Properties

Label 8001.2.a.w.1.8
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
Dimension $20$
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] = [8001,2,Mod(1,8001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8001, 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("8001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.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: \(63.8883066572\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 8 x^{19} + 152 x^{17} - 274 x^{16} - 1061 x^{15} + 3125 x^{14} + 2977 x^{13} - 15474 x^{12} - 56 x^{11} + 39579 x^{10} - 17664 x^{9} - 52271 x^{8} + 35701 x^{7} + \cdots + 31 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 889)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(1.72132\) of defining polynomial
Character \(\chi\) \(=\) 8001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.72132 q^{2} +0.962958 q^{4} -2.01516 q^{5} +1.00000 q^{7} +1.78509 q^{8} +O(q^{10})\) \(q-1.72132 q^{2} +0.962958 q^{4} -2.01516 q^{5} +1.00000 q^{7} +1.78509 q^{8} +3.46875 q^{10} -3.24791 q^{11} -2.61833 q^{13} -1.72132 q^{14} -4.99863 q^{16} +1.03520 q^{17} -7.18405 q^{19} -1.94052 q^{20} +5.59071 q^{22} +8.35185 q^{23} -0.939119 q^{25} +4.50700 q^{26} +0.962958 q^{28} +2.77158 q^{29} +7.77241 q^{31} +5.03409 q^{32} -1.78192 q^{34} -2.01516 q^{35} -0.894661 q^{37} +12.3661 q^{38} -3.59724 q^{40} -9.16796 q^{41} +11.1174 q^{43} -3.12760 q^{44} -14.3762 q^{46} -11.4796 q^{47} +1.00000 q^{49} +1.61653 q^{50} -2.52134 q^{52} +1.21297 q^{53} +6.54507 q^{55} +1.78509 q^{56} -4.77079 q^{58} +10.8406 q^{59} +1.59471 q^{61} -13.3788 q^{62} +1.33195 q^{64} +5.27637 q^{65} -6.57264 q^{67} +0.996855 q^{68} +3.46875 q^{70} -5.66866 q^{71} +16.1440 q^{73} +1.54000 q^{74} -6.91794 q^{76} -3.24791 q^{77} +6.54507 q^{79} +10.0730 q^{80} +15.7810 q^{82} +15.6260 q^{83} -2.08610 q^{85} -19.1366 q^{86} -5.79780 q^{88} +2.20445 q^{89} -2.61833 q^{91} +8.04248 q^{92} +19.7601 q^{94} +14.4770 q^{95} -17.3102 q^{97} -1.72132 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8} - 8 q^{10} - 26 q^{11} - 4 q^{13} - 8 q^{14} + 24 q^{16} - 4 q^{17} + q^{19} + 2 q^{20} + q^{22} - 31 q^{23} + 27 q^{25} - 4 q^{26} + 24 q^{28} - 16 q^{29} + 6 q^{31} - 41 q^{32} - 10 q^{34} - 3 q^{35} + 2 q^{37} - 3 q^{38} - 38 q^{40} - 25 q^{41} + 13 q^{43} - 66 q^{44} + 20 q^{46} - 19 q^{47} + 20 q^{49} + 4 q^{50} + 20 q^{52} - 24 q^{53} - 3 q^{55} - 24 q^{56} + 12 q^{58} - 23 q^{59} - 27 q^{61} - 7 q^{62} + 2 q^{64} - 26 q^{65} + 9 q^{67} + 25 q^{68} - 8 q^{70} - 63 q^{71} - 21 q^{73} - 21 q^{74} - 10 q^{76} - 26 q^{77} + 18 q^{79} + 23 q^{80} - 42 q^{82} + q^{83} - 41 q^{85} + 12 q^{86} + 57 q^{88} + 16 q^{89} - 4 q^{91} - 17 q^{92} + 7 q^{94} - 75 q^{95} - 32 q^{97} - 8 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.72132 −1.21716 −0.608580 0.793492i \(-0.708261\pi\)
−0.608580 + 0.793492i \(0.708261\pi\)
\(3\) 0 0
\(4\) 0.962958 0.481479
\(5\) −2.01516 −0.901208 −0.450604 0.892724i \(-0.648791\pi\)
−0.450604 + 0.892724i \(0.648791\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 1.78509 0.631123
\(9\) 0 0
\(10\) 3.46875 1.09691
\(11\) −3.24791 −0.979283 −0.489641 0.871924i \(-0.662872\pi\)
−0.489641 + 0.871924i \(0.662872\pi\)
\(12\) 0 0
\(13\) −2.61833 −0.726195 −0.363097 0.931751i \(-0.618281\pi\)
−0.363097 + 0.931751i \(0.618281\pi\)
\(14\) −1.72132 −0.460043
\(15\) 0 0
\(16\) −4.99863 −1.24966
\(17\) 1.03520 0.251073 0.125537 0.992089i \(-0.459935\pi\)
0.125537 + 0.992089i \(0.459935\pi\)
\(18\) 0 0
\(19\) −7.18405 −1.64813 −0.824067 0.566492i \(-0.808300\pi\)
−0.824067 + 0.566492i \(0.808300\pi\)
\(20\) −1.94052 −0.433913
\(21\) 0 0
\(22\) 5.59071 1.19194
\(23\) 8.35185 1.74148 0.870741 0.491742i \(-0.163640\pi\)
0.870741 + 0.491742i \(0.163640\pi\)
\(24\) 0 0
\(25\) −0.939119 −0.187824
\(26\) 4.50700 0.883895
\(27\) 0 0
\(28\) 0.962958 0.181982
\(29\) 2.77158 0.514670 0.257335 0.966322i \(-0.417156\pi\)
0.257335 + 0.966322i \(0.417156\pi\)
\(30\) 0 0
\(31\) 7.77241 1.39597 0.697983 0.716114i \(-0.254081\pi\)
0.697983 + 0.716114i \(0.254081\pi\)
\(32\) 5.03409 0.889910
\(33\) 0 0
\(34\) −1.78192 −0.305596
\(35\) −2.01516 −0.340625
\(36\) 0 0
\(37\) −0.894661 −0.147081 −0.0735407 0.997292i \(-0.523430\pi\)
−0.0735407 + 0.997292i \(0.523430\pi\)
\(38\) 12.3661 2.00604
\(39\) 0 0
\(40\) −3.59724 −0.568773
\(41\) −9.16796 −1.43179 −0.715897 0.698206i \(-0.753982\pi\)
−0.715897 + 0.698206i \(0.753982\pi\)
\(42\) 0 0
\(43\) 11.1174 1.69538 0.847691 0.530490i \(-0.177992\pi\)
0.847691 + 0.530490i \(0.177992\pi\)
\(44\) −3.12760 −0.471504
\(45\) 0 0
\(46\) −14.3762 −2.11966
\(47\) −11.4796 −1.67447 −0.837236 0.546841i \(-0.815830\pi\)
−0.837236 + 0.546841i \(0.815830\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.61653 0.228612
\(51\) 0 0
\(52\) −2.52134 −0.349648
\(53\) 1.21297 0.166614 0.0833072 0.996524i \(-0.473452\pi\)
0.0833072 + 0.996524i \(0.473452\pi\)
\(54\) 0 0
\(55\) 6.54507 0.882537
\(56\) 1.78509 0.238542
\(57\) 0 0
\(58\) −4.77079 −0.626436
\(59\) 10.8406 1.41133 0.705664 0.708547i \(-0.250649\pi\)
0.705664 + 0.708547i \(0.250649\pi\)
\(60\) 0 0
\(61\) 1.59471 0.204181 0.102091 0.994775i \(-0.467447\pi\)
0.102091 + 0.994775i \(0.467447\pi\)
\(62\) −13.3788 −1.69911
\(63\) 0 0
\(64\) 1.33195 0.166494
\(65\) 5.27637 0.654453
\(66\) 0 0
\(67\) −6.57264 −0.802976 −0.401488 0.915864i \(-0.631507\pi\)
−0.401488 + 0.915864i \(0.631507\pi\)
\(68\) 0.996855 0.120886
\(69\) 0 0
\(70\) 3.46875 0.414595
\(71\) −5.66866 −0.672746 −0.336373 0.941729i \(-0.609200\pi\)
−0.336373 + 0.941729i \(0.609200\pi\)
\(72\) 0 0
\(73\) 16.1440 1.88951 0.944754 0.327780i \(-0.106300\pi\)
0.944754 + 0.327780i \(0.106300\pi\)
\(74\) 1.54000 0.179022
\(75\) 0 0
\(76\) −6.91794 −0.793542
\(77\) −3.24791 −0.370134
\(78\) 0 0
\(79\) 6.54507 0.736378 0.368189 0.929751i \(-0.379978\pi\)
0.368189 + 0.929751i \(0.379978\pi\)
\(80\) 10.0730 1.12620
\(81\) 0 0
\(82\) 15.7810 1.74272
\(83\) 15.6260 1.71517 0.857587 0.514339i \(-0.171963\pi\)
0.857587 + 0.514339i \(0.171963\pi\)
\(84\) 0 0
\(85\) −2.08610 −0.226269
\(86\) −19.1366 −2.06355
\(87\) 0 0
\(88\) −5.79780 −0.618048
\(89\) 2.20445 0.233671 0.116835 0.993151i \(-0.462725\pi\)
0.116835 + 0.993151i \(0.462725\pi\)
\(90\) 0 0
\(91\) −2.61833 −0.274476
\(92\) 8.04248 0.838487
\(93\) 0 0
\(94\) 19.7601 2.03810
\(95\) 14.4770 1.48531
\(96\) 0 0
\(97\) −17.3102 −1.75758 −0.878792 0.477205i \(-0.841650\pi\)
−0.878792 + 0.477205i \(0.841650\pi\)
\(98\) −1.72132 −0.173880
\(99\) 0 0
\(100\) −0.904333 −0.0904333
\(101\) 0.549704 0.0546976 0.0273488 0.999626i \(-0.491294\pi\)
0.0273488 + 0.999626i \(0.491294\pi\)
\(102\) 0 0
\(103\) −3.42799 −0.337770 −0.168885 0.985636i \(-0.554017\pi\)
−0.168885 + 0.985636i \(0.554017\pi\)
\(104\) −4.67395 −0.458318
\(105\) 0 0
\(106\) −2.08792 −0.202796
\(107\) −2.93235 −0.283480 −0.141740 0.989904i \(-0.545270\pi\)
−0.141740 + 0.989904i \(0.545270\pi\)
\(108\) 0 0
\(109\) 5.63710 0.539936 0.269968 0.962869i \(-0.412987\pi\)
0.269968 + 0.962869i \(0.412987\pi\)
\(110\) −11.2662 −1.07419
\(111\) 0 0
\(112\) −4.99863 −0.472326
\(113\) −17.1714 −1.61535 −0.807675 0.589627i \(-0.799275\pi\)
−0.807675 + 0.589627i \(0.799275\pi\)
\(114\) 0 0
\(115\) −16.8303 −1.56944
\(116\) 2.66892 0.247803
\(117\) 0 0
\(118\) −18.6602 −1.71781
\(119\) 1.03520 0.0948967
\(120\) 0 0
\(121\) −0.451062 −0.0410056
\(122\) −2.74501 −0.248521
\(123\) 0 0
\(124\) 7.48451 0.672129
\(125\) 11.9683 1.07048
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) −12.3609 −1.09256
\(129\) 0 0
\(130\) −9.08234 −0.796574
\(131\) 13.6960 1.19663 0.598315 0.801261i \(-0.295837\pi\)
0.598315 + 0.801261i \(0.295837\pi\)
\(132\) 0 0
\(133\) −7.18405 −0.622936
\(134\) 11.3136 0.977350
\(135\) 0 0
\(136\) 1.84792 0.158458
\(137\) −14.1852 −1.21192 −0.605961 0.795495i \(-0.707211\pi\)
−0.605961 + 0.795495i \(0.707211\pi\)
\(138\) 0 0
\(139\) 17.5176 1.48582 0.742912 0.669389i \(-0.233444\pi\)
0.742912 + 0.669389i \(0.233444\pi\)
\(140\) −1.94052 −0.164004
\(141\) 0 0
\(142\) 9.75760 0.818839
\(143\) 8.50412 0.711150
\(144\) 0 0
\(145\) −5.58519 −0.463825
\(146\) −27.7890 −2.29983
\(147\) 0 0
\(148\) −0.861521 −0.0708166
\(149\) 2.82930 0.231785 0.115893 0.993262i \(-0.463027\pi\)
0.115893 + 0.993262i \(0.463027\pi\)
\(150\) 0 0
\(151\) −1.01483 −0.0825854 −0.0412927 0.999147i \(-0.513148\pi\)
−0.0412927 + 0.999147i \(0.513148\pi\)
\(152\) −12.8241 −1.04018
\(153\) 0 0
\(154\) 5.59071 0.450512
\(155\) −15.6627 −1.25806
\(156\) 0 0
\(157\) −8.81914 −0.703844 −0.351922 0.936029i \(-0.614472\pi\)
−0.351922 + 0.936029i \(0.614472\pi\)
\(158\) −11.2662 −0.896290
\(159\) 0 0
\(160\) −10.1445 −0.801994
\(161\) 8.35185 0.658218
\(162\) 0 0
\(163\) −18.0894 −1.41687 −0.708435 0.705776i \(-0.750599\pi\)
−0.708435 + 0.705776i \(0.750599\pi\)
\(164\) −8.82836 −0.689379
\(165\) 0 0
\(166\) −26.8974 −2.08764
\(167\) −2.26326 −0.175136 −0.0875682 0.996159i \(-0.527910\pi\)
−0.0875682 + 0.996159i \(0.527910\pi\)
\(168\) 0 0
\(169\) −6.14433 −0.472641
\(170\) 3.59085 0.275406
\(171\) 0 0
\(172\) 10.7056 0.816291
\(173\) 22.7872 1.73248 0.866241 0.499626i \(-0.166529\pi\)
0.866241 + 0.499626i \(0.166529\pi\)
\(174\) 0 0
\(175\) −0.939119 −0.0709908
\(176\) 16.2351 1.22377
\(177\) 0 0
\(178\) −3.79457 −0.284415
\(179\) −4.20646 −0.314405 −0.157203 0.987566i \(-0.550248\pi\)
−0.157203 + 0.987566i \(0.550248\pi\)
\(180\) 0 0
\(181\) 7.09021 0.527011 0.263505 0.964658i \(-0.415121\pi\)
0.263505 + 0.964658i \(0.415121\pi\)
\(182\) 4.50700 0.334081
\(183\) 0 0
\(184\) 14.9088 1.09909
\(185\) 1.80289 0.132551
\(186\) 0 0
\(187\) −3.36224 −0.245872
\(188\) −11.0544 −0.806223
\(189\) 0 0
\(190\) −24.9197 −1.80786
\(191\) −22.5450 −1.63130 −0.815650 0.578546i \(-0.803620\pi\)
−0.815650 + 0.578546i \(0.803620\pi\)
\(192\) 0 0
\(193\) −6.98495 −0.502788 −0.251394 0.967885i \(-0.580889\pi\)
−0.251394 + 0.967885i \(0.580889\pi\)
\(194\) 29.7965 2.13926
\(195\) 0 0
\(196\) 0.962958 0.0687827
\(197\) −13.5537 −0.965664 −0.482832 0.875713i \(-0.660392\pi\)
−0.482832 + 0.875713i \(0.660392\pi\)
\(198\) 0 0
\(199\) 27.1726 1.92621 0.963106 0.269123i \(-0.0867340\pi\)
0.963106 + 0.269123i \(0.0867340\pi\)
\(200\) −1.67641 −0.118540
\(201\) 0 0
\(202\) −0.946219 −0.0665757
\(203\) 2.77158 0.194527
\(204\) 0 0
\(205\) 18.4749 1.29034
\(206\) 5.90069 0.411120
\(207\) 0 0
\(208\) 13.0881 0.907494
\(209\) 23.3332 1.61399
\(210\) 0 0
\(211\) 8.72446 0.600616 0.300308 0.953842i \(-0.402910\pi\)
0.300308 + 0.953842i \(0.402910\pi\)
\(212\) 1.16804 0.0802213
\(213\) 0 0
\(214\) 5.04752 0.345041
\(215\) −22.4033 −1.52789
\(216\) 0 0
\(217\) 7.77241 0.527626
\(218\) −9.70327 −0.657189
\(219\) 0 0
\(220\) 6.30263 0.424923
\(221\) −2.71050 −0.182328
\(222\) 0 0
\(223\) −0.258796 −0.0173303 −0.00866514 0.999962i \(-0.502758\pi\)
−0.00866514 + 0.999962i \(0.502758\pi\)
\(224\) 5.03409 0.336354
\(225\) 0 0
\(226\) 29.5576 1.96614
\(227\) 5.53304 0.367240 0.183620 0.982997i \(-0.441218\pi\)
0.183620 + 0.982997i \(0.441218\pi\)
\(228\) 0 0
\(229\) −1.68175 −0.111133 −0.0555665 0.998455i \(-0.517696\pi\)
−0.0555665 + 0.998455i \(0.517696\pi\)
\(230\) 28.9705 1.91026
\(231\) 0 0
\(232\) 4.94751 0.324820
\(233\) −18.9196 −1.23946 −0.619732 0.784814i \(-0.712759\pi\)
−0.619732 + 0.784814i \(0.712759\pi\)
\(234\) 0 0
\(235\) 23.1333 1.50905
\(236\) 10.4391 0.679525
\(237\) 0 0
\(238\) −1.78192 −0.115505
\(239\) 21.0437 1.36121 0.680603 0.732653i \(-0.261718\pi\)
0.680603 + 0.732653i \(0.261718\pi\)
\(240\) 0 0
\(241\) 15.7612 1.01527 0.507634 0.861573i \(-0.330520\pi\)
0.507634 + 0.861573i \(0.330520\pi\)
\(242\) 0.776424 0.0499104
\(243\) 0 0
\(244\) 1.53564 0.0983090
\(245\) −2.01516 −0.128744
\(246\) 0 0
\(247\) 18.8102 1.19687
\(248\) 13.8744 0.881027
\(249\) 0 0
\(250\) −20.6013 −1.30294
\(251\) 7.62207 0.481101 0.240550 0.970637i \(-0.422672\pi\)
0.240550 + 0.970637i \(0.422672\pi\)
\(252\) 0 0
\(253\) −27.1261 −1.70540
\(254\) 1.72132 0.108006
\(255\) 0 0
\(256\) 18.6132 1.16333
\(257\) 10.8383 0.676076 0.338038 0.941132i \(-0.390237\pi\)
0.338038 + 0.941132i \(0.390237\pi\)
\(258\) 0 0
\(259\) −0.894661 −0.0555915
\(260\) 5.08092 0.315105
\(261\) 0 0
\(262\) −23.5753 −1.45649
\(263\) −4.06704 −0.250784 −0.125392 0.992107i \(-0.540019\pi\)
−0.125392 + 0.992107i \(0.540019\pi\)
\(264\) 0 0
\(265\) −2.44433 −0.150154
\(266\) 12.3661 0.758213
\(267\) 0 0
\(268\) −6.32918 −0.386616
\(269\) −16.5919 −1.01163 −0.505813 0.862643i \(-0.668808\pi\)
−0.505813 + 0.862643i \(0.668808\pi\)
\(270\) 0 0
\(271\) 16.2383 0.986406 0.493203 0.869914i \(-0.335826\pi\)
0.493203 + 0.869914i \(0.335826\pi\)
\(272\) −5.17459 −0.313755
\(273\) 0 0
\(274\) 24.4173 1.47510
\(275\) 3.05018 0.183933
\(276\) 0 0
\(277\) 20.6414 1.24022 0.620111 0.784514i \(-0.287088\pi\)
0.620111 + 0.784514i \(0.287088\pi\)
\(278\) −30.1535 −1.80849
\(279\) 0 0
\(280\) −3.59724 −0.214976
\(281\) −13.2288 −0.789163 −0.394581 0.918861i \(-0.629110\pi\)
−0.394581 + 0.918861i \(0.629110\pi\)
\(282\) 0 0
\(283\) −5.53483 −0.329011 −0.164506 0.986376i \(-0.552603\pi\)
−0.164506 + 0.986376i \(0.552603\pi\)
\(284\) −5.45868 −0.323913
\(285\) 0 0
\(286\) −14.6383 −0.865583
\(287\) −9.16796 −0.541167
\(288\) 0 0
\(289\) −15.9284 −0.936962
\(290\) 9.61392 0.564549
\(291\) 0 0
\(292\) 15.5460 0.909759
\(293\) 28.9827 1.69319 0.846593 0.532241i \(-0.178650\pi\)
0.846593 + 0.532241i \(0.178650\pi\)
\(294\) 0 0
\(295\) −21.8456 −1.27190
\(296\) −1.59705 −0.0928264
\(297\) 0 0
\(298\) −4.87014 −0.282120
\(299\) −21.8679 −1.26466
\(300\) 0 0
\(301\) 11.1174 0.640794
\(302\) 1.74684 0.100520
\(303\) 0 0
\(304\) 35.9104 2.05960
\(305\) −3.21359 −0.184010
\(306\) 0 0
\(307\) −15.1303 −0.863530 −0.431765 0.901986i \(-0.642109\pi\)
−0.431765 + 0.901986i \(0.642109\pi\)
\(308\) −3.12760 −0.178212
\(309\) 0 0
\(310\) 26.9605 1.53126
\(311\) 7.21480 0.409114 0.204557 0.978855i \(-0.434425\pi\)
0.204557 + 0.978855i \(0.434425\pi\)
\(312\) 0 0
\(313\) −5.70534 −0.322485 −0.161242 0.986915i \(-0.551550\pi\)
−0.161242 + 0.986915i \(0.551550\pi\)
\(314\) 15.1806 0.856690
\(315\) 0 0
\(316\) 6.30263 0.354550
\(317\) 5.85063 0.328604 0.164302 0.986410i \(-0.447463\pi\)
0.164302 + 0.986410i \(0.447463\pi\)
\(318\) 0 0
\(319\) −9.00186 −0.504007
\(320\) −2.68410 −0.150046
\(321\) 0 0
\(322\) −14.3762 −0.801157
\(323\) −7.43694 −0.413802
\(324\) 0 0
\(325\) 2.45893 0.136397
\(326\) 31.1377 1.72456
\(327\) 0 0
\(328\) −16.3656 −0.903638
\(329\) −11.4796 −0.632891
\(330\) 0 0
\(331\) −19.3118 −1.06147 −0.530736 0.847537i \(-0.678084\pi\)
−0.530736 + 0.847537i \(0.678084\pi\)
\(332\) 15.0472 0.825821
\(333\) 0 0
\(334\) 3.89581 0.213169
\(335\) 13.2449 0.723648
\(336\) 0 0
\(337\) 1.41941 0.0773203 0.0386602 0.999252i \(-0.487691\pi\)
0.0386602 + 0.999252i \(0.487691\pi\)
\(338\) 10.5764 0.575280
\(339\) 0 0
\(340\) −2.00883 −0.108944
\(341\) −25.2441 −1.36705
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 19.8455 1.07000
\(345\) 0 0
\(346\) −39.2243 −2.10871
\(347\) 14.9267 0.801307 0.400653 0.916230i \(-0.368783\pi\)
0.400653 + 0.916230i \(0.368783\pi\)
\(348\) 0 0
\(349\) −19.7344 −1.05636 −0.528180 0.849132i \(-0.677125\pi\)
−0.528180 + 0.849132i \(0.677125\pi\)
\(350\) 1.61653 0.0864071
\(351\) 0 0
\(352\) −16.3503 −0.871473
\(353\) −0.300628 −0.0160008 −0.00800040 0.999968i \(-0.502547\pi\)
−0.00800040 + 0.999968i \(0.502547\pi\)
\(354\) 0 0
\(355\) 11.4233 0.606284
\(356\) 2.12279 0.112508
\(357\) 0 0
\(358\) 7.24068 0.382682
\(359\) −29.5700 −1.56065 −0.780324 0.625376i \(-0.784946\pi\)
−0.780324 + 0.625376i \(0.784946\pi\)
\(360\) 0 0
\(361\) 32.6106 1.71635
\(362\) −12.2045 −0.641457
\(363\) 0 0
\(364\) −2.52134 −0.132154
\(365\) −32.5327 −1.70284
\(366\) 0 0
\(367\) −4.98916 −0.260432 −0.130216 0.991486i \(-0.541567\pi\)
−0.130216 + 0.991486i \(0.541567\pi\)
\(368\) −41.7478 −2.17625
\(369\) 0 0
\(370\) −3.10335 −0.161336
\(371\) 1.21297 0.0629743
\(372\) 0 0
\(373\) 25.4646 1.31851 0.659255 0.751920i \(-0.270872\pi\)
0.659255 + 0.751920i \(0.270872\pi\)
\(374\) 5.78751 0.299265
\(375\) 0 0
\(376\) −20.4921 −1.05680
\(377\) −7.25692 −0.373751
\(378\) 0 0
\(379\) −12.6469 −0.649630 −0.324815 0.945778i \(-0.605302\pi\)
−0.324815 + 0.945778i \(0.605302\pi\)
\(380\) 13.9408 0.715147
\(381\) 0 0
\(382\) 38.8073 1.98555
\(383\) 12.4457 0.635946 0.317973 0.948100i \(-0.396998\pi\)
0.317973 + 0.948100i \(0.396998\pi\)
\(384\) 0 0
\(385\) 6.54507 0.333568
\(386\) 12.0234 0.611973
\(387\) 0 0
\(388\) −16.6690 −0.846240
\(389\) 14.3780 0.728991 0.364496 0.931205i \(-0.381241\pi\)
0.364496 + 0.931205i \(0.381241\pi\)
\(390\) 0 0
\(391\) 8.64585 0.437239
\(392\) 1.78509 0.0901604
\(393\) 0 0
\(394\) 23.3304 1.17537
\(395\) −13.1894 −0.663629
\(396\) 0 0
\(397\) 4.33916 0.217776 0.108888 0.994054i \(-0.465271\pi\)
0.108888 + 0.994054i \(0.465271\pi\)
\(398\) −46.7728 −2.34451
\(399\) 0 0
\(400\) 4.69431 0.234715
\(401\) 5.64032 0.281664 0.140832 0.990033i \(-0.455022\pi\)
0.140832 + 0.990033i \(0.455022\pi\)
\(402\) 0 0
\(403\) −20.3508 −1.01374
\(404\) 0.529342 0.0263357
\(405\) 0 0
\(406\) −4.77079 −0.236770
\(407\) 2.90578 0.144034
\(408\) 0 0
\(409\) −1.85710 −0.0918276 −0.0459138 0.998945i \(-0.514620\pi\)
−0.0459138 + 0.998945i \(0.514620\pi\)
\(410\) −31.8013 −1.57056
\(411\) 0 0
\(412\) −3.30101 −0.162629
\(413\) 10.8406 0.533432
\(414\) 0 0
\(415\) −31.4889 −1.54573
\(416\) −13.1809 −0.646248
\(417\) 0 0
\(418\) −40.1640 −1.96448
\(419\) 1.24203 0.0606774 0.0303387 0.999540i \(-0.490341\pi\)
0.0303387 + 0.999540i \(0.490341\pi\)
\(420\) 0 0
\(421\) −10.4574 −0.509661 −0.254831 0.966986i \(-0.582020\pi\)
−0.254831 + 0.966986i \(0.582020\pi\)
\(422\) −15.0176 −0.731046
\(423\) 0 0
\(424\) 2.16526 0.105154
\(425\) −0.972178 −0.0471575
\(426\) 0 0
\(427\) 1.59471 0.0771733
\(428\) −2.82373 −0.136490
\(429\) 0 0
\(430\) 38.5634 1.85969
\(431\) −17.3256 −0.834543 −0.417272 0.908782i \(-0.637014\pi\)
−0.417272 + 0.908782i \(0.637014\pi\)
\(432\) 0 0
\(433\) −39.4038 −1.89362 −0.946812 0.321787i \(-0.895716\pi\)
−0.946812 + 0.321787i \(0.895716\pi\)
\(434\) −13.3788 −0.642205
\(435\) 0 0
\(436\) 5.42829 0.259968
\(437\) −60.0001 −2.87020
\(438\) 0 0
\(439\) −30.2773 −1.44506 −0.722529 0.691341i \(-0.757020\pi\)
−0.722529 + 0.691341i \(0.757020\pi\)
\(440\) 11.6835 0.556990
\(441\) 0 0
\(442\) 4.66565 0.221922
\(443\) −13.0145 −0.618338 −0.309169 0.951007i \(-0.600051\pi\)
−0.309169 + 0.951007i \(0.600051\pi\)
\(444\) 0 0
\(445\) −4.44232 −0.210586
\(446\) 0.445472 0.0210937
\(447\) 0 0
\(448\) 1.33195 0.0629289
\(449\) −12.3009 −0.580514 −0.290257 0.956949i \(-0.593741\pi\)
−0.290257 + 0.956949i \(0.593741\pi\)
\(450\) 0 0
\(451\) 29.7767 1.40213
\(452\) −16.5354 −0.777758
\(453\) 0 0
\(454\) −9.52415 −0.446990
\(455\) 5.27637 0.247360
\(456\) 0 0
\(457\) −10.9505 −0.512244 −0.256122 0.966644i \(-0.582445\pi\)
−0.256122 + 0.966644i \(0.582445\pi\)
\(458\) 2.89483 0.135267
\(459\) 0 0
\(460\) −16.2069 −0.755651
\(461\) −10.3995 −0.484355 −0.242177 0.970232i \(-0.577862\pi\)
−0.242177 + 0.970232i \(0.577862\pi\)
\(462\) 0 0
\(463\) 13.5446 0.629470 0.314735 0.949180i \(-0.398084\pi\)
0.314735 + 0.949180i \(0.398084\pi\)
\(464\) −13.8541 −0.643161
\(465\) 0 0
\(466\) 32.5668 1.50863
\(467\) −32.5621 −1.50679 −0.753397 0.657566i \(-0.771586\pi\)
−0.753397 + 0.657566i \(0.771586\pi\)
\(468\) 0 0
\(469\) −6.57264 −0.303496
\(470\) −39.8199 −1.83675
\(471\) 0 0
\(472\) 19.3514 0.890721
\(473\) −36.1082 −1.66026
\(474\) 0 0
\(475\) 6.74668 0.309559
\(476\) 0.996855 0.0456908
\(477\) 0 0
\(478\) −36.2231 −1.65681
\(479\) −16.0336 −0.732593 −0.366296 0.930498i \(-0.619374\pi\)
−0.366296 + 0.930498i \(0.619374\pi\)
\(480\) 0 0
\(481\) 2.34252 0.106810
\(482\) −27.1301 −1.23574
\(483\) 0 0
\(484\) −0.434354 −0.0197433
\(485\) 34.8828 1.58395
\(486\) 0 0
\(487\) 12.6892 0.575003 0.287502 0.957780i \(-0.407175\pi\)
0.287502 + 0.957780i \(0.407175\pi\)
\(488\) 2.84669 0.128864
\(489\) 0 0
\(490\) 3.46875 0.156702
\(491\) −20.3662 −0.919112 −0.459556 0.888149i \(-0.651991\pi\)
−0.459556 + 0.888149i \(0.651991\pi\)
\(492\) 0 0
\(493\) 2.86914 0.129220
\(494\) −32.3785 −1.45678
\(495\) 0 0
\(496\) −38.8514 −1.74448
\(497\) −5.66866 −0.254274
\(498\) 0 0
\(499\) −4.99576 −0.223641 −0.111820 0.993728i \(-0.535668\pi\)
−0.111820 + 0.993728i \(0.535668\pi\)
\(500\) 11.5250 0.515412
\(501\) 0 0
\(502\) −13.1201 −0.585577
\(503\) −10.9715 −0.489195 −0.244598 0.969625i \(-0.578656\pi\)
−0.244598 + 0.969625i \(0.578656\pi\)
\(504\) 0 0
\(505\) −1.10774 −0.0492939
\(506\) 46.6928 2.07575
\(507\) 0 0
\(508\) −0.962958 −0.0427244
\(509\) 38.3271 1.69882 0.849409 0.527734i \(-0.176958\pi\)
0.849409 + 0.527734i \(0.176958\pi\)
\(510\) 0 0
\(511\) 16.1440 0.714167
\(512\) −7.31758 −0.323394
\(513\) 0 0
\(514\) −18.6563 −0.822893
\(515\) 6.90796 0.304401
\(516\) 0 0
\(517\) 37.2848 1.63978
\(518\) 1.54000 0.0676638
\(519\) 0 0
\(520\) 9.41877 0.413040
\(521\) 21.3843 0.936862 0.468431 0.883500i \(-0.344819\pi\)
0.468431 + 0.883500i \(0.344819\pi\)
\(522\) 0 0
\(523\) 27.0915 1.18463 0.592315 0.805707i \(-0.298214\pi\)
0.592315 + 0.805707i \(0.298214\pi\)
\(524\) 13.1887 0.576152
\(525\) 0 0
\(526\) 7.00069 0.305245
\(527\) 8.04601 0.350490
\(528\) 0 0
\(529\) 46.7535 2.03276
\(530\) 4.20749 0.182762
\(531\) 0 0
\(532\) −6.91794 −0.299931
\(533\) 24.0048 1.03976
\(534\) 0 0
\(535\) 5.90915 0.255475
\(536\) −11.7327 −0.506777
\(537\) 0 0
\(538\) 28.5601 1.23131
\(539\) −3.24791 −0.139898
\(540\) 0 0
\(541\) −43.1632 −1.85573 −0.927865 0.372917i \(-0.878358\pi\)
−0.927865 + 0.372917i \(0.878358\pi\)
\(542\) −27.9514 −1.20061
\(543\) 0 0
\(544\) 5.21130 0.223432
\(545\) −11.3597 −0.486595
\(546\) 0 0
\(547\) 5.41592 0.231568 0.115784 0.993274i \(-0.463062\pi\)
0.115784 + 0.993274i \(0.463062\pi\)
\(548\) −13.6597 −0.583515
\(549\) 0 0
\(550\) −5.25035 −0.223876
\(551\) −19.9112 −0.848245
\(552\) 0 0
\(553\) 6.54507 0.278325
\(554\) −35.5305 −1.50955
\(555\) 0 0
\(556\) 16.8687 0.715393
\(557\) −14.4602 −0.612699 −0.306350 0.951919i \(-0.599108\pi\)
−0.306350 + 0.951919i \(0.599108\pi\)
\(558\) 0 0
\(559\) −29.1090 −1.23118
\(560\) 10.0730 0.425664
\(561\) 0 0
\(562\) 22.7710 0.960537
\(563\) 1.35164 0.0569648 0.0284824 0.999594i \(-0.490933\pi\)
0.0284824 + 0.999594i \(0.490933\pi\)
\(564\) 0 0
\(565\) 34.6032 1.45577
\(566\) 9.52724 0.400460
\(567\) 0 0
\(568\) −10.1190 −0.424585
\(569\) −19.4893 −0.817034 −0.408517 0.912751i \(-0.633954\pi\)
−0.408517 + 0.912751i \(0.633954\pi\)
\(570\) 0 0
\(571\) 1.57923 0.0660885 0.0330443 0.999454i \(-0.489480\pi\)
0.0330443 + 0.999454i \(0.489480\pi\)
\(572\) 8.18911 0.342404
\(573\) 0 0
\(574\) 15.7810 0.658687
\(575\) −7.84339 −0.327092
\(576\) 0 0
\(577\) −38.7980 −1.61518 −0.807591 0.589743i \(-0.799229\pi\)
−0.807591 + 0.589743i \(0.799229\pi\)
\(578\) 27.4179 1.14043
\(579\) 0 0
\(580\) −5.37830 −0.223322
\(581\) 15.6260 0.648275
\(582\) 0 0
\(583\) −3.93962 −0.163162
\(584\) 28.8184 1.19251
\(585\) 0 0
\(586\) −49.8886 −2.06088
\(587\) −7.94491 −0.327922 −0.163961 0.986467i \(-0.552427\pi\)
−0.163961 + 0.986467i \(0.552427\pi\)
\(588\) 0 0
\(589\) −55.8374 −2.30074
\(590\) 37.6034 1.54811
\(591\) 0 0
\(592\) 4.47208 0.183801
\(593\) −35.2002 −1.44550 −0.722749 0.691111i \(-0.757122\pi\)
−0.722749 + 0.691111i \(0.757122\pi\)
\(594\) 0 0
\(595\) −2.08610 −0.0855217
\(596\) 2.72450 0.111600
\(597\) 0 0
\(598\) 37.6418 1.53929
\(599\) −30.3691 −1.24085 −0.620425 0.784266i \(-0.713040\pi\)
−0.620425 + 0.784266i \(0.713040\pi\)
\(600\) 0 0
\(601\) −40.4918 −1.65170 −0.825848 0.563893i \(-0.809303\pi\)
−0.825848 + 0.563893i \(0.809303\pi\)
\(602\) −19.1366 −0.779950
\(603\) 0 0
\(604\) −0.977235 −0.0397631
\(605\) 0.908963 0.0369546
\(606\) 0 0
\(607\) 17.4539 0.708432 0.354216 0.935164i \(-0.384748\pi\)
0.354216 + 0.935164i \(0.384748\pi\)
\(608\) −36.1652 −1.46669
\(609\) 0 0
\(610\) 5.53164 0.223969
\(611\) 30.0574 1.21599
\(612\) 0 0
\(613\) 8.64856 0.349312 0.174656 0.984630i \(-0.444119\pi\)
0.174656 + 0.984630i \(0.444119\pi\)
\(614\) 26.0441 1.05105
\(615\) 0 0
\(616\) −5.79780 −0.233600
\(617\) 15.7048 0.632251 0.316125 0.948717i \(-0.397618\pi\)
0.316125 + 0.948717i \(0.397618\pi\)
\(618\) 0 0
\(619\) −10.6788 −0.429217 −0.214609 0.976700i \(-0.568848\pi\)
−0.214609 + 0.976700i \(0.568848\pi\)
\(620\) −15.0825 −0.605728
\(621\) 0 0
\(622\) −12.4190 −0.497957
\(623\) 2.20445 0.0883193
\(624\) 0 0
\(625\) −19.4225 −0.776898
\(626\) 9.82074 0.392516
\(627\) 0 0
\(628\) −8.49246 −0.338886
\(629\) −0.926154 −0.0369282
\(630\) 0 0
\(631\) −7.18368 −0.285978 −0.142989 0.989724i \(-0.545671\pi\)
−0.142989 + 0.989724i \(0.545671\pi\)
\(632\) 11.6835 0.464745
\(633\) 0 0
\(634\) −10.0708 −0.399964
\(635\) 2.01516 0.0799693
\(636\) 0 0
\(637\) −2.61833 −0.103742
\(638\) 15.4951 0.613457
\(639\) 0 0
\(640\) 24.9092 0.984624
\(641\) −0.0204183 −0.000806474 0 −0.000403237 1.00000i \(-0.500128\pi\)
−0.000403237 1.00000i \(0.500128\pi\)
\(642\) 0 0
\(643\) −39.9366 −1.57495 −0.787473 0.616349i \(-0.788611\pi\)
−0.787473 + 0.616349i \(0.788611\pi\)
\(644\) 8.04248 0.316918
\(645\) 0 0
\(646\) 12.8014 0.503664
\(647\) 16.2781 0.639957 0.319979 0.947425i \(-0.396324\pi\)
0.319979 + 0.947425i \(0.396324\pi\)
\(648\) 0 0
\(649\) −35.2094 −1.38209
\(650\) −4.23261 −0.166017
\(651\) 0 0
\(652\) −17.4193 −0.682194
\(653\) −42.9659 −1.68138 −0.840692 0.541514i \(-0.817851\pi\)
−0.840692 + 0.541514i \(0.817851\pi\)
\(654\) 0 0
\(655\) −27.5998 −1.07841
\(656\) 45.8272 1.78925
\(657\) 0 0
\(658\) 19.7601 0.770330
\(659\) −28.4514 −1.10831 −0.554156 0.832413i \(-0.686959\pi\)
−0.554156 + 0.832413i \(0.686959\pi\)
\(660\) 0 0
\(661\) 5.86659 0.228184 0.114092 0.993470i \(-0.463604\pi\)
0.114092 + 0.993470i \(0.463604\pi\)
\(662\) 33.2418 1.29198
\(663\) 0 0
\(664\) 27.8937 1.08249
\(665\) 14.4770 0.561395
\(666\) 0 0
\(667\) 23.1478 0.896288
\(668\) −2.17942 −0.0843245
\(669\) 0 0
\(670\) −22.7988 −0.880796
\(671\) −5.17947 −0.199951
\(672\) 0 0
\(673\) −22.2506 −0.857697 −0.428849 0.903376i \(-0.641081\pi\)
−0.428849 + 0.903376i \(0.641081\pi\)
\(674\) −2.44327 −0.0941112
\(675\) 0 0
\(676\) −5.91674 −0.227567
\(677\) 6.29073 0.241772 0.120886 0.992666i \(-0.461426\pi\)
0.120886 + 0.992666i \(0.461426\pi\)
\(678\) 0 0
\(679\) −17.3102 −0.664304
\(680\) −3.72387 −0.142804
\(681\) 0 0
\(682\) 43.4533 1.66391
\(683\) −1.06197 −0.0406352 −0.0203176 0.999794i \(-0.506468\pi\)
−0.0203176 + 0.999794i \(0.506468\pi\)
\(684\) 0 0
\(685\) 28.5854 1.09219
\(686\) −1.72132 −0.0657205
\(687\) 0 0
\(688\) −55.5716 −2.11865
\(689\) −3.17596 −0.120994
\(690\) 0 0
\(691\) −22.3544 −0.850402 −0.425201 0.905099i \(-0.639797\pi\)
−0.425201 + 0.905099i \(0.639797\pi\)
\(692\) 21.9432 0.834154
\(693\) 0 0
\(694\) −25.6937 −0.975319
\(695\) −35.3008 −1.33904
\(696\) 0 0
\(697\) −9.49068 −0.359485
\(698\) 33.9694 1.28576
\(699\) 0 0
\(700\) −0.904333 −0.0341806
\(701\) −10.9852 −0.414907 −0.207454 0.978245i \(-0.566518\pi\)
−0.207454 + 0.978245i \(0.566518\pi\)
\(702\) 0 0
\(703\) 6.42729 0.242410
\(704\) −4.32607 −0.163045
\(705\) 0 0
\(706\) 0.517478 0.0194755
\(707\) 0.549704 0.0206737
\(708\) 0 0
\(709\) −20.1642 −0.757283 −0.378641 0.925543i \(-0.623609\pi\)
−0.378641 + 0.925543i \(0.623609\pi\)
\(710\) −19.6631 −0.737945
\(711\) 0 0
\(712\) 3.93512 0.147475
\(713\) 64.9140 2.43105
\(714\) 0 0
\(715\) −17.1372 −0.640894
\(716\) −4.05064 −0.151380
\(717\) 0 0
\(718\) 50.8996 1.89956
\(719\) −6.75963 −0.252092 −0.126046 0.992024i \(-0.540229\pi\)
−0.126046 + 0.992024i \(0.540229\pi\)
\(720\) 0 0
\(721\) −3.42799 −0.127665
\(722\) −56.1334 −2.08907
\(723\) 0 0
\(724\) 6.82757 0.253745
\(725\) −2.60285 −0.0966673
\(726\) 0 0
\(727\) −2.42574 −0.0899657 −0.0449829 0.998988i \(-0.514323\pi\)
−0.0449829 + 0.998988i \(0.514323\pi\)
\(728\) −4.67395 −0.173228
\(729\) 0 0
\(730\) 55.9994 2.07263
\(731\) 11.5087 0.425665
\(732\) 0 0
\(733\) 41.0765 1.51720 0.758598 0.651559i \(-0.225885\pi\)
0.758598 + 0.651559i \(0.225885\pi\)
\(734\) 8.58796 0.316988
\(735\) 0 0
\(736\) 42.0440 1.54976
\(737\) 21.3474 0.786340
\(738\) 0 0
\(739\) 32.4278 1.19287 0.596437 0.802660i \(-0.296582\pi\)
0.596437 + 0.802660i \(0.296582\pi\)
\(740\) 1.73610 0.0638205
\(741\) 0 0
\(742\) −2.08792 −0.0766498
\(743\) 49.6144 1.82017 0.910087 0.414417i \(-0.136014\pi\)
0.910087 + 0.414417i \(0.136014\pi\)
\(744\) 0 0
\(745\) −5.70150 −0.208887
\(746\) −43.8329 −1.60484
\(747\) 0 0
\(748\) −3.23770 −0.118382
\(749\) −2.93235 −0.107146
\(750\) 0 0
\(751\) −10.6044 −0.386959 −0.193480 0.981104i \(-0.561977\pi\)
−0.193480 + 0.981104i \(0.561977\pi\)
\(752\) 57.3823 2.09252
\(753\) 0 0
\(754\) 12.4915 0.454914
\(755\) 2.04504 0.0744266
\(756\) 0 0
\(757\) −23.5334 −0.855336 −0.427668 0.903936i \(-0.640665\pi\)
−0.427668 + 0.903936i \(0.640665\pi\)
\(758\) 21.7695 0.790703
\(759\) 0 0
\(760\) 25.8427 0.937415
\(761\) −4.46951 −0.162020 −0.0810098 0.996713i \(-0.525815\pi\)
−0.0810098 + 0.996713i \(0.525815\pi\)
\(762\) 0 0
\(763\) 5.63710 0.204077
\(764\) −21.7099 −0.785436
\(765\) 0 0
\(766\) −21.4231 −0.774049
\(767\) −28.3843 −1.02490
\(768\) 0 0
\(769\) −27.0331 −0.974838 −0.487419 0.873168i \(-0.662062\pi\)
−0.487419 + 0.873168i \(0.662062\pi\)
\(770\) −11.2662 −0.406005
\(771\) 0 0
\(772\) −6.72622 −0.242082
\(773\) −45.1300 −1.62321 −0.811607 0.584204i \(-0.801407\pi\)
−0.811607 + 0.584204i \(0.801407\pi\)
\(774\) 0 0
\(775\) −7.29922 −0.262196
\(776\) −30.9002 −1.10925
\(777\) 0 0
\(778\) −24.7491 −0.887299
\(779\) 65.8631 2.35979
\(780\) 0 0
\(781\) 18.4113 0.658808
\(782\) −14.8823 −0.532190
\(783\) 0 0
\(784\) −4.99863 −0.178522
\(785\) 17.7720 0.634310
\(786\) 0 0
\(787\) −3.14036 −0.111942 −0.0559709 0.998432i \(-0.517825\pi\)
−0.0559709 + 0.998432i \(0.517825\pi\)
\(788\) −13.0517 −0.464947
\(789\) 0 0
\(790\) 22.7032 0.807743
\(791\) −17.1714 −0.610545
\(792\) 0 0
\(793\) −4.17547 −0.148275
\(794\) −7.46911 −0.265069
\(795\) 0 0
\(796\) 26.1660 0.927430
\(797\) 24.8520 0.880304 0.440152 0.897923i \(-0.354925\pi\)
0.440152 + 0.897923i \(0.354925\pi\)
\(798\) 0 0
\(799\) −11.8837 −0.420415
\(800\) −4.72761 −0.167146
\(801\) 0 0
\(802\) −9.70883 −0.342831
\(803\) −52.4342 −1.85036
\(804\) 0 0
\(805\) −16.8303 −0.593192
\(806\) 35.0303 1.23389
\(807\) 0 0
\(808\) 0.981268 0.0345209
\(809\) 18.8062 0.661191 0.330595 0.943773i \(-0.392750\pi\)
0.330595 + 0.943773i \(0.392750\pi\)
\(810\) 0 0
\(811\) 24.4579 0.858833 0.429416 0.903107i \(-0.358719\pi\)
0.429416 + 0.903107i \(0.358719\pi\)
\(812\) 2.66892 0.0936606
\(813\) 0 0
\(814\) −5.00179 −0.175313
\(815\) 36.4531 1.27690
\(816\) 0 0
\(817\) −79.8677 −2.79422
\(818\) 3.19667 0.111769
\(819\) 0 0
\(820\) 17.7906 0.621274
\(821\) −22.0065 −0.768031 −0.384015 0.923327i \(-0.625459\pi\)
−0.384015 + 0.923327i \(0.625459\pi\)
\(822\) 0 0
\(823\) 22.3836 0.780243 0.390122 0.920763i \(-0.372433\pi\)
0.390122 + 0.920763i \(0.372433\pi\)
\(824\) −6.11926 −0.213174
\(825\) 0 0
\(826\) −18.6602 −0.649272
\(827\) −42.9231 −1.49258 −0.746292 0.665619i \(-0.768167\pi\)
−0.746292 + 0.665619i \(0.768167\pi\)
\(828\) 0 0
\(829\) −35.3720 −1.22852 −0.614261 0.789103i \(-0.710546\pi\)
−0.614261 + 0.789103i \(0.710546\pi\)
\(830\) 54.2026 1.88140
\(831\) 0 0
\(832\) −3.48750 −0.120907
\(833\) 1.03520 0.0358676
\(834\) 0 0
\(835\) 4.56084 0.157834
\(836\) 22.4689 0.777102
\(837\) 0 0
\(838\) −2.13795 −0.0738541
\(839\) −33.4309 −1.15416 −0.577082 0.816686i \(-0.695809\pi\)
−0.577082 + 0.816686i \(0.695809\pi\)
\(840\) 0 0
\(841\) −21.3183 −0.735115
\(842\) 18.0005 0.620340
\(843\) 0 0
\(844\) 8.40129 0.289184
\(845\) 12.3818 0.425948
\(846\) 0 0
\(847\) −0.451062 −0.0154987
\(848\) −6.06319 −0.208211
\(849\) 0 0
\(850\) 1.67343 0.0573983
\(851\) −7.47207 −0.256139
\(852\) 0 0
\(853\) −27.3404 −0.936119 −0.468060 0.883697i \(-0.655047\pi\)
−0.468060 + 0.883697i \(0.655047\pi\)
\(854\) −2.74501 −0.0939322
\(855\) 0 0
\(856\) −5.23449 −0.178911
\(857\) 28.1895 0.962936 0.481468 0.876464i \(-0.340104\pi\)
0.481468 + 0.876464i \(0.340104\pi\)
\(858\) 0 0
\(859\) −26.0793 −0.889813 −0.444907 0.895577i \(-0.646763\pi\)
−0.444907 + 0.895577i \(0.646763\pi\)
\(860\) −21.5734 −0.735648
\(861\) 0 0
\(862\) 29.8229 1.01577
\(863\) −19.7358 −0.671813 −0.335907 0.941895i \(-0.609043\pi\)
−0.335907 + 0.941895i \(0.609043\pi\)
\(864\) 0 0
\(865\) −45.9200 −1.56133
\(866\) 67.8267 2.30484
\(867\) 0 0
\(868\) 7.48451 0.254041
\(869\) −21.2578 −0.721122
\(870\) 0 0
\(871\) 17.2094 0.583117
\(872\) 10.0627 0.340766
\(873\) 0 0
\(874\) 103.280 3.49349
\(875\) 11.9683 0.404602
\(876\) 0 0
\(877\) 6.40535 0.216293 0.108147 0.994135i \(-0.465508\pi\)
0.108147 + 0.994135i \(0.465508\pi\)
\(878\) 52.1171 1.75887
\(879\) 0 0
\(880\) −32.7164 −1.10287
\(881\) 52.8133 1.77933 0.889663 0.456618i \(-0.150939\pi\)
0.889663 + 0.456618i \(0.150939\pi\)
\(882\) 0 0
\(883\) 44.1835 1.48689 0.743446 0.668796i \(-0.233190\pi\)
0.743446 + 0.668796i \(0.233190\pi\)
\(884\) −2.61010 −0.0877871
\(885\) 0 0
\(886\) 22.4022 0.752616
\(887\) −39.5788 −1.32893 −0.664463 0.747322i \(-0.731340\pi\)
−0.664463 + 0.747322i \(0.731340\pi\)
\(888\) 0 0
\(889\) −1.00000 −0.0335389
\(890\) 7.64667 0.256317
\(891\) 0 0
\(892\) −0.249210 −0.00834417
\(893\) 82.4701 2.75976
\(894\) 0 0
\(895\) 8.47669 0.283345
\(896\) −12.3609 −0.412949
\(897\) 0 0
\(898\) 21.1738 0.706578
\(899\) 21.5419 0.718462
\(900\) 0 0
\(901\) 1.25567 0.0418324
\(902\) −51.2554 −1.70662
\(903\) 0 0
\(904\) −30.6524 −1.01949
\(905\) −14.2879 −0.474946
\(906\) 0 0
\(907\) −30.5378 −1.01399 −0.506995 0.861949i \(-0.669244\pi\)
−0.506995 + 0.861949i \(0.669244\pi\)
\(908\) 5.32808 0.176819
\(909\) 0 0
\(910\) −9.08234 −0.301077
\(911\) 27.6405 0.915770 0.457885 0.889012i \(-0.348607\pi\)
0.457885 + 0.889012i \(0.348607\pi\)
\(912\) 0 0
\(913\) −50.7518 −1.67964
\(914\) 18.8494 0.623483
\(915\) 0 0
\(916\) −1.61945 −0.0535082
\(917\) 13.6960 0.452283
\(918\) 0 0
\(919\) −55.2756 −1.82337 −0.911687 0.410885i \(-0.865220\pi\)
−0.911687 + 0.410885i \(0.865220\pi\)
\(920\) −30.0436 −0.990508
\(921\) 0 0
\(922\) 17.9010 0.589537
\(923\) 14.8424 0.488545
\(924\) 0 0
\(925\) 0.840193 0.0276254
\(926\) −23.3146 −0.766166
\(927\) 0 0
\(928\) 13.9524 0.458010
\(929\) −35.8329 −1.17564 −0.587820 0.808991i \(-0.700014\pi\)
−0.587820 + 0.808991i \(0.700014\pi\)
\(930\) 0 0
\(931\) −7.18405 −0.235448
\(932\) −18.2188 −0.596776
\(933\) 0 0
\(934\) 56.0499 1.83401
\(935\) 6.77547 0.221581
\(936\) 0 0
\(937\) −16.3521 −0.534199 −0.267100 0.963669i \(-0.586065\pi\)
−0.267100 + 0.963669i \(0.586065\pi\)
\(938\) 11.3136 0.369404
\(939\) 0 0
\(940\) 22.2764 0.726575
\(941\) 41.0319 1.33760 0.668801 0.743441i \(-0.266808\pi\)
0.668801 + 0.743441i \(0.266808\pi\)
\(942\) 0 0
\(943\) −76.5694 −2.49344
\(944\) −54.1882 −1.76368
\(945\) 0 0
\(946\) 62.1540 2.02080
\(947\) −18.8533 −0.612650 −0.306325 0.951927i \(-0.599099\pi\)
−0.306325 + 0.951927i \(0.599099\pi\)
\(948\) 0 0
\(949\) −42.2703 −1.37215
\(950\) −11.6132 −0.376783
\(951\) 0 0
\(952\) 1.84792 0.0598915
\(953\) 30.0702 0.974069 0.487034 0.873383i \(-0.338079\pi\)
0.487034 + 0.873383i \(0.338079\pi\)
\(954\) 0 0
\(955\) 45.4319 1.47014
\(956\) 20.2642 0.655392
\(957\) 0 0
\(958\) 27.5990 0.891683
\(959\) −14.1852 −0.458063
\(960\) 0 0
\(961\) 29.4104 0.948722
\(962\) −4.03224 −0.130005
\(963\) 0 0
\(964\) 15.1774 0.488830
\(965\) 14.0758 0.453117
\(966\) 0 0
\(967\) −18.0543 −0.580587 −0.290294 0.956938i \(-0.593753\pi\)
−0.290294 + 0.956938i \(0.593753\pi\)
\(968\) −0.805184 −0.0258796
\(969\) 0 0
\(970\) −60.0447 −1.92792
\(971\) −11.1615 −0.358189 −0.179094 0.983832i \(-0.557317\pi\)
−0.179094 + 0.983832i \(0.557317\pi\)
\(972\) 0 0
\(973\) 17.5176 0.561588
\(974\) −21.8423 −0.699871
\(975\) 0 0
\(976\) −7.97135 −0.255157
\(977\) −0.715429 −0.0228886 −0.0114443 0.999935i \(-0.503643\pi\)
−0.0114443 + 0.999935i \(0.503643\pi\)
\(978\) 0 0
\(979\) −7.15985 −0.228830
\(980\) −1.94052 −0.0619875
\(981\) 0 0
\(982\) 35.0568 1.11871
\(983\) −11.1984 −0.357175 −0.178587 0.983924i \(-0.557153\pi\)
−0.178587 + 0.983924i \(0.557153\pi\)
\(984\) 0 0
\(985\) 27.3130 0.870264
\(986\) −4.93873 −0.157281
\(987\) 0 0
\(988\) 18.1135 0.576266
\(989\) 92.8506 2.95248
\(990\) 0 0
\(991\) 15.0280 0.477381 0.238690 0.971096i \(-0.423282\pi\)
0.238690 + 0.971096i \(0.423282\pi\)
\(992\) 39.1270 1.24228
\(993\) 0 0
\(994\) 9.75760 0.309492
\(995\) −54.7571 −1.73592
\(996\) 0 0
\(997\) −12.8352 −0.406496 −0.203248 0.979127i \(-0.565150\pi\)
−0.203248 + 0.979127i \(0.565150\pi\)
\(998\) 8.59932 0.272207
\(999\) 0 0
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 8001.2.a.w.1.8 20
3.2 odd 2 889.2.a.d.1.13 20
21.20 even 2 6223.2.a.l.1.13 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
889.2.a.d.1.13 20 3.2 odd 2
6223.2.a.l.1.13 20 21.20 even 2
8001.2.a.w.1.8 20 1.1 even 1 trivial