Properties

Label 8020.2.a.f.1.17
Level $8020$
Weight $2$
Character 8020.1
Self dual yes
Analytic conductor $64.040$
Analytic rank $0$
Dimension $37$
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] = [8020,2,Mod(1,8020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8020, 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("8020.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8020 = 2^{2} \cdot 5 \cdot 401 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8020.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: \(64.0400224211\)
Analytic rank: \(0\)
Dimension: \(37\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.17
Character \(\chi\) \(=\) 8020.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.210422 q^{3} +1.00000 q^{5} -4.28610 q^{7} -2.95572 q^{9} +O(q^{10})\) \(q-0.210422 q^{3} +1.00000 q^{5} -4.28610 q^{7} -2.95572 q^{9} +1.06403 q^{11} +6.73502 q^{13} -0.210422 q^{15} -1.31084 q^{17} -3.85994 q^{19} +0.901891 q^{21} -1.51918 q^{23} +1.00000 q^{25} +1.25322 q^{27} -4.71498 q^{29} +9.73676 q^{31} -0.223896 q^{33} -4.28610 q^{35} -0.0446856 q^{37} -1.41720 q^{39} +2.52783 q^{41} -11.8904 q^{43} -2.95572 q^{45} -11.7086 q^{47} +11.3706 q^{49} +0.275830 q^{51} +12.1737 q^{53} +1.06403 q^{55} +0.812218 q^{57} -5.13146 q^{59} -10.2531 q^{61} +12.6685 q^{63} +6.73502 q^{65} +3.09659 q^{67} +0.319669 q^{69} -4.55846 q^{71} +10.6323 q^{73} -0.210422 q^{75} -4.56054 q^{77} +3.01867 q^{79} +8.60346 q^{81} +12.6498 q^{83} -1.31084 q^{85} +0.992137 q^{87} +10.1908 q^{89} -28.8670 q^{91} -2.04883 q^{93} -3.85994 q^{95} -14.2840 q^{97} -3.14498 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9} + 2 q^{11} + 27 q^{13} + 3 q^{15} + 36 q^{17} - 6 q^{19} + 20 q^{21} + 17 q^{23} + 37 q^{25} + 9 q^{27} + 29 q^{29} + 5 q^{31} + 36 q^{33} + 4 q^{35} + 35 q^{37} + 21 q^{39} + 24 q^{41} + 11 q^{43} + 50 q^{45} + 19 q^{47} + 57 q^{49} + 8 q^{51} + 65 q^{53} + 2 q^{55} + 62 q^{57} - 9 q^{59} + 13 q^{61} + 26 q^{63} + 27 q^{65} + 13 q^{67} + 20 q^{69} + 33 q^{71} + 67 q^{73} + 3 q^{75} + 62 q^{77} + 23 q^{79} + 97 q^{81} + 2 q^{83} + 36 q^{85} + 32 q^{87} + 34 q^{89} + q^{91} + 41 q^{93} - 6 q^{95} + 66 q^{97} - 7 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.210422 −0.121487 −0.0607437 0.998153i \(-0.519347\pi\)
−0.0607437 + 0.998153i \(0.519347\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −4.28610 −1.61999 −0.809996 0.586435i \(-0.800531\pi\)
−0.809996 + 0.586435i \(0.800531\pi\)
\(8\) 0 0
\(9\) −2.95572 −0.985241
\(10\) 0 0
\(11\) 1.06403 0.320817 0.160409 0.987051i \(-0.448719\pi\)
0.160409 + 0.987051i \(0.448719\pi\)
\(12\) 0 0
\(13\) 6.73502 1.86796 0.933980 0.357326i \(-0.116311\pi\)
0.933980 + 0.357326i \(0.116311\pi\)
\(14\) 0 0
\(15\) −0.210422 −0.0543308
\(16\) 0 0
\(17\) −1.31084 −0.317925 −0.158962 0.987285i \(-0.550815\pi\)
−0.158962 + 0.987285i \(0.550815\pi\)
\(18\) 0 0
\(19\) −3.85994 −0.885531 −0.442765 0.896637i \(-0.646003\pi\)
−0.442765 + 0.896637i \(0.646003\pi\)
\(20\) 0 0
\(21\) 0.901891 0.196809
\(22\) 0 0
\(23\) −1.51918 −0.316771 −0.158385 0.987377i \(-0.550629\pi\)
−0.158385 + 0.987377i \(0.550629\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.25322 0.241182
\(28\) 0 0
\(29\) −4.71498 −0.875550 −0.437775 0.899085i \(-0.644233\pi\)
−0.437775 + 0.899085i \(0.644233\pi\)
\(30\) 0 0
\(31\) 9.73676 1.74877 0.874387 0.485230i \(-0.161264\pi\)
0.874387 + 0.485230i \(0.161264\pi\)
\(32\) 0 0
\(33\) −0.223896 −0.0389753
\(34\) 0 0
\(35\) −4.28610 −0.724483
\(36\) 0 0
\(37\) −0.0446856 −0.00734627 −0.00367314 0.999993i \(-0.501169\pi\)
−0.00367314 + 0.999993i \(0.501169\pi\)
\(38\) 0 0
\(39\) −1.41720 −0.226934
\(40\) 0 0
\(41\) 2.52783 0.394781 0.197391 0.980325i \(-0.436753\pi\)
0.197391 + 0.980325i \(0.436753\pi\)
\(42\) 0 0
\(43\) −11.8904 −1.81326 −0.906632 0.421921i \(-0.861356\pi\)
−0.906632 + 0.421921i \(0.861356\pi\)
\(44\) 0 0
\(45\) −2.95572 −0.440613
\(46\) 0 0
\(47\) −11.7086 −1.70787 −0.853936 0.520378i \(-0.825791\pi\)
−0.853936 + 0.520378i \(0.825791\pi\)
\(48\) 0 0
\(49\) 11.3706 1.62437
\(50\) 0 0
\(51\) 0.275830 0.0386239
\(52\) 0 0
\(53\) 12.1737 1.67219 0.836095 0.548585i \(-0.184833\pi\)
0.836095 + 0.548585i \(0.184833\pi\)
\(54\) 0 0
\(55\) 1.06403 0.143474
\(56\) 0 0
\(57\) 0.812218 0.107581
\(58\) 0 0
\(59\) −5.13146 −0.668060 −0.334030 0.942563i \(-0.608409\pi\)
−0.334030 + 0.942563i \(0.608409\pi\)
\(60\) 0 0
\(61\) −10.2531 −1.31277 −0.656386 0.754425i \(-0.727916\pi\)
−0.656386 + 0.754425i \(0.727916\pi\)
\(62\) 0 0
\(63\) 12.6685 1.59608
\(64\) 0 0
\(65\) 6.73502 0.835377
\(66\) 0 0
\(67\) 3.09659 0.378308 0.189154 0.981947i \(-0.439425\pi\)
0.189154 + 0.981947i \(0.439425\pi\)
\(68\) 0 0
\(69\) 0.319669 0.0384836
\(70\) 0 0
\(71\) −4.55846 −0.540989 −0.270495 0.962721i \(-0.587187\pi\)
−0.270495 + 0.962721i \(0.587187\pi\)
\(72\) 0 0
\(73\) 10.6323 1.24442 0.622208 0.782852i \(-0.286236\pi\)
0.622208 + 0.782852i \(0.286236\pi\)
\(74\) 0 0
\(75\) −0.210422 −0.0242975
\(76\) 0 0
\(77\) −4.56054 −0.519722
\(78\) 0 0
\(79\) 3.01867 0.339627 0.169814 0.985476i \(-0.445683\pi\)
0.169814 + 0.985476i \(0.445683\pi\)
\(80\) 0 0
\(81\) 8.60346 0.955940
\(82\) 0 0
\(83\) 12.6498 1.38850 0.694250 0.719734i \(-0.255736\pi\)
0.694250 + 0.719734i \(0.255736\pi\)
\(84\) 0 0
\(85\) −1.31084 −0.142180
\(86\) 0 0
\(87\) 0.992137 0.106368
\(88\) 0 0
\(89\) 10.1908 1.08023 0.540113 0.841593i \(-0.318382\pi\)
0.540113 + 0.841593i \(0.318382\pi\)
\(90\) 0 0
\(91\) −28.8670 −3.02608
\(92\) 0 0
\(93\) −2.04883 −0.212454
\(94\) 0 0
\(95\) −3.85994 −0.396021
\(96\) 0 0
\(97\) −14.2840 −1.45032 −0.725162 0.688578i \(-0.758235\pi\)
−0.725162 + 0.688578i \(0.758235\pi\)
\(98\) 0 0
\(99\) −3.14498 −0.316082
\(100\) 0 0
\(101\) 6.77839 0.674475 0.337237 0.941420i \(-0.390508\pi\)
0.337237 + 0.941420i \(0.390508\pi\)
\(102\) 0 0
\(103\) 10.9190 1.07588 0.537940 0.842983i \(-0.319203\pi\)
0.537940 + 0.842983i \(0.319203\pi\)
\(104\) 0 0
\(105\) 0.901891 0.0880155
\(106\) 0 0
\(107\) −1.28967 −0.124677 −0.0623384 0.998055i \(-0.519856\pi\)
−0.0623384 + 0.998055i \(0.519856\pi\)
\(108\) 0 0
\(109\) −18.3087 −1.75365 −0.876827 0.480805i \(-0.840344\pi\)
−0.876827 + 0.480805i \(0.840344\pi\)
\(110\) 0 0
\(111\) 0.00940286 0.000892480 0
\(112\) 0 0
\(113\) −11.2950 −1.06254 −0.531272 0.847201i \(-0.678286\pi\)
−0.531272 + 0.847201i \(0.678286\pi\)
\(114\) 0 0
\(115\) −1.51918 −0.141664
\(116\) 0 0
\(117\) −19.9069 −1.84039
\(118\) 0 0
\(119\) 5.61838 0.515036
\(120\) 0 0
\(121\) −9.86784 −0.897076
\(122\) 0 0
\(123\) −0.531913 −0.0479610
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −8.88811 −0.788693 −0.394346 0.918962i \(-0.629029\pi\)
−0.394346 + 0.918962i \(0.629029\pi\)
\(128\) 0 0
\(129\) 2.50200 0.220289
\(130\) 0 0
\(131\) 18.1740 1.58787 0.793937 0.608000i \(-0.208028\pi\)
0.793937 + 0.608000i \(0.208028\pi\)
\(132\) 0 0
\(133\) 16.5441 1.43455
\(134\) 0 0
\(135\) 1.25322 0.107860
\(136\) 0 0
\(137\) 20.6208 1.76175 0.880876 0.473348i \(-0.156955\pi\)
0.880876 + 0.473348i \(0.156955\pi\)
\(138\) 0 0
\(139\) −4.20414 −0.356591 −0.178295 0.983977i \(-0.557058\pi\)
−0.178295 + 0.983977i \(0.557058\pi\)
\(140\) 0 0
\(141\) 2.46375 0.207485
\(142\) 0 0
\(143\) 7.16627 0.599274
\(144\) 0 0
\(145\) −4.71498 −0.391558
\(146\) 0 0
\(147\) −2.39263 −0.197341
\(148\) 0 0
\(149\) 12.6819 1.03894 0.519469 0.854489i \(-0.326130\pi\)
0.519469 + 0.854489i \(0.326130\pi\)
\(150\) 0 0
\(151\) −5.80764 −0.472619 −0.236309 0.971678i \(-0.575938\pi\)
−0.236309 + 0.971678i \(0.575938\pi\)
\(152\) 0 0
\(153\) 3.87447 0.313233
\(154\) 0 0
\(155\) 9.73676 0.782075
\(156\) 0 0
\(157\) 9.68576 0.773008 0.386504 0.922288i \(-0.373683\pi\)
0.386504 + 0.922288i \(0.373683\pi\)
\(158\) 0 0
\(159\) −2.56163 −0.203150
\(160\) 0 0
\(161\) 6.51134 0.513166
\(162\) 0 0
\(163\) 10.5682 0.827766 0.413883 0.910330i \(-0.364172\pi\)
0.413883 + 0.910330i \(0.364172\pi\)
\(164\) 0 0
\(165\) −0.223896 −0.0174303
\(166\) 0 0
\(167\) 3.95077 0.305720 0.152860 0.988248i \(-0.451152\pi\)
0.152860 + 0.988248i \(0.451152\pi\)
\(168\) 0 0
\(169\) 32.3605 2.48927
\(170\) 0 0
\(171\) 11.4089 0.872461
\(172\) 0 0
\(173\) −14.6528 −1.11403 −0.557016 0.830501i \(-0.688054\pi\)
−0.557016 + 0.830501i \(0.688054\pi\)
\(174\) 0 0
\(175\) −4.28610 −0.323998
\(176\) 0 0
\(177\) 1.07978 0.0811609
\(178\) 0 0
\(179\) −19.1044 −1.42793 −0.713966 0.700180i \(-0.753103\pi\)
−0.713966 + 0.700180i \(0.753103\pi\)
\(180\) 0 0
\(181\) 17.8763 1.32874 0.664368 0.747406i \(-0.268701\pi\)
0.664368 + 0.747406i \(0.268701\pi\)
\(182\) 0 0
\(183\) 2.15748 0.159485
\(184\) 0 0
\(185\) −0.0446856 −0.00328535
\(186\) 0 0
\(187\) −1.39477 −0.101996
\(188\) 0 0
\(189\) −5.37141 −0.390713
\(190\) 0 0
\(191\) −2.31803 −0.167727 −0.0838633 0.996477i \(-0.526726\pi\)
−0.0838633 + 0.996477i \(0.526726\pi\)
\(192\) 0 0
\(193\) 2.63579 0.189728 0.0948641 0.995490i \(-0.469758\pi\)
0.0948641 + 0.995490i \(0.469758\pi\)
\(194\) 0 0
\(195\) −1.41720 −0.101488
\(196\) 0 0
\(197\) 12.9127 0.919993 0.459997 0.887921i \(-0.347851\pi\)
0.459997 + 0.887921i \(0.347851\pi\)
\(198\) 0 0
\(199\) 9.46739 0.671125 0.335563 0.942018i \(-0.391074\pi\)
0.335563 + 0.942018i \(0.391074\pi\)
\(200\) 0 0
\(201\) −0.651592 −0.0459597
\(202\) 0 0
\(203\) 20.2089 1.41838
\(204\) 0 0
\(205\) 2.52783 0.176552
\(206\) 0 0
\(207\) 4.49027 0.312095
\(208\) 0 0
\(209\) −4.10709 −0.284094
\(210\) 0 0
\(211\) 24.0659 1.65677 0.828384 0.560161i \(-0.189261\pi\)
0.828384 + 0.560161i \(0.189261\pi\)
\(212\) 0 0
\(213\) 0.959202 0.0657234
\(214\) 0 0
\(215\) −11.8904 −0.810917
\(216\) 0 0
\(217\) −41.7327 −2.83300
\(218\) 0 0
\(219\) −2.23727 −0.151181
\(220\) 0 0
\(221\) −8.82853 −0.593871
\(222\) 0 0
\(223\) 2.29356 0.153588 0.0767942 0.997047i \(-0.475532\pi\)
0.0767942 + 0.997047i \(0.475532\pi\)
\(224\) 0 0
\(225\) −2.95572 −0.197048
\(226\) 0 0
\(227\) 5.91541 0.392619 0.196310 0.980542i \(-0.437104\pi\)
0.196310 + 0.980542i \(0.437104\pi\)
\(228\) 0 0
\(229\) 5.29024 0.349589 0.174794 0.984605i \(-0.444074\pi\)
0.174794 + 0.984605i \(0.444074\pi\)
\(230\) 0 0
\(231\) 0.959640 0.0631397
\(232\) 0 0
\(233\) 28.8272 1.88854 0.944268 0.329179i \(-0.106772\pi\)
0.944268 + 0.329179i \(0.106772\pi\)
\(234\) 0 0
\(235\) −11.7086 −0.763784
\(236\) 0 0
\(237\) −0.635196 −0.0412604
\(238\) 0 0
\(239\) −20.9836 −1.35732 −0.678658 0.734454i \(-0.737438\pi\)
−0.678658 + 0.734454i \(0.737438\pi\)
\(240\) 0 0
\(241\) 11.9505 0.769803 0.384901 0.922958i \(-0.374235\pi\)
0.384901 + 0.922958i \(0.374235\pi\)
\(242\) 0 0
\(243\) −5.57001 −0.357317
\(244\) 0 0
\(245\) 11.3706 0.726442
\(246\) 0 0
\(247\) −25.9968 −1.65414
\(248\) 0 0
\(249\) −2.66181 −0.168685
\(250\) 0 0
\(251\) 18.6227 1.17545 0.587727 0.809060i \(-0.300023\pi\)
0.587727 + 0.809060i \(0.300023\pi\)
\(252\) 0 0
\(253\) −1.61645 −0.101625
\(254\) 0 0
\(255\) 0.275830 0.0172731
\(256\) 0 0
\(257\) 16.1720 1.00878 0.504390 0.863476i \(-0.331717\pi\)
0.504390 + 0.863476i \(0.331717\pi\)
\(258\) 0 0
\(259\) 0.191527 0.0119009
\(260\) 0 0
\(261\) 13.9362 0.862627
\(262\) 0 0
\(263\) −1.82088 −0.112281 −0.0561403 0.998423i \(-0.517879\pi\)
−0.0561403 + 0.998423i \(0.517879\pi\)
\(264\) 0 0
\(265\) 12.1737 0.747826
\(266\) 0 0
\(267\) −2.14438 −0.131234
\(268\) 0 0
\(269\) 16.6642 1.01604 0.508018 0.861346i \(-0.330378\pi\)
0.508018 + 0.861346i \(0.330378\pi\)
\(270\) 0 0
\(271\) 20.5625 1.24908 0.624541 0.780992i \(-0.285286\pi\)
0.624541 + 0.780992i \(0.285286\pi\)
\(272\) 0 0
\(273\) 6.07426 0.367631
\(274\) 0 0
\(275\) 1.06403 0.0641635
\(276\) 0 0
\(277\) 20.7768 1.24836 0.624179 0.781282i \(-0.285434\pi\)
0.624179 + 0.781282i \(0.285434\pi\)
\(278\) 0 0
\(279\) −28.7792 −1.72296
\(280\) 0 0
\(281\) 9.09635 0.542643 0.271321 0.962489i \(-0.412539\pi\)
0.271321 + 0.962489i \(0.412539\pi\)
\(282\) 0 0
\(283\) 6.28342 0.373511 0.186755 0.982406i \(-0.440203\pi\)
0.186755 + 0.982406i \(0.440203\pi\)
\(284\) 0 0
\(285\) 0.812218 0.0481116
\(286\) 0 0
\(287\) −10.8345 −0.639543
\(288\) 0 0
\(289\) −15.2817 −0.898924
\(290\) 0 0
\(291\) 3.00568 0.176196
\(292\) 0 0
\(293\) −9.24930 −0.540350 −0.270175 0.962811i \(-0.587082\pi\)
−0.270175 + 0.962811i \(0.587082\pi\)
\(294\) 0 0
\(295\) −5.13146 −0.298765
\(296\) 0 0
\(297\) 1.33346 0.0773753
\(298\) 0 0
\(299\) −10.2317 −0.591715
\(300\) 0 0
\(301\) 50.9633 2.93747
\(302\) 0 0
\(303\) −1.42632 −0.0819402
\(304\) 0 0
\(305\) −10.2531 −0.587090
\(306\) 0 0
\(307\) 27.4362 1.56587 0.782934 0.622104i \(-0.213722\pi\)
0.782934 + 0.622104i \(0.213722\pi\)
\(308\) 0 0
\(309\) −2.29760 −0.130706
\(310\) 0 0
\(311\) 3.65992 0.207535 0.103768 0.994602i \(-0.466910\pi\)
0.103768 + 0.994602i \(0.466910\pi\)
\(312\) 0 0
\(313\) −11.3693 −0.642631 −0.321316 0.946972i \(-0.604125\pi\)
−0.321316 + 0.946972i \(0.604125\pi\)
\(314\) 0 0
\(315\) 12.6685 0.713790
\(316\) 0 0
\(317\) −5.15992 −0.289810 −0.144905 0.989446i \(-0.546288\pi\)
−0.144905 + 0.989446i \(0.546288\pi\)
\(318\) 0 0
\(319\) −5.01688 −0.280892
\(320\) 0 0
\(321\) 0.271375 0.0151467
\(322\) 0 0
\(323\) 5.05976 0.281532
\(324\) 0 0
\(325\) 6.73502 0.373592
\(326\) 0 0
\(327\) 3.85256 0.213047
\(328\) 0 0
\(329\) 50.1841 2.76674
\(330\) 0 0
\(331\) −24.0976 −1.32453 −0.662263 0.749271i \(-0.730404\pi\)
−0.662263 + 0.749271i \(0.730404\pi\)
\(332\) 0 0
\(333\) 0.132078 0.00723785
\(334\) 0 0
\(335\) 3.09659 0.169185
\(336\) 0 0
\(337\) 10.7255 0.584253 0.292126 0.956380i \(-0.405637\pi\)
0.292126 + 0.956380i \(0.405637\pi\)
\(338\) 0 0
\(339\) 2.37672 0.129086
\(340\) 0 0
\(341\) 10.3602 0.561037
\(342\) 0 0
\(343\) −18.7329 −1.01148
\(344\) 0 0
\(345\) 0.319669 0.0172104
\(346\) 0 0
\(347\) 7.94536 0.426529 0.213265 0.976994i \(-0.431590\pi\)
0.213265 + 0.976994i \(0.431590\pi\)
\(348\) 0 0
\(349\) 28.7190 1.53729 0.768645 0.639675i \(-0.220931\pi\)
0.768645 + 0.639675i \(0.220931\pi\)
\(350\) 0 0
\(351\) 8.44045 0.450518
\(352\) 0 0
\(353\) −4.33583 −0.230773 −0.115387 0.993321i \(-0.536811\pi\)
−0.115387 + 0.993321i \(0.536811\pi\)
\(354\) 0 0
\(355\) −4.55846 −0.241938
\(356\) 0 0
\(357\) −1.18223 −0.0625704
\(358\) 0 0
\(359\) −21.1095 −1.11412 −0.557058 0.830474i \(-0.688070\pi\)
−0.557058 + 0.830474i \(0.688070\pi\)
\(360\) 0 0
\(361\) −4.10087 −0.215835
\(362\) 0 0
\(363\) 2.07641 0.108984
\(364\) 0 0
\(365\) 10.6323 0.556520
\(366\) 0 0
\(367\) 35.3726 1.84643 0.923217 0.384279i \(-0.125550\pi\)
0.923217 + 0.384279i \(0.125550\pi\)
\(368\) 0 0
\(369\) −7.47157 −0.388955
\(370\) 0 0
\(371\) −52.1778 −2.70893
\(372\) 0 0
\(373\) 31.0835 1.60944 0.804720 0.593654i \(-0.202315\pi\)
0.804720 + 0.593654i \(0.202315\pi\)
\(374\) 0 0
\(375\) −0.210422 −0.0108662
\(376\) 0 0
\(377\) −31.7555 −1.63549
\(378\) 0 0
\(379\) 3.44745 0.177084 0.0885419 0.996072i \(-0.471779\pi\)
0.0885419 + 0.996072i \(0.471779\pi\)
\(380\) 0 0
\(381\) 1.87026 0.0958163
\(382\) 0 0
\(383\) −31.1927 −1.59387 −0.796935 0.604065i \(-0.793547\pi\)
−0.796935 + 0.604065i \(0.793547\pi\)
\(384\) 0 0
\(385\) −4.56054 −0.232427
\(386\) 0 0
\(387\) 35.1446 1.78650
\(388\) 0 0
\(389\) −24.4197 −1.23813 −0.619063 0.785341i \(-0.712487\pi\)
−0.619063 + 0.785341i \(0.712487\pi\)
\(390\) 0 0
\(391\) 1.99140 0.100709
\(392\) 0 0
\(393\) −3.82423 −0.192907
\(394\) 0 0
\(395\) 3.01867 0.151886
\(396\) 0 0
\(397\) −14.5362 −0.729550 −0.364775 0.931096i \(-0.618854\pi\)
−0.364775 + 0.931096i \(0.618854\pi\)
\(398\) 0 0
\(399\) −3.48124 −0.174280
\(400\) 0 0
\(401\) 1.00000 0.0499376
\(402\) 0 0
\(403\) 65.5773 3.26664
\(404\) 0 0
\(405\) 8.60346 0.427509
\(406\) 0 0
\(407\) −0.0475469 −0.00235681
\(408\) 0 0
\(409\) 16.9212 0.836697 0.418349 0.908287i \(-0.362609\pi\)
0.418349 + 0.908287i \(0.362609\pi\)
\(410\) 0 0
\(411\) −4.33907 −0.214031
\(412\) 0 0
\(413\) 21.9939 1.08225
\(414\) 0 0
\(415\) 12.6498 0.620956
\(416\) 0 0
\(417\) 0.884646 0.0433213
\(418\) 0 0
\(419\) −13.2221 −0.645944 −0.322972 0.946408i \(-0.604682\pi\)
−0.322972 + 0.946408i \(0.604682\pi\)
\(420\) 0 0
\(421\) 14.1125 0.687803 0.343901 0.939006i \(-0.388251\pi\)
0.343901 + 0.939006i \(0.388251\pi\)
\(422\) 0 0
\(423\) 34.6073 1.68267
\(424\) 0 0
\(425\) −1.31084 −0.0635850
\(426\) 0 0
\(427\) 43.9457 2.12668
\(428\) 0 0
\(429\) −1.50794 −0.0728043
\(430\) 0 0
\(431\) −8.90094 −0.428743 −0.214372 0.976752i \(-0.568770\pi\)
−0.214372 + 0.976752i \(0.568770\pi\)
\(432\) 0 0
\(433\) −7.01120 −0.336937 −0.168468 0.985707i \(-0.553882\pi\)
−0.168468 + 0.985707i \(0.553882\pi\)
\(434\) 0 0
\(435\) 0.992137 0.0475693
\(436\) 0 0
\(437\) 5.86394 0.280510
\(438\) 0 0
\(439\) −8.74796 −0.417517 −0.208759 0.977967i \(-0.566942\pi\)
−0.208759 + 0.977967i \(0.566942\pi\)
\(440\) 0 0
\(441\) −33.6084 −1.60040
\(442\) 0 0
\(443\) −1.29372 −0.0614667 −0.0307334 0.999528i \(-0.509784\pi\)
−0.0307334 + 0.999528i \(0.509784\pi\)
\(444\) 0 0
\(445\) 10.1908 0.483092
\(446\) 0 0
\(447\) −2.66855 −0.126218
\(448\) 0 0
\(449\) 7.55203 0.356402 0.178201 0.983994i \(-0.442972\pi\)
0.178201 + 0.983994i \(0.442972\pi\)
\(450\) 0 0
\(451\) 2.68969 0.126653
\(452\) 0 0
\(453\) 1.22206 0.0574173
\(454\) 0 0
\(455\) −28.8670 −1.35330
\(456\) 0 0
\(457\) 21.3717 0.999724 0.499862 0.866105i \(-0.333384\pi\)
0.499862 + 0.866105i \(0.333384\pi\)
\(458\) 0 0
\(459\) −1.64277 −0.0766777
\(460\) 0 0
\(461\) −11.9630 −0.557173 −0.278587 0.960411i \(-0.589866\pi\)
−0.278587 + 0.960411i \(0.589866\pi\)
\(462\) 0 0
\(463\) −2.58246 −0.120017 −0.0600086 0.998198i \(-0.519113\pi\)
−0.0600086 + 0.998198i \(0.519113\pi\)
\(464\) 0 0
\(465\) −2.04883 −0.0950123
\(466\) 0 0
\(467\) 4.95577 0.229326 0.114663 0.993404i \(-0.463421\pi\)
0.114663 + 0.993404i \(0.463421\pi\)
\(468\) 0 0
\(469\) −13.2723 −0.612857
\(470\) 0 0
\(471\) −2.03810 −0.0939107
\(472\) 0 0
\(473\) −12.6517 −0.581727
\(474\) 0 0
\(475\) −3.85994 −0.177106
\(476\) 0 0
\(477\) −35.9822 −1.64751
\(478\) 0 0
\(479\) 31.9505 1.45985 0.729927 0.683525i \(-0.239554\pi\)
0.729927 + 0.683525i \(0.239554\pi\)
\(480\) 0 0
\(481\) −0.300959 −0.0137225
\(482\) 0 0
\(483\) −1.37013 −0.0623432
\(484\) 0 0
\(485\) −14.2840 −0.648605
\(486\) 0 0
\(487\) −36.0593 −1.63400 −0.817002 0.576634i \(-0.804366\pi\)
−0.817002 + 0.576634i \(0.804366\pi\)
\(488\) 0 0
\(489\) −2.22379 −0.100563
\(490\) 0 0
\(491\) −15.0659 −0.679915 −0.339957 0.940441i \(-0.610413\pi\)
−0.339957 + 0.940441i \(0.610413\pi\)
\(492\) 0 0
\(493\) 6.18057 0.278359
\(494\) 0 0
\(495\) −3.14498 −0.141356
\(496\) 0 0
\(497\) 19.5380 0.876399
\(498\) 0 0
\(499\) 7.69725 0.344576 0.172288 0.985047i \(-0.444884\pi\)
0.172288 + 0.985047i \(0.444884\pi\)
\(500\) 0 0
\(501\) −0.831331 −0.0371411
\(502\) 0 0
\(503\) 37.4430 1.66950 0.834751 0.550627i \(-0.185611\pi\)
0.834751 + 0.550627i \(0.185611\pi\)
\(504\) 0 0
\(505\) 6.77839 0.301634
\(506\) 0 0
\(507\) −6.80938 −0.302415
\(508\) 0 0
\(509\) −33.7503 −1.49596 −0.747979 0.663723i \(-0.768976\pi\)
−0.747979 + 0.663723i \(0.768976\pi\)
\(510\) 0 0
\(511\) −45.5710 −2.01594
\(512\) 0 0
\(513\) −4.83734 −0.213574
\(514\) 0 0
\(515\) 10.9190 0.481148
\(516\) 0 0
\(517\) −12.4583 −0.547915
\(518\) 0 0
\(519\) 3.08328 0.135341
\(520\) 0 0
\(521\) −18.4889 −0.810013 −0.405006 0.914314i \(-0.632731\pi\)
−0.405006 + 0.914314i \(0.632731\pi\)
\(522\) 0 0
\(523\) −22.0049 −0.962206 −0.481103 0.876664i \(-0.659764\pi\)
−0.481103 + 0.876664i \(0.659764\pi\)
\(524\) 0 0
\(525\) 0.901891 0.0393617
\(526\) 0 0
\(527\) −12.7633 −0.555979
\(528\) 0 0
\(529\) −20.6921 −0.899656
\(530\) 0 0
\(531\) 15.1672 0.658200
\(532\) 0 0
\(533\) 17.0250 0.737435
\(534\) 0 0
\(535\) −1.28967 −0.0557571
\(536\) 0 0
\(537\) 4.02000 0.173476
\(538\) 0 0
\(539\) 12.0987 0.521128
\(540\) 0 0
\(541\) 5.47730 0.235487 0.117744 0.993044i \(-0.462434\pi\)
0.117744 + 0.993044i \(0.462434\pi\)
\(542\) 0 0
\(543\) −3.76158 −0.161425
\(544\) 0 0
\(545\) −18.3087 −0.784258
\(546\) 0 0
\(547\) 21.2221 0.907390 0.453695 0.891157i \(-0.350106\pi\)
0.453695 + 0.891157i \(0.350106\pi\)
\(548\) 0 0
\(549\) 30.3053 1.29340
\(550\) 0 0
\(551\) 18.1995 0.775326
\(552\) 0 0
\(553\) −12.9383 −0.550193
\(554\) 0 0
\(555\) 0.00940286 0.000399129 0
\(556\) 0 0
\(557\) 35.0557 1.48536 0.742679 0.669648i \(-0.233555\pi\)
0.742679 + 0.669648i \(0.233555\pi\)
\(558\) 0 0
\(559\) −80.0819 −3.38711
\(560\) 0 0
\(561\) 0.293491 0.0123912
\(562\) 0 0
\(563\) 34.8664 1.46944 0.734722 0.678368i \(-0.237313\pi\)
0.734722 + 0.678368i \(0.237313\pi\)
\(564\) 0 0
\(565\) −11.2950 −0.475184
\(566\) 0 0
\(567\) −36.8753 −1.54862
\(568\) 0 0
\(569\) 12.2725 0.514488 0.257244 0.966346i \(-0.417186\pi\)
0.257244 + 0.966346i \(0.417186\pi\)
\(570\) 0 0
\(571\) −41.7445 −1.74695 −0.873477 0.486866i \(-0.838140\pi\)
−0.873477 + 0.486866i \(0.838140\pi\)
\(572\) 0 0
\(573\) 0.487765 0.0203767
\(574\) 0 0
\(575\) −1.51918 −0.0633541
\(576\) 0 0
\(577\) −29.8412 −1.24231 −0.621153 0.783689i \(-0.713335\pi\)
−0.621153 + 0.783689i \(0.713335\pi\)
\(578\) 0 0
\(579\) −0.554629 −0.0230496
\(580\) 0 0
\(581\) −54.2184 −2.24936
\(582\) 0 0
\(583\) 12.9532 0.536468
\(584\) 0 0
\(585\) −19.9069 −0.823047
\(586\) 0 0
\(587\) 40.5901 1.67533 0.837666 0.546183i \(-0.183920\pi\)
0.837666 + 0.546183i \(0.183920\pi\)
\(588\) 0 0
\(589\) −37.5833 −1.54859
\(590\) 0 0
\(591\) −2.71713 −0.111768
\(592\) 0 0
\(593\) 5.36923 0.220488 0.110244 0.993905i \(-0.464837\pi\)
0.110244 + 0.993905i \(0.464837\pi\)
\(594\) 0 0
\(595\) 5.61838 0.230331
\(596\) 0 0
\(597\) −1.99215 −0.0815333
\(598\) 0 0
\(599\) −12.2663 −0.501187 −0.250594 0.968092i \(-0.580626\pi\)
−0.250594 + 0.968092i \(0.580626\pi\)
\(600\) 0 0
\(601\) 2.73817 0.111692 0.0558462 0.998439i \(-0.482214\pi\)
0.0558462 + 0.998439i \(0.482214\pi\)
\(602\) 0 0
\(603\) −9.15266 −0.372725
\(604\) 0 0
\(605\) −9.86784 −0.401185
\(606\) 0 0
\(607\) −1.52229 −0.0617880 −0.0308940 0.999523i \(-0.509835\pi\)
−0.0308940 + 0.999523i \(0.509835\pi\)
\(608\) 0 0
\(609\) −4.25240 −0.172316
\(610\) 0 0
\(611\) −78.8576 −3.19024
\(612\) 0 0
\(613\) 35.1669 1.42038 0.710188 0.704012i \(-0.248610\pi\)
0.710188 + 0.704012i \(0.248610\pi\)
\(614\) 0 0
\(615\) −0.531913 −0.0214488
\(616\) 0 0
\(617\) 41.5902 1.67436 0.837179 0.546929i \(-0.184203\pi\)
0.837179 + 0.546929i \(0.184203\pi\)
\(618\) 0 0
\(619\) −6.14656 −0.247051 −0.123525 0.992341i \(-0.539420\pi\)
−0.123525 + 0.992341i \(0.539420\pi\)
\(620\) 0 0
\(621\) −1.90386 −0.0763993
\(622\) 0 0
\(623\) −43.6789 −1.74996
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0.864225 0.0345138
\(628\) 0 0
\(629\) 0.0585757 0.00233556
\(630\) 0 0
\(631\) −27.4888 −1.09431 −0.547156 0.837031i \(-0.684289\pi\)
−0.547156 + 0.837031i \(0.684289\pi\)
\(632\) 0 0
\(633\) −5.06401 −0.201277
\(634\) 0 0
\(635\) −8.88811 −0.352714
\(636\) 0 0
\(637\) 76.5814 3.03427
\(638\) 0 0
\(639\) 13.4735 0.533005
\(640\) 0 0
\(641\) −22.6000 −0.892645 −0.446322 0.894872i \(-0.647267\pi\)
−0.446322 + 0.894872i \(0.647267\pi\)
\(642\) 0 0
\(643\) 33.3949 1.31697 0.658483 0.752596i \(-0.271199\pi\)
0.658483 + 0.752596i \(0.271199\pi\)
\(644\) 0 0
\(645\) 2.50200 0.0985162
\(646\) 0 0
\(647\) −7.42401 −0.291868 −0.145934 0.989294i \(-0.546619\pi\)
−0.145934 + 0.989294i \(0.546619\pi\)
\(648\) 0 0
\(649\) −5.46004 −0.214325
\(650\) 0 0
\(651\) 8.78149 0.344174
\(652\) 0 0
\(653\) −22.6074 −0.884697 −0.442348 0.896843i \(-0.645855\pi\)
−0.442348 + 0.896843i \(0.645855\pi\)
\(654\) 0 0
\(655\) 18.1740 0.710119
\(656\) 0 0
\(657\) −31.4261 −1.22605
\(658\) 0 0
\(659\) 18.4590 0.719061 0.359531 0.933133i \(-0.382937\pi\)
0.359531 + 0.933133i \(0.382937\pi\)
\(660\) 0 0
\(661\) −28.3737 −1.10361 −0.551805 0.833973i \(-0.686061\pi\)
−0.551805 + 0.833973i \(0.686061\pi\)
\(662\) 0 0
\(663\) 1.85772 0.0721479
\(664\) 0 0
\(665\) 16.5441 0.641552
\(666\) 0 0
\(667\) 7.16289 0.277348
\(668\) 0 0
\(669\) −0.482617 −0.0186591
\(670\) 0 0
\(671\) −10.9096 −0.421160
\(672\) 0 0
\(673\) −44.8614 −1.72928 −0.864639 0.502393i \(-0.832453\pi\)
−0.864639 + 0.502393i \(0.832453\pi\)
\(674\) 0 0
\(675\) 1.25322 0.0482364
\(676\) 0 0
\(677\) −42.9517 −1.65077 −0.825383 0.564573i \(-0.809041\pi\)
−0.825383 + 0.564573i \(0.809041\pi\)
\(678\) 0 0
\(679\) 61.2227 2.34951
\(680\) 0 0
\(681\) −1.24473 −0.0476983
\(682\) 0 0
\(683\) −2.12606 −0.0813516 −0.0406758 0.999172i \(-0.512951\pi\)
−0.0406758 + 0.999172i \(0.512951\pi\)
\(684\) 0 0
\(685\) 20.6208 0.787879
\(686\) 0 0
\(687\) −1.11318 −0.0424706
\(688\) 0 0
\(689\) 81.9903 3.12358
\(690\) 0 0
\(691\) −16.9580 −0.645112 −0.322556 0.946550i \(-0.604542\pi\)
−0.322556 + 0.946550i \(0.604542\pi\)
\(692\) 0 0
\(693\) 13.4797 0.512051
\(694\) 0 0
\(695\) −4.20414 −0.159472
\(696\) 0 0
\(697\) −3.31358 −0.125511
\(698\) 0 0
\(699\) −6.06590 −0.229433
\(700\) 0 0
\(701\) 18.1912 0.687073 0.343537 0.939139i \(-0.388375\pi\)
0.343537 + 0.939139i \(0.388375\pi\)
\(702\) 0 0
\(703\) 0.172484 0.00650535
\(704\) 0 0
\(705\) 2.46375 0.0927901
\(706\) 0 0
\(707\) −29.0528 −1.09264
\(708\) 0 0
\(709\) −11.6321 −0.436854 −0.218427 0.975853i \(-0.570092\pi\)
−0.218427 + 0.975853i \(0.570092\pi\)
\(710\) 0 0
\(711\) −8.92236 −0.334615
\(712\) 0 0
\(713\) −14.7919 −0.553960
\(714\) 0 0
\(715\) 7.16627 0.268003
\(716\) 0 0
\(717\) 4.41542 0.164897
\(718\) 0 0
\(719\) 25.4081 0.947563 0.473782 0.880642i \(-0.342889\pi\)
0.473782 + 0.880642i \(0.342889\pi\)
\(720\) 0 0
\(721\) −46.7999 −1.74292
\(722\) 0 0
\(723\) −2.51466 −0.0935214
\(724\) 0 0
\(725\) −4.71498 −0.175110
\(726\) 0 0
\(727\) 18.0922 0.671001 0.335501 0.942040i \(-0.391095\pi\)
0.335501 + 0.942040i \(0.391095\pi\)
\(728\) 0 0
\(729\) −24.6383 −0.912531
\(730\) 0 0
\(731\) 15.5864 0.576482
\(732\) 0 0
\(733\) 39.3044 1.45174 0.725870 0.687832i \(-0.241437\pi\)
0.725870 + 0.687832i \(0.241437\pi\)
\(734\) 0 0
\(735\) −2.39263 −0.0882536
\(736\) 0 0
\(737\) 3.29487 0.121368
\(738\) 0 0
\(739\) 40.3035 1.48259 0.741294 0.671180i \(-0.234212\pi\)
0.741294 + 0.671180i \(0.234212\pi\)
\(740\) 0 0
\(741\) 5.47031 0.200957
\(742\) 0 0
\(743\) 16.7950 0.616150 0.308075 0.951362i \(-0.400315\pi\)
0.308075 + 0.951362i \(0.400315\pi\)
\(744\) 0 0
\(745\) 12.6819 0.464627
\(746\) 0 0
\(747\) −37.3894 −1.36801
\(748\) 0 0
\(749\) 5.52763 0.201975
\(750\) 0 0
\(751\) 27.9908 1.02140 0.510699 0.859759i \(-0.329387\pi\)
0.510699 + 0.859759i \(0.329387\pi\)
\(752\) 0 0
\(753\) −3.91863 −0.142803
\(754\) 0 0
\(755\) −5.80764 −0.211362
\(756\) 0 0
\(757\) 0.0791012 0.00287498 0.00143749 0.999999i \(-0.499542\pi\)
0.00143749 + 0.999999i \(0.499542\pi\)
\(758\) 0 0
\(759\) 0.340138 0.0123462
\(760\) 0 0
\(761\) 28.7965 1.04387 0.521936 0.852985i \(-0.325210\pi\)
0.521936 + 0.852985i \(0.325210\pi\)
\(762\) 0 0
\(763\) 78.4728 2.84091
\(764\) 0 0
\(765\) 3.87447 0.140082
\(766\) 0 0
\(767\) −34.5605 −1.24791
\(768\) 0 0
\(769\) −32.3883 −1.16795 −0.583976 0.811771i \(-0.698504\pi\)
−0.583976 + 0.811771i \(0.698504\pi\)
\(770\) 0 0
\(771\) −3.40295 −0.122554
\(772\) 0 0
\(773\) 29.9188 1.07611 0.538053 0.842911i \(-0.319160\pi\)
0.538053 + 0.842911i \(0.319160\pi\)
\(774\) 0 0
\(775\) 9.73676 0.349755
\(776\) 0 0
\(777\) −0.0403016 −0.00144581
\(778\) 0 0
\(779\) −9.75728 −0.349591
\(780\) 0 0
\(781\) −4.85034 −0.173559
\(782\) 0 0
\(783\) −5.90889 −0.211167
\(784\) 0 0
\(785\) 9.68576 0.345700
\(786\) 0 0
\(787\) 35.5332 1.26662 0.633312 0.773897i \(-0.281695\pi\)
0.633312 + 0.773897i \(0.281695\pi\)
\(788\) 0 0
\(789\) 0.383155 0.0136407
\(790\) 0 0
\(791\) 48.4115 1.72131
\(792\) 0 0
\(793\) −69.0547 −2.45220
\(794\) 0 0
\(795\) −2.56163 −0.0908515
\(796\) 0 0
\(797\) 31.6009 1.11936 0.559681 0.828708i \(-0.310924\pi\)
0.559681 + 0.828708i \(0.310924\pi\)
\(798\) 0 0
\(799\) 15.3481 0.542975
\(800\) 0 0
\(801\) −30.1213 −1.06428
\(802\) 0 0
\(803\) 11.3131 0.399230
\(804\) 0 0
\(805\) 6.51134 0.229495
\(806\) 0 0
\(807\) −3.50653 −0.123436
\(808\) 0 0
\(809\) 4.23228 0.148799 0.0743995 0.997229i \(-0.476296\pi\)
0.0743995 + 0.997229i \(0.476296\pi\)
\(810\) 0 0
\(811\) −23.2699 −0.817118 −0.408559 0.912732i \(-0.633969\pi\)
−0.408559 + 0.912732i \(0.633969\pi\)
\(812\) 0 0
\(813\) −4.32681 −0.151748
\(814\) 0 0
\(815\) 10.5682 0.370188
\(816\) 0 0
\(817\) 45.8961 1.60570
\(818\) 0 0
\(819\) 85.3227 2.98142
\(820\) 0 0
\(821\) 15.2926 0.533715 0.266858 0.963736i \(-0.414015\pi\)
0.266858 + 0.963736i \(0.414015\pi\)
\(822\) 0 0
\(823\) 2.16187 0.0753581 0.0376790 0.999290i \(-0.488004\pi\)
0.0376790 + 0.999290i \(0.488004\pi\)
\(824\) 0 0
\(825\) −0.223896 −0.00779506
\(826\) 0 0
\(827\) −40.3349 −1.40258 −0.701291 0.712875i \(-0.747393\pi\)
−0.701291 + 0.712875i \(0.747393\pi\)
\(828\) 0 0
\(829\) −35.2799 −1.22532 −0.612661 0.790346i \(-0.709901\pi\)
−0.612661 + 0.790346i \(0.709901\pi\)
\(830\) 0 0
\(831\) −4.37191 −0.151660
\(832\) 0 0
\(833\) −14.9050 −0.516429
\(834\) 0 0
\(835\) 3.95077 0.136722
\(836\) 0 0
\(837\) 12.2023 0.421772
\(838\) 0 0
\(839\) −9.98484 −0.344715 −0.172358 0.985034i \(-0.555138\pi\)
−0.172358 + 0.985034i \(0.555138\pi\)
\(840\) 0 0
\(841\) −6.76898 −0.233413
\(842\) 0 0
\(843\) −1.91408 −0.0659243
\(844\) 0 0
\(845\) 32.3605 1.11324
\(846\) 0 0
\(847\) 42.2945 1.45326
\(848\) 0 0
\(849\) −1.32217 −0.0453769
\(850\) 0 0
\(851\) 0.0678855 0.00232708
\(852\) 0 0
\(853\) −50.1348 −1.71658 −0.858292 0.513162i \(-0.828474\pi\)
−0.858292 + 0.513162i \(0.828474\pi\)
\(854\) 0 0
\(855\) 11.4089 0.390176
\(856\) 0 0
\(857\) 35.4240 1.21006 0.605030 0.796203i \(-0.293161\pi\)
0.605030 + 0.796203i \(0.293161\pi\)
\(858\) 0 0
\(859\) 34.6681 1.18286 0.591429 0.806357i \(-0.298564\pi\)
0.591429 + 0.806357i \(0.298564\pi\)
\(860\) 0 0
\(861\) 2.27983 0.0776964
\(862\) 0 0
\(863\) 7.09376 0.241474 0.120737 0.992685i \(-0.461474\pi\)
0.120737 + 0.992685i \(0.461474\pi\)
\(864\) 0 0
\(865\) −14.6528 −0.498211
\(866\) 0 0
\(867\) 3.21561 0.109208
\(868\) 0 0
\(869\) 3.21196 0.108958
\(870\) 0 0
\(871\) 20.8556 0.706665
\(872\) 0 0
\(873\) 42.2196 1.42892
\(874\) 0 0
\(875\) −4.28610 −0.144897
\(876\) 0 0
\(877\) 36.8887 1.24564 0.622822 0.782363i \(-0.285986\pi\)
0.622822 + 0.782363i \(0.285986\pi\)
\(878\) 0 0
\(879\) 1.94626 0.0656457
\(880\) 0 0
\(881\) 49.6947 1.67426 0.837128 0.547007i \(-0.184233\pi\)
0.837128 + 0.547007i \(0.184233\pi\)
\(882\) 0 0
\(883\) −34.1640 −1.14971 −0.574856 0.818255i \(-0.694942\pi\)
−0.574856 + 0.818255i \(0.694942\pi\)
\(884\) 0 0
\(885\) 1.07978 0.0362962
\(886\) 0 0
\(887\) −16.1595 −0.542584 −0.271292 0.962497i \(-0.587451\pi\)
−0.271292 + 0.962497i \(0.587451\pi\)
\(888\) 0 0
\(889\) 38.0953 1.27768
\(890\) 0 0
\(891\) 9.15435 0.306682
\(892\) 0 0
\(893\) 45.1944 1.51237
\(894\) 0 0
\(895\) −19.1044 −0.638591
\(896\) 0 0
\(897\) 2.15298 0.0718859
\(898\) 0 0
\(899\) −45.9086 −1.53114
\(900\) 0 0
\(901\) −15.9578 −0.531631
\(902\) 0 0
\(903\) −10.7238 −0.356866
\(904\) 0 0
\(905\) 17.8763 0.594229
\(906\) 0 0
\(907\) 4.46670 0.148314 0.0741572 0.997247i \(-0.476373\pi\)
0.0741572 + 0.997247i \(0.476373\pi\)
\(908\) 0 0
\(909\) −20.0350 −0.664520
\(910\) 0 0
\(911\) −34.5461 −1.14456 −0.572282 0.820057i \(-0.693942\pi\)
−0.572282 + 0.820057i \(0.693942\pi\)
\(912\) 0 0
\(913\) 13.4598 0.445455
\(914\) 0 0
\(915\) 2.15748 0.0713240
\(916\) 0 0
\(917\) −77.8957 −2.57234
\(918\) 0 0
\(919\) 12.7790 0.421541 0.210771 0.977536i \(-0.432403\pi\)
0.210771 + 0.977536i \(0.432403\pi\)
\(920\) 0 0
\(921\) −5.77320 −0.190233
\(922\) 0 0
\(923\) −30.7013 −1.01055
\(924\) 0 0
\(925\) −0.0446856 −0.00146925
\(926\) 0 0
\(927\) −32.2735 −1.06000
\(928\) 0 0
\(929\) 33.2231 1.09002 0.545008 0.838431i \(-0.316527\pi\)
0.545008 + 0.838431i \(0.316527\pi\)
\(930\) 0 0
\(931\) −43.8899 −1.43843
\(932\) 0 0
\(933\) −0.770130 −0.0252129
\(934\) 0 0
\(935\) −1.39477 −0.0456139
\(936\) 0 0
\(937\) −32.1637 −1.05074 −0.525372 0.850873i \(-0.676074\pi\)
−0.525372 + 0.850873i \(0.676074\pi\)
\(938\) 0 0
\(939\) 2.39236 0.0780716
\(940\) 0 0
\(941\) 52.2577 1.70355 0.851776 0.523906i \(-0.175526\pi\)
0.851776 + 0.523906i \(0.175526\pi\)
\(942\) 0 0
\(943\) −3.84023 −0.125055
\(944\) 0 0
\(945\) −5.37141 −0.174732
\(946\) 0 0
\(947\) −29.4123 −0.955770 −0.477885 0.878422i \(-0.658596\pi\)
−0.477885 + 0.878422i \(0.658596\pi\)
\(948\) 0 0
\(949\) 71.6088 2.32452
\(950\) 0 0
\(951\) 1.08576 0.0352083
\(952\) 0 0
\(953\) −55.7473 −1.80583 −0.902916 0.429817i \(-0.858578\pi\)
−0.902916 + 0.429817i \(0.858578\pi\)
\(954\) 0 0
\(955\) −2.31803 −0.0750096
\(956\) 0 0
\(957\) 1.05566 0.0341248
\(958\) 0 0
\(959\) −88.3826 −2.85402
\(960\) 0 0
\(961\) 63.8045 2.05821
\(962\) 0 0
\(963\) 3.81190 0.122837
\(964\) 0 0
\(965\) 2.63579 0.0848490
\(966\) 0 0
\(967\) −4.22188 −0.135767 −0.0678833 0.997693i \(-0.521625\pi\)
−0.0678833 + 0.997693i \(0.521625\pi\)
\(968\) 0 0
\(969\) −1.06469 −0.0342026
\(970\) 0 0
\(971\) −34.5336 −1.10824 −0.554118 0.832438i \(-0.686944\pi\)
−0.554118 + 0.832438i \(0.686944\pi\)
\(972\) 0 0
\(973\) 18.0194 0.577674
\(974\) 0 0
\(975\) −1.41720 −0.0453867
\(976\) 0 0
\(977\) −9.83114 −0.314526 −0.157263 0.987557i \(-0.550267\pi\)
−0.157263 + 0.987557i \(0.550267\pi\)
\(978\) 0 0
\(979\) 10.8434 0.346555
\(980\) 0 0
\(981\) 54.1154 1.72777
\(982\) 0 0
\(983\) 56.2066 1.79271 0.896356 0.443334i \(-0.146204\pi\)
0.896356 + 0.443334i \(0.146204\pi\)
\(984\) 0 0
\(985\) 12.9127 0.411434
\(986\) 0 0
\(987\) −10.5599 −0.336124
\(988\) 0 0
\(989\) 18.0636 0.574389
\(990\) 0 0
\(991\) −54.8279 −1.74166 −0.870832 0.491580i \(-0.836420\pi\)
−0.870832 + 0.491580i \(0.836420\pi\)
\(992\) 0 0
\(993\) 5.07069 0.160913
\(994\) 0 0
\(995\) 9.46739 0.300136
\(996\) 0 0
\(997\) 8.51470 0.269663 0.134832 0.990869i \(-0.456951\pi\)
0.134832 + 0.990869i \(0.456951\pi\)
\(998\) 0 0
\(999\) −0.0560008 −0.00177179
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 8020.2.a.f.1.17 37
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8020.2.a.f.1.17 37 1.1 even 1 trivial