Properties

Label 2299.2.a.n.1.3
Level $2299$
Weight $2$
Character 2299.1
Self dual yes
Analytic conductor $18.358$
Analytic rank $1$
Dimension $5$
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] = [2299,2,Mod(1,2299)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2299, 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("2299.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2299 = 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2299.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: \(18.3576074247\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.246832.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 209)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.15351\) of defining polynomial
Character \(\chi\) \(=\) 2299.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.484093 q^{2} +2.26452 q^{3} -1.76565 q^{4} +0.637602 q^{5} -1.09624 q^{6} -2.66942 q^{7} +1.82293 q^{8} +2.12805 q^{9} +O(q^{10})\) \(q-0.484093 q^{2} +2.26452 q^{3} -1.76565 q^{4} +0.637602 q^{5} -1.09624 q^{6} -2.66942 q^{7} +1.82293 q^{8} +2.12805 q^{9} -0.308658 q^{10} -3.99836 q^{12} +1.20725 q^{13} +1.29225 q^{14} +1.44386 q^{15} +2.64884 q^{16} -0.222022 q^{17} -1.03017 q^{18} +1.00000 q^{19} -1.12578 q^{20} -6.04495 q^{21} -9.48717 q^{23} +4.12805 q^{24} -4.59346 q^{25} -0.584420 q^{26} -1.97454 q^{27} +4.71327 q^{28} -4.94518 q^{29} -0.698963 q^{30} -3.05563 q^{31} -4.92814 q^{32} +0.107479 q^{34} -1.70203 q^{35} -3.75740 q^{36} +7.18015 q^{37} -0.484093 q^{38} +2.73384 q^{39} +1.16230 q^{40} +6.92650 q^{41} +2.92632 q^{42} -1.53538 q^{43} +1.35685 q^{45} +4.59267 q^{46} +5.94581 q^{47} +5.99836 q^{48} +0.125785 q^{49} +2.22366 q^{50} -0.502774 q^{51} -2.13158 q^{52} -9.63879 q^{53} +0.955862 q^{54} -4.86615 q^{56} +2.26452 q^{57} +2.39392 q^{58} +2.65817 q^{59} -2.54936 q^{60} +0.809792 q^{61} +1.47921 q^{62} -5.68066 q^{63} -2.91201 q^{64} +0.769744 q^{65} -2.22447 q^{67} +0.392015 q^{68} -21.4839 q^{69} +0.823938 q^{70} +2.58912 q^{71} +3.87928 q^{72} -16.6108 q^{73} -3.47586 q^{74} -10.4020 q^{75} -1.76565 q^{76} -1.32343 q^{78} -5.69298 q^{79} +1.68891 q^{80} -10.8556 q^{81} -3.35307 q^{82} +2.93784 q^{83} +10.6733 q^{84} -0.141562 q^{85} +0.743267 q^{86} -11.1985 q^{87} -5.54136 q^{89} -0.656841 q^{90} -3.22265 q^{91} +16.7511 q^{92} -6.91954 q^{93} -2.87832 q^{94} +0.637602 q^{95} -11.1599 q^{96} +13.7728 q^{97} -0.0608914 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} + q^{3} + 6 q^{4} - 5 q^{5} + 2 q^{6} - 6 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{2} + q^{3} + 6 q^{4} - 5 q^{5} + 2 q^{6} - 6 q^{7} - 6 q^{8} + 4 q^{9} - 12 q^{10} + 6 q^{12} - 4 q^{13} - 14 q^{14} + 3 q^{15} + 8 q^{16} + 4 q^{17} + 20 q^{18} + 5 q^{19} - 8 q^{20} - 10 q^{21} + 3 q^{23} + 14 q^{24} + 6 q^{25} - 6 q^{26} - 11 q^{27} + 10 q^{28} - 10 q^{29} - 6 q^{30} + 11 q^{31} - 14 q^{32} - 4 q^{34} + 8 q^{35} - 26 q^{36} + q^{37} - 2 q^{38} - 2 q^{39} + 16 q^{40} - 2 q^{41} - 16 q^{42} - 20 q^{43} - 28 q^{45} + 4 q^{46} - 20 q^{47} + 4 q^{48} + 3 q^{49} + 32 q^{50} - 24 q^{51} - 6 q^{52} - 14 q^{53} - 16 q^{54} - 38 q^{56} + q^{57} - 6 q^{58} + 3 q^{59} - 40 q^{60} + 10 q^{61} + 6 q^{62} - 24 q^{63} + 9 q^{67} - 24 q^{68} - 5 q^{69} + 50 q^{70} + 23 q^{71} + 12 q^{72} - 8 q^{74} - 18 q^{75} + 6 q^{76} - 22 q^{78} - 44 q^{79} - 18 q^{80} + q^{81} - 30 q^{82} + 14 q^{83} - 14 q^{84} + 12 q^{85} + 52 q^{86} - 28 q^{87} - 27 q^{89} - 26 q^{90} + 24 q^{91} + 58 q^{92} - 27 q^{93} + 8 q^{94} - 5 q^{95} - 50 q^{96} + 15 q^{97} + 10 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.484093 −0.342305 −0.171153 0.985245i \(-0.554749\pi\)
−0.171153 + 0.985245i \(0.554749\pi\)
\(3\) 2.26452 1.30742 0.653711 0.756745i \(-0.273211\pi\)
0.653711 + 0.756745i \(0.273211\pi\)
\(4\) −1.76565 −0.882827
\(5\) 0.637602 0.285144 0.142572 0.989784i \(-0.454463\pi\)
0.142572 + 0.989784i \(0.454463\pi\)
\(6\) −1.09624 −0.447537
\(7\) −2.66942 −1.00894 −0.504472 0.863428i \(-0.668313\pi\)
−0.504472 + 0.863428i \(0.668313\pi\)
\(8\) 1.82293 0.644502
\(9\) 2.12805 0.709351
\(10\) −0.308658 −0.0976064
\(11\) 0 0
\(12\) −3.99836 −1.15423
\(13\) 1.20725 0.334831 0.167415 0.985886i \(-0.446458\pi\)
0.167415 + 0.985886i \(0.446458\pi\)
\(14\) 1.29225 0.345367
\(15\) 1.44386 0.372804
\(16\) 2.64884 0.662211
\(17\) −0.222022 −0.0538483 −0.0269242 0.999637i \(-0.508571\pi\)
−0.0269242 + 0.999637i \(0.508571\pi\)
\(18\) −1.03017 −0.242814
\(19\) 1.00000 0.229416
\(20\) −1.12578 −0.251733
\(21\) −6.04495 −1.31912
\(22\) 0 0
\(23\) −9.48717 −1.97821 −0.989106 0.147206i \(-0.952972\pi\)
−0.989106 + 0.147206i \(0.952972\pi\)
\(24\) 4.12805 0.842635
\(25\) −4.59346 −0.918693
\(26\) −0.584420 −0.114614
\(27\) −1.97454 −0.380001
\(28\) 4.71327 0.890724
\(29\) −4.94518 −0.918297 −0.459148 0.888360i \(-0.651845\pi\)
−0.459148 + 0.888360i \(0.651845\pi\)
\(30\) −0.698963 −0.127613
\(31\) −3.05563 −0.548808 −0.274404 0.961615i \(-0.588480\pi\)
−0.274404 + 0.961615i \(0.588480\pi\)
\(32\) −4.92814 −0.871180
\(33\) 0 0
\(34\) 0.107479 0.0184326
\(35\) −1.70203 −0.287695
\(36\) −3.75740 −0.626234
\(37\) 7.18015 1.18041 0.590205 0.807254i \(-0.299047\pi\)
0.590205 + 0.807254i \(0.299047\pi\)
\(38\) −0.484093 −0.0785302
\(39\) 2.73384 0.437765
\(40\) 1.16230 0.183776
\(41\) 6.92650 1.08174 0.540869 0.841107i \(-0.318096\pi\)
0.540869 + 0.841107i \(0.318096\pi\)
\(42\) 2.92632 0.451540
\(43\) −1.53538 −0.234144 −0.117072 0.993123i \(-0.537351\pi\)
−0.117072 + 0.993123i \(0.537351\pi\)
\(44\) 0 0
\(45\) 1.35685 0.202267
\(46\) 4.59267 0.677152
\(47\) 5.94581 0.867285 0.433642 0.901085i \(-0.357228\pi\)
0.433642 + 0.901085i \(0.357228\pi\)
\(48\) 5.99836 0.865789
\(49\) 0.125785 0.0179692
\(50\) 2.22366 0.314473
\(51\) −0.502774 −0.0704024
\(52\) −2.13158 −0.295598
\(53\) −9.63879 −1.32399 −0.661995 0.749509i \(-0.730290\pi\)
−0.661995 + 0.749509i \(0.730290\pi\)
\(54\) 0.955862 0.130076
\(55\) 0 0
\(56\) −4.86615 −0.650266
\(57\) 2.26452 0.299943
\(58\) 2.39392 0.314338
\(59\) 2.65817 0.346065 0.173032 0.984916i \(-0.444643\pi\)
0.173032 + 0.984916i \(0.444643\pi\)
\(60\) −2.54936 −0.329121
\(61\) 0.809792 0.103683 0.0518416 0.998655i \(-0.483491\pi\)
0.0518416 + 0.998655i \(0.483491\pi\)
\(62\) 1.47921 0.187860
\(63\) −5.68066 −0.715696
\(64\) −2.91201 −0.364001
\(65\) 0.769744 0.0954750
\(66\) 0 0
\(67\) −2.22447 −0.271763 −0.135881 0.990725i \(-0.543387\pi\)
−0.135881 + 0.990725i \(0.543387\pi\)
\(68\) 0.392015 0.0475387
\(69\) −21.4839 −2.58636
\(70\) 0.823938 0.0984794
\(71\) 2.58912 0.307272 0.153636 0.988127i \(-0.450902\pi\)
0.153636 + 0.988127i \(0.450902\pi\)
\(72\) 3.87928 0.457178
\(73\) −16.6108 −1.94414 −0.972071 0.234687i \(-0.924594\pi\)
−0.972071 + 0.234687i \(0.924594\pi\)
\(74\) −3.47586 −0.404060
\(75\) −10.4020 −1.20112
\(76\) −1.76565 −0.202534
\(77\) 0 0
\(78\) −1.32343 −0.149849
\(79\) −5.69298 −0.640510 −0.320255 0.947331i \(-0.603769\pi\)
−0.320255 + 0.947331i \(0.603769\pi\)
\(80\) 1.68891 0.188826
\(81\) −10.8556 −1.20617
\(82\) −3.35307 −0.370284
\(83\) 2.93784 0.322470 0.161235 0.986916i \(-0.448452\pi\)
0.161235 + 0.986916i \(0.448452\pi\)
\(84\) 10.6733 1.16455
\(85\) −0.141562 −0.0153545
\(86\) 0.743267 0.0801486
\(87\) −11.1985 −1.20060
\(88\) 0 0
\(89\) −5.54136 −0.587383 −0.293692 0.955900i \(-0.594884\pi\)
−0.293692 + 0.955900i \(0.594884\pi\)
\(90\) −0.656841 −0.0692372
\(91\) −3.22265 −0.337826
\(92\) 16.7511 1.74642
\(93\) −6.91954 −0.717523
\(94\) −2.87832 −0.296876
\(95\) 0.637602 0.0654166
\(96\) −11.1599 −1.13900
\(97\) 13.7728 1.39842 0.699209 0.714917i \(-0.253536\pi\)
0.699209 + 0.714917i \(0.253536\pi\)
\(98\) −0.0608914 −0.00615096
\(99\) 0 0
\(100\) 8.11047 0.811047
\(101\) −12.0308 −1.19711 −0.598555 0.801082i \(-0.704258\pi\)
−0.598555 + 0.801082i \(0.704258\pi\)
\(102\) 0.243389 0.0240991
\(103\) 8.53068 0.840553 0.420276 0.907396i \(-0.361933\pi\)
0.420276 + 0.907396i \(0.361933\pi\)
\(104\) 2.20073 0.215799
\(105\) −3.85427 −0.376138
\(106\) 4.66607 0.453208
\(107\) −1.64585 −0.159110 −0.0795552 0.996830i \(-0.525350\pi\)
−0.0795552 + 0.996830i \(0.525350\pi\)
\(108\) 3.48636 0.335475
\(109\) −16.8065 −1.60977 −0.804886 0.593430i \(-0.797774\pi\)
−0.804886 + 0.593430i \(0.797774\pi\)
\(110\) 0 0
\(111\) 16.2596 1.54329
\(112\) −7.07087 −0.668134
\(113\) −11.2947 −1.06252 −0.531258 0.847210i \(-0.678280\pi\)
−0.531258 + 0.847210i \(0.678280\pi\)
\(114\) −1.09624 −0.102672
\(115\) −6.04904 −0.564076
\(116\) 8.73148 0.810697
\(117\) 2.56909 0.237512
\(118\) −1.28680 −0.118460
\(119\) 0.592670 0.0543300
\(120\) 2.63205 0.240273
\(121\) 0 0
\(122\) −0.392015 −0.0354913
\(123\) 15.6852 1.41429
\(124\) 5.39519 0.484502
\(125\) −6.11681 −0.547104
\(126\) 2.74997 0.244986
\(127\) −18.5894 −1.64954 −0.824771 0.565467i \(-0.808696\pi\)
−0.824771 + 0.565467i \(0.808696\pi\)
\(128\) 11.2660 0.995779
\(129\) −3.47690 −0.306124
\(130\) −0.372628 −0.0326816
\(131\) 3.09098 0.270060 0.135030 0.990842i \(-0.456887\pi\)
0.135030 + 0.990842i \(0.456887\pi\)
\(132\) 0 0
\(133\) −2.66942 −0.231468
\(134\) 1.07685 0.0930257
\(135\) −1.25897 −0.108355
\(136\) −0.404730 −0.0347053
\(137\) −8.67619 −0.741257 −0.370629 0.928781i \(-0.620858\pi\)
−0.370629 + 0.928781i \(0.620858\pi\)
\(138\) 10.4002 0.885323
\(139\) −14.5426 −1.23348 −0.616742 0.787166i \(-0.711548\pi\)
−0.616742 + 0.787166i \(0.711548\pi\)
\(140\) 3.00519 0.253985
\(141\) 13.4644 1.13391
\(142\) −1.25338 −0.105181
\(143\) 0 0
\(144\) 5.63688 0.469740
\(145\) −3.15306 −0.261847
\(146\) 8.04115 0.665490
\(147\) 0.284842 0.0234933
\(148\) −12.6777 −1.04210
\(149\) 18.4716 1.51325 0.756625 0.653849i \(-0.226847\pi\)
0.756625 + 0.653849i \(0.226847\pi\)
\(150\) 5.03553 0.411149
\(151\) 0.913992 0.0743796 0.0371898 0.999308i \(-0.488159\pi\)
0.0371898 + 0.999308i \(0.488159\pi\)
\(152\) 1.82293 0.147859
\(153\) −0.472475 −0.0381973
\(154\) 0 0
\(155\) −1.94828 −0.156489
\(156\) −4.82702 −0.386471
\(157\) −2.39421 −0.191079 −0.0955395 0.995426i \(-0.530458\pi\)
−0.0955395 + 0.995426i \(0.530458\pi\)
\(158\) 2.75593 0.219250
\(159\) −21.8272 −1.73101
\(160\) −3.14219 −0.248412
\(161\) 25.3252 1.99591
\(162\) 5.25509 0.412879
\(163\) 14.1986 1.11212 0.556061 0.831141i \(-0.312312\pi\)
0.556061 + 0.831141i \(0.312312\pi\)
\(164\) −12.2298 −0.954987
\(165\) 0 0
\(166\) −1.42219 −0.110383
\(167\) −9.25056 −0.715830 −0.357915 0.933754i \(-0.616512\pi\)
−0.357915 + 0.933754i \(0.616512\pi\)
\(168\) −11.0195 −0.850172
\(169\) −11.5426 −0.887888
\(170\) 0.0685290 0.00525594
\(171\) 2.12805 0.162736
\(172\) 2.71095 0.206708
\(173\) −5.39376 −0.410080 −0.205040 0.978754i \(-0.565732\pi\)
−0.205040 + 0.978754i \(0.565732\pi\)
\(174\) 5.42109 0.410972
\(175\) 12.2619 0.926910
\(176\) 0 0
\(177\) 6.01949 0.452453
\(178\) 2.68253 0.201064
\(179\) 1.30720 0.0977048 0.0488524 0.998806i \(-0.484444\pi\)
0.0488524 + 0.998806i \(0.484444\pi\)
\(180\) −2.39573 −0.178567
\(181\) −10.7624 −0.799961 −0.399980 0.916524i \(-0.630983\pi\)
−0.399980 + 0.916524i \(0.630983\pi\)
\(182\) 1.56006 0.115639
\(183\) 1.83379 0.135558
\(184\) −17.2944 −1.27496
\(185\) 4.57808 0.336587
\(186\) 3.34970 0.245612
\(187\) 0 0
\(188\) −10.4982 −0.765663
\(189\) 5.27088 0.383400
\(190\) −0.308658 −0.0223924
\(191\) 15.3041 1.10737 0.553684 0.832727i \(-0.313222\pi\)
0.553684 + 0.832727i \(0.313222\pi\)
\(192\) −6.59431 −0.475903
\(193\) 18.3032 1.31749 0.658747 0.752364i \(-0.271087\pi\)
0.658747 + 0.752364i \(0.271087\pi\)
\(194\) −6.66732 −0.478686
\(195\) 1.74310 0.124826
\(196\) −0.222092 −0.0158637
\(197\) 0.576171 0.0410505 0.0205252 0.999789i \(-0.493466\pi\)
0.0205252 + 0.999789i \(0.493466\pi\)
\(198\) 0 0
\(199\) −10.1678 −0.720778 −0.360389 0.932802i \(-0.617356\pi\)
−0.360389 + 0.932802i \(0.617356\pi\)
\(200\) −8.37354 −0.592099
\(201\) −5.03736 −0.355308
\(202\) 5.82402 0.409777
\(203\) 13.2007 0.926510
\(204\) 0.887725 0.0621532
\(205\) 4.41635 0.308451
\(206\) −4.12964 −0.287726
\(207\) −20.1892 −1.40325
\(208\) 3.19781 0.221728
\(209\) 0 0
\(210\) 1.86582 0.128754
\(211\) 7.25837 0.499687 0.249843 0.968286i \(-0.419621\pi\)
0.249843 + 0.968286i \(0.419621\pi\)
\(212\) 17.0188 1.16885
\(213\) 5.86312 0.401734
\(214\) 0.796745 0.0544643
\(215\) −0.978963 −0.0667647
\(216\) −3.59945 −0.244911
\(217\) 8.15675 0.553716
\(218\) 8.13591 0.551033
\(219\) −37.6154 −2.54181
\(220\) 0 0
\(221\) −0.268036 −0.0180301
\(222\) −7.87115 −0.528277
\(223\) 16.2772 1.09000 0.545000 0.838436i \(-0.316529\pi\)
0.545000 + 0.838436i \(0.316529\pi\)
\(224\) 13.1553 0.878972
\(225\) −9.77513 −0.651675
\(226\) 5.46768 0.363705
\(227\) 20.9260 1.38890 0.694452 0.719539i \(-0.255647\pi\)
0.694452 + 0.719539i \(0.255647\pi\)
\(228\) −3.99836 −0.264798
\(229\) −10.8083 −0.714234 −0.357117 0.934060i \(-0.616240\pi\)
−0.357117 + 0.934060i \(0.616240\pi\)
\(230\) 2.92830 0.193086
\(231\) 0 0
\(232\) −9.01469 −0.591844
\(233\) 25.4173 1.66514 0.832570 0.553919i \(-0.186868\pi\)
0.832570 + 0.553919i \(0.186868\pi\)
\(234\) −1.24368 −0.0813017
\(235\) 3.79106 0.247301
\(236\) −4.69342 −0.305515
\(237\) −12.8919 −0.837417
\(238\) −0.286907 −0.0185974
\(239\) −0.352238 −0.0227844 −0.0113922 0.999935i \(-0.503626\pi\)
−0.0113922 + 0.999935i \(0.503626\pi\)
\(240\) 3.82457 0.246875
\(241\) 15.6832 1.01024 0.505122 0.863048i \(-0.331448\pi\)
0.505122 + 0.863048i \(0.331448\pi\)
\(242\) 0 0
\(243\) −18.6590 −1.19697
\(244\) −1.42981 −0.0915344
\(245\) 0.0802004 0.00512382
\(246\) −7.59309 −0.484118
\(247\) 1.20725 0.0768154
\(248\) −5.57019 −0.353707
\(249\) 6.65281 0.421605
\(250\) 2.96110 0.187277
\(251\) 1.16332 0.0734282 0.0367141 0.999326i \(-0.488311\pi\)
0.0367141 + 0.999326i \(0.488311\pi\)
\(252\) 10.0301 0.631836
\(253\) 0 0
\(254\) 8.99899 0.564647
\(255\) −0.320570 −0.0200748
\(256\) 0.370256 0.0231410
\(257\) −15.0510 −0.938857 −0.469428 0.882971i \(-0.655540\pi\)
−0.469428 + 0.882971i \(0.655540\pi\)
\(258\) 1.68314 0.104788
\(259\) −19.1668 −1.19097
\(260\) −1.35910 −0.0842879
\(261\) −10.5236 −0.651394
\(262\) −1.49632 −0.0924429
\(263\) 20.6609 1.27400 0.637002 0.770862i \(-0.280174\pi\)
0.637002 + 0.770862i \(0.280174\pi\)
\(264\) 0 0
\(265\) −6.14571 −0.377528
\(266\) 1.29225 0.0792326
\(267\) −12.5485 −0.767958
\(268\) 3.92765 0.239919
\(269\) −2.82662 −0.172342 −0.0861711 0.996280i \(-0.527463\pi\)
−0.0861711 + 0.996280i \(0.527463\pi\)
\(270\) 0.609459 0.0370905
\(271\) −23.5970 −1.43341 −0.716707 0.697375i \(-0.754351\pi\)
−0.716707 + 0.697375i \(0.754351\pi\)
\(272\) −0.588102 −0.0356589
\(273\) −7.29776 −0.441680
\(274\) 4.20008 0.253736
\(275\) 0 0
\(276\) 37.9331 2.28331
\(277\) 18.7773 1.12822 0.564110 0.825700i \(-0.309219\pi\)
0.564110 + 0.825700i \(0.309219\pi\)
\(278\) 7.03994 0.422228
\(279\) −6.50254 −0.389297
\(280\) −3.10267 −0.185420
\(281\) −27.2632 −1.62639 −0.813194 0.581993i \(-0.802273\pi\)
−0.813194 + 0.581993i \(0.802273\pi\)
\(282\) −6.51802 −0.388142
\(283\) 27.2601 1.62044 0.810221 0.586124i \(-0.199347\pi\)
0.810221 + 0.586124i \(0.199347\pi\)
\(284\) −4.57149 −0.271268
\(285\) 1.44386 0.0855270
\(286\) 0 0
\(287\) −18.4897 −1.09141
\(288\) −10.4873 −0.617972
\(289\) −16.9507 −0.997100
\(290\) 1.52637 0.0896316
\(291\) 31.1888 1.82832
\(292\) 29.3289 1.71634
\(293\) −30.9996 −1.81101 −0.905507 0.424332i \(-0.860509\pi\)
−0.905507 + 0.424332i \(0.860509\pi\)
\(294\) −0.137890 −0.00804189
\(295\) 1.69486 0.0986784
\(296\) 13.0889 0.760776
\(297\) 0 0
\(298\) −8.94195 −0.517993
\(299\) −11.4534 −0.662366
\(300\) 18.3663 1.06038
\(301\) 4.09858 0.236238
\(302\) −0.442457 −0.0254605
\(303\) −27.2440 −1.56513
\(304\) 2.64884 0.151922
\(305\) 0.516325 0.0295647
\(306\) 0.228722 0.0130751
\(307\) 18.2495 1.04156 0.520778 0.853692i \(-0.325642\pi\)
0.520778 + 0.853692i \(0.325642\pi\)
\(308\) 0 0
\(309\) 19.3179 1.09896
\(310\) 0.943146 0.0535671
\(311\) 30.8931 1.75179 0.875894 0.482503i \(-0.160272\pi\)
0.875894 + 0.482503i \(0.160272\pi\)
\(312\) 4.98359 0.282140
\(313\) 28.7973 1.62772 0.813859 0.581062i \(-0.197363\pi\)
0.813859 + 0.581062i \(0.197363\pi\)
\(314\) 1.15902 0.0654073
\(315\) −3.62200 −0.204076
\(316\) 10.0518 0.565460
\(317\) 5.96040 0.334769 0.167385 0.985892i \(-0.446468\pi\)
0.167385 + 0.985892i \(0.446468\pi\)
\(318\) 10.5664 0.592534
\(319\) 0 0
\(320\) −1.85670 −0.103793
\(321\) −3.72706 −0.208024
\(322\) −12.2597 −0.683209
\(323\) −0.222022 −0.0123536
\(324\) 19.1671 1.06484
\(325\) −5.54545 −0.307606
\(326\) −6.87345 −0.380685
\(327\) −38.0587 −2.10465
\(328\) 12.6265 0.697181
\(329\) −15.8718 −0.875043
\(330\) 0 0
\(331\) 33.4010 1.83589 0.917944 0.396711i \(-0.129849\pi\)
0.917944 + 0.396711i \(0.129849\pi\)
\(332\) −5.18722 −0.284686
\(333\) 15.2797 0.837325
\(334\) 4.47813 0.245032
\(335\) −1.41833 −0.0774915
\(336\) −16.0121 −0.873533
\(337\) −24.9988 −1.36177 −0.680885 0.732390i \(-0.738405\pi\)
−0.680885 + 0.732390i \(0.738405\pi\)
\(338\) 5.58766 0.303929
\(339\) −25.5771 −1.38916
\(340\) 0.249949 0.0135554
\(341\) 0 0
\(342\) −1.03017 −0.0557055
\(343\) 18.3501 0.990815
\(344\) −2.79889 −0.150906
\(345\) −13.6982 −0.737485
\(346\) 2.61108 0.140372
\(347\) 8.35577 0.448561 0.224281 0.974525i \(-0.427997\pi\)
0.224281 + 0.974525i \(0.427997\pi\)
\(348\) 19.7726 1.05992
\(349\) 28.3911 1.51974 0.759871 0.650074i \(-0.225262\pi\)
0.759871 + 0.650074i \(0.225262\pi\)
\(350\) −5.93588 −0.317286
\(351\) −2.38377 −0.127236
\(352\) 0 0
\(353\) −12.4082 −0.660423 −0.330212 0.943907i \(-0.607120\pi\)
−0.330212 + 0.943907i \(0.607120\pi\)
\(354\) −2.91399 −0.154877
\(355\) 1.65083 0.0876169
\(356\) 9.78413 0.518558
\(357\) 1.34211 0.0710322
\(358\) −0.632806 −0.0334448
\(359\) −12.1134 −0.639319 −0.319659 0.947533i \(-0.603568\pi\)
−0.319659 + 0.947533i \(0.603568\pi\)
\(360\) 2.47344 0.130362
\(361\) 1.00000 0.0526316
\(362\) 5.20999 0.273831
\(363\) 0 0
\(364\) 5.69009 0.298242
\(365\) −10.5911 −0.554361
\(366\) −0.887725 −0.0464021
\(367\) −4.06064 −0.211964 −0.105982 0.994368i \(-0.533799\pi\)
−0.105982 + 0.994368i \(0.533799\pi\)
\(368\) −25.1300 −1.30999
\(369\) 14.7399 0.767331
\(370\) −2.21621 −0.115216
\(371\) 25.7299 1.33583
\(372\) 12.2175 0.633449
\(373\) −35.4898 −1.83759 −0.918796 0.394732i \(-0.870837\pi\)
−0.918796 + 0.394732i \(0.870837\pi\)
\(374\) 0 0
\(375\) −13.8516 −0.715296
\(376\) 10.8388 0.558967
\(377\) −5.97006 −0.307474
\(378\) −2.55159 −0.131240
\(379\) 13.5584 0.696449 0.348224 0.937411i \(-0.386785\pi\)
0.348224 + 0.937411i \(0.386785\pi\)
\(380\) −1.12578 −0.0577515
\(381\) −42.0961 −2.15665
\(382\) −7.40862 −0.379058
\(383\) −23.8326 −1.21779 −0.608894 0.793251i \(-0.708387\pi\)
−0.608894 + 0.793251i \(0.708387\pi\)
\(384\) 25.5120 1.30190
\(385\) 0 0
\(386\) −8.86045 −0.450985
\(387\) −3.26737 −0.166090
\(388\) −24.3180 −1.23456
\(389\) −12.7567 −0.646789 −0.323395 0.946264i \(-0.604824\pi\)
−0.323395 + 0.946264i \(0.604824\pi\)
\(390\) −0.843823 −0.0427286
\(391\) 2.10636 0.106523
\(392\) 0.229296 0.0115812
\(393\) 6.99958 0.353082
\(394\) −0.278920 −0.0140518
\(395\) −3.62986 −0.182638
\(396\) 0 0
\(397\) −30.3169 −1.52156 −0.760780 0.649009i \(-0.775184\pi\)
−0.760780 + 0.649009i \(0.775184\pi\)
\(398\) 4.92217 0.246726
\(399\) −6.04495 −0.302626
\(400\) −12.1674 −0.608368
\(401\) 32.6370 1.62982 0.814908 0.579591i \(-0.196788\pi\)
0.814908 + 0.579591i \(0.196788\pi\)
\(402\) 2.43855 0.121624
\(403\) −3.68891 −0.183758
\(404\) 21.2422 1.05684
\(405\) −6.92152 −0.343933
\(406\) −6.39038 −0.317149
\(407\) 0 0
\(408\) −0.916520 −0.0453745
\(409\) 23.1587 1.14512 0.572562 0.819862i \(-0.305950\pi\)
0.572562 + 0.819862i \(0.305950\pi\)
\(410\) −2.13792 −0.105584
\(411\) −19.6474 −0.969135
\(412\) −15.0622 −0.742063
\(413\) −7.09578 −0.349160
\(414\) 9.77344 0.480338
\(415\) 1.87318 0.0919506
\(416\) −5.94949 −0.291698
\(417\) −32.9319 −1.61268
\(418\) 0 0
\(419\) −14.2213 −0.694755 −0.347377 0.937725i \(-0.612928\pi\)
−0.347377 + 0.937725i \(0.612928\pi\)
\(420\) 6.80531 0.332065
\(421\) 15.9605 0.777866 0.388933 0.921266i \(-0.372844\pi\)
0.388933 + 0.921266i \(0.372844\pi\)
\(422\) −3.51373 −0.171045
\(423\) 12.6530 0.615209
\(424\) −17.5708 −0.853313
\(425\) 1.01985 0.0494701
\(426\) −2.83829 −0.137516
\(427\) −2.16167 −0.104611
\(428\) 2.90600 0.140467
\(429\) 0 0
\(430\) 0.473909 0.0228539
\(431\) −24.5100 −1.18061 −0.590304 0.807181i \(-0.700992\pi\)
−0.590304 + 0.807181i \(0.700992\pi\)
\(432\) −5.23026 −0.251641
\(433\) −19.1839 −0.921921 −0.460961 0.887421i \(-0.652495\pi\)
−0.460961 + 0.887421i \(0.652495\pi\)
\(434\) −3.94862 −0.189540
\(435\) −7.14016 −0.342344
\(436\) 29.6745 1.42115
\(437\) −9.48717 −0.453833
\(438\) 18.2093 0.870076
\(439\) −13.7961 −0.658450 −0.329225 0.944251i \(-0.606788\pi\)
−0.329225 + 0.944251i \(0.606788\pi\)
\(440\) 0 0
\(441\) 0.267676 0.0127465
\(442\) 0.129754 0.00617178
\(443\) 14.1259 0.671140 0.335570 0.942015i \(-0.391071\pi\)
0.335570 + 0.942015i \(0.391071\pi\)
\(444\) −28.7088 −1.36246
\(445\) −3.53318 −0.167489
\(446\) −7.87967 −0.373113
\(447\) 41.8292 1.97846
\(448\) 7.77337 0.367257
\(449\) −21.0016 −0.991129 −0.495565 0.868571i \(-0.665039\pi\)
−0.495565 + 0.868571i \(0.665039\pi\)
\(450\) 4.73207 0.223072
\(451\) 0 0
\(452\) 19.9425 0.938018
\(453\) 2.06975 0.0972455
\(454\) −10.1301 −0.475429
\(455\) −2.05477 −0.0963290
\(456\) 4.12805 0.193314
\(457\) 22.4621 1.05073 0.525367 0.850876i \(-0.323928\pi\)
0.525367 + 0.850876i \(0.323928\pi\)
\(458\) 5.23223 0.244486
\(459\) 0.438393 0.0204624
\(460\) 10.6805 0.497981
\(461\) 31.1213 1.44946 0.724732 0.689031i \(-0.241964\pi\)
0.724732 + 0.689031i \(0.241964\pi\)
\(462\) 0 0
\(463\) 31.6932 1.47291 0.736455 0.676487i \(-0.236498\pi\)
0.736455 + 0.676487i \(0.236498\pi\)
\(464\) −13.0990 −0.608106
\(465\) −4.41191 −0.204597
\(466\) −12.3043 −0.569986
\(467\) −1.75474 −0.0811995 −0.0405997 0.999175i \(-0.512927\pi\)
−0.0405997 + 0.999175i \(0.512927\pi\)
\(468\) −4.53612 −0.209682
\(469\) 5.93804 0.274193
\(470\) −1.83522 −0.0846525
\(471\) −5.42174 −0.249821
\(472\) 4.84566 0.223039
\(473\) 0 0
\(474\) 6.24086 0.286652
\(475\) −4.59346 −0.210763
\(476\) −1.04645 −0.0479640
\(477\) −20.5118 −0.939173
\(478\) 0.170516 0.00779922
\(479\) 11.1294 0.508515 0.254257 0.967137i \(-0.418169\pi\)
0.254257 + 0.967137i \(0.418169\pi\)
\(480\) −7.11555 −0.324779
\(481\) 8.66823 0.395237
\(482\) −7.59212 −0.345812
\(483\) 57.3495 2.60949
\(484\) 0 0
\(485\) 8.78158 0.398751
\(486\) 9.03268 0.409731
\(487\) 3.27007 0.148181 0.0740905 0.997252i \(-0.476395\pi\)
0.0740905 + 0.997252i \(0.476395\pi\)
\(488\) 1.47619 0.0668240
\(489\) 32.1531 1.45401
\(490\) −0.0388244 −0.00175391
\(491\) −29.2686 −1.32087 −0.660437 0.750882i \(-0.729629\pi\)
−0.660437 + 0.750882i \(0.729629\pi\)
\(492\) −27.6946 −1.24857
\(493\) 1.09794 0.0494487
\(494\) −0.584420 −0.0262943
\(495\) 0 0
\(496\) −8.09389 −0.363426
\(497\) −6.91145 −0.310021
\(498\) −3.22058 −0.144317
\(499\) −1.04828 −0.0469275 −0.0234638 0.999725i \(-0.507469\pi\)
−0.0234638 + 0.999725i \(0.507469\pi\)
\(500\) 10.8002 0.482998
\(501\) −20.9481 −0.935891
\(502\) −0.563155 −0.0251348
\(503\) −28.4724 −1.26952 −0.634761 0.772709i \(-0.718901\pi\)
−0.634761 + 0.772709i \(0.718901\pi\)
\(504\) −10.3554 −0.461267
\(505\) −7.67086 −0.341349
\(506\) 0 0
\(507\) −26.1383 −1.16084
\(508\) 32.8224 1.45626
\(509\) 39.6801 1.75879 0.879396 0.476091i \(-0.157947\pi\)
0.879396 + 0.476091i \(0.157947\pi\)
\(510\) 0.155185 0.00687173
\(511\) 44.3410 1.96153
\(512\) −22.7112 −1.00370
\(513\) −1.97454 −0.0871782
\(514\) 7.28609 0.321376
\(515\) 5.43918 0.239679
\(516\) 6.13901 0.270255
\(517\) 0 0
\(518\) 9.27852 0.407675
\(519\) −12.2143 −0.536147
\(520\) 1.40319 0.0615338
\(521\) −15.3168 −0.671043 −0.335522 0.942033i \(-0.608913\pi\)
−0.335522 + 0.942033i \(0.608913\pi\)
\(522\) 5.09440 0.222976
\(523\) 4.17850 0.182713 0.0913566 0.995818i \(-0.470880\pi\)
0.0913566 + 0.995818i \(0.470880\pi\)
\(524\) −5.45760 −0.238416
\(525\) 27.7672 1.21186
\(526\) −10.0018 −0.436098
\(527\) 0.678418 0.0295524
\(528\) 0 0
\(529\) 67.0064 2.91332
\(530\) 2.97509 0.129230
\(531\) 5.65674 0.245481
\(532\) 4.71327 0.204346
\(533\) 8.36201 0.362199
\(534\) 6.07465 0.262876
\(535\) −1.04940 −0.0453694
\(536\) −4.05505 −0.175151
\(537\) 2.96018 0.127741
\(538\) 1.36835 0.0589937
\(539\) 0 0
\(540\) 2.22291 0.0956588
\(541\) 14.6201 0.628565 0.314283 0.949329i \(-0.398236\pi\)
0.314283 + 0.949329i \(0.398236\pi\)
\(542\) 11.4231 0.490665
\(543\) −24.3716 −1.04589
\(544\) 1.09416 0.0469116
\(545\) −10.7159 −0.459017
\(546\) 3.53279 0.151189
\(547\) 34.7220 1.48461 0.742303 0.670064i \(-0.233733\pi\)
0.742303 + 0.670064i \(0.233733\pi\)
\(548\) 15.3192 0.654402
\(549\) 1.72328 0.0735478
\(550\) 0 0
\(551\) −4.94518 −0.210672
\(552\) −39.1635 −1.66691
\(553\) 15.1969 0.646240
\(554\) −9.08997 −0.386196
\(555\) 10.3672 0.440061
\(556\) 25.6771 1.08895
\(557\) 7.07603 0.299821 0.149910 0.988700i \(-0.452101\pi\)
0.149910 + 0.988700i \(0.452101\pi\)
\(558\) 3.14783 0.133258
\(559\) −1.85359 −0.0783985
\(560\) −4.50840 −0.190515
\(561\) 0 0
\(562\) 13.1979 0.556721
\(563\) 32.5712 1.37271 0.686357 0.727265i \(-0.259209\pi\)
0.686357 + 0.727265i \(0.259209\pi\)
\(564\) −23.7735 −1.00104
\(565\) −7.20152 −0.302970
\(566\) −13.1964 −0.554686
\(567\) 28.9780 1.21696
\(568\) 4.71978 0.198037
\(569\) −32.6457 −1.36858 −0.684290 0.729210i \(-0.739888\pi\)
−0.684290 + 0.729210i \(0.739888\pi\)
\(570\) −0.698963 −0.0292763
\(571\) 35.8059 1.49843 0.749216 0.662326i \(-0.230431\pi\)
0.749216 + 0.662326i \(0.230431\pi\)
\(572\) 0 0
\(573\) 34.6565 1.44780
\(574\) 8.95073 0.373596
\(575\) 43.5790 1.81737
\(576\) −6.19691 −0.258205
\(577\) 15.7218 0.654509 0.327254 0.944936i \(-0.393877\pi\)
0.327254 + 0.944936i \(0.393877\pi\)
\(578\) 8.20571 0.341313
\(579\) 41.4480 1.72252
\(580\) 5.56721 0.231166
\(581\) −7.84233 −0.325355
\(582\) −15.0983 −0.625844
\(583\) 0 0
\(584\) −30.2802 −1.25300
\(585\) 1.63806 0.0677253
\(586\) 15.0067 0.619919
\(587\) 24.4810 1.01044 0.505220 0.862991i \(-0.331412\pi\)
0.505220 + 0.862991i \(0.331412\pi\)
\(588\) −0.502932 −0.0207406
\(589\) −3.05563 −0.125905
\(590\) −0.820468 −0.0337781
\(591\) 1.30475 0.0536703
\(592\) 19.0191 0.781680
\(593\) 21.2995 0.874666 0.437333 0.899300i \(-0.355923\pi\)
0.437333 + 0.899300i \(0.355923\pi\)
\(594\) 0 0
\(595\) 0.377887 0.0154919
\(596\) −32.6144 −1.33594
\(597\) −23.0253 −0.942361
\(598\) 5.54450 0.226731
\(599\) −8.90839 −0.363987 −0.181994 0.983300i \(-0.558255\pi\)
−0.181994 + 0.983300i \(0.558255\pi\)
\(600\) −18.9621 −0.774123
\(601\) 8.36794 0.341335 0.170668 0.985329i \(-0.445408\pi\)
0.170668 + 0.985329i \(0.445408\pi\)
\(602\) −1.98409 −0.0808655
\(603\) −4.73379 −0.192775
\(604\) −1.61379 −0.0656643
\(605\) 0 0
\(606\) 13.1886 0.535751
\(607\) −43.2321 −1.75474 −0.877368 0.479819i \(-0.840702\pi\)
−0.877368 + 0.479819i \(0.840702\pi\)
\(608\) −4.92814 −0.199862
\(609\) 29.8933 1.21134
\(610\) −0.249949 −0.0101201
\(611\) 7.17807 0.290394
\(612\) 0.834228 0.0337216
\(613\) 12.4893 0.504438 0.252219 0.967670i \(-0.418840\pi\)
0.252219 + 0.967670i \(0.418840\pi\)
\(614\) −8.83447 −0.356530
\(615\) 10.0009 0.403276
\(616\) 0 0
\(617\) −18.4079 −0.741073 −0.370536 0.928818i \(-0.620826\pi\)
−0.370536 + 0.928818i \(0.620826\pi\)
\(618\) −9.35165 −0.376179
\(619\) −7.30900 −0.293773 −0.146887 0.989153i \(-0.546925\pi\)
−0.146887 + 0.989153i \(0.546925\pi\)
\(620\) 3.43998 0.138153
\(621\) 18.7328 0.751722
\(622\) −14.9551 −0.599646
\(623\) 14.7922 0.592637
\(624\) 7.24151 0.289893
\(625\) 19.0672 0.762689
\(626\) −13.9406 −0.557177
\(627\) 0 0
\(628\) 4.22735 0.168690
\(629\) −1.59415 −0.0635631
\(630\) 1.75338 0.0698564
\(631\) −11.8397 −0.471329 −0.235665 0.971834i \(-0.575727\pi\)
−0.235665 + 0.971834i \(0.575727\pi\)
\(632\) −10.3779 −0.412810
\(633\) 16.4367 0.653301
\(634\) −2.88538 −0.114593
\(635\) −11.8526 −0.470357
\(636\) 38.5393 1.52818
\(637\) 0.151853 0.00601664
\(638\) 0 0
\(639\) 5.50979 0.217964
\(640\) 7.18320 0.283941
\(641\) −36.5974 −1.44551 −0.722756 0.691104i \(-0.757125\pi\)
−0.722756 + 0.691104i \(0.757125\pi\)
\(642\) 1.80424 0.0712078
\(643\) −14.0452 −0.553888 −0.276944 0.960886i \(-0.589322\pi\)
−0.276944 + 0.960886i \(0.589322\pi\)
\(644\) −44.7156 −1.76204
\(645\) −2.21688 −0.0872896
\(646\) 0.107479 0.00422872
\(647\) −43.5266 −1.71121 −0.855604 0.517631i \(-0.826814\pi\)
−0.855604 + 0.517631i \(0.826814\pi\)
\(648\) −19.7889 −0.777380
\(649\) 0 0
\(650\) 2.68451 0.105295
\(651\) 18.4711 0.723941
\(652\) −25.0699 −0.981812
\(653\) 44.0328 1.72314 0.861568 0.507642i \(-0.169483\pi\)
0.861568 + 0.507642i \(0.169483\pi\)
\(654\) 18.4239 0.720433
\(655\) 1.97081 0.0770060
\(656\) 18.3472 0.716338
\(657\) −35.3486 −1.37908
\(658\) 7.68344 0.299532
\(659\) 3.86784 0.150670 0.0753349 0.997158i \(-0.475997\pi\)
0.0753349 + 0.997158i \(0.475997\pi\)
\(660\) 0 0
\(661\) 7.96999 0.309997 0.154998 0.987915i \(-0.450463\pi\)
0.154998 + 0.987915i \(0.450463\pi\)
\(662\) −16.1692 −0.628434
\(663\) −0.606973 −0.0235729
\(664\) 5.35547 0.207833
\(665\) −1.70203 −0.0660017
\(666\) −7.39681 −0.286621
\(667\) 46.9158 1.81659
\(668\) 16.3333 0.631954
\(669\) 36.8600 1.42509
\(670\) 0.686602 0.0265258
\(671\) 0 0
\(672\) 29.7903 1.14919
\(673\) −21.7872 −0.839835 −0.419917 0.907562i \(-0.637941\pi\)
−0.419917 + 0.907562i \(0.637941\pi\)
\(674\) 12.1017 0.466141
\(675\) 9.06999 0.349104
\(676\) 20.3802 0.783852
\(677\) −33.2004 −1.27599 −0.637997 0.770039i \(-0.720237\pi\)
−0.637997 + 0.770039i \(0.720237\pi\)
\(678\) 12.3817 0.475515
\(679\) −36.7654 −1.41093
\(680\) −0.258057 −0.00989602
\(681\) 47.3873 1.81588
\(682\) 0 0
\(683\) −34.9595 −1.33769 −0.668843 0.743403i \(-0.733210\pi\)
−0.668843 + 0.743403i \(0.733210\pi\)
\(684\) −3.75740 −0.143668
\(685\) −5.53196 −0.211365
\(686\) −8.88317 −0.339161
\(687\) −24.4757 −0.933805
\(688\) −4.06699 −0.155052
\(689\) −11.6364 −0.443312
\(690\) 6.63118 0.252445
\(691\) −30.4626 −1.15885 −0.579426 0.815025i \(-0.696723\pi\)
−0.579426 + 0.815025i \(0.696723\pi\)
\(692\) 9.52351 0.362030
\(693\) 0 0
\(694\) −4.04497 −0.153545
\(695\) −9.27236 −0.351721
\(696\) −20.4140 −0.773789
\(697\) −1.53784 −0.0582497
\(698\) −13.7439 −0.520215
\(699\) 57.5579 2.17704
\(700\) −21.6502 −0.818301
\(701\) −10.3329 −0.390269 −0.195135 0.980776i \(-0.562514\pi\)
−0.195135 + 0.980776i \(0.562514\pi\)
\(702\) 1.15396 0.0435535
\(703\) 7.18015 0.270805
\(704\) 0 0
\(705\) 8.58493 0.323327
\(706\) 6.00673 0.226066
\(707\) 32.1152 1.20782
\(708\) −10.6283 −0.399437
\(709\) −45.4499 −1.70691 −0.853454 0.521169i \(-0.825496\pi\)
−0.853454 + 0.521169i \(0.825496\pi\)
\(710\) −0.799154 −0.0299917
\(711\) −12.1150 −0.454347
\(712\) −10.1015 −0.378570
\(713\) 28.9893 1.08566
\(714\) −0.649707 −0.0243147
\(715\) 0 0
\(716\) −2.30806 −0.0862564
\(717\) −0.797651 −0.0297888
\(718\) 5.86399 0.218842
\(719\) −7.51239 −0.280165 −0.140082 0.990140i \(-0.544737\pi\)
−0.140082 + 0.990140i \(0.544737\pi\)
\(720\) 3.59408 0.133944
\(721\) −22.7719 −0.848071
\(722\) −0.484093 −0.0180161
\(723\) 35.5149 1.32081
\(724\) 19.0026 0.706227
\(725\) 22.7155 0.843632
\(726\) 0 0
\(727\) 4.79898 0.177984 0.0889922 0.996032i \(-0.471635\pi\)
0.0889922 + 0.996032i \(0.471635\pi\)
\(728\) −5.87465 −0.217729
\(729\) −9.68700 −0.358778
\(730\) 5.12705 0.189761
\(731\) 0.340889 0.0126082
\(732\) −3.23784 −0.119674
\(733\) 8.55077 0.315830 0.157915 0.987453i \(-0.449523\pi\)
0.157915 + 0.987453i \(0.449523\pi\)
\(734\) 1.96572 0.0725562
\(735\) 0.181616 0.00669899
\(736\) 46.7541 1.72338
\(737\) 0 0
\(738\) −7.13550 −0.262661
\(739\) −38.8820 −1.43030 −0.715149 0.698972i \(-0.753641\pi\)
−0.715149 + 0.698972i \(0.753641\pi\)
\(740\) −8.08330 −0.297148
\(741\) 2.73384 0.100430
\(742\) −12.4557 −0.457262
\(743\) −28.4350 −1.04318 −0.521589 0.853197i \(-0.674660\pi\)
−0.521589 + 0.853197i \(0.674660\pi\)
\(744\) −12.6138 −0.462445
\(745\) 11.7775 0.431494
\(746\) 17.1804 0.629018
\(747\) 6.25189 0.228745
\(748\) 0 0
\(749\) 4.39346 0.160534
\(750\) 6.70548 0.244849
\(751\) 18.2732 0.666799 0.333400 0.942786i \(-0.391804\pi\)
0.333400 + 0.942786i \(0.391804\pi\)
\(752\) 15.7495 0.574326
\(753\) 2.63436 0.0960015
\(754\) 2.89006 0.105250
\(755\) 0.582763 0.0212089
\(756\) −9.30655 −0.338476
\(757\) 1.10637 0.0402117 0.0201058 0.999798i \(-0.493600\pi\)
0.0201058 + 0.999798i \(0.493600\pi\)
\(758\) −6.56352 −0.238398
\(759\) 0 0
\(760\) 1.16230 0.0421611
\(761\) −1.10110 −0.0399147 −0.0199573 0.999801i \(-0.506353\pi\)
−0.0199573 + 0.999801i \(0.506353\pi\)
\(762\) 20.3784 0.738231
\(763\) 44.8636 1.62417
\(764\) −27.0218 −0.977615
\(765\) −0.301251 −0.0108918
\(766\) 11.5372 0.416855
\(767\) 3.20908 0.115873
\(768\) 0.838451 0.0302550
\(769\) −30.2914 −1.09234 −0.546169 0.837675i \(-0.683914\pi\)
−0.546169 + 0.837675i \(0.683914\pi\)
\(770\) 0 0
\(771\) −34.0833 −1.22748
\(772\) −32.3171 −1.16312
\(773\) −43.9677 −1.58141 −0.790704 0.612199i \(-0.790285\pi\)
−0.790704 + 0.612199i \(0.790285\pi\)
\(774\) 1.58171 0.0568535
\(775\) 14.0359 0.504186
\(776\) 25.1068 0.901283
\(777\) −43.4036 −1.55710
\(778\) 6.17542 0.221399
\(779\) 6.92650 0.248168
\(780\) −3.07771 −0.110200
\(781\) 0 0
\(782\) −1.01967 −0.0364635
\(783\) 9.76447 0.348954
\(784\) 0.333183 0.0118994
\(785\) −1.52655 −0.0544851
\(786\) −3.38845 −0.120862
\(787\) 37.8350 1.34867 0.674337 0.738424i \(-0.264430\pi\)
0.674337 + 0.738424i \(0.264430\pi\)
\(788\) −1.01732 −0.0362405
\(789\) 46.7870 1.66566
\(790\) 1.75719 0.0625179
\(791\) 30.1502 1.07202
\(792\) 0 0
\(793\) 0.977621 0.0347163
\(794\) 14.6762 0.520838
\(795\) −13.9171 −0.493588
\(796\) 17.9529 0.636323
\(797\) 11.2902 0.399918 0.199959 0.979804i \(-0.435919\pi\)
0.199959 + 0.979804i \(0.435919\pi\)
\(798\) 2.92632 0.103590
\(799\) −1.32010 −0.0467018
\(800\) 22.6372 0.800347
\(801\) −11.7923 −0.416661
\(802\) −15.7993 −0.557894
\(803\) 0 0
\(804\) 8.89424 0.313676
\(805\) 16.1474 0.569121
\(806\) 1.78577 0.0629012
\(807\) −6.40095 −0.225324
\(808\) −21.9313 −0.771539
\(809\) −17.8104 −0.626180 −0.313090 0.949723i \(-0.601364\pi\)
−0.313090 + 0.949723i \(0.601364\pi\)
\(810\) 3.35066 0.117730
\(811\) −28.1638 −0.988965 −0.494482 0.869188i \(-0.664642\pi\)
−0.494482 + 0.869188i \(0.664642\pi\)
\(812\) −23.3079 −0.817949
\(813\) −53.4358 −1.87408
\(814\) 0 0
\(815\) 9.05307 0.317115
\(816\) −1.33177 −0.0466213
\(817\) −1.53538 −0.0537162
\(818\) −11.2110 −0.391982
\(819\) −6.85797 −0.239637
\(820\) −7.79774 −0.272309
\(821\) 44.2644 1.54484 0.772418 0.635114i \(-0.219047\pi\)
0.772418 + 0.635114i \(0.219047\pi\)
\(822\) 9.51117 0.331740
\(823\) −22.4258 −0.781716 −0.390858 0.920451i \(-0.627822\pi\)
−0.390858 + 0.920451i \(0.627822\pi\)
\(824\) 15.5508 0.541738
\(825\) 0 0
\(826\) 3.43501 0.119519
\(827\) −19.7303 −0.686090 −0.343045 0.939319i \(-0.611458\pi\)
−0.343045 + 0.939319i \(0.611458\pi\)
\(828\) 35.6471 1.23882
\(829\) 5.70836 0.198259 0.0991297 0.995075i \(-0.468394\pi\)
0.0991297 + 0.995075i \(0.468394\pi\)
\(830\) −0.906791 −0.0314752
\(831\) 42.5216 1.47506
\(832\) −3.51552 −0.121879
\(833\) −0.0279270 −0.000967612 0
\(834\) 15.9421 0.552030
\(835\) −5.89817 −0.204115
\(836\) 0 0
\(837\) 6.03348 0.208547
\(838\) 6.88441 0.237818
\(839\) −0.0666011 −0.00229932 −0.00114966 0.999999i \(-0.500366\pi\)
−0.00114966 + 0.999999i \(0.500366\pi\)
\(840\) −7.02605 −0.242422
\(841\) −4.54521 −0.156731
\(842\) −7.72635 −0.266268
\(843\) −61.7381 −2.12637
\(844\) −12.8158 −0.441137
\(845\) −7.35955 −0.253176
\(846\) −6.12522 −0.210589
\(847\) 0 0
\(848\) −25.5316 −0.876760
\(849\) 61.7310 2.11860
\(850\) −0.493703 −0.0169339
\(851\) −68.1193 −2.33510
\(852\) −10.3522 −0.354662
\(853\) 14.8615 0.508848 0.254424 0.967093i \(-0.418114\pi\)
0.254424 + 0.967093i \(0.418114\pi\)
\(854\) 1.04645 0.0358088
\(855\) 1.35685 0.0464033
\(856\) −3.00026 −0.102547
\(857\) −12.5169 −0.427569 −0.213785 0.976881i \(-0.568579\pi\)
−0.213785 + 0.976881i \(0.568579\pi\)
\(858\) 0 0
\(859\) −21.2738 −0.725854 −0.362927 0.931818i \(-0.618223\pi\)
−0.362927 + 0.931818i \(0.618223\pi\)
\(860\) 1.72851 0.0589417
\(861\) −41.8703 −1.42694
\(862\) 11.8651 0.404128
\(863\) −34.4402 −1.17236 −0.586179 0.810181i \(-0.699369\pi\)
−0.586179 + 0.810181i \(0.699369\pi\)
\(864\) 9.73082 0.331049
\(865\) −3.43907 −0.116932
\(866\) 9.28680 0.315578
\(867\) −38.3852 −1.30363
\(868\) −14.4020 −0.488836
\(869\) 0 0
\(870\) 3.45650 0.117186
\(871\) −2.68549 −0.0909944
\(872\) −30.6370 −1.03750
\(873\) 29.3093 0.991969
\(874\) 4.59267 0.155349
\(875\) 16.3283 0.551998
\(876\) 66.4158 2.24398
\(877\) 0.910260 0.0307373 0.0153687 0.999882i \(-0.495108\pi\)
0.0153687 + 0.999882i \(0.495108\pi\)
\(878\) 6.67857 0.225391
\(879\) −70.1991 −2.36776
\(880\) 0 0
\(881\) 33.7502 1.13707 0.568537 0.822658i \(-0.307510\pi\)
0.568537 + 0.822658i \(0.307510\pi\)
\(882\) −0.129580 −0.00436319
\(883\) 39.8800 1.34207 0.671035 0.741426i \(-0.265850\pi\)
0.671035 + 0.741426i \(0.265850\pi\)
\(884\) 0.473259 0.0159174
\(885\) 3.83804 0.129014
\(886\) −6.83822 −0.229735
\(887\) 16.1868 0.543500 0.271750 0.962368i \(-0.412398\pi\)
0.271750 + 0.962368i \(0.412398\pi\)
\(888\) 29.6400 0.994655
\(889\) 49.6228 1.66430
\(890\) 1.71039 0.0573324
\(891\) 0 0
\(892\) −28.7399 −0.962282
\(893\) 5.94581 0.198969
\(894\) −20.2492 −0.677236
\(895\) 0.833474 0.0278599
\(896\) −30.0735 −1.00469
\(897\) −25.9364 −0.865991
\(898\) 10.1667 0.339269
\(899\) 15.1106 0.503968
\(900\) 17.2595 0.575317
\(901\) 2.14003 0.0712946
\(902\) 0 0
\(903\) 9.28131 0.308863
\(904\) −20.5894 −0.684793
\(905\) −6.86211 −0.228104
\(906\) −1.00195 −0.0332876
\(907\) −14.3005 −0.474839 −0.237419 0.971407i \(-0.576302\pi\)
−0.237419 + 0.971407i \(0.576302\pi\)
\(908\) −36.9480 −1.22616
\(909\) −25.6022 −0.849171
\(910\) 0.994698 0.0329739
\(911\) −8.50108 −0.281653 −0.140827 0.990034i \(-0.544976\pi\)
−0.140827 + 0.990034i \(0.544976\pi\)
\(912\) 5.99836 0.198626
\(913\) 0 0
\(914\) −10.8738 −0.359672
\(915\) 1.16923 0.0386535
\(916\) 19.0838 0.630545
\(917\) −8.25111 −0.272476
\(918\) −0.212223 −0.00700439
\(919\) 26.0156 0.858177 0.429088 0.903263i \(-0.358835\pi\)
0.429088 + 0.903263i \(0.358835\pi\)
\(920\) −11.0269 −0.363548
\(921\) 41.3265 1.36175
\(922\) −15.0656 −0.496159
\(923\) 3.12571 0.102884
\(924\) 0 0
\(925\) −32.9818 −1.08443
\(926\) −15.3425 −0.504185
\(927\) 18.1537 0.596247
\(928\) 24.3705 0.800001
\(929\) 5.99773 0.196779 0.0983896 0.995148i \(-0.468631\pi\)
0.0983896 + 0.995148i \(0.468631\pi\)
\(930\) 2.13577 0.0700348
\(931\) 0.125785 0.00412242
\(932\) −44.8781 −1.47003
\(933\) 69.9581 2.29033
\(934\) 0.849454 0.0277950
\(935\) 0 0
\(936\) 4.68326 0.153077
\(937\) 50.5847 1.65253 0.826265 0.563282i \(-0.190461\pi\)
0.826265 + 0.563282i \(0.190461\pi\)
\(938\) −2.87456 −0.0938578
\(939\) 65.2120 2.12811
\(940\) −6.69370 −0.218324
\(941\) 21.8640 0.712747 0.356373 0.934344i \(-0.384013\pi\)
0.356373 + 0.934344i \(0.384013\pi\)
\(942\) 2.62463 0.0855149
\(943\) −65.7129 −2.13991
\(944\) 7.04109 0.229168
\(945\) 3.36072 0.109324
\(946\) 0 0
\(947\) −33.1722 −1.07795 −0.538976 0.842321i \(-0.681188\pi\)
−0.538976 + 0.842321i \(0.681188\pi\)
\(948\) 22.7626 0.739295
\(949\) −20.0533 −0.650958
\(950\) 2.22366 0.0721451
\(951\) 13.4974 0.437685
\(952\) 1.08039 0.0350157
\(953\) −22.4494 −0.727209 −0.363604 0.931553i \(-0.618454\pi\)
−0.363604 + 0.931553i \(0.618454\pi\)
\(954\) 9.92963 0.321484
\(955\) 9.75794 0.315760
\(956\) 0.621931 0.0201147
\(957\) 0 0
\(958\) −5.38765 −0.174067
\(959\) 23.1604 0.747887
\(960\) −4.20454 −0.135701
\(961\) −21.6631 −0.698810
\(962\) −4.19623 −0.135292
\(963\) −3.50246 −0.112865
\(964\) −27.6911 −0.891870
\(965\) 11.6702 0.375676
\(966\) −27.7625 −0.893242
\(967\) 0.790013 0.0254051 0.0127026 0.999919i \(-0.495957\pi\)
0.0127026 + 0.999919i \(0.495957\pi\)
\(968\) 0 0
\(969\) −0.502774 −0.0161514
\(970\) −4.25110 −0.136495
\(971\) 51.6970 1.65904 0.829518 0.558479i \(-0.188615\pi\)
0.829518 + 0.558479i \(0.188615\pi\)
\(972\) 32.9453 1.05672
\(973\) 38.8201 1.24452
\(974\) −1.58302 −0.0507231
\(975\) −12.5578 −0.402171
\(976\) 2.14501 0.0686602
\(977\) 21.4536 0.686360 0.343180 0.939270i \(-0.388496\pi\)
0.343180 + 0.939270i \(0.388496\pi\)
\(978\) −15.5651 −0.497716
\(979\) 0 0
\(980\) −0.141606 −0.00452345
\(981\) −35.7651 −1.14189
\(982\) 14.1687 0.452142
\(983\) −42.9916 −1.37122 −0.685610 0.727969i \(-0.740464\pi\)
−0.685610 + 0.727969i \(0.740464\pi\)
\(984\) 28.5929 0.911510
\(985\) 0.367368 0.0117053
\(986\) −0.531505 −0.0169266
\(987\) −35.9421 −1.14405
\(988\) −2.13158 −0.0678147
\(989\) 14.5664 0.463186
\(990\) 0 0
\(991\) 57.7319 1.83392 0.916958 0.398985i \(-0.130637\pi\)
0.916958 + 0.398985i \(0.130637\pi\)
\(992\) 15.0586 0.478110
\(993\) 75.6373 2.40028
\(994\) 3.34578 0.106122
\(995\) −6.48303 −0.205526
\(996\) −11.7466 −0.372204
\(997\) 30.3487 0.961152 0.480576 0.876953i \(-0.340428\pi\)
0.480576 + 0.876953i \(0.340428\pi\)
\(998\) 0.507466 0.0160635
\(999\) −14.1775 −0.448557
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 2299.2.a.n.1.3 5
11.10 odd 2 209.2.a.c.1.3 5
33.32 even 2 1881.2.a.k.1.3 5
44.43 even 2 3344.2.a.t.1.1 5
55.54 odd 2 5225.2.a.h.1.3 5
209.208 even 2 3971.2.a.h.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.c.1.3 5 11.10 odd 2
1881.2.a.k.1.3 5 33.32 even 2
2299.2.a.n.1.3 5 1.1 even 1 trivial
3344.2.a.t.1.1 5 44.43 even 2
3971.2.a.h.1.3 5 209.208 even 2
5225.2.a.h.1.3 5 55.54 odd 2