Properties

Label 2010.2.a.r.1.3
Level $2010$
Weight $2$
Character 2010.1
Self dual yes
Analytic conductor $16.050$
Analytic rank $1$
Dimension $4$
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] = [2010,2,Mod(1,2010)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2010, 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("2010.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2010 = 2 \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2010.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: \(16.0499308063\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.70292.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} + 10x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.46640\) of defining polynomial
Character \(\chi\) \(=\) 2010.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +2.08313 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +2.08313 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +2.20503 q^{11} -1.00000 q^{12} -4.28816 q^{13} -2.08313 q^{14} +1.00000 q^{15} +1.00000 q^{16} -4.93280 q^{17} -1.00000 q^{18} +4.93280 q^{19} -1.00000 q^{20} -2.08313 q^{21} -2.20503 q^{22} -3.35536 q^{23} +1.00000 q^{24} +1.00000 q^{25} +4.28816 q^{26} -1.00000 q^{27} +2.08313 q^{28} -2.81090 q^{29} -1.00000 q^{30} -2.28816 q^{31} -1.00000 q^{32} -2.20503 q^{33} +4.93280 q^{34} -2.08313 q^{35} +1.00000 q^{36} -0.727768 q^{37} -4.93280 q^{38} +4.28816 q^{39} +1.00000 q^{40} -1.47726 q^{41} +2.08313 q^{42} +4.41006 q^{43} +2.20503 q^{44} -1.00000 q^{45} +3.35536 q^{46} +9.38722 q^{47} -1.00000 q^{48} -2.66057 q^{49} -1.00000 q^{50} +4.93280 q^{51} -4.28816 q^{52} -6.16626 q^{53} +1.00000 q^{54} -2.20503 q^{55} -2.08313 q^{56} -4.93280 q^{57} +2.81090 q^{58} -1.47726 q^{59} +1.00000 q^{60} +4.08313 q^{61} +2.28816 q^{62} +2.08313 q^{63} +1.00000 q^{64} +4.28816 q^{65} +2.20503 q^{66} -1.00000 q^{67} -4.93280 q^{68} +3.35536 q^{69} +2.08313 q^{70} -4.84967 q^{71} -1.00000 q^{72} -0.544464 q^{73} +0.727768 q^{74} -1.00000 q^{75} +4.93280 q^{76} +4.59337 q^{77} -4.28816 q^{78} +4.83263 q^{79} -1.00000 q^{80} +1.00000 q^{81} +1.47726 q^{82} -16.5934 q^{83} -2.08313 q^{84} +4.93280 q^{85} -4.41006 q^{86} +2.81090 q^{87} -2.20503 q^{88} +2.18330 q^{89} +1.00000 q^{90} -8.93280 q^{91} -3.35536 q^{92} +2.28816 q^{93} -9.38722 q^{94} -4.93280 q^{95} +1.00000 q^{96} +2.20503 q^{97} +2.66057 q^{98} +2.20503 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{5} + 4 q^{6} + q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{5} + 4 q^{6} + q^{7} - 4 q^{8} + 4 q^{9} + 4 q^{10} - 3 q^{11} - 4 q^{12} + 2 q^{13} - q^{14} + 4 q^{15} + 4 q^{16} - 2 q^{17} - 4 q^{18} + 2 q^{19} - 4 q^{20} - q^{21} + 3 q^{22} - 12 q^{23} + 4 q^{24} + 4 q^{25} - 2 q^{26} - 4 q^{27} + q^{28} + 2 q^{29} - 4 q^{30} + 10 q^{31} - 4 q^{32} + 3 q^{33} + 2 q^{34} - q^{35} + 4 q^{36} + 3 q^{37} - 2 q^{38} - 2 q^{39} + 4 q^{40} + q^{42} - 6 q^{43} - 3 q^{44} - 4 q^{45} + 12 q^{46} - 14 q^{47} - 4 q^{48} + 13 q^{49} - 4 q^{50} + 2 q^{51} + 2 q^{52} - 10 q^{53} + 4 q^{54} + 3 q^{55} - q^{56} - 2 q^{57} - 2 q^{58} + 4 q^{60} + 9 q^{61} - 10 q^{62} + q^{63} + 4 q^{64} - 2 q^{65} - 3 q^{66} - 4 q^{67} - 2 q^{68} + 12 q^{69} + q^{70} - 9 q^{71} - 4 q^{72} - 14 q^{73} - 3 q^{74} - 4 q^{75} + 2 q^{76} - 23 q^{77} + 2 q^{78} + 12 q^{79} - 4 q^{80} + 4 q^{81} - 25 q^{83} - q^{84} + 2 q^{85} + 6 q^{86} - 2 q^{87} + 3 q^{88} - 9 q^{89} + 4 q^{90} - 18 q^{91} - 12 q^{92} - 10 q^{93} + 14 q^{94} - 2 q^{95} + 4 q^{96} - 3 q^{97} - 13 q^{98} - 3 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) 2.08313 0.787349 0.393675 0.919250i \(-0.371204\pi\)
0.393675 + 0.919250i \(0.371204\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 2.20503 0.664842 0.332421 0.943131i \(-0.392134\pi\)
0.332421 + 0.943131i \(0.392134\pi\)
\(12\) −1.00000 −0.288675
\(13\) −4.28816 −1.18932 −0.594661 0.803976i \(-0.702714\pi\)
−0.594661 + 0.803976i \(0.702714\pi\)
\(14\) −2.08313 −0.556740
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −4.93280 −1.19638 −0.598190 0.801354i \(-0.704113\pi\)
−0.598190 + 0.801354i \(0.704113\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.93280 1.13166 0.565831 0.824521i \(-0.308555\pi\)
0.565831 + 0.824521i \(0.308555\pi\)
\(20\) −1.00000 −0.223607
\(21\) −2.08313 −0.454576
\(22\) −2.20503 −0.470114
\(23\) −3.35536 −0.699641 −0.349821 0.936817i \(-0.613757\pi\)
−0.349821 + 0.936817i \(0.613757\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 4.28816 0.840978
\(27\) −1.00000 −0.192450
\(28\) 2.08313 0.393675
\(29\) −2.81090 −0.521971 −0.260985 0.965343i \(-0.584047\pi\)
−0.260985 + 0.965343i \(0.584047\pi\)
\(30\) −1.00000 −0.182574
\(31\) −2.28816 −0.410966 −0.205483 0.978661i \(-0.565877\pi\)
−0.205483 + 0.978661i \(0.565877\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.20503 −0.383847
\(34\) 4.93280 0.845968
\(35\) −2.08313 −0.352113
\(36\) 1.00000 0.166667
\(37\) −0.727768 −0.119644 −0.0598222 0.998209i \(-0.519053\pi\)
−0.0598222 + 0.998209i \(0.519053\pi\)
\(38\) −4.93280 −0.800206
\(39\) 4.28816 0.686656
\(40\) 1.00000 0.158114
\(41\) −1.47726 −0.230710 −0.115355 0.993324i \(-0.536801\pi\)
−0.115355 + 0.993324i \(0.536801\pi\)
\(42\) 2.08313 0.321434
\(43\) 4.41006 0.672529 0.336264 0.941768i \(-0.390836\pi\)
0.336264 + 0.941768i \(0.390836\pi\)
\(44\) 2.20503 0.332421
\(45\) −1.00000 −0.149071
\(46\) 3.35536 0.494721
\(47\) 9.38722 1.36927 0.684634 0.728887i \(-0.259962\pi\)
0.684634 + 0.728887i \(0.259962\pi\)
\(48\) −1.00000 −0.144338
\(49\) −2.66057 −0.380081
\(50\) −1.00000 −0.141421
\(51\) 4.93280 0.690730
\(52\) −4.28816 −0.594661
\(53\) −6.16626 −0.847001 −0.423500 0.905896i \(-0.639199\pi\)
−0.423500 + 0.905896i \(0.639199\pi\)
\(54\) 1.00000 0.136083
\(55\) −2.20503 −0.297326
\(56\) −2.08313 −0.278370
\(57\) −4.93280 −0.653365
\(58\) 2.81090 0.369089
\(59\) −1.47726 −0.192323 −0.0961617 0.995366i \(-0.530657\pi\)
−0.0961617 + 0.995366i \(0.530657\pi\)
\(60\) 1.00000 0.129099
\(61\) 4.08313 0.522791 0.261396 0.965232i \(-0.415817\pi\)
0.261396 + 0.965232i \(0.415817\pi\)
\(62\) 2.28816 0.290597
\(63\) 2.08313 0.262450
\(64\) 1.00000 0.125000
\(65\) 4.28816 0.531881
\(66\) 2.20503 0.271421
\(67\) −1.00000 −0.122169
\(68\) −4.93280 −0.598190
\(69\) 3.35536 0.403938
\(70\) 2.08313 0.248982
\(71\) −4.84967 −0.575550 −0.287775 0.957698i \(-0.592916\pi\)
−0.287775 + 0.957698i \(0.592916\pi\)
\(72\) −1.00000 −0.117851
\(73\) −0.544464 −0.0637246 −0.0318623 0.999492i \(-0.510144\pi\)
−0.0318623 + 0.999492i \(0.510144\pi\)
\(74\) 0.727768 0.0846013
\(75\) −1.00000 −0.115470
\(76\) 4.93280 0.565831
\(77\) 4.59337 0.523463
\(78\) −4.28816 −0.485539
\(79\) 4.83263 0.543713 0.271856 0.962338i \(-0.412362\pi\)
0.271856 + 0.962338i \(0.412362\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 1.47726 0.163137
\(83\) −16.5934 −1.82136 −0.910679 0.413114i \(-0.864441\pi\)
−0.910679 + 0.413114i \(0.864441\pi\)
\(84\) −2.08313 −0.227288
\(85\) 4.93280 0.535037
\(86\) −4.41006 −0.475549
\(87\) 2.81090 0.301360
\(88\) −2.20503 −0.235057
\(89\) 2.18330 0.231430 0.115715 0.993282i \(-0.463084\pi\)
0.115715 + 0.993282i \(0.463084\pi\)
\(90\) 1.00000 0.105409
\(91\) −8.93280 −0.936412
\(92\) −3.35536 −0.349821
\(93\) 2.28816 0.237271
\(94\) −9.38722 −0.968218
\(95\) −4.93280 −0.506095
\(96\) 1.00000 0.102062
\(97\) 2.20503 0.223887 0.111944 0.993715i \(-0.464292\pi\)
0.111944 + 0.993715i \(0.464292\pi\)
\(98\) 2.66057 0.268758
\(99\) 2.20503 0.221614
\(100\) 1.00000 0.100000
\(101\) −9.13783 −0.909248 −0.454624 0.890683i \(-0.650226\pi\)
−0.454624 + 0.890683i \(0.650226\pi\)
\(102\) −4.93280 −0.488420
\(103\) 0.410065 0.0404049 0.0202024 0.999796i \(-0.493569\pi\)
0.0202024 + 0.999796i \(0.493569\pi\)
\(104\) 4.28816 0.420489
\(105\) 2.08313 0.203293
\(106\) 6.16626 0.598920
\(107\) 0.576325 0.0557154 0.0278577 0.999612i \(-0.491131\pi\)
0.0278577 + 0.999612i \(0.491131\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 5.78247 0.553860 0.276930 0.960890i \(-0.410683\pi\)
0.276930 + 0.960890i \(0.410683\pi\)
\(110\) 2.20503 0.210242
\(111\) 0.727768 0.0690767
\(112\) 2.08313 0.196837
\(113\) −13.8268 −1.30072 −0.650359 0.759627i \(-0.725382\pi\)
−0.650359 + 0.759627i \(0.725382\pi\)
\(114\) 4.93280 0.461999
\(115\) 3.35536 0.312889
\(116\) −2.81090 −0.260985
\(117\) −4.28816 −0.396441
\(118\) 1.47726 0.135993
\(119\) −10.2757 −0.941969
\(120\) −1.00000 −0.0912871
\(121\) −6.13783 −0.557985
\(122\) −4.08313 −0.369669
\(123\) 1.47726 0.133200
\(124\) −2.28816 −0.205483
\(125\) −1.00000 −0.0894427
\(126\) −2.08313 −0.185580
\(127\) −11.1208 −0.986810 −0.493405 0.869800i \(-0.664248\pi\)
−0.493405 + 0.869800i \(0.664248\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.41006 −0.388285
\(130\) −4.28816 −0.376097
\(131\) −17.6435 −1.54152 −0.770761 0.637124i \(-0.780124\pi\)
−0.770761 + 0.637124i \(0.780124\pi\)
\(132\) −2.20503 −0.191923
\(133\) 10.2757 0.891013
\(134\) 1.00000 0.0863868
\(135\) 1.00000 0.0860663
\(136\) 4.93280 0.422984
\(137\) −4.72777 −0.403920 −0.201960 0.979394i \(-0.564731\pi\)
−0.201960 + 0.979394i \(0.564731\pi\)
\(138\) −3.35536 −0.285627
\(139\) −20.0855 −1.70363 −0.851813 0.523846i \(-0.824497\pi\)
−0.851813 + 0.523846i \(0.824497\pi\)
\(140\) −2.08313 −0.176057
\(141\) −9.38722 −0.790547
\(142\) 4.84967 0.406975
\(143\) −9.45554 −0.790712
\(144\) 1.00000 0.0833333
\(145\) 2.81090 0.233432
\(146\) 0.544464 0.0450601
\(147\) 2.66057 0.219440
\(148\) −0.727768 −0.0598222
\(149\) −2.81090 −0.230278 −0.115139 0.993349i \(-0.536731\pi\)
−0.115139 + 0.993349i \(0.536731\pi\)
\(150\) 1.00000 0.0816497
\(151\) 11.7142 0.953285 0.476642 0.879097i \(-0.341854\pi\)
0.476642 + 0.879097i \(0.341854\pi\)
\(152\) −4.93280 −0.400103
\(153\) −4.93280 −0.398793
\(154\) −4.59337 −0.370144
\(155\) 2.28816 0.183790
\(156\) 4.28816 0.343328
\(157\) −20.1981 −1.61199 −0.805993 0.591925i \(-0.798368\pi\)
−0.805993 + 0.591925i \(0.798368\pi\)
\(158\) −4.83263 −0.384463
\(159\) 6.16626 0.489016
\(160\) 1.00000 0.0790569
\(161\) −6.98966 −0.550862
\(162\) −1.00000 −0.0785674
\(163\) −10.7278 −0.840264 −0.420132 0.907463i \(-0.638016\pi\)
−0.420132 + 0.907463i \(0.638016\pi\)
\(164\) −1.47726 −0.115355
\(165\) 2.20503 0.171662
\(166\) 16.5934 1.28790
\(167\) 3.93169 0.304243 0.152122 0.988362i \(-0.451389\pi\)
0.152122 + 0.988362i \(0.451389\pi\)
\(168\) 2.08313 0.160717
\(169\) 5.38834 0.414487
\(170\) −4.93280 −0.378329
\(171\) 4.93280 0.377221
\(172\) 4.41006 0.336264
\(173\) 9.04779 0.687891 0.343945 0.938990i \(-0.388236\pi\)
0.343945 + 0.938990i \(0.388236\pi\)
\(174\) −2.81090 −0.213094
\(175\) 2.08313 0.157470
\(176\) 2.20503 0.166211
\(177\) 1.47726 0.111038
\(178\) −2.18330 −0.163646
\(179\) 13.6754 1.02215 0.511073 0.859537i \(-0.329248\pi\)
0.511073 + 0.859537i \(0.329248\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 0.876984 0.0651857 0.0325929 0.999469i \(-0.489624\pi\)
0.0325929 + 0.999469i \(0.489624\pi\)
\(182\) 8.93280 0.662143
\(183\) −4.08313 −0.301834
\(184\) 3.35536 0.247361
\(185\) 0.727768 0.0535066
\(186\) −2.28816 −0.167776
\(187\) −10.8770 −0.795404
\(188\) 9.38722 0.684634
\(189\) −2.08313 −0.151525
\(190\) 4.93280 0.357863
\(191\) −16.4101 −1.18739 −0.593695 0.804690i \(-0.702332\pi\)
−0.593695 + 0.804690i \(0.702332\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −3.17660 −0.228657 −0.114329 0.993443i \(-0.536472\pi\)
−0.114329 + 0.993443i \(0.536472\pi\)
\(194\) −2.20503 −0.158312
\(195\) −4.28816 −0.307082
\(196\) −2.66057 −0.190041
\(197\) −10.5763 −0.753532 −0.376766 0.926308i \(-0.622964\pi\)
−0.376766 + 0.926308i \(0.622964\pi\)
\(198\) −2.20503 −0.156705
\(199\) 8.59337 0.609168 0.304584 0.952486i \(-0.401483\pi\)
0.304584 + 0.952486i \(0.401483\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 1.00000 0.0705346
\(202\) 9.13783 0.642936
\(203\) −5.85547 −0.410973
\(204\) 4.93280 0.345365
\(205\) 1.47726 0.103177
\(206\) −0.410065 −0.0285705
\(207\) −3.35536 −0.233214
\(208\) −4.28816 −0.297331
\(209\) 10.8770 0.752377
\(210\) −2.08313 −0.143750
\(211\) −12.8553 −0.884992 −0.442496 0.896770i \(-0.645907\pi\)
−0.442496 + 0.896770i \(0.645907\pi\)
\(212\) −6.16626 −0.423500
\(213\) 4.84967 0.332294
\(214\) −0.576325 −0.0393968
\(215\) −4.41006 −0.300764
\(216\) 1.00000 0.0680414
\(217\) −4.76654 −0.323574
\(218\) −5.78247 −0.391638
\(219\) 0.544464 0.0367914
\(220\) −2.20503 −0.148663
\(221\) 21.1526 1.42288
\(222\) −0.727768 −0.0488446
\(223\) 18.8520 1.26242 0.631211 0.775611i \(-0.282558\pi\)
0.631211 + 0.775611i \(0.282558\pi\)
\(224\) −2.08313 −0.139185
\(225\) 1.00000 0.0666667
\(226\) 13.8268 0.919747
\(227\) −22.1094 −1.46745 −0.733726 0.679445i \(-0.762221\pi\)
−0.733726 + 0.679445i \(0.762221\pi\)
\(228\) −4.93280 −0.326683
\(229\) 0.326934 0.0216044 0.0108022 0.999942i \(-0.496561\pi\)
0.0108022 + 0.999942i \(0.496561\pi\)
\(230\) −3.35536 −0.221246
\(231\) −4.59337 −0.302222
\(232\) 2.81090 0.184545
\(233\) −2.14922 −0.140800 −0.0703999 0.997519i \(-0.522428\pi\)
−0.0703999 + 0.997519i \(0.522428\pi\)
\(234\) 4.28816 0.280326
\(235\) −9.38722 −0.612355
\(236\) −1.47726 −0.0961617
\(237\) −4.83263 −0.313913
\(238\) 10.2757 0.666072
\(239\) 11.1208 0.719344 0.359672 0.933079i \(-0.382889\pi\)
0.359672 + 0.933079i \(0.382889\pi\)
\(240\) 1.00000 0.0645497
\(241\) −7.15956 −0.461188 −0.230594 0.973050i \(-0.574067\pi\)
−0.230594 + 0.973050i \(0.574067\pi\)
\(242\) 6.13783 0.394555
\(243\) −1.00000 −0.0641500
\(244\) 4.08313 0.261396
\(245\) 2.66057 0.169978
\(246\) −1.47726 −0.0941869
\(247\) −21.1526 −1.34591
\(248\) 2.28816 0.145298
\(249\) 16.5934 1.05156
\(250\) 1.00000 0.0632456
\(251\) −10.7278 −0.677131 −0.338565 0.940943i \(-0.609942\pi\)
−0.338565 + 0.940943i \(0.609942\pi\)
\(252\) 2.08313 0.131225
\(253\) −7.39868 −0.465151
\(254\) 11.1208 0.697780
\(255\) −4.93280 −0.308904
\(256\) 1.00000 0.0625000
\(257\) −20.4202 −1.27378 −0.636888 0.770956i \(-0.719779\pi\)
−0.636888 + 0.770956i \(0.719779\pi\)
\(258\) 4.41006 0.274559
\(259\) −1.51604 −0.0942019
\(260\) 4.28816 0.265941
\(261\) −2.81090 −0.173990
\(262\) 17.6435 1.09002
\(263\) −3.59917 −0.221934 −0.110967 0.993824i \(-0.535395\pi\)
−0.110967 + 0.993824i \(0.535395\pi\)
\(264\) 2.20503 0.135710
\(265\) 6.16626 0.378790
\(266\) −10.2757 −0.630041
\(267\) −2.18330 −0.133616
\(268\) −1.00000 −0.0610847
\(269\) 8.94530 0.545404 0.272702 0.962098i \(-0.412083\pi\)
0.272702 + 0.962098i \(0.412083\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −7.41118 −0.450197 −0.225099 0.974336i \(-0.572270\pi\)
−0.225099 + 0.974336i \(0.572270\pi\)
\(272\) −4.93280 −0.299095
\(273\) 8.93280 0.540638
\(274\) 4.72777 0.285615
\(275\) 2.20503 0.132968
\(276\) 3.35536 0.201969
\(277\) 16.2586 0.976886 0.488443 0.872596i \(-0.337565\pi\)
0.488443 + 0.872596i \(0.337565\pi\)
\(278\) 20.0855 1.20465
\(279\) −2.28816 −0.136989
\(280\) 2.08313 0.124491
\(281\) 19.1526 1.14255 0.571276 0.820758i \(-0.306449\pi\)
0.571276 + 0.820758i \(0.306449\pi\)
\(282\) 9.38722 0.559001
\(283\) −24.2905 −1.44392 −0.721960 0.691935i \(-0.756758\pi\)
−0.721960 + 0.691935i \(0.756758\pi\)
\(284\) −4.84967 −0.287775
\(285\) 4.93280 0.292194
\(286\) 9.45554 0.559118
\(287\) −3.07733 −0.181649
\(288\) −1.00000 −0.0589256
\(289\) 7.33252 0.431325
\(290\) −2.81090 −0.165062
\(291\) −2.20503 −0.129261
\(292\) −0.544464 −0.0318623
\(293\) 13.9705 0.816163 0.408081 0.912946i \(-0.366198\pi\)
0.408081 + 0.912946i \(0.366198\pi\)
\(294\) −2.66057 −0.155168
\(295\) 1.47726 0.0860096
\(296\) 0.727768 0.0423007
\(297\) −2.20503 −0.127949
\(298\) 2.81090 0.162831
\(299\) 14.3883 0.832099
\(300\) −1.00000 −0.0577350
\(301\) 9.18674 0.529515
\(302\) −11.7142 −0.674074
\(303\) 9.13783 0.524955
\(304\) 4.93280 0.282916
\(305\) −4.08313 −0.233799
\(306\) 4.93280 0.281989
\(307\) −20.3144 −1.15941 −0.579703 0.814828i \(-0.696831\pi\)
−0.579703 + 0.814828i \(0.696831\pi\)
\(308\) 4.59337 0.261732
\(309\) −0.410065 −0.0233278
\(310\) −2.28816 −0.129959
\(311\) −7.75620 −0.439814 −0.219907 0.975521i \(-0.570575\pi\)
−0.219907 + 0.975521i \(0.570575\pi\)
\(312\) −4.28816 −0.242769
\(313\) −20.8132 −1.17643 −0.588216 0.808704i \(-0.700170\pi\)
−0.588216 + 0.808704i \(0.700170\pi\)
\(314\) 20.1981 1.13985
\(315\) −2.08313 −0.117371
\(316\) 4.83263 0.271856
\(317\) 0.488511 0.0274375 0.0137188 0.999906i \(-0.495633\pi\)
0.0137188 + 0.999906i \(0.495633\pi\)
\(318\) −6.16626 −0.345787
\(319\) −6.19812 −0.347028
\(320\) −1.00000 −0.0559017
\(321\) −0.576325 −0.0321673
\(322\) 6.98966 0.389518
\(323\) −24.3325 −1.35390
\(324\) 1.00000 0.0555556
\(325\) −4.28816 −0.237864
\(326\) 10.7278 0.594156
\(327\) −5.78247 −0.319771
\(328\) 1.47726 0.0815683
\(329\) 19.5548 1.07809
\(330\) −2.20503 −0.121383
\(331\) −23.0570 −1.26733 −0.633664 0.773608i \(-0.718450\pi\)
−0.633664 + 0.773608i \(0.718450\pi\)
\(332\) −16.5934 −0.910679
\(333\) −0.727768 −0.0398815
\(334\) −3.93169 −0.215132
\(335\) 1.00000 0.0546358
\(336\) −2.08313 −0.113644
\(337\) 30.3293 1.65214 0.826070 0.563568i \(-0.190572\pi\)
0.826070 + 0.563568i \(0.190572\pi\)
\(338\) −5.38834 −0.293087
\(339\) 13.8268 0.750970
\(340\) 4.93280 0.267519
\(341\) −5.04547 −0.273228
\(342\) −4.93280 −0.266735
\(343\) −20.1242 −1.08661
\(344\) −4.41006 −0.237775
\(345\) −3.35536 −0.180647
\(346\) −9.04779 −0.486412
\(347\) 17.5432 0.941769 0.470885 0.882195i \(-0.343935\pi\)
0.470885 + 0.882195i \(0.343935\pi\)
\(348\) 2.81090 0.150680
\(349\) 14.8201 0.793303 0.396652 0.917969i \(-0.370172\pi\)
0.396652 + 0.917969i \(0.370172\pi\)
\(350\) −2.08313 −0.111348
\(351\) 4.28816 0.228885
\(352\) −2.20503 −0.117529
\(353\) 18.7665 0.998842 0.499421 0.866359i \(-0.333546\pi\)
0.499421 + 0.866359i \(0.333546\pi\)
\(354\) −1.47726 −0.0785157
\(355\) 4.84967 0.257394
\(356\) 2.18330 0.115715
\(357\) 10.2757 0.543846
\(358\) −13.6754 −0.722767
\(359\) −6.43738 −0.339752 −0.169876 0.985465i \(-0.554337\pi\)
−0.169876 + 0.985465i \(0.554337\pi\)
\(360\) 1.00000 0.0527046
\(361\) 5.33252 0.280659
\(362\) −0.876984 −0.0460933
\(363\) 6.13783 0.322153
\(364\) −8.93280 −0.468206
\(365\) 0.544464 0.0284985
\(366\) 4.08313 0.213429
\(367\) −18.1298 −0.946368 −0.473184 0.880964i \(-0.656895\pi\)
−0.473184 + 0.880964i \(0.656895\pi\)
\(368\) −3.35536 −0.174910
\(369\) −1.47726 −0.0769033
\(370\) −0.727768 −0.0378349
\(371\) −12.8451 −0.666886
\(372\) 2.28816 0.118636
\(373\) 12.6207 0.653474 0.326737 0.945115i \(-0.394051\pi\)
0.326737 + 0.945115i \(0.394051\pi\)
\(374\) 10.8770 0.562435
\(375\) 1.00000 0.0516398
\(376\) −9.38722 −0.484109
\(377\) 12.0536 0.620791
\(378\) 2.08313 0.107145
\(379\) 28.4180 1.45973 0.729867 0.683590i \(-0.239582\pi\)
0.729867 + 0.683590i \(0.239582\pi\)
\(380\) −4.93280 −0.253047
\(381\) 11.1208 0.569735
\(382\) 16.4101 0.839612
\(383\) 13.6048 0.695170 0.347585 0.937648i \(-0.387002\pi\)
0.347585 + 0.937648i \(0.387002\pi\)
\(384\) 1.00000 0.0510310
\(385\) −4.59337 −0.234100
\(386\) 3.17660 0.161685
\(387\) 4.41006 0.224176
\(388\) 2.20503 0.111944
\(389\) −23.6948 −1.20137 −0.600686 0.799485i \(-0.705106\pi\)
−0.600686 + 0.799485i \(0.705106\pi\)
\(390\) 4.28816 0.217140
\(391\) 16.5513 0.837037
\(392\) 2.66057 0.134379
\(393\) 17.6435 0.889998
\(394\) 10.5763 0.532828
\(395\) −4.83263 −0.243156
\(396\) 2.20503 0.110807
\(397\) −4.65022 −0.233388 −0.116694 0.993168i \(-0.537230\pi\)
−0.116694 + 0.993168i \(0.537230\pi\)
\(398\) −8.59337 −0.430747
\(399\) −10.2757 −0.514427
\(400\) 1.00000 0.0500000
\(401\) 23.5193 1.17450 0.587248 0.809407i \(-0.300211\pi\)
0.587248 + 0.809407i \(0.300211\pi\)
\(402\) −1.00000 −0.0498755
\(403\) 9.81201 0.488771
\(404\) −9.13783 −0.454624
\(405\) −1.00000 −0.0496904
\(406\) 5.85547 0.290602
\(407\) −1.60475 −0.0795446
\(408\) −4.93280 −0.244210
\(409\) 18.5763 0.918540 0.459270 0.888297i \(-0.348111\pi\)
0.459270 + 0.888297i \(0.348111\pi\)
\(410\) −1.47726 −0.0729569
\(411\) 4.72777 0.233204
\(412\) 0.410065 0.0202024
\(413\) −3.07733 −0.151426
\(414\) 3.35536 0.164907
\(415\) 16.5934 0.814536
\(416\) 4.28816 0.210244
\(417\) 20.0855 0.983589
\(418\) −10.8770 −0.532011
\(419\) −1.97605 −0.0965361 −0.0482681 0.998834i \(-0.515370\pi\)
−0.0482681 + 0.998834i \(0.515370\pi\)
\(420\) 2.08313 0.101646
\(421\) 9.92246 0.483591 0.241795 0.970327i \(-0.422264\pi\)
0.241795 + 0.970327i \(0.422264\pi\)
\(422\) 12.8553 0.625784
\(423\) 9.38722 0.456422
\(424\) 6.16626 0.299460
\(425\) −4.93280 −0.239276
\(426\) −4.84967 −0.234967
\(427\) 8.50569 0.411619
\(428\) 0.576325 0.0278577
\(429\) 9.45554 0.456518
\(430\) 4.41006 0.212672
\(431\) 2.38275 0.114773 0.0573865 0.998352i \(-0.481723\pi\)
0.0573865 + 0.998352i \(0.481723\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −20.9965 −1.00903 −0.504514 0.863403i \(-0.668328\pi\)
−0.504514 + 0.863403i \(0.668328\pi\)
\(434\) 4.76654 0.228801
\(435\) −2.81090 −0.134772
\(436\) 5.78247 0.276930
\(437\) −16.5513 −0.791758
\(438\) −0.544464 −0.0260155
\(439\) 17.0034 0.811530 0.405765 0.913978i \(-0.367005\pi\)
0.405765 + 0.913978i \(0.367005\pi\)
\(440\) 2.20503 0.105121
\(441\) −2.66057 −0.126694
\(442\) −21.1526 −1.00613
\(443\) −13.8656 −0.658775 −0.329387 0.944195i \(-0.606842\pi\)
−0.329387 + 0.944195i \(0.606842\pi\)
\(444\) 0.727768 0.0345384
\(445\) −2.18330 −0.103499
\(446\) −18.8520 −0.892668
\(447\) 2.81090 0.132951
\(448\) 2.08313 0.0984186
\(449\) 39.0922 1.84487 0.922436 0.386149i \(-0.126195\pi\)
0.922436 + 0.386149i \(0.126195\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −3.25741 −0.153386
\(452\) −13.8268 −0.650359
\(453\) −11.7142 −0.550379
\(454\) 22.1094 1.03765
\(455\) 8.93280 0.418776
\(456\) 4.93280 0.231000
\(457\) −6.68900 −0.312898 −0.156449 0.987686i \(-0.550005\pi\)
−0.156449 + 0.987686i \(0.550005\pi\)
\(458\) −0.326934 −0.0152766
\(459\) 4.93280 0.230243
\(460\) 3.35536 0.156445
\(461\) −22.0545 −1.02718 −0.513590 0.858036i \(-0.671685\pi\)
−0.513590 + 0.858036i \(0.671685\pi\)
\(462\) 4.59337 0.213703
\(463\) 39.4191 1.83196 0.915980 0.401224i \(-0.131415\pi\)
0.915980 + 0.401224i \(0.131415\pi\)
\(464\) −2.81090 −0.130493
\(465\) −2.28816 −0.106111
\(466\) 2.14922 0.0995605
\(467\) 10.0148 0.463430 0.231715 0.972784i \(-0.425566\pi\)
0.231715 + 0.972784i \(0.425566\pi\)
\(468\) −4.28816 −0.198220
\(469\) −2.08313 −0.0961900
\(470\) 9.38722 0.433000
\(471\) 20.1981 0.930680
\(472\) 1.47726 0.0679966
\(473\) 9.72433 0.447125
\(474\) 4.83263 0.221970
\(475\) 4.93280 0.226332
\(476\) −10.2757 −0.470984
\(477\) −6.16626 −0.282334
\(478\) −11.1208 −0.508653
\(479\) 22.8474 1.04393 0.521963 0.852968i \(-0.325200\pi\)
0.521963 + 0.852968i \(0.325200\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 3.12079 0.142296
\(482\) 7.15956 0.326109
\(483\) 6.98966 0.318040
\(484\) −6.13783 −0.278992
\(485\) −2.20503 −0.100125
\(486\) 1.00000 0.0453609
\(487\) 35.8896 1.62632 0.813158 0.582044i \(-0.197747\pi\)
0.813158 + 0.582044i \(0.197747\pi\)
\(488\) −4.08313 −0.184835
\(489\) 10.7278 0.485126
\(490\) −2.66057 −0.120192
\(491\) −35.2403 −1.59037 −0.795187 0.606364i \(-0.792627\pi\)
−0.795187 + 0.606364i \(0.792627\pi\)
\(492\) 1.47726 0.0666002
\(493\) 13.8656 0.624475
\(494\) 21.1526 0.951703
\(495\) −2.20503 −0.0991088
\(496\) −2.28816 −0.102742
\(497\) −10.1025 −0.453159
\(498\) −16.5934 −0.743567
\(499\) 8.17094 0.365782 0.182891 0.983133i \(-0.441455\pi\)
0.182891 + 0.983133i \(0.441455\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −3.93169 −0.175655
\(502\) 10.7278 0.478804
\(503\) −0.470564 −0.0209814 −0.0104907 0.999945i \(-0.503339\pi\)
−0.0104907 + 0.999945i \(0.503339\pi\)
\(504\) −2.08313 −0.0927900
\(505\) 9.13783 0.406628
\(506\) 7.39868 0.328912
\(507\) −5.38834 −0.239304
\(508\) −11.1208 −0.493405
\(509\) 1.02284 0.0453366 0.0226683 0.999743i \(-0.492784\pi\)
0.0226683 + 0.999743i \(0.492784\pi\)
\(510\) 4.93280 0.218428
\(511\) −1.13419 −0.0501735
\(512\) −1.00000 −0.0441942
\(513\) −4.93280 −0.217788
\(514\) 20.4202 0.900696
\(515\) −0.410065 −0.0180696
\(516\) −4.41006 −0.194142
\(517\) 20.6991 0.910347
\(518\) 1.51604 0.0666108
\(519\) −9.04779 −0.397154
\(520\) −4.28816 −0.188048
\(521\) −0.841857 −0.0368824 −0.0184412 0.999830i \(-0.505870\pi\)
−0.0184412 + 0.999830i \(0.505870\pi\)
\(522\) 2.81090 0.123030
\(523\) 38.8987 1.70092 0.850460 0.526040i \(-0.176324\pi\)
0.850460 + 0.526040i \(0.176324\pi\)
\(524\) −17.6435 −0.770761
\(525\) −2.08313 −0.0909153
\(526\) 3.59917 0.156931
\(527\) 11.2870 0.491672
\(528\) −2.20503 −0.0959617
\(529\) −11.7415 −0.510502
\(530\) −6.16626 −0.267845
\(531\) −1.47726 −0.0641078
\(532\) 10.2757 0.445507
\(533\) 6.33475 0.274388
\(534\) 2.18330 0.0944808
\(535\) −0.576325 −0.0249167
\(536\) 1.00000 0.0431934
\(537\) −13.6754 −0.590136
\(538\) −8.94530 −0.385659
\(539\) −5.86664 −0.252694
\(540\) 1.00000 0.0430331
\(541\) −11.2644 −0.484295 −0.242148 0.970239i \(-0.577852\pi\)
−0.242148 + 0.970239i \(0.577852\pi\)
\(542\) 7.41118 0.318337
\(543\) −0.876984 −0.0376350
\(544\) 4.93280 0.211492
\(545\) −5.78247 −0.247694
\(546\) −8.93280 −0.382289
\(547\) 33.1092 1.41565 0.707823 0.706389i \(-0.249677\pi\)
0.707823 + 0.706389i \(0.249677\pi\)
\(548\) −4.72777 −0.201960
\(549\) 4.08313 0.174264
\(550\) −2.20503 −0.0940229
\(551\) −13.8656 −0.590694
\(552\) −3.35536 −0.142814
\(553\) 10.0670 0.428092
\(554\) −16.2586 −0.690763
\(555\) −0.727768 −0.0308920
\(556\) −20.0855 −0.851813
\(557\) 14.1103 0.597873 0.298936 0.954273i \(-0.403368\pi\)
0.298936 + 0.954273i \(0.403368\pi\)
\(558\) 2.28816 0.0968656
\(559\) −18.9111 −0.799853
\(560\) −2.08313 −0.0880283
\(561\) 10.8770 0.459227
\(562\) −19.1526 −0.807906
\(563\) −16.4101 −0.691602 −0.345801 0.938308i \(-0.612393\pi\)
−0.345801 + 0.938308i \(0.612393\pi\)
\(564\) −9.38722 −0.395273
\(565\) 13.8268 0.581699
\(566\) 24.2905 1.02101
\(567\) 2.08313 0.0874832
\(568\) 4.84967 0.203488
\(569\) 30.7255 1.28808 0.644041 0.764991i \(-0.277257\pi\)
0.644041 + 0.764991i \(0.277257\pi\)
\(570\) −4.93280 −0.206612
\(571\) −9.86560 −0.412863 −0.206431 0.978461i \(-0.566185\pi\)
−0.206431 + 0.978461i \(0.566185\pi\)
\(572\) −9.45554 −0.395356
\(573\) 16.4101 0.685540
\(574\) 3.07733 0.128445
\(575\) −3.35536 −0.139928
\(576\) 1.00000 0.0416667
\(577\) −2.86685 −0.119349 −0.0596743 0.998218i \(-0.519006\pi\)
−0.0596743 + 0.998218i \(0.519006\pi\)
\(578\) −7.33252 −0.304993
\(579\) 3.17660 0.132015
\(580\) 2.81090 0.116716
\(581\) −34.5661 −1.43405
\(582\) 2.20503 0.0914015
\(583\) −13.5968 −0.563122
\(584\) 0.544464 0.0225301
\(585\) 4.28816 0.177294
\(586\) −13.9705 −0.577114
\(587\) −6.26553 −0.258606 −0.129303 0.991605i \(-0.541274\pi\)
−0.129303 + 0.991605i \(0.541274\pi\)
\(588\) 2.66057 0.109720
\(589\) −11.2870 −0.465075
\(590\) −1.47726 −0.0608180
\(591\) 10.5763 0.435052
\(592\) −0.727768 −0.0299111
\(593\) −9.77324 −0.401339 −0.200669 0.979659i \(-0.564312\pi\)
−0.200669 + 0.979659i \(0.564312\pi\)
\(594\) 2.20503 0.0904736
\(595\) 10.2757 0.421261
\(596\) −2.81090 −0.115139
\(597\) −8.59337 −0.351703
\(598\) −14.3883 −0.588383
\(599\) −15.1845 −0.620422 −0.310211 0.950668i \(-0.600400\pi\)
−0.310211 + 0.950668i \(0.600400\pi\)
\(600\) 1.00000 0.0408248
\(601\) −20.2688 −0.826780 −0.413390 0.910554i \(-0.635655\pi\)
−0.413390 + 0.910554i \(0.635655\pi\)
\(602\) −9.18674 −0.374424
\(603\) −1.00000 −0.0407231
\(604\) 11.7142 0.476642
\(605\) 6.13783 0.249538
\(606\) −9.13783 −0.371199
\(607\) −5.31891 −0.215888 −0.107944 0.994157i \(-0.534427\pi\)
−0.107944 + 0.994157i \(0.534427\pi\)
\(608\) −4.93280 −0.200051
\(609\) 5.85547 0.237276
\(610\) 4.08313 0.165321
\(611\) −40.2539 −1.62850
\(612\) −4.93280 −0.199397
\(613\) −10.6650 −0.430757 −0.215378 0.976531i \(-0.569099\pi\)
−0.215378 + 0.976531i \(0.569099\pi\)
\(614\) 20.3144 0.819824
\(615\) −1.47726 −0.0595690
\(616\) −4.59337 −0.185072
\(617\) 37.1628 1.49612 0.748059 0.663633i \(-0.230986\pi\)
0.748059 + 0.663633i \(0.230986\pi\)
\(618\) 0.410065 0.0164952
\(619\) −15.0672 −0.605602 −0.302801 0.953054i \(-0.597922\pi\)
−0.302801 + 0.953054i \(0.597922\pi\)
\(620\) 2.28816 0.0918948
\(621\) 3.35536 0.134646
\(622\) 7.75620 0.310995
\(623\) 4.54811 0.182216
\(624\) 4.28816 0.171664
\(625\) 1.00000 0.0400000
\(626\) 20.8132 0.831864
\(627\) −10.8770 −0.434385
\(628\) −20.1981 −0.805993
\(629\) 3.58994 0.143140
\(630\) 2.08313 0.0829939
\(631\) −12.4226 −0.494534 −0.247267 0.968947i \(-0.579533\pi\)
−0.247267 + 0.968947i \(0.579533\pi\)
\(632\) −4.83263 −0.192232
\(633\) 12.8553 0.510951
\(634\) −0.488511 −0.0194013
\(635\) 11.1208 0.441315
\(636\) 6.16626 0.244508
\(637\) 11.4090 0.452039
\(638\) 6.19812 0.245386
\(639\) −4.84967 −0.191850
\(640\) 1.00000 0.0395285
\(641\) 31.8177 1.25672 0.628362 0.777921i \(-0.283726\pi\)
0.628362 + 0.777921i \(0.283726\pi\)
\(642\) 0.576325 0.0227457
\(643\) −40.3427 −1.59096 −0.795479 0.605981i \(-0.792781\pi\)
−0.795479 + 0.605981i \(0.792781\pi\)
\(644\) −6.98966 −0.275431
\(645\) 4.41006 0.173646
\(646\) 24.3325 0.957350
\(647\) 43.6662 1.71670 0.858349 0.513067i \(-0.171491\pi\)
0.858349 + 0.513067i \(0.171491\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −3.25741 −0.127865
\(650\) 4.28816 0.168196
\(651\) 4.76654 0.186815
\(652\) −10.7278 −0.420132
\(653\) 33.1411 1.29691 0.648455 0.761253i \(-0.275416\pi\)
0.648455 + 0.761253i \(0.275416\pi\)
\(654\) 5.78247 0.226113
\(655\) 17.6435 0.689390
\(656\) −1.47726 −0.0576775
\(657\) −0.544464 −0.0212415
\(658\) −19.5548 −0.762326
\(659\) 34.6640 1.35032 0.675159 0.737672i \(-0.264075\pi\)
0.675159 + 0.737672i \(0.264075\pi\)
\(660\) 2.20503 0.0858308
\(661\) −10.8884 −0.423511 −0.211756 0.977323i \(-0.567918\pi\)
−0.211756 + 0.977323i \(0.567918\pi\)
\(662\) 23.0570 0.896137
\(663\) −21.1526 −0.821501
\(664\) 16.5934 0.643948
\(665\) −10.2757 −0.398473
\(666\) 0.727768 0.0282004
\(667\) 9.43158 0.365192
\(668\) 3.93169 0.152122
\(669\) −18.8520 −0.728860
\(670\) −1.00000 −0.0386334
\(671\) 9.00343 0.347574
\(672\) 2.08313 0.0803585
\(673\) −8.48865 −0.327213 −0.163607 0.986526i \(-0.552313\pi\)
−0.163607 + 0.986526i \(0.552313\pi\)
\(674\) −30.3293 −1.16824
\(675\) −1.00000 −0.0384900
\(676\) 5.38834 0.207244
\(677\) −15.1526 −0.582364 −0.291182 0.956668i \(-0.594048\pi\)
−0.291182 + 0.956668i \(0.594048\pi\)
\(678\) −13.8268 −0.531016
\(679\) 4.59337 0.176277
\(680\) −4.93280 −0.189164
\(681\) 22.1094 0.847234
\(682\) 5.04547 0.193201
\(683\) 5.01138 0.191755 0.0958776 0.995393i \(-0.469434\pi\)
0.0958776 + 0.995393i \(0.469434\pi\)
\(684\) 4.93280 0.188610
\(685\) 4.72777 0.180639
\(686\) 20.1242 0.768346
\(687\) −0.326934 −0.0124733
\(688\) 4.41006 0.168132
\(689\) 26.4419 1.00736
\(690\) 3.35536 0.127736
\(691\) −9.89969 −0.376602 −0.188301 0.982111i \(-0.560298\pi\)
−0.188301 + 0.982111i \(0.560298\pi\)
\(692\) 9.04779 0.343945
\(693\) 4.59337 0.174488
\(694\) −17.5432 −0.665931
\(695\) 20.0855 0.761885
\(696\) −2.81090 −0.106547
\(697\) 7.28705 0.276017
\(698\) −14.8201 −0.560950
\(699\) 2.14922 0.0812908
\(700\) 2.08313 0.0787349
\(701\) 11.3609 0.429097 0.214549 0.976713i \(-0.431172\pi\)
0.214549 + 0.976713i \(0.431172\pi\)
\(702\) −4.28816 −0.161846
\(703\) −3.58994 −0.135397
\(704\) 2.20503 0.0831053
\(705\) 9.38722 0.353543
\(706\) −18.7665 −0.706288
\(707\) −19.0353 −0.715896
\(708\) 1.47726 0.0555190
\(709\) −49.5170 −1.85965 −0.929826 0.368001i \(-0.880042\pi\)
−0.929826 + 0.368001i \(0.880042\pi\)
\(710\) −4.84967 −0.182005
\(711\) 4.83263 0.181238
\(712\) −2.18330 −0.0818228
\(713\) 7.67761 0.287529
\(714\) −10.2757 −0.384557
\(715\) 9.45554 0.353617
\(716\) 13.6754 0.511073
\(717\) −11.1208 −0.415313
\(718\) 6.43738 0.240241
\(719\) −28.4863 −1.06236 −0.531180 0.847259i \(-0.678251\pi\)
−0.531180 + 0.847259i \(0.678251\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0.854218 0.0318127
\(722\) −5.33252 −0.198456
\(723\) 7.15956 0.266267
\(724\) 0.876984 0.0325929
\(725\) −2.81090 −0.104394
\(726\) −6.13783 −0.227796
\(727\) −19.4191 −0.720214 −0.360107 0.932911i \(-0.617260\pi\)
−0.360107 + 0.932911i \(0.617260\pi\)
\(728\) 8.93280 0.331072
\(729\) 1.00000 0.0370370
\(730\) −0.544464 −0.0201515
\(731\) −21.7540 −0.804600
\(732\) −4.08313 −0.150917
\(733\) 20.0628 0.741037 0.370519 0.928825i \(-0.379180\pi\)
0.370519 + 0.928825i \(0.379180\pi\)
\(734\) 18.1298 0.669183
\(735\) −2.66057 −0.0981366
\(736\) 3.35536 0.123680
\(737\) −2.20503 −0.0812234
\(738\) 1.47726 0.0543788
\(739\) 6.21985 0.228801 0.114400 0.993435i \(-0.463505\pi\)
0.114400 + 0.993435i \(0.463505\pi\)
\(740\) 0.727768 0.0267533
\(741\) 21.1526 0.777062
\(742\) 12.8451 0.471559
\(743\) 51.6925 1.89641 0.948207 0.317652i \(-0.102894\pi\)
0.948207 + 0.317652i \(0.102894\pi\)
\(744\) −2.28816 −0.0838881
\(745\) 2.81090 0.102983
\(746\) −12.6207 −0.462076
\(747\) −16.5934 −0.607120
\(748\) −10.8770 −0.397702
\(749\) 1.20056 0.0438675
\(750\) −1.00000 −0.0365148
\(751\) −12.8201 −0.467813 −0.233907 0.972259i \(-0.575151\pi\)
−0.233907 + 0.972259i \(0.575151\pi\)
\(752\) 9.38722 0.342317
\(753\) 10.7278 0.390942
\(754\) −12.0536 −0.438966
\(755\) −11.7142 −0.426322
\(756\) −2.08313 −0.0757627
\(757\) −30.4613 −1.10713 −0.553567 0.832805i \(-0.686734\pi\)
−0.553567 + 0.832805i \(0.686734\pi\)
\(758\) −28.4180 −1.03219
\(759\) 7.39868 0.268555
\(760\) 4.93280 0.178931
\(761\) 0.937484 0.0339838 0.0169919 0.999856i \(-0.494591\pi\)
0.0169919 + 0.999856i \(0.494591\pi\)
\(762\) −11.1208 −0.402864
\(763\) 12.0456 0.436081
\(764\) −16.4101 −0.593695
\(765\) 4.93280 0.178346
\(766\) −13.6048 −0.491560
\(767\) 6.33475 0.228734
\(768\) −1.00000 −0.0360844
\(769\) −30.2734 −1.09169 −0.545844 0.837887i \(-0.683791\pi\)
−0.545844 + 0.837887i \(0.683791\pi\)
\(770\) 4.59337 0.165534
\(771\) 20.4202 0.735415
\(772\) −3.17660 −0.114329
\(773\) −26.8949 −0.967343 −0.483672 0.875249i \(-0.660697\pi\)
−0.483672 + 0.875249i \(0.660697\pi\)
\(774\) −4.41006 −0.158516
\(775\) −2.28816 −0.0821932
\(776\) −2.20503 −0.0791560
\(777\) 1.51604 0.0543875
\(778\) 23.6948 0.849498
\(779\) −7.28705 −0.261086
\(780\) −4.28816 −0.153541
\(781\) −10.6937 −0.382650
\(782\) −16.5513 −0.591874
\(783\) 2.81090 0.100453
\(784\) −2.66057 −0.0950203
\(785\) 20.1981 0.720902
\(786\) −17.6435 −0.629324
\(787\) 33.0936 1.17966 0.589829 0.807528i \(-0.299195\pi\)
0.589829 + 0.807528i \(0.299195\pi\)
\(788\) −10.5763 −0.376766
\(789\) 3.59917 0.128134
\(790\) 4.83263 0.171937
\(791\) −28.8031 −1.02412
\(792\) −2.20503 −0.0783524
\(793\) −17.5091 −0.621767
\(794\) 4.65022 0.165030
\(795\) −6.16626 −0.218695
\(796\) 8.59337 0.304584
\(797\) 22.6243 0.801395 0.400697 0.916210i \(-0.368768\pi\)
0.400697 + 0.916210i \(0.368768\pi\)
\(798\) 10.2757 0.363755
\(799\) −46.3053 −1.63816
\(800\) −1.00000 −0.0353553
\(801\) 2.18330 0.0771433
\(802\) −23.5193 −0.830494
\(803\) −1.20056 −0.0423668
\(804\) 1.00000 0.0352673
\(805\) 6.98966 0.246353
\(806\) −9.81201 −0.345613
\(807\) −8.94530 −0.314889
\(808\) 9.13783 0.321468
\(809\) 26.0000 0.914111 0.457056 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) 1.00000 0.0351364
\(811\) −43.5217 −1.52825 −0.764127 0.645066i \(-0.776830\pi\)
−0.764127 + 0.645066i \(0.776830\pi\)
\(812\) −5.85547 −0.205487
\(813\) 7.41118 0.259921
\(814\) 1.60475 0.0562465
\(815\) 10.7278 0.375777
\(816\) 4.93280 0.172683
\(817\) 21.7540 0.761075
\(818\) −18.5763 −0.649506
\(819\) −8.93280 −0.312137
\(820\) 1.47726 0.0515883
\(821\) 10.9431 0.381916 0.190958 0.981598i \(-0.438841\pi\)
0.190958 + 0.981598i \(0.438841\pi\)
\(822\) −4.72777 −0.164900
\(823\) 8.63541 0.301011 0.150506 0.988609i \(-0.451910\pi\)
0.150506 + 0.988609i \(0.451910\pi\)
\(824\) −0.410065 −0.0142853
\(825\) −2.20503 −0.0767694
\(826\) 3.07733 0.107074
\(827\) −23.9395 −0.832458 −0.416229 0.909260i \(-0.636649\pi\)
−0.416229 + 0.909260i \(0.636649\pi\)
\(828\) −3.35536 −0.116607
\(829\) −15.6993 −0.545261 −0.272630 0.962119i \(-0.587894\pi\)
−0.272630 + 0.962119i \(0.587894\pi\)
\(830\) −16.5934 −0.575964
\(831\) −16.2586 −0.564005
\(832\) −4.28816 −0.148665
\(833\) 13.1241 0.454722
\(834\) −20.0855 −0.695502
\(835\) −3.93169 −0.136062
\(836\) 10.8770 0.376188
\(837\) 2.28816 0.0790905
\(838\) 1.97605 0.0682613
\(839\) 25.1742 0.869111 0.434556 0.900645i \(-0.356905\pi\)
0.434556 + 0.900645i \(0.356905\pi\)
\(840\) −2.08313 −0.0718748
\(841\) −21.0989 −0.727547
\(842\) −9.92246 −0.341950
\(843\) −19.1526 −0.659652
\(844\) −12.8553 −0.442496
\(845\) −5.38834 −0.185364
\(846\) −9.38722 −0.322739
\(847\) −12.7859 −0.439329
\(848\) −6.16626 −0.211750
\(849\) 24.2905 0.833647
\(850\) 4.93280 0.169194
\(851\) 2.44193 0.0837081
\(852\) 4.84967 0.166147
\(853\) 8.33029 0.285224 0.142612 0.989779i \(-0.454450\pi\)
0.142612 + 0.989779i \(0.454450\pi\)
\(854\) −8.50569 −0.291059
\(855\) −4.93280 −0.168698
\(856\) −0.576325 −0.0196984
\(857\) −35.7562 −1.22141 −0.610703 0.791859i \(-0.709113\pi\)
−0.610703 + 0.791859i \(0.709113\pi\)
\(858\) −9.45554 −0.322807
\(859\) −51.5410 −1.75856 −0.879278 0.476309i \(-0.841974\pi\)
−0.879278 + 0.476309i \(0.841974\pi\)
\(860\) −4.41006 −0.150382
\(861\) 3.07733 0.104875
\(862\) −2.38275 −0.0811568
\(863\) −3.65825 −0.124528 −0.0622641 0.998060i \(-0.519832\pi\)
−0.0622641 + 0.998060i \(0.519832\pi\)
\(864\) 1.00000 0.0340207
\(865\) −9.04779 −0.307634
\(866\) 20.9965 0.713491
\(867\) −7.33252 −0.249025
\(868\) −4.76654 −0.161787
\(869\) 10.6561 0.361483
\(870\) 2.81090 0.0952984
\(871\) 4.28816 0.145299
\(872\) −5.78247 −0.195819
\(873\) 2.20503 0.0746290
\(874\) 16.5513 0.559857
\(875\) −2.08313 −0.0704227
\(876\) 0.544464 0.0183957
\(877\) 10.0185 0.338299 0.169150 0.985590i \(-0.445898\pi\)
0.169150 + 0.985590i \(0.445898\pi\)
\(878\) −17.0034 −0.573838
\(879\) −13.9705 −0.471212
\(880\) −2.20503 −0.0743316
\(881\) 13.9145 0.468792 0.234396 0.972141i \(-0.424689\pi\)
0.234396 + 0.972141i \(0.424689\pi\)
\(882\) 2.66057 0.0895860
\(883\) −19.7562 −0.664849 −0.332424 0.943130i \(-0.607867\pi\)
−0.332424 + 0.943130i \(0.607867\pi\)
\(884\) 21.1526 0.711441
\(885\) −1.47726 −0.0496577
\(886\) 13.8656 0.465824
\(887\) −8.75182 −0.293857 −0.146929 0.989147i \(-0.546939\pi\)
−0.146929 + 0.989147i \(0.546939\pi\)
\(888\) −0.727768 −0.0244223
\(889\) −23.1660 −0.776964
\(890\) 2.18330 0.0731845
\(891\) 2.20503 0.0738714
\(892\) 18.8520 0.631211
\(893\) 46.3053 1.54955
\(894\) −2.81090 −0.0940105
\(895\) −13.6754 −0.457118
\(896\) −2.08313 −0.0695925
\(897\) −14.3883 −0.480413
\(898\) −39.0922 −1.30452
\(899\) 6.43179 0.214512
\(900\) 1.00000 0.0333333
\(901\) 30.4169 1.01333
\(902\) 3.25741 0.108460
\(903\) −9.18674 −0.305716
\(904\) 13.8268 0.459873
\(905\) −0.876984 −0.0291519
\(906\) 11.7142 0.389177
\(907\) −24.3920 −0.809922 −0.404961 0.914334i \(-0.632715\pi\)
−0.404961 + 0.914334i \(0.632715\pi\)
\(908\) −22.1094 −0.733726
\(909\) −9.13783 −0.303083
\(910\) −8.93280 −0.296119
\(911\) −6.94061 −0.229953 −0.114976 0.993368i \(-0.536679\pi\)
−0.114976 + 0.993368i \(0.536679\pi\)
\(912\) −4.93280 −0.163341
\(913\) −36.5889 −1.21092
\(914\) 6.68900 0.221252
\(915\) 4.08313 0.134984
\(916\) 0.326934 0.0108022
\(917\) −36.7538 −1.21372
\(918\) −4.93280 −0.162807
\(919\) −17.0764 −0.563299 −0.281650 0.959517i \(-0.590882\pi\)
−0.281650 + 0.959517i \(0.590882\pi\)
\(920\) −3.35536 −0.110623
\(921\) 20.3144 0.669383
\(922\) 22.0545 0.726326
\(923\) 20.7962 0.684514
\(924\) −4.59337 −0.151111
\(925\) −0.727768 −0.0239289
\(926\) −39.4191 −1.29539
\(927\) 0.410065 0.0134683
\(928\) 2.81090 0.0922723
\(929\) 16.6433 0.546049 0.273025 0.962007i \(-0.411976\pi\)
0.273025 + 0.962007i \(0.411976\pi\)
\(930\) 2.28816 0.0750318
\(931\) −13.1241 −0.430124
\(932\) −2.14922 −0.0703999
\(933\) 7.75620 0.253926
\(934\) −10.0148 −0.327695
\(935\) 10.8770 0.355715
\(936\) 4.28816 0.140163
\(937\) 9.72652 0.317752 0.158876 0.987299i \(-0.449213\pi\)
0.158876 + 0.987299i \(0.449213\pi\)
\(938\) 2.08313 0.0680166
\(939\) 20.8132 0.679214
\(940\) −9.38722 −0.306177
\(941\) −7.20056 −0.234732 −0.117366 0.993089i \(-0.537445\pi\)
−0.117366 + 0.993089i \(0.537445\pi\)
\(942\) −20.1981 −0.658090
\(943\) 4.95676 0.161414
\(944\) −1.47726 −0.0480808
\(945\) 2.08313 0.0677642
\(946\) −9.72433 −0.316165
\(947\) 30.1561 0.979941 0.489971 0.871739i \(-0.337008\pi\)
0.489971 + 0.871739i \(0.337008\pi\)
\(948\) −4.83263 −0.156956
\(949\) 2.33475 0.0757891
\(950\) −4.93280 −0.160041
\(951\) −0.488511 −0.0158411
\(952\) 10.2757 0.333036
\(953\) −19.0193 −0.616095 −0.308048 0.951371i \(-0.599676\pi\)
−0.308048 + 0.951371i \(0.599676\pi\)
\(954\) 6.16626 0.199640
\(955\) 16.4101 0.531017
\(956\) 11.1208 0.359672
\(957\) 6.19812 0.200357
\(958\) −22.8474 −0.738167
\(959\) −9.84856 −0.318026
\(960\) 1.00000 0.0322749
\(961\) −25.7643 −0.831107
\(962\) −3.12079 −0.100618
\(963\) 0.576325 0.0185718
\(964\) −7.15956 −0.230594
\(965\) 3.17660 0.102259
\(966\) −6.98966 −0.224888
\(967\) 40.1865 1.29231 0.646156 0.763206i \(-0.276376\pi\)
0.646156 + 0.763206i \(0.276376\pi\)
\(968\) 6.13783 0.197277
\(969\) 24.3325 0.781673
\(970\) 2.20503 0.0707993
\(971\) −50.4387 −1.61865 −0.809327 0.587359i \(-0.800168\pi\)
−0.809327 + 0.587359i \(0.800168\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −41.8406 −1.34135
\(974\) −35.8896 −1.14998
\(975\) 4.28816 0.137331
\(976\) 4.08313 0.130698
\(977\) 58.8142 1.88163 0.940817 0.338916i \(-0.110060\pi\)
0.940817 + 0.338916i \(0.110060\pi\)
\(978\) −10.7278 −0.343036
\(979\) 4.81426 0.153864
\(980\) 2.66057 0.0849888
\(981\) 5.78247 0.184620
\(982\) 35.2403 1.12456
\(983\) 13.2643 0.423065 0.211532 0.977371i \(-0.432155\pi\)
0.211532 + 0.977371i \(0.432155\pi\)
\(984\) −1.47726 −0.0470935
\(985\) 10.5763 0.336990
\(986\) −13.8656 −0.441571
\(987\) −19.5548 −0.622436
\(988\) −21.1526 −0.672955
\(989\) −14.7974 −0.470529
\(990\) 2.20503 0.0700805
\(991\) −15.6071 −0.495775 −0.247887 0.968789i \(-0.579736\pi\)
−0.247887 + 0.968789i \(0.579736\pi\)
\(992\) 2.28816 0.0726492
\(993\) 23.0570 0.731692
\(994\) 10.1025 0.320432
\(995\) −8.59337 −0.272428
\(996\) 16.5934 0.525781
\(997\) −20.5911 −0.652128 −0.326064 0.945348i \(-0.605722\pi\)
−0.326064 + 0.945348i \(0.605722\pi\)
\(998\) −8.17094 −0.258647
\(999\) 0.727768 0.0230256
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 2010.2.a.r.1.3 4
3.2 odd 2 6030.2.a.bu.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2010.2.a.r.1.3 4 1.1 even 1 trivial
6030.2.a.bu.1.3 4 3.2 odd 2