Properties

Label 8002.2.a.e.1.24
Level $8002$
Weight $2$
Character 8002.1
Self dual yes
Analytic conductor $63.896$
Analytic rank $0$
Dimension $77$
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] = [8002,2,Mod(1,8002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8002, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8002.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8002 = 2 \cdot 4001 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8002.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: \(63.8962916974\)
Analytic rank: \(0\)
Dimension: \(77\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.24
Character \(\chi\) \(=\) 8002.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.28149 q^{3} +1.00000 q^{4} -0.444196 q^{5} +1.28149 q^{6} +1.92855 q^{7} -1.00000 q^{8} -1.35779 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.28149 q^{3} +1.00000 q^{4} -0.444196 q^{5} +1.28149 q^{6} +1.92855 q^{7} -1.00000 q^{8} -1.35779 q^{9} +0.444196 q^{10} +2.61160 q^{11} -1.28149 q^{12} -2.34755 q^{13} -1.92855 q^{14} +0.569232 q^{15} +1.00000 q^{16} -4.89602 q^{17} +1.35779 q^{18} +3.17903 q^{19} -0.444196 q^{20} -2.47141 q^{21} -2.61160 q^{22} -5.99830 q^{23} +1.28149 q^{24} -4.80269 q^{25} +2.34755 q^{26} +5.58445 q^{27} +1.92855 q^{28} -4.96047 q^{29} -0.569232 q^{30} -6.67509 q^{31} -1.00000 q^{32} -3.34674 q^{33} +4.89602 q^{34} -0.856654 q^{35} -1.35779 q^{36} +8.13959 q^{37} -3.17903 q^{38} +3.00835 q^{39} +0.444196 q^{40} +1.24289 q^{41} +2.47141 q^{42} -6.72451 q^{43} +2.61160 q^{44} +0.603126 q^{45} +5.99830 q^{46} -8.23219 q^{47} -1.28149 q^{48} -3.28070 q^{49} +4.80269 q^{50} +6.27418 q^{51} -2.34755 q^{52} +12.6320 q^{53} -5.58445 q^{54} -1.16007 q^{55} -1.92855 q^{56} -4.07388 q^{57} +4.96047 q^{58} +11.3965 q^{59} +0.569232 q^{60} +3.01624 q^{61} +6.67509 q^{62} -2.61857 q^{63} +1.00000 q^{64} +1.04277 q^{65} +3.34674 q^{66} -4.38366 q^{67} -4.89602 q^{68} +7.68674 q^{69} +0.856654 q^{70} +2.29127 q^{71} +1.35779 q^{72} -4.14794 q^{73} -8.13959 q^{74} +6.15458 q^{75} +3.17903 q^{76} +5.03661 q^{77} -3.00835 q^{78} -13.7362 q^{79} -0.444196 q^{80} -3.08302 q^{81} -1.24289 q^{82} +6.48484 q^{83} -2.47141 q^{84} +2.17479 q^{85} +6.72451 q^{86} +6.35678 q^{87} -2.61160 q^{88} +2.80274 q^{89} -0.603126 q^{90} -4.52735 q^{91} -5.99830 q^{92} +8.55404 q^{93} +8.23219 q^{94} -1.41211 q^{95} +1.28149 q^{96} +11.1750 q^{97} +3.28070 q^{98} -3.54602 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19} + 18 q^{20} + 10 q^{21} - 30 q^{22} + 53 q^{23} - 10 q^{24} + 59 q^{25} + 2 q^{26} + 43 q^{27} + 21 q^{28} + 30 q^{29} - 21 q^{30} + 22 q^{31} - 77 q^{32} + 31 q^{33} - 60 q^{34} + 41 q^{35} + 71 q^{36} - 3 q^{37} + 3 q^{38} + 44 q^{39} - 18 q^{40} + 48 q^{41} - 10 q^{42} + 21 q^{43} + 30 q^{44} + 33 q^{45} - 53 q^{46} + 107 q^{47} + 10 q^{48} + 24 q^{49} - 59 q^{50} + 18 q^{51} - 2 q^{52} + 42 q^{53} - 43 q^{54} + 49 q^{55} - 21 q^{56} + 32 q^{57} - 30 q^{58} + 42 q^{59} + 21 q^{60} - 31 q^{61} - 22 q^{62} + 109 q^{63} + 77 q^{64} + 39 q^{65} - 31 q^{66} - q^{67} + 60 q^{68} - 33 q^{69} - 41 q^{70} + 58 q^{71} - 71 q^{72} + 35 q^{73} + 3 q^{74} + 34 q^{75} - 3 q^{76} + 86 q^{77} - 44 q^{78} + 25 q^{79} + 18 q^{80} + 53 q^{81} - 48 q^{82} + 107 q^{83} + 10 q^{84} + 21 q^{85} - 21 q^{86} + 100 q^{87} - 30 q^{88} + 34 q^{89} - 33 q^{90} - 51 q^{91} + 53 q^{92} + 48 q^{93} - 107 q^{94} + 118 q^{95} - 10 q^{96} - 13 q^{97} - 24 q^{98} + 63 q^{99}+O(q^{100}) \) 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
\(3\) −1.28149 −0.739867 −0.369933 0.929058i \(-0.620619\pi\)
−0.369933 + 0.929058i \(0.620619\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.444196 −0.198651 −0.0993253 0.995055i \(-0.531668\pi\)
−0.0993253 + 0.995055i \(0.531668\pi\)
\(6\) 1.28149 0.523165
\(7\) 1.92855 0.728923 0.364461 0.931218i \(-0.381253\pi\)
0.364461 + 0.931218i \(0.381253\pi\)
\(8\) −1.00000 −0.353553
\(9\) −1.35779 −0.452597
\(10\) 0.444196 0.140467
\(11\) 2.61160 0.787428 0.393714 0.919233i \(-0.371190\pi\)
0.393714 + 0.919233i \(0.371190\pi\)
\(12\) −1.28149 −0.369933
\(13\) −2.34755 −0.651092 −0.325546 0.945526i \(-0.605548\pi\)
−0.325546 + 0.945526i \(0.605548\pi\)
\(14\) −1.92855 −0.515426
\(15\) 0.569232 0.146975
\(16\) 1.00000 0.250000
\(17\) −4.89602 −1.18746 −0.593730 0.804665i \(-0.702345\pi\)
−0.593730 + 0.804665i \(0.702345\pi\)
\(18\) 1.35779 0.320035
\(19\) 3.17903 0.729319 0.364659 0.931141i \(-0.381185\pi\)
0.364659 + 0.931141i \(0.381185\pi\)
\(20\) −0.444196 −0.0993253
\(21\) −2.47141 −0.539306
\(22\) −2.61160 −0.556796
\(23\) −5.99830 −1.25073 −0.625366 0.780332i \(-0.715050\pi\)
−0.625366 + 0.780332i \(0.715050\pi\)
\(24\) 1.28149 0.261582
\(25\) −4.80269 −0.960538
\(26\) 2.34755 0.460391
\(27\) 5.58445 1.07473
\(28\) 1.92855 0.364461
\(29\) −4.96047 −0.921136 −0.460568 0.887624i \(-0.652354\pi\)
−0.460568 + 0.887624i \(0.652354\pi\)
\(30\) −0.569232 −0.103927
\(31\) −6.67509 −1.19888 −0.599441 0.800419i \(-0.704610\pi\)
−0.599441 + 0.800419i \(0.704610\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.34674 −0.582592
\(34\) 4.89602 0.839660
\(35\) −0.856654 −0.144801
\(36\) −1.35779 −0.226299
\(37\) 8.13959 1.33814 0.669070 0.743199i \(-0.266693\pi\)
0.669070 + 0.743199i \(0.266693\pi\)
\(38\) −3.17903 −0.515706
\(39\) 3.00835 0.481721
\(40\) 0.444196 0.0702336
\(41\) 1.24289 0.194107 0.0970536 0.995279i \(-0.469058\pi\)
0.0970536 + 0.995279i \(0.469058\pi\)
\(42\) 2.47141 0.381347
\(43\) −6.72451 −1.02548 −0.512739 0.858545i \(-0.671369\pi\)
−0.512739 + 0.858545i \(0.671369\pi\)
\(44\) 2.61160 0.393714
\(45\) 0.603126 0.0899088
\(46\) 5.99830 0.884401
\(47\) −8.23219 −1.20079 −0.600394 0.799704i \(-0.704990\pi\)
−0.600394 + 0.799704i \(0.704990\pi\)
\(48\) −1.28149 −0.184967
\(49\) −3.28070 −0.468671
\(50\) 4.80269 0.679203
\(51\) 6.27418 0.878561
\(52\) −2.34755 −0.325546
\(53\) 12.6320 1.73514 0.867568 0.497318i \(-0.165682\pi\)
0.867568 + 0.497318i \(0.165682\pi\)
\(54\) −5.58445 −0.759948
\(55\) −1.16007 −0.156423
\(56\) −1.92855 −0.257713
\(57\) −4.07388 −0.539599
\(58\) 4.96047 0.651342
\(59\) 11.3965 1.48370 0.741851 0.670565i \(-0.233948\pi\)
0.741851 + 0.670565i \(0.233948\pi\)
\(60\) 0.569232 0.0734875
\(61\) 3.01624 0.386190 0.193095 0.981180i \(-0.438147\pi\)
0.193095 + 0.981180i \(0.438147\pi\)
\(62\) 6.67509 0.847737
\(63\) −2.61857 −0.329909
\(64\) 1.00000 0.125000
\(65\) 1.04277 0.129340
\(66\) 3.34674 0.411955
\(67\) −4.38366 −0.535550 −0.267775 0.963482i \(-0.586288\pi\)
−0.267775 + 0.963482i \(0.586288\pi\)
\(68\) −4.89602 −0.593730
\(69\) 7.68674 0.925375
\(70\) 0.856654 0.102390
\(71\) 2.29127 0.271923 0.135962 0.990714i \(-0.456588\pi\)
0.135962 + 0.990714i \(0.456588\pi\)
\(72\) 1.35779 0.160017
\(73\) −4.14794 −0.485480 −0.242740 0.970091i \(-0.578046\pi\)
−0.242740 + 0.970091i \(0.578046\pi\)
\(74\) −8.13959 −0.946208
\(75\) 6.15458 0.710670
\(76\) 3.17903 0.364659
\(77\) 5.03661 0.573974
\(78\) −3.00835 −0.340628
\(79\) −13.7362 −1.54544 −0.772719 0.634748i \(-0.781104\pi\)
−0.772719 + 0.634748i \(0.781104\pi\)
\(80\) −0.444196 −0.0496627
\(81\) −3.08302 −0.342558
\(82\) −1.24289 −0.137255
\(83\) 6.48484 0.711804 0.355902 0.934523i \(-0.384174\pi\)
0.355902 + 0.934523i \(0.384174\pi\)
\(84\) −2.47141 −0.269653
\(85\) 2.17479 0.235890
\(86\) 6.72451 0.725122
\(87\) 6.35678 0.681518
\(88\) −2.61160 −0.278398
\(89\) 2.80274 0.297090 0.148545 0.988906i \(-0.452541\pi\)
0.148545 + 0.988906i \(0.452541\pi\)
\(90\) −0.603126 −0.0635751
\(91\) −4.52735 −0.474596
\(92\) −5.99830 −0.625366
\(93\) 8.55404 0.887013
\(94\) 8.23219 0.849085
\(95\) −1.41211 −0.144880
\(96\) 1.28149 0.130791
\(97\) 11.1750 1.13465 0.567325 0.823494i \(-0.307978\pi\)
0.567325 + 0.823494i \(0.307978\pi\)
\(98\) 3.28070 0.331401
\(99\) −3.54602 −0.356388
\(100\) −4.80269 −0.480269
\(101\) 0.0783921 0.00780030 0.00390015 0.999992i \(-0.498759\pi\)
0.00390015 + 0.999992i \(0.498759\pi\)
\(102\) −6.27418 −0.621237
\(103\) 5.32059 0.524254 0.262127 0.965033i \(-0.415576\pi\)
0.262127 + 0.965033i \(0.415576\pi\)
\(104\) 2.34755 0.230196
\(105\) 1.09779 0.107133
\(106\) −12.6320 −1.22693
\(107\) 16.0609 1.55266 0.776331 0.630325i \(-0.217078\pi\)
0.776331 + 0.630325i \(0.217078\pi\)
\(108\) 5.58445 0.537364
\(109\) −0.313546 −0.0300323 −0.0150161 0.999887i \(-0.504780\pi\)
−0.0150161 + 0.999887i \(0.504780\pi\)
\(110\) 1.16007 0.110608
\(111\) −10.4308 −0.990045
\(112\) 1.92855 0.182231
\(113\) 13.8659 1.30440 0.652199 0.758048i \(-0.273847\pi\)
0.652199 + 0.758048i \(0.273847\pi\)
\(114\) 4.07388 0.381554
\(115\) 2.66442 0.248459
\(116\) −4.96047 −0.460568
\(117\) 3.18748 0.294682
\(118\) −11.3965 −1.04914
\(119\) −9.44221 −0.865566
\(120\) −0.569232 −0.0519635
\(121\) −4.17952 −0.379957
\(122\) −3.01624 −0.273078
\(123\) −1.59275 −0.143613
\(124\) −6.67509 −0.599441
\(125\) 4.35432 0.389462
\(126\) 2.61857 0.233281
\(127\) −18.3525 −1.62852 −0.814261 0.580499i \(-0.802858\pi\)
−0.814261 + 0.580499i \(0.802858\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 8.61737 0.758717
\(130\) −1.04277 −0.0914571
\(131\) −9.86495 −0.861905 −0.430952 0.902375i \(-0.641822\pi\)
−0.430952 + 0.902375i \(0.641822\pi\)
\(132\) −3.34674 −0.291296
\(133\) 6.13091 0.531617
\(134\) 4.38366 0.378691
\(135\) −2.48059 −0.213496
\(136\) 4.89602 0.419830
\(137\) 4.15166 0.354700 0.177350 0.984148i \(-0.443247\pi\)
0.177350 + 0.984148i \(0.443247\pi\)
\(138\) −7.68674 −0.654339
\(139\) −17.0535 −1.44646 −0.723229 0.690608i \(-0.757343\pi\)
−0.723229 + 0.690608i \(0.757343\pi\)
\(140\) −0.856654 −0.0724005
\(141\) 10.5494 0.888423
\(142\) −2.29127 −0.192279
\(143\) −6.13086 −0.512688
\(144\) −1.35779 −0.113149
\(145\) 2.20342 0.182984
\(146\) 4.14794 0.343286
\(147\) 4.20417 0.346754
\(148\) 8.13959 0.669070
\(149\) 8.75424 0.717175 0.358588 0.933496i \(-0.383258\pi\)
0.358588 + 0.933496i \(0.383258\pi\)
\(150\) −6.15458 −0.502520
\(151\) 14.7565 1.20086 0.600432 0.799675i \(-0.294995\pi\)
0.600432 + 0.799675i \(0.294995\pi\)
\(152\) −3.17903 −0.257853
\(153\) 6.64778 0.537441
\(154\) −5.03661 −0.405861
\(155\) 2.96505 0.238159
\(156\) 3.00835 0.240861
\(157\) −11.1030 −0.886118 −0.443059 0.896492i \(-0.646107\pi\)
−0.443059 + 0.896492i \(0.646107\pi\)
\(158\) 13.7362 1.09279
\(159\) −16.1877 −1.28377
\(160\) 0.444196 0.0351168
\(161\) −11.5680 −0.911687
\(162\) 3.08302 0.242225
\(163\) 10.1334 0.793709 0.396855 0.917881i \(-0.370102\pi\)
0.396855 + 0.917881i \(0.370102\pi\)
\(164\) 1.24289 0.0970536
\(165\) 1.48661 0.115732
\(166\) −6.48484 −0.503321
\(167\) 6.99047 0.540939 0.270469 0.962729i \(-0.412821\pi\)
0.270469 + 0.962729i \(0.412821\pi\)
\(168\) 2.47141 0.190673
\(169\) −7.48903 −0.576079
\(170\) −2.17479 −0.166799
\(171\) −4.31646 −0.330088
\(172\) −6.72451 −0.512739
\(173\) 22.5518 1.71458 0.857290 0.514833i \(-0.172146\pi\)
0.857290 + 0.514833i \(0.172146\pi\)
\(174\) −6.35678 −0.481906
\(175\) −9.26222 −0.700158
\(176\) 2.61160 0.196857
\(177\) −14.6045 −1.09774
\(178\) −2.80274 −0.210074
\(179\) −14.8361 −1.10890 −0.554451 0.832216i \(-0.687072\pi\)
−0.554451 + 0.832216i \(0.687072\pi\)
\(180\) 0.603126 0.0449544
\(181\) −5.05705 −0.375888 −0.187944 0.982180i \(-0.560182\pi\)
−0.187944 + 0.982180i \(0.560182\pi\)
\(182\) 4.52735 0.335590
\(183\) −3.86527 −0.285729
\(184\) 5.99830 0.442200
\(185\) −3.61558 −0.265822
\(186\) −8.55404 −0.627213
\(187\) −12.7865 −0.935039
\(188\) −8.23219 −0.600394
\(189\) 10.7699 0.783394
\(190\) 1.41211 0.102445
\(191\) 4.15023 0.300300 0.150150 0.988663i \(-0.452024\pi\)
0.150150 + 0.988663i \(0.452024\pi\)
\(192\) −1.28149 −0.0924833
\(193\) 13.2285 0.952205 0.476103 0.879390i \(-0.342049\pi\)
0.476103 + 0.879390i \(0.342049\pi\)
\(194\) −11.1750 −0.802319
\(195\) −1.33630 −0.0956942
\(196\) −3.28070 −0.234336
\(197\) −5.25500 −0.374403 −0.187202 0.982322i \(-0.559942\pi\)
−0.187202 + 0.982322i \(0.559942\pi\)
\(198\) 3.54602 0.252004
\(199\) 0.250345 0.0177465 0.00887323 0.999961i \(-0.497176\pi\)
0.00887323 + 0.999961i \(0.497176\pi\)
\(200\) 4.80269 0.339601
\(201\) 5.61761 0.396235
\(202\) −0.0783921 −0.00551565
\(203\) −9.56651 −0.671437
\(204\) 6.27418 0.439281
\(205\) −0.552088 −0.0385595
\(206\) −5.32059 −0.370703
\(207\) 8.14444 0.566078
\(208\) −2.34755 −0.162773
\(209\) 8.30236 0.574286
\(210\) −1.09779 −0.0757548
\(211\) 8.95055 0.616181 0.308091 0.951357i \(-0.400310\pi\)
0.308091 + 0.951357i \(0.400310\pi\)
\(212\) 12.6320 0.867568
\(213\) −2.93623 −0.201187
\(214\) −16.0609 −1.09790
\(215\) 2.98700 0.203712
\(216\) −5.58445 −0.379974
\(217\) −12.8732 −0.873892
\(218\) 0.313546 0.0212360
\(219\) 5.31553 0.359190
\(220\) −1.16007 −0.0782116
\(221\) 11.4936 0.773145
\(222\) 10.4308 0.700068
\(223\) 13.5470 0.907176 0.453588 0.891212i \(-0.350144\pi\)
0.453588 + 0.891212i \(0.350144\pi\)
\(224\) −1.92855 −0.128857
\(225\) 6.52105 0.434737
\(226\) −13.8659 −0.922349
\(227\) 11.5920 0.769388 0.384694 0.923044i \(-0.374307\pi\)
0.384694 + 0.923044i \(0.374307\pi\)
\(228\) −4.07388 −0.269799
\(229\) −4.42095 −0.292144 −0.146072 0.989274i \(-0.546663\pi\)
−0.146072 + 0.989274i \(0.546663\pi\)
\(230\) −2.66442 −0.175687
\(231\) −6.45434 −0.424665
\(232\) 4.96047 0.325671
\(233\) −2.05348 −0.134528 −0.0672641 0.997735i \(-0.521427\pi\)
−0.0672641 + 0.997735i \(0.521427\pi\)
\(234\) −3.18748 −0.208372
\(235\) 3.65671 0.238537
\(236\) 11.3965 0.741851
\(237\) 17.6027 1.14342
\(238\) 9.44221 0.612048
\(239\) −8.04445 −0.520352 −0.260176 0.965561i \(-0.583781\pi\)
−0.260176 + 0.965561i \(0.583781\pi\)
\(240\) 0.569232 0.0367438
\(241\) −23.3128 −1.50171 −0.750853 0.660469i \(-0.770358\pi\)
−0.750853 + 0.660469i \(0.770358\pi\)
\(242\) 4.17952 0.268670
\(243\) −12.8025 −0.821281
\(244\) 3.01624 0.193095
\(245\) 1.45728 0.0931019
\(246\) 1.59275 0.101550
\(247\) −7.46291 −0.474854
\(248\) 6.67509 0.423869
\(249\) −8.31024 −0.526640
\(250\) −4.35432 −0.275391
\(251\) 8.67424 0.547513 0.273757 0.961799i \(-0.411734\pi\)
0.273757 + 0.961799i \(0.411734\pi\)
\(252\) −2.61857 −0.164954
\(253\) −15.6652 −0.984861
\(254\) 18.3525 1.15154
\(255\) −2.78697 −0.174527
\(256\) 1.00000 0.0625000
\(257\) −30.9625 −1.93138 −0.965692 0.259688i \(-0.916380\pi\)
−0.965692 + 0.259688i \(0.916380\pi\)
\(258\) −8.61737 −0.536494
\(259\) 15.6976 0.975401
\(260\) 1.04277 0.0646699
\(261\) 6.73529 0.416904
\(262\) 9.86495 0.609459
\(263\) 15.4625 0.953461 0.476730 0.879050i \(-0.341822\pi\)
0.476730 + 0.879050i \(0.341822\pi\)
\(264\) 3.34674 0.205977
\(265\) −5.61108 −0.344686
\(266\) −6.13091 −0.375910
\(267\) −3.59167 −0.219807
\(268\) −4.38366 −0.267775
\(269\) 14.6441 0.892865 0.446432 0.894817i \(-0.352694\pi\)
0.446432 + 0.894817i \(0.352694\pi\)
\(270\) 2.48059 0.150964
\(271\) 19.9045 1.20911 0.604555 0.796564i \(-0.293351\pi\)
0.604555 + 0.796564i \(0.293351\pi\)
\(272\) −4.89602 −0.296865
\(273\) 5.80174 0.351138
\(274\) −4.15166 −0.250811
\(275\) −12.5427 −0.756355
\(276\) 7.68674 0.462687
\(277\) 4.25582 0.255708 0.127854 0.991793i \(-0.459191\pi\)
0.127854 + 0.991793i \(0.459191\pi\)
\(278\) 17.0535 1.02280
\(279\) 9.06339 0.542611
\(280\) 0.856654 0.0511949
\(281\) −6.91566 −0.412554 −0.206277 0.978494i \(-0.566135\pi\)
−0.206277 + 0.978494i \(0.566135\pi\)
\(282\) −10.5494 −0.628210
\(283\) −3.17045 −0.188464 −0.0942320 0.995550i \(-0.530040\pi\)
−0.0942320 + 0.995550i \(0.530040\pi\)
\(284\) 2.29127 0.135962
\(285\) 1.80960 0.107192
\(286\) 6.13086 0.362525
\(287\) 2.39698 0.141489
\(288\) 1.35779 0.0800087
\(289\) 6.97100 0.410059
\(290\) −2.20342 −0.129389
\(291\) −14.3206 −0.839490
\(292\) −4.14794 −0.242740
\(293\) 18.8052 1.09861 0.549304 0.835622i \(-0.314893\pi\)
0.549304 + 0.835622i \(0.314893\pi\)
\(294\) −4.20417 −0.245192
\(295\) −5.06230 −0.294738
\(296\) −8.13959 −0.473104
\(297\) 14.5844 0.846271
\(298\) −8.75424 −0.507120
\(299\) 14.0813 0.814341
\(300\) 6.15458 0.355335
\(301\) −12.9685 −0.747494
\(302\) −14.7565 −0.849140
\(303\) −0.100458 −0.00577118
\(304\) 3.17903 0.182330
\(305\) −1.33980 −0.0767169
\(306\) −6.64778 −0.380028
\(307\) −10.6421 −0.607379 −0.303690 0.952771i \(-0.598219\pi\)
−0.303690 + 0.952771i \(0.598219\pi\)
\(308\) 5.03661 0.286987
\(309\) −6.81827 −0.387878
\(310\) −2.96505 −0.168404
\(311\) −13.1201 −0.743974 −0.371987 0.928238i \(-0.621323\pi\)
−0.371987 + 0.928238i \(0.621323\pi\)
\(312\) −3.00835 −0.170314
\(313\) −1.47324 −0.0832725 −0.0416363 0.999133i \(-0.513257\pi\)
−0.0416363 + 0.999133i \(0.513257\pi\)
\(314\) 11.1030 0.626580
\(315\) 1.16316 0.0655366
\(316\) −13.7362 −0.772719
\(317\) 4.44230 0.249505 0.124752 0.992188i \(-0.460186\pi\)
0.124752 + 0.992188i \(0.460186\pi\)
\(318\) 16.1877 0.907762
\(319\) −12.9548 −0.725329
\(320\) −0.444196 −0.0248313
\(321\) −20.5818 −1.14876
\(322\) 11.5680 0.644660
\(323\) −15.5646 −0.866036
\(324\) −3.08302 −0.171279
\(325\) 11.2745 0.625398
\(326\) −10.1334 −0.561237
\(327\) 0.401805 0.0222199
\(328\) −1.24289 −0.0686273
\(329\) −15.8762 −0.875282
\(330\) −1.48661 −0.0818351
\(331\) 13.2676 0.729251 0.364625 0.931154i \(-0.381197\pi\)
0.364625 + 0.931154i \(0.381197\pi\)
\(332\) 6.48484 0.355902
\(333\) −11.0519 −0.605639
\(334\) −6.99047 −0.382501
\(335\) 1.94721 0.106387
\(336\) −2.47141 −0.134826
\(337\) 2.94659 0.160511 0.0802554 0.996774i \(-0.474426\pi\)
0.0802554 + 0.996774i \(0.474426\pi\)
\(338\) 7.48903 0.407350
\(339\) −17.7690 −0.965081
\(340\) 2.17479 0.117945
\(341\) −17.4327 −0.944033
\(342\) 4.31646 0.233407
\(343\) −19.8268 −1.07055
\(344\) 6.72451 0.362561
\(345\) −3.41442 −0.183826
\(346\) −22.5518 −1.21239
\(347\) 35.3434 1.89733 0.948666 0.316281i \(-0.102434\pi\)
0.948666 + 0.316281i \(0.102434\pi\)
\(348\) 6.35678 0.340759
\(349\) 26.4974 1.41837 0.709187 0.705020i \(-0.249062\pi\)
0.709187 + 0.705020i \(0.249062\pi\)
\(350\) 9.26222 0.495086
\(351\) −13.1098 −0.699747
\(352\) −2.61160 −0.139199
\(353\) 14.2395 0.757893 0.378947 0.925419i \(-0.376286\pi\)
0.378947 + 0.925419i \(0.376286\pi\)
\(354\) 14.6045 0.776220
\(355\) −1.01777 −0.0540177
\(356\) 2.80274 0.148545
\(357\) 12.1001 0.640403
\(358\) 14.8361 0.784112
\(359\) −29.0838 −1.53498 −0.767491 0.641059i \(-0.778495\pi\)
−0.767491 + 0.641059i \(0.778495\pi\)
\(360\) −0.603126 −0.0317876
\(361\) −8.89379 −0.468094
\(362\) 5.05705 0.265793
\(363\) 5.35600 0.281117
\(364\) −4.52735 −0.237298
\(365\) 1.84250 0.0964409
\(366\) 3.86527 0.202041
\(367\) −17.1077 −0.893013 −0.446506 0.894780i \(-0.647332\pi\)
−0.446506 + 0.894780i \(0.647332\pi\)
\(368\) −5.99830 −0.312683
\(369\) −1.68759 −0.0878524
\(370\) 3.61558 0.187965
\(371\) 24.3614 1.26478
\(372\) 8.55404 0.443506
\(373\) 28.5179 1.47660 0.738300 0.674473i \(-0.235629\pi\)
0.738300 + 0.674473i \(0.235629\pi\)
\(374\) 12.7865 0.661172
\(375\) −5.58000 −0.288150
\(376\) 8.23219 0.424543
\(377\) 11.6449 0.599744
\(378\) −10.7699 −0.553943
\(379\) −2.11558 −0.108670 −0.0543351 0.998523i \(-0.517304\pi\)
−0.0543351 + 0.998523i \(0.517304\pi\)
\(380\) −1.41211 −0.0724398
\(381\) 23.5185 1.20489
\(382\) −4.15023 −0.212344
\(383\) −3.74914 −0.191572 −0.0957860 0.995402i \(-0.530536\pi\)
−0.0957860 + 0.995402i \(0.530536\pi\)
\(384\) 1.28149 0.0653956
\(385\) −2.23724 −0.114020
\(386\) −13.2285 −0.673311
\(387\) 9.13048 0.464129
\(388\) 11.1750 0.567325
\(389\) 1.59074 0.0806539 0.0403269 0.999187i \(-0.487160\pi\)
0.0403269 + 0.999187i \(0.487160\pi\)
\(390\) 1.33630 0.0676660
\(391\) 29.3678 1.48519
\(392\) 3.28070 0.165700
\(393\) 12.6418 0.637694
\(394\) 5.25500 0.264743
\(395\) 6.10155 0.307002
\(396\) −3.54602 −0.178194
\(397\) −13.1087 −0.657906 −0.328953 0.944346i \(-0.606696\pi\)
−0.328953 + 0.944346i \(0.606696\pi\)
\(398\) −0.250345 −0.0125486
\(399\) −7.85668 −0.393326
\(400\) −4.80269 −0.240134
\(401\) −34.2894 −1.71233 −0.856165 0.516703i \(-0.827159\pi\)
−0.856165 + 0.516703i \(0.827159\pi\)
\(402\) −5.61761 −0.280181
\(403\) 15.6701 0.780582
\(404\) 0.0783921 0.00390015
\(405\) 1.36947 0.0680494
\(406\) 9.56651 0.474778
\(407\) 21.2574 1.05369
\(408\) −6.27418 −0.310618
\(409\) 30.1685 1.49174 0.745868 0.666094i \(-0.232035\pi\)
0.745868 + 0.666094i \(0.232035\pi\)
\(410\) 0.552088 0.0272657
\(411\) −5.32030 −0.262431
\(412\) 5.32059 0.262127
\(413\) 21.9788 1.08150
\(414\) −8.14444 −0.400277
\(415\) −2.88054 −0.141400
\(416\) 2.34755 0.115098
\(417\) 21.8538 1.07019
\(418\) −8.30236 −0.406082
\(419\) −11.5348 −0.563513 −0.281757 0.959486i \(-0.590917\pi\)
−0.281757 + 0.959486i \(0.590917\pi\)
\(420\) 1.09779 0.0535667
\(421\) 9.40504 0.458374 0.229187 0.973382i \(-0.426393\pi\)
0.229187 + 0.973382i \(0.426393\pi\)
\(422\) −8.95055 −0.435706
\(423\) 11.1776 0.543473
\(424\) −12.6320 −0.613464
\(425\) 23.5141 1.14060
\(426\) 2.93623 0.142261
\(427\) 5.81697 0.281503
\(428\) 16.0609 0.776331
\(429\) 7.85661 0.379321
\(430\) −2.98700 −0.144046
\(431\) −19.1709 −0.923431 −0.461716 0.887028i \(-0.652766\pi\)
−0.461716 + 0.887028i \(0.652766\pi\)
\(432\) 5.58445 0.268682
\(433\) 11.5169 0.553465 0.276732 0.960947i \(-0.410748\pi\)
0.276732 + 0.960947i \(0.410748\pi\)
\(434\) 12.8732 0.617935
\(435\) −2.82366 −0.135384
\(436\) −0.313546 −0.0150161
\(437\) −19.0688 −0.912182
\(438\) −5.31553 −0.253986
\(439\) −8.05954 −0.384661 −0.192330 0.981330i \(-0.561605\pi\)
−0.192330 + 0.981330i \(0.561605\pi\)
\(440\) 1.16007 0.0553039
\(441\) 4.45451 0.212119
\(442\) −11.4936 −0.546696
\(443\) 40.5401 1.92612 0.963060 0.269286i \(-0.0867878\pi\)
0.963060 + 0.269286i \(0.0867878\pi\)
\(444\) −10.4308 −0.495023
\(445\) −1.24497 −0.0590171
\(446\) −13.5470 −0.641470
\(447\) −11.2184 −0.530614
\(448\) 1.92855 0.0911154
\(449\) 1.90466 0.0898865 0.0449433 0.998990i \(-0.485689\pi\)
0.0449433 + 0.998990i \(0.485689\pi\)
\(450\) −6.52105 −0.307405
\(451\) 3.24594 0.152845
\(452\) 13.8659 0.652199
\(453\) −18.9102 −0.888480
\(454\) −11.5920 −0.544039
\(455\) 2.01103 0.0942788
\(456\) 4.07388 0.190777
\(457\) −2.20960 −0.103361 −0.0516803 0.998664i \(-0.516458\pi\)
−0.0516803 + 0.998664i \(0.516458\pi\)
\(458\) 4.42095 0.206577
\(459\) −27.3416 −1.27620
\(460\) 2.66442 0.124229
\(461\) −16.2813 −0.758298 −0.379149 0.925336i \(-0.623783\pi\)
−0.379149 + 0.925336i \(0.623783\pi\)
\(462\) 6.45434 0.300283
\(463\) 11.0963 0.515690 0.257845 0.966186i \(-0.416988\pi\)
0.257845 + 0.966186i \(0.416988\pi\)
\(464\) −4.96047 −0.230284
\(465\) −3.79967 −0.176206
\(466\) 2.05348 0.0951258
\(467\) −41.3828 −1.91497 −0.957484 0.288485i \(-0.906849\pi\)
−0.957484 + 0.288485i \(0.906849\pi\)
\(468\) 3.18748 0.147341
\(469\) −8.45411 −0.390374
\(470\) −3.65671 −0.168671
\(471\) 14.2284 0.655609
\(472\) −11.3965 −0.524568
\(473\) −17.5618 −0.807490
\(474\) −17.6027 −0.808519
\(475\) −15.2679 −0.700538
\(476\) −9.44221 −0.432783
\(477\) −17.1516 −0.785318
\(478\) 8.04445 0.367945
\(479\) −21.5887 −0.986413 −0.493206 0.869912i \(-0.664175\pi\)
−0.493206 + 0.869912i \(0.664175\pi\)
\(480\) −0.569232 −0.0259818
\(481\) −19.1081 −0.871252
\(482\) 23.3128 1.06187
\(483\) 14.8242 0.674527
\(484\) −4.17952 −0.189978
\(485\) −4.96390 −0.225399
\(486\) 12.8025 0.580733
\(487\) 20.3941 0.924145 0.462072 0.886842i \(-0.347106\pi\)
0.462072 + 0.886842i \(0.347106\pi\)
\(488\) −3.01624 −0.136539
\(489\) −12.9858 −0.587239
\(490\) −1.45728 −0.0658330
\(491\) 35.7075 1.61146 0.805728 0.592286i \(-0.201774\pi\)
0.805728 + 0.592286i \(0.201774\pi\)
\(492\) −1.59275 −0.0718067
\(493\) 24.2866 1.09381
\(494\) 7.46291 0.335772
\(495\) 1.57513 0.0707967
\(496\) −6.67509 −0.299720
\(497\) 4.41882 0.198211
\(498\) 8.31024 0.372391
\(499\) 5.46065 0.244452 0.122226 0.992502i \(-0.460997\pi\)
0.122226 + 0.992502i \(0.460997\pi\)
\(500\) 4.35432 0.194731
\(501\) −8.95819 −0.400222
\(502\) −8.67424 −0.387150
\(503\) 34.3443 1.53134 0.765668 0.643236i \(-0.222409\pi\)
0.765668 + 0.643236i \(0.222409\pi\)
\(504\) 2.61857 0.116640
\(505\) −0.0348215 −0.00154954
\(506\) 15.6652 0.696402
\(507\) 9.59709 0.426222
\(508\) −18.3525 −0.814261
\(509\) 31.3862 1.39117 0.695584 0.718445i \(-0.255146\pi\)
0.695584 + 0.718445i \(0.255146\pi\)
\(510\) 2.78697 0.123409
\(511\) −7.99950 −0.353877
\(512\) −1.00000 −0.0441942
\(513\) 17.7531 0.783820
\(514\) 30.9625 1.36570
\(515\) −2.36339 −0.104143
\(516\) 8.61737 0.379358
\(517\) −21.4992 −0.945534
\(518\) −15.6976 −0.689713
\(519\) −28.8998 −1.26856
\(520\) −1.04277 −0.0457285
\(521\) −1.21298 −0.0531416 −0.0265708 0.999647i \(-0.508459\pi\)
−0.0265708 + 0.999647i \(0.508459\pi\)
\(522\) −6.73529 −0.294795
\(523\) 37.2660 1.62953 0.814764 0.579793i \(-0.196867\pi\)
0.814764 + 0.579793i \(0.196867\pi\)
\(524\) −9.86495 −0.430952
\(525\) 11.8694 0.518024
\(526\) −15.4625 −0.674199
\(527\) 32.6814 1.42362
\(528\) −3.34674 −0.145648
\(529\) 12.9796 0.564330
\(530\) 5.61108 0.243730
\(531\) −15.4741 −0.671519
\(532\) 6.13091 0.265809
\(533\) −2.91775 −0.126382
\(534\) 3.59167 0.155427
\(535\) −7.13418 −0.308437
\(536\) 4.38366 0.189345
\(537\) 19.0123 0.820440
\(538\) −14.6441 −0.631351
\(539\) −8.56789 −0.369045
\(540\) −2.48059 −0.106748
\(541\) 36.4018 1.56504 0.782518 0.622628i \(-0.213935\pi\)
0.782518 + 0.622628i \(0.213935\pi\)
\(542\) −19.9045 −0.854970
\(543\) 6.48054 0.278107
\(544\) 4.89602 0.209915
\(545\) 0.139276 0.00596594
\(546\) −5.80174 −0.248292
\(547\) 19.9794 0.854259 0.427130 0.904190i \(-0.359525\pi\)
0.427130 + 0.904190i \(0.359525\pi\)
\(548\) 4.15166 0.177350
\(549\) −4.09543 −0.174789
\(550\) 12.5427 0.534824
\(551\) −15.7695 −0.671802
\(552\) −7.68674 −0.327169
\(553\) −26.4908 −1.12651
\(554\) −4.25582 −0.180813
\(555\) 4.63331 0.196673
\(556\) −17.0535 −0.723229
\(557\) −5.23383 −0.221764 −0.110882 0.993834i \(-0.535368\pi\)
−0.110882 + 0.993834i \(0.535368\pi\)
\(558\) −9.06339 −0.383684
\(559\) 15.7861 0.667680
\(560\) −0.856654 −0.0362003
\(561\) 16.3857 0.691804
\(562\) 6.91566 0.291720
\(563\) 12.6524 0.533236 0.266618 0.963802i \(-0.414094\pi\)
0.266618 + 0.963802i \(0.414094\pi\)
\(564\) 10.5494 0.444211
\(565\) −6.15920 −0.259120
\(566\) 3.17045 0.133264
\(567\) −5.94576 −0.249699
\(568\) −2.29127 −0.0961394
\(569\) −2.61190 −0.109496 −0.0547482 0.998500i \(-0.517436\pi\)
−0.0547482 + 0.998500i \(0.517436\pi\)
\(570\) −1.80960 −0.0757959
\(571\) 5.87976 0.246060 0.123030 0.992403i \(-0.460739\pi\)
0.123030 + 0.992403i \(0.460739\pi\)
\(572\) −6.13086 −0.256344
\(573\) −5.31846 −0.222182
\(574\) −2.39698 −0.100048
\(575\) 28.8080 1.20138
\(576\) −1.35779 −0.0565747
\(577\) −15.2383 −0.634381 −0.317190 0.948362i \(-0.602739\pi\)
−0.317190 + 0.948362i \(0.602739\pi\)
\(578\) −6.97100 −0.289955
\(579\) −16.9521 −0.704505
\(580\) 2.20342 0.0914922
\(581\) 12.5063 0.518850
\(582\) 14.3206 0.593609
\(583\) 32.9897 1.36630
\(584\) 4.14794 0.171643
\(585\) −1.41587 −0.0585389
\(586\) −18.8052 −0.776834
\(587\) 9.77351 0.403396 0.201698 0.979448i \(-0.435354\pi\)
0.201698 + 0.979448i \(0.435354\pi\)
\(588\) 4.20417 0.173377
\(589\) −21.2203 −0.874367
\(590\) 5.06230 0.208411
\(591\) 6.73421 0.277009
\(592\) 8.13959 0.334535
\(593\) 11.4282 0.469298 0.234649 0.972080i \(-0.424606\pi\)
0.234649 + 0.972080i \(0.424606\pi\)
\(594\) −14.5844 −0.598404
\(595\) 4.19420 0.171945
\(596\) 8.75424 0.358588
\(597\) −0.320813 −0.0131300
\(598\) −14.0813 −0.575826
\(599\) 28.5897 1.16814 0.584072 0.811702i \(-0.301459\pi\)
0.584072 + 0.811702i \(0.301459\pi\)
\(600\) −6.15458 −0.251260
\(601\) 19.3689 0.790076 0.395038 0.918665i \(-0.370731\pi\)
0.395038 + 0.918665i \(0.370731\pi\)
\(602\) 12.9685 0.528558
\(603\) 5.95210 0.242388
\(604\) 14.7565 0.600432
\(605\) 1.85653 0.0754787
\(606\) 0.100458 0.00408084
\(607\) 1.49904 0.0608441 0.0304221 0.999537i \(-0.490315\pi\)
0.0304221 + 0.999537i \(0.490315\pi\)
\(608\) −3.17903 −0.128927
\(609\) 12.2594 0.496774
\(610\) 1.33980 0.0542471
\(611\) 19.3254 0.781823
\(612\) 6.64778 0.268720
\(613\) −25.0088 −1.01010 −0.505048 0.863091i \(-0.668525\pi\)
−0.505048 + 0.863091i \(0.668525\pi\)
\(614\) 10.6421 0.429482
\(615\) 0.707494 0.0285289
\(616\) −5.03661 −0.202931
\(617\) −0.274941 −0.0110687 −0.00553436 0.999985i \(-0.501762\pi\)
−0.00553436 + 0.999985i \(0.501762\pi\)
\(618\) 6.81827 0.274271
\(619\) 42.6333 1.71358 0.856789 0.515668i \(-0.172456\pi\)
0.856789 + 0.515668i \(0.172456\pi\)
\(620\) 2.96505 0.119079
\(621\) −33.4972 −1.34420
\(622\) 13.1201 0.526069
\(623\) 5.40522 0.216556
\(624\) 3.00835 0.120430
\(625\) 22.0793 0.883171
\(626\) 1.47324 0.0588826
\(627\) −10.6394 −0.424895
\(628\) −11.1030 −0.443059
\(629\) −39.8516 −1.58899
\(630\) −1.16316 −0.0463413
\(631\) 9.11841 0.362998 0.181499 0.983391i \(-0.441905\pi\)
0.181499 + 0.983391i \(0.441905\pi\)
\(632\) 13.7362 0.546395
\(633\) −11.4700 −0.455892
\(634\) −4.44230 −0.176426
\(635\) 8.15212 0.323507
\(636\) −16.1877 −0.641885
\(637\) 7.70159 0.305148
\(638\) 12.9548 0.512885
\(639\) −3.11106 −0.123072
\(640\) 0.444196 0.0175584
\(641\) 33.9575 1.34124 0.670620 0.741801i \(-0.266028\pi\)
0.670620 + 0.741801i \(0.266028\pi\)
\(642\) 20.5818 0.812298
\(643\) −18.7186 −0.738190 −0.369095 0.929392i \(-0.620332\pi\)
−0.369095 + 0.929392i \(0.620332\pi\)
\(644\) −11.5680 −0.455843
\(645\) −3.82780 −0.150720
\(646\) 15.5646 0.612380
\(647\) −29.4523 −1.15789 −0.578945 0.815366i \(-0.696536\pi\)
−0.578945 + 0.815366i \(0.696536\pi\)
\(648\) 3.08302 0.121113
\(649\) 29.7632 1.16831
\(650\) −11.2745 −0.442223
\(651\) 16.4969 0.646564
\(652\) 10.1334 0.396855
\(653\) 9.54567 0.373551 0.186775 0.982403i \(-0.440196\pi\)
0.186775 + 0.982403i \(0.440196\pi\)
\(654\) −0.401805 −0.0157118
\(655\) 4.38198 0.171218
\(656\) 1.24289 0.0485268
\(657\) 5.63204 0.219727
\(658\) 15.8762 0.618918
\(659\) 10.0451 0.391301 0.195650 0.980674i \(-0.437318\pi\)
0.195650 + 0.980674i \(0.437318\pi\)
\(660\) 1.48661 0.0578661
\(661\) 0.515682 0.0200577 0.0100289 0.999950i \(-0.496808\pi\)
0.0100289 + 0.999950i \(0.496808\pi\)
\(662\) −13.2676 −0.515658
\(663\) −14.7289 −0.572024
\(664\) −6.48484 −0.251661
\(665\) −2.72333 −0.105606
\(666\) 11.0519 0.428251
\(667\) 29.7544 1.15209
\(668\) 6.99047 0.270469
\(669\) −17.3603 −0.671189
\(670\) −1.94721 −0.0752272
\(671\) 7.87723 0.304097
\(672\) 2.47141 0.0953367
\(673\) −50.0967 −1.93109 −0.965544 0.260240i \(-0.916198\pi\)
−0.965544 + 0.260240i \(0.916198\pi\)
\(674\) −2.94659 −0.113498
\(675\) −26.8204 −1.03232
\(676\) −7.48903 −0.288040
\(677\) 43.0916 1.65614 0.828072 0.560622i \(-0.189438\pi\)
0.828072 + 0.560622i \(0.189438\pi\)
\(678\) 17.7690 0.682415
\(679\) 21.5515 0.827073
\(680\) −2.17479 −0.0833996
\(681\) −14.8550 −0.569244
\(682\) 17.4327 0.667532
\(683\) −37.9121 −1.45067 −0.725334 0.688398i \(-0.758314\pi\)
−0.725334 + 0.688398i \(0.758314\pi\)
\(684\) −4.31646 −0.165044
\(685\) −1.84415 −0.0704615
\(686\) 19.8268 0.756992
\(687\) 5.66538 0.216148
\(688\) −6.72451 −0.256369
\(689\) −29.6542 −1.12973
\(690\) 3.41442 0.129985
\(691\) 50.5948 1.92472 0.962359 0.271781i \(-0.0876125\pi\)
0.962359 + 0.271781i \(0.0876125\pi\)
\(692\) 22.5518 0.857290
\(693\) −6.83866 −0.259779
\(694\) −35.3434 −1.34162
\(695\) 7.57510 0.287340
\(696\) −6.35678 −0.240953
\(697\) −6.08522 −0.230494
\(698\) −26.4974 −1.00294
\(699\) 2.63151 0.0995329
\(700\) −9.26222 −0.350079
\(701\) 13.0332 0.492256 0.246128 0.969237i \(-0.420842\pi\)
0.246128 + 0.969237i \(0.420842\pi\)
\(702\) 13.1098 0.494796
\(703\) 25.8760 0.975931
\(704\) 2.61160 0.0984285
\(705\) −4.68602 −0.176486
\(706\) −14.2395 −0.535912
\(707\) 0.151183 0.00568582
\(708\) −14.6045 −0.548871
\(709\) −33.3029 −1.25072 −0.625358 0.780338i \(-0.715047\pi\)
−0.625358 + 0.780338i \(0.715047\pi\)
\(710\) 1.01777 0.0381963
\(711\) 18.6508 0.699461
\(712\) −2.80274 −0.105037
\(713\) 40.0392 1.49948
\(714\) −12.1001 −0.452834
\(715\) 2.72331 0.101846
\(716\) −14.8361 −0.554451
\(717\) 10.3089 0.384991
\(718\) 29.0838 1.08540
\(719\) 18.3802 0.685464 0.342732 0.939433i \(-0.388648\pi\)
0.342732 + 0.939433i \(0.388648\pi\)
\(720\) 0.603126 0.0224772
\(721\) 10.2610 0.382140
\(722\) 8.89379 0.330993
\(723\) 29.8750 1.11106
\(724\) −5.05705 −0.187944
\(725\) 23.8236 0.884786
\(726\) −5.35600 −0.198780
\(727\) 29.5150 1.09465 0.547325 0.836920i \(-0.315646\pi\)
0.547325 + 0.836920i \(0.315646\pi\)
\(728\) 4.52735 0.167795
\(729\) 25.6553 0.950197
\(730\) −1.84250 −0.0681940
\(731\) 32.9233 1.21771
\(732\) −3.86527 −0.142865
\(733\) −20.5853 −0.760335 −0.380167 0.924918i \(-0.624134\pi\)
−0.380167 + 0.924918i \(0.624134\pi\)
\(734\) 17.1077 0.631455
\(735\) −1.86748 −0.0688830
\(736\) 5.99830 0.221100
\(737\) −11.4484 −0.421707
\(738\) 1.68759 0.0621210
\(739\) −45.7499 −1.68294 −0.841468 0.540307i \(-0.818308\pi\)
−0.841468 + 0.540307i \(0.818308\pi\)
\(740\) −3.61558 −0.132911
\(741\) 9.56362 0.351328
\(742\) −24.3614 −0.894335
\(743\) 4.75347 0.174388 0.0871940 0.996191i \(-0.472210\pi\)
0.0871940 + 0.996191i \(0.472210\pi\)
\(744\) −8.55404 −0.313606
\(745\) −3.88860 −0.142467
\(746\) −28.5179 −1.04411
\(747\) −8.80507 −0.322161
\(748\) −12.7865 −0.467519
\(749\) 30.9742 1.13177
\(750\) 5.58000 0.203753
\(751\) 3.65764 0.133469 0.0667346 0.997771i \(-0.478742\pi\)
0.0667346 + 0.997771i \(0.478742\pi\)
\(752\) −8.23219 −0.300197
\(753\) −11.1159 −0.405087
\(754\) −11.6449 −0.424083
\(755\) −6.55477 −0.238553
\(756\) 10.7699 0.391697
\(757\) −1.61821 −0.0588149 −0.0294075 0.999568i \(-0.509362\pi\)
−0.0294075 + 0.999568i \(0.509362\pi\)
\(758\) 2.11558 0.0768414
\(759\) 20.0747 0.728666
\(760\) 1.41211 0.0512227
\(761\) 26.9525 0.977027 0.488514 0.872556i \(-0.337539\pi\)
0.488514 + 0.872556i \(0.337539\pi\)
\(762\) −23.5185 −0.851985
\(763\) −0.604689 −0.0218912
\(764\) 4.15023 0.150150
\(765\) −2.95292 −0.106763
\(766\) 3.74914 0.135462
\(767\) −26.7539 −0.966026
\(768\) −1.28149 −0.0462417
\(769\) −12.2465 −0.441621 −0.220810 0.975317i \(-0.570870\pi\)
−0.220810 + 0.975317i \(0.570870\pi\)
\(770\) 2.23724 0.0806246
\(771\) 39.6780 1.42897
\(772\) 13.2285 0.476103
\(773\) −19.6075 −0.705235 −0.352617 0.935768i \(-0.614708\pi\)
−0.352617 + 0.935768i \(0.614708\pi\)
\(774\) −9.13048 −0.328188
\(775\) 32.0584 1.15157
\(776\) −11.1750 −0.401159
\(777\) −20.1163 −0.721667
\(778\) −1.59074 −0.0570309
\(779\) 3.95119 0.141566
\(780\) −1.33630 −0.0478471
\(781\) 5.98388 0.214120
\(782\) −29.3678 −1.05019
\(783\) −27.7015 −0.989971
\(784\) −3.28070 −0.117168
\(785\) 4.93193 0.176028
\(786\) −12.6418 −0.450918
\(787\) −0.501678 −0.0178829 −0.00894145 0.999960i \(-0.502846\pi\)
−0.00894145 + 0.999960i \(0.502846\pi\)
\(788\) −5.25500 −0.187202
\(789\) −19.8150 −0.705434
\(790\) −6.10155 −0.217083
\(791\) 26.7411 0.950805
\(792\) 3.54602 0.126002
\(793\) −7.08076 −0.251445
\(794\) 13.1087 0.465210
\(795\) 7.19053 0.255022
\(796\) 0.250345 0.00887323
\(797\) −13.6881 −0.484859 −0.242430 0.970169i \(-0.577944\pi\)
−0.242430 + 0.970169i \(0.577944\pi\)
\(798\) 7.85668 0.278123
\(799\) 40.3049 1.42589
\(800\) 4.80269 0.169801
\(801\) −3.80554 −0.134462
\(802\) 34.2894 1.21080
\(803\) −10.8328 −0.382280
\(804\) 5.61761 0.198118
\(805\) 5.13847 0.181107
\(806\) −15.6701 −0.551955
\(807\) −18.7662 −0.660601
\(808\) −0.0783921 −0.00275782
\(809\) −13.8045 −0.485340 −0.242670 0.970109i \(-0.578023\pi\)
−0.242670 + 0.970109i \(0.578023\pi\)
\(810\) −1.36947 −0.0481182
\(811\) 24.0332 0.843921 0.421960 0.906614i \(-0.361342\pi\)
0.421960 + 0.906614i \(0.361342\pi\)
\(812\) −9.56651 −0.335719
\(813\) −25.5073 −0.894580
\(814\) −21.2574 −0.745071
\(815\) −4.50122 −0.157671
\(816\) 6.27418 0.219640
\(817\) −21.3774 −0.747900
\(818\) −30.1685 −1.05482
\(819\) 6.14721 0.214801
\(820\) −0.552088 −0.0192798
\(821\) 47.2729 1.64983 0.824917 0.565253i \(-0.191222\pi\)
0.824917 + 0.565253i \(0.191222\pi\)
\(822\) 5.32030 0.185567
\(823\) 42.7677 1.49079 0.745395 0.666623i \(-0.232261\pi\)
0.745395 + 0.666623i \(0.232261\pi\)
\(824\) −5.32059 −0.185352
\(825\) 16.0733 0.559602
\(826\) −21.9788 −0.764739
\(827\) −7.16581 −0.249180 −0.124590 0.992208i \(-0.539762\pi\)
−0.124590 + 0.992208i \(0.539762\pi\)
\(828\) 8.14444 0.283039
\(829\) 6.01419 0.208881 0.104441 0.994531i \(-0.466695\pi\)
0.104441 + 0.994531i \(0.466695\pi\)
\(830\) 2.88054 0.0999852
\(831\) −5.45378 −0.189189
\(832\) −2.34755 −0.0813865
\(833\) 16.0624 0.556528
\(834\) −21.8538 −0.756736
\(835\) −3.10514 −0.107458
\(836\) 8.30236 0.287143
\(837\) −37.2767 −1.28847
\(838\) 11.5348 0.398464
\(839\) 53.1519 1.83501 0.917503 0.397728i \(-0.130201\pi\)
0.917503 + 0.397728i \(0.130201\pi\)
\(840\) −1.09779 −0.0378774
\(841\) −4.39374 −0.151508
\(842\) −9.40504 −0.324119
\(843\) 8.86233 0.305235
\(844\) 8.95055 0.308091
\(845\) 3.32660 0.114439
\(846\) −11.1776 −0.384294
\(847\) −8.06042 −0.276959
\(848\) 12.6320 0.433784
\(849\) 4.06290 0.139438
\(850\) −23.5141 −0.806526
\(851\) −48.8237 −1.67365
\(852\) −2.93623 −0.100593
\(853\) 22.4049 0.767130 0.383565 0.923514i \(-0.374696\pi\)
0.383565 + 0.923514i \(0.374696\pi\)
\(854\) −5.81697 −0.199053
\(855\) 1.91736 0.0655722
\(856\) −16.0609 −0.548949
\(857\) 53.7524 1.83615 0.918073 0.396412i \(-0.129745\pi\)
0.918073 + 0.396412i \(0.129745\pi\)
\(858\) −7.85661 −0.268220
\(859\) −41.4151 −1.41306 −0.706532 0.707681i \(-0.749741\pi\)
−0.706532 + 0.707681i \(0.749741\pi\)
\(860\) 2.98700 0.101856
\(861\) −3.07170 −0.104683
\(862\) 19.1709 0.652964
\(863\) 47.6234 1.62112 0.810560 0.585656i \(-0.199163\pi\)
0.810560 + 0.585656i \(0.199163\pi\)
\(864\) −5.58445 −0.189987
\(865\) −10.0174 −0.340603
\(866\) −11.5169 −0.391359
\(867\) −8.93325 −0.303389
\(868\) −12.8732 −0.436946
\(869\) −35.8734 −1.21692
\(870\) 2.82366 0.0957309
\(871\) 10.2908 0.348692
\(872\) 0.313546 0.0106180
\(873\) −15.1733 −0.513540
\(874\) 19.0688 0.645010
\(875\) 8.39752 0.283888
\(876\) 5.31553 0.179595
\(877\) −8.03858 −0.271444 −0.135722 0.990747i \(-0.543335\pi\)
−0.135722 + 0.990747i \(0.543335\pi\)
\(878\) 8.05954 0.271996
\(879\) −24.0986 −0.812824
\(880\) −1.16007 −0.0391058
\(881\) −15.3881 −0.518439 −0.259220 0.965818i \(-0.583465\pi\)
−0.259220 + 0.965818i \(0.583465\pi\)
\(882\) −4.45451 −0.149991
\(883\) 56.4447 1.89952 0.949758 0.312985i \(-0.101329\pi\)
0.949758 + 0.312985i \(0.101329\pi\)
\(884\) 11.4936 0.386572
\(885\) 6.48727 0.218067
\(886\) −40.5401 −1.36197
\(887\) 3.80241 0.127672 0.0638362 0.997960i \(-0.479666\pi\)
0.0638362 + 0.997960i \(0.479666\pi\)
\(888\) 10.4308 0.350034
\(889\) −35.3937 −1.18707
\(890\) 1.24497 0.0417314
\(891\) −8.05164 −0.269740
\(892\) 13.5470 0.453588
\(893\) −26.1703 −0.875757
\(894\) 11.2184 0.375201
\(895\) 6.59014 0.220284
\(896\) −1.92855 −0.0644283
\(897\) −18.0450 −0.602504
\(898\) −1.90466 −0.0635594
\(899\) 33.1116 1.10433
\(900\) 6.52105 0.217368
\(901\) −61.8464 −2.06040
\(902\) −3.24594 −0.108078
\(903\) 16.6190 0.553046
\(904\) −13.8659 −0.461174
\(905\) 2.24632 0.0746703
\(906\) 18.9102 0.628250
\(907\) −22.5071 −0.747337 −0.373669 0.927562i \(-0.621900\pi\)
−0.373669 + 0.927562i \(0.621900\pi\)
\(908\) 11.5920 0.384694
\(909\) −0.106440 −0.00353040
\(910\) −2.01103 −0.0666652
\(911\) 13.0053 0.430884 0.215442 0.976517i \(-0.430881\pi\)
0.215442 + 0.976517i \(0.430881\pi\)
\(912\) −4.07388 −0.134900
\(913\) 16.9358 0.560495
\(914\) 2.20960 0.0730870
\(915\) 1.71694 0.0567603
\(916\) −4.42095 −0.146072
\(917\) −19.0250 −0.628262
\(918\) 27.3416 0.902407
\(919\) −8.14130 −0.268557 −0.134278 0.990944i \(-0.542872\pi\)
−0.134278 + 0.990944i \(0.542872\pi\)
\(920\) −2.66442 −0.0878434
\(921\) 13.6378 0.449379
\(922\) 16.2813 0.536197
\(923\) −5.37885 −0.177047
\(924\) −6.45434 −0.212332
\(925\) −39.0919 −1.28533
\(926\) −11.0963 −0.364648
\(927\) −7.22426 −0.237276
\(928\) 4.96047 0.162835
\(929\) −4.08394 −0.133990 −0.0669948 0.997753i \(-0.521341\pi\)
−0.0669948 + 0.997753i \(0.521341\pi\)
\(930\) 3.79967 0.124596
\(931\) −10.4294 −0.341811
\(932\) −2.05348 −0.0672641
\(933\) 16.8133 0.550441
\(934\) 41.3828 1.35409
\(935\) 5.67970 0.185746
\(936\) −3.18748 −0.104186
\(937\) −2.04053 −0.0666612 −0.0333306 0.999444i \(-0.510611\pi\)
−0.0333306 + 0.999444i \(0.510611\pi\)
\(938\) 8.45411 0.276036
\(939\) 1.88794 0.0616105
\(940\) 3.65671 0.119269
\(941\) 38.3103 1.24888 0.624439 0.781073i \(-0.285327\pi\)
0.624439 + 0.781073i \(0.285327\pi\)
\(942\) −14.2284 −0.463586
\(943\) −7.45524 −0.242776
\(944\) 11.3965 0.370925
\(945\) −4.78395 −0.155622
\(946\) 17.5618 0.570982
\(947\) −3.35828 −0.109129 −0.0545647 0.998510i \(-0.517377\pi\)
−0.0545647 + 0.998510i \(0.517377\pi\)
\(948\) 17.6027 0.571709
\(949\) 9.73748 0.316092
\(950\) 15.2679 0.495355
\(951\) −5.69275 −0.184600
\(952\) 9.44221 0.306024
\(953\) 3.07212 0.0995158 0.0497579 0.998761i \(-0.484155\pi\)
0.0497579 + 0.998761i \(0.484155\pi\)
\(954\) 17.1516 0.555304
\(955\) −1.84352 −0.0596548
\(956\) −8.04445 −0.260176
\(957\) 16.6014 0.536646
\(958\) 21.5887 0.697499
\(959\) 8.00668 0.258549
\(960\) 0.569232 0.0183719
\(961\) 13.5568 0.437318
\(962\) 19.1081 0.616068
\(963\) −21.8073 −0.702731
\(964\) −23.3128 −0.750853
\(965\) −5.87603 −0.189156
\(966\) −14.8242 −0.476962
\(967\) 43.4270 1.39652 0.698259 0.715846i \(-0.253959\pi\)
0.698259 + 0.715846i \(0.253959\pi\)
\(968\) 4.17952 0.134335
\(969\) 19.9458 0.640751
\(970\) 4.96390 0.159381
\(971\) 12.8866 0.413552 0.206776 0.978388i \(-0.433703\pi\)
0.206776 + 0.978388i \(0.433703\pi\)
\(972\) −12.8025 −0.410640
\(973\) −32.8885 −1.05436
\(974\) −20.3941 −0.653469
\(975\) −14.4482 −0.462711
\(976\) 3.01624 0.0965475
\(977\) −24.3447 −0.778855 −0.389427 0.921057i \(-0.627327\pi\)
−0.389427 + 0.921057i \(0.627327\pi\)
\(978\) 12.9858 0.415241
\(979\) 7.31965 0.233937
\(980\) 1.45728 0.0465510
\(981\) 0.425731 0.0135925
\(982\) −35.7075 −1.13947
\(983\) 26.7994 0.854769 0.427385 0.904070i \(-0.359435\pi\)
0.427385 + 0.904070i \(0.359435\pi\)
\(984\) 1.59275 0.0507750
\(985\) 2.33425 0.0743755
\(986\) −24.2866 −0.773441
\(987\) 20.3451 0.647592
\(988\) −7.46291 −0.237427
\(989\) 40.3356 1.28260
\(990\) −1.57513 −0.0500608
\(991\) −21.1945 −0.673264 −0.336632 0.941636i \(-0.609288\pi\)
−0.336632 + 0.941636i \(0.609288\pi\)
\(992\) 6.67509 0.211934
\(993\) −17.0022 −0.539548
\(994\) −4.41882 −0.140156
\(995\) −0.111202 −0.00352535
\(996\) −8.31024 −0.263320
\(997\) −3.80139 −0.120391 −0.0601956 0.998187i \(-0.519172\pi\)
−0.0601956 + 0.998187i \(0.519172\pi\)
\(998\) −5.46065 −0.172854
\(999\) 45.4551 1.43814
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 8002.2.a.e.1.24 77
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8002.2.a.e.1.24 77 1.1 even 1 trivial