Properties

Label 8045.2.a.b.1.3
Level $8045$
Weight $2$
Character 8045.1
Self dual yes
Analytic conductor $64.240$
Analytic rank $1$
Dimension $126$
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] = [8045,2,Mod(1,8045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8045, 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("8045.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8045 = 5 \cdot 1609 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8045.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: \(64.2396484261\)
Analytic rank: \(1\)
Dimension: \(126\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Character \(\chi\) \(=\) 8045.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.59235 q^{2} -0.476573 q^{3} +4.72028 q^{4} -1.00000 q^{5} +1.23544 q^{6} -4.74079 q^{7} -7.05191 q^{8} -2.77288 q^{9} +O(q^{10})\) \(q-2.59235 q^{2} -0.476573 q^{3} +4.72028 q^{4} -1.00000 q^{5} +1.23544 q^{6} -4.74079 q^{7} -7.05191 q^{8} -2.77288 q^{9} +2.59235 q^{10} -6.33301 q^{11} -2.24956 q^{12} -0.258183 q^{13} +12.2898 q^{14} +0.476573 q^{15} +8.84045 q^{16} -5.75668 q^{17} +7.18827 q^{18} -5.13370 q^{19} -4.72028 q^{20} +2.25933 q^{21} +16.4174 q^{22} +8.08141 q^{23} +3.36075 q^{24} +1.00000 q^{25} +0.669301 q^{26} +2.75120 q^{27} -22.3778 q^{28} +1.19129 q^{29} -1.23544 q^{30} -9.86768 q^{31} -8.81373 q^{32} +3.01814 q^{33} +14.9233 q^{34} +4.74079 q^{35} -13.0887 q^{36} +9.79477 q^{37} +13.3083 q^{38} +0.123043 q^{39} +7.05191 q^{40} +2.89622 q^{41} -5.85698 q^{42} -9.66952 q^{43} -29.8936 q^{44} +2.77288 q^{45} -20.9498 q^{46} +1.79413 q^{47} -4.21312 q^{48} +15.4751 q^{49} -2.59235 q^{50} +2.74348 q^{51} -1.21870 q^{52} +6.02549 q^{53} -7.13207 q^{54} +6.33301 q^{55} +33.4316 q^{56} +2.44659 q^{57} -3.08825 q^{58} -2.26955 q^{59} +2.24956 q^{60} +7.81763 q^{61} +25.5805 q^{62} +13.1456 q^{63} +5.16737 q^{64} +0.258183 q^{65} -7.82409 q^{66} +6.71712 q^{67} -27.1731 q^{68} -3.85138 q^{69} -12.2898 q^{70} -4.41165 q^{71} +19.5541 q^{72} -12.6241 q^{73} -25.3915 q^{74} -0.476573 q^{75} -24.2325 q^{76} +30.0235 q^{77} -0.318971 q^{78} -4.41978 q^{79} -8.84045 q^{80} +7.00749 q^{81} -7.50801 q^{82} +11.4842 q^{83} +10.6647 q^{84} +5.75668 q^{85} +25.0668 q^{86} -0.567739 q^{87} +44.6598 q^{88} -10.2915 q^{89} -7.18827 q^{90} +1.22399 q^{91} +38.1465 q^{92} +4.70267 q^{93} -4.65101 q^{94} +5.13370 q^{95} +4.20039 q^{96} +13.4837 q^{97} -40.1168 q^{98} +17.5607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9} - 5 q^{10} - 44 q^{11} - 11 q^{12} - 35 q^{13} - 14 q^{14} + 9 q^{15} + 75 q^{16} + 11 q^{17} - 15 q^{18} - 130 q^{19} - 109 q^{20} - 44 q^{21} - 14 q^{22} + 75 q^{23} - 63 q^{24} + 126 q^{25} - 43 q^{26} - 42 q^{27} - 77 q^{28} - 24 q^{29} + 21 q^{30} - 78 q^{31} + 24 q^{32} - 29 q^{33} - 57 q^{34} + 23 q^{35} + 50 q^{36} - 31 q^{37} - 3 q^{38} - 57 q^{39} - 12 q^{40} - 38 q^{41} - 10 q^{42} - 100 q^{43} - 90 q^{44} - 109 q^{45} - 96 q^{46} + 12 q^{47} - 22 q^{48} + 65 q^{49} + 5 q^{50} - 74 q^{51} - 112 q^{52} + 20 q^{53} - 90 q^{54} + 44 q^{55} - 57 q^{56} + 6 q^{57} - 35 q^{58} - 97 q^{59} + 11 q^{60} - 102 q^{61} - 16 q^{62} - 15 q^{63} + 4 q^{64} + 35 q^{65} - 83 q^{66} - 121 q^{67} + 41 q^{68} - 71 q^{69} + 14 q^{70} - 32 q^{71} - 32 q^{72} - 85 q^{73} - 42 q^{74} - 9 q^{75} - 275 q^{76} + 13 q^{77} + 10 q^{78} - 97 q^{79} - 75 q^{80} + 86 q^{81} - 55 q^{82} - 73 q^{83} - 111 q^{84} - 11 q^{85} - 56 q^{86} - q^{87} - 37 q^{88} - 67 q^{89} + 15 q^{90} - 180 q^{91} + 98 q^{92} - 44 q^{93} - 86 q^{94} + 130 q^{95} - 179 q^{96} - 50 q^{97} + 18 q^{98} - 217 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.59235 −1.83307 −0.916534 0.399957i \(-0.869025\pi\)
−0.916534 + 0.399957i \(0.869025\pi\)
\(3\) −0.476573 −0.275150 −0.137575 0.990491i \(-0.543931\pi\)
−0.137575 + 0.990491i \(0.543931\pi\)
\(4\) 4.72028 2.36014
\(5\) −1.00000 −0.447214
\(6\) 1.23544 0.504368
\(7\) −4.74079 −1.79185 −0.895925 0.444206i \(-0.853486\pi\)
−0.895925 + 0.444206i \(0.853486\pi\)
\(8\) −7.05191 −2.49323
\(9\) −2.77288 −0.924293
\(10\) 2.59235 0.819773
\(11\) −6.33301 −1.90948 −0.954738 0.297450i \(-0.903864\pi\)
−0.954738 + 0.297450i \(0.903864\pi\)
\(12\) −2.24956 −0.649391
\(13\) −0.258183 −0.0716071 −0.0358036 0.999359i \(-0.511399\pi\)
−0.0358036 + 0.999359i \(0.511399\pi\)
\(14\) 12.2898 3.28458
\(15\) 0.476573 0.123051
\(16\) 8.84045 2.21011
\(17\) −5.75668 −1.39620 −0.698100 0.716001i \(-0.745971\pi\)
−0.698100 + 0.716001i \(0.745971\pi\)
\(18\) 7.18827 1.69429
\(19\) −5.13370 −1.17775 −0.588876 0.808223i \(-0.700429\pi\)
−0.588876 + 0.808223i \(0.700429\pi\)
\(20\) −4.72028 −1.05549
\(21\) 2.25933 0.493027
\(22\) 16.4174 3.50020
\(23\) 8.08141 1.68509 0.842545 0.538626i \(-0.181056\pi\)
0.842545 + 0.538626i \(0.181056\pi\)
\(24\) 3.36075 0.686010
\(25\) 1.00000 0.200000
\(26\) 0.669301 0.131261
\(27\) 2.75120 0.529469
\(28\) −22.3778 −4.22901
\(29\) 1.19129 0.221218 0.110609 0.993864i \(-0.464720\pi\)
0.110609 + 0.993864i \(0.464720\pi\)
\(30\) −1.23544 −0.225560
\(31\) −9.86768 −1.77229 −0.886143 0.463411i \(-0.846625\pi\)
−0.886143 + 0.463411i \(0.846625\pi\)
\(32\) −8.81373 −1.55806
\(33\) 3.01814 0.525392
\(34\) 14.9233 2.55933
\(35\) 4.74079 0.801340
\(36\) −13.0887 −2.18146
\(37\) 9.79477 1.61025 0.805125 0.593105i \(-0.202098\pi\)
0.805125 + 0.593105i \(0.202098\pi\)
\(38\) 13.3083 2.15890
\(39\) 0.123043 0.0197027
\(40\) 7.05191 1.11500
\(41\) 2.89622 0.452313 0.226157 0.974091i \(-0.427384\pi\)
0.226157 + 0.974091i \(0.427384\pi\)
\(42\) −5.85698 −0.903752
\(43\) −9.66952 −1.47459 −0.737294 0.675572i \(-0.763897\pi\)
−0.737294 + 0.675572i \(0.763897\pi\)
\(44\) −29.8936 −4.50662
\(45\) 2.77288 0.413356
\(46\) −20.9498 −3.08888
\(47\) 1.79413 0.261701 0.130850 0.991402i \(-0.458229\pi\)
0.130850 + 0.991402i \(0.458229\pi\)
\(48\) −4.21312 −0.608112
\(49\) 15.4751 2.21073
\(50\) −2.59235 −0.366614
\(51\) 2.74348 0.384164
\(52\) −1.21870 −0.169003
\(53\) 6.02549 0.827665 0.413832 0.910353i \(-0.364190\pi\)
0.413832 + 0.910353i \(0.364190\pi\)
\(54\) −7.13207 −0.970552
\(55\) 6.33301 0.853943
\(56\) 33.4316 4.46749
\(57\) 2.44659 0.324058
\(58\) −3.08825 −0.405507
\(59\) −2.26955 −0.295470 −0.147735 0.989027i \(-0.547198\pi\)
−0.147735 + 0.989027i \(0.547198\pi\)
\(60\) 2.24956 0.290417
\(61\) 7.81763 1.00094 0.500472 0.865753i \(-0.333160\pi\)
0.500472 + 0.865753i \(0.333160\pi\)
\(62\) 25.5805 3.24872
\(63\) 13.1456 1.65619
\(64\) 5.16737 0.645921
\(65\) 0.258183 0.0320237
\(66\) −7.82409 −0.963078
\(67\) 6.71712 0.820627 0.410313 0.911945i \(-0.365419\pi\)
0.410313 + 0.911945i \(0.365419\pi\)
\(68\) −27.1731 −3.29522
\(69\) −3.85138 −0.463652
\(70\) −12.2898 −1.46891
\(71\) −4.41165 −0.523566 −0.261783 0.965127i \(-0.584311\pi\)
−0.261783 + 0.965127i \(0.584311\pi\)
\(72\) 19.5541 2.30447
\(73\) −12.6241 −1.47754 −0.738772 0.673955i \(-0.764594\pi\)
−0.738772 + 0.673955i \(0.764594\pi\)
\(74\) −25.3915 −2.95170
\(75\) −0.476573 −0.0550299
\(76\) −24.2325 −2.77966
\(77\) 30.0235 3.42149
\(78\) −0.318971 −0.0361164
\(79\) −4.41978 −0.497264 −0.248632 0.968598i \(-0.579981\pi\)
−0.248632 + 0.968598i \(0.579981\pi\)
\(80\) −8.84045 −0.988393
\(81\) 7.00749 0.778610
\(82\) −7.50801 −0.829121
\(83\) 11.4842 1.26055 0.630277 0.776371i \(-0.282941\pi\)
0.630277 + 0.776371i \(0.282941\pi\)
\(84\) 10.6647 1.16361
\(85\) 5.75668 0.624399
\(86\) 25.0668 2.70302
\(87\) −0.567739 −0.0608680
\(88\) 44.6598 4.76075
\(89\) −10.2915 −1.09089 −0.545446 0.838146i \(-0.683640\pi\)
−0.545446 + 0.838146i \(0.683640\pi\)
\(90\) −7.18827 −0.757710
\(91\) 1.22399 0.128309
\(92\) 38.1465 3.97705
\(93\) 4.70267 0.487644
\(94\) −4.65101 −0.479715
\(95\) 5.13370 0.526707
\(96\) 4.20039 0.428700
\(97\) 13.4837 1.36906 0.684529 0.728985i \(-0.260008\pi\)
0.684529 + 0.728985i \(0.260008\pi\)
\(98\) −40.1168 −4.05241
\(99\) 17.5607 1.76491
\(100\) 4.72028 0.472028
\(101\) −0.341340 −0.0339646 −0.0169823 0.999856i \(-0.505406\pi\)
−0.0169823 + 0.999856i \(0.505406\pi\)
\(102\) −7.11205 −0.704198
\(103\) −5.83074 −0.574520 −0.287260 0.957853i \(-0.592744\pi\)
−0.287260 + 0.957853i \(0.592744\pi\)
\(104\) 1.82068 0.178533
\(105\) −2.25933 −0.220488
\(106\) −15.6202 −1.51717
\(107\) 15.1834 1.46784 0.733918 0.679238i \(-0.237690\pi\)
0.733918 + 0.679238i \(0.237690\pi\)
\(108\) 12.9864 1.24962
\(109\) 8.46282 0.810591 0.405295 0.914186i \(-0.367169\pi\)
0.405295 + 0.914186i \(0.367169\pi\)
\(110\) −16.4174 −1.56534
\(111\) −4.66793 −0.443060
\(112\) −41.9107 −3.96019
\(113\) 17.8557 1.67972 0.839859 0.542804i \(-0.182637\pi\)
0.839859 + 0.542804i \(0.182637\pi\)
\(114\) −6.34240 −0.594021
\(115\) −8.08141 −0.753595
\(116\) 5.62324 0.522105
\(117\) 0.715910 0.0661859
\(118\) 5.88347 0.541617
\(119\) 27.2912 2.50178
\(120\) −3.36075 −0.306793
\(121\) 29.1070 2.64609
\(122\) −20.2660 −1.83480
\(123\) −1.38026 −0.124454
\(124\) −46.5782 −4.18284
\(125\) −1.00000 −0.0894427
\(126\) −34.0781 −3.03592
\(127\) −5.11518 −0.453899 −0.226949 0.973907i \(-0.572875\pi\)
−0.226949 + 0.973907i \(0.572875\pi\)
\(128\) 4.23184 0.374045
\(129\) 4.60823 0.405732
\(130\) −0.669301 −0.0587016
\(131\) −4.28861 −0.374697 −0.187349 0.982293i \(-0.559989\pi\)
−0.187349 + 0.982293i \(0.559989\pi\)
\(132\) 14.2465 1.24000
\(133\) 24.3378 2.11035
\(134\) −17.4131 −1.50426
\(135\) −2.75120 −0.236786
\(136\) 40.5955 3.48104
\(137\) 0.992308 0.0847786 0.0423893 0.999101i \(-0.486503\pi\)
0.0423893 + 0.999101i \(0.486503\pi\)
\(138\) 9.98413 0.849906
\(139\) −5.08248 −0.431090 −0.215545 0.976494i \(-0.569153\pi\)
−0.215545 + 0.976494i \(0.569153\pi\)
\(140\) 22.3778 1.89127
\(141\) −0.855034 −0.0720069
\(142\) 11.4365 0.959733
\(143\) 1.63508 0.136732
\(144\) −24.5135 −2.04279
\(145\) −1.19129 −0.0989316
\(146\) 32.7262 2.70844
\(147\) −7.37501 −0.608281
\(148\) 46.2340 3.80041
\(149\) 0.292431 0.0239569 0.0119784 0.999928i \(-0.496187\pi\)
0.0119784 + 0.999928i \(0.496187\pi\)
\(150\) 1.23544 0.100874
\(151\) −7.43589 −0.605124 −0.302562 0.953130i \(-0.597842\pi\)
−0.302562 + 0.953130i \(0.597842\pi\)
\(152\) 36.2024 2.93640
\(153\) 15.9626 1.29050
\(154\) −77.8313 −6.27183
\(155\) 9.86768 0.792591
\(156\) 0.580798 0.0465011
\(157\) −12.4726 −0.995425 −0.497712 0.867342i \(-0.665827\pi\)
−0.497712 + 0.867342i \(0.665827\pi\)
\(158\) 11.4576 0.911518
\(159\) −2.87159 −0.227732
\(160\) 8.81373 0.696787
\(161\) −38.3123 −3.01943
\(162\) −18.1659 −1.42724
\(163\) −21.4251 −1.67814 −0.839071 0.544023i \(-0.816901\pi\)
−0.839071 + 0.544023i \(0.816901\pi\)
\(164\) 13.6709 1.06752
\(165\) −3.01814 −0.234962
\(166\) −29.7710 −2.31068
\(167\) −19.3497 −1.49732 −0.748662 0.662952i \(-0.769303\pi\)
−0.748662 + 0.662952i \(0.769303\pi\)
\(168\) −15.9326 −1.22923
\(169\) −12.9333 −0.994872
\(170\) −14.9233 −1.14457
\(171\) 14.2351 1.08859
\(172\) −45.6428 −3.48023
\(173\) 1.33845 0.101761 0.0508804 0.998705i \(-0.483797\pi\)
0.0508804 + 0.998705i \(0.483797\pi\)
\(174\) 1.47178 0.111575
\(175\) −4.74079 −0.358370
\(176\) −55.9867 −4.22016
\(177\) 1.08161 0.0812986
\(178\) 26.6790 1.99968
\(179\) 15.7794 1.17941 0.589706 0.807618i \(-0.299244\pi\)
0.589706 + 0.807618i \(0.299244\pi\)
\(180\) 13.0887 0.975578
\(181\) 22.1852 1.64901 0.824505 0.565855i \(-0.191454\pi\)
0.824505 + 0.565855i \(0.191454\pi\)
\(182\) −3.17302 −0.235200
\(183\) −3.72567 −0.275410
\(184\) −56.9893 −4.20131
\(185\) −9.79477 −0.720126
\(186\) −12.1910 −0.893885
\(187\) 36.4571 2.66601
\(188\) 8.46879 0.617650
\(189\) −13.0429 −0.948728
\(190\) −13.3083 −0.965489
\(191\) 5.44896 0.394273 0.197137 0.980376i \(-0.436836\pi\)
0.197137 + 0.980376i \(0.436836\pi\)
\(192\) −2.46263 −0.177725
\(193\) −24.6678 −1.77563 −0.887815 0.460200i \(-0.847778\pi\)
−0.887815 + 0.460200i \(0.847778\pi\)
\(194\) −34.9544 −2.50958
\(195\) −0.123043 −0.00881131
\(196\) 73.0467 5.21762
\(197\) 2.88676 0.205673 0.102837 0.994698i \(-0.467208\pi\)
0.102837 + 0.994698i \(0.467208\pi\)
\(198\) −45.5234 −3.23521
\(199\) −23.6574 −1.67703 −0.838515 0.544879i \(-0.816576\pi\)
−0.838515 + 0.544879i \(0.816576\pi\)
\(200\) −7.05191 −0.498645
\(201\) −3.20120 −0.225795
\(202\) 0.884873 0.0622595
\(203\) −5.64768 −0.396389
\(204\) 12.9500 0.906680
\(205\) −2.89622 −0.202281
\(206\) 15.1153 1.05313
\(207\) −22.4088 −1.55752
\(208\) −2.28246 −0.158260
\(209\) 32.5118 2.24889
\(210\) 5.85698 0.404170
\(211\) −1.51484 −0.104286 −0.0521430 0.998640i \(-0.516605\pi\)
−0.0521430 + 0.998640i \(0.516605\pi\)
\(212\) 28.4420 1.95340
\(213\) 2.10247 0.144059
\(214\) −39.3607 −2.69064
\(215\) 9.66952 0.659456
\(216\) −19.4012 −1.32008
\(217\) 46.7806 3.17567
\(218\) −21.9386 −1.48587
\(219\) 6.01633 0.406546
\(220\) 29.8936 2.01542
\(221\) 1.48628 0.0999778
\(222\) 12.1009 0.812159
\(223\) 0.946409 0.0633762 0.0316881 0.999498i \(-0.489912\pi\)
0.0316881 + 0.999498i \(0.489912\pi\)
\(224\) 41.7840 2.79181
\(225\) −2.77288 −0.184859
\(226\) −46.2881 −3.07904
\(227\) 2.87218 0.190633 0.0953166 0.995447i \(-0.469614\pi\)
0.0953166 + 0.995447i \(0.469614\pi\)
\(228\) 11.5486 0.764822
\(229\) −13.7717 −0.910060 −0.455030 0.890476i \(-0.650371\pi\)
−0.455030 + 0.890476i \(0.650371\pi\)
\(230\) 20.9498 1.38139
\(231\) −14.3084 −0.941423
\(232\) −8.40090 −0.551546
\(233\) −9.47782 −0.620912 −0.310456 0.950588i \(-0.600482\pi\)
−0.310456 + 0.950588i \(0.600482\pi\)
\(234\) −1.85589 −0.121323
\(235\) −1.79413 −0.117036
\(236\) −10.7129 −0.697351
\(237\) 2.10635 0.136822
\(238\) −70.7483 −4.58593
\(239\) −17.8262 −1.15308 −0.576541 0.817068i \(-0.695598\pi\)
−0.576541 + 0.817068i \(0.695598\pi\)
\(240\) 4.21312 0.271956
\(241\) −27.0386 −1.74171 −0.870855 0.491540i \(-0.836434\pi\)
−0.870855 + 0.491540i \(0.836434\pi\)
\(242\) −75.4556 −4.85047
\(243\) −11.5932 −0.743703
\(244\) 36.9014 2.36237
\(245\) −15.4751 −0.988667
\(246\) 3.57812 0.228132
\(247\) 1.32544 0.0843354
\(248\) 69.5859 4.41871
\(249\) −5.47306 −0.346841
\(250\) 2.59235 0.163955
\(251\) −17.1993 −1.08561 −0.542805 0.839859i \(-0.682638\pi\)
−0.542805 + 0.839859i \(0.682638\pi\)
\(252\) 62.0510 3.90885
\(253\) −51.1797 −3.21764
\(254\) 13.2603 0.832027
\(255\) −2.74348 −0.171803
\(256\) −21.3051 −1.33157
\(257\) 31.0798 1.93870 0.969352 0.245678i \(-0.0790105\pi\)
0.969352 + 0.245678i \(0.0790105\pi\)
\(258\) −11.9462 −0.743735
\(259\) −46.4350 −2.88533
\(260\) 1.21870 0.0755803
\(261\) −3.30331 −0.204470
\(262\) 11.1176 0.686846
\(263\) 21.8505 1.34736 0.673680 0.739023i \(-0.264712\pi\)
0.673680 + 0.739023i \(0.264712\pi\)
\(264\) −21.2837 −1.30992
\(265\) −6.02549 −0.370143
\(266\) −63.0921 −3.86842
\(267\) 4.90463 0.300159
\(268\) 31.7067 1.93679
\(269\) −2.15027 −0.131104 −0.0655521 0.997849i \(-0.520881\pi\)
−0.0655521 + 0.997849i \(0.520881\pi\)
\(270\) 7.13207 0.434044
\(271\) 15.6683 0.951780 0.475890 0.879505i \(-0.342126\pi\)
0.475890 + 0.879505i \(0.342126\pi\)
\(272\) −50.8916 −3.08576
\(273\) −0.583322 −0.0353042
\(274\) −2.57241 −0.155405
\(275\) −6.33301 −0.381895
\(276\) −18.1796 −1.09428
\(277\) 27.8285 1.67205 0.836025 0.548691i \(-0.184874\pi\)
0.836025 + 0.548691i \(0.184874\pi\)
\(278\) 13.1756 0.790218
\(279\) 27.3619 1.63811
\(280\) −33.4316 −1.99792
\(281\) 15.5529 0.927808 0.463904 0.885885i \(-0.346448\pi\)
0.463904 + 0.885885i \(0.346448\pi\)
\(282\) 2.21655 0.131993
\(283\) −4.21635 −0.250636 −0.125318 0.992117i \(-0.539995\pi\)
−0.125318 + 0.992117i \(0.539995\pi\)
\(284\) −20.8242 −1.23569
\(285\) −2.44659 −0.144923
\(286\) −4.23869 −0.250639
\(287\) −13.7304 −0.810477
\(288\) 24.4394 1.44011
\(289\) 16.1393 0.949372
\(290\) 3.08825 0.181348
\(291\) −6.42595 −0.376696
\(292\) −59.5894 −3.48721
\(293\) −23.0035 −1.34388 −0.671941 0.740605i \(-0.734539\pi\)
−0.671941 + 0.740605i \(0.734539\pi\)
\(294\) 19.1186 1.11502
\(295\) 2.26955 0.132138
\(296\) −69.0718 −4.01472
\(297\) −17.4234 −1.01101
\(298\) −0.758084 −0.0439146
\(299\) −2.08648 −0.120664
\(300\) −2.24956 −0.129878
\(301\) 45.8411 2.64224
\(302\) 19.2764 1.10923
\(303\) 0.162674 0.00934536
\(304\) −45.3842 −2.60297
\(305\) −7.81763 −0.447636
\(306\) −41.3805 −2.36557
\(307\) 19.0514 1.08732 0.543660 0.839305i \(-0.317038\pi\)
0.543660 + 0.839305i \(0.317038\pi\)
\(308\) 141.719 8.07519
\(309\) 2.77877 0.158079
\(310\) −25.5805 −1.45287
\(311\) −14.6502 −0.830738 −0.415369 0.909653i \(-0.636348\pi\)
−0.415369 + 0.909653i \(0.636348\pi\)
\(312\) −0.867689 −0.0491232
\(313\) 5.86356 0.331428 0.165714 0.986174i \(-0.447007\pi\)
0.165714 + 0.986174i \(0.447007\pi\)
\(314\) 32.3334 1.82468
\(315\) −13.1456 −0.740672
\(316\) −20.8626 −1.17361
\(317\) −1.22051 −0.0685505 −0.0342752 0.999412i \(-0.510912\pi\)
−0.0342752 + 0.999412i \(0.510912\pi\)
\(318\) 7.44416 0.417448
\(319\) −7.54448 −0.422410
\(320\) −5.16737 −0.288865
\(321\) −7.23601 −0.403875
\(322\) 99.3188 5.53482
\(323\) 29.5531 1.64438
\(324\) 33.0773 1.83763
\(325\) −0.258183 −0.0143214
\(326\) 55.5413 3.07615
\(327\) −4.03315 −0.223034
\(328\) −20.4239 −1.12772
\(329\) −8.50559 −0.468928
\(330\) 7.82409 0.430702
\(331\) 28.5721 1.57046 0.785231 0.619203i \(-0.212544\pi\)
0.785231 + 0.619203i \(0.212544\pi\)
\(332\) 54.2085 2.97508
\(333\) −27.1597 −1.48834
\(334\) 50.1612 2.74470
\(335\) −6.71712 −0.366996
\(336\) 19.9735 1.08965
\(337\) −6.24100 −0.339969 −0.169984 0.985447i \(-0.554372\pi\)
−0.169984 + 0.985447i \(0.554372\pi\)
\(338\) 33.5277 1.82367
\(339\) −8.50953 −0.462174
\(340\) 27.1731 1.47367
\(341\) 62.4921 3.38414
\(342\) −36.9024 −1.99545
\(343\) −40.1786 −2.16944
\(344\) 68.1885 3.67648
\(345\) 3.85138 0.207352
\(346\) −3.46974 −0.186534
\(347\) 26.4712 1.42105 0.710523 0.703674i \(-0.248458\pi\)
0.710523 + 0.703674i \(0.248458\pi\)
\(348\) −2.67989 −0.143657
\(349\) −24.5298 −1.31305 −0.656525 0.754304i \(-0.727974\pi\)
−0.656525 + 0.754304i \(0.727974\pi\)
\(350\) 12.2898 0.656916
\(351\) −0.710313 −0.0379137
\(352\) 55.8175 2.97508
\(353\) 0.137490 0.00731786 0.00365893 0.999993i \(-0.498835\pi\)
0.00365893 + 0.999993i \(0.498835\pi\)
\(354\) −2.80390 −0.149026
\(355\) 4.41165 0.234146
\(356\) −48.5785 −2.57465
\(357\) −13.0063 −0.688364
\(358\) −40.9058 −2.16194
\(359\) 30.9444 1.63318 0.816591 0.577217i \(-0.195861\pi\)
0.816591 + 0.577217i \(0.195861\pi\)
\(360\) −19.5541 −1.03059
\(361\) 7.35489 0.387100
\(362\) −57.5117 −3.02275
\(363\) −13.8716 −0.728072
\(364\) 5.77758 0.302827
\(365\) 12.6241 0.660778
\(366\) 9.65824 0.504845
\(367\) 13.4789 0.703594 0.351797 0.936076i \(-0.385571\pi\)
0.351797 + 0.936076i \(0.385571\pi\)
\(368\) 71.4433 3.72424
\(369\) −8.03086 −0.418070
\(370\) 25.3915 1.32004
\(371\) −28.5656 −1.48305
\(372\) 22.1979 1.15091
\(373\) −9.25520 −0.479216 −0.239608 0.970870i \(-0.577019\pi\)
−0.239608 + 0.970870i \(0.577019\pi\)
\(374\) −94.5096 −4.88697
\(375\) 0.476573 0.0246101
\(376\) −12.6520 −0.652479
\(377\) −0.307572 −0.0158408
\(378\) 33.8116 1.73908
\(379\) 26.4010 1.35613 0.678064 0.735003i \(-0.262819\pi\)
0.678064 + 0.735003i \(0.262819\pi\)
\(380\) 24.2325 1.24310
\(381\) 2.43776 0.124890
\(382\) −14.1256 −0.722729
\(383\) −23.3400 −1.19262 −0.596310 0.802754i \(-0.703367\pi\)
−0.596310 + 0.802754i \(0.703367\pi\)
\(384\) −2.01678 −0.102918
\(385\) −30.0235 −1.53014
\(386\) 63.9477 3.25485
\(387\) 26.8124 1.36295
\(388\) 63.6466 3.23117
\(389\) −19.9581 −1.01192 −0.505958 0.862558i \(-0.668861\pi\)
−0.505958 + 0.862558i \(0.668861\pi\)
\(390\) 0.318971 0.0161517
\(391\) −46.5221 −2.35272
\(392\) −109.129 −5.51184
\(393\) 2.04384 0.103098
\(394\) −7.48349 −0.377013
\(395\) 4.41978 0.222383
\(396\) 82.8912 4.16544
\(397\) 1.69642 0.0851408 0.0425704 0.999093i \(-0.486445\pi\)
0.0425704 + 0.999093i \(0.486445\pi\)
\(398\) 61.3283 3.07411
\(399\) −11.5987 −0.580663
\(400\) 8.84045 0.442023
\(401\) −19.8208 −0.989802 −0.494901 0.868949i \(-0.664796\pi\)
−0.494901 + 0.868949i \(0.664796\pi\)
\(402\) 8.29863 0.413898
\(403\) 2.54767 0.126908
\(404\) −1.61122 −0.0801612
\(405\) −7.00749 −0.348205
\(406\) 14.6408 0.726608
\(407\) −62.0304 −3.07473
\(408\) −19.3468 −0.957807
\(409\) 6.66554 0.329590 0.164795 0.986328i \(-0.447304\pi\)
0.164795 + 0.986328i \(0.447304\pi\)
\(410\) 7.50801 0.370794
\(411\) −0.472907 −0.0233268
\(412\) −27.5227 −1.35595
\(413\) 10.7595 0.529438
\(414\) 58.0913 2.85503
\(415\) −11.4842 −0.563737
\(416\) 2.27556 0.111568
\(417\) 2.42217 0.118614
\(418\) −84.2819 −4.12236
\(419\) 17.0950 0.835145 0.417573 0.908644i \(-0.362881\pi\)
0.417573 + 0.908644i \(0.362881\pi\)
\(420\) −10.6647 −0.520383
\(421\) 22.8432 1.11331 0.556654 0.830745i \(-0.312085\pi\)
0.556654 + 0.830745i \(0.312085\pi\)
\(422\) 3.92700 0.191163
\(423\) −4.97490 −0.241888
\(424\) −42.4912 −2.06355
\(425\) −5.75668 −0.279240
\(426\) −5.45035 −0.264070
\(427\) −37.0617 −1.79354
\(428\) 71.6699 3.46429
\(429\) −0.779234 −0.0376218
\(430\) −25.0668 −1.20883
\(431\) −9.18431 −0.442393 −0.221196 0.975229i \(-0.570996\pi\)
−0.221196 + 0.975229i \(0.570996\pi\)
\(432\) 24.3219 1.17019
\(433\) −9.92294 −0.476866 −0.238433 0.971159i \(-0.576634\pi\)
−0.238433 + 0.971159i \(0.576634\pi\)
\(434\) −121.272 −5.82122
\(435\) 0.567739 0.0272210
\(436\) 39.9468 1.91311
\(437\) −41.4875 −1.98462
\(438\) −15.5964 −0.745226
\(439\) −11.0486 −0.527322 −0.263661 0.964615i \(-0.584930\pi\)
−0.263661 + 0.964615i \(0.584930\pi\)
\(440\) −44.6598 −2.12907
\(441\) −42.9105 −2.04336
\(442\) −3.85295 −0.183266
\(443\) −9.99508 −0.474881 −0.237440 0.971402i \(-0.576308\pi\)
−0.237440 + 0.971402i \(0.576308\pi\)
\(444\) −22.0339 −1.04568
\(445\) 10.2915 0.487862
\(446\) −2.45342 −0.116173
\(447\) −0.139365 −0.00659173
\(448\) −24.4974 −1.15739
\(449\) −1.24600 −0.0588026 −0.0294013 0.999568i \(-0.509360\pi\)
−0.0294013 + 0.999568i \(0.509360\pi\)
\(450\) 7.18827 0.338858
\(451\) −18.3418 −0.863681
\(452\) 84.2836 3.96437
\(453\) 3.54375 0.166500
\(454\) −7.44569 −0.349444
\(455\) −1.22399 −0.0573816
\(456\) −17.2531 −0.807950
\(457\) 10.1025 0.472576 0.236288 0.971683i \(-0.424069\pi\)
0.236288 + 0.971683i \(0.424069\pi\)
\(458\) 35.7011 1.66820
\(459\) −15.8378 −0.739244
\(460\) −38.1465 −1.77859
\(461\) −6.74221 −0.314016 −0.157008 0.987597i \(-0.550185\pi\)
−0.157008 + 0.987597i \(0.550185\pi\)
\(462\) 37.0923 1.72569
\(463\) 11.2097 0.520957 0.260479 0.965480i \(-0.416120\pi\)
0.260479 + 0.965480i \(0.416120\pi\)
\(464\) 10.5316 0.488917
\(465\) −4.70267 −0.218081
\(466\) 24.5698 1.13817
\(467\) −14.2759 −0.660612 −0.330306 0.943874i \(-0.607152\pi\)
−0.330306 + 0.943874i \(0.607152\pi\)
\(468\) 3.37930 0.156208
\(469\) −31.8445 −1.47044
\(470\) 4.65101 0.214535
\(471\) 5.94413 0.273891
\(472\) 16.0047 0.736674
\(473\) 61.2372 2.81569
\(474\) −5.46039 −0.250804
\(475\) −5.13370 −0.235550
\(476\) 128.822 5.90454
\(477\) −16.7080 −0.765004
\(478\) 46.2118 2.11368
\(479\) 21.2100 0.969109 0.484554 0.874761i \(-0.338982\pi\)
0.484554 + 0.874761i \(0.338982\pi\)
\(480\) −4.20039 −0.191721
\(481\) −2.52885 −0.115305
\(482\) 70.0935 3.19267
\(483\) 18.2586 0.830795
\(484\) 137.393 6.24515
\(485\) −13.4837 −0.612262
\(486\) 30.0536 1.36326
\(487\) 29.8026 1.35049 0.675243 0.737595i \(-0.264039\pi\)
0.675243 + 0.737595i \(0.264039\pi\)
\(488\) −55.1292 −2.49558
\(489\) 10.2106 0.461740
\(490\) 40.1168 1.81229
\(491\) 7.21911 0.325794 0.162897 0.986643i \(-0.447916\pi\)
0.162897 + 0.986643i \(0.447916\pi\)
\(492\) −6.51521 −0.293728
\(493\) −6.85790 −0.308864
\(494\) −3.43599 −0.154593
\(495\) −17.5607 −0.789293
\(496\) −87.2347 −3.91696
\(497\) 20.9147 0.938152
\(498\) 14.1881 0.635783
\(499\) 27.0547 1.21114 0.605568 0.795793i \(-0.292946\pi\)
0.605568 + 0.795793i \(0.292946\pi\)
\(500\) −4.72028 −0.211097
\(501\) 9.22155 0.411988
\(502\) 44.5866 1.99000
\(503\) 25.9863 1.15867 0.579336 0.815089i \(-0.303312\pi\)
0.579336 + 0.815089i \(0.303312\pi\)
\(504\) −92.7017 −4.12926
\(505\) 0.341340 0.0151894
\(506\) 132.676 5.89815
\(507\) 6.16369 0.273739
\(508\) −24.1451 −1.07126
\(509\) 12.9395 0.573535 0.286767 0.958000i \(-0.407419\pi\)
0.286767 + 0.958000i \(0.407419\pi\)
\(510\) 7.11205 0.314927
\(511\) 59.8484 2.64754
\(512\) 46.7667 2.06682
\(513\) −14.1238 −0.623583
\(514\) −80.5696 −3.55377
\(515\) 5.83074 0.256933
\(516\) 21.7521 0.957584
\(517\) −11.3622 −0.499711
\(518\) 120.376 5.28900
\(519\) −0.637872 −0.0279995
\(520\) −1.82068 −0.0798423
\(521\) 6.60962 0.289573 0.144786 0.989463i \(-0.453750\pi\)
0.144786 + 0.989463i \(0.453750\pi\)
\(522\) 8.56335 0.374807
\(523\) −14.3861 −0.629062 −0.314531 0.949247i \(-0.601847\pi\)
−0.314531 + 0.949247i \(0.601847\pi\)
\(524\) −20.2434 −0.884338
\(525\) 2.25933 0.0986054
\(526\) −56.6442 −2.46980
\(527\) 56.8050 2.47447
\(528\) 26.6818 1.16117
\(529\) 42.3092 1.83953
\(530\) 15.6202 0.678497
\(531\) 6.29319 0.273101
\(532\) 114.881 4.98073
\(533\) −0.747755 −0.0323889
\(534\) −12.7145 −0.550211
\(535\) −15.1834 −0.656436
\(536\) −47.3685 −2.04601
\(537\) −7.52006 −0.324515
\(538\) 5.57425 0.240323
\(539\) −98.0039 −4.22133
\(540\) −12.9864 −0.558847
\(541\) −6.07273 −0.261087 −0.130544 0.991443i \(-0.541672\pi\)
−0.130544 + 0.991443i \(0.541672\pi\)
\(542\) −40.6176 −1.74468
\(543\) −10.5729 −0.453724
\(544\) 50.7378 2.17537
\(545\) −8.46282 −0.362507
\(546\) 1.51217 0.0647151
\(547\) −12.6019 −0.538818 −0.269409 0.963026i \(-0.586828\pi\)
−0.269409 + 0.963026i \(0.586828\pi\)
\(548\) 4.68397 0.200089
\(549\) −21.6773 −0.925166
\(550\) 16.4174 0.700039
\(551\) −6.11575 −0.260540
\(552\) 27.1596 1.15599
\(553\) 20.9532 0.891022
\(554\) −72.1411 −3.06498
\(555\) 4.66793 0.198142
\(556\) −23.9907 −1.01743
\(557\) 11.4955 0.487081 0.243541 0.969891i \(-0.421691\pi\)
0.243541 + 0.969891i \(0.421691\pi\)
\(558\) −70.9315 −3.00277
\(559\) 2.49651 0.105591
\(560\) 41.9107 1.77105
\(561\) −17.3745 −0.733551
\(562\) −40.3186 −1.70074
\(563\) 24.4086 1.02870 0.514351 0.857580i \(-0.328033\pi\)
0.514351 + 0.857580i \(0.328033\pi\)
\(564\) −4.03600 −0.169946
\(565\) −17.8557 −0.751193
\(566\) 10.9302 0.459432
\(567\) −33.2210 −1.39515
\(568\) 31.1105 1.30537
\(569\) 30.5123 1.27914 0.639572 0.768731i \(-0.279112\pi\)
0.639572 + 0.768731i \(0.279112\pi\)
\(570\) 6.34240 0.265654
\(571\) 1.60704 0.0672524 0.0336262 0.999434i \(-0.489294\pi\)
0.0336262 + 0.999434i \(0.489294\pi\)
\(572\) 7.71802 0.322706
\(573\) −2.59683 −0.108484
\(574\) 35.5939 1.48566
\(575\) 8.08141 0.337018
\(576\) −14.3285 −0.597020
\(577\) 23.8958 0.994795 0.497397 0.867523i \(-0.334289\pi\)
0.497397 + 0.867523i \(0.334289\pi\)
\(578\) −41.8388 −1.74026
\(579\) 11.7560 0.488564
\(580\) −5.62324 −0.233492
\(581\) −54.4441 −2.25872
\(582\) 16.6583 0.690509
\(583\) −38.1595 −1.58041
\(584\) 89.0243 3.68385
\(585\) −0.715910 −0.0295993
\(586\) 59.6332 2.46343
\(587\) 44.2954 1.82827 0.914133 0.405415i \(-0.132873\pi\)
0.914133 + 0.405415i \(0.132873\pi\)
\(588\) −34.8121 −1.43563
\(589\) 50.6577 2.08731
\(590\) −5.88347 −0.242219
\(591\) −1.37575 −0.0565909
\(592\) 86.5902 3.55884
\(593\) 6.99096 0.287084 0.143542 0.989644i \(-0.454151\pi\)
0.143542 + 0.989644i \(0.454151\pi\)
\(594\) 45.1675 1.85324
\(595\) −27.2912 −1.11883
\(596\) 1.38036 0.0565416
\(597\) 11.2745 0.461434
\(598\) 5.40890 0.221186
\(599\) 0.223153 0.00911779 0.00455890 0.999990i \(-0.498549\pi\)
0.00455890 + 0.999990i \(0.498549\pi\)
\(600\) 3.36075 0.137202
\(601\) −29.6741 −1.21043 −0.605217 0.796061i \(-0.706914\pi\)
−0.605217 + 0.796061i \(0.706914\pi\)
\(602\) −118.836 −4.84340
\(603\) −18.6258 −0.758499
\(604\) −35.0994 −1.42818
\(605\) −29.1070 −1.18337
\(606\) −0.421707 −0.0171307
\(607\) −44.9240 −1.82341 −0.911704 0.410848i \(-0.865233\pi\)
−0.911704 + 0.410848i \(0.865233\pi\)
\(608\) 45.2471 1.83501
\(609\) 2.69153 0.109066
\(610\) 20.2660 0.820547
\(611\) −0.463214 −0.0187396
\(612\) 75.3477 3.04575
\(613\) 15.2195 0.614712 0.307356 0.951595i \(-0.400556\pi\)
0.307356 + 0.951595i \(0.400556\pi\)
\(614\) −49.3879 −1.99313
\(615\) 1.38026 0.0556575
\(616\) −211.723 −8.53055
\(617\) −14.8585 −0.598181 −0.299090 0.954225i \(-0.596683\pi\)
−0.299090 + 0.954225i \(0.596683\pi\)
\(618\) −7.20356 −0.289769
\(619\) −6.82364 −0.274265 −0.137133 0.990553i \(-0.543789\pi\)
−0.137133 + 0.990553i \(0.543789\pi\)
\(620\) 46.5782 1.87062
\(621\) 22.2336 0.892202
\(622\) 37.9785 1.52280
\(623\) 48.7896 1.95471
\(624\) 1.08776 0.0435452
\(625\) 1.00000 0.0400000
\(626\) −15.2004 −0.607530
\(627\) −15.4943 −0.618781
\(628\) −58.8743 −2.34934
\(629\) −56.3853 −2.24823
\(630\) 34.0781 1.35770
\(631\) −9.75479 −0.388332 −0.194166 0.980969i \(-0.562200\pi\)
−0.194166 + 0.980969i \(0.562200\pi\)
\(632\) 31.1678 1.23979
\(633\) 0.721933 0.0286943
\(634\) 3.16398 0.125658
\(635\) 5.11518 0.202990
\(636\) −13.5547 −0.537478
\(637\) −3.99541 −0.158304
\(638\) 19.5579 0.774306
\(639\) 12.2330 0.483929
\(640\) −4.23184 −0.167278
\(641\) 48.0353 1.89728 0.948640 0.316356i \(-0.102459\pi\)
0.948640 + 0.316356i \(0.102459\pi\)
\(642\) 18.7583 0.740329
\(643\) −20.8122 −0.820753 −0.410377 0.911916i \(-0.634603\pi\)
−0.410377 + 0.911916i \(0.634603\pi\)
\(644\) −180.844 −7.12627
\(645\) −4.60823 −0.181449
\(646\) −76.6119 −3.01425
\(647\) −3.39284 −0.133386 −0.0666932 0.997774i \(-0.521245\pi\)
−0.0666932 + 0.997774i \(0.521245\pi\)
\(648\) −49.4161 −1.94125
\(649\) 14.3731 0.564193
\(650\) 0.669301 0.0262521
\(651\) −22.2944 −0.873785
\(652\) −101.132 −3.96064
\(653\) 1.38477 0.0541904 0.0270952 0.999633i \(-0.491374\pi\)
0.0270952 + 0.999633i \(0.491374\pi\)
\(654\) 10.4553 0.408836
\(655\) 4.28861 0.167570
\(656\) 25.6039 0.999664
\(657\) 35.0052 1.36568
\(658\) 22.0495 0.859578
\(659\) 5.73217 0.223293 0.111647 0.993748i \(-0.464387\pi\)
0.111647 + 0.993748i \(0.464387\pi\)
\(660\) −14.2465 −0.554543
\(661\) 23.3052 0.906466 0.453233 0.891392i \(-0.350271\pi\)
0.453233 + 0.891392i \(0.350271\pi\)
\(662\) −74.0688 −2.87876
\(663\) −0.708320 −0.0275089
\(664\) −80.9854 −3.14284
\(665\) −24.3378 −0.943779
\(666\) 70.4075 2.72823
\(667\) 9.62734 0.372772
\(668\) −91.3359 −3.53389
\(669\) −0.451033 −0.0174380
\(670\) 17.4131 0.672728
\(671\) −49.5091 −1.91128
\(672\) −19.9132 −0.768167
\(673\) −12.0448 −0.464292 −0.232146 0.972681i \(-0.574575\pi\)
−0.232146 + 0.972681i \(0.574575\pi\)
\(674\) 16.1789 0.623186
\(675\) 2.75120 0.105894
\(676\) −61.0489 −2.34804
\(677\) −7.63676 −0.293504 −0.146752 0.989173i \(-0.546882\pi\)
−0.146752 + 0.989173i \(0.546882\pi\)
\(678\) 22.0597 0.847197
\(679\) −63.9232 −2.45315
\(680\) −40.5955 −1.55677
\(681\) −1.36880 −0.0524527
\(682\) −162.001 −6.20335
\(683\) −0.450082 −0.0172219 −0.00861095 0.999963i \(-0.502741\pi\)
−0.00861095 + 0.999963i \(0.502741\pi\)
\(684\) 67.1937 2.56922
\(685\) −0.992308 −0.0379141
\(686\) 104.157 3.97673
\(687\) 6.56323 0.250403
\(688\) −85.4829 −3.25901
\(689\) −1.55568 −0.0592667
\(690\) −9.98413 −0.380089
\(691\) 27.9171 1.06202 0.531009 0.847366i \(-0.321813\pi\)
0.531009 + 0.847366i \(0.321813\pi\)
\(692\) 6.31788 0.240170
\(693\) −83.2514 −3.16246
\(694\) −68.6225 −2.60487
\(695\) 5.08248 0.192789
\(696\) 4.00364 0.151758
\(697\) −16.6726 −0.631519
\(698\) 63.5898 2.40691
\(699\) 4.51687 0.170844
\(700\) −22.3778 −0.845803
\(701\) 2.21888 0.0838059 0.0419030 0.999122i \(-0.486658\pi\)
0.0419030 + 0.999122i \(0.486658\pi\)
\(702\) 1.84138 0.0694984
\(703\) −50.2834 −1.89648
\(704\) −32.7250 −1.23337
\(705\) 0.855034 0.0322025
\(706\) −0.356422 −0.0134141
\(707\) 1.61822 0.0608595
\(708\) 5.10548 0.191876
\(709\) 45.3720 1.70398 0.851990 0.523558i \(-0.175396\pi\)
0.851990 + 0.523558i \(0.175396\pi\)
\(710\) −11.4365 −0.429206
\(711\) 12.2555 0.459617
\(712\) 72.5744 2.71984
\(713\) −79.7447 −2.98646
\(714\) 33.7168 1.26182
\(715\) −1.63508 −0.0611484
\(716\) 74.4833 2.78357
\(717\) 8.49550 0.317270
\(718\) −80.2187 −2.99373
\(719\) 1.56136 0.0582289 0.0291144 0.999576i \(-0.490731\pi\)
0.0291144 + 0.999576i \(0.490731\pi\)
\(720\) 24.5135 0.913564
\(721\) 27.6423 1.02945
\(722\) −19.0664 −0.709580
\(723\) 12.8859 0.479231
\(724\) 104.720 3.89189
\(725\) 1.19129 0.0442436
\(726\) 35.9601 1.33461
\(727\) 8.01315 0.297191 0.148596 0.988898i \(-0.452525\pi\)
0.148596 + 0.988898i \(0.452525\pi\)
\(728\) −8.63148 −0.319904
\(729\) −15.4975 −0.573980
\(730\) −32.7262 −1.21125
\(731\) 55.6643 2.05882
\(732\) −17.5862 −0.650005
\(733\) 18.8716 0.697037 0.348519 0.937302i \(-0.386685\pi\)
0.348519 + 0.937302i \(0.386685\pi\)
\(734\) −34.9421 −1.28973
\(735\) 7.37501 0.272031
\(736\) −71.2274 −2.62548
\(737\) −42.5396 −1.56697
\(738\) 20.8188 0.766350
\(739\) −20.9337 −0.770060 −0.385030 0.922904i \(-0.625809\pi\)
−0.385030 + 0.922904i \(0.625809\pi\)
\(740\) −46.2340 −1.69960
\(741\) −0.631667 −0.0232049
\(742\) 74.0520 2.71853
\(743\) −12.2430 −0.449152 −0.224576 0.974457i \(-0.572100\pi\)
−0.224576 + 0.974457i \(0.572100\pi\)
\(744\) −33.1628 −1.21581
\(745\) −0.292431 −0.0107138
\(746\) 23.9927 0.878436
\(747\) −31.8442 −1.16512
\(748\) 172.088 6.29215
\(749\) −71.9813 −2.63014
\(750\) −1.23544 −0.0451121
\(751\) −29.5282 −1.07750 −0.538750 0.842466i \(-0.681103\pi\)
−0.538750 + 0.842466i \(0.681103\pi\)
\(752\) 15.8609 0.578388
\(753\) 8.19673 0.298705
\(754\) 0.797335 0.0290372
\(755\) 7.43589 0.270620
\(756\) −61.5659 −2.23913
\(757\) −29.7713 −1.08206 −0.541029 0.841004i \(-0.681965\pi\)
−0.541029 + 0.841004i \(0.681965\pi\)
\(758\) −68.4406 −2.48588
\(759\) 24.3909 0.885332
\(760\) −36.2024 −1.31320
\(761\) 21.9275 0.794871 0.397435 0.917630i \(-0.369900\pi\)
0.397435 + 0.917630i \(0.369900\pi\)
\(762\) −6.31952 −0.228932
\(763\) −40.1204 −1.45246
\(764\) 25.7206 0.930539
\(765\) −15.9626 −0.577128
\(766\) 60.5055 2.18615
\(767\) 0.585960 0.0211578
\(768\) 10.1535 0.366382
\(769\) 15.9077 0.573647 0.286823 0.957983i \(-0.407401\pi\)
0.286823 + 0.957983i \(0.407401\pi\)
\(770\) 77.8313 2.80485
\(771\) −14.8118 −0.533434
\(772\) −116.439 −4.19073
\(773\) −27.7464 −0.997969 −0.498984 0.866611i \(-0.666293\pi\)
−0.498984 + 0.866611i \(0.666293\pi\)
\(774\) −69.5071 −2.49838
\(775\) −9.86768 −0.354457
\(776\) −95.0855 −3.41337
\(777\) 22.1297 0.793897
\(778\) 51.7384 1.85491
\(779\) −14.8683 −0.532713
\(780\) −0.580798 −0.0207959
\(781\) 27.9390 0.999737
\(782\) 120.601 4.31270
\(783\) 3.27749 0.117128
\(784\) 136.807 4.88595
\(785\) 12.4726 0.445168
\(786\) −5.29834 −0.188985
\(787\) 6.87006 0.244891 0.122446 0.992475i \(-0.460926\pi\)
0.122446 + 0.992475i \(0.460926\pi\)
\(788\) 13.6263 0.485417
\(789\) −10.4134 −0.370726
\(790\) −11.4576 −0.407643
\(791\) −84.6499 −3.00980
\(792\) −123.836 −4.40033
\(793\) −2.01838 −0.0716748
\(794\) −4.39771 −0.156069
\(795\) 2.87159 0.101845
\(796\) −111.670 −3.95802
\(797\) −11.4810 −0.406677 −0.203338 0.979109i \(-0.565179\pi\)
−0.203338 + 0.979109i \(0.565179\pi\)
\(798\) 30.0680 1.06440
\(799\) −10.3282 −0.365386
\(800\) −8.81373 −0.311613
\(801\) 28.5369 1.00830
\(802\) 51.3824 1.81437
\(803\) 79.9488 2.82133
\(804\) −15.1105 −0.532908
\(805\) 38.3123 1.35033
\(806\) −6.60445 −0.232632
\(807\) 1.02476 0.0360733
\(808\) 2.40710 0.0846815
\(809\) −50.9641 −1.79180 −0.895901 0.444253i \(-0.853469\pi\)
−0.895901 + 0.444253i \(0.853469\pi\)
\(810\) 18.1659 0.638283
\(811\) 4.26572 0.149790 0.0748948 0.997191i \(-0.476138\pi\)
0.0748948 + 0.997191i \(0.476138\pi\)
\(812\) −26.6586 −0.935533
\(813\) −7.46708 −0.261882
\(814\) 160.805 5.63620
\(815\) 21.4251 0.750488
\(816\) 24.2536 0.849046
\(817\) 49.6404 1.73670
\(818\) −17.2794 −0.604160
\(819\) −3.39398 −0.118595
\(820\) −13.6709 −0.477410
\(821\) −8.04193 −0.280665 −0.140333 0.990104i \(-0.544817\pi\)
−0.140333 + 0.990104i \(0.544817\pi\)
\(822\) 1.22594 0.0427596
\(823\) 34.6286 1.20708 0.603539 0.797334i \(-0.293757\pi\)
0.603539 + 0.797334i \(0.293757\pi\)
\(824\) 41.1178 1.43241
\(825\) 3.01814 0.105078
\(826\) −27.8923 −0.970497
\(827\) 31.1771 1.08413 0.542066 0.840336i \(-0.317642\pi\)
0.542066 + 0.840336i \(0.317642\pi\)
\(828\) −105.776 −3.67595
\(829\) 24.3141 0.844463 0.422231 0.906488i \(-0.361247\pi\)
0.422231 + 0.906488i \(0.361247\pi\)
\(830\) 29.7710 1.03337
\(831\) −13.2623 −0.460064
\(832\) −1.33413 −0.0462526
\(833\) −89.0850 −3.08661
\(834\) −6.27912 −0.217428
\(835\) 19.3497 0.669624
\(836\) 153.465 5.30769
\(837\) −27.1479 −0.938370
\(838\) −44.3162 −1.53088
\(839\) 24.0474 0.830210 0.415105 0.909774i \(-0.363745\pi\)
0.415105 + 0.909774i \(0.363745\pi\)
\(840\) 15.9326 0.549727
\(841\) −27.5808 −0.951063
\(842\) −59.2174 −2.04077
\(843\) −7.41210 −0.255286
\(844\) −7.15047 −0.246129
\(845\) 12.9333 0.444920
\(846\) 12.8967 0.443397
\(847\) −137.990 −4.74140
\(848\) 53.2681 1.82923
\(849\) 2.00940 0.0689624
\(850\) 14.9233 0.511866
\(851\) 79.1556 2.71342
\(852\) 9.92426 0.339999
\(853\) −23.6269 −0.808970 −0.404485 0.914545i \(-0.632549\pi\)
−0.404485 + 0.914545i \(0.632549\pi\)
\(854\) 96.0769 3.28768
\(855\) −14.2351 −0.486831
\(856\) −107.072 −3.65964
\(857\) 6.14053 0.209757 0.104878 0.994485i \(-0.466555\pi\)
0.104878 + 0.994485i \(0.466555\pi\)
\(858\) 2.02005 0.0689633
\(859\) −55.2066 −1.88362 −0.941812 0.336141i \(-0.890878\pi\)
−0.941812 + 0.336141i \(0.890878\pi\)
\(860\) 45.6428 1.55641
\(861\) 6.54352 0.223003
\(862\) 23.8089 0.810936
\(863\) 7.93344 0.270057 0.135029 0.990842i \(-0.456887\pi\)
0.135029 + 0.990842i \(0.456887\pi\)
\(864\) −24.2483 −0.824945
\(865\) −1.33845 −0.0455088
\(866\) 25.7237 0.874128
\(867\) −7.69157 −0.261220
\(868\) 220.817 7.49502
\(869\) 27.9905 0.949513
\(870\) −1.47178 −0.0498980
\(871\) −1.73425 −0.0587627
\(872\) −59.6790 −2.02099
\(873\) −37.3885 −1.26541
\(874\) 107.550 3.63794
\(875\) 4.74079 0.160268
\(876\) 28.3987 0.959504
\(877\) −12.6137 −0.425933 −0.212966 0.977060i \(-0.568313\pi\)
−0.212966 + 0.977060i \(0.568313\pi\)
\(878\) 28.6419 0.966616
\(879\) 10.9629 0.369769
\(880\) 55.9867 1.88731
\(881\) −12.4845 −0.420615 −0.210307 0.977635i \(-0.567446\pi\)
−0.210307 + 0.977635i \(0.567446\pi\)
\(882\) 111.239 3.74561
\(883\) −49.0722 −1.65141 −0.825705 0.564102i \(-0.809222\pi\)
−0.825705 + 0.564102i \(0.809222\pi\)
\(884\) 7.01564 0.235961
\(885\) −1.08161 −0.0363578
\(886\) 25.9107 0.870489
\(887\) 39.7658 1.33520 0.667602 0.744519i \(-0.267321\pi\)
0.667602 + 0.744519i \(0.267321\pi\)
\(888\) 32.9178 1.10465
\(889\) 24.2500 0.813319
\(890\) −26.6790 −0.894283
\(891\) −44.3785 −1.48674
\(892\) 4.46731 0.149577
\(893\) −9.21053 −0.308219
\(894\) 0.361283 0.0120831
\(895\) −15.7794 −0.527449
\(896\) −20.0623 −0.670233
\(897\) 0.994363 0.0332008
\(898\) 3.23008 0.107789
\(899\) −11.7553 −0.392062
\(900\) −13.0887 −0.436292
\(901\) −34.6868 −1.15558
\(902\) 47.5483 1.58319
\(903\) −21.8467 −0.727011
\(904\) −125.916 −4.18792
\(905\) −22.1852 −0.737459
\(906\) −9.18663 −0.305205
\(907\) −11.1211 −0.369270 −0.184635 0.982807i \(-0.559110\pi\)
−0.184635 + 0.982807i \(0.559110\pi\)
\(908\) 13.5575 0.449921
\(909\) 0.946495 0.0313933
\(910\) 3.17302 0.105184
\(911\) −18.4124 −0.610031 −0.305015 0.952347i \(-0.598662\pi\)
−0.305015 + 0.952347i \(0.598662\pi\)
\(912\) 21.6289 0.716205
\(913\) −72.7295 −2.40699
\(914\) −26.1893 −0.866263
\(915\) 3.72567 0.123167
\(916\) −65.0063 −2.14787
\(917\) 20.3314 0.671402
\(918\) 41.0570 1.35508
\(919\) 11.8351 0.390405 0.195203 0.980763i \(-0.437464\pi\)
0.195203 + 0.980763i \(0.437464\pi\)
\(920\) 56.9893 1.87888
\(921\) −9.07938 −0.299176
\(922\) 17.4782 0.575613
\(923\) 1.13901 0.0374911
\(924\) −67.5395 −2.22189
\(925\) 9.79477 0.322050
\(926\) −29.0594 −0.954950
\(927\) 16.1679 0.531024
\(928\) −10.4998 −0.344671
\(929\) 47.1164 1.54584 0.772919 0.634505i \(-0.218796\pi\)
0.772919 + 0.634505i \(0.218796\pi\)
\(930\) 12.1910 0.399758
\(931\) −79.4444 −2.60369
\(932\) −44.7379 −1.46544
\(933\) 6.98191 0.228577
\(934\) 37.0082 1.21095
\(935\) −36.4571 −1.19227
\(936\) −5.04853 −0.165016
\(937\) −12.3302 −0.402809 −0.201404 0.979508i \(-0.564551\pi\)
−0.201404 + 0.979508i \(0.564551\pi\)
\(938\) 82.5519 2.69542
\(939\) −2.79441 −0.0911923
\(940\) −8.46879 −0.276221
\(941\) −0.432330 −0.0140935 −0.00704677 0.999975i \(-0.502243\pi\)
−0.00704677 + 0.999975i \(0.502243\pi\)
\(942\) −15.4093 −0.502061
\(943\) 23.4055 0.762189
\(944\) −20.0639 −0.653023
\(945\) 13.0429 0.424284
\(946\) −158.748 −5.16135
\(947\) 13.0107 0.422790 0.211395 0.977401i \(-0.432199\pi\)
0.211395 + 0.977401i \(0.432199\pi\)
\(948\) 9.94254 0.322919
\(949\) 3.25934 0.105803
\(950\) 13.3083 0.431780
\(951\) 0.581661 0.0188616
\(952\) −192.455 −6.23750
\(953\) −29.6517 −0.960513 −0.480257 0.877128i \(-0.659456\pi\)
−0.480257 + 0.877128i \(0.659456\pi\)
\(954\) 43.3128 1.40231
\(955\) −5.44896 −0.176324
\(956\) −84.1447 −2.72143
\(957\) 3.59550 0.116226
\(958\) −54.9837 −1.77644
\(959\) −4.70432 −0.151910
\(960\) 2.46263 0.0794810
\(961\) 66.3710 2.14100
\(962\) 6.55565 0.211363
\(963\) −42.1017 −1.35671
\(964\) −127.630 −4.11068
\(965\) 24.6678 0.794086
\(966\) −47.3327 −1.52290
\(967\) 47.1633 1.51667 0.758335 0.651865i \(-0.226013\pi\)
0.758335 + 0.651865i \(0.226013\pi\)
\(968\) −205.260 −6.59731
\(969\) −14.0842 −0.452450
\(970\) 34.9544 1.12232
\(971\) −48.1479 −1.54514 −0.772570 0.634929i \(-0.781029\pi\)
−0.772570 + 0.634929i \(0.781029\pi\)
\(972\) −54.7230 −1.75524
\(973\) 24.0950 0.772449
\(974\) −77.2589 −2.47553
\(975\) 0.123043 0.00394054
\(976\) 69.1114 2.21220
\(977\) 14.7257 0.471116 0.235558 0.971860i \(-0.424308\pi\)
0.235558 + 0.971860i \(0.424308\pi\)
\(978\) −26.4695 −0.846401
\(979\) 65.1759 2.08303
\(980\) −73.0467 −2.33339
\(981\) −23.4664 −0.749223
\(982\) −18.7145 −0.597203
\(983\) −0.987474 −0.0314955 −0.0157478 0.999876i \(-0.505013\pi\)
−0.0157478 + 0.999876i \(0.505013\pi\)
\(984\) 9.73347 0.310292
\(985\) −2.88676 −0.0919798
\(986\) 17.7781 0.566169
\(987\) 4.05354 0.129026
\(988\) 6.25642 0.199043
\(989\) −78.1433 −2.48481
\(990\) 45.5234 1.44683
\(991\) −0.0597053 −0.00189660 −0.000948301 1.00000i \(-0.500302\pi\)
−0.000948301 1.00000i \(0.500302\pi\)
\(992\) 86.9711 2.76133
\(993\) −13.6167 −0.432112
\(994\) −54.2182 −1.71970
\(995\) 23.6574 0.749990
\(996\) −25.8343 −0.818592
\(997\) −46.2885 −1.46597 −0.732986 0.680243i \(-0.761874\pi\)
−0.732986 + 0.680243i \(0.761874\pi\)
\(998\) −70.1354 −2.22010
\(999\) 26.9474 0.852577
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 8045.2.a.b.1.3 126
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8045.2.a.b.1.3 126 1.1 even 1 trivial