Properties

Label 6039.2.a.m.1.18
Level $6039$
Weight $2$
Character 6039.1
Self dual yes
Analytic conductor $48.222$
Analytic rank $1$
Dimension $25$
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] = [6039,2,Mod(1,6039)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6039, 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("6039.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6039.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: \(48.2216577807\)
Analytic rank: \(1\)
Dimension: \(25\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.18
Character \(\chi\) \(=\) 6039.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.04320 q^{2} -0.911728 q^{4} +1.64856 q^{5} +5.01087 q^{7} -3.03752 q^{8} +O(q^{10})\) \(q+1.04320 q^{2} -0.911728 q^{4} +1.64856 q^{5} +5.01087 q^{7} -3.03752 q^{8} +1.71978 q^{10} +1.00000 q^{11} -3.01798 q^{13} +5.22736 q^{14} -1.34530 q^{16} -3.18804 q^{17} -6.03685 q^{19} -1.50303 q^{20} +1.04320 q^{22} -4.89559 q^{23} -2.28226 q^{25} -3.14836 q^{26} -4.56855 q^{28} -0.904141 q^{29} +5.23680 q^{31} +4.67163 q^{32} -3.32577 q^{34} +8.26071 q^{35} -5.38300 q^{37} -6.29766 q^{38} -5.00753 q^{40} -11.7027 q^{41} -11.2235 q^{43} -0.911728 q^{44} -5.10709 q^{46} -7.10722 q^{47} +18.1089 q^{49} -2.38086 q^{50} +2.75158 q^{52} -13.0696 q^{53} +1.64856 q^{55} -15.2206 q^{56} -0.943203 q^{58} +0.468335 q^{59} +1.00000 q^{61} +5.46305 q^{62} +7.56405 q^{64} -4.97531 q^{65} -3.15644 q^{67} +2.90662 q^{68} +8.61759 q^{70} +9.71441 q^{71} -3.32256 q^{73} -5.61556 q^{74} +5.50397 q^{76} +5.01087 q^{77} -5.59101 q^{79} -2.21780 q^{80} -12.2083 q^{82} -0.444183 q^{83} -5.25566 q^{85} -11.7084 q^{86} -3.03752 q^{88} +3.27073 q^{89} -15.1227 q^{91} +4.46345 q^{92} -7.41427 q^{94} -9.95209 q^{95} +7.38020 q^{97} +18.8912 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8} - 12 q^{10} + 25 q^{11} - 4 q^{13} - 14 q^{14} + 21 q^{16} - 16 q^{17} - 18 q^{19} - 28 q^{20} - 5 q^{22} - 8 q^{23} + 29 q^{25} - 16 q^{26} + 18 q^{28} - 28 q^{29} - 8 q^{31} - 35 q^{32} + 6 q^{34} - 22 q^{35} + 4 q^{37} + 4 q^{38} - 12 q^{40} - 58 q^{41} - 26 q^{43} + 25 q^{44} + 8 q^{46} - 20 q^{47} + 23 q^{49} - 27 q^{50} - 2 q^{52} - 36 q^{53} - 12 q^{55} - 70 q^{56} + 12 q^{58} - 18 q^{59} + 25 q^{61} - 42 q^{62} + 35 q^{64} - 76 q^{65} - 8 q^{67} - 28 q^{68} + 76 q^{70} - 24 q^{71} + 2 q^{73} - 40 q^{74} - 64 q^{76} - 4 q^{77} - 22 q^{79} - 36 q^{80} + 30 q^{82} - 14 q^{83} - 70 q^{86} - 15 q^{88} - 82 q^{89} - 6 q^{91} - 48 q^{92} - 16 q^{94} - 34 q^{95} + 16 q^{97} - 35 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.04320 0.737656 0.368828 0.929498i \(-0.379759\pi\)
0.368828 + 0.929498i \(0.379759\pi\)
\(3\) 0 0
\(4\) −0.911728 −0.455864
\(5\) 1.64856 0.737257 0.368628 0.929577i \(-0.379828\pi\)
0.368628 + 0.929577i \(0.379828\pi\)
\(6\) 0 0
\(7\) 5.01087 1.89393 0.946966 0.321334i \(-0.104131\pi\)
0.946966 + 0.321334i \(0.104131\pi\)
\(8\) −3.03752 −1.07393
\(9\) 0 0
\(10\) 1.71978 0.543842
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −3.01798 −0.837037 −0.418518 0.908208i \(-0.637450\pi\)
−0.418518 + 0.908208i \(0.637450\pi\)
\(14\) 5.22736 1.39707
\(15\) 0 0
\(16\) −1.34530 −0.336324
\(17\) −3.18804 −0.773213 −0.386606 0.922245i \(-0.626353\pi\)
−0.386606 + 0.922245i \(0.626353\pi\)
\(18\) 0 0
\(19\) −6.03685 −1.38495 −0.692475 0.721442i \(-0.743480\pi\)
−0.692475 + 0.721442i \(0.743480\pi\)
\(20\) −1.50303 −0.336089
\(21\) 0 0
\(22\) 1.04320 0.222412
\(23\) −4.89559 −1.02080 −0.510401 0.859937i \(-0.670503\pi\)
−0.510401 + 0.859937i \(0.670503\pi\)
\(24\) 0 0
\(25\) −2.28226 −0.456453
\(26\) −3.14836 −0.617445
\(27\) 0 0
\(28\) −4.56855 −0.863375
\(29\) −0.904141 −0.167895 −0.0839474 0.996470i \(-0.526753\pi\)
−0.0839474 + 0.996470i \(0.526753\pi\)
\(30\) 0 0
\(31\) 5.23680 0.940558 0.470279 0.882518i \(-0.344153\pi\)
0.470279 + 0.882518i \(0.344153\pi\)
\(32\) 4.67163 0.825835
\(33\) 0 0
\(34\) −3.32577 −0.570365
\(35\) 8.26071 1.39631
\(36\) 0 0
\(37\) −5.38300 −0.884960 −0.442480 0.896778i \(-0.645901\pi\)
−0.442480 + 0.896778i \(0.645901\pi\)
\(38\) −6.29766 −1.02162
\(39\) 0 0
\(40\) −5.00753 −0.791759
\(41\) −11.7027 −1.82766 −0.913828 0.406101i \(-0.866888\pi\)
−0.913828 + 0.406101i \(0.866888\pi\)
\(42\) 0 0
\(43\) −11.2235 −1.71157 −0.855785 0.517331i \(-0.826926\pi\)
−0.855785 + 0.517331i \(0.826926\pi\)
\(44\) −0.911728 −0.137448
\(45\) 0 0
\(46\) −5.10709 −0.753000
\(47\) −7.10722 −1.03669 −0.518347 0.855170i \(-0.673452\pi\)
−0.518347 + 0.855170i \(0.673452\pi\)
\(48\) 0 0
\(49\) 18.1089 2.58698
\(50\) −2.38086 −0.336705
\(51\) 0 0
\(52\) 2.75158 0.381575
\(53\) −13.0696 −1.79525 −0.897625 0.440760i \(-0.854709\pi\)
−0.897625 + 0.440760i \(0.854709\pi\)
\(54\) 0 0
\(55\) 1.64856 0.222291
\(56\) −15.2206 −2.03394
\(57\) 0 0
\(58\) −0.943203 −0.123849
\(59\) 0.468335 0.0609720 0.0304860 0.999535i \(-0.490295\pi\)
0.0304860 + 0.999535i \(0.490295\pi\)
\(60\) 0 0
\(61\) 1.00000 0.128037
\(62\) 5.46305 0.693808
\(63\) 0 0
\(64\) 7.56405 0.945506
\(65\) −4.97531 −0.617111
\(66\) 0 0
\(67\) −3.15644 −0.385621 −0.192810 0.981236i \(-0.561760\pi\)
−0.192810 + 0.981236i \(0.561760\pi\)
\(68\) 2.90662 0.352480
\(69\) 0 0
\(70\) 8.61759 1.03000
\(71\) 9.71441 1.15289 0.576444 0.817136i \(-0.304440\pi\)
0.576444 + 0.817136i \(0.304440\pi\)
\(72\) 0 0
\(73\) −3.32256 −0.388877 −0.194438 0.980915i \(-0.562288\pi\)
−0.194438 + 0.980915i \(0.562288\pi\)
\(74\) −5.61556 −0.652796
\(75\) 0 0
\(76\) 5.50397 0.631348
\(77\) 5.01087 0.571042
\(78\) 0 0
\(79\) −5.59101 −0.629038 −0.314519 0.949251i \(-0.601843\pi\)
−0.314519 + 0.949251i \(0.601843\pi\)
\(80\) −2.21780 −0.247957
\(81\) 0 0
\(82\) −12.2083 −1.34818
\(83\) −0.444183 −0.0487554 −0.0243777 0.999703i \(-0.507760\pi\)
−0.0243777 + 0.999703i \(0.507760\pi\)
\(84\) 0 0
\(85\) −5.25566 −0.570056
\(86\) −11.7084 −1.26255
\(87\) 0 0
\(88\) −3.03752 −0.323801
\(89\) 3.27073 0.346697 0.173348 0.984861i \(-0.444541\pi\)
0.173348 + 0.984861i \(0.444541\pi\)
\(90\) 0 0
\(91\) −15.1227 −1.58529
\(92\) 4.46345 0.465347
\(93\) 0 0
\(94\) −7.41427 −0.764723
\(95\) −9.95209 −1.02106
\(96\) 0 0
\(97\) 7.38020 0.749346 0.374673 0.927157i \(-0.377755\pi\)
0.374673 + 0.927157i \(0.377755\pi\)
\(98\) 18.8912 1.90830
\(99\) 0 0
\(100\) 2.08080 0.208080
\(101\) 1.95613 0.194643 0.0973213 0.995253i \(-0.468973\pi\)
0.0973213 + 0.995253i \(0.468973\pi\)
\(102\) 0 0
\(103\) 4.90048 0.482858 0.241429 0.970418i \(-0.422384\pi\)
0.241429 + 0.970418i \(0.422384\pi\)
\(104\) 9.16718 0.898916
\(105\) 0 0
\(106\) −13.6343 −1.32428
\(107\) −1.67145 −0.161585 −0.0807927 0.996731i \(-0.525745\pi\)
−0.0807927 + 0.996731i \(0.525745\pi\)
\(108\) 0 0
\(109\) 17.3299 1.65991 0.829954 0.557832i \(-0.188367\pi\)
0.829954 + 0.557832i \(0.188367\pi\)
\(110\) 1.71978 0.163974
\(111\) 0 0
\(112\) −6.74111 −0.636975
\(113\) 8.43203 0.793219 0.396609 0.917987i \(-0.370187\pi\)
0.396609 + 0.917987i \(0.370187\pi\)
\(114\) 0 0
\(115\) −8.07066 −0.752593
\(116\) 0.824331 0.0765372
\(117\) 0 0
\(118\) 0.488568 0.0449763
\(119\) −15.9748 −1.46441
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 1.04320 0.0944471
\(123\) 0 0
\(124\) −4.77454 −0.428766
\(125\) −12.0052 −1.07378
\(126\) 0 0
\(127\) 14.7903 1.31242 0.656212 0.754577i \(-0.272158\pi\)
0.656212 + 0.754577i \(0.272158\pi\)
\(128\) −1.45242 −0.128377
\(129\) 0 0
\(130\) −5.19025 −0.455215
\(131\) −0.343726 −0.0300315 −0.0150157 0.999887i \(-0.504780\pi\)
−0.0150157 + 0.999887i \(0.504780\pi\)
\(132\) 0 0
\(133\) −30.2499 −2.62300
\(134\) −3.29281 −0.284455
\(135\) 0 0
\(136\) 9.68373 0.830373
\(137\) 13.7289 1.17294 0.586469 0.809972i \(-0.300518\pi\)
0.586469 + 0.809972i \(0.300518\pi\)
\(138\) 0 0
\(139\) −16.3286 −1.38498 −0.692488 0.721429i \(-0.743486\pi\)
−0.692488 + 0.721429i \(0.743486\pi\)
\(140\) −7.53152 −0.636529
\(141\) 0 0
\(142\) 10.1341 0.850435
\(143\) −3.01798 −0.252376
\(144\) 0 0
\(145\) −1.49053 −0.123782
\(146\) −3.46611 −0.286857
\(147\) 0 0
\(148\) 4.90783 0.403421
\(149\) −0.633092 −0.0518649 −0.0259325 0.999664i \(-0.508255\pi\)
−0.0259325 + 0.999664i \(0.508255\pi\)
\(150\) 0 0
\(151\) −10.3808 −0.844777 −0.422388 0.906415i \(-0.638808\pi\)
−0.422388 + 0.906415i \(0.638808\pi\)
\(152\) 18.3371 1.48733
\(153\) 0 0
\(154\) 5.22736 0.421232
\(155\) 8.63316 0.693432
\(156\) 0 0
\(157\) 20.2141 1.61326 0.806631 0.591056i \(-0.201289\pi\)
0.806631 + 0.591056i \(0.201289\pi\)
\(158\) −5.83256 −0.464013
\(159\) 0 0
\(160\) 7.70144 0.608852
\(161\) −24.5312 −1.93333
\(162\) 0 0
\(163\) 11.6042 0.908909 0.454455 0.890770i \(-0.349834\pi\)
0.454455 + 0.890770i \(0.349834\pi\)
\(164\) 10.6697 0.833163
\(165\) 0 0
\(166\) −0.463373 −0.0359647
\(167\) −12.8356 −0.993252 −0.496626 0.867965i \(-0.665428\pi\)
−0.496626 + 0.867965i \(0.665428\pi\)
\(168\) 0 0
\(169\) −3.89181 −0.299370
\(170\) −5.48272 −0.420505
\(171\) 0 0
\(172\) 10.2328 0.780243
\(173\) 13.6455 1.03745 0.518726 0.854941i \(-0.326407\pi\)
0.518726 + 0.854941i \(0.326407\pi\)
\(174\) 0 0
\(175\) −11.4361 −0.864490
\(176\) −1.34530 −0.101406
\(177\) 0 0
\(178\) 3.41204 0.255743
\(179\) −4.63889 −0.346727 −0.173364 0.984858i \(-0.555464\pi\)
−0.173364 + 0.984858i \(0.555464\pi\)
\(180\) 0 0
\(181\) 16.0481 1.19284 0.596422 0.802671i \(-0.296589\pi\)
0.596422 + 0.802671i \(0.296589\pi\)
\(182\) −15.7761 −1.16940
\(183\) 0 0
\(184\) 14.8705 1.09627
\(185\) −8.87418 −0.652443
\(186\) 0 0
\(187\) −3.18804 −0.233132
\(188\) 6.47985 0.472591
\(189\) 0 0
\(190\) −10.3821 −0.753193
\(191\) 18.4234 1.33307 0.666536 0.745473i \(-0.267776\pi\)
0.666536 + 0.745473i \(0.267776\pi\)
\(192\) 0 0
\(193\) −17.6427 −1.26995 −0.634976 0.772532i \(-0.718990\pi\)
−0.634976 + 0.772532i \(0.718990\pi\)
\(194\) 7.69904 0.552759
\(195\) 0 0
\(196\) −16.5103 −1.17931
\(197\) 17.6732 1.25916 0.629581 0.776935i \(-0.283227\pi\)
0.629581 + 0.776935i \(0.283227\pi\)
\(198\) 0 0
\(199\) −13.5925 −0.963544 −0.481772 0.876297i \(-0.660007\pi\)
−0.481772 + 0.876297i \(0.660007\pi\)
\(200\) 6.93242 0.490196
\(201\) 0 0
\(202\) 2.04065 0.143579
\(203\) −4.53054 −0.317981
\(204\) 0 0
\(205\) −19.2926 −1.34745
\(206\) 5.11219 0.356183
\(207\) 0 0
\(208\) 4.06008 0.281516
\(209\) −6.03685 −0.417578
\(210\) 0 0
\(211\) 19.5411 1.34527 0.672633 0.739976i \(-0.265163\pi\)
0.672633 + 0.739976i \(0.265163\pi\)
\(212\) 11.9159 0.818390
\(213\) 0 0
\(214\) −1.74366 −0.119194
\(215\) −18.5026 −1.26187
\(216\) 0 0
\(217\) 26.2410 1.78135
\(218\) 18.0786 1.22444
\(219\) 0 0
\(220\) −1.50303 −0.101335
\(221\) 9.62143 0.647207
\(222\) 0 0
\(223\) −11.7071 −0.783966 −0.391983 0.919972i \(-0.628211\pi\)
−0.391983 + 0.919972i \(0.628211\pi\)
\(224\) 23.4089 1.56408
\(225\) 0 0
\(226\) 8.79632 0.585123
\(227\) 8.66900 0.575382 0.287691 0.957723i \(-0.407112\pi\)
0.287691 + 0.957723i \(0.407112\pi\)
\(228\) 0 0
\(229\) 1.24221 0.0820876 0.0410438 0.999157i \(-0.486932\pi\)
0.0410438 + 0.999157i \(0.486932\pi\)
\(230\) −8.41933 −0.555154
\(231\) 0 0
\(232\) 2.74635 0.180307
\(233\) 8.15946 0.534544 0.267272 0.963621i \(-0.413878\pi\)
0.267272 + 0.963621i \(0.413878\pi\)
\(234\) 0 0
\(235\) −11.7166 −0.764310
\(236\) −0.426994 −0.0277949
\(237\) 0 0
\(238\) −16.6650 −1.08023
\(239\) −29.0899 −1.88167 −0.940833 0.338870i \(-0.889955\pi\)
−0.940833 + 0.338870i \(0.889955\pi\)
\(240\) 0 0
\(241\) −14.5444 −0.936887 −0.468443 0.883493i \(-0.655185\pi\)
−0.468443 + 0.883493i \(0.655185\pi\)
\(242\) 1.04320 0.0670596
\(243\) 0 0
\(244\) −0.911728 −0.0583674
\(245\) 29.8535 1.90727
\(246\) 0 0
\(247\) 18.2191 1.15925
\(248\) −15.9069 −1.01009
\(249\) 0 0
\(250\) −12.5239 −0.792080
\(251\) 7.59268 0.479246 0.239623 0.970866i \(-0.422976\pi\)
0.239623 + 0.970866i \(0.422976\pi\)
\(252\) 0 0
\(253\) −4.89559 −0.307783
\(254\) 15.4292 0.968117
\(255\) 0 0
\(256\) −16.6433 −1.04020
\(257\) −9.70834 −0.605590 −0.302795 0.953056i \(-0.597920\pi\)
−0.302795 + 0.953056i \(0.597920\pi\)
\(258\) 0 0
\(259\) −26.9735 −1.67605
\(260\) 4.53613 0.281319
\(261\) 0 0
\(262\) −0.358576 −0.0221529
\(263\) −27.9737 −1.72493 −0.862465 0.506116i \(-0.831081\pi\)
−0.862465 + 0.506116i \(0.831081\pi\)
\(264\) 0 0
\(265\) −21.5460 −1.32356
\(266\) −31.5568 −1.93487
\(267\) 0 0
\(268\) 2.87782 0.175791
\(269\) −5.48461 −0.334403 −0.167201 0.985923i \(-0.553473\pi\)
−0.167201 + 0.985923i \(0.553473\pi\)
\(270\) 0 0
\(271\) −8.15441 −0.495345 −0.247673 0.968844i \(-0.579666\pi\)
−0.247673 + 0.968844i \(0.579666\pi\)
\(272\) 4.28885 0.260050
\(273\) 0 0
\(274\) 14.3220 0.865225
\(275\) −2.28226 −0.137626
\(276\) 0 0
\(277\) 15.1489 0.910209 0.455105 0.890438i \(-0.349602\pi\)
0.455105 + 0.890438i \(0.349602\pi\)
\(278\) −17.0341 −1.02164
\(279\) 0 0
\(280\) −25.0921 −1.49954
\(281\) −28.0473 −1.67316 −0.836581 0.547843i \(-0.815449\pi\)
−0.836581 + 0.547843i \(0.815449\pi\)
\(282\) 0 0
\(283\) 12.2114 0.725892 0.362946 0.931810i \(-0.381771\pi\)
0.362946 + 0.931810i \(0.381771\pi\)
\(284\) −8.85690 −0.525560
\(285\) 0 0
\(286\) −3.14836 −0.186167
\(287\) −58.6408 −3.46146
\(288\) 0 0
\(289\) −6.83642 −0.402142
\(290\) −1.55492 −0.0913082
\(291\) 0 0
\(292\) 3.02927 0.177275
\(293\) −10.3409 −0.604122 −0.302061 0.953289i \(-0.597675\pi\)
−0.302061 + 0.953289i \(0.597675\pi\)
\(294\) 0 0
\(295\) 0.772076 0.0449520
\(296\) 16.3510 0.950382
\(297\) 0 0
\(298\) −0.660443 −0.0382584
\(299\) 14.7748 0.854448
\(300\) 0 0
\(301\) −56.2396 −3.24160
\(302\) −10.8293 −0.623154
\(303\) 0 0
\(304\) 8.12136 0.465792
\(305\) 1.64856 0.0943960
\(306\) 0 0
\(307\) 27.4302 1.56553 0.782763 0.622320i \(-0.213810\pi\)
0.782763 + 0.622320i \(0.213810\pi\)
\(308\) −4.56855 −0.260317
\(309\) 0 0
\(310\) 9.00614 0.511514
\(311\) −26.8048 −1.51996 −0.759982 0.649944i \(-0.774792\pi\)
−0.759982 + 0.649944i \(0.774792\pi\)
\(312\) 0 0
\(313\) 18.6081 1.05179 0.525897 0.850548i \(-0.323730\pi\)
0.525897 + 0.850548i \(0.323730\pi\)
\(314\) 21.0874 1.19003
\(315\) 0 0
\(316\) 5.09748 0.286756
\(317\) −27.1197 −1.52319 −0.761597 0.648051i \(-0.775584\pi\)
−0.761597 + 0.648051i \(0.775584\pi\)
\(318\) 0 0
\(319\) −0.904141 −0.0506222
\(320\) 12.4698 0.697081
\(321\) 0 0
\(322\) −25.5910 −1.42613
\(323\) 19.2457 1.07086
\(324\) 0 0
\(325\) 6.88782 0.382067
\(326\) 12.1055 0.670462
\(327\) 0 0
\(328\) 35.5473 1.96277
\(329\) −35.6134 −1.96343
\(330\) 0 0
\(331\) −14.5870 −0.801774 −0.400887 0.916128i \(-0.631298\pi\)
−0.400887 + 0.916128i \(0.631298\pi\)
\(332\) 0.404974 0.0222258
\(333\) 0 0
\(334\) −13.3902 −0.732678
\(335\) −5.20357 −0.284302
\(336\) 0 0
\(337\) −2.98497 −0.162602 −0.0813008 0.996690i \(-0.525907\pi\)
−0.0813008 + 0.996690i \(0.525907\pi\)
\(338\) −4.05994 −0.220832
\(339\) 0 0
\(340\) 4.79173 0.259868
\(341\) 5.23680 0.283589
\(342\) 0 0
\(343\) 55.6650 3.00563
\(344\) 34.0917 1.83810
\(345\) 0 0
\(346\) 14.2351 0.765282
\(347\) −23.5999 −1.26691 −0.633455 0.773780i \(-0.718364\pi\)
−0.633455 + 0.773780i \(0.718364\pi\)
\(348\) 0 0
\(349\) 23.9001 1.27934 0.639672 0.768648i \(-0.279070\pi\)
0.639672 + 0.768648i \(0.279070\pi\)
\(350\) −11.9302 −0.637696
\(351\) 0 0
\(352\) 4.67163 0.248999
\(353\) −34.4485 −1.83351 −0.916755 0.399449i \(-0.869201\pi\)
−0.916755 + 0.399449i \(0.869201\pi\)
\(354\) 0 0
\(355\) 16.0148 0.849975
\(356\) −2.98202 −0.158047
\(357\) 0 0
\(358\) −4.83931 −0.255765
\(359\) −8.98660 −0.474294 −0.237147 0.971474i \(-0.576212\pi\)
−0.237147 + 0.971474i \(0.576212\pi\)
\(360\) 0 0
\(361\) 17.4436 0.918085
\(362\) 16.7414 0.879908
\(363\) 0 0
\(364\) 13.7878 0.722677
\(365\) −5.47743 −0.286702
\(366\) 0 0
\(367\) 12.8108 0.668717 0.334359 0.942446i \(-0.391480\pi\)
0.334359 + 0.942446i \(0.391480\pi\)
\(368\) 6.58602 0.343320
\(369\) 0 0
\(370\) −9.25757 −0.481278
\(371\) −65.4902 −3.40008
\(372\) 0 0
\(373\) 13.0797 0.677241 0.338621 0.940923i \(-0.390040\pi\)
0.338621 + 0.940923i \(0.390040\pi\)
\(374\) −3.32577 −0.171971
\(375\) 0 0
\(376\) 21.5883 1.11333
\(377\) 2.72868 0.140534
\(378\) 0 0
\(379\) 2.92093 0.150038 0.0750192 0.997182i \(-0.476098\pi\)
0.0750192 + 0.997182i \(0.476098\pi\)
\(380\) 9.07360 0.465466
\(381\) 0 0
\(382\) 19.2194 0.983348
\(383\) 9.46576 0.483678 0.241839 0.970316i \(-0.422249\pi\)
0.241839 + 0.970316i \(0.422249\pi\)
\(384\) 0 0
\(385\) 8.26071 0.421005
\(386\) −18.4049 −0.936787
\(387\) 0 0
\(388\) −6.72873 −0.341600
\(389\) 16.2760 0.825226 0.412613 0.910907i \(-0.364616\pi\)
0.412613 + 0.910907i \(0.364616\pi\)
\(390\) 0 0
\(391\) 15.6073 0.789296
\(392\) −55.0060 −2.77822
\(393\) 0 0
\(394\) 18.4367 0.928828
\(395\) −9.21709 −0.463762
\(396\) 0 0
\(397\) 21.0178 1.05486 0.527428 0.849600i \(-0.323157\pi\)
0.527428 + 0.849600i \(0.323157\pi\)
\(398\) −14.1797 −0.710764
\(399\) 0 0
\(400\) 3.07032 0.153516
\(401\) −1.15040 −0.0574484 −0.0287242 0.999587i \(-0.509144\pi\)
−0.0287242 + 0.999587i \(0.509144\pi\)
\(402\) 0 0
\(403\) −15.8046 −0.787281
\(404\) −1.78346 −0.0887306
\(405\) 0 0
\(406\) −4.72627 −0.234561
\(407\) −5.38300 −0.266826
\(408\) 0 0
\(409\) −17.9590 −0.888015 −0.444007 0.896023i \(-0.646444\pi\)
−0.444007 + 0.896023i \(0.646444\pi\)
\(410\) −20.1261 −0.993956
\(411\) 0 0
\(412\) −4.46790 −0.220118
\(413\) 2.34677 0.115477
\(414\) 0 0
\(415\) −0.732260 −0.0359453
\(416\) −14.0989 −0.691254
\(417\) 0 0
\(418\) −6.29766 −0.308029
\(419\) 30.0432 1.46771 0.733853 0.679308i \(-0.237720\pi\)
0.733853 + 0.679308i \(0.237720\pi\)
\(420\) 0 0
\(421\) −6.11518 −0.298036 −0.149018 0.988834i \(-0.547611\pi\)
−0.149018 + 0.988834i \(0.547611\pi\)
\(422\) 20.3854 0.992343
\(423\) 0 0
\(424\) 39.6993 1.92797
\(425\) 7.27594 0.352935
\(426\) 0 0
\(427\) 5.01087 0.242493
\(428\) 1.52391 0.0736610
\(429\) 0 0
\(430\) −19.3020 −0.930824
\(431\) −32.2060 −1.55131 −0.775654 0.631158i \(-0.782580\pi\)
−0.775654 + 0.631158i \(0.782580\pi\)
\(432\) 0 0
\(433\) −28.5878 −1.37384 −0.686921 0.726733i \(-0.741038\pi\)
−0.686921 + 0.726733i \(0.741038\pi\)
\(434\) 27.3746 1.31402
\(435\) 0 0
\(436\) −15.8002 −0.756692
\(437\) 29.5540 1.41376
\(438\) 0 0
\(439\) −19.8151 −0.945723 −0.472861 0.881137i \(-0.656779\pi\)
−0.472861 + 0.881137i \(0.656779\pi\)
\(440\) −5.00753 −0.238724
\(441\) 0 0
\(442\) 10.0371 0.477416
\(443\) 27.8588 1.32361 0.661806 0.749675i \(-0.269790\pi\)
0.661806 + 0.749675i \(0.269790\pi\)
\(444\) 0 0
\(445\) 5.39199 0.255605
\(446\) −12.2129 −0.578297
\(447\) 0 0
\(448\) 37.9025 1.79072
\(449\) 29.3342 1.38437 0.692183 0.721722i \(-0.256649\pi\)
0.692183 + 0.721722i \(0.256649\pi\)
\(450\) 0 0
\(451\) −11.7027 −0.551059
\(452\) −7.68772 −0.361600
\(453\) 0 0
\(454\) 9.04352 0.424434
\(455\) −24.9306 −1.16877
\(456\) 0 0
\(457\) −20.0757 −0.939101 −0.469550 0.882906i \(-0.655584\pi\)
−0.469550 + 0.882906i \(0.655584\pi\)
\(458\) 1.29588 0.0605524
\(459\) 0 0
\(460\) 7.35824 0.343080
\(461\) −37.3900 −1.74143 −0.870713 0.491792i \(-0.836342\pi\)
−0.870713 + 0.491792i \(0.836342\pi\)
\(462\) 0 0
\(463\) 3.21469 0.149399 0.0746996 0.997206i \(-0.476200\pi\)
0.0746996 + 0.997206i \(0.476200\pi\)
\(464\) 1.21634 0.0564671
\(465\) 0 0
\(466\) 8.51197 0.394309
\(467\) −32.0410 −1.48268 −0.741341 0.671129i \(-0.765810\pi\)
−0.741341 + 0.671129i \(0.765810\pi\)
\(468\) 0 0
\(469\) −15.8165 −0.730340
\(470\) −12.2228 −0.563797
\(471\) 0 0
\(472\) −1.42258 −0.0654794
\(473\) −11.2235 −0.516058
\(474\) 0 0
\(475\) 13.7777 0.632164
\(476\) 14.5647 0.667573
\(477\) 0 0
\(478\) −30.3466 −1.38802
\(479\) 35.8435 1.63773 0.818867 0.573984i \(-0.194603\pi\)
0.818867 + 0.573984i \(0.194603\pi\)
\(480\) 0 0
\(481\) 16.2458 0.740744
\(482\) −15.1728 −0.691100
\(483\) 0 0
\(484\) −0.911728 −0.0414422
\(485\) 12.1667 0.552460
\(486\) 0 0
\(487\) 27.6269 1.25190 0.625948 0.779865i \(-0.284712\pi\)
0.625948 + 0.779865i \(0.284712\pi\)
\(488\) −3.03752 −0.137502
\(489\) 0 0
\(490\) 31.1432 1.40691
\(491\) −1.38118 −0.0623318 −0.0311659 0.999514i \(-0.509922\pi\)
−0.0311659 + 0.999514i \(0.509922\pi\)
\(492\) 0 0
\(493\) 2.88244 0.129818
\(494\) 19.0062 0.855130
\(495\) 0 0
\(496\) −7.04505 −0.316332
\(497\) 48.6777 2.18349
\(498\) 0 0
\(499\) 32.7390 1.46560 0.732799 0.680445i \(-0.238213\pi\)
0.732799 + 0.680445i \(0.238213\pi\)
\(500\) 10.9455 0.489497
\(501\) 0 0
\(502\) 7.92071 0.353519
\(503\) −28.8264 −1.28530 −0.642652 0.766158i \(-0.722166\pi\)
−0.642652 + 0.766158i \(0.722166\pi\)
\(504\) 0 0
\(505\) 3.22480 0.143502
\(506\) −5.10709 −0.227038
\(507\) 0 0
\(508\) −13.4847 −0.598287
\(509\) −22.0483 −0.977275 −0.488638 0.872487i \(-0.662506\pi\)
−0.488638 + 0.872487i \(0.662506\pi\)
\(510\) 0 0
\(511\) −16.6489 −0.736506
\(512\) −14.4575 −0.638936
\(513\) 0 0
\(514\) −10.1278 −0.446717
\(515\) 8.07871 0.355991
\(516\) 0 0
\(517\) −7.10722 −0.312575
\(518\) −28.1389 −1.23635
\(519\) 0 0
\(520\) 15.1126 0.662732
\(521\) −0.419315 −0.0183705 −0.00918527 0.999958i \(-0.502924\pi\)
−0.00918527 + 0.999958i \(0.502924\pi\)
\(522\) 0 0
\(523\) 32.9826 1.44223 0.721114 0.692816i \(-0.243630\pi\)
0.721114 + 0.692816i \(0.243630\pi\)
\(524\) 0.313384 0.0136903
\(525\) 0 0
\(526\) −29.1822 −1.27240
\(527\) −16.6951 −0.727251
\(528\) 0 0
\(529\) 0.966817 0.0420355
\(530\) −22.4768 −0.976332
\(531\) 0 0
\(532\) 27.5797 1.19573
\(533\) 35.3185 1.52982
\(534\) 0 0
\(535\) −2.75548 −0.119130
\(536\) 9.58777 0.414128
\(537\) 0 0
\(538\) −5.72156 −0.246674
\(539\) 18.1089 0.780003
\(540\) 0 0
\(541\) −21.7042 −0.933136 −0.466568 0.884485i \(-0.654510\pi\)
−0.466568 + 0.884485i \(0.654510\pi\)
\(542\) −8.50670 −0.365394
\(543\) 0 0
\(544\) −14.8933 −0.638546
\(545\) 28.5694 1.22378
\(546\) 0 0
\(547\) −35.6177 −1.52290 −0.761451 0.648223i \(-0.775513\pi\)
−0.761451 + 0.648223i \(0.775513\pi\)
\(548\) −12.5170 −0.534700
\(549\) 0 0
\(550\) −2.38086 −0.101520
\(551\) 5.45817 0.232526
\(552\) 0 0
\(553\) −28.0158 −1.19135
\(554\) 15.8034 0.671421
\(555\) 0 0
\(556\) 14.8873 0.631361
\(557\) −2.05471 −0.0870607 −0.0435303 0.999052i \(-0.513861\pi\)
−0.0435303 + 0.999052i \(0.513861\pi\)
\(558\) 0 0
\(559\) 33.8723 1.43265
\(560\) −11.1131 −0.469614
\(561\) 0 0
\(562\) −29.2590 −1.23422
\(563\) 11.9315 0.502854 0.251427 0.967876i \(-0.419100\pi\)
0.251427 + 0.967876i \(0.419100\pi\)
\(564\) 0 0
\(565\) 13.9007 0.584806
\(566\) 12.7390 0.535459
\(567\) 0 0
\(568\) −29.5078 −1.23812
\(569\) −39.3292 −1.64877 −0.824384 0.566031i \(-0.808478\pi\)
−0.824384 + 0.566031i \(0.808478\pi\)
\(570\) 0 0
\(571\) 41.1563 1.72234 0.861168 0.508320i \(-0.169733\pi\)
0.861168 + 0.508320i \(0.169733\pi\)
\(572\) 2.75158 0.115049
\(573\) 0 0
\(574\) −61.1742 −2.55336
\(575\) 11.1730 0.465947
\(576\) 0 0
\(577\) 21.1444 0.880253 0.440127 0.897936i \(-0.354934\pi\)
0.440127 + 0.897936i \(0.354934\pi\)
\(578\) −7.13177 −0.296643
\(579\) 0 0
\(580\) 1.35896 0.0564275
\(581\) −2.22574 −0.0923394
\(582\) 0 0
\(583\) −13.0696 −0.541288
\(584\) 10.0924 0.417625
\(585\) 0 0
\(586\) −10.7877 −0.445634
\(587\) −16.6914 −0.688927 −0.344464 0.938800i \(-0.611939\pi\)
−0.344464 + 0.938800i \(0.611939\pi\)
\(588\) 0 0
\(589\) −31.6138 −1.30262
\(590\) 0.805432 0.0331591
\(591\) 0 0
\(592\) 7.24173 0.297633
\(593\) −11.4836 −0.471573 −0.235787 0.971805i \(-0.575767\pi\)
−0.235787 + 0.971805i \(0.575767\pi\)
\(594\) 0 0
\(595\) −26.3354 −1.07965
\(596\) 0.577207 0.0236433
\(597\) 0 0
\(598\) 15.4131 0.630289
\(599\) −36.1765 −1.47813 −0.739067 0.673632i \(-0.764733\pi\)
−0.739067 + 0.673632i \(0.764733\pi\)
\(600\) 0 0
\(601\) 17.6510 0.719999 0.359999 0.932953i \(-0.382777\pi\)
0.359999 + 0.932953i \(0.382777\pi\)
\(602\) −58.6694 −2.39118
\(603\) 0 0
\(604\) 9.46446 0.385103
\(605\) 1.64856 0.0670233
\(606\) 0 0
\(607\) 28.3146 1.14925 0.574627 0.818415i \(-0.305147\pi\)
0.574627 + 0.818415i \(0.305147\pi\)
\(608\) −28.2019 −1.14374
\(609\) 0 0
\(610\) 1.71978 0.0696318
\(611\) 21.4494 0.867751
\(612\) 0 0
\(613\) 3.45999 0.139747 0.0698737 0.997556i \(-0.477740\pi\)
0.0698737 + 0.997556i \(0.477740\pi\)
\(614\) 28.6153 1.15482
\(615\) 0 0
\(616\) −15.2206 −0.613257
\(617\) −13.0700 −0.526180 −0.263090 0.964771i \(-0.584742\pi\)
−0.263090 + 0.964771i \(0.584742\pi\)
\(618\) 0 0
\(619\) 8.11491 0.326166 0.163083 0.986612i \(-0.447856\pi\)
0.163083 + 0.986612i \(0.447856\pi\)
\(620\) −7.87110 −0.316111
\(621\) 0 0
\(622\) −27.9629 −1.12121
\(623\) 16.3892 0.656620
\(624\) 0 0
\(625\) −8.37996 −0.335199
\(626\) 19.4121 0.775862
\(627\) 0 0
\(628\) −18.4298 −0.735428
\(629\) 17.1612 0.684262
\(630\) 0 0
\(631\) 1.68038 0.0668947 0.0334474 0.999440i \(-0.489351\pi\)
0.0334474 + 0.999440i \(0.489351\pi\)
\(632\) 16.9828 0.675540
\(633\) 0 0
\(634\) −28.2913 −1.12359
\(635\) 24.3826 0.967593
\(636\) 0 0
\(637\) −54.6521 −2.16540
\(638\) −0.943203 −0.0373417
\(639\) 0 0
\(640\) −2.39440 −0.0946468
\(641\) −39.1788 −1.54747 −0.773735 0.633509i \(-0.781614\pi\)
−0.773735 + 0.633509i \(0.781614\pi\)
\(642\) 0 0
\(643\) 6.09680 0.240434 0.120217 0.992748i \(-0.461641\pi\)
0.120217 + 0.992748i \(0.461641\pi\)
\(644\) 22.3658 0.881335
\(645\) 0 0
\(646\) 20.0772 0.789926
\(647\) −41.2844 −1.62306 −0.811529 0.584312i \(-0.801364\pi\)
−0.811529 + 0.584312i \(0.801364\pi\)
\(648\) 0 0
\(649\) 0.468335 0.0183837
\(650\) 7.18539 0.281834
\(651\) 0 0
\(652\) −10.5798 −0.414339
\(653\) 16.4695 0.644500 0.322250 0.946655i \(-0.395561\pi\)
0.322250 + 0.946655i \(0.395561\pi\)
\(654\) 0 0
\(655\) −0.566651 −0.0221409
\(656\) 15.7436 0.614685
\(657\) 0 0
\(658\) −37.1520 −1.44833
\(659\) 46.5604 1.81374 0.906868 0.421415i \(-0.138466\pi\)
0.906868 + 0.421415i \(0.138466\pi\)
\(660\) 0 0
\(661\) −28.4805 −1.10776 −0.553882 0.832595i \(-0.686854\pi\)
−0.553882 + 0.832595i \(0.686854\pi\)
\(662\) −15.2172 −0.591433
\(663\) 0 0
\(664\) 1.34922 0.0523597
\(665\) −49.8687 −1.93382
\(666\) 0 0
\(667\) 4.42631 0.171387
\(668\) 11.7026 0.452788
\(669\) 0 0
\(670\) −5.42838 −0.209717
\(671\) 1.00000 0.0386046
\(672\) 0 0
\(673\) 31.6459 1.21986 0.609930 0.792456i \(-0.291198\pi\)
0.609930 + 0.792456i \(0.291198\pi\)
\(674\) −3.11393 −0.119944
\(675\) 0 0
\(676\) 3.54827 0.136472
\(677\) 40.6813 1.56351 0.781755 0.623586i \(-0.214325\pi\)
0.781755 + 0.623586i \(0.214325\pi\)
\(678\) 0 0
\(679\) 36.9812 1.41921
\(680\) 15.9642 0.612198
\(681\) 0 0
\(682\) 5.46305 0.209191
\(683\) 3.16227 0.121001 0.0605005 0.998168i \(-0.480730\pi\)
0.0605005 + 0.998168i \(0.480730\pi\)
\(684\) 0 0
\(685\) 22.6328 0.864757
\(686\) 58.0699 2.21712
\(687\) 0 0
\(688\) 15.0990 0.575643
\(689\) 39.4438 1.50269
\(690\) 0 0
\(691\) −5.23615 −0.199192 −0.0995962 0.995028i \(-0.531755\pi\)
−0.0995962 + 0.995028i \(0.531755\pi\)
\(692\) −12.4410 −0.472937
\(693\) 0 0
\(694\) −24.6195 −0.934543
\(695\) −26.9187 −1.02108
\(696\) 0 0
\(697\) 37.3087 1.41317
\(698\) 24.9327 0.943716
\(699\) 0 0
\(700\) 10.4266 0.394090
\(701\) 26.1863 0.989043 0.494521 0.869165i \(-0.335343\pi\)
0.494521 + 0.869165i \(0.335343\pi\)
\(702\) 0 0
\(703\) 32.4964 1.22562
\(704\) 7.56405 0.285081
\(705\) 0 0
\(706\) −35.9368 −1.35250
\(707\) 9.80194 0.368640
\(708\) 0 0
\(709\) 11.6344 0.436938 0.218469 0.975844i \(-0.429894\pi\)
0.218469 + 0.975844i \(0.429894\pi\)
\(710\) 16.7066 0.626989
\(711\) 0 0
\(712\) −9.93492 −0.372327
\(713\) −25.6373 −0.960123
\(714\) 0 0
\(715\) −4.97531 −0.186066
\(716\) 4.22941 0.158060
\(717\) 0 0
\(718\) −9.37484 −0.349866
\(719\) −10.3289 −0.385205 −0.192602 0.981277i \(-0.561693\pi\)
−0.192602 + 0.981277i \(0.561693\pi\)
\(720\) 0 0
\(721\) 24.5557 0.914501
\(722\) 18.1972 0.677231
\(723\) 0 0
\(724\) −14.6315 −0.543774
\(725\) 2.06349 0.0766360
\(726\) 0 0
\(727\) 1.38619 0.0514109 0.0257055 0.999670i \(-0.491817\pi\)
0.0257055 + 0.999670i \(0.491817\pi\)
\(728\) 45.9356 1.70249
\(729\) 0 0
\(730\) −5.71407 −0.211487
\(731\) 35.7810 1.32341
\(732\) 0 0
\(733\) 20.8739 0.770996 0.385498 0.922709i \(-0.374030\pi\)
0.385498 + 0.922709i \(0.374030\pi\)
\(734\) 13.3642 0.493283
\(735\) 0 0
\(736\) −22.8704 −0.843014
\(737\) −3.15644 −0.116269
\(738\) 0 0
\(739\) −48.2442 −1.77469 −0.887347 0.461103i \(-0.847454\pi\)
−0.887347 + 0.461103i \(0.847454\pi\)
\(740\) 8.09084 0.297425
\(741\) 0 0
\(742\) −68.3196 −2.50809
\(743\) 10.1761 0.373325 0.186663 0.982424i \(-0.440233\pi\)
0.186663 + 0.982424i \(0.440233\pi\)
\(744\) 0 0
\(745\) −1.04369 −0.0382377
\(746\) 13.6448 0.499571
\(747\) 0 0
\(748\) 2.90662 0.106277
\(749\) −8.37544 −0.306032
\(750\) 0 0
\(751\) 9.83366 0.358835 0.179418 0.983773i \(-0.442579\pi\)
0.179418 + 0.983773i \(0.442579\pi\)
\(752\) 9.56131 0.348665
\(753\) 0 0
\(754\) 2.84656 0.103666
\(755\) −17.1133 −0.622817
\(756\) 0 0
\(757\) 21.7125 0.789153 0.394576 0.918863i \(-0.370891\pi\)
0.394576 + 0.918863i \(0.370891\pi\)
\(758\) 3.04713 0.110677
\(759\) 0 0
\(760\) 30.2297 1.09655
\(761\) 28.6061 1.03697 0.518484 0.855087i \(-0.326496\pi\)
0.518484 + 0.855087i \(0.326496\pi\)
\(762\) 0 0
\(763\) 86.8382 3.14375
\(764\) −16.7971 −0.607699
\(765\) 0 0
\(766\) 9.87471 0.356788
\(767\) −1.41342 −0.0510358
\(768\) 0 0
\(769\) 18.3697 0.662429 0.331214 0.943556i \(-0.392542\pi\)
0.331214 + 0.943556i \(0.392542\pi\)
\(770\) 8.61759 0.310556
\(771\) 0 0
\(772\) 16.0854 0.578925
\(773\) −28.0562 −1.00911 −0.504556 0.863379i \(-0.668344\pi\)
−0.504556 + 0.863379i \(0.668344\pi\)
\(774\) 0 0
\(775\) −11.9518 −0.429320
\(776\) −22.4175 −0.804742
\(777\) 0 0
\(778\) 16.9792 0.608732
\(779\) 70.6476 2.53121
\(780\) 0 0
\(781\) 9.71441 0.347609
\(782\) 16.2816 0.582229
\(783\) 0 0
\(784\) −24.3618 −0.870063
\(785\) 33.3241 1.18939
\(786\) 0 0
\(787\) −17.8283 −0.635511 −0.317756 0.948173i \(-0.602929\pi\)
−0.317756 + 0.948173i \(0.602929\pi\)
\(788\) −16.1131 −0.574007
\(789\) 0 0
\(790\) −9.61530 −0.342097
\(791\) 42.2518 1.50230
\(792\) 0 0
\(793\) −3.01798 −0.107172
\(794\) 21.9259 0.778120
\(795\) 0 0
\(796\) 12.3926 0.439245
\(797\) 18.9394 0.670867 0.335434 0.942064i \(-0.391117\pi\)
0.335434 + 0.942064i \(0.391117\pi\)
\(798\) 0 0
\(799\) 22.6581 0.801585
\(800\) −10.6619 −0.376954
\(801\) 0 0
\(802\) −1.20010 −0.0423772
\(803\) −3.32256 −0.117251
\(804\) 0 0
\(805\) −40.4410 −1.42536
\(806\) −16.4874 −0.580743
\(807\) 0 0
\(808\) −5.94180 −0.209032
\(809\) 25.7275 0.904529 0.452265 0.891884i \(-0.350616\pi\)
0.452265 + 0.891884i \(0.350616\pi\)
\(810\) 0 0
\(811\) −46.4287 −1.63033 −0.815166 0.579227i \(-0.803355\pi\)
−0.815166 + 0.579227i \(0.803355\pi\)
\(812\) 4.13062 0.144956
\(813\) 0 0
\(814\) −5.61556 −0.196825
\(815\) 19.1301 0.670099
\(816\) 0 0
\(817\) 67.7548 2.37044
\(818\) −18.7349 −0.655049
\(819\) 0 0
\(820\) 17.5896 0.614255
\(821\) −24.6784 −0.861281 −0.430641 0.902524i \(-0.641712\pi\)
−0.430641 + 0.902524i \(0.641712\pi\)
\(822\) 0 0
\(823\) 4.02139 0.140177 0.0700885 0.997541i \(-0.477672\pi\)
0.0700885 + 0.997541i \(0.477672\pi\)
\(824\) −14.8853 −0.518554
\(825\) 0 0
\(826\) 2.44815 0.0851821
\(827\) −0.943492 −0.0328084 −0.0164042 0.999865i \(-0.505222\pi\)
−0.0164042 + 0.999865i \(0.505222\pi\)
\(828\) 0 0
\(829\) 9.46783 0.328831 0.164416 0.986391i \(-0.447426\pi\)
0.164416 + 0.986391i \(0.447426\pi\)
\(830\) −0.763896 −0.0265152
\(831\) 0 0
\(832\) −22.8281 −0.791423
\(833\) −57.7317 −2.00028
\(834\) 0 0
\(835\) −21.1603 −0.732282
\(836\) 5.50397 0.190359
\(837\) 0 0
\(838\) 31.3412 1.08266
\(839\) 19.2334 0.664011 0.332005 0.943278i \(-0.392275\pi\)
0.332005 + 0.943278i \(0.392275\pi\)
\(840\) 0 0
\(841\) −28.1825 −0.971811
\(842\) −6.37938 −0.219848
\(843\) 0 0
\(844\) −17.8162 −0.613258
\(845\) −6.41586 −0.220712
\(846\) 0 0
\(847\) 5.01087 0.172176
\(848\) 17.5825 0.603786
\(849\) 0 0
\(850\) 7.59028 0.260344
\(851\) 26.3530 0.903369
\(852\) 0 0
\(853\) −42.0074 −1.43831 −0.719153 0.694852i \(-0.755470\pi\)
−0.719153 + 0.694852i \(0.755470\pi\)
\(854\) 5.22736 0.178876
\(855\) 0 0
\(856\) 5.07708 0.173531
\(857\) 25.4294 0.868652 0.434326 0.900756i \(-0.356987\pi\)
0.434326 + 0.900756i \(0.356987\pi\)
\(858\) 0 0
\(859\) 5.15518 0.175893 0.0879463 0.996125i \(-0.471970\pi\)
0.0879463 + 0.996125i \(0.471970\pi\)
\(860\) 16.8693 0.575240
\(861\) 0 0
\(862\) −33.5974 −1.14433
\(863\) 24.1318 0.821457 0.410729 0.911758i \(-0.365274\pi\)
0.410729 + 0.911758i \(0.365274\pi\)
\(864\) 0 0
\(865\) 22.4955 0.764868
\(866\) −29.8229 −1.01342
\(867\) 0 0
\(868\) −23.9246 −0.812054
\(869\) −5.59101 −0.189662
\(870\) 0 0
\(871\) 9.52607 0.322779
\(872\) −52.6401 −1.78262
\(873\) 0 0
\(874\) 30.8308 1.04287
\(875\) −60.1566 −2.03367
\(876\) 0 0
\(877\) 11.3805 0.384291 0.192146 0.981366i \(-0.438455\pi\)
0.192146 + 0.981366i \(0.438455\pi\)
\(878\) −20.6712 −0.697618
\(879\) 0 0
\(880\) −2.21780 −0.0747619
\(881\) 34.9613 1.17788 0.588938 0.808178i \(-0.299546\pi\)
0.588938 + 0.808178i \(0.299546\pi\)
\(882\) 0 0
\(883\) −44.0098 −1.48105 −0.740523 0.672031i \(-0.765422\pi\)
−0.740523 + 0.672031i \(0.765422\pi\)
\(884\) −8.77212 −0.295038
\(885\) 0 0
\(886\) 29.0624 0.976370
\(887\) −6.63679 −0.222842 −0.111421 0.993773i \(-0.535540\pi\)
−0.111421 + 0.993773i \(0.535540\pi\)
\(888\) 0 0
\(889\) 74.1122 2.48564
\(890\) 5.62493 0.188548
\(891\) 0 0
\(892\) 10.6737 0.357382
\(893\) 42.9052 1.43577
\(894\) 0 0
\(895\) −7.64748 −0.255627
\(896\) −7.27789 −0.243137
\(897\) 0 0
\(898\) 30.6015 1.02119
\(899\) −4.73481 −0.157915
\(900\) 0 0
\(901\) 41.6664 1.38811
\(902\) −12.2083 −0.406492
\(903\) 0 0
\(904\) −25.6125 −0.851859
\(905\) 26.4561 0.879432
\(906\) 0 0
\(907\) −49.3540 −1.63877 −0.819386 0.573242i \(-0.805686\pi\)
−0.819386 + 0.573242i \(0.805686\pi\)
\(908\) −7.90377 −0.262296
\(909\) 0 0
\(910\) −26.0077 −0.862147
\(911\) −46.6081 −1.54420 −0.772098 0.635504i \(-0.780792\pi\)
−0.772098 + 0.635504i \(0.780792\pi\)
\(912\) 0 0
\(913\) −0.444183 −0.0147003
\(914\) −20.9430 −0.692733
\(915\) 0 0
\(916\) −1.13256 −0.0374208
\(917\) −1.72237 −0.0568775
\(918\) 0 0
\(919\) −0.500503 −0.0165101 −0.00825504 0.999966i \(-0.502628\pi\)
−0.00825504 + 0.999966i \(0.502628\pi\)
\(920\) 24.5148 0.808229
\(921\) 0 0
\(922\) −39.0053 −1.28457
\(923\) −29.3179 −0.965010
\(924\) 0 0
\(925\) 12.2854 0.403942
\(926\) 3.35357 0.110205
\(927\) 0 0
\(928\) −4.22381 −0.138653
\(929\) −30.9867 −1.01664 −0.508321 0.861168i \(-0.669734\pi\)
−0.508321 + 0.861168i \(0.669734\pi\)
\(930\) 0 0
\(931\) −109.320 −3.58283
\(932\) −7.43921 −0.243679
\(933\) 0 0
\(934\) −33.4253 −1.09371
\(935\) −5.25566 −0.171878
\(936\) 0 0
\(937\) 18.8113 0.614538 0.307269 0.951623i \(-0.400585\pi\)
0.307269 + 0.951623i \(0.400585\pi\)
\(938\) −16.4999 −0.538739
\(939\) 0 0
\(940\) 10.6824 0.348421
\(941\) 13.0851 0.426561 0.213281 0.976991i \(-0.431585\pi\)
0.213281 + 0.976991i \(0.431585\pi\)
\(942\) 0 0
\(943\) 57.2917 1.86567
\(944\) −0.630049 −0.0205063
\(945\) 0 0
\(946\) −11.7084 −0.380673
\(947\) 10.6578 0.346331 0.173165 0.984893i \(-0.444600\pi\)
0.173165 + 0.984893i \(0.444600\pi\)
\(948\) 0 0
\(949\) 10.0274 0.325504
\(950\) 14.3729 0.466319
\(951\) 0 0
\(952\) 48.5240 1.57267
\(953\) 27.5269 0.891684 0.445842 0.895112i \(-0.352904\pi\)
0.445842 + 0.895112i \(0.352904\pi\)
\(954\) 0 0
\(955\) 30.3720 0.982816
\(956\) 26.5220 0.857784
\(957\) 0 0
\(958\) 37.3921 1.20808
\(959\) 68.7937 2.22147
\(960\) 0 0
\(961\) −3.57589 −0.115351
\(962\) 16.9476 0.546414
\(963\) 0 0
\(964\) 13.2605 0.427093
\(965\) −29.0850 −0.936280
\(966\) 0 0
\(967\) −7.68136 −0.247016 −0.123508 0.992344i \(-0.539414\pi\)
−0.123508 + 0.992344i \(0.539414\pi\)
\(968\) −3.03752 −0.0976297
\(969\) 0 0
\(970\) 12.6923 0.407525
\(971\) 42.1602 1.35299 0.676493 0.736449i \(-0.263499\pi\)
0.676493 + 0.736449i \(0.263499\pi\)
\(972\) 0 0
\(973\) −81.8207 −2.62305
\(974\) 28.8205 0.923469
\(975\) 0 0
\(976\) −1.34530 −0.0430619
\(977\) 2.82716 0.0904488 0.0452244 0.998977i \(-0.485600\pi\)
0.0452244 + 0.998977i \(0.485600\pi\)
\(978\) 0 0
\(979\) 3.27073 0.104533
\(980\) −27.2182 −0.869454
\(981\) 0 0
\(982\) −1.44085 −0.0459794
\(983\) −24.3097 −0.775359 −0.387680 0.921794i \(-0.626723\pi\)
−0.387680 + 0.921794i \(0.626723\pi\)
\(984\) 0 0
\(985\) 29.1352 0.928326
\(986\) 3.00696 0.0957613
\(987\) 0 0
\(988\) −16.6109 −0.528462
\(989\) 54.9458 1.74717
\(990\) 0 0
\(991\) 39.6676 1.26008 0.630042 0.776561i \(-0.283038\pi\)
0.630042 + 0.776561i \(0.283038\pi\)
\(992\) 24.4644 0.776746
\(993\) 0 0
\(994\) 50.7807 1.61067
\(995\) −22.4079 −0.710379
\(996\) 0 0
\(997\) 27.7285 0.878170 0.439085 0.898445i \(-0.355303\pi\)
0.439085 + 0.898445i \(0.355303\pi\)
\(998\) 34.1534 1.08111
\(999\) 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 6039.2.a.m.1.18 25
3.2 odd 2 6039.2.a.p.1.8 yes 25
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6039.2.a.m.1.18 25 1.1 even 1 trivial
6039.2.a.p.1.8 yes 25 3.2 odd 2