0.201417\pi\)
0.806392 + 0.591381i \(0.201417\pi\)
\(332\) 5.78212 0.317335
\(333\) 11.5468 0.632762
\(334\) −18.6937 −1.02287
\(335\) −0.394069 −0.0215303
\(336\) 3.78636 0.206563
\(337\) −30.6442 −1.66929 −0.834647 0.550785i \(-0.814328\pi\)
−0.834647 + 0.550785i \(0.814328\pi\)
\(338\) −14.9941 −0.815573
\(339\) −4.40620 −0.239312
\(340\) 0.0718708 0.00389774
\(341\) −5.81979 −0.315159
\(342\) 11.8036 0.638264
\(343\) 6.13681 0.331356
\(344\) 9.70871 0.523458
\(345\) −0.0704122 −0.00379087
\(346\) −25.1360 −1.35132
\(347\) 15.9639 0.856987 0.428494 0.903545i \(-0.359044\pi\)
0.428494 + 0.903545i \(0.359044\pi\)
\(348\) −0.216240 −0.0115917
\(349\) −20.3707 −1.09042 −0.545209 0.838300i \(-0.683550\pi\)
−0.545209 + 0.838300i \(0.683550\pi\)
\(350\) −23.5730 −1.26003
\(351\) 1.50303 0.0802257
\(352\) 5.84951 0.311780
\(353\) −17.6373 −0.938738 −0.469369 0.883002i \(-0.655519\pi\)
−0.469369 + 0.883002i \(0.655519\pi\)
\(354\) 3.54081 0.188192
\(355\) 0.0476981 0.00253155
\(356\) 4.38528 0.232420
\(357\) 6.22746 0.329592
\(358\) 2.28485 0.120758
\(359\) 31.2874 1.65129 0.825644 0.564192i \(-0.190812\pi\)
0.825644 + 0.564192i \(0.190812\pi\)
\(360\) 0.265392 0.0139874
\(361\) −7.05349 −0.371236
\(362\) 24.2703 1.27562
\(363\) 2.83490 0.148794
\(364\) −1.52411 −0.0798851
\(365\) 0.0233448 0.00122192
\(366\) −2.92578 −0.152933
\(367\) 0.0122125 0.000637487 0 0.000318744 1.00000i \(-0.499899\pi\)
0.000318744 1.00000i \(0.499899\pi\)
\(368\) −15.5689 −0.811582
\(369\) 18.9422 0.986091
\(370\) −0.146111 −0.00759593
\(371\) 23.0426 1.19631
\(372\) −0.670824 −0.0347806
\(373\) −35.0219 −1.81336 −0.906682 0.421814i \(-0.861394\pi\)
−0.906682 + 0.421814i \(0.861394\pi\)
\(374\) −9.34483 −0.483210
\(375\) 0.114557 0.00591571
\(376\) −5.50758 −0.284032
\(377\) −0.677227 −0.0348790
\(378\) 10.4654 0.538284
\(379\) 11.0529 0.567749 0.283874 0.958861i \(-0.408380\pi\)
0.283874 + 0.958861i \(0.408380\pi\)
\(380\) 0.0596230 0.00305859
\(381\) 1.70200 0.0871961
\(382\) −15.7017 −0.803370
\(383\) −2.19567 −0.112194 −0.0560968 0.998425i \(-0.517866\pi\)
−0.0560968 + 0.998425i \(0.517866\pi\)
\(384\) −1.62127 −0.0827349
\(385\) 0.223688 0.0114002
\(386\) 18.9768 0.965892
\(387\) 9.02332 0.458681
\(388\) −2.68360 −0.136239
\(389\) 32.7752 1.66177 0.830884 0.556446i \(-0.187835\pi\)
0.830884 + 0.556446i \(0.187835\pi\)
\(390\) −0.00927627 −0.000469722 0
\(391\) −25.6062 −1.29496
\(392\) −26.2954 −1.32812
\(393\) −3.19423 −0.161127
\(394\) −22.9977 −1.15861
\(395\) −0.240856 −0.0121188
\(396\) 3.05764 0.153652
\(397\) −35.4390 −1.77863 −0.889316 0.457294i \(-0.848819\pi\)
−0.889316 + 0.457294i \(0.848819\pi\)
\(398\) 1.94772 0.0976305
\(399\) 5.16621 0.258634
\(400\) 12.6637 0.633186
\(401\) −6.03871 −0.301559 −0.150779 0.988567i \(-0.548178\pi\)
−0.150779 + 0.988567i \(0.548178\pi\)
\(402\) 5.90599 0.294564
\(403\) −2.10090 −0.104654
\(404\) 5.93354 0.295205
\(405\) 0.233631 0.0116092
\(406\) −4.71547 −0.234025
\(407\) −7.58370 −0.375910
\(408\) −4.85259 −0.240239
\(409\) 6.65664 0.329149 0.164575 0.986365i \(-0.447375\pi\)
0.164575 + 0.986365i \(0.447375\pi\)
\(410\) −0.239690 −0.0118374
\(411\) 8.03499 0.396337
\(412\) −6.86424 −0.338177
\(413\) −30.8228 −1.51669
\(414\) −20.9884 −1.03152
\(415\) −0.306346 −0.0150379
\(416\) 2.11163 0.103531
\(417\) 0.378969 0.0185582
\(418\) −7.75234 −0.379179
\(419\) −23.1709 −1.13197 −0.565987 0.824414i \(-0.691505\pi\)
−0.565987 + 0.824414i \(0.691505\pi\)
\(420\) 0.0257836 0.00125811
\(421\) −6.12806 −0.298663 −0.149332 0.988787i \(-0.547712\pi\)
−0.149332 + 0.988787i \(0.547712\pi\)
\(422\) 14.0199 0.682477
\(423\) −5.11877 −0.248883
\(424\) −17.9553 −0.871989
\(425\) 20.8281 1.01031
\(426\) −0.714861 −0.0346351
\(427\) 25.4689 1.23253
\(428\) 10.7449 0.519376
\(429\) −0.481474 −0.0232458
\(430\) −0.114179 −0.00550619
\(431\) 14.7821 0.712031 0.356015 0.934480i \(-0.384135\pi\)
0.356015 + 0.934480i \(0.384135\pi\)
\(432\) −5.62216 −0.270496
\(433\) −2.25372 −0.108307 −0.0541534 0.998533i \(-0.517246\pi\)
−0.0541534 + 0.998533i \(0.517246\pi\)
\(434\) −14.6284 −0.702185
\(435\) 0.0114568 0.000549310 0
\(436\) 0.367779 0.0176134
\(437\) −21.2425 −1.01617
\(438\) −0.349873 −0.0167176
\(439\) 13.8210 0.659640 0.329820 0.944044i \(-0.393012\pi\)
0.329820 + 0.944044i \(0.393012\pi\)
\(440\) −0.174304 −0.00830960
\(441\) −24.4391 −1.16376
\(442\) −3.37342 −0.160457
\(443\) −12.5364 −0.595621 −0.297811 0.954625i \(-0.596256\pi\)
−0.297811 + 0.954625i \(0.596256\pi\)
\(444\) −0.874143 −0.0414850
\(445\) −0.232340 −0.0110140
\(446\) −27.0820 −1.28237
\(447\) −1.88715 −0.0892590
\(448\) 34.6855 1.63874
\(449\) 3.08640 0.145656 0.0728280 0.997345i \(-0.476798\pi\)
0.0728280 + 0.997345i \(0.476798\pi\)
\(450\) 17.0720 0.804780
\(451\) −12.4408 −0.585816
\(452\) −6.63430 −0.312051
\(453\) 7.00393 0.329073
\(454\) 12.8561 0.603365
\(455\) 0.0807499 0.00378561
\(456\) −4.02564 −0.188518
\(457\) 4.70253 0.219975 0.109987 0.993933i \(-0.464919\pi\)
0.109987 + 0.993933i \(0.464919\pi\)
\(458\) 8.82803 0.412507
\(459\) −9.24681 −0.431604
\(460\) −0.106018 −0.00494311
\(461\) 37.7987 1.76046 0.880231 0.474545i \(-0.157387\pi\)
0.880231 + 0.474545i \(0.157387\pi\)
\(462\) −3.35246 −0.155970
\(463\) 10.8225 0.502966 0.251483 0.967862i \(-0.419082\pi\)
0.251483 + 0.967862i \(0.419082\pi\)
\(464\) 2.53321 0.117601
\(465\) 0.0355414 0.00164819
\(466\) 19.0258 0.881353
\(467\) −10.5603 −0.488671 −0.244335 0.969691i \(-0.578570\pi\)
−0.244335 + 0.969691i \(0.578570\pi\)
\(468\) 1.10379 0.0510225
\(469\) −51.4116 −2.37397
\(470\) 0.0647717 0.00298770
\(471\) −2.22262 −0.102413
\(472\) 24.0179 1.10551
\(473\) −5.92633 −0.272493
\(474\) 3.60975 0.165802
\(475\) 17.2787 0.792801
\(476\) 9.37651 0.429772
\(477\) −16.6878 −0.764081
\(478\) −28.8541 −1.31976
\(479\) −13.5504 −0.619134 −0.309567 0.950878i \(-0.600184\pi\)
−0.309567 + 0.950878i \(0.600184\pi\)
\(480\) −0.0357229 −0.00163052
\(481\) −2.73766 −0.124827
\(482\) −21.7247 −0.989533
\(483\) −9.18623 −0.417988
\(484\) 4.26844 0.194020
\(485\) 0.142181 0.00645613
\(486\) −11.4618 −0.519917
\(487\) 22.5007 1.01960 0.509801 0.860292i \(-0.329719\pi\)
0.509801 + 0.860292i \(0.329719\pi\)
\(488\) −19.8460 −0.898387
\(489\) −3.50759 −0.158619
\(490\) 0.309246 0.0139703
\(491\) 34.7873 1.56993 0.784965 0.619540i \(-0.212681\pi\)
0.784965 + 0.619540i \(0.212681\pi\)
\(492\) −1.43400 −0.0646499
\(493\) 4.16638 0.187645
\(494\) −2.79854 −0.125912
\(495\) −0.161999 −0.00728130
\(496\) 7.85855 0.352860
\(497\) 6.22287 0.279134
\(498\) 4.59127 0.205740
\(499\) 4.67908 0.209464 0.104732 0.994500i \(-0.466601\pi\)
0.104732 + 0.994500i \(0.466601\pi\)
\(500\) 0.172486 0.00771380
\(501\) 5.92545 0.264730
\(502\) −10.8708 −0.485186
\(503\) 3.31304 0.147721 0.0738606 0.997269i \(-0.476468\pi\)
0.0738606 + 0.997269i \(0.476468\pi\)
\(504\) 34.6239 1.54227
\(505\) −0.314369 −0.0139892
\(506\) 13.7847 0.612805
\(507\) 4.75278 0.211078
\(508\) 2.56266 0.113700
\(509\) 21.6383 0.959099 0.479550 0.877515i \(-0.340800\pi\)
0.479550 + 0.877515i \(0.340800\pi\)
\(510\) 0.0570687 0.00252705
\(511\) 3.04564 0.134731
\(512\) −23.4695 −1.03722
\(513\) −7.67102 −0.338684
\(514\) 24.6829 1.08872
\(515\) 0.363679 0.0160256
\(516\) −0.683104 −0.0300720
\(517\) 3.36190 0.147856
\(518\) −19.0621 −0.837540
\(519\) 7.96752 0.349735
\(520\) −0.0629224 −0.00275933
\(521\) 18.6397 0.816619 0.408310 0.912843i \(-0.366118\pi\)
0.408310 + 0.912843i \(0.366118\pi\)
\(522\) 3.41502 0.149471
\(523\) −17.3550 −0.758881 −0.379441 0.925216i \(-0.623884\pi\)
−0.379441 + 0.925216i \(0.623884\pi\)
\(524\) −4.80946 −0.210102
\(525\) 7.47209 0.326109
\(526\) −10.6232 −0.463192
\(527\) 12.9250 0.563023
\(528\) 1.80098 0.0783777
\(529\) 14.7721 0.642267
\(530\) 0.211163 0.00917234
\(531\) 22.3224 0.968708
\(532\) 7.77862 0.337246
\(533\) −4.49105 −0.194529
\(534\) 3.48212 0.150686
\(535\) −0.569284 −0.0246123
\(536\) 40.0612 1.73038
\(537\) −0.724243 −0.0312534
\(538\) 11.2293 0.484130
\(539\) 16.0511 0.691368
\(540\) −0.0382847 −0.00164751
\(541\) 13.1985 0.567447 0.283724 0.958906i \(-0.408430\pi\)
0.283724 + 0.958906i \(0.408430\pi\)
\(542\) −14.3603 −0.616826
\(543\) −7.69311 −0.330143
\(544\) −12.9910 −0.556985
\(545\) −0.0194856 −0.000834670 0
\(546\) −1.21021 −0.0517924
\(547\) 0.526453 0.0225095 0.0112547 0.999937i \(-0.496417\pi\)
0.0112547 + 0.999937i \(0.496417\pi\)
\(548\) 12.0981 0.516804
\(549\) −18.4450 −0.787213
\(550\) −11.2125 −0.478103
\(551\) 3.45637 0.147246
\(552\) 7.15814 0.304671
\(553\) −31.4229 −1.33624
\(554\) −18.5530 −0.788241
\(555\) 0.0463136 0.00196590
\(556\) 0.570603 0.0241989
\(557\) −28.2681 −1.19776 −0.598878 0.800840i \(-0.704387\pi\)
−0.598878 + 0.800840i \(0.704387\pi\)
\(558\) 10.5941 0.448485
\(559\) −2.13936 −0.0904854
\(560\) −0.302050 −0.0127639
\(561\) 2.96209 0.125059
\(562\) 24.3061 1.02529
\(563\) 19.1250 0.806023 0.403012 0.915195i \(-0.367963\pi\)
0.403012 + 0.915195i \(0.367963\pi\)
\(564\) 0.387513 0.0163172
\(565\) 0.351496 0.0147876
\(566\) 10.3146 0.433554
\(567\) 30.4803 1.28005
\(568\) −4.84902 −0.203460
\(569\) −10.0202 −0.420067 −0.210033 0.977694i \(-0.567357\pi\)
−0.210033 + 0.977694i \(0.567357\pi\)
\(570\) 0.0473434 0.00198300
\(571\) −32.3114 −1.35219 −0.676095 0.736815i \(-0.736329\pi\)
−0.676095 + 0.736815i \(0.736329\pi\)
\(572\) −0.724943 −0.0303114
\(573\) 4.97707 0.207920
\(574\) −31.2708 −1.30522
\(575\) −30.7239 −1.28128
\(576\) −25.1198 −1.04666
\(577\) −31.8860 −1.32743 −0.663716 0.747984i \(-0.731022\pi\)
−0.663716 + 0.747984i \(0.731022\pi\)
\(578\) 0.428921 0.0178408
\(579\) −6.01518 −0.249982
\(580\) 0.0172502 0.000716274 0
\(581\) −39.9670 −1.65811
\(582\) −2.13090 −0.0883287
\(583\) 10.9602 0.453925
\(584\) −2.37324 −0.0982055
\(585\) −0.0584804 −0.00241787
\(586\) 18.3862 0.759529
\(587\) 26.9417 1.11200 0.556001 0.831182i \(-0.312335\pi\)
0.556001 + 0.831182i \(0.312335\pi\)
\(588\) 1.85014 0.0762985
\(589\) 10.7224 0.441809
\(590\) −0.282462 −0.0116288
\(591\) 7.28974 0.299860
\(592\) 10.2404 0.420877
\(593\) 4.92949 0.202430 0.101215 0.994865i \(-0.467727\pi\)
0.101215 + 0.994865i \(0.467727\pi\)
\(594\) 4.97788 0.204245
\(595\) −0.496783 −0.0203661
\(596\) −2.84143 −0.116389
\(597\) −0.617382 −0.0252677
\(598\) 4.97619 0.203491
\(599\) 1.56201 0.0638219 0.0319110 0.999491i \(-0.489841\pi\)
0.0319110 + 0.999491i \(0.489841\pi\)
\(600\) −5.82244 −0.237700
\(601\) −0.516815 −0.0210813 −0.0105407 0.999944i \(-0.503355\pi\)
−0.0105407 + 0.999944i \(0.503355\pi\)
\(602\) −14.8962 −0.607123
\(603\) 37.2331 1.51625
\(604\) 10.5456 0.429096
\(605\) −0.226149 −0.00919426
\(606\) 4.71151 0.191392
\(607\) 20.1693 0.818648 0.409324 0.912389i \(-0.365765\pi\)
0.409324 + 0.912389i \(0.365765\pi\)
\(608\) −10.7772 −0.437071
\(609\) 1.49469 0.0605679
\(610\) 0.233398 0.00945002
\(611\) 1.21362 0.0490979
\(612\) −6.79062 −0.274495
\(613\) 31.3265 1.26527 0.632633 0.774452i \(-0.281974\pi\)
0.632633 + 0.774452i \(0.281974\pi\)
\(614\) −4.42121 −0.178426
\(615\) 0.0759759 0.00306365
\(616\) −22.7402 −0.916231
\(617\) 26.6442 1.07265 0.536327 0.844010i \(-0.319811\pi\)
0.536327 + 0.844010i \(0.319811\pi\)
\(618\) −5.45053 −0.219252
\(619\) 21.6375 0.869686 0.434843 0.900506i \(-0.356804\pi\)
0.434843 + 0.900506i \(0.356804\pi\)
\(620\) 0.0535137 0.00214916
\(621\) 13.6401 0.547359
\(622\) 30.0266 1.20396
\(623\) −30.3118 −1.21442
\(624\) 0.650142 0.0260265
\(625\) 24.9863 0.999452
\(626\) 38.2573 1.52907
\(627\) 2.45731 0.0981353
\(628\) −3.34654 −0.133541
\(629\) 16.8424 0.671552
\(630\) −0.407193 −0.0162230
\(631\) 28.5404 1.13618 0.568088 0.822968i \(-0.307683\pi\)
0.568088 + 0.822968i \(0.307683\pi\)
\(632\) 24.4855 0.973982
\(633\) −4.44397 −0.176632
\(634\) 38.7594 1.53933
\(635\) −0.135774 −0.00538802
\(636\) 1.26334 0.0500946
\(637\) 5.79432 0.229579
\(638\) −2.24291 −0.0887977
\(639\) −4.50670 −0.178282
\(640\) 0.129333 0.00511235
\(641\) 30.7200 1.21337 0.606684 0.794943i \(-0.292499\pi\)
0.606684 + 0.794943i \(0.292499\pi\)
\(642\) 8.53197 0.336730
\(643\) −25.9826 −1.02465 −0.512327 0.858791i \(-0.671216\pi\)
−0.512327 + 0.858791i \(0.671216\pi\)
\(644\) −13.8315 −0.545036
\(645\) 0.0361920 0.00142506
\(646\) 17.2170 0.677392
\(647\) 12.8682 0.505903 0.252951 0.967479i \(-0.418599\pi\)
0.252951 + 0.967479i \(0.418599\pi\)
\(648\) −23.7510 −0.933029
\(649\) −14.6608 −0.575489
\(650\) −4.04764 −0.158761
\(651\) 4.63685 0.181733
\(652\) −5.28128 −0.206831
\(653\) 41.8671 1.63839 0.819194 0.573517i \(-0.194421\pi\)
0.819194 + 0.573517i \(0.194421\pi\)
\(654\) 0.292034 0.0114194
\(655\) 0.254813 0.00995638
\(656\) 16.7990 0.655892
\(657\) −2.20570 −0.0860527
\(658\) 8.45034 0.329429
\(659\) −10.1048 −0.393625 −0.196813 0.980441i \(-0.563059\pi\)
−0.196813 + 0.980441i \(0.563059\pi\)
\(660\) 0.0122640 0.000477375 0
\(661\) 2.13762 0.0831438 0.0415719 0.999136i \(-0.486763\pi\)
0.0415719 + 0.999136i \(0.486763\pi\)
\(662\) 35.0806 1.36345
\(663\) 1.06929 0.0415279
\(664\) 31.1433 1.20859
\(665\) −0.412124 −0.0159815
\(666\) 13.8051 0.534936
\(667\) −6.14590 −0.237970
\(668\) 8.92179 0.345195
\(669\) 8.58435 0.331890
\(670\) −0.471139 −0.0182017
\(671\) 12.1143 0.467666
\(672\) −4.66053 −0.179784
\(673\) 41.5513 1.60169 0.800843 0.598874i \(-0.204385\pi\)
0.800843 + 0.598874i \(0.204385\pi\)
\(674\) −36.6374 −1.41122
\(675\) −11.0949 −0.427043
\(676\) 7.15614 0.275236
\(677\) −17.3113 −0.665326 −0.332663 0.943046i \(-0.607947\pi\)
−0.332663 + 0.943046i \(0.607947\pi\)
\(678\) −5.26794 −0.202314
\(679\) 18.5495 0.711864
\(680\) 0.387106 0.0148448
\(681\) −4.07507 −0.156157
\(682\) −6.95799 −0.266435
\(683\) 26.2315 1.00372 0.501861 0.864948i \(-0.332649\pi\)
0.501861 + 0.864948i \(0.332649\pi\)
\(684\) −5.63340 −0.215399
\(685\) −0.640976 −0.0244904
\(686\) 7.33701 0.280128
\(687\) −2.79828 −0.106761
\(688\) 8.00241 0.305089
\(689\) 3.95655 0.150733
\(690\) −0.0841831 −0.00320479
\(691\) 35.4542 1.34874 0.674371 0.738393i \(-0.264415\pi\)
0.674371 + 0.738393i \(0.264415\pi\)
\(692\) 11.9965 0.456037
\(693\) −21.1349 −0.802848
\(694\) 19.0860 0.724496
\(695\) −0.0302315 −0.00114675
\(696\) −1.16470 −0.0441479
\(697\) 27.6295 1.04654
\(698\) −24.3547 −0.921837
\(699\) −6.03072 −0.228103
\(700\) 11.2505 0.425230
\(701\) −25.3920 −0.959042 −0.479521 0.877530i \(-0.659190\pi\)
−0.479521 + 0.877530i \(0.659190\pi\)
\(702\) 1.79698 0.0678227
\(703\) 13.9722 0.526973
\(704\) 16.4982 0.621798
\(705\) −0.0205311 −0.000773245 0
\(706\) −21.0867 −0.793608
\(707\) −41.0136 −1.54248
\(708\) −1.68990 −0.0635102
\(709\) −25.3437 −0.951801 −0.475901 0.879499i \(-0.657878\pi\)
−0.475901 + 0.879499i \(0.657878\pi\)
\(710\) 0.0570267 0.00214017
\(711\) 22.7570 0.853453
\(712\) 23.6197 0.885188
\(713\) −19.0659 −0.714024
\(714\) 7.44538 0.278637
\(715\) 0.0384087 0.00143640
\(716\) −1.09047 −0.0407529
\(717\) 9.14607 0.341566
\(718\) 37.4064 1.39600
\(719\) 25.0797 0.935313 0.467657 0.883910i \(-0.345098\pi\)
0.467657 + 0.883910i \(0.345098\pi\)
\(720\) 0.218749 0.00815230
\(721\) 47.4468 1.76701
\(722\) −8.43297 −0.313843
\(723\) 6.88621 0.256101
\(724\) −11.5833 −0.430490
\(725\) 4.99909 0.185661
\(726\) 3.38934 0.125790
\(727\) 51.2949 1.90242 0.951211 0.308540i \(-0.0998405\pi\)
0.951211 + 0.308540i \(0.0998405\pi\)
\(728\) −8.20907 −0.304248
\(729\) −19.5511 −0.724115
\(730\) 0.0279104 0.00103301
\(731\) 13.1616 0.486800
\(732\) 1.39636 0.0516111
\(733\) −32.0511 −1.18384 −0.591918 0.805998i \(-0.701629\pi\)
−0.591918 + 0.805998i \(0.701629\pi\)
\(734\) 0.0146009 0.000538931 0
\(735\) −0.0980235 −0.00361565
\(736\) 19.1633 0.706367
\(737\) −24.4539 −0.900772
\(738\) 22.6468 0.833640
\(739\) 9.53180 0.350633 0.175316 0.984512i \(-0.443905\pi\)
0.175316 + 0.984512i \(0.443905\pi\)
\(740\) 0.0697331 0.00256344
\(741\) 0.887070 0.0325873
\(742\) 27.5491 1.01136
\(743\) 15.8842 0.582736 0.291368 0.956611i \(-0.405890\pi\)
0.291368 + 0.956611i \(0.405890\pi\)
\(744\) −3.61315 −0.132465
\(745\) 0.150543 0.00551548
\(746\) −41.8713 −1.53302
\(747\) 28.9447 1.05903
\(748\) 4.45994 0.163071
\(749\) −74.2708 −2.71379
\(750\) 0.136962 0.00500113
\(751\) 8.01121 0.292333 0.146167 0.989260i \(-0.453306\pi\)
0.146167 + 0.989260i \(0.453306\pi\)
\(752\) −4.53963 −0.165543
\(753\) 3.44578 0.125571
\(754\) −0.809675 −0.0294866
\(755\) −0.558725 −0.0203341
\(756\) −4.99476 −0.181658
\(757\) 2.88914 0.105007 0.0525037 0.998621i \(-0.483280\pi\)
0.0525037 + 0.998621i \(0.483280\pi\)
\(758\) 13.2146 0.479974
\(759\) −4.36943 −0.158600
\(760\) 0.321138 0.0116489
\(761\) −7.83742 −0.284106 −0.142053 0.989859i \(-0.545370\pi\)
−0.142053 + 0.989859i \(0.545370\pi\)
\(762\) 2.03487 0.0737155
\(763\) −2.54215 −0.0920321
\(764\) 7.49384 0.271118
\(765\) 0.359778 0.0130078
\(766\) −2.62509 −0.0948483
\(767\) −5.29246 −0.191100
\(768\) 4.72717 0.170577
\(769\) −21.1905 −0.764150 −0.382075 0.924131i \(-0.624790\pi\)
−0.382075 + 0.924131i \(0.624790\pi\)
\(770\) 0.267436 0.00963772
\(771\) −7.82389 −0.281771
\(772\) −9.05690 −0.325965
\(773\) 44.9789 1.61778 0.808890 0.587961i \(-0.200069\pi\)
0.808890 + 0.587961i \(0.200069\pi\)
\(774\) 10.7880 0.387768
\(775\) 15.5082 0.557072
\(776\) −14.4542 −0.518877
\(777\) 6.04223 0.216764
\(778\) 39.1852 1.40486
\(779\) 22.9210 0.821231
\(780\) 0.00442721 0.000158520 0
\(781\) 2.95991 0.105914
\(782\) −30.6141 −1.09476
\(783\) −2.21938 −0.0793143
\(784\) −21.6740 −0.774071
\(785\) 0.177305 0.00632829
\(786\) −3.81894 −0.136217
\(787\) −45.1149 −1.60817 −0.804087 0.594512i \(-0.797345\pi\)
−0.804087 + 0.594512i \(0.797345\pi\)
\(788\) 10.9760 0.391002
\(789\) 3.36729 0.119879
\(790\) −0.287961 −0.0102452
\(791\) 45.8574 1.63050
\(792\) 16.4689 0.585195
\(793\) 4.37317 0.155296
\(794\) −42.3699 −1.50365
\(795\) −0.0669337 −0.00237389
\(796\) −0.929575 −0.0329479
\(797\) 32.7739 1.16091 0.580455 0.814292i \(-0.302875\pi\)
0.580455 + 0.814292i \(0.302875\pi\)
\(798\) 6.17658 0.218649
\(799\) −7.46636 −0.264141
\(800\) −15.5874 −0.551099
\(801\) 21.9523 0.775647
\(802\) −7.21972 −0.254937
\(803\) 1.44866 0.0511221
\(804\) −2.81871 −0.0994081
\(805\) 0.732813 0.0258283
\(806\) −2.51179 −0.0884739
\(807\) −3.55943 −0.125298
\(808\) 31.9589 1.12431
\(809\) −17.5841 −0.618223 −0.309112 0.951026i \(-0.600032\pi\)
−0.309112 + 0.951026i \(0.600032\pi\)
\(810\) 0.279323 0.00981441
\(811\) −39.0338 −1.37066 −0.685330 0.728232i \(-0.740342\pi\)
−0.685330 + 0.728232i \(0.740342\pi\)
\(812\) 2.25052 0.0789776
\(813\) 4.55186 0.159641
\(814\) −9.06688 −0.317794
\(815\) 0.279811 0.00980135
\(816\) −3.99975 −0.140019
\(817\) 10.9187 0.381997
\(818\) 7.95850 0.278262
\(819\) −7.62955 −0.266598
\(820\) 0.114395 0.00399484
\(821\) 0.794572 0.0277307 0.0138654 0.999904i \(-0.495586\pi\)
0.0138654 + 0.999904i \(0.495586\pi\)
\(822\) 9.60643 0.335063
\(823\) −37.6273 −1.31161 −0.655803 0.754932i \(-0.727670\pi\)
−0.655803 + 0.754932i \(0.727670\pi\)
\(824\) −36.9718 −1.28797
\(825\) 3.55410 0.123738
\(826\) −36.8509 −1.28221
\(827\) −40.2318 −1.39900 −0.699499 0.714634i \(-0.746593\pi\)
−0.699499 + 0.714634i \(0.746593\pi\)
\(828\) 10.0170 0.348113
\(829\) −13.0473 −0.453152 −0.226576 0.973994i \(-0.572753\pi\)
−0.226576 + 0.973994i \(0.572753\pi\)
\(830\) −0.366259 −0.0127130
\(831\) 5.88086 0.204005
\(832\) 5.95573 0.206478
\(833\) −35.6474 −1.23511
\(834\) 0.453085 0.0156890
\(835\) −0.472691 −0.0163582
\(836\) 3.69990 0.127964
\(837\) −6.88501 −0.237981
\(838\) −27.7026 −0.956970
\(839\) −32.5161 −1.12258 −0.561289 0.827620i \(-0.689695\pi\)
−0.561289 + 0.827620i \(0.689695\pi\)
\(840\) 0.138874 0.00479162
\(841\) 1.00000 0.0344828
\(842\) −7.32655 −0.252490
\(843\) −7.70445 −0.265355
\(844\) −6.69117 −0.230319
\(845\) −0.379144 −0.0130429
\(846\) −6.11987 −0.210406
\(847\) −29.5042 −1.01378
\(848\) −14.7997 −0.508224
\(849\) −3.26948 −0.112208
\(850\) 24.9016 0.854116
\(851\) −24.8446 −0.851661
\(852\) 0.341176 0.0116885
\(853\) 22.5792 0.773097 0.386549 0.922269i \(-0.373667\pi\)
0.386549 + 0.922269i \(0.373667\pi\)
\(854\) 30.4499 1.04198
\(855\) 0.298467 0.0102074
\(856\) 57.8737 1.97808
\(857\) −2.88091 −0.0984100 −0.0492050 0.998789i \(-0.515669\pi\)
−0.0492050 + 0.998789i \(0.515669\pi\)
\(858\) −0.575638 −0.0196520
\(859\) −29.2296 −0.997302 −0.498651 0.866803i \(-0.666171\pi\)
−0.498651 + 0.866803i \(0.666171\pi\)
\(860\) 0.0544933 0.00185821
\(861\) 9.91208 0.337803
\(862\) 17.6731 0.601950
\(863\) −13.0246 −0.443362 −0.221681 0.975119i \(-0.571154\pi\)
−0.221681 + 0.975119i \(0.571154\pi\)
\(864\) 6.92016 0.235429
\(865\) −0.635593 −0.0216108
\(866\) −2.69449 −0.0915624
\(867\) −0.135958 −0.00461737
\(868\) 6.98158 0.236970
\(869\) −14.9463 −0.507018
\(870\) 0.0136974 0.000464386 0
\(871\) −8.82770 −0.299115
\(872\) 1.98091 0.0670821
\(873\) −13.4338 −0.454667
\(874\) −25.3970 −0.859067
\(875\) −1.19225 −0.0403054
\(876\) 0.166981 0.00564177
\(877\) −21.9791 −0.742182 −0.371091 0.928597i \(-0.621016\pi\)
−0.371091 + 0.928597i \(0.621016\pi\)
\(878\) 16.5240 0.557659
\(879\) −5.82800 −0.196574
\(880\) −0.143670 −0.00484311
\(881\) 23.9173 0.805794 0.402897 0.915245i \(-0.368003\pi\)
0.402897 + 0.915245i \(0.368003\pi\)
\(882\) −29.2187 −0.983845
\(883\) 1.71200 0.0576133 0.0288067 0.999585i \(-0.490829\pi\)
0.0288067 + 0.999585i \(0.490829\pi\)
\(884\) 1.61001 0.0541504
\(885\) 0.0895336 0.00300964
\(886\) −14.9882 −0.503537
\(887\) −16.5012 −0.554056 −0.277028 0.960862i \(-0.589350\pi\)
−0.277028 + 0.960862i \(0.589350\pi\)
\(888\) −4.70826 −0.157999
\(889\) −17.7135 −0.594092
\(890\) −0.277779 −0.00931118
\(891\) 14.4980 0.485700
\(892\) 12.9252 0.432769
\(893\) −6.19398 −0.207274
\(894\) −2.25622 −0.0754594
\(895\) 0.0577750 0.00193121
\(896\) 16.8733 0.563697
\(897\) −1.57733 −0.0526656
\(898\) 3.69002 0.123137
\(899\) 3.10222 0.103465
\(900\) −8.14781 −0.271594
\(901\) −24.3412 −0.810922
\(902\) −14.8739 −0.495248
\(903\) 4.72173 0.157129
\(904\) −35.7333 −1.18847
\(905\) 0.613703 0.0204002
\(906\) 8.37372 0.278198
\(907\) −50.3022 −1.67026 −0.835128 0.550055i \(-0.814607\pi\)
−0.835128 + 0.550055i \(0.814607\pi\)
\(908\) −6.13572 −0.203621
\(909\) 29.7027 0.985178
\(910\) 0.0965425 0.00320035
\(911\) −45.1616 −1.49627 −0.748135 0.663546i \(-0.769051\pi\)
−0.748135 + 0.663546i \(0.769051\pi\)
\(912\) −3.31814 −0.109875
\(913\) −19.0103 −0.629148
\(914\) 5.62222 0.185966
\(915\) −0.0739817 −0.00244576
\(916\) −4.21329 −0.139211
\(917\) 33.2438 1.09781
\(918\) −11.0552 −0.364877
\(919\) −21.3789 −0.705226 −0.352613 0.935769i \(-0.614707\pi\)
−0.352613 + 0.935769i \(0.614707\pi\)
\(920\) −0.571027 −0.0188262
\(921\) 1.40142 0.0461783
\(922\) 45.1912 1.48829
\(923\) 1.06851 0.0351703
\(924\) 1.60000 0.0526362
\(925\) 20.2086 0.664455
\(926\) 12.9392 0.425207
\(927\) −34.3618 −1.12859
\(928\) −3.11805 −0.102355
\(929\) 32.0168 1.05044 0.525218 0.850968i \(-0.323984\pi\)
0.525218 + 0.850968i \(0.323984\pi\)
\(930\) 0.0424924 0.00139338
\(931\) −29.5725 −0.969201
\(932\) −9.08029 −0.297435
\(933\) −9.51771 −0.311596
\(934\) −12.6256 −0.413121
\(935\) −0.236295 −0.00772767
\(936\) 5.94514 0.194323
\(937\) −43.4092 −1.41812 −0.709058 0.705150i \(-0.750880\pi\)
−0.709058 + 0.705150i \(0.750880\pi\)
\(938\) −61.4664 −2.00695
\(939\) −12.1266 −0.395738
\(940\) −0.0309131 −0.00100827
\(941\) −50.2495 −1.63809 −0.819043 0.573732i \(-0.805495\pi\)
−0.819043 + 0.573732i \(0.805495\pi\)
\(942\) −2.65731 −0.0865797
\(943\) −40.7567 −1.32722
\(944\) 19.7968 0.644330
\(945\) 0.264631 0.00860843
\(946\) −7.08536 −0.230365
\(947\) 3.84195 0.124847 0.0624233 0.998050i \(-0.480117\pi\)
0.0624233 + 0.998050i \(0.480117\pi\)
\(948\) −1.72280 −0.0559540
\(949\) 0.522956 0.0169759
\(950\) 20.6580 0.670233
\(951\) −12.2858 −0.398394
\(952\) 50.5032 1.63682
\(953\) −25.9928 −0.841991 −0.420995 0.907063i \(-0.638319\pi\)
−0.420995 + 0.907063i \(0.638319\pi\)
\(954\) −19.9515 −0.645953
\(955\) −0.397036 −0.0128478
\(956\) 13.7710 0.445386
\(957\) 0.710949 0.0229817
\(958\) −16.2005 −0.523415
\(959\) −83.6239 −2.70036
\(960\) −0.100754 −0.00325182
\(961\) −21.3763 −0.689557
\(962\) −3.27308 −0.105528
\(963\) 53.7881 1.73330
\(964\) 10.3684 0.333943
\(965\) 0.479849 0.0154469
\(966\) −10.9828 −0.353366
\(967\) 45.5252 1.46399 0.731995 0.681309i \(-0.238589\pi\)
0.731995 + 0.681309i \(0.238589\pi\)
\(968\) 22.9904 0.738940
\(969\) −5.45736 −0.175316
\(970\) 0.169988 0.00545800
\(971\) 7.92777 0.254414 0.127207 0.991876i \(-0.459399\pi\)
0.127207 + 0.991876i \(0.459399\pi\)
\(972\) 5.47028 0.175459
\(973\) −3.94410 −0.126442
\(974\) 26.9012 0.861971
\(975\) 1.28300 0.0410890
\(976\) −16.3581 −0.523610
\(977\) 17.7730 0.568608 0.284304 0.958734i \(-0.408238\pi\)
0.284304 + 0.958734i \(0.408238\pi\)
\(978\) −4.19358 −0.134096
\(979\) −14.4178 −0.460795
\(980\) −0.147591 −0.00471463
\(981\) 1.84107 0.0587808
\(982\) 41.5908 1.32722
\(983\) −12.1055 −0.386105 −0.193052 0.981188i \(-0.561839\pi\)
−0.193052 + 0.981188i \(0.561839\pi\)
\(984\) −7.72375 −0.246224
\(985\) −0.581525 −0.0185289
\(986\) 4.98122 0.158634
\(987\) −2.67856 −0.0852594
\(988\) 1.33564 0.0424923
\(989\) −19.4149 −0.617359
\(990\) −0.193681 −0.00615560
\(991\) 33.0534 1.04998 0.524988 0.851110i \(-0.324070\pi\)
0.524988 + 0.851110i \(0.324070\pi\)
\(992\) −9.67288 −0.307114
\(993\) −11.1197 −0.352873
\(994\) 7.43990 0.235979
\(995\) 0.0492504 0.00156134
\(996\) −2.19124 −0.0694321
\(997\) −16.9351 −0.536339 −0.268169 0.963372i \(-0.586419\pi\)
−0.268169 + 0.963372i \(0.586419\pi\)
\(998\) 5.59418 0.177081
\(999\) −8.97178 −0.283854
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 4031.2.a.c.1.43 61
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4031.2.a.c.1.43 61 1.1 even 1 trivial