Properties

Label 4031.2.a.c.1.40
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 1.40
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)
 
\(f(q)\) \(=\) \(q+0.932122 q^{2} -1.74033 q^{3} -1.13115 q^{4} +4.00156 q^{5} -1.62220 q^{6} -2.15132 q^{7} -2.91861 q^{8} +0.0287417 q^{9} +O(q^{10})\) \(q+0.932122 q^{2} -1.74033 q^{3} -1.13115 q^{4} +4.00156 q^{5} -1.62220 q^{6} -2.15132 q^{7} -2.91861 q^{8} +0.0287417 q^{9} +3.72994 q^{10} -1.10471 q^{11} +1.96857 q^{12} -1.13002 q^{13} -2.00530 q^{14} -6.96403 q^{15} -0.458207 q^{16} +0.641226 q^{17} +0.0267907 q^{18} +5.81681 q^{19} -4.52636 q^{20} +3.74401 q^{21} -1.02972 q^{22} -0.711332 q^{23} +5.07934 q^{24} +11.0125 q^{25} -1.05332 q^{26} +5.17096 q^{27} +2.43347 q^{28} -1.00000 q^{29} -6.49132 q^{30} +4.05496 q^{31} +5.41012 q^{32} +1.92255 q^{33} +0.597701 q^{34} -8.60865 q^{35} -0.0325111 q^{36} -5.22581 q^{37} +5.42198 q^{38} +1.96661 q^{39} -11.6790 q^{40} -5.27988 q^{41} +3.48987 q^{42} +1.20920 q^{43} +1.24959 q^{44} +0.115011 q^{45} -0.663049 q^{46} -3.92896 q^{47} +0.797430 q^{48} -2.37180 q^{49} +10.2650 q^{50} -1.11594 q^{51} +1.27823 q^{52} -9.33718 q^{53} +4.81997 q^{54} -4.42054 q^{55} +6.27888 q^{56} -10.1232 q^{57} -0.932122 q^{58} +8.54483 q^{59} +7.87735 q^{60} -12.3405 q^{61} +3.77972 q^{62} -0.0618327 q^{63} +5.95931 q^{64} -4.52186 q^{65} +1.79205 q^{66} -10.2367 q^{67} -0.725322 q^{68} +1.23795 q^{69} -8.02431 q^{70} -0.482849 q^{71} -0.0838858 q^{72} +2.63848 q^{73} -4.87109 q^{74} -19.1653 q^{75} -6.57967 q^{76} +2.37658 q^{77} +1.83312 q^{78} -6.08175 q^{79} -1.83354 q^{80} -9.08540 q^{81} -4.92149 q^{82} +15.9310 q^{83} -4.23503 q^{84} +2.56590 q^{85} +1.12712 q^{86} +1.74033 q^{87} +3.22421 q^{88} -9.49288 q^{89} +0.107205 q^{90} +2.43105 q^{91} +0.804622 q^{92} -7.05697 q^{93} -3.66228 q^{94} +23.2763 q^{95} -9.41538 q^{96} -12.9710 q^{97} -2.21081 q^{98} -0.0317511 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.932122 0.659110 0.329555 0.944136i \(-0.393101\pi\)
0.329555 + 0.944136i \(0.393101\pi\)
\(3\) −1.74033 −1.00478 −0.502389 0.864641i \(-0.667546\pi\)
−0.502389 + 0.864641i \(0.667546\pi\)
\(4\) −1.13115 −0.565574
\(5\) 4.00156 1.78955 0.894776 0.446516i \(-0.147335\pi\)
0.894776 + 0.446516i \(0.147335\pi\)
\(6\) −1.62220 −0.662260
\(7\) −2.15132 −0.813124 −0.406562 0.913623i \(-0.633273\pi\)
−0.406562 + 0.913623i \(0.633273\pi\)
\(8\) −2.91861 −1.03189
\(9\) 0.0287417 0.00958056
\(10\) 3.72994 1.17951
\(11\) −1.10471 −0.333081 −0.166541 0.986035i \(-0.553260\pi\)
−0.166541 + 0.986035i \(0.553260\pi\)
\(12\) 1.96857 0.568277
\(13\) −1.13002 −0.313412 −0.156706 0.987645i \(-0.550088\pi\)
−0.156706 + 0.987645i \(0.550088\pi\)
\(14\) −2.00530 −0.535938
\(15\) −6.96403 −1.79810
\(16\) −0.458207 −0.114552
\(17\) 0.641226 0.155520 0.0777601 0.996972i \(-0.475223\pi\)
0.0777601 + 0.996972i \(0.475223\pi\)
\(18\) 0.0267907 0.00631464
\(19\) 5.81681 1.33447 0.667234 0.744849i \(-0.267478\pi\)
0.667234 + 0.744849i \(0.267478\pi\)
\(20\) −4.52636 −1.01212
\(21\) 3.74401 0.817010
\(22\) −1.02972 −0.219537
\(23\) −0.711332 −0.148323 −0.0741615 0.997246i \(-0.523628\pi\)
−0.0741615 + 0.997246i \(0.523628\pi\)
\(24\) 5.07934 1.03682
\(25\) 11.0125 2.20249
\(26\) −1.05332 −0.206573
\(27\) 5.17096 0.995153
\(28\) 2.43347 0.459882
\(29\) −1.00000 −0.185695
\(30\) −6.49132 −1.18515
\(31\) 4.05496 0.728293 0.364146 0.931342i \(-0.381361\pi\)
0.364146 + 0.931342i \(0.381361\pi\)
\(32\) 5.41012 0.956383
\(33\) 1.92255 0.334673
\(34\) 0.597701 0.102505
\(35\) −8.60865 −1.45513
\(36\) −0.0325111 −0.00541852
\(37\) −5.22581 −0.859118 −0.429559 0.903039i \(-0.641331\pi\)
−0.429559 + 0.903039i \(0.641331\pi\)
\(38\) 5.42198 0.879560
\(39\) 1.96661 0.314910
\(40\) −11.6790 −1.84661
\(41\) −5.27988 −0.824579 −0.412289 0.911053i \(-0.635271\pi\)
−0.412289 + 0.911053i \(0.635271\pi\)
\(42\) 3.48987 0.538499
\(43\) 1.20920 0.184402 0.0922008 0.995740i \(-0.470610\pi\)
0.0922008 + 0.995740i \(0.470610\pi\)
\(44\) 1.24959 0.188382
\(45\) 0.115011 0.0171449
\(46\) −0.663049 −0.0977612
\(47\) −3.92896 −0.573098 −0.286549 0.958066i \(-0.592508\pi\)
−0.286549 + 0.958066i \(0.592508\pi\)
\(48\) 0.797430 0.115099
\(49\) −2.37180 −0.338829
\(50\) 10.2650 1.45169
\(51\) −1.11594 −0.156263
\(52\) 1.27823 0.177258
\(53\) −9.33718 −1.28256 −0.641280 0.767307i \(-0.721596\pi\)
−0.641280 + 0.767307i \(0.721596\pi\)
\(54\) 4.81997 0.655915
\(55\) −4.42054 −0.596066
\(56\) 6.27888 0.839051
\(57\) −10.1232 −1.34084
\(58\) −0.932122 −0.122394
\(59\) 8.54483 1.11244 0.556221 0.831034i \(-0.312251\pi\)
0.556221 + 0.831034i \(0.312251\pi\)
\(60\) 7.87735 1.01696
\(61\) −12.3405 −1.58004 −0.790018 0.613084i \(-0.789929\pi\)
−0.790018 + 0.613084i \(0.789929\pi\)
\(62\) 3.77972 0.480025
\(63\) −0.0618327 −0.00779018
\(64\) 5.95931 0.744913
\(65\) −4.52186 −0.560868
\(66\) 1.79205 0.220586
\(67\) −10.2367 −1.25062 −0.625308 0.780378i \(-0.715027\pi\)
−0.625308 + 0.780378i \(0.715027\pi\)
\(68\) −0.725322 −0.0879582
\(69\) 1.23795 0.149032
\(70\) −8.02431 −0.959089
\(71\) −0.482849 −0.0573036 −0.0286518 0.999589i \(-0.509121\pi\)
−0.0286518 + 0.999589i \(0.509121\pi\)
\(72\) −0.0838858 −0.00988604
\(73\) 2.63848 0.308811 0.154405 0.988008i \(-0.450654\pi\)
0.154405 + 0.988008i \(0.450654\pi\)
\(74\) −4.87109 −0.566253
\(75\) −19.1653 −2.21302
\(76\) −6.57967 −0.754740
\(77\) 2.37658 0.270836
\(78\) 1.83312 0.207560
\(79\) −6.08175 −0.684251 −0.342125 0.939654i \(-0.611147\pi\)
−0.342125 + 0.939654i \(0.611147\pi\)
\(80\) −1.83354 −0.204996
\(81\) −9.08540 −1.00949
\(82\) −4.92149 −0.543488
\(83\) 15.9310 1.74866 0.874330 0.485333i \(-0.161301\pi\)
0.874330 + 0.485333i \(0.161301\pi\)
\(84\) −4.23503 −0.462080
\(85\) 2.56590 0.278311
\(86\) 1.12712 0.121541
\(87\) 1.74033 0.186583
\(88\) 3.22421 0.343702
\(89\) −9.49288 −1.00624 −0.503122 0.864216i \(-0.667815\pi\)
−0.503122 + 0.864216i \(0.667815\pi\)
\(90\) 0.107205 0.0113004
\(91\) 2.43105 0.254843
\(92\) 0.804622 0.0838877
\(93\) −7.05697 −0.731773
\(94\) −3.66228 −0.377735
\(95\) 23.2763 2.38810
\(96\) −9.41538 −0.960954
\(97\) −12.9710 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(98\) −2.21081 −0.223326
\(99\) −0.0317511 −0.00319110
\(100\) −12.4567 −1.24567
\(101\) −7.82868 −0.778983 −0.389492 0.921030i \(-0.627349\pi\)
−0.389492 + 0.921030i \(0.627349\pi\)
\(102\) −1.04020 −0.102995
\(103\) −6.75006 −0.665103 −0.332551 0.943085i \(-0.607910\pi\)
−0.332551 + 0.943085i \(0.607910\pi\)
\(104\) 3.29810 0.323406
\(105\) 14.9819 1.46208
\(106\) −8.70339 −0.845348
\(107\) −0.698664 −0.0675424 −0.0337712 0.999430i \(-0.510752\pi\)
−0.0337712 + 0.999430i \(0.510752\pi\)
\(108\) −5.84913 −0.562833
\(109\) −12.5917 −1.20607 −0.603033 0.797716i \(-0.706041\pi\)
−0.603033 + 0.797716i \(0.706041\pi\)
\(110\) −4.12049 −0.392873
\(111\) 9.09462 0.863223
\(112\) 0.985751 0.0931447
\(113\) 7.55301 0.710528 0.355264 0.934766i \(-0.384391\pi\)
0.355264 + 0.934766i \(0.384391\pi\)
\(114\) −9.43602 −0.883764
\(115\) −2.84644 −0.265432
\(116\) 1.13115 0.105024
\(117\) −0.0324788 −0.00300267
\(118\) 7.96483 0.733222
\(119\) −1.37949 −0.126457
\(120\) 20.3253 1.85544
\(121\) −9.77963 −0.889057
\(122\) −11.5028 −1.04142
\(123\) 9.18872 0.828519
\(124\) −4.58677 −0.411904
\(125\) 24.0593 2.15193
\(126\) −0.0576356 −0.00513459
\(127\) −5.78891 −0.513683 −0.256841 0.966454i \(-0.582682\pi\)
−0.256841 + 0.966454i \(0.582682\pi\)
\(128\) −5.26544 −0.465404
\(129\) −2.10441 −0.185283
\(130\) −4.21492 −0.369673
\(131\) −9.86685 −0.862071 −0.431035 0.902335i \(-0.641852\pi\)
−0.431035 + 0.902335i \(0.641852\pi\)
\(132\) −2.17469 −0.189282
\(133\) −12.5138 −1.08509
\(134\) −9.54189 −0.824294
\(135\) 20.6919 1.78088
\(136\) −1.87149 −0.160479
\(137\) 0.368103 0.0314491 0.0157246 0.999876i \(-0.494995\pi\)
0.0157246 + 0.999876i \(0.494995\pi\)
\(138\) 1.15392 0.0982284
\(139\) −1.00000 −0.0848189
\(140\) 9.73766 0.822983
\(141\) 6.83769 0.575837
\(142\) −0.450074 −0.0377694
\(143\) 1.24834 0.104392
\(144\) −0.0131696 −0.00109747
\(145\) −4.00156 −0.332311
\(146\) 2.45939 0.203540
\(147\) 4.12772 0.340448
\(148\) 5.91117 0.485895
\(149\) −7.86801 −0.644572 −0.322286 0.946642i \(-0.604451\pi\)
−0.322286 + 0.946642i \(0.604451\pi\)
\(150\) −17.8644 −1.45862
\(151\) 17.5929 1.43169 0.715845 0.698259i \(-0.246042\pi\)
0.715845 + 0.698259i \(0.246042\pi\)
\(152\) −16.9770 −1.37702
\(153\) 0.0184299 0.00148997
\(154\) 2.21526 0.178511
\(155\) 16.2262 1.30332
\(156\) −2.22453 −0.178105
\(157\) 17.3072 1.38127 0.690634 0.723204i \(-0.257332\pi\)
0.690634 + 0.723204i \(0.257332\pi\)
\(158\) −5.66894 −0.450996
\(159\) 16.2497 1.28869
\(160\) 21.6489 1.71150
\(161\) 1.53031 0.120605
\(162\) −8.46870 −0.665364
\(163\) 21.7632 1.70462 0.852312 0.523034i \(-0.175200\pi\)
0.852312 + 0.523034i \(0.175200\pi\)
\(164\) 5.97233 0.466360
\(165\) 7.69320 0.598914
\(166\) 14.8497 1.15256
\(167\) −6.84289 −0.529519 −0.264759 0.964315i \(-0.585293\pi\)
−0.264759 + 0.964315i \(0.585293\pi\)
\(168\) −10.9273 −0.843061
\(169\) −11.7230 −0.901773
\(170\) 2.39174 0.183438
\(171\) 0.167185 0.0127849
\(172\) −1.36779 −0.104293
\(173\) −18.1776 −1.38202 −0.691008 0.722847i \(-0.742833\pi\)
−0.691008 + 0.722847i \(0.742833\pi\)
\(174\) 1.62220 0.122979
\(175\) −23.6914 −1.79090
\(176\) 0.506183 0.0381550
\(177\) −14.8708 −1.11776
\(178\) −8.84852 −0.663225
\(179\) 10.3943 0.776904 0.388452 0.921469i \(-0.373010\pi\)
0.388452 + 0.921469i \(0.373010\pi\)
\(180\) −0.130095 −0.00969671
\(181\) 4.53084 0.336775 0.168387 0.985721i \(-0.446144\pi\)
0.168387 + 0.985721i \(0.446144\pi\)
\(182\) 2.26603 0.167970
\(183\) 21.4765 1.58759
\(184\) 2.07610 0.153052
\(185\) −20.9114 −1.53744
\(186\) −6.57795 −0.482319
\(187\) −0.708366 −0.0518009
\(188\) 4.44424 0.324130
\(189\) −11.1244 −0.809183
\(190\) 21.6964 1.57402
\(191\) −9.57482 −0.692810 −0.346405 0.938085i \(-0.612598\pi\)
−0.346405 + 0.938085i \(0.612598\pi\)
\(192\) −10.3711 −0.748473
\(193\) 9.45774 0.680783 0.340392 0.940284i \(-0.389440\pi\)
0.340392 + 0.940284i \(0.389440\pi\)
\(194\) −12.0906 −0.868054
\(195\) 7.86952 0.563548
\(196\) 2.68286 0.191633
\(197\) 0.0556912 0.00396783 0.00198392 0.999998i \(-0.499368\pi\)
0.00198392 + 0.999998i \(0.499368\pi\)
\(198\) −0.0295959 −0.00210329
\(199\) −8.82244 −0.625407 −0.312703 0.949851i \(-0.601235\pi\)
−0.312703 + 0.949851i \(0.601235\pi\)
\(200\) −32.1411 −2.27272
\(201\) 17.8153 1.25659
\(202\) −7.29729 −0.513435
\(203\) 2.15132 0.150993
\(204\) 1.26230 0.0883786
\(205\) −21.1277 −1.47563
\(206\) −6.29188 −0.438376
\(207\) −0.0204449 −0.00142102
\(208\) 0.517785 0.0359019
\(209\) −6.42586 −0.444486
\(210\) 13.9649 0.963672
\(211\) −11.0226 −0.758825 −0.379413 0.925228i \(-0.623874\pi\)
−0.379413 + 0.925228i \(0.623874\pi\)
\(212\) 10.5617 0.725383
\(213\) 0.840315 0.0575775
\(214\) −0.651240 −0.0445179
\(215\) 4.83869 0.329996
\(216\) −15.0920 −1.02688
\(217\) −8.72354 −0.592193
\(218\) −11.7370 −0.794930
\(219\) −4.59182 −0.310287
\(220\) 5.00029 0.337120
\(221\) −0.724601 −0.0487420
\(222\) 8.47730 0.568959
\(223\) −3.45379 −0.231283 −0.115642 0.993291i \(-0.536892\pi\)
−0.115642 + 0.993291i \(0.536892\pi\)
\(224\) −11.6389 −0.777658
\(225\) 0.316517 0.0211011
\(226\) 7.04033 0.468316
\(227\) −14.8267 −0.984081 −0.492040 0.870572i \(-0.663749\pi\)
−0.492040 + 0.870572i \(0.663749\pi\)
\(228\) 11.4508 0.758347
\(229\) −25.5740 −1.68998 −0.844989 0.534783i \(-0.820393\pi\)
−0.844989 + 0.534783i \(0.820393\pi\)
\(230\) −2.65323 −0.174949
\(231\) −4.13603 −0.272131
\(232\) 2.91861 0.191616
\(233\) −12.2051 −0.799582 −0.399791 0.916606i \(-0.630917\pi\)
−0.399791 + 0.916606i \(0.630917\pi\)
\(234\) −0.0302742 −0.00197909
\(235\) −15.7220 −1.02559
\(236\) −9.66547 −0.629169
\(237\) 10.5842 0.687521
\(238\) −1.28585 −0.0833492
\(239\) −8.96006 −0.579578 −0.289789 0.957090i \(-0.593585\pi\)
−0.289789 + 0.957090i \(0.593585\pi\)
\(240\) 3.19096 0.205976
\(241\) 12.5515 0.808511 0.404255 0.914646i \(-0.367531\pi\)
0.404255 + 0.914646i \(0.367531\pi\)
\(242\) −9.11581 −0.585986
\(243\) 0.298682 0.0191605
\(244\) 13.9589 0.893628
\(245\) −9.49091 −0.606352
\(246\) 8.56501 0.546085
\(247\) −6.57313 −0.418239
\(248\) −11.8349 −0.751515
\(249\) −27.7252 −1.75702
\(250\) 22.4262 1.41836
\(251\) −10.3939 −0.656060 −0.328030 0.944667i \(-0.606385\pi\)
−0.328030 + 0.944667i \(0.606385\pi\)
\(252\) 0.0699419 0.00440593
\(253\) 0.785813 0.0494036
\(254\) −5.39597 −0.338573
\(255\) −4.46552 −0.279641
\(256\) −16.8266 −1.05167
\(257\) 10.8018 0.673800 0.336900 0.941540i \(-0.390622\pi\)
0.336900 + 0.941540i \(0.390622\pi\)
\(258\) −1.96156 −0.122122
\(259\) 11.2424 0.698569
\(260\) 5.11489 0.317212
\(261\) −0.0287417 −0.00177906
\(262\) −9.19711 −0.568199
\(263\) 2.02945 0.125141 0.0625705 0.998041i \(-0.480070\pi\)
0.0625705 + 0.998041i \(0.480070\pi\)
\(264\) −5.61118 −0.345344
\(265\) −37.3633 −2.29521
\(266\) −11.6644 −0.715192
\(267\) 16.5207 1.01105
\(268\) 11.5793 0.707317
\(269\) −0.426791 −0.0260219 −0.0130110 0.999915i \(-0.504142\pi\)
−0.0130110 + 0.999915i \(0.504142\pi\)
\(270\) 19.2874 1.17379
\(271\) 28.5499 1.73429 0.867143 0.498060i \(-0.165954\pi\)
0.867143 + 0.498060i \(0.165954\pi\)
\(272\) −0.293814 −0.0178151
\(273\) −4.23082 −0.256061
\(274\) 0.343117 0.0207284
\(275\) −12.1655 −0.733610
\(276\) −1.40031 −0.0842886
\(277\) 7.04733 0.423433 0.211716 0.977331i \(-0.432095\pi\)
0.211716 + 0.977331i \(0.432095\pi\)
\(278\) −0.932122 −0.0559050
\(279\) 0.116546 0.00697745
\(280\) 25.1253 1.50152
\(281\) −10.1970 −0.608300 −0.304150 0.952624i \(-0.598372\pi\)
−0.304150 + 0.952624i \(0.598372\pi\)
\(282\) 6.37356 0.379540
\(283\) −17.1356 −1.01861 −0.509304 0.860587i \(-0.670097\pi\)
−0.509304 + 0.860587i \(0.670097\pi\)
\(284\) 0.546174 0.0324094
\(285\) −40.5084 −2.39951
\(286\) 1.16361 0.0688056
\(287\) 11.3587 0.670485
\(288\) 0.155496 0.00916268
\(289\) −16.5888 −0.975813
\(290\) −3.72994 −0.219030
\(291\) 22.5739 1.32330
\(292\) −2.98451 −0.174655
\(293\) −2.37102 −0.138517 −0.0692583 0.997599i \(-0.522063\pi\)
−0.0692583 + 0.997599i \(0.522063\pi\)
\(294\) 3.84753 0.224393
\(295\) 34.1926 1.99077
\(296\) 15.2521 0.886511
\(297\) −5.71239 −0.331467
\(298\) −7.33394 −0.424844
\(299\) 0.803823 0.0464863
\(300\) 21.6788 1.25163
\(301\) −2.60139 −0.149941
\(302\) 16.3987 0.943641
\(303\) 13.6245 0.782706
\(304\) −2.66530 −0.152865
\(305\) −49.3811 −2.82756
\(306\) 0.0171789 0.000982054 0
\(307\) −2.97983 −0.170068 −0.0850341 0.996378i \(-0.527100\pi\)
−0.0850341 + 0.996378i \(0.527100\pi\)
\(308\) −2.68826 −0.153178
\(309\) 11.7473 0.668281
\(310\) 15.1248 0.859030
\(311\) −25.2041 −1.42919 −0.714596 0.699538i \(-0.753389\pi\)
−0.714596 + 0.699538i \(0.753389\pi\)
\(312\) −5.73978 −0.324951
\(313\) 11.6696 0.659605 0.329802 0.944050i \(-0.393018\pi\)
0.329802 + 0.944050i \(0.393018\pi\)
\(314\) 16.1325 0.910407
\(315\) −0.247427 −0.0139409
\(316\) 6.87937 0.386995
\(317\) 13.5433 0.760667 0.380334 0.924849i \(-0.375809\pi\)
0.380334 + 0.924849i \(0.375809\pi\)
\(318\) 15.1467 0.849388
\(319\) 1.10471 0.0618516
\(320\) 23.8465 1.33306
\(321\) 1.21591 0.0678652
\(322\) 1.42643 0.0794920
\(323\) 3.72989 0.207537
\(324\) 10.2769 0.570941
\(325\) −12.4444 −0.690289
\(326\) 20.2859 1.12353
\(327\) 21.9137 1.21183
\(328\) 15.4099 0.850871
\(329\) 8.45248 0.466000
\(330\) 7.17100 0.394750
\(331\) −14.1403 −0.777219 −0.388610 0.921402i \(-0.627045\pi\)
−0.388610 + 0.921402i \(0.627045\pi\)
\(332\) −18.0204 −0.988997
\(333\) −0.150198 −0.00823083
\(334\) −6.37841 −0.349011
\(335\) −40.9629 −2.23804
\(336\) −1.71553 −0.0935899
\(337\) −0.644408 −0.0351032 −0.0175516 0.999846i \(-0.505587\pi\)
−0.0175516 + 0.999846i \(0.505587\pi\)
\(338\) −10.9273 −0.594367
\(339\) −13.1447 −0.713923
\(340\) −2.90242 −0.157406
\(341\) −4.47954 −0.242581
\(342\) 0.155837 0.00842668
\(343\) 20.1618 1.08863
\(344\) −3.52919 −0.190281
\(345\) 4.95374 0.266700
\(346\) −16.9437 −0.910900
\(347\) −5.27509 −0.283182 −0.141591 0.989925i \(-0.545222\pi\)
−0.141591 + 0.989925i \(0.545222\pi\)
\(348\) −1.96857 −0.105526
\(349\) −30.0555 −1.60883 −0.804417 0.594065i \(-0.797522\pi\)
−0.804417 + 0.594065i \(0.797522\pi\)
\(350\) −22.0833 −1.18040
\(351\) −5.84332 −0.311893
\(352\) −5.97659 −0.318553
\(353\) −5.09939 −0.271413 −0.135707 0.990749i \(-0.543330\pi\)
−0.135707 + 0.990749i \(0.543330\pi\)
\(354\) −13.8614 −0.736726
\(355\) −1.93215 −0.102548
\(356\) 10.7379 0.569105
\(357\) 2.40076 0.127062
\(358\) 9.68873 0.512065
\(359\) −6.83839 −0.360916 −0.180458 0.983583i \(-0.557758\pi\)
−0.180458 + 0.983583i \(0.557758\pi\)
\(360\) −0.335674 −0.0176916
\(361\) 14.8353 0.780803
\(362\) 4.22329 0.221971
\(363\) 17.0198 0.893306
\(364\) −2.74988 −0.144133
\(365\) 10.5580 0.552633
\(366\) 20.0187 1.04639
\(367\) 14.6652 0.765519 0.382759 0.923848i \(-0.374974\pi\)
0.382759 + 0.923848i \(0.374974\pi\)
\(368\) 0.325937 0.0169907
\(369\) −0.151753 −0.00789992
\(370\) −19.4920 −1.01334
\(371\) 20.0873 1.04288
\(372\) 7.98248 0.413872
\(373\) 22.5622 1.16823 0.584114 0.811671i \(-0.301442\pi\)
0.584114 + 0.811671i \(0.301442\pi\)
\(374\) −0.660284 −0.0341425
\(375\) −41.8710 −2.16221
\(376\) 11.4671 0.591372
\(377\) 1.13002 0.0581992
\(378\) −10.3693 −0.533340
\(379\) 0.139131 0.00714670 0.00357335 0.999994i \(-0.498863\pi\)
0.00357335 + 0.999994i \(0.498863\pi\)
\(380\) −26.3289 −1.35065
\(381\) 10.0746 0.516138
\(382\) −8.92490 −0.456638
\(383\) 22.1460 1.13161 0.565804 0.824540i \(-0.308566\pi\)
0.565804 + 0.824540i \(0.308566\pi\)
\(384\) 9.16359 0.467628
\(385\) 9.51002 0.484676
\(386\) 8.81577 0.448711
\(387\) 0.0347545 0.00176667
\(388\) 14.6722 0.744867
\(389\) 21.9416 1.11248 0.556241 0.831021i \(-0.312243\pi\)
0.556241 + 0.831021i \(0.312243\pi\)
\(390\) 7.33535 0.371440
\(391\) −0.456125 −0.0230672
\(392\) 6.92237 0.349633
\(393\) 17.1716 0.866191
\(394\) 0.0519110 0.00261524
\(395\) −24.3365 −1.22450
\(396\) 0.0359152 0.00180481
\(397\) 18.1890 0.912882 0.456441 0.889754i \(-0.349124\pi\)
0.456441 + 0.889754i \(0.349124\pi\)
\(398\) −8.22360 −0.412212
\(399\) 21.7782 1.09027
\(400\) −5.04599 −0.252299
\(401\) 8.95242 0.447062 0.223531 0.974697i \(-0.428242\pi\)
0.223531 + 0.974697i \(0.428242\pi\)
\(402\) 16.6060 0.828233
\(403\) −4.58221 −0.228256
\(404\) 8.85540 0.440573
\(405\) −36.3558 −1.80653
\(406\) 2.00530 0.0995212
\(407\) 5.77298 0.286156
\(408\) 3.25701 0.161246
\(409\) −22.9374 −1.13418 −0.567090 0.823656i \(-0.691931\pi\)
−0.567090 + 0.823656i \(0.691931\pi\)
\(410\) −19.6936 −0.972600
\(411\) −0.640620 −0.0315994
\(412\) 7.63532 0.376165
\(413\) −18.3827 −0.904554
\(414\) −0.0190571 −0.000936607 0
\(415\) 63.7490 3.12932
\(416\) −6.11357 −0.299742
\(417\) 1.74033 0.0852242
\(418\) −5.98968 −0.292965
\(419\) −22.2129 −1.08517 −0.542585 0.840001i \(-0.682554\pi\)
−0.542585 + 0.840001i \(0.682554\pi\)
\(420\) −16.9467 −0.826916
\(421\) 15.1352 0.737643 0.368821 0.929500i \(-0.379761\pi\)
0.368821 + 0.929500i \(0.379761\pi\)
\(422\) −10.2744 −0.500149
\(423\) −0.112925 −0.00549060
\(424\) 27.2516 1.32345
\(425\) 7.06149 0.342532
\(426\) 0.783277 0.0379499
\(427\) 26.5484 1.28477
\(428\) 0.790293 0.0382003
\(429\) −2.17253 −0.104891
\(430\) 4.51025 0.217504
\(431\) 2.95172 0.142179 0.0710897 0.997470i \(-0.477352\pi\)
0.0710897 + 0.997470i \(0.477352\pi\)
\(432\) −2.36937 −0.113996
\(433\) −18.7423 −0.900698 −0.450349 0.892853i \(-0.648700\pi\)
−0.450349 + 0.892853i \(0.648700\pi\)
\(434\) −8.13141 −0.390320
\(435\) 6.96403 0.333899
\(436\) 14.2431 0.682120
\(437\) −4.13768 −0.197932
\(438\) −4.28014 −0.204513
\(439\) −14.5131 −0.692671 −0.346335 0.938111i \(-0.612574\pi\)
−0.346335 + 0.938111i \(0.612574\pi\)
\(440\) 12.9019 0.615072
\(441\) −0.0681696 −0.00324617
\(442\) −0.675417 −0.0321263
\(443\) −8.46908 −0.402378 −0.201189 0.979552i \(-0.564481\pi\)
−0.201189 + 0.979552i \(0.564481\pi\)
\(444\) −10.2874 −0.488217
\(445\) −37.9863 −1.80072
\(446\) −3.21936 −0.152441
\(447\) 13.6929 0.647652
\(448\) −12.8204 −0.605707
\(449\) 11.3371 0.535031 0.267515 0.963554i \(-0.413797\pi\)
0.267515 + 0.963554i \(0.413797\pi\)
\(450\) 0.295032 0.0139080
\(451\) 5.83271 0.274652
\(452\) −8.54358 −0.401856
\(453\) −30.6174 −1.43853
\(454\) −13.8203 −0.648617
\(455\) 9.72799 0.456055
\(456\) 29.5456 1.38360
\(457\) 9.56474 0.447420 0.223710 0.974656i \(-0.428183\pi\)
0.223710 + 0.974656i \(0.428183\pi\)
\(458\) −23.8381 −1.11388
\(459\) 3.31576 0.154766
\(460\) 3.21974 0.150121
\(461\) 7.30349 0.340158 0.170079 0.985430i \(-0.445598\pi\)
0.170079 + 0.985430i \(0.445598\pi\)
\(462\) −3.85528 −0.179364
\(463\) −7.18072 −0.333716 −0.166858 0.985981i \(-0.553362\pi\)
−0.166858 + 0.985981i \(0.553362\pi\)
\(464\) 0.458207 0.0212717
\(465\) −28.2389 −1.30955
\(466\) −11.3766 −0.527012
\(467\) −21.3523 −0.988067 −0.494033 0.869443i \(-0.664478\pi\)
−0.494033 + 0.869443i \(0.664478\pi\)
\(468\) 0.0367383 0.00169823
\(469\) 22.0225 1.01691
\(470\) −14.6548 −0.675976
\(471\) −30.1203 −1.38787
\(472\) −24.9391 −1.14791
\(473\) −1.33581 −0.0614207
\(474\) 9.86581 0.453152
\(475\) 64.0574 2.93916
\(476\) 1.56040 0.0715210
\(477\) −0.268366 −0.0122876
\(478\) −8.35187 −0.382006
\(479\) 6.88264 0.314476 0.157238 0.987561i \(-0.449741\pi\)
0.157238 + 0.987561i \(0.449741\pi\)
\(480\) −37.6762 −1.71968
\(481\) 5.90529 0.269258
\(482\) 11.6995 0.532897
\(483\) −2.66324 −0.121181
\(484\) 11.0622 0.502828
\(485\) −51.9044 −2.35686
\(486\) 0.278408 0.0126288
\(487\) 7.44550 0.337388 0.168694 0.985668i \(-0.446045\pi\)
0.168694 + 0.985668i \(0.446045\pi\)
\(488\) 36.0171 1.63042
\(489\) −37.8751 −1.71277
\(490\) −8.84669 −0.399653
\(491\) 9.05789 0.408777 0.204389 0.978890i \(-0.434479\pi\)
0.204389 + 0.978890i \(0.434479\pi\)
\(492\) −10.3938 −0.468589
\(493\) −0.641226 −0.0288794
\(494\) −6.12696 −0.275665
\(495\) −0.127054 −0.00571064
\(496\) −1.85801 −0.0834272
\(497\) 1.03876 0.0465950
\(498\) −25.8433 −1.15807
\(499\) 6.32078 0.282957 0.141478 0.989941i \(-0.454814\pi\)
0.141478 + 0.989941i \(0.454814\pi\)
\(500\) −27.2146 −1.21707
\(501\) 11.9089 0.532049
\(502\) −9.68843 −0.432416
\(503\) −4.52009 −0.201541 −0.100770 0.994910i \(-0.532131\pi\)
−0.100770 + 0.994910i \(0.532131\pi\)
\(504\) 0.180466 0.00803858
\(505\) −31.3269 −1.39403
\(506\) 0.732473 0.0325624
\(507\) 20.4019 0.906082
\(508\) 6.54812 0.290526
\(509\) 20.5404 0.910436 0.455218 0.890380i \(-0.349561\pi\)
0.455218 + 0.890380i \(0.349561\pi\)
\(510\) −4.16241 −0.184314
\(511\) −5.67623 −0.251102
\(512\) −5.15361 −0.227760
\(513\) 30.0785 1.32800
\(514\) 10.0686 0.444108
\(515\) −27.0108 −1.19024
\(516\) 2.38040 0.104791
\(517\) 4.34035 0.190888
\(518\) 10.4793 0.460434
\(519\) 31.6349 1.38862
\(520\) 13.1976 0.578751
\(521\) 16.2114 0.710234 0.355117 0.934822i \(-0.384441\pi\)
0.355117 + 0.934822i \(0.384441\pi\)
\(522\) −0.0267907 −0.00117260
\(523\) 7.64964 0.334495 0.167248 0.985915i \(-0.446512\pi\)
0.167248 + 0.985915i \(0.446512\pi\)
\(524\) 11.1609 0.487565
\(525\) 41.2308 1.79946
\(526\) 1.89169 0.0824817
\(527\) 2.60015 0.113264
\(528\) −0.880925 −0.0383373
\(529\) −22.4940 −0.978000
\(530\) −34.8271 −1.51279
\(531\) 0.245593 0.0106578
\(532\) 14.1550 0.613698
\(533\) 5.96639 0.258433
\(534\) 15.3993 0.666394
\(535\) −2.79575 −0.120871
\(536\) 29.8771 1.29049
\(537\) −18.0894 −0.780617
\(538\) −0.397822 −0.0171513
\(539\) 2.62014 0.112858
\(540\) −23.4056 −1.00722
\(541\) 46.0016 1.97776 0.988882 0.148705i \(-0.0475105\pi\)
0.988882 + 0.148705i \(0.0475105\pi\)
\(542\) 26.6120 1.14308
\(543\) −7.88515 −0.338384
\(544\) 3.46911 0.148737
\(545\) −50.3864 −2.15832
\(546\) −3.94364 −0.168772
\(547\) 16.3088 0.697312 0.348656 0.937251i \(-0.386638\pi\)
0.348656 + 0.937251i \(0.386638\pi\)
\(548\) −0.416379 −0.0177868
\(549\) −0.354686 −0.0151376
\(550\) −11.3398 −0.483529
\(551\) −5.81681 −0.247804
\(552\) −3.61310 −0.153784
\(553\) 13.0838 0.556381
\(554\) 6.56897 0.279089
\(555\) 36.3927 1.54478
\(556\) 1.13115 0.0479714
\(557\) −44.2058 −1.87306 −0.936531 0.350585i \(-0.885983\pi\)
−0.936531 + 0.350585i \(0.885983\pi\)
\(558\) 0.108636 0.00459891
\(559\) −1.36643 −0.0577937
\(560\) 3.94454 0.166687
\(561\) 1.23279 0.0520484
\(562\) −9.50482 −0.400937
\(563\) 19.2680 0.812049 0.406025 0.913862i \(-0.366915\pi\)
0.406025 + 0.913862i \(0.366915\pi\)
\(564\) −7.73444 −0.325679
\(565\) 30.2238 1.27153
\(566\) −15.9725 −0.671375
\(567\) 19.5456 0.820840
\(568\) 1.40925 0.0591308
\(569\) 11.5693 0.485009 0.242504 0.970150i \(-0.422031\pi\)
0.242504 + 0.970150i \(0.422031\pi\)
\(570\) −37.7588 −1.58154
\(571\) −15.4021 −0.644557 −0.322279 0.946645i \(-0.604449\pi\)
−0.322279 + 0.946645i \(0.604449\pi\)
\(572\) −1.41206 −0.0590413
\(573\) 16.6633 0.696120
\(574\) 10.5877 0.441923
\(575\) −7.83353 −0.326681
\(576\) 0.171280 0.00713669
\(577\) −45.3665 −1.88863 −0.944315 0.329042i \(-0.893274\pi\)
−0.944315 + 0.329042i \(0.893274\pi\)
\(578\) −15.4628 −0.643168
\(579\) −16.4596 −0.684037
\(580\) 4.52636 0.187947
\(581\) −34.2728 −1.42188
\(582\) 21.0416 0.872202
\(583\) 10.3148 0.427197
\(584\) −7.70071 −0.318657
\(585\) −0.129966 −0.00537342
\(586\) −2.21008 −0.0912977
\(587\) −43.3121 −1.78768 −0.893842 0.448383i \(-0.852000\pi\)
−0.893842 + 0.448383i \(0.852000\pi\)
\(588\) −4.66906 −0.192549
\(589\) 23.5869 0.971883
\(590\) 31.8717 1.31214
\(591\) −0.0969209 −0.00398679
\(592\) 2.39450 0.0984134
\(593\) 17.8552 0.733227 0.366614 0.930373i \(-0.380517\pi\)
0.366614 + 0.930373i \(0.380517\pi\)
\(594\) −5.32465 −0.218473
\(595\) −5.52009 −0.226302
\(596\) 8.89988 0.364553
\(597\) 15.3539 0.628395
\(598\) 0.749261 0.0306396
\(599\) 19.9486 0.815079 0.407540 0.913188i \(-0.366387\pi\)
0.407540 + 0.913188i \(0.366387\pi\)
\(600\) 55.9361 2.28358
\(601\) 17.9018 0.730230 0.365115 0.930962i \(-0.381030\pi\)
0.365115 + 0.930962i \(0.381030\pi\)
\(602\) −2.42481 −0.0988278
\(603\) −0.294221 −0.0119816
\(604\) −19.9002 −0.809727
\(605\) −39.1337 −1.59101
\(606\) 12.6997 0.515889
\(607\) 13.6079 0.552327 0.276164 0.961111i \(-0.410937\pi\)
0.276164 + 0.961111i \(0.410937\pi\)
\(608\) 31.4696 1.27626
\(609\) −3.74401 −0.151715
\(610\) −46.0293 −1.86367
\(611\) 4.43983 0.179616
\(612\) −0.0208470 −0.000842689 0
\(613\) −23.7023 −0.957329 −0.478664 0.877998i \(-0.658879\pi\)
−0.478664 + 0.877998i \(0.658879\pi\)
\(614\) −2.77757 −0.112094
\(615\) 36.7692 1.48268
\(616\) −6.93632 −0.279472
\(617\) 34.7167 1.39764 0.698822 0.715296i \(-0.253708\pi\)
0.698822 + 0.715296i \(0.253708\pi\)
\(618\) 10.9499 0.440471
\(619\) −38.4144 −1.54400 −0.772002 0.635620i \(-0.780744\pi\)
−0.772002 + 0.635620i \(0.780744\pi\)
\(620\) −18.3542 −0.737123
\(621\) −3.67827 −0.147604
\(622\) −23.4933 −0.941994
\(623\) 20.4223 0.818201
\(624\) −0.901115 −0.0360735
\(625\) 41.2122 1.64849
\(626\) 10.8775 0.434752
\(627\) 11.1831 0.446610
\(628\) −19.5771 −0.781210
\(629\) −3.35093 −0.133610
\(630\) −0.230632 −0.00918861
\(631\) 6.36670 0.253454 0.126727 0.991938i \(-0.459553\pi\)
0.126727 + 0.991938i \(0.459553\pi\)
\(632\) 17.7503 0.706068
\(633\) 19.1829 0.762451
\(634\) 12.6240 0.501363
\(635\) −23.1647 −0.919262
\(636\) −18.3809 −0.728849
\(637\) 2.68020 0.106193
\(638\) 1.02972 0.0407670
\(639\) −0.0138779 −0.000549001 0
\(640\) −21.0700 −0.832864
\(641\) 30.7291 1.21373 0.606863 0.794806i \(-0.292428\pi\)
0.606863 + 0.794806i \(0.292428\pi\)
\(642\) 1.13337 0.0447306
\(643\) −40.0953 −1.58120 −0.790602 0.612330i \(-0.790232\pi\)
−0.790602 + 0.612330i \(0.790232\pi\)
\(644\) −1.73100 −0.0682111
\(645\) −8.42091 −0.331573
\(646\) 3.47671 0.136789
\(647\) 16.7772 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(648\) 26.5168 1.04168
\(649\) −9.43952 −0.370534
\(650\) −11.5997 −0.454976
\(651\) 15.1818 0.595023
\(652\) −24.6174 −0.964091
\(653\) 50.0384 1.95816 0.979078 0.203486i \(-0.0652272\pi\)
0.979078 + 0.203486i \(0.0652272\pi\)
\(654\) 20.4262 0.798729
\(655\) −39.4828 −1.54272
\(656\) 2.41928 0.0944569
\(657\) 0.0758344 0.00295858
\(658\) 7.87874 0.307145
\(659\) −29.7121 −1.15742 −0.578709 0.815534i \(-0.696443\pi\)
−0.578709 + 0.815534i \(0.696443\pi\)
\(660\) −8.70215 −0.338731
\(661\) 35.2423 1.37077 0.685384 0.728182i \(-0.259635\pi\)
0.685384 + 0.728182i \(0.259635\pi\)
\(662\) −13.1805 −0.512273
\(663\) 1.26104 0.0489749
\(664\) −46.4965 −1.80442
\(665\) −50.0749 −1.94182
\(666\) −0.140003 −0.00542502
\(667\) 0.711332 0.0275429
\(668\) 7.74032 0.299482
\(669\) 6.01073 0.232388
\(670\) −38.1824 −1.47512
\(671\) 13.6326 0.526280
\(672\) 20.2555 0.781375
\(673\) 12.7537 0.491618 0.245809 0.969318i \(-0.420946\pi\)
0.245809 + 0.969318i \(0.420946\pi\)
\(674\) −0.600667 −0.0231368
\(675\) 56.9451 2.19182
\(676\) 13.2605 0.510019
\(677\) −21.0530 −0.809132 −0.404566 0.914509i \(-0.632577\pi\)
−0.404566 + 0.914509i \(0.632577\pi\)
\(678\) −12.2525 −0.470554
\(679\) 27.9049 1.07089
\(680\) −7.48888 −0.287185
\(681\) 25.8033 0.988784
\(682\) −4.17548 −0.159887
\(683\) 12.3201 0.471416 0.235708 0.971824i \(-0.424259\pi\)
0.235708 + 0.971824i \(0.424259\pi\)
\(684\) −0.189111 −0.00723083
\(685\) 1.47298 0.0562799
\(686\) 18.7932 0.717530
\(687\) 44.5072 1.69805
\(688\) −0.554064 −0.0211235
\(689\) 10.5512 0.401970
\(690\) 4.61749 0.175785
\(691\) −14.1582 −0.538604 −0.269302 0.963056i \(-0.586793\pi\)
−0.269302 + 0.963056i \(0.586793\pi\)
\(692\) 20.5615 0.781632
\(693\) 0.0683069 0.00259476
\(694\) −4.91703 −0.186648
\(695\) −4.00156 −0.151788
\(696\) −5.07934 −0.192532
\(697\) −3.38560 −0.128239
\(698\) −28.0154 −1.06040
\(699\) 21.2409 0.803403
\(700\) 26.7985 1.01289
\(701\) 4.73015 0.178655 0.0893276 0.996002i \(-0.471528\pi\)
0.0893276 + 0.996002i \(0.471528\pi\)
\(702\) −5.44668 −0.205572
\(703\) −30.3975 −1.14646
\(704\) −6.58328 −0.248117
\(705\) 27.3614 1.03049
\(706\) −4.75326 −0.178891
\(707\) 16.8420 0.633410
\(708\) 16.8211 0.632175
\(709\) 3.71267 0.139432 0.0697161 0.997567i \(-0.477791\pi\)
0.0697161 + 0.997567i \(0.477791\pi\)
\(710\) −1.80100 −0.0675903
\(711\) −0.174800 −0.00655550
\(712\) 27.7060 1.03833
\(713\) −2.88443 −0.108023
\(714\) 2.23780 0.0837475
\(715\) 4.99532 0.186814
\(716\) −11.7575 −0.439397
\(717\) 15.5935 0.582348
\(718\) −6.37421 −0.237884
\(719\) 29.5106 1.10056 0.550280 0.834980i \(-0.314521\pi\)
0.550280 + 0.834980i \(0.314521\pi\)
\(720\) −0.0526990 −0.00196398
\(721\) 14.5216 0.540811
\(722\) 13.8283 0.514635
\(723\) −21.8437 −0.812374
\(724\) −5.12505 −0.190471
\(725\) −11.0125 −0.408993
\(726\) 15.8645 0.588787
\(727\) −1.05323 −0.0390623 −0.0195311 0.999809i \(-0.506217\pi\)
−0.0195311 + 0.999809i \(0.506217\pi\)
\(728\) −7.09529 −0.262969
\(729\) 26.7364 0.990237
\(730\) 9.84138 0.364246
\(731\) 0.775372 0.0286782
\(732\) −24.2931 −0.897898
\(733\) 11.2175 0.414327 0.207163 0.978306i \(-0.433577\pi\)
0.207163 + 0.978306i \(0.433577\pi\)
\(734\) 13.6698 0.504561
\(735\) 16.5173 0.609250
\(736\) −3.84839 −0.141854
\(737\) 11.3086 0.416557
\(738\) −0.141452 −0.00520692
\(739\) −14.1682 −0.521186 −0.260593 0.965449i \(-0.583918\pi\)
−0.260593 + 0.965449i \(0.583918\pi\)
\(740\) 23.6539 0.869534
\(741\) 11.4394 0.420237
\(742\) 18.7238 0.687373
\(743\) 25.8220 0.947318 0.473659 0.880708i \(-0.342933\pi\)
0.473659 + 0.880708i \(0.342933\pi\)
\(744\) 20.5966 0.755106
\(745\) −31.4843 −1.15350
\(746\) 21.0308 0.769991
\(747\) 0.457885 0.0167531
\(748\) 0.801267 0.0292972
\(749\) 1.50305 0.0549204
\(750\) −39.0289 −1.42513
\(751\) −0.579153 −0.0211336 −0.0105668 0.999944i \(-0.503364\pi\)
−0.0105668 + 0.999944i \(0.503364\pi\)
\(752\) 1.80028 0.0656494
\(753\) 18.0889 0.659195
\(754\) 1.05332 0.0383597
\(755\) 70.3990 2.56208
\(756\) 12.5834 0.457653
\(757\) 51.1370 1.85861 0.929303 0.369318i \(-0.120409\pi\)
0.929303 + 0.369318i \(0.120409\pi\)
\(758\) 0.129688 0.00471046
\(759\) −1.36757 −0.0496397
\(760\) −67.9345 −2.46424
\(761\) 0.854785 0.0309860 0.0154930 0.999880i \(-0.495068\pi\)
0.0154930 + 0.999880i \(0.495068\pi\)
\(762\) 9.39076 0.340191
\(763\) 27.0888 0.980682
\(764\) 10.8305 0.391835
\(765\) 0.0737484 0.00266638
\(766\) 20.6428 0.745854
\(767\) −9.65587 −0.348653
\(768\) 29.2839 1.05669
\(769\) −15.6921 −0.565872 −0.282936 0.959139i \(-0.591308\pi\)
−0.282936 + 0.959139i \(0.591308\pi\)
\(770\) 8.86450 0.319455
\(771\) −18.7987 −0.677020
\(772\) −10.6981 −0.385033
\(773\) −27.8533 −1.00182 −0.500908 0.865501i \(-0.667000\pi\)
−0.500908 + 0.865501i \(0.667000\pi\)
\(774\) 0.0323954 0.00116443
\(775\) 44.6552 1.60406
\(776\) 37.8574 1.35900
\(777\) −19.5655 −0.701908
\(778\) 20.4522 0.733248
\(779\) −30.7120 −1.10037
\(780\) −8.90159 −0.318728
\(781\) 0.533406 0.0190868
\(782\) −0.425164 −0.0152038
\(783\) −5.17096 −0.184795
\(784\) 1.08678 0.0388134
\(785\) 69.2559 2.47185
\(786\) 16.0060 0.570915
\(787\) 36.2276 1.29138 0.645688 0.763602i \(-0.276571\pi\)
0.645688 + 0.763602i \(0.276571\pi\)
\(788\) −0.0629950 −0.00224410
\(789\) −3.53190 −0.125739
\(790\) −22.6846 −0.807081
\(791\) −16.2490 −0.577747
\(792\) 0.0926691 0.00329285
\(793\) 13.9450 0.495203
\(794\) 16.9544 0.601689
\(795\) 65.0243 2.30618
\(796\) 9.97949 0.353714
\(797\) −50.1246 −1.77550 −0.887752 0.460323i \(-0.847734\pi\)
−0.887752 + 0.460323i \(0.847734\pi\)
\(798\) 20.2999 0.718610
\(799\) −2.51936 −0.0891284
\(800\) 59.5788 2.10643
\(801\) −0.272841 −0.00964037
\(802\) 8.34475 0.294663
\(803\) −2.91474 −0.102859
\(804\) −20.1517 −0.710697
\(805\) 6.12361 0.215829
\(806\) −4.27118 −0.150446
\(807\) 0.742757 0.0261463
\(808\) 22.8489 0.803821
\(809\) −40.8190 −1.43512 −0.717560 0.696497i \(-0.754741\pi\)
−0.717560 + 0.696497i \(0.754741\pi\)
\(810\) −33.8880 −1.19070
\(811\) −44.3385 −1.55694 −0.778468 0.627684i \(-0.784003\pi\)
−0.778468 + 0.627684i \(0.784003\pi\)
\(812\) −2.43347 −0.0853980
\(813\) −49.6863 −1.74257
\(814\) 5.38112 0.188608
\(815\) 87.0867 3.05051
\(816\) 0.511333 0.0179002
\(817\) 7.03369 0.246078
\(818\) −21.3804 −0.747549
\(819\) 0.0698724 0.00244154
\(820\) 23.8986 0.834576
\(821\) −50.0851 −1.74798 −0.873992 0.485941i \(-0.838477\pi\)
−0.873992 + 0.485941i \(0.838477\pi\)
\(822\) −0.597136 −0.0208275
\(823\) 7.71373 0.268884 0.134442 0.990921i \(-0.457076\pi\)
0.134442 + 0.990921i \(0.457076\pi\)
\(824\) 19.7008 0.686310
\(825\) 21.1720 0.737115
\(826\) −17.1349 −0.596200
\(827\) −52.0043 −1.80837 −0.904183 0.427146i \(-0.859519\pi\)
−0.904183 + 0.427146i \(0.859519\pi\)
\(828\) 0.0231262 0.000803691 0
\(829\) 41.0699 1.42642 0.713209 0.700952i \(-0.247241\pi\)
0.713209 + 0.700952i \(0.247241\pi\)
\(830\) 59.4219 2.06256
\(831\) −12.2647 −0.425456
\(832\) −6.73416 −0.233465
\(833\) −1.52086 −0.0526948
\(834\) 1.62220 0.0561721
\(835\) −27.3822 −0.947601
\(836\) 7.26860 0.251390
\(837\) 20.9681 0.724763
\(838\) −20.7051 −0.715247
\(839\) −5.53293 −0.191018 −0.0955090 0.995429i \(-0.530448\pi\)
−0.0955090 + 0.995429i \(0.530448\pi\)
\(840\) −43.7263 −1.50870
\(841\) 1.00000 0.0344828
\(842\) 14.1078 0.486188
\(843\) 17.7461 0.611207
\(844\) 12.4682 0.429172
\(845\) −46.9105 −1.61377
\(846\) −0.105260 −0.00361891
\(847\) 21.0391 0.722914
\(848\) 4.27836 0.146919
\(849\) 29.8216 1.02348
\(850\) 6.58217 0.225766
\(851\) 3.71729 0.127427
\(852\) −0.950521 −0.0325643
\(853\) 35.3025 1.20873 0.604367 0.796706i \(-0.293426\pi\)
0.604367 + 0.796706i \(0.293426\pi\)
\(854\) 24.7463 0.846802
\(855\) 0.669000 0.0228793
\(856\) 2.03913 0.0696960
\(857\) −0.769176 −0.0262745 −0.0131373 0.999914i \(-0.504182\pi\)
−0.0131373 + 0.999914i \(0.504182\pi\)
\(858\) −2.02506 −0.0691345
\(859\) −2.41712 −0.0824710 −0.0412355 0.999149i \(-0.513129\pi\)
−0.0412355 + 0.999149i \(0.513129\pi\)
\(860\) −5.47328 −0.186637
\(861\) −19.7679 −0.673689
\(862\) 2.75137 0.0937119
\(863\) 34.0792 1.16007 0.580035 0.814591i \(-0.303039\pi\)
0.580035 + 0.814591i \(0.303039\pi\)
\(864\) 27.9755 0.951747
\(865\) −72.7386 −2.47319
\(866\) −17.4701 −0.593659
\(867\) 28.8700 0.980477
\(868\) 9.86762 0.334929
\(869\) 6.71855 0.227911
\(870\) 6.49132 0.220076
\(871\) 11.5678 0.391959
\(872\) 36.7503 1.24452
\(873\) −0.372809 −0.0126177
\(874\) −3.85683 −0.130459
\(875\) −51.7593 −1.74978
\(876\) 5.19403 0.175490
\(877\) 34.8514 1.17685 0.588424 0.808553i \(-0.299749\pi\)
0.588424 + 0.808553i \(0.299749\pi\)
\(878\) −13.5279 −0.456546
\(879\) 4.12636 0.139179
\(880\) 2.02552 0.0682804
\(881\) 17.1367 0.577351 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(882\) −0.0635424 −0.00213958
\(883\) 55.6457 1.87263 0.936314 0.351163i \(-0.114214\pi\)
0.936314 + 0.351163i \(0.114214\pi\)
\(884\) 0.819632 0.0275672
\(885\) −59.5064 −2.00029
\(886\) −7.89422 −0.265211
\(887\) 37.1510 1.24741 0.623704 0.781661i \(-0.285627\pi\)
0.623704 + 0.781661i \(0.285627\pi\)
\(888\) −26.5437 −0.890748
\(889\) 12.4538 0.417688
\(890\) −35.4079 −1.18688
\(891\) 10.0367 0.336242
\(892\) 3.90675 0.130808
\(893\) −22.8540 −0.764781
\(894\) 12.7635 0.426874
\(895\) 41.5933 1.39031
\(896\) 11.3277 0.378431
\(897\) −1.39892 −0.0467084
\(898\) 10.5676 0.352644
\(899\) −4.05496 −0.135241
\(900\) −0.358028 −0.0119343
\(901\) −5.98724 −0.199464
\(902\) 5.43680 0.181026
\(903\) 4.52726 0.150658
\(904\) −22.0443 −0.733183
\(905\) 18.1304 0.602675
\(906\) −28.5392 −0.948151
\(907\) −11.0585 −0.367192 −0.183596 0.983002i \(-0.558774\pi\)
−0.183596 + 0.983002i \(0.558774\pi\)
\(908\) 16.7712 0.556571
\(909\) −0.225009 −0.00746309
\(910\) 9.06767 0.300590
\(911\) 34.6013 1.14639 0.573196 0.819418i \(-0.305703\pi\)
0.573196 + 0.819418i \(0.305703\pi\)
\(912\) 4.63850 0.153596
\(913\) −17.5991 −0.582446
\(914\) 8.91551 0.294899
\(915\) 85.9394 2.84107
\(916\) 28.9280 0.955808
\(917\) 21.2268 0.700971
\(918\) 3.09069 0.102008
\(919\) 12.2658 0.404612 0.202306 0.979322i \(-0.435156\pi\)
0.202306 + 0.979322i \(0.435156\pi\)
\(920\) 8.30765 0.273895
\(921\) 5.18589 0.170881
\(922\) 6.80775 0.224201
\(923\) 0.545631 0.0179597
\(924\) 4.67846 0.153910
\(925\) −57.5491 −1.89220
\(926\) −6.69331 −0.219956
\(927\) −0.194008 −0.00637206
\(928\) −5.41012 −0.177596
\(929\) −3.00251 −0.0985091 −0.0492546 0.998786i \(-0.515685\pi\)
−0.0492546 + 0.998786i \(0.515685\pi\)
\(930\) −26.3221 −0.863135
\(931\) −13.7963 −0.452156
\(932\) 13.8058 0.452223
\(933\) 43.8633 1.43602
\(934\) −19.9030 −0.651245
\(935\) −2.83457 −0.0927003
\(936\) 0.0947930 0.00309841
\(937\) −9.65067 −0.315274 −0.157637 0.987497i \(-0.550388\pi\)
−0.157637 + 0.987497i \(0.550388\pi\)
\(938\) 20.5277 0.670253
\(939\) −20.3089 −0.662757
\(940\) 17.7839 0.580047
\(941\) −18.8891 −0.615766 −0.307883 0.951424i \(-0.599620\pi\)
−0.307883 + 0.951424i \(0.599620\pi\)
\(942\) −28.0758 −0.914758
\(943\) 3.75575 0.122304
\(944\) −3.91530 −0.127432
\(945\) −44.5150 −1.44807
\(946\) −1.24514 −0.0404830
\(947\) −20.4373 −0.664121 −0.332061 0.943258i \(-0.607744\pi\)
−0.332061 + 0.943258i \(0.607744\pi\)
\(948\) −11.9724 −0.388844
\(949\) −2.98155 −0.0967852
\(950\) 59.7094 1.93723
\(951\) −23.5698 −0.764302
\(952\) 4.02618 0.130489
\(953\) −24.8443 −0.804786 −0.402393 0.915467i \(-0.631821\pi\)
−0.402393 + 0.915467i \(0.631821\pi\)
\(954\) −0.250150 −0.00809890
\(955\) −38.3142 −1.23982
\(956\) 10.1352 0.327795
\(957\) −1.92255 −0.0621472
\(958\) 6.41546 0.207274
\(959\) −0.791908 −0.0255721
\(960\) −41.5008 −1.33943
\(961\) −14.5573 −0.469589
\(962\) 5.50445 0.177471
\(963\) −0.0200808 −0.000647094 0
\(964\) −14.1976 −0.457273
\(965\) 37.8457 1.21830
\(966\) −2.48246 −0.0798719
\(967\) −0.618562 −0.0198916 −0.00994581 0.999951i \(-0.503166\pi\)
−0.00994581 + 0.999951i \(0.503166\pi\)
\(968\) 28.5429 0.917405
\(969\) −6.49123 −0.208528
\(970\) −48.3812 −1.55343
\(971\) 33.9845 1.09062 0.545308 0.838236i \(-0.316413\pi\)
0.545308 + 0.838236i \(0.316413\pi\)
\(972\) −0.337854 −0.0108367
\(973\) 2.15132 0.0689683
\(974\) 6.94011 0.222376
\(975\) 21.6573 0.693588
\(976\) 5.65449 0.180996
\(977\) −59.5556 −1.90535 −0.952676 0.303987i \(-0.901682\pi\)
−0.952676 + 0.303987i \(0.901682\pi\)
\(978\) −35.3042 −1.12890
\(979\) 10.4868 0.335161
\(980\) 10.7356 0.342937
\(981\) −0.361907 −0.0115548
\(982\) 8.44306 0.269429
\(983\) −33.8426 −1.07941 −0.539706 0.841853i \(-0.681465\pi\)
−0.539706 + 0.841853i \(0.681465\pi\)
\(984\) −26.8183 −0.854937
\(985\) 0.222852 0.00710064
\(986\) −0.597701 −0.0190347
\(987\) −14.7101 −0.468227
\(988\) 7.43519 0.236545
\(989\) −0.860144 −0.0273510
\(990\) −0.118430 −0.00376394
\(991\) −34.8968 −1.10853 −0.554267 0.832339i \(-0.687001\pi\)
−0.554267 + 0.832339i \(0.687001\pi\)
\(992\) 21.9378 0.696527
\(993\) 24.6087 0.780934
\(994\) 0.968255 0.0307112
\(995\) −35.3035 −1.11920
\(996\) 31.3614 0.993723
\(997\) 52.6927 1.66879 0.834397 0.551164i \(-0.185816\pi\)
0.834397 + 0.551164i \(0.185816\pi\)
\(998\) 5.89174 0.186500
\(999\) −27.0225 −0.854953
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.1.40 61
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4031.2.a.c.1.40 61 1.1 even 1 trivial