Properties

Label 8002.2.a.d.1.21
Level $8002$
Weight $2$
Character 8002.1
Self dual yes
Analytic conductor $63.896$
Analytic rank $1$
Dimension $69$
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: \(1\)
Dimension: \(69\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.21
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.86236 q^{3} +1.00000 q^{4} -0.878383 q^{5} -1.86236 q^{6} +0.141603 q^{7} +1.00000 q^{8} +0.468368 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.86236 q^{3} +1.00000 q^{4} -0.878383 q^{5} -1.86236 q^{6} +0.141603 q^{7} +1.00000 q^{8} +0.468368 q^{9} -0.878383 q^{10} -3.52538 q^{11} -1.86236 q^{12} +0.284387 q^{13} +0.141603 q^{14} +1.63586 q^{15} +1.00000 q^{16} +4.97859 q^{17} +0.468368 q^{18} -5.84511 q^{19} -0.878383 q^{20} -0.263716 q^{21} -3.52538 q^{22} +2.56520 q^{23} -1.86236 q^{24} -4.22844 q^{25} +0.284387 q^{26} +4.71480 q^{27} +0.141603 q^{28} +7.07784 q^{29} +1.63586 q^{30} +9.16073 q^{31} +1.00000 q^{32} +6.56551 q^{33} +4.97859 q^{34} -0.124382 q^{35} +0.468368 q^{36} -3.98793 q^{37} -5.84511 q^{38} -0.529629 q^{39} -0.878383 q^{40} -3.25631 q^{41} -0.263716 q^{42} -0.968504 q^{43} -3.52538 q^{44} -0.411407 q^{45} +2.56520 q^{46} +4.49473 q^{47} -1.86236 q^{48} -6.97995 q^{49} -4.22844 q^{50} -9.27191 q^{51} +0.284387 q^{52} +2.63159 q^{53} +4.71480 q^{54} +3.09664 q^{55} +0.141603 q^{56} +10.8857 q^{57} +7.07784 q^{58} -11.8218 q^{59} +1.63586 q^{60} +0.646205 q^{61} +9.16073 q^{62} +0.0663224 q^{63} +1.00000 q^{64} -0.249801 q^{65} +6.56551 q^{66} +13.2199 q^{67} +4.97859 q^{68} -4.77732 q^{69} -0.124382 q^{70} +11.0119 q^{71} +0.468368 q^{72} +5.90064 q^{73} -3.98793 q^{74} +7.87486 q^{75} -5.84511 q^{76} -0.499205 q^{77} -0.529629 q^{78} -4.49078 q^{79} -0.878383 q^{80} -10.1857 q^{81} -3.25631 q^{82} -14.8597 q^{83} -0.263716 q^{84} -4.37311 q^{85} -0.968504 q^{86} -13.1815 q^{87} -3.52538 q^{88} -11.9727 q^{89} -0.411407 q^{90} +0.0402701 q^{91} +2.56520 q^{92} -17.0605 q^{93} +4.49473 q^{94} +5.13425 q^{95} -1.86236 q^{96} +7.98287 q^{97} -6.97995 q^{98} -1.65118 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 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
\(3\) −1.86236 −1.07523 −0.537616 0.843190i \(-0.680675\pi\)
−0.537616 + 0.843190i \(0.680675\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.878383 −0.392825 −0.196412 0.980521i \(-0.562929\pi\)
−0.196412 + 0.980521i \(0.562929\pi\)
\(6\) −1.86236 −0.760303
\(7\) 0.141603 0.0535210 0.0267605 0.999642i \(-0.491481\pi\)
0.0267605 + 0.999642i \(0.491481\pi\)
\(8\) 1.00000 0.353553
\(9\) 0.468368 0.156123
\(10\) −0.878383 −0.277769
\(11\) −3.52538 −1.06294 −0.531471 0.847076i \(-0.678361\pi\)
−0.531471 + 0.847076i \(0.678361\pi\)
\(12\) −1.86236 −0.537616
\(13\) 0.284387 0.0788747 0.0394373 0.999222i \(-0.487443\pi\)
0.0394373 + 0.999222i \(0.487443\pi\)
\(14\) 0.141603 0.0378450
\(15\) 1.63586 0.422378
\(16\) 1.00000 0.250000
\(17\) 4.97859 1.20749 0.603743 0.797179i \(-0.293675\pi\)
0.603743 + 0.797179i \(0.293675\pi\)
\(18\) 0.468368 0.110395
\(19\) −5.84511 −1.34096 −0.670480 0.741927i \(-0.733912\pi\)
−0.670480 + 0.741927i \(0.733912\pi\)
\(20\) −0.878383 −0.196412
\(21\) −0.263716 −0.0575474
\(22\) −3.52538 −0.751614
\(23\) 2.56520 0.534882 0.267441 0.963574i \(-0.413822\pi\)
0.267441 + 0.963574i \(0.413822\pi\)
\(24\) −1.86236 −0.380152
\(25\) −4.22844 −0.845689
\(26\) 0.284387 0.0557728
\(27\) 4.71480 0.907363
\(28\) 0.141603 0.0267605
\(29\) 7.07784 1.31432 0.657161 0.753750i \(-0.271757\pi\)
0.657161 + 0.753750i \(0.271757\pi\)
\(30\) 1.63586 0.298666
\(31\) 9.16073 1.64532 0.822658 0.568536i \(-0.192490\pi\)
0.822658 + 0.568536i \(0.192490\pi\)
\(32\) 1.00000 0.176777
\(33\) 6.56551 1.14291
\(34\) 4.97859 0.853822
\(35\) −0.124382 −0.0210244
\(36\) 0.468368 0.0780614
\(37\) −3.98793 −0.655611 −0.327805 0.944745i \(-0.606309\pi\)
−0.327805 + 0.944745i \(0.606309\pi\)
\(38\) −5.84511 −0.948202
\(39\) −0.529629 −0.0848085
\(40\) −0.878383 −0.138885
\(41\) −3.25631 −0.508550 −0.254275 0.967132i \(-0.581837\pi\)
−0.254275 + 0.967132i \(0.581837\pi\)
\(42\) −0.263716 −0.0406922
\(43\) −0.968504 −0.147696 −0.0738478 0.997270i \(-0.523528\pi\)
−0.0738478 + 0.997270i \(0.523528\pi\)
\(44\) −3.52538 −0.531471
\(45\) −0.411407 −0.0613289
\(46\) 2.56520 0.378218
\(47\) 4.49473 0.655623 0.327812 0.944743i \(-0.393689\pi\)
0.327812 + 0.944743i \(0.393689\pi\)
\(48\) −1.86236 −0.268808
\(49\) −6.97995 −0.997136
\(50\) −4.22844 −0.597992
\(51\) −9.27191 −1.29833
\(52\) 0.284387 0.0394373
\(53\) 2.63159 0.361477 0.180739 0.983531i \(-0.442151\pi\)
0.180739 + 0.983531i \(0.442151\pi\)
\(54\) 4.71480 0.641603
\(55\) 3.09664 0.417550
\(56\) 0.141603 0.0189225
\(57\) 10.8857 1.44184
\(58\) 7.07784 0.929366
\(59\) −11.8218 −1.53906 −0.769531 0.638609i \(-0.779510\pi\)
−0.769531 + 0.638609i \(0.779510\pi\)
\(60\) 1.63586 0.211189
\(61\) 0.646205 0.0827380 0.0413690 0.999144i \(-0.486828\pi\)
0.0413690 + 0.999144i \(0.486828\pi\)
\(62\) 9.16073 1.16341
\(63\) 0.0663224 0.00835584
\(64\) 1.00000 0.125000
\(65\) −0.249801 −0.0309839
\(66\) 6.56551 0.808159
\(67\) 13.2199 1.61507 0.807533 0.589822i \(-0.200802\pi\)
0.807533 + 0.589822i \(0.200802\pi\)
\(68\) 4.97859 0.603743
\(69\) −4.77732 −0.575122
\(70\) −0.124382 −0.0148665
\(71\) 11.0119 1.30687 0.653436 0.756981i \(-0.273327\pi\)
0.653436 + 0.756981i \(0.273327\pi\)
\(72\) 0.468368 0.0551977
\(73\) 5.90064 0.690618 0.345309 0.938489i \(-0.387774\pi\)
0.345309 + 0.938489i \(0.387774\pi\)
\(74\) −3.98793 −0.463587
\(75\) 7.87486 0.909311
\(76\) −5.84511 −0.670480
\(77\) −0.499205 −0.0568897
\(78\) −0.529629 −0.0599687
\(79\) −4.49078 −0.505252 −0.252626 0.967564i \(-0.581294\pi\)
−0.252626 + 0.967564i \(0.581294\pi\)
\(80\) −0.878383 −0.0982062
\(81\) −10.1857 −1.13175
\(82\) −3.25631 −0.359599
\(83\) −14.8597 −1.63107 −0.815533 0.578710i \(-0.803556\pi\)
−0.815533 + 0.578710i \(0.803556\pi\)
\(84\) −0.263716 −0.0287737
\(85\) −4.37311 −0.474331
\(86\) −0.968504 −0.104437
\(87\) −13.1815 −1.41320
\(88\) −3.52538 −0.375807
\(89\) −11.9727 −1.26910 −0.634550 0.772882i \(-0.718814\pi\)
−0.634550 + 0.772882i \(0.718814\pi\)
\(90\) −0.411407 −0.0433661
\(91\) 0.0402701 0.00422145
\(92\) 2.56520 0.267441
\(93\) −17.0605 −1.76910
\(94\) 4.49473 0.463596
\(95\) 5.13425 0.526763
\(96\) −1.86236 −0.190076
\(97\) 7.98287 0.810538 0.405269 0.914198i \(-0.367178\pi\)
0.405269 + 0.914198i \(0.367178\pi\)
\(98\) −6.97995 −0.705081
\(99\) −1.65118 −0.165950
\(100\) −4.22844 −0.422844
\(101\) −11.6296 −1.15719 −0.578595 0.815615i \(-0.696399\pi\)
−0.578595 + 0.815615i \(0.696399\pi\)
\(102\) −9.27191 −0.918056
\(103\) 5.86002 0.577405 0.288702 0.957419i \(-0.406776\pi\)
0.288702 + 0.957419i \(0.406776\pi\)
\(104\) 0.284387 0.0278864
\(105\) 0.231643 0.0226061
\(106\) 2.63159 0.255603
\(107\) −0.654070 −0.0632313 −0.0316157 0.999500i \(-0.510065\pi\)
−0.0316157 + 0.999500i \(0.510065\pi\)
\(108\) 4.71480 0.453682
\(109\) 1.24286 0.119044 0.0595220 0.998227i \(-0.481042\pi\)
0.0595220 + 0.998227i \(0.481042\pi\)
\(110\) 3.09664 0.295253
\(111\) 7.42694 0.704934
\(112\) 0.141603 0.0133802
\(113\) −6.47849 −0.609445 −0.304723 0.952441i \(-0.598564\pi\)
−0.304723 + 0.952441i \(0.598564\pi\)
\(114\) 10.8857 1.01954
\(115\) −2.25323 −0.210115
\(116\) 7.07784 0.657161
\(117\) 0.133198 0.0123141
\(118\) −11.8218 −1.08828
\(119\) 0.704985 0.0646258
\(120\) 1.63586 0.149333
\(121\) 1.42831 0.129847
\(122\) 0.646205 0.0585046
\(123\) 6.06441 0.546809
\(124\) 9.16073 0.822658
\(125\) 8.10611 0.725033
\(126\) 0.0663224 0.00590847
\(127\) −5.90292 −0.523799 −0.261900 0.965095i \(-0.584349\pi\)
−0.261900 + 0.965095i \(0.584349\pi\)
\(128\) 1.00000 0.0883883
\(129\) 1.80370 0.158807
\(130\) −0.249801 −0.0219090
\(131\) 13.4225 1.17273 0.586366 0.810046i \(-0.300558\pi\)
0.586366 + 0.810046i \(0.300558\pi\)
\(132\) 6.56551 0.571455
\(133\) −0.827686 −0.0717695
\(134\) 13.2199 1.14202
\(135\) −4.14140 −0.356435
\(136\) 4.97859 0.426911
\(137\) 4.17159 0.356403 0.178201 0.983994i \(-0.442972\pi\)
0.178201 + 0.983994i \(0.442972\pi\)
\(138\) −4.77732 −0.406672
\(139\) −12.2987 −1.04316 −0.521581 0.853202i \(-0.674658\pi\)
−0.521581 + 0.853202i \(0.674658\pi\)
\(140\) −0.124382 −0.0105122
\(141\) −8.37078 −0.704947
\(142\) 11.0119 0.924099
\(143\) −1.00257 −0.0838392
\(144\) 0.468368 0.0390307
\(145\) −6.21706 −0.516299
\(146\) 5.90064 0.488341
\(147\) 12.9991 1.07215
\(148\) −3.98793 −0.327805
\(149\) −15.4457 −1.26536 −0.632682 0.774412i \(-0.718046\pi\)
−0.632682 + 0.774412i \(0.718046\pi\)
\(150\) 7.87486 0.642980
\(151\) −9.89714 −0.805418 −0.402709 0.915328i \(-0.631931\pi\)
−0.402709 + 0.915328i \(0.631931\pi\)
\(152\) −5.84511 −0.474101
\(153\) 2.33181 0.188516
\(154\) −0.499205 −0.0402271
\(155\) −8.04664 −0.646321
\(156\) −0.529629 −0.0424043
\(157\) −18.7245 −1.49437 −0.747187 0.664614i \(-0.768596\pi\)
−0.747187 + 0.664614i \(0.768596\pi\)
\(158\) −4.49078 −0.357267
\(159\) −4.90096 −0.388672
\(160\) −0.878383 −0.0694423
\(161\) 0.363241 0.0286274
\(162\) −10.1857 −0.800267
\(163\) 8.96605 0.702275 0.351138 0.936324i \(-0.385795\pi\)
0.351138 + 0.936324i \(0.385795\pi\)
\(164\) −3.25631 −0.254275
\(165\) −5.76704 −0.448963
\(166\) −14.8597 −1.15334
\(167\) −12.1450 −0.939807 −0.469904 0.882718i \(-0.655711\pi\)
−0.469904 + 0.882718i \(0.655711\pi\)
\(168\) −0.263716 −0.0203461
\(169\) −12.9191 −0.993779
\(170\) −4.37311 −0.335402
\(171\) −2.73766 −0.209354
\(172\) −0.968504 −0.0738478
\(173\) −1.44330 −0.109732 −0.0548662 0.998494i \(-0.517473\pi\)
−0.0548662 + 0.998494i \(0.517473\pi\)
\(174\) −13.1815 −0.999284
\(175\) −0.598761 −0.0452621
\(176\) −3.52538 −0.265736
\(177\) 22.0163 1.65485
\(178\) −11.9727 −0.897389
\(179\) −4.37680 −0.327137 −0.163569 0.986532i \(-0.552301\pi\)
−0.163569 + 0.986532i \(0.552301\pi\)
\(180\) −0.411407 −0.0306645
\(181\) −23.9988 −1.78382 −0.891910 0.452214i \(-0.850634\pi\)
−0.891910 + 0.452214i \(0.850634\pi\)
\(182\) 0.0402701 0.00298502
\(183\) −1.20346 −0.0889626
\(184\) 2.56520 0.189109
\(185\) 3.50293 0.257540
\(186\) −17.0605 −1.25094
\(187\) −17.5514 −1.28349
\(188\) 4.49473 0.327812
\(189\) 0.667631 0.0485630
\(190\) 5.13425 0.372478
\(191\) 10.5894 0.766221 0.383110 0.923703i \(-0.374853\pi\)
0.383110 + 0.923703i \(0.374853\pi\)
\(192\) −1.86236 −0.134404
\(193\) −8.01743 −0.577108 −0.288554 0.957464i \(-0.593174\pi\)
−0.288554 + 0.957464i \(0.593174\pi\)
\(194\) 7.98287 0.573137
\(195\) 0.465217 0.0333149
\(196\) −6.97995 −0.498568
\(197\) −16.6672 −1.18749 −0.593744 0.804654i \(-0.702351\pi\)
−0.593744 + 0.804654i \(0.702351\pi\)
\(198\) −1.65118 −0.117344
\(199\) 26.9698 1.91184 0.955919 0.293631i \(-0.0948636\pi\)
0.955919 + 0.293631i \(0.0948636\pi\)
\(200\) −4.22844 −0.298996
\(201\) −24.6201 −1.73657
\(202\) −11.6296 −0.818257
\(203\) 1.00225 0.0703438
\(204\) −9.27191 −0.649164
\(205\) 2.86029 0.199771
\(206\) 5.86002 0.408287
\(207\) 1.20146 0.0835072
\(208\) 0.284387 0.0197187
\(209\) 20.6062 1.42536
\(210\) 0.231643 0.0159849
\(211\) −22.0766 −1.51982 −0.759910 0.650029i \(-0.774757\pi\)
−0.759910 + 0.650029i \(0.774757\pi\)
\(212\) 2.63159 0.180739
\(213\) −20.5081 −1.40519
\(214\) −0.654070 −0.0447113
\(215\) 0.850718 0.0580185
\(216\) 4.71480 0.320801
\(217\) 1.29719 0.0880589
\(218\) 1.24286 0.0841768
\(219\) −10.9891 −0.742574
\(220\) 3.09664 0.208775
\(221\) 1.41585 0.0952401
\(222\) 7.42694 0.498463
\(223\) −9.63646 −0.645305 −0.322653 0.946517i \(-0.604575\pi\)
−0.322653 + 0.946517i \(0.604575\pi\)
\(224\) 0.141603 0.00946126
\(225\) −1.98047 −0.132031
\(226\) −6.47849 −0.430943
\(227\) 17.6804 1.17349 0.586744 0.809773i \(-0.300410\pi\)
0.586744 + 0.809773i \(0.300410\pi\)
\(228\) 10.8857 0.720921
\(229\) −0.450830 −0.0297917 −0.0148958 0.999889i \(-0.504742\pi\)
−0.0148958 + 0.999889i \(0.504742\pi\)
\(230\) −2.25323 −0.148574
\(231\) 0.929698 0.0611696
\(232\) 7.07784 0.464683
\(233\) −0.302059 −0.0197886 −0.00989428 0.999951i \(-0.503149\pi\)
−0.00989428 + 0.999951i \(0.503149\pi\)
\(234\) 0.133198 0.00870741
\(235\) −3.94809 −0.257545
\(236\) −11.8218 −0.769531
\(237\) 8.36343 0.543263
\(238\) 0.704985 0.0456974
\(239\) 3.38848 0.219183 0.109591 0.993977i \(-0.465046\pi\)
0.109591 + 0.993977i \(0.465046\pi\)
\(240\) 1.63586 0.105594
\(241\) 18.1980 1.17224 0.586118 0.810226i \(-0.300656\pi\)
0.586118 + 0.810226i \(0.300656\pi\)
\(242\) 1.42831 0.0918155
\(243\) 4.82507 0.309528
\(244\) 0.646205 0.0413690
\(245\) 6.13107 0.391700
\(246\) 6.06441 0.386652
\(247\) −1.66227 −0.105768
\(248\) 9.16073 0.581707
\(249\) 27.6741 1.75377
\(250\) 8.10611 0.512675
\(251\) 23.2298 1.46625 0.733127 0.680092i \(-0.238060\pi\)
0.733127 + 0.680092i \(0.238060\pi\)
\(252\) 0.0663224 0.00417792
\(253\) −9.04332 −0.568548
\(254\) −5.90292 −0.370382
\(255\) 8.14429 0.510015
\(256\) 1.00000 0.0625000
\(257\) 20.8328 1.29952 0.649758 0.760141i \(-0.274870\pi\)
0.649758 + 0.760141i \(0.274870\pi\)
\(258\) 1.80370 0.112293
\(259\) −0.564703 −0.0350889
\(260\) −0.249801 −0.0154920
\(261\) 3.31504 0.205196
\(262\) 13.4225 0.829247
\(263\) 0.718720 0.0443182 0.0221591 0.999754i \(-0.492946\pi\)
0.0221591 + 0.999754i \(0.492946\pi\)
\(264\) 6.56551 0.404079
\(265\) −2.31155 −0.141997
\(266\) −0.827686 −0.0507487
\(267\) 22.2974 1.36458
\(268\) 13.2199 0.807533
\(269\) −20.3809 −1.24264 −0.621322 0.783555i \(-0.713404\pi\)
−0.621322 + 0.783555i \(0.713404\pi\)
\(270\) −4.14140 −0.252038
\(271\) −28.1812 −1.71189 −0.855943 0.517070i \(-0.827023\pi\)
−0.855943 + 0.517070i \(0.827023\pi\)
\(272\) 4.97859 0.301872
\(273\) −0.0749972 −0.00453904
\(274\) 4.17159 0.252015
\(275\) 14.9069 0.898918
\(276\) −4.77732 −0.287561
\(277\) 12.0953 0.726734 0.363367 0.931646i \(-0.381627\pi\)
0.363367 + 0.931646i \(0.381627\pi\)
\(278\) −12.2987 −0.737627
\(279\) 4.29060 0.256871
\(280\) −0.124382 −0.00743324
\(281\) −30.1348 −1.79769 −0.898846 0.438265i \(-0.855593\pi\)
−0.898846 + 0.438265i \(0.855593\pi\)
\(282\) −8.37078 −0.498473
\(283\) 15.1822 0.902490 0.451245 0.892400i \(-0.350980\pi\)
0.451245 + 0.892400i \(0.350980\pi\)
\(284\) 11.0119 0.653436
\(285\) −9.56180 −0.566392
\(286\) −1.00257 −0.0592833
\(287\) −0.461104 −0.0272181
\(288\) 0.468368 0.0275989
\(289\) 7.78638 0.458023
\(290\) −6.21706 −0.365078
\(291\) −14.8669 −0.871515
\(292\) 5.90064 0.345309
\(293\) 20.7875 1.21442 0.607209 0.794542i \(-0.292289\pi\)
0.607209 + 0.794542i \(0.292289\pi\)
\(294\) 12.9991 0.758126
\(295\) 10.3840 0.604582
\(296\) −3.98793 −0.231793
\(297\) −16.6215 −0.964475
\(298\) −15.4457 −0.894747
\(299\) 0.729509 0.0421886
\(300\) 7.87486 0.454655
\(301\) −0.137143 −0.00790481
\(302\) −9.89714 −0.569517
\(303\) 21.6585 1.24425
\(304\) −5.84511 −0.335240
\(305\) −0.567616 −0.0325016
\(306\) 2.33181 0.133301
\(307\) 6.97316 0.397979 0.198990 0.980002i \(-0.436234\pi\)
0.198990 + 0.980002i \(0.436234\pi\)
\(308\) −0.499205 −0.0284449
\(309\) −10.9134 −0.620844
\(310\) −8.04664 −0.457018
\(311\) −8.73597 −0.495371 −0.247686 0.968840i \(-0.579670\pi\)
−0.247686 + 0.968840i \(0.579670\pi\)
\(312\) −0.529629 −0.0299843
\(313\) 10.0540 0.568284 0.284142 0.958782i \(-0.408291\pi\)
0.284142 + 0.958782i \(0.408291\pi\)
\(314\) −18.7245 −1.05668
\(315\) −0.0582565 −0.00328238
\(316\) −4.49078 −0.252626
\(317\) −17.8156 −1.00062 −0.500311 0.865845i \(-0.666781\pi\)
−0.500311 + 0.865845i \(0.666781\pi\)
\(318\) −4.90096 −0.274832
\(319\) −24.9521 −1.39705
\(320\) −0.878383 −0.0491031
\(321\) 1.21811 0.0679883
\(322\) 0.363241 0.0202426
\(323\) −29.1004 −1.61919
\(324\) −10.1857 −0.565874
\(325\) −1.20251 −0.0667034
\(326\) 8.96605 0.496584
\(327\) −2.31464 −0.128000
\(328\) −3.25631 −0.179800
\(329\) 0.636468 0.0350896
\(330\) −5.76704 −0.317465
\(331\) −0.772988 −0.0424872 −0.0212436 0.999774i \(-0.506763\pi\)
−0.0212436 + 0.999774i \(0.506763\pi\)
\(332\) −14.8597 −0.815533
\(333\) −1.86782 −0.102356
\(334\) −12.1450 −0.664544
\(335\) −11.6121 −0.634438
\(336\) −0.263716 −0.0143869
\(337\) −11.7517 −0.640154 −0.320077 0.947391i \(-0.603709\pi\)
−0.320077 + 0.947391i \(0.603709\pi\)
\(338\) −12.9191 −0.702708
\(339\) 12.0653 0.655295
\(340\) −4.37311 −0.237165
\(341\) −32.2951 −1.74888
\(342\) −2.73766 −0.148036
\(343\) −1.97961 −0.106889
\(344\) −0.968504 −0.0522183
\(345\) 4.19632 0.225922
\(346\) −1.44330 −0.0775925
\(347\) 36.6012 1.96486 0.982428 0.186644i \(-0.0597610\pi\)
0.982428 + 0.186644i \(0.0597610\pi\)
\(348\) −13.1815 −0.706600
\(349\) −34.0850 −1.82453 −0.912265 0.409602i \(-0.865668\pi\)
−0.912265 + 0.409602i \(0.865668\pi\)
\(350\) −0.598761 −0.0320051
\(351\) 1.34083 0.0715680
\(352\) −3.52538 −0.187903
\(353\) −3.87943 −0.206481 −0.103241 0.994656i \(-0.532921\pi\)
−0.103241 + 0.994656i \(0.532921\pi\)
\(354\) 22.0163 1.17015
\(355\) −9.67268 −0.513372
\(356\) −11.9727 −0.634550
\(357\) −1.31293 −0.0694877
\(358\) −4.37680 −0.231321
\(359\) −1.45585 −0.0768366 −0.0384183 0.999262i \(-0.512232\pi\)
−0.0384183 + 0.999262i \(0.512232\pi\)
\(360\) −0.411407 −0.0216830
\(361\) 15.1653 0.798175
\(362\) −23.9988 −1.26135
\(363\) −2.66003 −0.139615
\(364\) 0.0402701 0.00211072
\(365\) −5.18303 −0.271292
\(366\) −1.20346 −0.0629060
\(367\) −33.4927 −1.74831 −0.874153 0.485652i \(-0.838582\pi\)
−0.874153 + 0.485652i \(0.838582\pi\)
\(368\) 2.56520 0.133720
\(369\) −1.52515 −0.0793962
\(370\) 3.50293 0.182109
\(371\) 0.372642 0.0193466
\(372\) −17.0605 −0.884548
\(373\) 7.64489 0.395837 0.197919 0.980218i \(-0.436582\pi\)
0.197919 + 0.980218i \(0.436582\pi\)
\(374\) −17.5514 −0.907563
\(375\) −15.0965 −0.779578
\(376\) 4.49473 0.231798
\(377\) 2.01284 0.103667
\(378\) 0.667631 0.0343392
\(379\) −38.0059 −1.95223 −0.976117 0.217246i \(-0.930293\pi\)
−0.976117 + 0.217246i \(0.930293\pi\)
\(380\) 5.13425 0.263381
\(381\) 10.9933 0.563205
\(382\) 10.5894 0.541800
\(383\) 16.3762 0.836784 0.418392 0.908267i \(-0.362594\pi\)
0.418392 + 0.908267i \(0.362594\pi\)
\(384\) −1.86236 −0.0950379
\(385\) 0.438494 0.0223477
\(386\) −8.01743 −0.408077
\(387\) −0.453617 −0.0230586
\(388\) 7.98287 0.405269
\(389\) −26.7429 −1.35592 −0.677960 0.735099i \(-0.737136\pi\)
−0.677960 + 0.735099i \(0.737136\pi\)
\(390\) 0.465217 0.0235572
\(391\) 12.7711 0.645862
\(392\) −6.97995 −0.352541
\(393\) −24.9975 −1.26096
\(394\) −16.6672 −0.839680
\(395\) 3.94463 0.198476
\(396\) −1.65118 −0.0829748
\(397\) −27.2404 −1.36715 −0.683577 0.729878i \(-0.739577\pi\)
−0.683577 + 0.729878i \(0.739577\pi\)
\(398\) 26.9698 1.35187
\(399\) 1.54145 0.0771688
\(400\) −4.22844 −0.211422
\(401\) 34.3241 1.71406 0.857031 0.515264i \(-0.172306\pi\)
0.857031 + 0.515264i \(0.172306\pi\)
\(402\) −24.6201 −1.22794
\(403\) 2.60519 0.129774
\(404\) −11.6296 −0.578595
\(405\) 8.94698 0.444579
\(406\) 1.00225 0.0497406
\(407\) 14.0590 0.696877
\(408\) −9.27191 −0.459028
\(409\) −4.95539 −0.245028 −0.122514 0.992467i \(-0.539096\pi\)
−0.122514 + 0.992467i \(0.539096\pi\)
\(410\) 2.86029 0.141260
\(411\) −7.76898 −0.383215
\(412\) 5.86002 0.288702
\(413\) −1.67400 −0.0823721
\(414\) 1.20146 0.0590485
\(415\) 13.0525 0.640724
\(416\) 0.284387 0.0139432
\(417\) 22.9045 1.12164
\(418\) 20.6062 1.00788
\(419\) −26.0862 −1.27440 −0.637198 0.770700i \(-0.719907\pi\)
−0.637198 + 0.770700i \(0.719907\pi\)
\(420\) 0.231643 0.0113030
\(421\) 28.2421 1.37644 0.688218 0.725504i \(-0.258393\pi\)
0.688218 + 0.725504i \(0.258393\pi\)
\(422\) −22.0766 −1.07467
\(423\) 2.10519 0.102358
\(424\) 2.63159 0.127801
\(425\) −21.0517 −1.02116
\(426\) −20.5081 −0.993620
\(427\) 0.0915047 0.00442822
\(428\) −0.654070 −0.0316157
\(429\) 1.86714 0.0901466
\(430\) 0.850718 0.0410253
\(431\) −10.9748 −0.528639 −0.264320 0.964435i \(-0.585147\pi\)
−0.264320 + 0.964435i \(0.585147\pi\)
\(432\) 4.71480 0.226841
\(433\) 5.53798 0.266138 0.133069 0.991107i \(-0.457517\pi\)
0.133069 + 0.991107i \(0.457517\pi\)
\(434\) 1.29719 0.0622671
\(435\) 11.5784 0.555141
\(436\) 1.24286 0.0595220
\(437\) −14.9939 −0.717255
\(438\) −10.9891 −0.525079
\(439\) −28.8937 −1.37902 −0.689511 0.724276i \(-0.742174\pi\)
−0.689511 + 0.724276i \(0.742174\pi\)
\(440\) 3.09664 0.147626
\(441\) −3.26919 −0.155676
\(442\) 1.41585 0.0673449
\(443\) −20.1205 −0.955953 −0.477976 0.878373i \(-0.658630\pi\)
−0.477976 + 0.878373i \(0.658630\pi\)
\(444\) 7.42694 0.352467
\(445\) 10.5166 0.498534
\(446\) −9.63646 −0.456300
\(447\) 28.7654 1.36056
\(448\) 0.141603 0.00669012
\(449\) −30.1798 −1.42427 −0.712137 0.702041i \(-0.752272\pi\)
−0.712137 + 0.702041i \(0.752272\pi\)
\(450\) −1.98047 −0.0933602
\(451\) 11.4797 0.540559
\(452\) −6.47849 −0.304723
\(453\) 18.4320 0.866011
\(454\) 17.6804 0.829781
\(455\) −0.0353726 −0.00165829
\(456\) 10.8857 0.509768
\(457\) −36.6265 −1.71332 −0.856659 0.515884i \(-0.827464\pi\)
−0.856659 + 0.515884i \(0.827464\pi\)
\(458\) −0.450830 −0.0210659
\(459\) 23.4731 1.09563
\(460\) −2.25323 −0.105057
\(461\) −31.4929 −1.46677 −0.733385 0.679814i \(-0.762061\pi\)
−0.733385 + 0.679814i \(0.762061\pi\)
\(462\) 0.929698 0.0432535
\(463\) 13.9316 0.647457 0.323728 0.946150i \(-0.395064\pi\)
0.323728 + 0.946150i \(0.395064\pi\)
\(464\) 7.07784 0.328581
\(465\) 14.9857 0.694945
\(466\) −0.302059 −0.0139926
\(467\) 30.7439 1.42266 0.711330 0.702858i \(-0.248093\pi\)
0.711330 + 0.702858i \(0.248093\pi\)
\(468\) 0.133198 0.00615707
\(469\) 1.87198 0.0864399
\(470\) −3.94809 −0.182112
\(471\) 34.8716 1.60680
\(472\) −11.8218 −0.544141
\(473\) 3.41435 0.156992
\(474\) 8.36343 0.384145
\(475\) 24.7157 1.13403
\(476\) 0.704985 0.0323129
\(477\) 1.23255 0.0564348
\(478\) 3.38848 0.154986
\(479\) −25.8778 −1.18238 −0.591192 0.806531i \(-0.701343\pi\)
−0.591192 + 0.806531i \(0.701343\pi\)
\(480\) 1.63586 0.0746665
\(481\) −1.13411 −0.0517111
\(482\) 18.1980 0.828896
\(483\) −0.676484 −0.0307811
\(484\) 1.42831 0.0649234
\(485\) −7.01202 −0.318399
\(486\) 4.82507 0.218869
\(487\) −19.7582 −0.895329 −0.447664 0.894202i \(-0.647744\pi\)
−0.447664 + 0.894202i \(0.647744\pi\)
\(488\) 0.646205 0.0292523
\(489\) −16.6980 −0.755108
\(490\) 6.13107 0.276974
\(491\) −7.90154 −0.356592 −0.178296 0.983977i \(-0.557058\pi\)
−0.178296 + 0.983977i \(0.557058\pi\)
\(492\) 6.06441 0.273405
\(493\) 35.2377 1.58703
\(494\) −1.66227 −0.0747891
\(495\) 1.45037 0.0651891
\(496\) 9.16073 0.411329
\(497\) 1.55932 0.0699451
\(498\) 27.6741 1.24011
\(499\) 10.9025 0.488062 0.244031 0.969767i \(-0.421530\pi\)
0.244031 + 0.969767i \(0.421530\pi\)
\(500\) 8.10611 0.362516
\(501\) 22.6183 1.01051
\(502\) 23.2298 1.03680
\(503\) −33.3057 −1.48503 −0.742514 0.669831i \(-0.766367\pi\)
−0.742514 + 0.669831i \(0.766367\pi\)
\(504\) 0.0663224 0.00295424
\(505\) 10.2153 0.454573
\(506\) −9.04332 −0.402024
\(507\) 24.0600 1.06854
\(508\) −5.90292 −0.261900
\(509\) −14.2511 −0.631670 −0.315835 0.948814i \(-0.602285\pi\)
−0.315835 + 0.948814i \(0.602285\pi\)
\(510\) 8.14429 0.360635
\(511\) 0.835550 0.0369625
\(512\) 1.00000 0.0441942
\(513\) −27.5585 −1.21674
\(514\) 20.8328 0.918896
\(515\) −5.14734 −0.226819
\(516\) 1.80370 0.0794034
\(517\) −15.8456 −0.696890
\(518\) −0.564703 −0.0248116
\(519\) 2.68795 0.117988
\(520\) −0.249801 −0.0109545
\(521\) 24.6385 1.07943 0.539717 0.841847i \(-0.318531\pi\)
0.539717 + 0.841847i \(0.318531\pi\)
\(522\) 3.31504 0.145095
\(523\) −27.7214 −1.21217 −0.606085 0.795400i \(-0.707261\pi\)
−0.606085 + 0.795400i \(0.707261\pi\)
\(524\) 13.4225 0.586366
\(525\) 1.11511 0.0486672
\(526\) 0.718720 0.0313377
\(527\) 45.6076 1.98670
\(528\) 6.56551 0.285727
\(529\) −16.4197 −0.713902
\(530\) −2.31155 −0.100407
\(531\) −5.53694 −0.240283
\(532\) −0.827686 −0.0358848
\(533\) −0.926051 −0.0401117
\(534\) 22.2974 0.964901
\(535\) 0.574524 0.0248388
\(536\) 13.2199 0.571012
\(537\) 8.15116 0.351749
\(538\) −20.3809 −0.878682
\(539\) 24.6070 1.05990
\(540\) −4.14140 −0.178218
\(541\) −6.27573 −0.269815 −0.134907 0.990858i \(-0.543074\pi\)
−0.134907 + 0.990858i \(0.543074\pi\)
\(542\) −28.1812 −1.21049
\(543\) 44.6944 1.91802
\(544\) 4.97859 0.213455
\(545\) −1.09170 −0.0467635
\(546\) −0.0749972 −0.00320958
\(547\) 29.2716 1.25156 0.625781 0.779998i \(-0.284780\pi\)
0.625781 + 0.779998i \(0.284780\pi\)
\(548\) 4.17159 0.178201
\(549\) 0.302662 0.0129173
\(550\) 14.9069 0.635631
\(551\) −41.3708 −1.76245
\(552\) −4.77732 −0.203336
\(553\) −0.635909 −0.0270416
\(554\) 12.0953 0.513879
\(555\) −6.52370 −0.276915
\(556\) −12.2987 −0.521581
\(557\) −31.5199 −1.33554 −0.667771 0.744367i \(-0.732751\pi\)
−0.667771 + 0.744367i \(0.732751\pi\)
\(558\) 4.29060 0.181635
\(559\) −0.275430 −0.0116494
\(560\) −0.124382 −0.00525609
\(561\) 32.6870 1.38005
\(562\) −30.1348 −1.27116
\(563\) −27.1655 −1.14489 −0.572444 0.819943i \(-0.694005\pi\)
−0.572444 + 0.819943i \(0.694005\pi\)
\(564\) −8.37078 −0.352473
\(565\) 5.69060 0.239405
\(566\) 15.1822 0.638157
\(567\) −1.44233 −0.0605723
\(568\) 11.0119 0.462049
\(569\) 15.7815 0.661593 0.330796 0.943702i \(-0.392683\pi\)
0.330796 + 0.943702i \(0.392683\pi\)
\(570\) −9.56180 −0.400500
\(571\) −25.6775 −1.07457 −0.537285 0.843401i \(-0.680550\pi\)
−0.537285 + 0.843401i \(0.680550\pi\)
\(572\) −1.00257 −0.0419196
\(573\) −19.7212 −0.823865
\(574\) −0.461104 −0.0192461
\(575\) −10.8468 −0.452343
\(576\) 0.468368 0.0195153
\(577\) 8.05862 0.335485 0.167742 0.985831i \(-0.446352\pi\)
0.167742 + 0.985831i \(0.446352\pi\)
\(578\) 7.78638 0.323871
\(579\) 14.9313 0.620524
\(580\) −6.21706 −0.258149
\(581\) −2.10418 −0.0872963
\(582\) −14.8669 −0.616255
\(583\) −9.27737 −0.384229
\(584\) 5.90064 0.244170
\(585\) −0.116999 −0.00483730
\(586\) 20.7875 0.858723
\(587\) −27.6986 −1.14324 −0.571622 0.820517i \(-0.693686\pi\)
−0.571622 + 0.820517i \(0.693686\pi\)
\(588\) 12.9991 0.536076
\(589\) −53.5455 −2.20630
\(590\) 10.3840 0.427504
\(591\) 31.0402 1.27682
\(592\) −3.98793 −0.163903
\(593\) −5.66975 −0.232829 −0.116414 0.993201i \(-0.537140\pi\)
−0.116414 + 0.993201i \(0.537140\pi\)
\(594\) −16.6215 −0.681987
\(595\) −0.619247 −0.0253866
\(596\) −15.4457 −0.632682
\(597\) −50.2273 −2.05567
\(598\) 0.729509 0.0298319
\(599\) 45.5563 1.86138 0.930690 0.365808i \(-0.119207\pi\)
0.930690 + 0.365808i \(0.119207\pi\)
\(600\) 7.87486 0.321490
\(601\) 16.2865 0.664340 0.332170 0.943220i \(-0.392219\pi\)
0.332170 + 0.943220i \(0.392219\pi\)
\(602\) −0.137143 −0.00558954
\(603\) 6.19178 0.252149
\(604\) −9.89714 −0.402709
\(605\) −1.25461 −0.0510071
\(606\) 21.6585 0.879815
\(607\) −11.6060 −0.471074 −0.235537 0.971865i \(-0.575685\pi\)
−0.235537 + 0.971865i \(0.575685\pi\)
\(608\) −5.84511 −0.237051
\(609\) −1.86654 −0.0756359
\(610\) −0.567616 −0.0229821
\(611\) 1.27824 0.0517121
\(612\) 2.33181 0.0942580
\(613\) 17.9087 0.723324 0.361662 0.932309i \(-0.382209\pi\)
0.361662 + 0.932309i \(0.382209\pi\)
\(614\) 6.97316 0.281414
\(615\) −5.32687 −0.214800
\(616\) −0.499205 −0.0201136
\(617\) −3.41505 −0.137485 −0.0687423 0.997634i \(-0.521899\pi\)
−0.0687423 + 0.997634i \(0.521899\pi\)
\(618\) −10.9134 −0.439003
\(619\) −32.7170 −1.31501 −0.657504 0.753451i \(-0.728388\pi\)
−0.657504 + 0.753451i \(0.728388\pi\)
\(620\) −8.04664 −0.323161
\(621\) 12.0944 0.485332
\(622\) −8.73597 −0.350280
\(623\) −1.69537 −0.0679234
\(624\) −0.529629 −0.0212021
\(625\) 14.0219 0.560878
\(626\) 10.0540 0.401837
\(627\) −38.3762 −1.53260
\(628\) −18.7245 −0.747187
\(629\) −19.8543 −0.791641
\(630\) −0.0582565 −0.00232100
\(631\) 39.3751 1.56750 0.783748 0.621079i \(-0.213305\pi\)
0.783748 + 0.621079i \(0.213305\pi\)
\(632\) −4.49078 −0.178634
\(633\) 41.1146 1.63416
\(634\) −17.8156 −0.707547
\(635\) 5.18502 0.205761
\(636\) −4.90096 −0.194336
\(637\) −1.98500 −0.0786487
\(638\) −24.9521 −0.987863
\(639\) 5.15763 0.204033
\(640\) −0.878383 −0.0347212
\(641\) 6.77066 0.267425 0.133713 0.991020i \(-0.457310\pi\)
0.133713 + 0.991020i \(0.457310\pi\)
\(642\) 1.21811 0.0480750
\(643\) 0.281208 0.0110897 0.00554487 0.999985i \(-0.498235\pi\)
0.00554487 + 0.999985i \(0.498235\pi\)
\(644\) 0.363241 0.0143137
\(645\) −1.58434 −0.0623833
\(646\) −29.1004 −1.14494
\(647\) 48.9842 1.92577 0.962883 0.269918i \(-0.0869967\pi\)
0.962883 + 0.269918i \(0.0869967\pi\)
\(648\) −10.1857 −0.400133
\(649\) 41.6762 1.63594
\(650\) −1.20251 −0.0471664
\(651\) −2.41583 −0.0946837
\(652\) 8.96605 0.351138
\(653\) 23.4517 0.917734 0.458867 0.888505i \(-0.348255\pi\)
0.458867 + 0.888505i \(0.348255\pi\)
\(654\) −2.31464 −0.0905096
\(655\) −11.7901 −0.460678
\(656\) −3.25631 −0.127138
\(657\) 2.76367 0.107821
\(658\) 0.636468 0.0248121
\(659\) −29.5520 −1.15118 −0.575591 0.817738i \(-0.695228\pi\)
−0.575591 + 0.817738i \(0.695228\pi\)
\(660\) −5.76704 −0.224482
\(661\) 3.84627 0.149603 0.0748014 0.997198i \(-0.476168\pi\)
0.0748014 + 0.997198i \(0.476168\pi\)
\(662\) −0.772988 −0.0300430
\(663\) −2.63681 −0.102405
\(664\) −14.8597 −0.576669
\(665\) 0.727026 0.0281929
\(666\) −1.86782 −0.0723765
\(667\) 18.1561 0.703007
\(668\) −12.1450 −0.469904
\(669\) 17.9465 0.693853
\(670\) −11.6121 −0.448616
\(671\) −2.27812 −0.0879458
\(672\) −0.263716 −0.0101730
\(673\) 48.1622 1.85652 0.928259 0.371935i \(-0.121306\pi\)
0.928259 + 0.371935i \(0.121306\pi\)
\(674\) −11.7517 −0.452657
\(675\) −19.9363 −0.767347
\(676\) −12.9191 −0.496889
\(677\) −15.9300 −0.612238 −0.306119 0.951993i \(-0.599031\pi\)
−0.306119 + 0.951993i \(0.599031\pi\)
\(678\) 12.0653 0.463363
\(679\) 1.13040 0.0433808
\(680\) −4.37311 −0.167701
\(681\) −32.9272 −1.26177
\(682\) −32.2951 −1.23664
\(683\) 32.6067 1.24766 0.623830 0.781560i \(-0.285576\pi\)
0.623830 + 0.781560i \(0.285576\pi\)
\(684\) −2.73766 −0.104677
\(685\) −3.66425 −0.140004
\(686\) −1.97961 −0.0755817
\(687\) 0.839605 0.0320329
\(688\) −0.968504 −0.0369239
\(689\) 0.748390 0.0285114
\(690\) 4.19632 0.159751
\(691\) 45.3110 1.72371 0.861856 0.507154i \(-0.169302\pi\)
0.861856 + 0.507154i \(0.169302\pi\)
\(692\) −1.44330 −0.0548662
\(693\) −0.233812 −0.00888178
\(694\) 36.6012 1.38936
\(695\) 10.8030 0.409780
\(696\) −13.1815 −0.499642
\(697\) −16.2118 −0.614067
\(698\) −34.0850 −1.29014
\(699\) 0.562541 0.0212773
\(700\) −0.598761 −0.0226310
\(701\) −39.7274 −1.50048 −0.750241 0.661164i \(-0.770063\pi\)
−0.750241 + 0.661164i \(0.770063\pi\)
\(702\) 1.34083 0.0506062
\(703\) 23.3099 0.879148
\(704\) −3.52538 −0.132868
\(705\) 7.35275 0.276921
\(706\) −3.87943 −0.146004
\(707\) −1.64679 −0.0619339
\(708\) 22.0163 0.827424
\(709\) 24.4279 0.917408 0.458704 0.888589i \(-0.348314\pi\)
0.458704 + 0.888589i \(0.348314\pi\)
\(710\) −9.67268 −0.363009
\(711\) −2.10334 −0.0788814
\(712\) −11.9727 −0.448694
\(713\) 23.4991 0.880049
\(714\) −1.31293 −0.0491352
\(715\) 0.880642 0.0329341
\(716\) −4.37680 −0.163569
\(717\) −6.31056 −0.235672
\(718\) −1.45585 −0.0543317
\(719\) 13.0240 0.485713 0.242856 0.970062i \(-0.421916\pi\)
0.242856 + 0.970062i \(0.421916\pi\)
\(720\) −0.411407 −0.0153322
\(721\) 0.829798 0.0309033
\(722\) 15.1653 0.564395
\(723\) −33.8911 −1.26042
\(724\) −23.9988 −0.891910
\(725\) −29.9282 −1.11151
\(726\) −2.66003 −0.0987229
\(727\) 30.5655 1.13361 0.566806 0.823852i \(-0.308179\pi\)
0.566806 + 0.823852i \(0.308179\pi\)
\(728\) 0.0402701 0.00149251
\(729\) 21.5712 0.798934
\(730\) −5.18303 −0.191832
\(731\) −4.82179 −0.178340
\(732\) −1.20346 −0.0444813
\(733\) −14.8471 −0.548391 −0.274195 0.961674i \(-0.588411\pi\)
−0.274195 + 0.961674i \(0.588411\pi\)
\(734\) −33.4927 −1.23624
\(735\) −11.4182 −0.421168
\(736\) 2.56520 0.0945546
\(737\) −46.6051 −1.71672
\(738\) −1.52515 −0.0561416
\(739\) 34.8636 1.28248 0.641239 0.767341i \(-0.278421\pi\)
0.641239 + 0.767341i \(0.278421\pi\)
\(740\) 3.50293 0.128770
\(741\) 3.09574 0.113725
\(742\) 0.372642 0.0136801
\(743\) −50.8248 −1.86458 −0.932291 0.361708i \(-0.882194\pi\)
−0.932291 + 0.361708i \(0.882194\pi\)
\(744\) −17.0605 −0.625470
\(745\) 13.5673 0.497066
\(746\) 7.64489 0.279899
\(747\) −6.95982 −0.254647
\(748\) −17.5514 −0.641744
\(749\) −0.0926184 −0.00338420
\(750\) −15.0965 −0.551245
\(751\) 32.0136 1.16819 0.584097 0.811684i \(-0.301449\pi\)
0.584097 + 0.811684i \(0.301449\pi\)
\(752\) 4.49473 0.163906
\(753\) −43.2622 −1.57656
\(754\) 2.01284 0.0733035
\(755\) 8.69349 0.316388
\(756\) 0.667631 0.0242815
\(757\) 15.1634 0.551125 0.275562 0.961283i \(-0.411136\pi\)
0.275562 + 0.961283i \(0.411136\pi\)
\(758\) −38.0059 −1.38044
\(759\) 16.8419 0.611321
\(760\) 5.13425 0.186239
\(761\) −31.5300 −1.14296 −0.571481 0.820615i \(-0.693631\pi\)
−0.571481 + 0.820615i \(0.693631\pi\)
\(762\) 10.9933 0.398246
\(763\) 0.175992 0.00637135
\(764\) 10.5894 0.383110
\(765\) −2.04823 −0.0740538
\(766\) 16.3762 0.591696
\(767\) −3.36195 −0.121393
\(768\) −1.86236 −0.0672020
\(769\) 12.5218 0.451548 0.225774 0.974180i \(-0.427509\pi\)
0.225774 + 0.974180i \(0.427509\pi\)
\(770\) 0.438494 0.0158022
\(771\) −38.7981 −1.39728
\(772\) −8.01743 −0.288554
\(773\) −1.82043 −0.0654763 −0.0327382 0.999464i \(-0.510423\pi\)
−0.0327382 + 0.999464i \(0.510423\pi\)
\(774\) −0.453617 −0.0163049
\(775\) −38.7356 −1.39143
\(776\) 7.98287 0.286568
\(777\) 1.05168 0.0377287
\(778\) −26.7429 −0.958780
\(779\) 19.0335 0.681946
\(780\) 0.465217 0.0166575
\(781\) −38.8212 −1.38913
\(782\) 12.7711 0.456693
\(783\) 33.3706 1.19257
\(784\) −6.97995 −0.249284
\(785\) 16.4472 0.587027
\(786\) −24.9975 −0.891632
\(787\) 35.1032 1.25129 0.625646 0.780107i \(-0.284835\pi\)
0.625646 + 0.780107i \(0.284835\pi\)
\(788\) −16.6672 −0.593744
\(789\) −1.33851 −0.0476523
\(790\) 3.94463 0.140343
\(791\) −0.917375 −0.0326181
\(792\) −1.65118 −0.0586720
\(793\) 0.183772 0.00652594
\(794\) −27.2404 −0.966724
\(795\) 4.30492 0.152680
\(796\) 26.9698 0.955919
\(797\) −45.3458 −1.60623 −0.803116 0.595823i \(-0.796826\pi\)
−0.803116 + 0.595823i \(0.796826\pi\)
\(798\) 1.54145 0.0545666
\(799\) 22.3774 0.791656
\(800\) −4.22844 −0.149498
\(801\) −5.60761 −0.198135
\(802\) 34.3241 1.21203
\(803\) −20.8020 −0.734087
\(804\) −24.6201 −0.868285
\(805\) −0.319065 −0.0112456
\(806\) 2.60519 0.0917639
\(807\) 37.9564 1.33613
\(808\) −11.6296 −0.409128
\(809\) 17.5703 0.617738 0.308869 0.951105i \(-0.400050\pi\)
0.308869 + 0.951105i \(0.400050\pi\)
\(810\) 8.94698 0.314365
\(811\) −45.4995 −1.59770 −0.798851 0.601528i \(-0.794559\pi\)
−0.798851 + 0.601528i \(0.794559\pi\)
\(812\) 1.00225 0.0351719
\(813\) 52.4834 1.84067
\(814\) 14.0590 0.492766
\(815\) −7.87563 −0.275871
\(816\) −9.27191 −0.324582
\(817\) 5.66102 0.198054
\(818\) −4.95539 −0.173261
\(819\) 0.0188612 0.000659064 0
\(820\) 2.86029 0.0998856
\(821\) 48.5889 1.69576 0.847881 0.530186i \(-0.177878\pi\)
0.847881 + 0.530186i \(0.177878\pi\)
\(822\) −7.76898 −0.270974
\(823\) −27.7645 −0.967809 −0.483905 0.875121i \(-0.660782\pi\)
−0.483905 + 0.875121i \(0.660782\pi\)
\(824\) 5.86002 0.204143
\(825\) −27.7619 −0.966545
\(826\) −1.67400 −0.0582459
\(827\) −27.9723 −0.972693 −0.486347 0.873766i \(-0.661671\pi\)
−0.486347 + 0.873766i \(0.661671\pi\)
\(828\) 1.20146 0.0417536
\(829\) −24.2882 −0.843563 −0.421782 0.906697i \(-0.638595\pi\)
−0.421782 + 0.906697i \(0.638595\pi\)
\(830\) 13.0525 0.453060
\(831\) −22.5257 −0.781407
\(832\) 0.284387 0.00985933
\(833\) −34.7503 −1.20403
\(834\) 22.9045 0.793120
\(835\) 10.6680 0.369180
\(836\) 20.6062 0.712682
\(837\) 43.1910 1.49290
\(838\) −26.0862 −0.901134
\(839\) −27.7153 −0.956837 −0.478419 0.878132i \(-0.658790\pi\)
−0.478419 + 0.878132i \(0.658790\pi\)
\(840\) 0.231643 0.00799245
\(841\) 21.0958 0.727443
\(842\) 28.2421 0.973287
\(843\) 56.1217 1.93293
\(844\) −22.0766 −0.759910
\(845\) 11.3479 0.390381
\(846\) 2.10519 0.0723778
\(847\) 0.202254 0.00694953
\(848\) 2.63159 0.0903693
\(849\) −28.2747 −0.970386
\(850\) −21.0517 −0.722067
\(851\) −10.2298 −0.350674
\(852\) −20.5081 −0.702595
\(853\) 0.857427 0.0293577 0.0146789 0.999892i \(-0.495327\pi\)
0.0146789 + 0.999892i \(0.495327\pi\)
\(854\) 0.0915047 0.00313123
\(855\) 2.40472 0.0822397
\(856\) −0.654070 −0.0223556
\(857\) −51.9181 −1.77349 −0.886745 0.462259i \(-0.847039\pi\)
−0.886745 + 0.462259i \(0.847039\pi\)
\(858\) 1.86714 0.0637433
\(859\) 9.01728 0.307666 0.153833 0.988097i \(-0.450838\pi\)
0.153833 + 0.988097i \(0.450838\pi\)
\(860\) 0.850718 0.0290092
\(861\) 0.858739 0.0292658
\(862\) −10.9748 −0.373804
\(863\) −5.87738 −0.200068 −0.100034 0.994984i \(-0.531895\pi\)
−0.100034 + 0.994984i \(0.531895\pi\)
\(864\) 4.71480 0.160401
\(865\) 1.26777 0.0431056
\(866\) 5.53798 0.188188
\(867\) −14.5010 −0.492480
\(868\) 1.29719 0.0440295
\(869\) 15.8317 0.537054
\(870\) 11.5784 0.392544
\(871\) 3.75956 0.127388
\(872\) 1.24286 0.0420884
\(873\) 3.73892 0.126543
\(874\) −14.9939 −0.507176
\(875\) 1.14785 0.0388045
\(876\) −10.9891 −0.371287
\(877\) −12.2548 −0.413816 −0.206908 0.978360i \(-0.566340\pi\)
−0.206908 + 0.978360i \(0.566340\pi\)
\(878\) −28.8937 −0.975115
\(879\) −38.7137 −1.30578
\(880\) 3.09664 0.104388
\(881\) −0.837034 −0.0282004 −0.0141002 0.999901i \(-0.504488\pi\)
−0.0141002 + 0.999901i \(0.504488\pi\)
\(882\) −3.26919 −0.110079
\(883\) 44.6016 1.50096 0.750482 0.660891i \(-0.229821\pi\)
0.750482 + 0.660891i \(0.229821\pi\)
\(884\) 1.41585 0.0476200
\(885\) −19.3388 −0.650066
\(886\) −20.1205 −0.675961
\(887\) −17.3231 −0.581652 −0.290826 0.956776i \(-0.593930\pi\)
−0.290826 + 0.956776i \(0.593930\pi\)
\(888\) 7.42694 0.249232
\(889\) −0.835872 −0.0280342
\(890\) 10.5166 0.352517
\(891\) 35.9086 1.20298
\(892\) −9.63646 −0.322653
\(893\) −26.2722 −0.879165
\(894\) 28.7654 0.962060
\(895\) 3.84451 0.128508
\(896\) 0.141603 0.00473063
\(897\) −1.35861 −0.0453625
\(898\) −30.1798 −1.00711
\(899\) 64.8382 2.16248
\(900\) −1.98047 −0.0660156
\(901\) 13.1016 0.436479
\(902\) 11.4797 0.382233
\(903\) 0.255410 0.00849950
\(904\) −6.47849 −0.215471
\(905\) 21.0802 0.700729
\(906\) 18.4320 0.612362
\(907\) −28.6392 −0.950950 −0.475475 0.879729i \(-0.657724\pi\)
−0.475475 + 0.879729i \(0.657724\pi\)
\(908\) 17.6804 0.586744
\(909\) −5.44694 −0.180664
\(910\) −0.0353726 −0.00117259
\(911\) −49.9955 −1.65642 −0.828212 0.560415i \(-0.810641\pi\)
−0.828212 + 0.560415i \(0.810641\pi\)
\(912\) 10.8857 0.360461
\(913\) 52.3862 1.73373
\(914\) −36.6265 −1.21150
\(915\) 1.05710 0.0349467
\(916\) −0.450830 −0.0148958
\(917\) 1.90067 0.0627657
\(918\) 23.4731 0.774726
\(919\) −23.8951 −0.788226 −0.394113 0.919062i \(-0.628948\pi\)
−0.394113 + 0.919062i \(0.628948\pi\)
\(920\) −2.25323 −0.0742868
\(921\) −12.9865 −0.427920
\(922\) −31.4929 −1.03716
\(923\) 3.13164 0.103079
\(924\) 0.929698 0.0305848
\(925\) 16.8627 0.554443
\(926\) 13.9316 0.457821
\(927\) 2.74465 0.0901460
\(928\) 7.07784 0.232342
\(929\) 45.1902 1.48264 0.741321 0.671151i \(-0.234200\pi\)
0.741321 + 0.671151i \(0.234200\pi\)
\(930\) 14.9857 0.491400
\(931\) 40.7986 1.33712
\(932\) −0.302059 −0.00989428
\(933\) 16.2695 0.532639
\(934\) 30.7439 1.00597
\(935\) 15.4169 0.504186
\(936\) 0.133198 0.00435370
\(937\) −36.3889 −1.18877 −0.594387 0.804179i \(-0.702605\pi\)
−0.594387 + 0.804179i \(0.702605\pi\)
\(938\) 1.87198 0.0611223
\(939\) −18.7241 −0.611036
\(940\) −3.94809 −0.128773
\(941\) 56.3622 1.83736 0.918678 0.395008i \(-0.129258\pi\)
0.918678 + 0.395008i \(0.129258\pi\)
\(942\) 34.8716 1.13618
\(943\) −8.35309 −0.272014
\(944\) −11.8218 −0.384766
\(945\) −0.586436 −0.0190768
\(946\) 3.41435 0.111010
\(947\) 50.0176 1.62535 0.812676 0.582715i \(-0.198010\pi\)
0.812676 + 0.582715i \(0.198010\pi\)
\(948\) 8.36343 0.271632
\(949\) 1.67806 0.0544723
\(950\) 24.7157 0.801884
\(951\) 33.1790 1.07590
\(952\) 0.704985 0.0228487
\(953\) 0.128291 0.00415575 0.00207788 0.999998i \(-0.499339\pi\)
0.00207788 + 0.999998i \(0.499339\pi\)
\(954\) 1.23255 0.0399054
\(955\) −9.30154 −0.300991
\(956\) 3.38848 0.109591
\(957\) 46.4697 1.50215
\(958\) −25.8778 −0.836072
\(959\) 0.590710 0.0190750
\(960\) 1.63586 0.0527972
\(961\) 52.9190 1.70707
\(962\) −1.13411 −0.0365653
\(963\) −0.306346 −0.00987185
\(964\) 18.1980 0.586118
\(965\) 7.04238 0.226702
\(966\) −0.676484 −0.0217655
\(967\) −3.43643 −0.110508 −0.0552541 0.998472i \(-0.517597\pi\)
−0.0552541 + 0.998472i \(0.517597\pi\)
\(968\) 1.42831 0.0459078
\(969\) 54.1953 1.74101
\(970\) −7.01202 −0.225142
\(971\) −29.3235 −0.941035 −0.470517 0.882391i \(-0.655933\pi\)
−0.470517 + 0.882391i \(0.655933\pi\)
\(972\) 4.82507 0.154764
\(973\) −1.74153 −0.0558310
\(974\) −19.7582 −0.633093
\(975\) 2.23951 0.0717216
\(976\) 0.646205 0.0206845
\(977\) −54.4195 −1.74104 −0.870518 0.492137i \(-0.836216\pi\)
−0.870518 + 0.492137i \(0.836216\pi\)
\(978\) −16.6980 −0.533942
\(979\) 42.2082 1.34898
\(980\) 6.13107 0.195850
\(981\) 0.582114 0.0185855
\(982\) −7.90154 −0.252148
\(983\) 50.5793 1.61323 0.806614 0.591078i \(-0.201298\pi\)
0.806614 + 0.591078i \(0.201298\pi\)
\(984\) 6.06441 0.193326
\(985\) 14.6402 0.466475
\(986\) 35.2377 1.12220
\(987\) −1.18533 −0.0377294
\(988\) −1.66227 −0.0528839
\(989\) −2.48441 −0.0789996
\(990\) 1.45037 0.0460957
\(991\) 58.8726 1.87015 0.935074 0.354452i \(-0.115332\pi\)
0.935074 + 0.354452i \(0.115332\pi\)
\(992\) 9.16073 0.290854
\(993\) 1.43958 0.0456836
\(994\) 1.55932 0.0494587
\(995\) −23.6898 −0.751018
\(996\) 27.6741 0.876887
\(997\) 17.3078 0.548143 0.274071 0.961709i \(-0.411630\pi\)
0.274071 + 0.961709i \(0.411630\pi\)
\(998\) 10.9025 0.345112
\(999\) −18.8023 −0.594877
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.d.1.21 69
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8002.2.a.d.1.21 69 1.1 even 1 trivial