Properties

Label 2013.2.a.e.1.3
Level $2013$
Weight $2$
Character 2013.1
Self dual yes
Analytic conductor $16.074$
Analytic rank $0$
Dimension $13$
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] = [2013,2,Mod(1,2013)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2013, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2013.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2013 = 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2013.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: \(16.0738859269\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} - 760 x^{4} - 742 x^{3} + 366 x^{2} + 236 x - 47 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.46794\) of defining polynomial
Character \(\chi\) \(=\) 2013.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.46794 q^{2} -1.00000 q^{3} +0.154851 q^{4} +3.84216 q^{5} +1.46794 q^{6} +2.46223 q^{7} +2.70857 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.46794 q^{2} -1.00000 q^{3} +0.154851 q^{4} +3.84216 q^{5} +1.46794 q^{6} +2.46223 q^{7} +2.70857 q^{8} +1.00000 q^{9} -5.64006 q^{10} +1.00000 q^{11} -0.154851 q^{12} -4.64696 q^{13} -3.61441 q^{14} -3.84216 q^{15} -4.28572 q^{16} +5.70692 q^{17} -1.46794 q^{18} +6.71196 q^{19} +0.594961 q^{20} -2.46223 q^{21} -1.46794 q^{22} +2.93787 q^{23} -2.70857 q^{24} +9.76218 q^{25} +6.82147 q^{26} -1.00000 q^{27} +0.381279 q^{28} +5.67137 q^{29} +5.64006 q^{30} +0.232595 q^{31} +0.874048 q^{32} -1.00000 q^{33} -8.37742 q^{34} +9.46029 q^{35} +0.154851 q^{36} +5.55034 q^{37} -9.85276 q^{38} +4.64696 q^{39} +10.4068 q^{40} -4.35928 q^{41} +3.61441 q^{42} -9.35832 q^{43} +0.154851 q^{44} +3.84216 q^{45} -4.31262 q^{46} +10.7428 q^{47} +4.28572 q^{48} -0.937404 q^{49} -14.3303 q^{50} -5.70692 q^{51} -0.719586 q^{52} -11.3834 q^{53} +1.46794 q^{54} +3.84216 q^{55} +6.66913 q^{56} -6.71196 q^{57} -8.32523 q^{58} -13.2037 q^{59} -0.594961 q^{60} -1.00000 q^{61} -0.341436 q^{62} +2.46223 q^{63} +7.28839 q^{64} -17.8544 q^{65} +1.46794 q^{66} +12.5899 q^{67} +0.883721 q^{68} -2.93787 q^{69} -13.8871 q^{70} -0.299770 q^{71} +2.70857 q^{72} -2.65468 q^{73} -8.14757 q^{74} -9.76218 q^{75} +1.03935 q^{76} +2.46223 q^{77} -6.82147 q^{78} -12.9557 q^{79} -16.4664 q^{80} +1.00000 q^{81} +6.39917 q^{82} -7.42505 q^{83} -0.381279 q^{84} +21.9269 q^{85} +13.7375 q^{86} -5.67137 q^{87} +2.70857 q^{88} -3.40420 q^{89} -5.64006 q^{90} -11.4419 q^{91} +0.454931 q^{92} -0.232595 q^{93} -15.7698 q^{94} +25.7884 q^{95} -0.874048 q^{96} -15.8330 q^{97} +1.37605 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 2 q^{2} - 13 q^{3} + 16 q^{4} + 3 q^{5} - 2 q^{6} + 11 q^{7} + 9 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 2 q^{2} - 13 q^{3} + 16 q^{4} + 3 q^{5} - 2 q^{6} + 11 q^{7} + 9 q^{8} + 13 q^{9} + 6 q^{10} + 13 q^{11} - 16 q^{12} + 13 q^{13} + q^{14} - 3 q^{15} + 18 q^{16} + 17 q^{17} + 2 q^{18} + 14 q^{19} - 7 q^{20} - 11 q^{21} + 2 q^{22} + 7 q^{23} - 9 q^{24} + 18 q^{25} - 10 q^{26} - 13 q^{27} + 19 q^{28} - 6 q^{29} - 6 q^{30} + 27 q^{31} + 5 q^{32} - 13 q^{33} + 6 q^{34} + 14 q^{35} + 16 q^{36} + 10 q^{37} + 2 q^{38} - 13 q^{39} + 8 q^{40} + 3 q^{41} - q^{42} + 29 q^{43} + 16 q^{44} + 3 q^{45} - 24 q^{46} + 8 q^{47} - 18 q^{48} + 8 q^{49} - 27 q^{50} - 17 q^{51} + 37 q^{52} - 24 q^{53} - 2 q^{54} + 3 q^{55} + 24 q^{56} - 14 q^{57} - 5 q^{58} + 13 q^{59} + 7 q^{60} - 13 q^{61} + 39 q^{62} + 11 q^{63} + 47 q^{64} - 11 q^{65} - 2 q^{66} + 44 q^{67} - 8 q^{68} - 7 q^{69} - 12 q^{70} + 3 q^{71} + 9 q^{72} + 48 q^{73} - 22 q^{74} - 18 q^{75} + 47 q^{76} + 11 q^{77} + 10 q^{78} - 17 q^{79} - 26 q^{80} + 13 q^{81} + 56 q^{82} + 50 q^{83} - 19 q^{84} + 8 q^{85} + 18 q^{86} + 6 q^{87} + 9 q^{88} - 15 q^{89} + 6 q^{90} + 47 q^{91} + 14 q^{92} - 27 q^{93} + 45 q^{94} - q^{95} - 5 q^{96} + 27 q^{97} + 47 q^{98} + 13 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.46794 −1.03799 −0.518996 0.854777i \(-0.673694\pi\)
−0.518996 + 0.854777i \(0.673694\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.154851 0.0774254
\(5\) 3.84216 1.71827 0.859133 0.511753i \(-0.171004\pi\)
0.859133 + 0.511753i \(0.171004\pi\)
\(6\) 1.46794 0.599284
\(7\) 2.46223 0.930637 0.465318 0.885143i \(-0.345940\pi\)
0.465318 + 0.885143i \(0.345940\pi\)
\(8\) 2.70857 0.957624
\(9\) 1.00000 0.333333
\(10\) −5.64006 −1.78354
\(11\) 1.00000 0.301511
\(12\) −0.154851 −0.0447016
\(13\) −4.64696 −1.28884 −0.644418 0.764674i \(-0.722900\pi\)
−0.644418 + 0.764674i \(0.722900\pi\)
\(14\) −3.61441 −0.965993
\(15\) −3.84216 −0.992041
\(16\) −4.28572 −1.07143
\(17\) 5.70692 1.38413 0.692066 0.721834i \(-0.256701\pi\)
0.692066 + 0.721834i \(0.256701\pi\)
\(18\) −1.46794 −0.345997
\(19\) 6.71196 1.53983 0.769914 0.638147i \(-0.220299\pi\)
0.769914 + 0.638147i \(0.220299\pi\)
\(20\) 0.594961 0.133037
\(21\) −2.46223 −0.537304
\(22\) −1.46794 −0.312966
\(23\) 2.93787 0.612588 0.306294 0.951937i \(-0.400911\pi\)
0.306294 + 0.951937i \(0.400911\pi\)
\(24\) −2.70857 −0.552885
\(25\) 9.76218 1.95244
\(26\) 6.82147 1.33780
\(27\) −1.00000 −0.192450
\(28\) 0.381279 0.0720549
\(29\) 5.67137 1.05315 0.526573 0.850130i \(-0.323477\pi\)
0.526573 + 0.850130i \(0.323477\pi\)
\(30\) 5.64006 1.02973
\(31\) 0.232595 0.0417754 0.0208877 0.999782i \(-0.493351\pi\)
0.0208877 + 0.999782i \(0.493351\pi\)
\(32\) 0.874048 0.154511
\(33\) −1.00000 −0.174078
\(34\) −8.37742 −1.43672
\(35\) 9.46029 1.59908
\(36\) 0.154851 0.0258085
\(37\) 5.55034 0.912470 0.456235 0.889859i \(-0.349198\pi\)
0.456235 + 0.889859i \(0.349198\pi\)
\(38\) −9.85276 −1.59833
\(39\) 4.64696 0.744109
\(40\) 10.4068 1.64545
\(41\) −4.35928 −0.680806 −0.340403 0.940280i \(-0.610563\pi\)
−0.340403 + 0.940280i \(0.610563\pi\)
\(42\) 3.61441 0.557716
\(43\) −9.35832 −1.42713 −0.713565 0.700589i \(-0.752921\pi\)
−0.713565 + 0.700589i \(0.752921\pi\)
\(44\) 0.154851 0.0233446
\(45\) 3.84216 0.572755
\(46\) −4.31262 −0.635861
\(47\) 10.7428 1.56699 0.783497 0.621395i \(-0.213434\pi\)
0.783497 + 0.621395i \(0.213434\pi\)
\(48\) 4.28572 0.618591
\(49\) −0.937404 −0.133915
\(50\) −14.3303 −2.02661
\(51\) −5.70692 −0.799129
\(52\) −0.719586 −0.0997886
\(53\) −11.3834 −1.56363 −0.781816 0.623510i \(-0.785706\pi\)
−0.781816 + 0.623510i \(0.785706\pi\)
\(54\) 1.46794 0.199761
\(55\) 3.84216 0.518076
\(56\) 6.66913 0.891200
\(57\) −6.71196 −0.889020
\(58\) −8.32523 −1.09316
\(59\) −13.2037 −1.71897 −0.859485 0.511161i \(-0.829216\pi\)
−0.859485 + 0.511161i \(0.829216\pi\)
\(60\) −0.594961 −0.0768091
\(61\) −1.00000 −0.128037
\(62\) −0.341436 −0.0433624
\(63\) 2.46223 0.310212
\(64\) 7.28839 0.911049
\(65\) −17.8544 −2.21456
\(66\) 1.46794 0.180691
\(67\) 12.5899 1.53810 0.769049 0.639190i \(-0.220730\pi\)
0.769049 + 0.639190i \(0.220730\pi\)
\(68\) 0.883721 0.107167
\(69\) −2.93787 −0.353678
\(70\) −13.8871 −1.65983
\(71\) −0.299770 −0.0355761 −0.0177881 0.999842i \(-0.505662\pi\)
−0.0177881 + 0.999842i \(0.505662\pi\)
\(72\) 2.70857 0.319208
\(73\) −2.65468 −0.310707 −0.155353 0.987859i \(-0.549652\pi\)
−0.155353 + 0.987859i \(0.549652\pi\)
\(74\) −8.14757 −0.947135
\(75\) −9.76218 −1.12724
\(76\) 1.03935 0.119222
\(77\) 2.46223 0.280598
\(78\) −6.82147 −0.772379
\(79\) −12.9557 −1.45763 −0.728815 0.684710i \(-0.759929\pi\)
−0.728815 + 0.684710i \(0.759929\pi\)
\(80\) −16.4664 −1.84100
\(81\) 1.00000 0.111111
\(82\) 6.39917 0.706670
\(83\) −7.42505 −0.815005 −0.407502 0.913204i \(-0.633600\pi\)
−0.407502 + 0.913204i \(0.633600\pi\)
\(84\) −0.381279 −0.0416009
\(85\) 21.9269 2.37831
\(86\) 13.7375 1.48135
\(87\) −5.67137 −0.608035
\(88\) 2.70857 0.288735
\(89\) −3.40420 −0.360845 −0.180422 0.983589i \(-0.557746\pi\)
−0.180422 + 0.983589i \(0.557746\pi\)
\(90\) −5.64006 −0.594515
\(91\) −11.4419 −1.19944
\(92\) 0.454931 0.0474299
\(93\) −0.232595 −0.0241190
\(94\) −15.7698 −1.62653
\(95\) 25.7884 2.64583
\(96\) −0.874048 −0.0892071
\(97\) −15.8330 −1.60760 −0.803798 0.594902i \(-0.797191\pi\)
−0.803798 + 0.594902i \(0.797191\pi\)
\(98\) 1.37605 0.139002
\(99\) 1.00000 0.100504
\(100\) 1.51168 0.151168
\(101\) 7.72797 0.768962 0.384481 0.923133i \(-0.374380\pi\)
0.384481 + 0.923133i \(0.374380\pi\)
\(102\) 8.37742 0.829489
\(103\) 16.1339 1.58972 0.794860 0.606793i \(-0.207544\pi\)
0.794860 + 0.606793i \(0.207544\pi\)
\(104\) −12.5866 −1.23422
\(105\) −9.46029 −0.923230
\(106\) 16.7102 1.62304
\(107\) 1.14677 0.110863 0.0554314 0.998462i \(-0.482347\pi\)
0.0554314 + 0.998462i \(0.482347\pi\)
\(108\) −0.154851 −0.0149005
\(109\) 11.7557 1.12600 0.562998 0.826458i \(-0.309648\pi\)
0.562998 + 0.826458i \(0.309648\pi\)
\(110\) −5.64006 −0.537759
\(111\) −5.55034 −0.526815
\(112\) −10.5525 −0.997113
\(113\) −16.4246 −1.54509 −0.772546 0.634959i \(-0.781017\pi\)
−0.772546 + 0.634959i \(0.781017\pi\)
\(114\) 9.85276 0.922795
\(115\) 11.2878 1.05259
\(116\) 0.878216 0.0815403
\(117\) −4.64696 −0.429612
\(118\) 19.3822 1.78428
\(119\) 14.0518 1.28812
\(120\) −10.4068 −0.950002
\(121\) 1.00000 0.0909091
\(122\) 1.46794 0.132901
\(123\) 4.35928 0.393063
\(124\) 0.0360176 0.00323447
\(125\) 18.2970 1.63654
\(126\) −3.61441 −0.321998
\(127\) 11.9297 1.05859 0.529295 0.848438i \(-0.322457\pi\)
0.529295 + 0.848438i \(0.322457\pi\)
\(128\) −12.4470 −1.10017
\(129\) 9.35832 0.823954
\(130\) 26.2091 2.29869
\(131\) 20.8633 1.82283 0.911417 0.411485i \(-0.134990\pi\)
0.911417 + 0.411485i \(0.134990\pi\)
\(132\) −0.154851 −0.0134780
\(133\) 16.5264 1.43302
\(134\) −18.4812 −1.59653
\(135\) −3.84216 −0.330680
\(136\) 15.4576 1.32548
\(137\) −19.1597 −1.63692 −0.818460 0.574564i \(-0.805172\pi\)
−0.818460 + 0.574564i \(0.805172\pi\)
\(138\) 4.31262 0.367115
\(139\) −2.58202 −0.219004 −0.109502 0.993987i \(-0.534926\pi\)
−0.109502 + 0.993987i \(0.534926\pi\)
\(140\) 1.46493 0.123809
\(141\) −10.7428 −0.904705
\(142\) 0.440044 0.0369277
\(143\) −4.64696 −0.388598
\(144\) −4.28572 −0.357144
\(145\) 21.7903 1.80959
\(146\) 3.89691 0.322511
\(147\) 0.937404 0.0773158
\(148\) 0.859474 0.0706483
\(149\) 9.45719 0.774763 0.387382 0.921919i \(-0.373380\pi\)
0.387382 + 0.921919i \(0.373380\pi\)
\(150\) 14.3303 1.17006
\(151\) 6.16589 0.501773 0.250887 0.968016i \(-0.419278\pi\)
0.250887 + 0.968016i \(0.419278\pi\)
\(152\) 18.1798 1.47458
\(153\) 5.70692 0.461377
\(154\) −3.61441 −0.291258
\(155\) 0.893668 0.0717811
\(156\) 0.719586 0.0576130
\(157\) 0.475244 0.0379286 0.0189643 0.999820i \(-0.493963\pi\)
0.0189643 + 0.999820i \(0.493963\pi\)
\(158\) 19.0182 1.51301
\(159\) 11.3834 0.902763
\(160\) 3.35823 0.265491
\(161\) 7.23372 0.570097
\(162\) −1.46794 −0.115332
\(163\) 17.6623 1.38341 0.691707 0.722178i \(-0.256859\pi\)
0.691707 + 0.722178i \(0.256859\pi\)
\(164\) −0.675038 −0.0527116
\(165\) −3.84216 −0.299112
\(166\) 10.8995 0.845968
\(167\) −15.9865 −1.23707 −0.618536 0.785756i \(-0.712274\pi\)
−0.618536 + 0.785756i \(0.712274\pi\)
\(168\) −6.66913 −0.514535
\(169\) 8.59425 0.661096
\(170\) −32.1874 −2.46866
\(171\) 6.71196 0.513276
\(172\) −1.44914 −0.110496
\(173\) −2.22261 −0.168982 −0.0844909 0.996424i \(-0.526926\pi\)
−0.0844909 + 0.996424i \(0.526926\pi\)
\(174\) 8.32523 0.631134
\(175\) 24.0368 1.81701
\(176\) −4.28572 −0.323049
\(177\) 13.2037 0.992448
\(178\) 4.99717 0.374553
\(179\) −24.5110 −1.83204 −0.916020 0.401132i \(-0.868617\pi\)
−0.916020 + 0.401132i \(0.868617\pi\)
\(180\) 0.594961 0.0443458
\(181\) −14.1308 −1.05034 −0.525168 0.850999i \(-0.675997\pi\)
−0.525168 + 0.850999i \(0.675997\pi\)
\(182\) 16.7960 1.24501
\(183\) 1.00000 0.0739221
\(184\) 7.95743 0.586629
\(185\) 21.3253 1.56787
\(186\) 0.341436 0.0250353
\(187\) 5.70692 0.417331
\(188\) 1.66353 0.121325
\(189\) −2.46223 −0.179101
\(190\) −37.8558 −2.74635
\(191\) −21.1658 −1.53151 −0.765753 0.643135i \(-0.777633\pi\)
−0.765753 + 0.643135i \(0.777633\pi\)
\(192\) −7.28839 −0.525995
\(193\) −10.1194 −0.728414 −0.364207 0.931318i \(-0.618660\pi\)
−0.364207 + 0.931318i \(0.618660\pi\)
\(194\) 23.2419 1.66867
\(195\) 17.8544 1.27858
\(196\) −0.145158 −0.0103684
\(197\) 4.80836 0.342582 0.171291 0.985221i \(-0.445206\pi\)
0.171291 + 0.985221i \(0.445206\pi\)
\(198\) −1.46794 −0.104322
\(199\) 13.0978 0.928480 0.464240 0.885710i \(-0.346328\pi\)
0.464240 + 0.885710i \(0.346328\pi\)
\(200\) 26.4415 1.86970
\(201\) −12.5899 −0.888021
\(202\) −11.3442 −0.798176
\(203\) 13.9642 0.980097
\(204\) −0.883721 −0.0618729
\(205\) −16.7491 −1.16980
\(206\) −23.6836 −1.65011
\(207\) 2.93787 0.204196
\(208\) 19.9156 1.38090
\(209\) 6.71196 0.464276
\(210\) 13.8871 0.958304
\(211\) −6.36135 −0.437933 −0.218967 0.975732i \(-0.570269\pi\)
−0.218967 + 0.975732i \(0.570269\pi\)
\(212\) −1.76273 −0.121065
\(213\) 0.299770 0.0205399
\(214\) −1.68340 −0.115075
\(215\) −35.9561 −2.45219
\(216\) −2.70857 −0.184295
\(217\) 0.572704 0.0388777
\(218\) −17.2567 −1.16877
\(219\) 2.65468 0.179387
\(220\) 0.594961 0.0401123
\(221\) −26.5198 −1.78392
\(222\) 8.14757 0.546829
\(223\) 22.5356 1.50910 0.754548 0.656244i \(-0.227856\pi\)
0.754548 + 0.656244i \(0.227856\pi\)
\(224\) 2.15211 0.143794
\(225\) 9.76218 0.650812
\(226\) 24.1103 1.60379
\(227\) 27.1685 1.80323 0.901617 0.432535i \(-0.142381\pi\)
0.901617 + 0.432535i \(0.142381\pi\)
\(228\) −1.03935 −0.0688327
\(229\) −24.6096 −1.62625 −0.813124 0.582090i \(-0.802235\pi\)
−0.813124 + 0.582090i \(0.802235\pi\)
\(230\) −16.5698 −1.09258
\(231\) −2.46223 −0.162003
\(232\) 15.3613 1.00852
\(233\) 24.0471 1.57538 0.787690 0.616071i \(-0.211277\pi\)
0.787690 + 0.616071i \(0.211277\pi\)
\(234\) 6.82147 0.445933
\(235\) 41.2754 2.69251
\(236\) −2.04460 −0.133092
\(237\) 12.9557 0.841564
\(238\) −20.6272 −1.33706
\(239\) 16.2704 1.05245 0.526224 0.850346i \(-0.323607\pi\)
0.526224 + 0.850346i \(0.323607\pi\)
\(240\) 16.4664 1.06290
\(241\) 1.53799 0.0990706 0.0495353 0.998772i \(-0.484226\pi\)
0.0495353 + 0.998772i \(0.484226\pi\)
\(242\) −1.46794 −0.0943628
\(243\) −1.00000 −0.0641500
\(244\) −0.154851 −0.00991330
\(245\) −3.60165 −0.230101
\(246\) −6.39917 −0.407996
\(247\) −31.1902 −1.98458
\(248\) 0.630001 0.0400051
\(249\) 7.42505 0.470543
\(250\) −26.8590 −1.69871
\(251\) −22.8738 −1.44378 −0.721891 0.692006i \(-0.756727\pi\)
−0.721891 + 0.692006i \(0.756727\pi\)
\(252\) 0.381279 0.0240183
\(253\) 2.93787 0.184702
\(254\) −17.5121 −1.09881
\(255\) −21.9269 −1.37312
\(256\) 3.69471 0.230920
\(257\) −10.1026 −0.630186 −0.315093 0.949061i \(-0.602036\pi\)
−0.315093 + 0.949061i \(0.602036\pi\)
\(258\) −13.7375 −0.855257
\(259\) 13.6662 0.849178
\(260\) −2.76476 −0.171463
\(261\) 5.67137 0.351049
\(262\) −30.6261 −1.89208
\(263\) −11.1552 −0.687859 −0.343929 0.938996i \(-0.611758\pi\)
−0.343929 + 0.938996i \(0.611758\pi\)
\(264\) −2.70857 −0.166701
\(265\) −43.7368 −2.68673
\(266\) −24.2598 −1.48746
\(267\) 3.40420 0.208334
\(268\) 1.94955 0.119088
\(269\) 6.03952 0.368236 0.184118 0.982904i \(-0.441057\pi\)
0.184118 + 0.982904i \(0.441057\pi\)
\(270\) 5.64006 0.343243
\(271\) 4.26042 0.258802 0.129401 0.991592i \(-0.458695\pi\)
0.129401 + 0.991592i \(0.458695\pi\)
\(272\) −24.4583 −1.48300
\(273\) 11.4419 0.692496
\(274\) 28.1252 1.69911
\(275\) 9.76218 0.588681
\(276\) −0.454931 −0.0273837
\(277\) 8.39998 0.504706 0.252353 0.967635i \(-0.418796\pi\)
0.252353 + 0.967635i \(0.418796\pi\)
\(278\) 3.79026 0.227325
\(279\) 0.232595 0.0139251
\(280\) 25.6239 1.53132
\(281\) 12.3489 0.736671 0.368335 0.929693i \(-0.379928\pi\)
0.368335 + 0.929693i \(0.379928\pi\)
\(282\) 15.7698 0.939075
\(283\) 7.31966 0.435109 0.217554 0.976048i \(-0.430192\pi\)
0.217554 + 0.976048i \(0.430192\pi\)
\(284\) −0.0464196 −0.00275450
\(285\) −25.7884 −1.52757
\(286\) 6.82147 0.403362
\(287\) −10.7336 −0.633583
\(288\) 0.874048 0.0515038
\(289\) 15.5689 0.915820
\(290\) −31.9869 −1.87833
\(291\) 15.8330 0.928147
\(292\) −0.411079 −0.0240566
\(293\) 16.8649 0.985255 0.492628 0.870240i \(-0.336036\pi\)
0.492628 + 0.870240i \(0.336036\pi\)
\(294\) −1.37605 −0.0802531
\(295\) −50.7305 −2.95365
\(296\) 15.0335 0.873803
\(297\) −1.00000 −0.0580259
\(298\) −13.8826 −0.804197
\(299\) −13.6522 −0.789525
\(300\) −1.51168 −0.0872769
\(301\) −23.0424 −1.32814
\(302\) −9.05117 −0.520836
\(303\) −7.72797 −0.443961
\(304\) −28.7656 −1.64982
\(305\) −3.84216 −0.220001
\(306\) −8.37742 −0.478905
\(307\) 20.2128 1.15360 0.576802 0.816884i \(-0.304300\pi\)
0.576802 + 0.816884i \(0.304300\pi\)
\(308\) 0.381279 0.0217254
\(309\) −16.1339 −0.917825
\(310\) −1.31185 −0.0745082
\(311\) −27.5538 −1.56243 −0.781215 0.624262i \(-0.785400\pi\)
−0.781215 + 0.624262i \(0.785400\pi\)
\(312\) 12.5866 0.712577
\(313\) 18.0023 1.01755 0.508776 0.860899i \(-0.330098\pi\)
0.508776 + 0.860899i \(0.330098\pi\)
\(314\) −0.697630 −0.0393695
\(315\) 9.46029 0.533027
\(316\) −2.00620 −0.112858
\(317\) 29.5066 1.65726 0.828628 0.559799i \(-0.189122\pi\)
0.828628 + 0.559799i \(0.189122\pi\)
\(318\) −16.7102 −0.937060
\(319\) 5.67137 0.317536
\(320\) 28.0032 1.56542
\(321\) −1.14677 −0.0640066
\(322\) −10.6187 −0.591756
\(323\) 38.3046 2.13133
\(324\) 0.154851 0.00860282
\(325\) −45.3645 −2.51637
\(326\) −25.9271 −1.43597
\(327\) −11.7557 −0.650094
\(328\) −11.8074 −0.651956
\(329\) 26.4512 1.45830
\(330\) 5.64006 0.310475
\(331\) 17.8450 0.980852 0.490426 0.871483i \(-0.336841\pi\)
0.490426 + 0.871483i \(0.336841\pi\)
\(332\) −1.14977 −0.0631021
\(333\) 5.55034 0.304157
\(334\) 23.4672 1.28407
\(335\) 48.3723 2.64286
\(336\) 10.5525 0.575683
\(337\) −15.1235 −0.823832 −0.411916 0.911222i \(-0.635140\pi\)
−0.411916 + 0.911222i \(0.635140\pi\)
\(338\) −12.6159 −0.686212
\(339\) 16.4246 0.892059
\(340\) 3.39540 0.184141
\(341\) 0.232595 0.0125957
\(342\) −9.85276 −0.532776
\(343\) −19.5437 −1.05526
\(344\) −25.3477 −1.36665
\(345\) −11.2878 −0.607713
\(346\) 3.26266 0.175402
\(347\) 11.2684 0.604920 0.302460 0.953162i \(-0.402192\pi\)
0.302460 + 0.953162i \(0.402192\pi\)
\(348\) −0.878216 −0.0470773
\(349\) 6.44520 0.345004 0.172502 0.985009i \(-0.444815\pi\)
0.172502 + 0.985009i \(0.444815\pi\)
\(350\) −35.2845 −1.88604
\(351\) 4.64696 0.248036
\(352\) 0.874048 0.0465869
\(353\) −22.9363 −1.22078 −0.610389 0.792102i \(-0.708987\pi\)
−0.610389 + 0.792102i \(0.708987\pi\)
\(354\) −19.3822 −1.03015
\(355\) −1.15176 −0.0611292
\(356\) −0.527143 −0.0279385
\(357\) −14.0518 −0.743699
\(358\) 35.9807 1.90164
\(359\) 11.2684 0.594724 0.297362 0.954765i \(-0.403893\pi\)
0.297362 + 0.954765i \(0.403893\pi\)
\(360\) 10.4068 0.548484
\(361\) 26.0504 1.37107
\(362\) 20.7432 1.09024
\(363\) −1.00000 −0.0524864
\(364\) −1.77179 −0.0928669
\(365\) −10.1997 −0.533877
\(366\) −1.46794 −0.0767305
\(367\) 10.1541 0.530039 0.265020 0.964243i \(-0.414622\pi\)
0.265020 + 0.964243i \(0.414622\pi\)
\(368\) −12.5909 −0.656346
\(369\) −4.35928 −0.226935
\(370\) −31.3042 −1.62743
\(371\) −28.0286 −1.45517
\(372\) −0.0360176 −0.00186742
\(373\) −24.8297 −1.28563 −0.642817 0.766020i \(-0.722234\pi\)
−0.642817 + 0.766020i \(0.722234\pi\)
\(374\) −8.37742 −0.433186
\(375\) −18.2970 −0.944855
\(376\) 29.0976 1.50059
\(377\) −26.3546 −1.35733
\(378\) 3.61441 0.185905
\(379\) 12.2701 0.630271 0.315136 0.949047i \(-0.397950\pi\)
0.315136 + 0.949047i \(0.397950\pi\)
\(380\) 3.99335 0.204855
\(381\) −11.9297 −0.611177
\(382\) 31.0702 1.58969
\(383\) −22.3129 −1.14014 −0.570068 0.821597i \(-0.693083\pi\)
−0.570068 + 0.821597i \(0.693083\pi\)
\(384\) 12.4470 0.635185
\(385\) 9.46029 0.482141
\(386\) 14.8547 0.756087
\(387\) −9.35832 −0.475710
\(388\) −2.45175 −0.124469
\(389\) 24.6635 1.25049 0.625245 0.780429i \(-0.284999\pi\)
0.625245 + 0.780429i \(0.284999\pi\)
\(390\) −26.2091 −1.32715
\(391\) 16.7662 0.847903
\(392\) −2.53902 −0.128240
\(393\) −20.8633 −1.05241
\(394\) −7.05839 −0.355597
\(395\) −49.7779 −2.50460
\(396\) 0.154851 0.00778154
\(397\) −3.99794 −0.200651 −0.100325 0.994955i \(-0.531988\pi\)
−0.100325 + 0.994955i \(0.531988\pi\)
\(398\) −19.2268 −0.963754
\(399\) −16.5264 −0.827355
\(400\) −41.8380 −2.09190
\(401\) 1.45172 0.0724952 0.0362476 0.999343i \(-0.488460\pi\)
0.0362476 + 0.999343i \(0.488460\pi\)
\(402\) 18.4812 0.921758
\(403\) −1.08086 −0.0538416
\(404\) 1.19668 0.0595372
\(405\) 3.84216 0.190918
\(406\) −20.4987 −1.01733
\(407\) 5.55034 0.275120
\(408\) −15.4576 −0.765265
\(409\) 13.2091 0.653147 0.326574 0.945172i \(-0.394106\pi\)
0.326574 + 0.945172i \(0.394106\pi\)
\(410\) 24.5866 1.21425
\(411\) 19.1597 0.945076
\(412\) 2.49835 0.123085
\(413\) −32.5105 −1.59974
\(414\) −4.31262 −0.211954
\(415\) −28.5282 −1.40039
\(416\) −4.06167 −0.199140
\(417\) 2.58202 0.126442
\(418\) −9.85276 −0.481914
\(419\) −12.6761 −0.619267 −0.309634 0.950856i \(-0.600206\pi\)
−0.309634 + 0.950856i \(0.600206\pi\)
\(420\) −1.46493 −0.0714814
\(421\) −4.36098 −0.212541 −0.106271 0.994337i \(-0.533891\pi\)
−0.106271 + 0.994337i \(0.533891\pi\)
\(422\) 9.33809 0.454571
\(423\) 10.7428 0.522332
\(424\) −30.8328 −1.49737
\(425\) 55.7120 2.70243
\(426\) −0.440044 −0.0213202
\(427\) −2.46223 −0.119156
\(428\) 0.177579 0.00858359
\(429\) 4.64696 0.224357
\(430\) 52.7815 2.54535
\(431\) −1.72411 −0.0830477 −0.0415238 0.999138i \(-0.513221\pi\)
−0.0415238 + 0.999138i \(0.513221\pi\)
\(432\) 4.28572 0.206197
\(433\) −6.68214 −0.321123 −0.160561 0.987026i \(-0.551330\pi\)
−0.160561 + 0.987026i \(0.551330\pi\)
\(434\) −0.840696 −0.0403547
\(435\) −21.7903 −1.04476
\(436\) 1.82039 0.0871807
\(437\) 19.7189 0.943281
\(438\) −3.89691 −0.186202
\(439\) −4.62223 −0.220607 −0.110303 0.993898i \(-0.535182\pi\)
−0.110303 + 0.993898i \(0.535182\pi\)
\(440\) 10.4068 0.496123
\(441\) −0.937404 −0.0446383
\(442\) 38.9296 1.85169
\(443\) 24.7151 1.17425 0.587125 0.809497i \(-0.300260\pi\)
0.587125 + 0.809497i \(0.300260\pi\)
\(444\) −0.859474 −0.0407888
\(445\) −13.0795 −0.620027
\(446\) −33.0810 −1.56643
\(447\) −9.45719 −0.447310
\(448\) 17.9457 0.847856
\(449\) −17.0015 −0.802350 −0.401175 0.916002i \(-0.631398\pi\)
−0.401175 + 0.916002i \(0.631398\pi\)
\(450\) −14.3303 −0.675537
\(451\) −4.35928 −0.205271
\(452\) −2.54335 −0.119629
\(453\) −6.16589 −0.289699
\(454\) −39.8817 −1.87174
\(455\) −43.9616 −2.06095
\(456\) −18.1798 −0.851347
\(457\) −31.3936 −1.46853 −0.734265 0.678863i \(-0.762473\pi\)
−0.734265 + 0.678863i \(0.762473\pi\)
\(458\) 36.1254 1.68803
\(459\) −5.70692 −0.266376
\(460\) 1.74792 0.0814971
\(461\) 4.61978 0.215165 0.107582 0.994196i \(-0.465689\pi\)
0.107582 + 0.994196i \(0.465689\pi\)
\(462\) 3.61441 0.168158
\(463\) 27.6857 1.28666 0.643331 0.765588i \(-0.277552\pi\)
0.643331 + 0.765588i \(0.277552\pi\)
\(464\) −24.3059 −1.12837
\(465\) −0.893668 −0.0414429
\(466\) −35.2998 −1.63523
\(467\) 3.83606 0.177512 0.0887558 0.996053i \(-0.471711\pi\)
0.0887558 + 0.996053i \(0.471711\pi\)
\(468\) −0.719586 −0.0332629
\(469\) 30.9992 1.43141
\(470\) −60.5899 −2.79480
\(471\) −0.475244 −0.0218981
\(472\) −35.7630 −1.64613
\(473\) −9.35832 −0.430296
\(474\) −19.0182 −0.873536
\(475\) 65.5233 3.00641
\(476\) 2.17593 0.0997335
\(477\) −11.3834 −0.521210
\(478\) −23.8841 −1.09243
\(479\) 19.8463 0.906801 0.453400 0.891307i \(-0.350211\pi\)
0.453400 + 0.891307i \(0.350211\pi\)
\(480\) −3.35823 −0.153282
\(481\) −25.7922 −1.17602
\(482\) −2.25768 −0.102834
\(483\) −7.23372 −0.329146
\(484\) 0.154851 0.00703867
\(485\) −60.8329 −2.76228
\(486\) 1.46794 0.0665872
\(487\) −18.7149 −0.848054 −0.424027 0.905650i \(-0.639384\pi\)
−0.424027 + 0.905650i \(0.639384\pi\)
\(488\) −2.70857 −0.122611
\(489\) −17.6623 −0.798715
\(490\) 5.28701 0.238843
\(491\) −33.1187 −1.49463 −0.747313 0.664472i \(-0.768656\pi\)
−0.747313 + 0.664472i \(0.768656\pi\)
\(492\) 0.675038 0.0304331
\(493\) 32.3661 1.45769
\(494\) 45.7854 2.05998
\(495\) 3.84216 0.172692
\(496\) −0.996839 −0.0447594
\(497\) −0.738103 −0.0331085
\(498\) −10.8995 −0.488420
\(499\) 3.36699 0.150727 0.0753635 0.997156i \(-0.475988\pi\)
0.0753635 + 0.997156i \(0.475988\pi\)
\(500\) 2.83331 0.126709
\(501\) 15.9865 0.714224
\(502\) 33.5774 1.49863
\(503\) 15.7130 0.700609 0.350305 0.936636i \(-0.386078\pi\)
0.350305 + 0.936636i \(0.386078\pi\)
\(504\) 6.66913 0.297067
\(505\) 29.6921 1.32128
\(506\) −4.31262 −0.191719
\(507\) −8.59425 −0.381684
\(508\) 1.84732 0.0819617
\(509\) −7.97241 −0.353371 −0.176685 0.984267i \(-0.556538\pi\)
−0.176685 + 0.984267i \(0.556538\pi\)
\(510\) 32.1874 1.42528
\(511\) −6.53644 −0.289155
\(512\) 19.4704 0.860480
\(513\) −6.71196 −0.296340
\(514\) 14.8301 0.654127
\(515\) 61.9889 2.73156
\(516\) 1.44914 0.0637950
\(517\) 10.7428 0.472467
\(518\) −20.0612 −0.881439
\(519\) 2.22261 0.0975617
\(520\) −48.3598 −2.12072
\(521\) −19.8637 −0.870244 −0.435122 0.900372i \(-0.643295\pi\)
−0.435122 + 0.900372i \(0.643295\pi\)
\(522\) −8.32523 −0.364386
\(523\) 10.5930 0.463197 0.231599 0.972811i \(-0.425604\pi\)
0.231599 + 0.972811i \(0.425604\pi\)
\(524\) 3.23070 0.141134
\(525\) −24.0368 −1.04905
\(526\) 16.3752 0.713991
\(527\) 1.32740 0.0578226
\(528\) 4.28572 0.186512
\(529\) −14.3689 −0.624736
\(530\) 64.2031 2.78880
\(531\) −13.2037 −0.572990
\(532\) 2.55913 0.110952
\(533\) 20.2574 0.877446
\(534\) −4.99717 −0.216249
\(535\) 4.40608 0.190492
\(536\) 34.1006 1.47292
\(537\) 24.5110 1.05773
\(538\) −8.86566 −0.382226
\(539\) −0.937404 −0.0403768
\(540\) −0.594961 −0.0256030
\(541\) 0.967556 0.0415985 0.0207993 0.999784i \(-0.493379\pi\)
0.0207993 + 0.999784i \(0.493379\pi\)
\(542\) −6.25404 −0.268634
\(543\) 14.1308 0.606412
\(544\) 4.98812 0.213864
\(545\) 45.1674 1.93476
\(546\) −16.7960 −0.718804
\(547\) 2.94593 0.125959 0.0629794 0.998015i \(-0.479940\pi\)
0.0629794 + 0.998015i \(0.479940\pi\)
\(548\) −2.96689 −0.126739
\(549\) −1.00000 −0.0426790
\(550\) −14.3303 −0.611046
\(551\) 38.0660 1.62167
\(552\) −7.95743 −0.338691
\(553\) −31.9000 −1.35653
\(554\) −12.3307 −0.523880
\(555\) −21.3253 −0.905207
\(556\) −0.399828 −0.0169565
\(557\) 0.0922439 0.00390850 0.00195425 0.999998i \(-0.499378\pi\)
0.00195425 + 0.999998i \(0.499378\pi\)
\(558\) −0.341436 −0.0144541
\(559\) 43.4877 1.83934
\(560\) −40.5442 −1.71330
\(561\) −5.70692 −0.240946
\(562\) −18.1274 −0.764658
\(563\) 16.5637 0.698079 0.349039 0.937108i \(-0.386508\pi\)
0.349039 + 0.937108i \(0.386508\pi\)
\(564\) −1.66353 −0.0700471
\(565\) −63.1057 −2.65488
\(566\) −10.7448 −0.451639
\(567\) 2.46223 0.103404
\(568\) −0.811948 −0.0340686
\(569\) −44.9171 −1.88302 −0.941512 0.336980i \(-0.890594\pi\)
−0.941512 + 0.336980i \(0.890594\pi\)
\(570\) 37.8558 1.58561
\(571\) 11.9694 0.500903 0.250452 0.968129i \(-0.419421\pi\)
0.250452 + 0.968129i \(0.419421\pi\)
\(572\) −0.719586 −0.0300874
\(573\) 21.1658 0.884215
\(574\) 15.7563 0.657653
\(575\) 28.6800 1.19604
\(576\) 7.28839 0.303683
\(577\) 1.73895 0.0723935 0.0361967 0.999345i \(-0.488476\pi\)
0.0361967 + 0.999345i \(0.488476\pi\)
\(578\) −22.8543 −0.950613
\(579\) 10.1194 0.420550
\(580\) 3.37424 0.140108
\(581\) −18.2822 −0.758474
\(582\) −23.2419 −0.963408
\(583\) −11.3834 −0.471453
\(584\) −7.19039 −0.297540
\(585\) −17.8544 −0.738187
\(586\) −24.7566 −1.02269
\(587\) 34.0791 1.40660 0.703298 0.710895i \(-0.251710\pi\)
0.703298 + 0.710895i \(0.251710\pi\)
\(588\) 0.145158 0.00598620
\(589\) 1.56117 0.0643269
\(590\) 74.4694 3.06586
\(591\) −4.80836 −0.197790
\(592\) −23.7872 −0.977648
\(593\) −19.1334 −0.785714 −0.392857 0.919600i \(-0.628513\pi\)
−0.392857 + 0.919600i \(0.628513\pi\)
\(594\) 1.46794 0.0602303
\(595\) 53.9891 2.21334
\(596\) 1.46445 0.0599863
\(597\) −13.0978 −0.536058
\(598\) 20.0406 0.819520
\(599\) 11.7116 0.478521 0.239261 0.970955i \(-0.423095\pi\)
0.239261 + 0.970955i \(0.423095\pi\)
\(600\) −26.4415 −1.07947
\(601\) −20.3403 −0.829698 −0.414849 0.909890i \(-0.636166\pi\)
−0.414849 + 0.909890i \(0.636166\pi\)
\(602\) 33.8248 1.37860
\(603\) 12.5899 0.512699
\(604\) 0.954793 0.0388500
\(605\) 3.84216 0.156206
\(606\) 11.3442 0.460827
\(607\) 18.2830 0.742085 0.371042 0.928616i \(-0.379000\pi\)
0.371042 + 0.928616i \(0.379000\pi\)
\(608\) 5.86657 0.237921
\(609\) −13.9642 −0.565859
\(610\) 5.64006 0.228359
\(611\) −49.9213 −2.01960
\(612\) 0.883721 0.0357223
\(613\) −33.4060 −1.34925 −0.674627 0.738158i \(-0.735696\pi\)
−0.674627 + 0.738158i \(0.735696\pi\)
\(614\) −29.6712 −1.19743
\(615\) 16.7491 0.675387
\(616\) 6.66913 0.268707
\(617\) −37.0969 −1.49347 −0.746733 0.665124i \(-0.768379\pi\)
−0.746733 + 0.665124i \(0.768379\pi\)
\(618\) 23.6836 0.952694
\(619\) −35.5774 −1.42998 −0.714988 0.699137i \(-0.753568\pi\)
−0.714988 + 0.699137i \(0.753568\pi\)
\(620\) 0.138385 0.00555768
\(621\) −2.93787 −0.117893
\(622\) 40.4473 1.62179
\(623\) −8.38194 −0.335815
\(624\) −19.9156 −0.797262
\(625\) 21.4892 0.859568
\(626\) −26.4263 −1.05621
\(627\) −6.71196 −0.268050
\(628\) 0.0735919 0.00293663
\(629\) 31.6753 1.26298
\(630\) −13.8871 −0.553277
\(631\) 3.96808 0.157967 0.0789834 0.996876i \(-0.474833\pi\)
0.0789834 + 0.996876i \(0.474833\pi\)
\(632\) −35.0914 −1.39586
\(633\) 6.36135 0.252841
\(634\) −43.3140 −1.72022
\(635\) 45.8358 1.81894
\(636\) 1.76273 0.0698968
\(637\) 4.35608 0.172594
\(638\) −8.32523 −0.329599
\(639\) −0.299770 −0.0118587
\(640\) −47.8235 −1.89039
\(641\) −4.20491 −0.166084 −0.0830419 0.996546i \(-0.526464\pi\)
−0.0830419 + 0.996546i \(0.526464\pi\)
\(642\) 1.68340 0.0664383
\(643\) 29.2183 1.15226 0.576129 0.817359i \(-0.304563\pi\)
0.576129 + 0.817359i \(0.304563\pi\)
\(644\) 1.12015 0.0441400
\(645\) 35.9561 1.41577
\(646\) −56.2289 −2.21230
\(647\) 8.11234 0.318929 0.159464 0.987204i \(-0.449023\pi\)
0.159464 + 0.987204i \(0.449023\pi\)
\(648\) 2.70857 0.106403
\(649\) −13.2037 −0.518289
\(650\) 66.5923 2.61197
\(651\) −0.572704 −0.0224460
\(652\) 2.73501 0.107111
\(653\) −23.5399 −0.921189 −0.460595 0.887611i \(-0.652364\pi\)
−0.460595 + 0.887611i \(0.652364\pi\)
\(654\) 17.2567 0.674792
\(655\) 80.1600 3.13211
\(656\) 18.6827 0.729436
\(657\) −2.65468 −0.103569
\(658\) −38.8288 −1.51371
\(659\) −46.8281 −1.82416 −0.912082 0.410007i \(-0.865526\pi\)
−0.912082 + 0.410007i \(0.865526\pi\)
\(660\) −0.594961 −0.0231588
\(661\) −45.2831 −1.76131 −0.880655 0.473758i \(-0.842897\pi\)
−0.880655 + 0.473758i \(0.842897\pi\)
\(662\) −26.1955 −1.01812
\(663\) 26.5198 1.02995
\(664\) −20.1113 −0.780468
\(665\) 63.4971 2.46231
\(666\) −8.14757 −0.315712
\(667\) 16.6617 0.645145
\(668\) −2.47552 −0.0957808
\(669\) −22.5356 −0.871277
\(670\) −71.0077 −2.74327
\(671\) −1.00000 −0.0386046
\(672\) −2.15211 −0.0830195
\(673\) 42.7822 1.64913 0.824566 0.565765i \(-0.191419\pi\)
0.824566 + 0.565765i \(0.191419\pi\)
\(674\) 22.2005 0.855130
\(675\) −9.76218 −0.375746
\(676\) 1.33083 0.0511856
\(677\) 32.8918 1.26414 0.632068 0.774913i \(-0.282206\pi\)
0.632068 + 0.774913i \(0.282206\pi\)
\(678\) −24.1103 −0.925950
\(679\) −38.9845 −1.49609
\(680\) 59.3905 2.27752
\(681\) −27.1685 −1.04110
\(682\) −0.341436 −0.0130743
\(683\) −29.9574 −1.14629 −0.573144 0.819455i \(-0.694276\pi\)
−0.573144 + 0.819455i \(0.694276\pi\)
\(684\) 1.03935 0.0397406
\(685\) −73.6144 −2.81266
\(686\) 28.6891 1.09535
\(687\) 24.6096 0.938915
\(688\) 40.1072 1.52907
\(689\) 52.8983 2.01526
\(690\) 16.5698 0.630800
\(691\) −46.0051 −1.75011 −0.875057 0.484019i \(-0.839177\pi\)
−0.875057 + 0.484019i \(0.839177\pi\)
\(692\) −0.344173 −0.0130835
\(693\) 2.46223 0.0935325
\(694\) −16.5414 −0.627901
\(695\) −9.92054 −0.376308
\(696\) −15.3613 −0.582269
\(697\) −24.8781 −0.942324
\(698\) −9.46117 −0.358111
\(699\) −24.0471 −0.909547
\(700\) 3.72211 0.140683
\(701\) −4.97041 −0.187730 −0.0938650 0.995585i \(-0.529922\pi\)
−0.0938650 + 0.995585i \(0.529922\pi\)
\(702\) −6.82147 −0.257460
\(703\) 37.2536 1.40505
\(704\) 7.28839 0.274692
\(705\) −41.2754 −1.55452
\(706\) 33.6692 1.26716
\(707\) 19.0281 0.715625
\(708\) 2.04460 0.0768407
\(709\) −31.0750 −1.16704 −0.583522 0.812097i \(-0.698326\pi\)
−0.583522 + 0.812097i \(0.698326\pi\)
\(710\) 1.69072 0.0634516
\(711\) −12.9557 −0.485877
\(712\) −9.22052 −0.345554
\(713\) 0.683335 0.0255911
\(714\) 20.6272 0.771953
\(715\) −17.8544 −0.667715
\(716\) −3.79555 −0.141846
\(717\) −16.2704 −0.607631
\(718\) −16.5414 −0.617318
\(719\) −37.4717 −1.39746 −0.698729 0.715386i \(-0.746251\pi\)
−0.698729 + 0.715386i \(0.746251\pi\)
\(720\) −16.4664 −0.613667
\(721\) 39.7254 1.47945
\(722\) −38.2404 −1.42316
\(723\) −1.53799 −0.0571985
\(724\) −2.18817 −0.0813226
\(725\) 55.3649 2.05620
\(726\) 1.46794 0.0544804
\(727\) −7.56311 −0.280500 −0.140250 0.990116i \(-0.544791\pi\)
−0.140250 + 0.990116i \(0.544791\pi\)
\(728\) −30.9912 −1.14861
\(729\) 1.00000 0.0370370
\(730\) 14.9726 0.554159
\(731\) −53.4072 −1.97534
\(732\) 0.154851 0.00572345
\(733\) −12.6942 −0.468869 −0.234435 0.972132i \(-0.575324\pi\)
−0.234435 + 0.972132i \(0.575324\pi\)
\(734\) −14.9056 −0.550176
\(735\) 3.60165 0.132849
\(736\) 2.56784 0.0946518
\(737\) 12.5899 0.463754
\(738\) 6.39917 0.235557
\(739\) 13.3928 0.492662 0.246331 0.969186i \(-0.420775\pi\)
0.246331 + 0.969186i \(0.420775\pi\)
\(740\) 3.30223 0.121393
\(741\) 31.1902 1.14580
\(742\) 41.1444 1.51046
\(743\) −3.77240 −0.138396 −0.0691980 0.997603i \(-0.522044\pi\)
−0.0691980 + 0.997603i \(0.522044\pi\)
\(744\) −0.630001 −0.0230970
\(745\) 36.3360 1.33125
\(746\) 36.4486 1.33448
\(747\) −7.42505 −0.271668
\(748\) 0.883721 0.0323120
\(749\) 2.82362 0.103173
\(750\) 26.8590 0.980751
\(751\) −9.51843 −0.347332 −0.173666 0.984805i \(-0.555561\pi\)
−0.173666 + 0.984805i \(0.555561\pi\)
\(752\) −46.0406 −1.67893
\(753\) 22.8738 0.833568
\(754\) 38.6870 1.40890
\(755\) 23.6903 0.862179
\(756\) −0.381279 −0.0138670
\(757\) −10.7916 −0.392229 −0.196114 0.980581i \(-0.562832\pi\)
−0.196114 + 0.980581i \(0.562832\pi\)
\(758\) −18.0117 −0.654216
\(759\) −2.93787 −0.106638
\(760\) 69.8497 2.53371
\(761\) −22.1374 −0.802480 −0.401240 0.915973i \(-0.631421\pi\)
−0.401240 + 0.915973i \(0.631421\pi\)
\(762\) 17.5121 0.634396
\(763\) 28.9454 1.04789
\(764\) −3.27755 −0.118577
\(765\) 21.9269 0.792768
\(766\) 32.7540 1.18345
\(767\) 61.3569 2.21547
\(768\) −3.69471 −0.133322
\(769\) 19.4521 0.701460 0.350730 0.936477i \(-0.385934\pi\)
0.350730 + 0.936477i \(0.385934\pi\)
\(770\) −13.8871 −0.500458
\(771\) 10.1026 0.363838
\(772\) −1.56700 −0.0563977
\(773\) 39.4207 1.41786 0.708932 0.705277i \(-0.249177\pi\)
0.708932 + 0.705277i \(0.249177\pi\)
\(774\) 13.7375 0.493783
\(775\) 2.27064 0.0815637
\(776\) −42.8848 −1.53947
\(777\) −13.6662 −0.490273
\(778\) −36.2046 −1.29800
\(779\) −29.2593 −1.04832
\(780\) 2.76476 0.0989943
\(781\) −0.299770 −0.0107266
\(782\) −24.6118 −0.880116
\(783\) −5.67137 −0.202678
\(784\) 4.01745 0.143480
\(785\) 1.82596 0.0651714
\(786\) 30.6261 1.09240
\(787\) −37.7502 −1.34565 −0.672824 0.739802i \(-0.734919\pi\)
−0.672824 + 0.739802i \(0.734919\pi\)
\(788\) 0.744578 0.0265245
\(789\) 11.1552 0.397135
\(790\) 73.0710 2.59975
\(791\) −40.4411 −1.43792
\(792\) 2.70857 0.0962449
\(793\) 4.64696 0.165018
\(794\) 5.86874 0.208274
\(795\) 43.7368 1.55119
\(796\) 2.02821 0.0718879
\(797\) −20.3463 −0.720705 −0.360352 0.932816i \(-0.617344\pi\)
−0.360352 + 0.932816i \(0.617344\pi\)
\(798\) 24.2598 0.858787
\(799\) 61.3082 2.16893
\(800\) 8.53261 0.301673
\(801\) −3.40420 −0.120282
\(802\) −2.13103 −0.0752494
\(803\) −2.65468 −0.0936816
\(804\) −1.94955 −0.0687554
\(805\) 27.7931 0.979578
\(806\) 1.58664 0.0558870
\(807\) −6.03952 −0.212601
\(808\) 20.9318 0.736377
\(809\) −3.16766 −0.111369 −0.0556844 0.998448i \(-0.517734\pi\)
−0.0556844 + 0.998448i \(0.517734\pi\)
\(810\) −5.64006 −0.198172
\(811\) −14.3129 −0.502593 −0.251296 0.967910i \(-0.580857\pi\)
−0.251296 + 0.967910i \(0.580857\pi\)
\(812\) 2.16237 0.0758844
\(813\) −4.26042 −0.149419
\(814\) −8.14757 −0.285572
\(815\) 67.8612 2.37707
\(816\) 24.4583 0.856211
\(817\) −62.8126 −2.19754
\(818\) −19.3902 −0.677961
\(819\) −11.4419 −0.399813
\(820\) −2.59360 −0.0905726
\(821\) −14.3547 −0.500982 −0.250491 0.968119i \(-0.580592\pi\)
−0.250491 + 0.968119i \(0.580592\pi\)
\(822\) −28.1252 −0.980980
\(823\) 21.5880 0.752509 0.376255 0.926516i \(-0.377212\pi\)
0.376255 + 0.926516i \(0.377212\pi\)
\(824\) 43.6998 1.52235
\(825\) −9.76218 −0.339875
\(826\) 47.7235 1.66051
\(827\) 27.7946 0.966513 0.483257 0.875479i \(-0.339454\pi\)
0.483257 + 0.875479i \(0.339454\pi\)
\(828\) 0.454931 0.0158100
\(829\) 18.2361 0.633368 0.316684 0.948531i \(-0.397431\pi\)
0.316684 + 0.948531i \(0.397431\pi\)
\(830\) 41.8777 1.45360
\(831\) −8.39998 −0.291392
\(832\) −33.8689 −1.17419
\(833\) −5.34969 −0.185356
\(834\) −3.79026 −0.131246
\(835\) −61.4227 −2.12562
\(836\) 1.03935 0.0359467
\(837\) −0.232595 −0.00803967
\(838\) 18.6078 0.642794
\(839\) 25.3815 0.876268 0.438134 0.898910i \(-0.355640\pi\)
0.438134 + 0.898910i \(0.355640\pi\)
\(840\) −25.6239 −0.884107
\(841\) 3.16442 0.109118
\(842\) 6.40166 0.220616
\(843\) −12.3489 −0.425317
\(844\) −0.985060 −0.0339072
\(845\) 33.0205 1.13594
\(846\) −15.7698 −0.542175
\(847\) 2.46223 0.0846034
\(848\) 48.7861 1.67532
\(849\) −7.31966 −0.251210
\(850\) −81.7819 −2.80510
\(851\) 16.3062 0.558968
\(852\) 0.0464196 0.00159031
\(853\) −50.9440 −1.74429 −0.872144 0.489249i \(-0.837271\pi\)
−0.872144 + 0.489249i \(0.837271\pi\)
\(854\) 3.61441 0.123683
\(855\) 25.7884 0.881944
\(856\) 3.10612 0.106165
\(857\) 25.5195 0.871729 0.435865 0.900012i \(-0.356443\pi\)
0.435865 + 0.900012i \(0.356443\pi\)
\(858\) −6.82147 −0.232881
\(859\) 19.5049 0.665497 0.332748 0.943016i \(-0.392024\pi\)
0.332748 + 0.943016i \(0.392024\pi\)
\(860\) −5.56784 −0.189862
\(861\) 10.7336 0.365799
\(862\) 2.53090 0.0862027
\(863\) −23.2335 −0.790876 −0.395438 0.918493i \(-0.629407\pi\)
−0.395438 + 0.918493i \(0.629407\pi\)
\(864\) −0.874048 −0.0297357
\(865\) −8.53962 −0.290356
\(866\) 9.80898 0.333323
\(867\) −15.5689 −0.528749
\(868\) 0.0886837 0.00301012
\(869\) −12.9557 −0.439492
\(870\) 31.9869 1.08446
\(871\) −58.5047 −1.98235
\(872\) 31.8413 1.07828
\(873\) −15.8330 −0.535866
\(874\) −28.9461 −0.979117
\(875\) 45.0516 1.52302
\(876\) 0.411079 0.0138891
\(877\) 7.94166 0.268171 0.134085 0.990970i \(-0.457190\pi\)
0.134085 + 0.990970i \(0.457190\pi\)
\(878\) 6.78516 0.228988
\(879\) −16.8649 −0.568837
\(880\) −16.4664 −0.555083
\(881\) 8.19164 0.275983 0.137992 0.990433i \(-0.455935\pi\)
0.137992 + 0.990433i \(0.455935\pi\)
\(882\) 1.37605 0.0463341
\(883\) 16.4576 0.553844 0.276922 0.960892i \(-0.410686\pi\)
0.276922 + 0.960892i \(0.410686\pi\)
\(884\) −4.10662 −0.138121
\(885\) 50.7305 1.70529
\(886\) −36.2803 −1.21886
\(887\) 13.3032 0.446679 0.223340 0.974741i \(-0.428304\pi\)
0.223340 + 0.974741i \(0.428304\pi\)
\(888\) −15.0335 −0.504490
\(889\) 29.3737 0.985162
\(890\) 19.1999 0.643582
\(891\) 1.00000 0.0335013
\(892\) 3.48966 0.116842
\(893\) 72.1050 2.41290
\(894\) 13.8826 0.464303
\(895\) −94.1752 −3.14793
\(896\) −30.6475 −1.02386
\(897\) 13.6522 0.455833
\(898\) 24.9572 0.832832
\(899\) 1.31913 0.0439956
\(900\) 1.51168 0.0503893
\(901\) −64.9642 −2.16427
\(902\) 6.39917 0.213069
\(903\) 23.0424 0.766802
\(904\) −44.4871 −1.47962
\(905\) −54.2928 −1.80476
\(906\) 9.05117 0.300705
\(907\) 6.59934 0.219128 0.109564 0.993980i \(-0.465055\pi\)
0.109564 + 0.993980i \(0.465055\pi\)
\(908\) 4.20706 0.139616
\(909\) 7.72797 0.256321
\(910\) 64.5330 2.13925
\(911\) −45.1362 −1.49543 −0.747714 0.664021i \(-0.768848\pi\)
−0.747714 + 0.664021i \(0.768848\pi\)
\(912\) 28.7656 0.952524
\(913\) −7.42505 −0.245733
\(914\) 46.0840 1.52432
\(915\) 3.84216 0.127018
\(916\) −3.81082 −0.125913
\(917\) 51.3703 1.69640
\(918\) 8.37742 0.276496
\(919\) 22.6447 0.746979 0.373490 0.927634i \(-0.378161\pi\)
0.373490 + 0.927634i \(0.378161\pi\)
\(920\) 30.5737 1.00798
\(921\) −20.2128 −0.666034
\(922\) −6.78156 −0.223339
\(923\) 1.39302 0.0458518
\(924\) −0.381279 −0.0125432
\(925\) 54.1834 1.78154
\(926\) −40.6409 −1.33554
\(927\) 16.1339 0.529906
\(928\) 4.95705 0.162723
\(929\) −11.5176 −0.377879 −0.188940 0.981989i \(-0.560505\pi\)
−0.188940 + 0.981989i \(0.560505\pi\)
\(930\) 1.31185 0.0430173
\(931\) −6.29181 −0.206206
\(932\) 3.72372 0.121974
\(933\) 27.5538 0.902070
\(934\) −5.63111 −0.184255
\(935\) 21.9269 0.717086
\(936\) −12.5866 −0.411407
\(937\) −7.74043 −0.252869 −0.126434 0.991975i \(-0.540353\pi\)
−0.126434 + 0.991975i \(0.540353\pi\)
\(938\) −45.5050 −1.48579
\(939\) −18.0023 −0.587484
\(940\) 6.39153 0.208469
\(941\) −31.6675 −1.03233 −0.516166 0.856489i \(-0.672641\pi\)
−0.516166 + 0.856489i \(0.672641\pi\)
\(942\) 0.697630 0.0227300
\(943\) −12.8070 −0.417053
\(944\) 56.5872 1.84176
\(945\) −9.46029 −0.307743
\(946\) 13.7375 0.446643
\(947\) 32.2734 1.04874 0.524372 0.851489i \(-0.324300\pi\)
0.524372 + 0.851489i \(0.324300\pi\)
\(948\) 2.00620 0.0651584
\(949\) 12.3362 0.400450
\(950\) −96.1843 −3.12063
\(951\) −29.5066 −0.956818
\(952\) 38.0602 1.23354
\(953\) 32.3994 1.04952 0.524759 0.851251i \(-0.324155\pi\)
0.524759 + 0.851251i \(0.324155\pi\)
\(954\) 16.7102 0.541012
\(955\) −81.3225 −2.63153
\(956\) 2.51949 0.0814862
\(957\) −5.67137 −0.183329
\(958\) −29.1332 −0.941251
\(959\) −47.1755 −1.52338
\(960\) −28.0032 −0.903798
\(961\) −30.9459 −0.998255
\(962\) 37.8614 1.22070
\(963\) 1.14677 0.0369542
\(964\) 0.238159 0.00767058
\(965\) −38.8805 −1.25161
\(966\) 10.6187 0.341650
\(967\) −54.5762 −1.75505 −0.877526 0.479529i \(-0.840808\pi\)
−0.877526 + 0.479529i \(0.840808\pi\)
\(968\) 2.70857 0.0870567
\(969\) −38.3046 −1.23052
\(970\) 89.2990 2.86722
\(971\) 3.84654 0.123441 0.0617207 0.998093i \(-0.480341\pi\)
0.0617207 + 0.998093i \(0.480341\pi\)
\(972\) −0.154851 −0.00496684
\(973\) −6.35755 −0.203814
\(974\) 27.4724 0.880272
\(975\) 45.3645 1.45283
\(976\) 4.28572 0.137183
\(977\) −3.97466 −0.127161 −0.0635804 0.997977i \(-0.520252\pi\)
−0.0635804 + 0.997977i \(0.520252\pi\)
\(978\) 25.9271 0.829059
\(979\) −3.40420 −0.108799
\(980\) −0.557719 −0.0178157
\(981\) 11.7557 0.375332
\(982\) 48.6163 1.55141
\(983\) 3.16990 0.101104 0.0505521 0.998721i \(-0.483902\pi\)
0.0505521 + 0.998721i \(0.483902\pi\)
\(984\) 11.8074 0.376407
\(985\) 18.4745 0.588646
\(986\) −47.5115 −1.51307
\(987\) −26.4512 −0.841952
\(988\) −4.82983 −0.153657
\(989\) −27.4935 −0.874243
\(990\) −5.64006 −0.179253
\(991\) 25.3534 0.805378 0.402689 0.915337i \(-0.368076\pi\)
0.402689 + 0.915337i \(0.368076\pi\)
\(992\) 0.203299 0.00645476
\(993\) −17.8450 −0.566295
\(994\) 1.08349 0.0343663
\(995\) 50.3239 1.59537
\(996\) 1.14977 0.0364320
\(997\) 5.71711 0.181063 0.0905314 0.995894i \(-0.471143\pi\)
0.0905314 + 0.995894i \(0.471143\pi\)
\(998\) −4.94254 −0.156453
\(999\) −5.55034 −0.175605
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 2013.2.a.e.1.3 13
3.2 odd 2 6039.2.a.i.1.11 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.e.1.3 13 1.1 even 1 trivial
6039.2.a.i.1.11 13 3.2 odd 2