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Properties

Label 2070.2.a.l.1.1
Level $2070$
Weight $2$
Character 2070.1
Self dual yes
Analytic conductor $16.529$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2070,2,Mod(1,2070)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2070, 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("2070.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2070.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: \(16.5290332184\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2070.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -2.00000 q^{11} +4.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} -4.00000 q^{19} -1.00000 q^{20} -2.00000 q^{22} +1.00000 q^{23} +1.00000 q^{25} +4.00000 q^{26} -2.00000 q^{28} -2.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} -6.00000 q^{34} +2.00000 q^{35} -4.00000 q^{37} -4.00000 q^{38} -1.00000 q^{40} -2.00000 q^{44} +1.00000 q^{46} -8.00000 q^{47} -3.00000 q^{49} +1.00000 q^{50} +4.00000 q^{52} -10.0000 q^{53} +2.00000 q^{55} -2.00000 q^{56} -2.00000 q^{58} +2.00000 q^{59} +6.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} -4.00000 q^{67} -6.00000 q^{68} +2.00000 q^{70} -4.00000 q^{71} +14.0000 q^{73} -4.00000 q^{74} -4.00000 q^{76} +4.00000 q^{77} -8.00000 q^{79} -1.00000 q^{80} -8.00000 q^{83} +6.00000 q^{85} -2.00000 q^{88} -8.00000 q^{91} +1.00000 q^{92} -8.00000 q^{94} +4.00000 q^{95} +10.0000 q^{97} -3.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) −4.00000 −0.648886
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 4.00000 0.554700
\(53\) −10.0000 −1.37361 −0.686803 − 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) 2.00000 0.260378 0.130189 − 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) −4.00000 −0.474713 −0.237356 − 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) 14.0000 1.63858 0.819288 − 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 0 0
\(83\) −8.00000 −0.878114 −0.439057 − 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) 0 0
\(87\) 0 0
\(88\) −2.00000 −0.213201
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) −3.00000 −0.303046
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −10.0000 −0.995037 −0.497519 − 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) −2.00000 −0.197066 −0.0985329 − 0.995134i \(-0.531415\pi\)
−0.0985329 + 0.995134i \(0.531415\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) −8.00000 −0.773389 −0.386695 − 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 − 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 2.00000 0.190693
\(111\) 0 0
\(112\) −2.00000 −0.188982
\(113\) −2.00000 −0.188144 −0.0940721 − 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) 2.00000 0.184115
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 6.00000 0.543214
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 − 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −4.00000 −0.350823
\(131\) −10.0000 −0.873704 −0.436852 − 0.899533i \(-0.643907\pi\)
−0.436852 + 0.899533i \(0.643907\pi\)
\(132\) 0 0
\(133\) 8.00000 0.693688
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 − 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) −4.00000 −0.335673
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 14.0000 1.15865
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) 10.0000 0.819232 0.409616 − 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 − 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −4.00000 −0.324443
\(153\) 0 0
\(154\) 4.00000 0.322329
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) 16.0000 1.27694 0.638470 − 0.769647i \(-0.279568\pi\)
0.638470 + 0.769647i \(0.279568\pi\)
\(158\) −8.00000 −0.636446
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 − 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −8.00000 −0.620920
\(167\) 8.00000 0.619059 0.309529 − 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 6.00000 0.460179
\(171\) 0 0
\(172\) 0 0
\(173\) 14.0000 1.06440 0.532200 − 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) −2.00000 −0.151186
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) 0 0
\(179\) 10.0000 0.747435 0.373718 − 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 − 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) −8.00000 −0.592999
\(183\) 0 0
\(184\) 1.00000 0.0737210
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) 12.0000 0.877527
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) −20.0000 −1.44715 −0.723575 − 0.690246i \(-0.757502\pi\)
−0.723575 + 0.690246i \(0.757502\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 − 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −14.0000 −0.997459 −0.498729 − 0.866758i \(-0.666200\pi\)
−0.498729 + 0.866758i \(0.666200\pi\)
\(198\) 0 0
\(199\) 28.0000 1.98487 0.992434 − 0.122782i \(-0.0391815\pi\)
0.992434 + 0.122782i \(0.0391815\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −10.0000 −0.703598
\(203\) 4.00000 0.280745
\(204\) 0 0
\(205\) 0 0
\(206\) −2.00000 −0.139347
\(207\) 0 0
\(208\) 4.00000 0.277350
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 − 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −10.0000 −0.686803
\(213\) 0 0
\(214\) −8.00000 −0.546869
\(215\) 0 0
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) −2.00000 −0.135457
\(219\) 0 0
\(220\) 2.00000 0.134840
\(221\) −24.0000 −1.61441
\(222\) 0 0
\(223\) 26.0000 1.74109 0.870544 − 0.492090i \(-0.163767\pi\)
0.870544 + 0.492090i \(0.163767\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) 4.00000 0.265489 0.132745 − 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 0 0
\(229\) −22.0000 −1.45380 −0.726900 − 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) −18.0000 −1.17922 −0.589610 − 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 2.00000 0.130189
\(237\) 0 0
\(238\) 12.0000 0.777844
\(239\) −20.0000 −1.29369 −0.646846 − 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 − 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) −7.00000 −0.449977
\(243\) 0 0
\(244\) 6.00000 0.384111
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) −16.0000 −1.01806
\(248\) −4.00000 −0.254000
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 2.00000 0.126239 0.0631194 − 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) 0 0
\(253\) −2.00000 −0.125739
\(254\) 2.00000 0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 − 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) 8.00000 0.497096
\(260\) −4.00000 −0.248069
\(261\) 0 0
\(262\) −10.0000 −0.617802
\(263\) 8.00000 0.493301 0.246651 − 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) 0 0
\(265\) 10.0000 0.614295
\(266\) 8.00000 0.490511
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) −18.0000 −1.09748 −0.548740 − 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) −32.0000 −1.94386 −0.971931 − 0.235267i \(-0.924404\pi\)
−0.971931 + 0.235267i \(0.924404\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) −2.00000 −0.120605
\(276\) 0 0
\(277\) 24.0000 1.44202 0.721010 − 0.692925i \(-0.243678\pi\)
0.721010 + 0.692925i \(0.243678\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) 2.00000 0.119523
\(281\) 16.0000 0.954480 0.477240 − 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 − 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −4.00000 −0.237356
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 2.00000 0.117444
\(291\) 0 0
\(292\) 14.0000 0.819288
\(293\) 10.0000 0.584206 0.292103 − 0.956387i \(-0.405645\pi\)
0.292103 + 0.956387i \(0.405645\pi\)
\(294\) 0 0
\(295\) −2.00000 −0.116445
\(296\) −4.00000 −0.232495
\(297\) 0 0
\(298\) 10.0000 0.579284
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 0 0
\(302\) 8.00000 0.460348
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) −6.00000 −0.343559
\(306\) 0 0
\(307\) −32.0000 −1.82634 −0.913168 − 0.407583i \(-0.866372\pi\)
−0.913168 + 0.407583i \(0.866372\pi\)
\(308\) 4.00000 0.227921
\(309\) 0 0
\(310\) 4.00000 0.227185
\(311\) −4.00000 −0.226819 −0.113410 − 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 − 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 16.0000 0.902932
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 18.0000 1.01098 0.505490 − 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −2.00000 −0.111456
\(323\) 24.0000 1.33540
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) 0 0
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 − 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) −8.00000 −0.439057
\(333\) 0 0
\(334\) 8.00000 0.437741
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) −26.0000 −1.41631 −0.708155 − 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) 3.00000 0.163178
\(339\) 0 0
\(340\) 6.00000 0.325396
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) 24.0000 1.28839 0.644194 − 0.764862i \(-0.277193\pi\)
0.644194 + 0.764862i \(0.277193\pi\)
\(348\) 0 0
\(349\) 18.0000 0.963518 0.481759 − 0.876304i \(-0.339998\pi\)
0.481759 + 0.876304i \(0.339998\pi\)
\(350\) −2.00000 −0.106904
\(351\) 0 0
\(352\) −2.00000 −0.106600
\(353\) −2.00000 −0.106449 −0.0532246 − 0.998583i \(-0.516950\pi\)
−0.0532246 + 0.998583i \(0.516950\pi\)
\(354\) 0 0
\(355\) 4.00000 0.212298
\(356\) 0 0
\(357\) 0 0
\(358\) 10.0000 0.528516
\(359\) −8.00000 −0.422224 −0.211112 − 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −2.00000 −0.105118
\(363\) 0 0
\(364\) −8.00000 −0.419314
\(365\) −14.0000 −0.732793
\(366\) 0 0
\(367\) −18.0000 −0.939592 −0.469796 − 0.882775i \(-0.655673\pi\)
−0.469796 + 0.882775i \(0.655673\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) 4.00000 0.207950
\(371\) 20.0000 1.03835
\(372\) 0 0
\(373\) −24.0000 −1.24267 −0.621336 − 0.783544i \(-0.713410\pi\)
−0.621336 + 0.783544i \(0.713410\pi\)
\(374\) 12.0000 0.620505
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) −8.00000 −0.412021
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 − 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 4.00000 0.205196
\(381\) 0 0
\(382\) −20.0000 −1.02329
\(383\) −24.0000 −1.22634 −0.613171 − 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) −4.00000 −0.203859
\(386\) 14.0000 0.712581
\(387\) 0 0
\(388\) 10.0000 0.507673
\(389\) 26.0000 1.31825 0.659126 − 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) −3.00000 −0.151523
\(393\) 0 0
\(394\) −14.0000 −0.705310
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) 8.00000 0.401508 0.200754 − 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) 28.0000 1.40351
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −32.0000 −1.59800 −0.799002 − 0.601329i \(-0.794638\pi\)
−0.799002 + 0.601329i \(0.794638\pi\)
\(402\) 0 0
\(403\) −16.0000 −0.797017
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) 4.00000 0.198517
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) −22.0000 −1.08783 −0.543915 − 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −2.00000 −0.0985329
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) 4.00000 0.196116
\(417\) 0 0
\(418\) 8.00000 0.391293
\(419\) 18.0000 0.879358 0.439679 − 0.898155i \(-0.355092\pi\)
0.439679 + 0.898155i \(0.355092\pi\)
\(420\) 0 0
\(421\) 18.0000 0.877266 0.438633 − 0.898666i \(-0.355463\pi\)
0.438633 + 0.898666i \(0.355463\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) −10.0000 −0.485643
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) −12.0000 −0.580721
\(428\) −8.00000 −0.386695
\(429\) 0 0
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 − 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 0 0
\(433\) 38.0000 1.82616 0.913082 − 0.407777i \(-0.133696\pi\)
0.913082 + 0.407777i \(0.133696\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) −4.00000 −0.191346
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 − 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 2.00000 0.0953463
\(441\) 0 0
\(442\) −24.0000 −1.14156
\(443\) −8.00000 −0.380091 −0.190046 − 0.981775i \(-0.560864\pi\)
−0.190046 + 0.981775i \(0.560864\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 26.0000 1.23114
\(447\) 0 0
\(448\) −2.00000 −0.0944911
\(449\) 12.0000 0.566315 0.283158 − 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −2.00000 −0.0940721
\(453\) 0 0
\(454\) 4.00000 0.187729
\(455\) 8.00000 0.375046
\(456\) 0 0
\(457\) −38.0000 −1.77757 −0.888783 − 0.458329i \(-0.848448\pi\)
−0.888783 + 0.458329i \(0.848448\pi\)
\(458\) −22.0000 −1.02799
\(459\) 0 0
\(460\) −1.00000 −0.0466252
\(461\) 18.0000 0.838344 0.419172 − 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) −10.0000 −0.464739 −0.232370 − 0.972628i \(-0.574648\pi\)
−0.232370 + 0.972628i \(0.574648\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) 24.0000 1.11059 0.555294 − 0.831654i \(-0.312606\pi\)
0.555294 + 0.831654i \(0.312606\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 8.00000 0.369012
\(471\) 0 0
\(472\) 2.00000 0.0920575
\(473\) 0 0
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 12.0000 0.550019
\(477\) 0 0
\(478\) −20.0000 −0.914779
\(479\) 36.0000 1.64488 0.822441 − 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) −2.00000 −0.0910975
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) −10.0000 −0.454077
\(486\) 0 0
\(487\) 14.0000 0.634401 0.317200 − 0.948359i \(-0.397257\pi\)
0.317200 + 0.948359i \(0.397257\pi\)
\(488\) 6.00000 0.271607
\(489\) 0 0
\(490\) 3.00000 0.135526
\(491\) −42.0000 −1.89543 −0.947717 − 0.319113i \(-0.896615\pi\)
−0.947717 + 0.319113i \(0.896615\pi\)
\(492\) 0 0
\(493\) 12.0000 0.540453
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 − 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 2.00000 0.0892644
\(503\) 24.0000 1.07011 0.535054 − 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) 10.0000 0.444994
\(506\) −2.00000 −0.0889108
\(507\) 0 0
\(508\) 2.00000 0.0887357
\(509\) 6.00000 0.265945 0.132973 − 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) −28.0000 −1.23865
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 18.0000 0.793946
\(515\) 2.00000 0.0881305
\(516\) 0 0
\(517\) 16.0000 0.703679
\(518\) 8.00000 0.351500
\(519\) 0 0
\(520\) −4.00000 −0.175412
\(521\) 24.0000 1.05146 0.525730 − 0.850652i \(-0.323792\pi\)
0.525730 + 0.850652i \(0.323792\pi\)
\(522\) 0 0
\(523\) 28.0000 1.22435 0.612177 − 0.790721i \(-0.290294\pi\)
0.612177 + 0.790721i \(0.290294\pi\)
\(524\) −10.0000 −0.436852
\(525\) 0 0
\(526\) 8.00000 0.348817
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 10.0000 0.434372
\(531\) 0 0
\(532\) 8.00000 0.346844
\(533\) 0 0
\(534\) 0 0
\(535\) 8.00000 0.345870
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) −18.0000 −0.776035
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) −14.0000 −0.601907 −0.300954 − 0.953639i \(-0.597305\pi\)
−0.300954 + 0.953639i \(0.597305\pi\)
\(542\) −32.0000 −1.37452
\(543\) 0 0
\(544\) −6.00000 −0.257248
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 − 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 6.00000 0.256307
\(549\) 0 0
\(550\) −2.00000 −0.0852803
\(551\) 8.00000 0.340811
\(552\) 0 0
\(553\) 16.0000 0.680389
\(554\) 24.0000 1.01966
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) 14.0000 0.593199 0.296600 − 0.955002i \(-0.404147\pi\)
0.296600 + 0.955002i \(0.404147\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) 16.0000 0.674919
\(563\) −36.0000 −1.51722 −0.758610 − 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 0 0
\(565\) 2.00000 0.0841406
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) −4.00000 −0.167836
\(569\) 8.00000 0.335377 0.167689 − 0.985840i \(-0.446370\pi\)
0.167689 + 0.985840i \(0.446370\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 − 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) −8.00000 −0.334497
\(573\) 0 0
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) 26.0000 1.08239 0.541197 − 0.840896i \(-0.317971\pi\)
0.541197 + 0.840896i \(0.317971\pi\)
\(578\) 19.0000 0.790296
\(579\) 0 0
\(580\) 2.00000 0.0830455
\(581\) 16.0000 0.663792
\(582\) 0 0
\(583\) 20.0000 0.828315
\(584\) 14.0000 0.579324
\(585\) 0 0
\(586\) 10.0000 0.413096
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) −2.00000 −0.0823387
\(591\) 0 0
\(592\) −4.00000 −0.164399
\(593\) −6.00000 −0.246390 −0.123195 − 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) 10.0000 0.409616
\(597\) 0 0
\(598\) 4.00000 0.163572
\(599\) −8.00000 −0.326871 −0.163436 − 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 − 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 8.00000 0.325515
\(605\) 7.00000 0.284590
\(606\) 0 0
\(607\) −42.0000 −1.70473 −0.852364 − 0.522949i \(-0.824832\pi\)
−0.852364 + 0.522949i \(0.824832\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) −6.00000 −0.242933
\(611\) −32.0000 −1.29458
\(612\) 0 0
\(613\) 4.00000 0.161558 0.0807792 − 0.996732i \(-0.474259\pi\)
0.0807792 + 0.996732i \(0.474259\pi\)
\(614\) −32.0000 −1.29141
\(615\) 0 0
\(616\) 4.00000 0.161165
\(617\) 18.0000 0.724653 0.362326 − 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 − 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 4.00000 0.160644
\(621\) 0 0
\(622\) −4.00000 −0.160385
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 14.0000 0.559553
\(627\) 0 0
\(628\) 16.0000 0.638470
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 − 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) −8.00000 −0.318223
\(633\) 0 0
\(634\) 18.0000 0.714871
\(635\) −2.00000 −0.0793676
\(636\) 0 0
\(637\) −12.0000 −0.475457
\(638\) 4.00000 0.158362
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −12.0000 −0.473972 −0.236986 − 0.971513i \(-0.576159\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) 0 0
\(643\) −20.0000 −0.788723 −0.394362 − 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) −2.00000 −0.0788110
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) −32.0000 −1.25805 −0.629025 − 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 0 0
\(649\) −4.00000 −0.157014
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) 18.0000 0.704394 0.352197 − 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 0 0
\(655\) 10.0000 0.390732
\(656\) 0 0
\(657\) 0 0
\(658\) 16.0000 0.623745
\(659\) 18.0000 0.701180 0.350590 − 0.936529i \(-0.385981\pi\)
0.350590 + 0.936529i \(0.385981\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 − 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −12.0000 −0.466393
\(663\) 0 0
\(664\) −8.00000 −0.310460
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) −2.00000 −0.0774403
\(668\) 8.00000 0.309529
\(669\) 0 0
\(670\) 4.00000 0.154533
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 − 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) −26.0000 −1.00148
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) −22.0000 −0.845529 −0.422764 − 0.906240i \(-0.638940\pi\)
−0.422764 + 0.906240i \(0.638940\pi\)
\(678\) 0 0
\(679\) −20.0000 −0.767530
\(680\) 6.00000 0.230089
\(681\) 0 0
\(682\) 8.00000 0.306336
\(683\) −36.0000 −1.37750 −0.688751 − 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) −6.00000 −0.229248
\(686\) 20.0000 0.763604
\(687\) 0 0
\(688\) 0 0
\(689\) −40.0000 −1.52388
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 − 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 14.0000 0.532200
\(693\) 0 0
\(694\) 24.0000 0.911028
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) 0 0
\(698\) 18.0000 0.681310
\(699\) 0 0
\(700\) −2.00000 −0.0755929
\(701\) 6.00000 0.226617 0.113308 − 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) 16.0000 0.603451
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) −2.00000 −0.0752710
\(707\) 20.0000 0.752177
\(708\) 0 0
\(709\) −14.0000 −0.525781 −0.262891 − 0.964826i \(-0.584676\pi\)
−0.262891 + 0.964826i \(0.584676\pi\)
\(710\) 4.00000 0.150117
\(711\) 0 0
\(712\) 0 0
\(713\) −4.00000 −0.149801
\(714\) 0 0
\(715\) 8.00000 0.299183
\(716\) 10.0000 0.373718
\(717\) 0 0
\(718\) −8.00000 −0.298557
\(719\) −40.0000 −1.49175 −0.745874 − 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) −3.00000 −0.111648
\(723\) 0 0
\(724\) −2.00000 −0.0743294
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) 2.00000 0.0741759 0.0370879 − 0.999312i \(-0.488192\pi\)
0.0370879 + 0.999312i \(0.488192\pi\)
\(728\) −8.00000 −0.296500
\(729\) 0 0
\(730\) −14.0000 −0.518163
\(731\) 0 0
\(732\) 0 0
\(733\) 44.0000 1.62518 0.812589 − 0.582838i \(-0.198058\pi\)
0.812589 + 0.582838i \(0.198058\pi\)
\(734\) −18.0000 −0.664392
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) −52.0000 −1.91285 −0.956425 − 0.291977i \(-0.905687\pi\)
−0.956425 + 0.291977i \(0.905687\pi\)
\(740\) 4.00000 0.147043
\(741\) 0 0
\(742\) 20.0000 0.734223
\(743\) 40.0000 1.46746 0.733729 − 0.679442i \(-0.237778\pi\)
0.733729 + 0.679442i \(0.237778\pi\)
\(744\) 0 0
\(745\) −10.0000 −0.366372
\(746\) −24.0000 −0.878702
\(747\) 0 0
\(748\) 12.0000 0.438763
\(749\) 16.0000 0.584627
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 − 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) −8.00000 −0.291730
\(753\) 0 0
\(754\) −8.00000 −0.291343
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) −20.0000 −0.726912 −0.363456 − 0.931611i \(-0.618403\pi\)
−0.363456 + 0.931611i \(0.618403\pi\)
\(758\) −4.00000 −0.145287
\(759\) 0 0
\(760\) 4.00000 0.145095
\(761\) −12.0000 −0.435000 −0.217500 − 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(762\) 0 0
\(763\) 4.00000 0.144810
\(764\) −20.0000 −0.723575
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) −34.0000 −1.22607 −0.613036 − 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) −4.00000 −0.144150
\(771\) 0 0
\(772\) 14.0000 0.503871
\(773\) 18.0000 0.647415 0.323708 − 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) 26.0000 0.932145
\(779\) 0 0
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) −6.00000 −0.214560
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) −16.0000 −0.571064
\(786\) 0 0
\(787\) −28.0000 −0.998092 −0.499046 − 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) −14.0000 −0.498729
\(789\) 0 0
\(790\) 8.00000 0.284627
\(791\) 4.00000 0.142224
\(792\) 0 0
\(793\) 24.0000 0.852265
\(794\) 8.00000 0.283909
\(795\) 0 0
\(796\) 28.0000 0.992434
\(797\) −42.0000 −1.48772 −0.743858 − 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) 48.0000 1.69812
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −32.0000 −1.12996
\(803\) −28.0000 −0.988099
\(804\) 0 0
\(805\) 2.00000 0.0704907
\(806\) −16.0000 −0.563576
\(807\) 0 0
\(808\) −10.0000 −0.351799
\(809\) −48.0000 −1.68759 −0.843795 − 0.536666i \(-0.819684\pi\)
−0.843795 + 0.536666i \(0.819684\pi\)
\(810\) 0 0
\(811\) −12.0000 −0.421377 −0.210688 − 0.977553i \(-0.567571\pi\)
−0.210688 + 0.977553i \(0.567571\pi\)
\(812\) 4.00000 0.140372
\(813\) 0 0
\(814\) 8.00000 0.280400
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) 0 0
\(818\) −22.0000 −0.769212
\(819\) 0 0
\(820\) 0 0
\(821\) 30.0000 1.04701 0.523504 − 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) −42.0000 −1.46403 −0.732014 − 0.681290i \(-0.761419\pi\)
−0.732014 + 0.681290i \(0.761419\pi\)
\(824\) −2.00000 −0.0696733
\(825\) 0 0
\(826\) −4.00000 −0.139178
\(827\) −12.0000 −0.417281 −0.208640 − 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) 34.0000 1.18087 0.590434 − 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 8.00000 0.277684
\(831\) 0 0
\(832\) 4.00000 0.138675
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) −8.00000 −0.276851
\(836\) 8.00000 0.276686
\(837\) 0 0
\(838\) 18.0000 0.621800
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 18.0000 0.620321
\(843\) 0 0
\(844\) 4.00000 0.137686
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) 14.0000 0.481046
\(848\) −10.0000 −0.343401
\(849\) 0 0
\(850\) −6.00000 −0.205798
\(851\) −4.00000 −0.137118
\(852\) 0 0
\(853\) 32.0000 1.09566 0.547830 − 0.836590i \(-0.315454\pi\)
0.547830 + 0.836590i \(0.315454\pi\)
\(854\) −12.0000 −0.410632
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) 30.0000 1.02478 0.512390 − 0.858753i \(-0.328760\pi\)
0.512390 + 0.858753i \(0.328760\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 − 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 16.0000 0.544962
\(863\) 24.0000 0.816970 0.408485 − 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) −14.0000 −0.476014
\(866\) 38.0000 1.29129
\(867\) 0 0
\(868\) 8.00000 0.271538
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) −2.00000 −0.0677285
\(873\) 0 0
\(874\) −4.00000 −0.135302
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) −28.0000 −0.945493 −0.472746 − 0.881199i \(-0.656737\pi\)
−0.472746 + 0.881199i \(0.656737\pi\)
\(878\) −8.00000 −0.269987
\(879\) 0 0
\(880\) 2.00000 0.0674200
\(881\) −48.0000 −1.61716 −0.808581 − 0.588386i \(-0.799764\pi\)
−0.808581 + 0.588386i \(0.799764\pi\)
\(882\) 0 0
\(883\) −4.00000 −0.134611 −0.0673054 − 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) −24.0000 −0.807207
\(885\) 0 0
\(886\) −8.00000 −0.268765
\(887\) −24.0000 −0.805841 −0.402921 − 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 0 0
\(889\) −4.00000 −0.134156
\(890\) 0 0
\(891\) 0 0
\(892\) 26.0000 0.870544
\(893\) 32.0000 1.07084
\(894\) 0 0
\(895\) −10.0000 −0.334263
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) 12.0000 0.400445
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) 60.0000 1.99889
\(902\) 0 0
\(903\) 0 0
\(904\) −2.00000 −0.0665190
\(905\) 2.00000 0.0664822
\(906\) 0 0
\(907\) 16.0000 0.531271 0.265636 − 0.964073i \(-0.414418\pi\)
0.265636 + 0.964073i \(0.414418\pi\)
\(908\) 4.00000 0.132745
\(909\) 0 0
\(910\) 8.00000 0.265197
\(911\) −48.0000 −1.59031 −0.795155 − 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 0 0
\(913\) 16.0000 0.529523
\(914\) −38.0000 −1.25693
\(915\) 0 0
\(916\) −22.0000 −0.726900
\(917\) 20.0000 0.660458
\(918\) 0 0
\(919\) −40.0000 −1.31948 −0.659739 − 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 0 0
\(922\) 18.0000 0.592798
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) −10.0000 −0.328620
\(927\) 0 0
\(928\) −2.00000 −0.0656532
\(929\) −36.0000 −1.18112 −0.590561 − 0.806993i \(-0.701093\pi\)
−0.590561 + 0.806993i \(0.701093\pi\)
\(930\) 0 0
\(931\) 12.0000 0.393284
\(932\) −18.0000 −0.589610
\(933\) 0 0
\(934\) 24.0000 0.785304
\(935\) −12.0000 −0.392442
\(936\) 0 0
\(937\) −54.0000 −1.76410 −0.882052 − 0.471153i \(-0.843838\pi\)
−0.882052 + 0.471153i \(0.843838\pi\)
\(938\) 8.00000 0.261209
\(939\) 0 0
\(940\) 8.00000 0.260931
\(941\) 34.0000 1.10837 0.554184 − 0.832394i \(-0.313030\pi\)
0.554184 + 0.832394i \(0.313030\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 2.00000 0.0650945
\(945\) 0 0
\(946\) 0 0
\(947\) 32.0000 1.03986 0.519930 − 0.854209i \(-0.325958\pi\)
0.519930 + 0.854209i \(0.325958\pi\)
\(948\) 0 0
\(949\) 56.0000 1.81784
\(950\) −4.00000 −0.129777
\(951\) 0 0
\(952\) 12.0000 0.388922
\(953\) 22.0000 0.712650 0.356325 − 0.934362i \(-0.384030\pi\)
0.356325 + 0.934362i \(0.384030\pi\)
\(954\) 0 0
\(955\) 20.0000 0.647185
\(956\) −20.0000 −0.646846
\(957\) 0 0
\(958\) 36.0000 1.16311
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −16.0000 −0.515861
\(963\) 0 0
\(964\) −2.00000 −0.0644157
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) 58.0000 1.86515 0.932577 − 0.360971i \(-0.117555\pi\)
0.932577 + 0.360971i \(0.117555\pi\)
\(968\) −7.00000 −0.224989
\(969\) 0 0
\(970\) −10.0000 −0.321081
\(971\) −30.0000 −0.962746 −0.481373 − 0.876516i \(-0.659862\pi\)
−0.481373 + 0.876516i \(0.659862\pi\)
\(972\) 0 0
\(973\) −8.00000 −0.256468
\(974\) 14.0000 0.448589
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) 2.00000 0.0639857 0.0319928 − 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 3.00000 0.0958315
\(981\) 0 0
\(982\) −42.0000 −1.34027
\(983\) 24.0000 0.765481 0.382741 − 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) 14.0000 0.446077
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) −16.0000 −0.509028
\(989\) 0 0
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 − 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) −4.00000 −0.127000
\(993\) 0 0
\(994\) 8.00000 0.253745
\(995\) −28.0000 −0.887660
\(996\) 0 0
\(997\) 28.0000 0.886769 0.443384 − 0.896332i \(-0.353778\pi\)
0.443384 + 0.896332i \(0.353778\pi\)
\(998\) −4.00000 −0.126618
\(999\) 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 2070.2.a.l.1.1 yes 1
3.2 odd 2 2070.2.a.f.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2070.2.a.f.1.1 ✓ 1 3.2 odd 2
2070.2.a.l.1.1 yes 1 1.1 even 1 trivial