Properties

Label 8020.2.a.e.1.17
Level $8020$
Weight $2$
Character 8020.1
Self dual yes
Analytic conductor $64.040$
Analytic rank $0$
Dimension $35$
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: \(35\)
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)
 
\(f(q)\) \(=\) \(q-0.478839 q^{3} -1.00000 q^{5} +1.59155 q^{7} -2.77071 q^{9} +O(q^{10})\) \(q-0.478839 q^{3} -1.00000 q^{5} +1.59155 q^{7} -2.77071 q^{9} -6.51847 q^{11} -3.37511 q^{13} +0.478839 q^{15} -3.23762 q^{17} -0.00948756 q^{19} -0.762097 q^{21} +8.46543 q^{23} +1.00000 q^{25} +2.76324 q^{27} +0.667812 q^{29} -2.17937 q^{31} +3.12130 q^{33} -1.59155 q^{35} -9.25603 q^{37} +1.61613 q^{39} -5.04803 q^{41} -5.75194 q^{43} +2.77071 q^{45} +3.87332 q^{47} -4.46697 q^{49} +1.55030 q^{51} +3.05072 q^{53} +6.51847 q^{55} +0.00454302 q^{57} -10.2701 q^{59} -13.0510 q^{61} -4.40973 q^{63} +3.37511 q^{65} +2.12502 q^{67} -4.05358 q^{69} +3.04241 q^{71} +1.95988 q^{73} -0.478839 q^{75} -10.3745 q^{77} +8.09588 q^{79} +6.98899 q^{81} -1.98216 q^{83} +3.23762 q^{85} -0.319775 q^{87} -9.89121 q^{89} -5.37165 q^{91} +1.04357 q^{93} +0.00948756 q^{95} +3.02485 q^{97} +18.0608 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9} - 2 q^{11} - q^{13} + q^{15} + 18 q^{17} + 2 q^{19} + 12 q^{21} + 13 q^{23} + 35 q^{25} - 7 q^{27} + 25 q^{29} + 13 q^{31} + 14 q^{33} - 6 q^{35} - 19 q^{37} - 3 q^{39} + 24 q^{41} - 5 q^{43} - 52 q^{45} + 19 q^{47} + 55 q^{49} + 41 q^{53} + 2 q^{55} + 14 q^{57} + 3 q^{59} + 13 q^{61} + 70 q^{63} + q^{65} - 17 q^{67} + 64 q^{69} + 17 q^{71} - 63 q^{73} - q^{75} + 54 q^{77} + 11 q^{79} + 107 q^{81} - 8 q^{83} - 18 q^{85} + 36 q^{87} + 38 q^{89} - 27 q^{91} + q^{93} - 2 q^{95} - 54 q^{97} - 51 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.478839 −0.276458 −0.138229 0.990400i \(-0.544141\pi\)
−0.138229 + 0.990400i \(0.544141\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.59155 0.601550 0.300775 0.953695i \(-0.402755\pi\)
0.300775 + 0.953695i \(0.402755\pi\)
\(8\) 0 0
\(9\) −2.77071 −0.923571
\(10\) 0 0
\(11\) −6.51847 −1.96539 −0.982697 0.185221i \(-0.940700\pi\)
−0.982697 + 0.185221i \(0.940700\pi\)
\(12\) 0 0
\(13\) −3.37511 −0.936086 −0.468043 0.883706i \(-0.655041\pi\)
−0.468043 + 0.883706i \(0.655041\pi\)
\(14\) 0 0
\(15\) 0.478839 0.123636
\(16\) 0 0
\(17\) −3.23762 −0.785237 −0.392619 0.919701i \(-0.628431\pi\)
−0.392619 + 0.919701i \(0.628431\pi\)
\(18\) 0 0
\(19\) −0.00948756 −0.00217660 −0.00108830 0.999999i \(-0.500346\pi\)
−0.00108830 + 0.999999i \(0.500346\pi\)
\(20\) 0 0
\(21\) −0.762097 −0.166303
\(22\) 0 0
\(23\) 8.46543 1.76516 0.882582 0.470158i \(-0.155803\pi\)
0.882582 + 0.470158i \(0.155803\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 2.76324 0.531787
\(28\) 0 0
\(29\) 0.667812 0.124010 0.0620048 0.998076i \(-0.480251\pi\)
0.0620048 + 0.998076i \(0.480251\pi\)
\(30\) 0 0
\(31\) −2.17937 −0.391426 −0.195713 0.980661i \(-0.562702\pi\)
−0.195713 + 0.980661i \(0.562702\pi\)
\(32\) 0 0
\(33\) 3.12130 0.543349
\(34\) 0 0
\(35\) −1.59155 −0.269021
\(36\) 0 0
\(37\) −9.25603 −1.52168 −0.760841 0.648938i \(-0.775213\pi\)
−0.760841 + 0.648938i \(0.775213\pi\)
\(38\) 0 0
\(39\) 1.61613 0.258789
\(40\) 0 0
\(41\) −5.04803 −0.788369 −0.394185 0.919031i \(-0.628973\pi\)
−0.394185 + 0.919031i \(0.628973\pi\)
\(42\) 0 0
\(43\) −5.75194 −0.877163 −0.438582 0.898691i \(-0.644519\pi\)
−0.438582 + 0.898691i \(0.644519\pi\)
\(44\) 0 0
\(45\) 2.77071 0.413033
\(46\) 0 0
\(47\) 3.87332 0.564982 0.282491 0.959270i \(-0.408839\pi\)
0.282491 + 0.959270i \(0.408839\pi\)
\(48\) 0 0
\(49\) −4.46697 −0.638138
\(50\) 0 0
\(51\) 1.55030 0.217085
\(52\) 0 0
\(53\) 3.05072 0.419049 0.209525 0.977803i \(-0.432808\pi\)
0.209525 + 0.977803i \(0.432808\pi\)
\(54\) 0 0
\(55\) 6.51847 0.878951
\(56\) 0 0
\(57\) 0.00454302 0.000601738 0
\(58\) 0 0
\(59\) −10.2701 −1.33705 −0.668524 0.743690i \(-0.733074\pi\)
−0.668524 + 0.743690i \(0.733074\pi\)
\(60\) 0 0
\(61\) −13.0510 −1.67101 −0.835505 0.549483i \(-0.814825\pi\)
−0.835505 + 0.549483i \(0.814825\pi\)
\(62\) 0 0
\(63\) −4.40973 −0.555574
\(64\) 0 0
\(65\) 3.37511 0.418630
\(66\) 0 0
\(67\) 2.12502 0.259612 0.129806 0.991539i \(-0.458564\pi\)
0.129806 + 0.991539i \(0.458564\pi\)
\(68\) 0 0
\(69\) −4.05358 −0.487994
\(70\) 0 0
\(71\) 3.04241 0.361067 0.180534 0.983569i \(-0.442218\pi\)
0.180534 + 0.983569i \(0.442218\pi\)
\(72\) 0 0
\(73\) 1.95988 0.229387 0.114693 0.993401i \(-0.463411\pi\)
0.114693 + 0.993401i \(0.463411\pi\)
\(74\) 0 0
\(75\) −0.478839 −0.0552916
\(76\) 0 0
\(77\) −10.3745 −1.18228
\(78\) 0 0
\(79\) 8.09588 0.910857 0.455429 0.890272i \(-0.349486\pi\)
0.455429 + 0.890272i \(0.349486\pi\)
\(80\) 0 0
\(81\) 6.98899 0.776554
\(82\) 0 0
\(83\) −1.98216 −0.217570 −0.108785 0.994065i \(-0.534696\pi\)
−0.108785 + 0.994065i \(0.534696\pi\)
\(84\) 0 0
\(85\) 3.23762 0.351169
\(86\) 0 0
\(87\) −0.319775 −0.0342835
\(88\) 0 0
\(89\) −9.89121 −1.04847 −0.524233 0.851575i \(-0.675648\pi\)
−0.524233 + 0.851575i \(0.675648\pi\)
\(90\) 0 0
\(91\) −5.37165 −0.563102
\(92\) 0 0
\(93\) 1.04357 0.108213
\(94\) 0 0
\(95\) 0.00948756 0.000973403 0
\(96\) 0 0
\(97\) 3.02485 0.307127 0.153564 0.988139i \(-0.450925\pi\)
0.153564 + 0.988139i \(0.450925\pi\)
\(98\) 0 0
\(99\) 18.0608 1.81518
\(100\) 0 0
\(101\) −11.6742 −1.16162 −0.580812 0.814038i \(-0.697265\pi\)
−0.580812 + 0.814038i \(0.697265\pi\)
\(102\) 0 0
\(103\) −8.03823 −0.792031 −0.396015 0.918244i \(-0.629607\pi\)
−0.396015 + 0.918244i \(0.629607\pi\)
\(104\) 0 0
\(105\) 0.762097 0.0743731
\(106\) 0 0
\(107\) 19.8919 1.92302 0.961511 0.274766i \(-0.0886004\pi\)
0.961511 + 0.274766i \(0.0886004\pi\)
\(108\) 0 0
\(109\) 13.5363 1.29654 0.648270 0.761410i \(-0.275493\pi\)
0.648270 + 0.761410i \(0.275493\pi\)
\(110\) 0 0
\(111\) 4.43215 0.420681
\(112\) 0 0
\(113\) 7.38594 0.694810 0.347405 0.937715i \(-0.387063\pi\)
0.347405 + 0.937715i \(0.387063\pi\)
\(114\) 0 0
\(115\) −8.46543 −0.789405
\(116\) 0 0
\(117\) 9.35145 0.864542
\(118\) 0 0
\(119\) −5.15283 −0.472359
\(120\) 0 0
\(121\) 31.4905 2.86277
\(122\) 0 0
\(123\) 2.41719 0.217951
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 1.93979 0.172128 0.0860642 0.996290i \(-0.472571\pi\)
0.0860642 + 0.996290i \(0.472571\pi\)
\(128\) 0 0
\(129\) 2.75426 0.242499
\(130\) 0 0
\(131\) −12.2579 −1.07098 −0.535490 0.844541i \(-0.679873\pi\)
−0.535490 + 0.844541i \(0.679873\pi\)
\(132\) 0 0
\(133\) −0.0150999 −0.00130933
\(134\) 0 0
\(135\) −2.76324 −0.237822
\(136\) 0 0
\(137\) 12.3710 1.05693 0.528463 0.848956i \(-0.322768\pi\)
0.528463 + 0.848956i \(0.322768\pi\)
\(138\) 0 0
\(139\) 2.21075 0.187513 0.0937565 0.995595i \(-0.470112\pi\)
0.0937565 + 0.995595i \(0.470112\pi\)
\(140\) 0 0
\(141\) −1.85470 −0.156194
\(142\) 0 0
\(143\) 22.0005 1.83978
\(144\) 0 0
\(145\) −0.667812 −0.0554588
\(146\) 0 0
\(147\) 2.13896 0.176418
\(148\) 0 0
\(149\) 2.91615 0.238900 0.119450 0.992840i \(-0.461887\pi\)
0.119450 + 0.992840i \(0.461887\pi\)
\(150\) 0 0
\(151\) −21.1186 −1.71861 −0.859303 0.511466i \(-0.829102\pi\)
−0.859303 + 0.511466i \(0.829102\pi\)
\(152\) 0 0
\(153\) 8.97050 0.725222
\(154\) 0 0
\(155\) 2.17937 0.175051
\(156\) 0 0
\(157\) −11.3435 −0.905314 −0.452657 0.891685i \(-0.649524\pi\)
−0.452657 + 0.891685i \(0.649524\pi\)
\(158\) 0 0
\(159\) −1.46081 −0.115849
\(160\) 0 0
\(161\) 13.4732 1.06183
\(162\) 0 0
\(163\) 14.0049 1.09695 0.548473 0.836168i \(-0.315209\pi\)
0.548473 + 0.836168i \(0.315209\pi\)
\(164\) 0 0
\(165\) −3.12130 −0.242993
\(166\) 0 0
\(167\) 0.315144 0.0243866 0.0121933 0.999926i \(-0.496119\pi\)
0.0121933 + 0.999926i \(0.496119\pi\)
\(168\) 0 0
\(169\) −1.60865 −0.123743
\(170\) 0 0
\(171\) 0.0262873 0.00201024
\(172\) 0 0
\(173\) 17.9627 1.36568 0.682840 0.730568i \(-0.260744\pi\)
0.682840 + 0.730568i \(0.260744\pi\)
\(174\) 0 0
\(175\) 1.59155 0.120310
\(176\) 0 0
\(177\) 4.91771 0.369638
\(178\) 0 0
\(179\) −12.5698 −0.939514 −0.469757 0.882796i \(-0.655659\pi\)
−0.469757 + 0.882796i \(0.655659\pi\)
\(180\) 0 0
\(181\) 7.08922 0.526938 0.263469 0.964668i \(-0.415133\pi\)
0.263469 + 0.964668i \(0.415133\pi\)
\(182\) 0 0
\(183\) 6.24934 0.461964
\(184\) 0 0
\(185\) 9.25603 0.680517
\(186\) 0 0
\(187\) 21.1043 1.54330
\(188\) 0 0
\(189\) 4.39784 0.319896
\(190\) 0 0
\(191\) 16.9385 1.22563 0.612815 0.790227i \(-0.290037\pi\)
0.612815 + 0.790227i \(0.290037\pi\)
\(192\) 0 0
\(193\) 4.43727 0.319402 0.159701 0.987165i \(-0.448947\pi\)
0.159701 + 0.987165i \(0.448947\pi\)
\(194\) 0 0
\(195\) −1.61613 −0.115734
\(196\) 0 0
\(197\) −20.0520 −1.42864 −0.714321 0.699818i \(-0.753265\pi\)
−0.714321 + 0.699818i \(0.753265\pi\)
\(198\) 0 0
\(199\) 24.0566 1.70532 0.852662 0.522463i \(-0.174987\pi\)
0.852662 + 0.522463i \(0.174987\pi\)
\(200\) 0 0
\(201\) −1.01754 −0.0717720
\(202\) 0 0
\(203\) 1.06286 0.0745980
\(204\) 0 0
\(205\) 5.04803 0.352569
\(206\) 0 0
\(207\) −23.4553 −1.63025
\(208\) 0 0
\(209\) 0.0618444 0.00427787
\(210\) 0 0
\(211\) 26.6275 1.83311 0.916555 0.399908i \(-0.130958\pi\)
0.916555 + 0.399908i \(0.130958\pi\)
\(212\) 0 0
\(213\) −1.45683 −0.0998200
\(214\) 0 0
\(215\) 5.75194 0.392279
\(216\) 0 0
\(217\) −3.46858 −0.235462
\(218\) 0 0
\(219\) −0.938469 −0.0634158
\(220\) 0 0
\(221\) 10.9273 0.735050
\(222\) 0 0
\(223\) 12.5388 0.839661 0.419831 0.907602i \(-0.362089\pi\)
0.419831 + 0.907602i \(0.362089\pi\)
\(224\) 0 0
\(225\) −2.77071 −0.184714
\(226\) 0 0
\(227\) 12.9761 0.861257 0.430628 0.902529i \(-0.358292\pi\)
0.430628 + 0.902529i \(0.358292\pi\)
\(228\) 0 0
\(229\) −25.8603 −1.70890 −0.854448 0.519537i \(-0.826105\pi\)
−0.854448 + 0.519537i \(0.826105\pi\)
\(230\) 0 0
\(231\) 4.96771 0.326851
\(232\) 0 0
\(233\) 18.6146 1.21948 0.609741 0.792601i \(-0.291274\pi\)
0.609741 + 0.792601i \(0.291274\pi\)
\(234\) 0 0
\(235\) −3.87332 −0.252668
\(236\) 0 0
\(237\) −3.87663 −0.251814
\(238\) 0 0
\(239\) −12.5495 −0.811762 −0.405881 0.913926i \(-0.633035\pi\)
−0.405881 + 0.913926i \(0.633035\pi\)
\(240\) 0 0
\(241\) 15.8773 1.02275 0.511374 0.859358i \(-0.329137\pi\)
0.511374 + 0.859358i \(0.329137\pi\)
\(242\) 0 0
\(243\) −11.6363 −0.746471
\(244\) 0 0
\(245\) 4.46697 0.285384
\(246\) 0 0
\(247\) 0.0320215 0.00203748
\(248\) 0 0
\(249\) 0.949135 0.0601490
\(250\) 0 0
\(251\) 23.9721 1.51311 0.756554 0.653931i \(-0.226881\pi\)
0.756554 + 0.653931i \(0.226881\pi\)
\(252\) 0 0
\(253\) −55.1817 −3.46924
\(254\) 0 0
\(255\) −1.55030 −0.0970834
\(256\) 0 0
\(257\) 5.06832 0.316153 0.158077 0.987427i \(-0.449471\pi\)
0.158077 + 0.987427i \(0.449471\pi\)
\(258\) 0 0
\(259\) −14.7314 −0.915367
\(260\) 0 0
\(261\) −1.85032 −0.114532
\(262\) 0 0
\(263\) −11.2226 −0.692016 −0.346008 0.938232i \(-0.612463\pi\)
−0.346008 + 0.938232i \(0.612463\pi\)
\(264\) 0 0
\(265\) −3.05072 −0.187404
\(266\) 0 0
\(267\) 4.73630 0.289857
\(268\) 0 0
\(269\) 11.0188 0.671826 0.335913 0.941893i \(-0.390955\pi\)
0.335913 + 0.941893i \(0.390955\pi\)
\(270\) 0 0
\(271\) −2.62774 −0.159624 −0.0798120 0.996810i \(-0.525432\pi\)
−0.0798120 + 0.996810i \(0.525432\pi\)
\(272\) 0 0
\(273\) 2.57216 0.155674
\(274\) 0 0
\(275\) −6.51847 −0.393079
\(276\) 0 0
\(277\) 21.3754 1.28433 0.642163 0.766569i \(-0.278037\pi\)
0.642163 + 0.766569i \(0.278037\pi\)
\(278\) 0 0
\(279\) 6.03841 0.361510
\(280\) 0 0
\(281\) 9.70369 0.578873 0.289437 0.957197i \(-0.406532\pi\)
0.289437 + 0.957197i \(0.406532\pi\)
\(282\) 0 0
\(283\) −21.1599 −1.25783 −0.628914 0.777475i \(-0.716500\pi\)
−0.628914 + 0.777475i \(0.716500\pi\)
\(284\) 0 0
\(285\) −0.00454302 −0.000269105 0
\(286\) 0 0
\(287\) −8.03419 −0.474243
\(288\) 0 0
\(289\) −6.51785 −0.383403
\(290\) 0 0
\(291\) −1.44842 −0.0849078
\(292\) 0 0
\(293\) −22.1188 −1.29219 −0.646096 0.763256i \(-0.723599\pi\)
−0.646096 + 0.763256i \(0.723599\pi\)
\(294\) 0 0
\(295\) 10.2701 0.597946
\(296\) 0 0
\(297\) −18.0121 −1.04517
\(298\) 0 0
\(299\) −28.5717 −1.65235
\(300\) 0 0
\(301\) −9.15451 −0.527657
\(302\) 0 0
\(303\) 5.59006 0.321140
\(304\) 0 0
\(305\) 13.0510 0.747298
\(306\) 0 0
\(307\) −14.5312 −0.829340 −0.414670 0.909972i \(-0.636103\pi\)
−0.414670 + 0.909972i \(0.636103\pi\)
\(308\) 0 0
\(309\) 3.84902 0.218963
\(310\) 0 0
\(311\) −17.7278 −1.00525 −0.502625 0.864505i \(-0.667632\pi\)
−0.502625 + 0.864505i \(0.667632\pi\)
\(312\) 0 0
\(313\) 25.5951 1.44672 0.723360 0.690471i \(-0.242597\pi\)
0.723360 + 0.690471i \(0.242597\pi\)
\(314\) 0 0
\(315\) 4.40973 0.248460
\(316\) 0 0
\(317\) 16.4433 0.923546 0.461773 0.886998i \(-0.347213\pi\)
0.461773 + 0.886998i \(0.347213\pi\)
\(318\) 0 0
\(319\) −4.35312 −0.243728
\(320\) 0 0
\(321\) −9.52502 −0.531635
\(322\) 0 0
\(323\) 0.0307171 0.00170914
\(324\) 0 0
\(325\) −3.37511 −0.187217
\(326\) 0 0
\(327\) −6.48171 −0.358439
\(328\) 0 0
\(329\) 6.16459 0.339865
\(330\) 0 0
\(331\) −7.06819 −0.388503 −0.194251 0.980952i \(-0.562228\pi\)
−0.194251 + 0.980952i \(0.562228\pi\)
\(332\) 0 0
\(333\) 25.6458 1.40538
\(334\) 0 0
\(335\) −2.12502 −0.116102
\(336\) 0 0
\(337\) −1.49664 −0.0815274 −0.0407637 0.999169i \(-0.512979\pi\)
−0.0407637 + 0.999169i \(0.512979\pi\)
\(338\) 0 0
\(339\) −3.53668 −0.192086
\(340\) 0 0
\(341\) 14.2062 0.769307
\(342\) 0 0
\(343\) −18.2503 −0.985421
\(344\) 0 0
\(345\) 4.05358 0.218238
\(346\) 0 0
\(347\) 20.1701 1.08279 0.541395 0.840768i \(-0.317896\pi\)
0.541395 + 0.840768i \(0.317896\pi\)
\(348\) 0 0
\(349\) −1.25271 −0.0670563 −0.0335281 0.999438i \(-0.510674\pi\)
−0.0335281 + 0.999438i \(0.510674\pi\)
\(350\) 0 0
\(351\) −9.32625 −0.497798
\(352\) 0 0
\(353\) 1.07028 0.0569654 0.0284827 0.999594i \(-0.490932\pi\)
0.0284827 + 0.999594i \(0.490932\pi\)
\(354\) 0 0
\(355\) −3.04241 −0.161474
\(356\) 0 0
\(357\) 2.46738 0.130587
\(358\) 0 0
\(359\) −7.35021 −0.387929 −0.193965 0.981009i \(-0.562135\pi\)
−0.193965 + 0.981009i \(0.562135\pi\)
\(360\) 0 0
\(361\) −18.9999 −0.999995
\(362\) 0 0
\(363\) −15.0789 −0.791437
\(364\) 0 0
\(365\) −1.95988 −0.102585
\(366\) 0 0
\(367\) −32.5066 −1.69683 −0.848417 0.529329i \(-0.822444\pi\)
−0.848417 + 0.529329i \(0.822444\pi\)
\(368\) 0 0
\(369\) 13.9866 0.728115
\(370\) 0 0
\(371\) 4.85538 0.252079
\(372\) 0 0
\(373\) −16.5269 −0.855729 −0.427864 0.903843i \(-0.640734\pi\)
−0.427864 + 0.903843i \(0.640734\pi\)
\(374\) 0 0
\(375\) 0.478839 0.0247272
\(376\) 0 0
\(377\) −2.25394 −0.116084
\(378\) 0 0
\(379\) −7.95294 −0.408515 −0.204258 0.978917i \(-0.565478\pi\)
−0.204258 + 0.978917i \(0.565478\pi\)
\(380\) 0 0
\(381\) −0.928848 −0.0475863
\(382\) 0 0
\(383\) 35.3368 1.80563 0.902813 0.430033i \(-0.141498\pi\)
0.902813 + 0.430033i \(0.141498\pi\)
\(384\) 0 0
\(385\) 10.3745 0.528733
\(386\) 0 0
\(387\) 15.9370 0.810122
\(388\) 0 0
\(389\) −5.32277 −0.269875 −0.134937 0.990854i \(-0.543083\pi\)
−0.134937 + 0.990854i \(0.543083\pi\)
\(390\) 0 0
\(391\) −27.4078 −1.38607
\(392\) 0 0
\(393\) 5.86958 0.296081
\(394\) 0 0
\(395\) −8.09588 −0.407348
\(396\) 0 0
\(397\) −0.276063 −0.0138552 −0.00692761 0.999976i \(-0.502205\pi\)
−0.00692761 + 0.999976i \(0.502205\pi\)
\(398\) 0 0
\(399\) 0.00723044 0.000361975 0
\(400\) 0 0
\(401\) −1.00000 −0.0499376
\(402\) 0 0
\(403\) 7.35561 0.366409
\(404\) 0 0
\(405\) −6.98899 −0.347286
\(406\) 0 0
\(407\) 60.3352 2.99070
\(408\) 0 0
\(409\) 9.42243 0.465909 0.232955 0.972488i \(-0.425161\pi\)
0.232955 + 0.972488i \(0.425161\pi\)
\(410\) 0 0
\(411\) −5.92373 −0.292196
\(412\) 0 0
\(413\) −16.3453 −0.804301
\(414\) 0 0
\(415\) 1.98216 0.0973002
\(416\) 0 0
\(417\) −1.05859 −0.0518395
\(418\) 0 0
\(419\) −19.3124 −0.943473 −0.471737 0.881739i \(-0.656373\pi\)
−0.471737 + 0.881739i \(0.656373\pi\)
\(420\) 0 0
\(421\) 31.8066 1.55016 0.775079 0.631865i \(-0.217710\pi\)
0.775079 + 0.631865i \(0.217710\pi\)
\(422\) 0 0
\(423\) −10.7319 −0.521801
\(424\) 0 0
\(425\) −3.23762 −0.157047
\(426\) 0 0
\(427\) −20.7713 −1.00520
\(428\) 0 0
\(429\) −10.5347 −0.508621
\(430\) 0 0
\(431\) 23.3343 1.12398 0.561988 0.827145i \(-0.310037\pi\)
0.561988 + 0.827145i \(0.310037\pi\)
\(432\) 0 0
\(433\) −36.3387 −1.74632 −0.873162 0.487429i \(-0.837935\pi\)
−0.873162 + 0.487429i \(0.837935\pi\)
\(434\) 0 0
\(435\) 0.319775 0.0153320
\(436\) 0 0
\(437\) −0.0803163 −0.00384205
\(438\) 0 0
\(439\) −21.5069 −1.02647 −0.513234 0.858249i \(-0.671553\pi\)
−0.513234 + 0.858249i \(0.671553\pi\)
\(440\) 0 0
\(441\) 12.3767 0.589366
\(442\) 0 0
\(443\) −6.38206 −0.303221 −0.151610 0.988440i \(-0.548446\pi\)
−0.151610 + 0.988440i \(0.548446\pi\)
\(444\) 0 0
\(445\) 9.89121 0.468888
\(446\) 0 0
\(447\) −1.39637 −0.0660459
\(448\) 0 0
\(449\) 14.6905 0.693288 0.346644 0.937997i \(-0.387321\pi\)
0.346644 + 0.937997i \(0.387321\pi\)
\(450\) 0 0
\(451\) 32.9054 1.54946
\(452\) 0 0
\(453\) 10.1124 0.475123
\(454\) 0 0
\(455\) 5.37165 0.251827
\(456\) 0 0
\(457\) −10.1232 −0.473544 −0.236772 0.971565i \(-0.576089\pi\)
−0.236772 + 0.971565i \(0.576089\pi\)
\(458\) 0 0
\(459\) −8.94632 −0.417579
\(460\) 0 0
\(461\) 33.3021 1.55103 0.775516 0.631328i \(-0.217490\pi\)
0.775516 + 0.631328i \(0.217490\pi\)
\(462\) 0 0
\(463\) 32.3006 1.50113 0.750567 0.660794i \(-0.229780\pi\)
0.750567 + 0.660794i \(0.229780\pi\)
\(464\) 0 0
\(465\) −1.04357 −0.0483943
\(466\) 0 0
\(467\) 36.2767 1.67869 0.839343 0.543602i \(-0.182940\pi\)
0.839343 + 0.543602i \(0.182940\pi\)
\(468\) 0 0
\(469\) 3.38208 0.156170
\(470\) 0 0
\(471\) 5.43174 0.250281
\(472\) 0 0
\(473\) 37.4939 1.72397
\(474\) 0 0
\(475\) −0.00948756 −0.000435319 0
\(476\) 0 0
\(477\) −8.45268 −0.387022
\(478\) 0 0
\(479\) 4.42794 0.202318 0.101159 0.994870i \(-0.467745\pi\)
0.101159 + 0.994870i \(0.467745\pi\)
\(480\) 0 0
\(481\) 31.2401 1.42443
\(482\) 0 0
\(483\) −6.45148 −0.293553
\(484\) 0 0
\(485\) −3.02485 −0.137351
\(486\) 0 0
\(487\) 16.1553 0.732066 0.366033 0.930602i \(-0.380716\pi\)
0.366033 + 0.930602i \(0.380716\pi\)
\(488\) 0 0
\(489\) −6.70609 −0.303260
\(490\) 0 0
\(491\) 28.4388 1.28343 0.641713 0.766945i \(-0.278224\pi\)
0.641713 + 0.766945i \(0.278224\pi\)
\(492\) 0 0
\(493\) −2.16212 −0.0973770
\(494\) 0 0
\(495\) −18.0608 −0.811773
\(496\) 0 0
\(497\) 4.84215 0.217200
\(498\) 0 0
\(499\) −41.8276 −1.87246 −0.936231 0.351385i \(-0.885711\pi\)
−0.936231 + 0.351385i \(0.885711\pi\)
\(500\) 0 0
\(501\) −0.150903 −0.00674186
\(502\) 0 0
\(503\) −15.8979 −0.708854 −0.354427 0.935084i \(-0.615324\pi\)
−0.354427 + 0.935084i \(0.615324\pi\)
\(504\) 0 0
\(505\) 11.6742 0.519494
\(506\) 0 0
\(507\) 0.770287 0.0342097
\(508\) 0 0
\(509\) 25.5133 1.13086 0.565428 0.824798i \(-0.308711\pi\)
0.565428 + 0.824798i \(0.308711\pi\)
\(510\) 0 0
\(511\) 3.11925 0.137988
\(512\) 0 0
\(513\) −0.0262165 −0.00115748
\(514\) 0 0
\(515\) 8.03823 0.354207
\(516\) 0 0
\(517\) −25.2482 −1.11041
\(518\) 0 0
\(519\) −8.60126 −0.377554
\(520\) 0 0
\(521\) 17.3889 0.761822 0.380911 0.924612i \(-0.375610\pi\)
0.380911 + 0.924612i \(0.375610\pi\)
\(522\) 0 0
\(523\) −1.27388 −0.0557031 −0.0278516 0.999612i \(-0.508867\pi\)
−0.0278516 + 0.999612i \(0.508867\pi\)
\(524\) 0 0
\(525\) −0.762097 −0.0332606
\(526\) 0 0
\(527\) 7.05596 0.307362
\(528\) 0 0
\(529\) 48.6635 2.11581
\(530\) 0 0
\(531\) 28.4554 1.23486
\(532\) 0 0
\(533\) 17.0376 0.737982
\(534\) 0 0
\(535\) −19.8919 −0.860002
\(536\) 0 0
\(537\) 6.01894 0.259736
\(538\) 0 0
\(539\) 29.1178 1.25419
\(540\) 0 0
\(541\) −0.532136 −0.0228783 −0.0114392 0.999935i \(-0.503641\pi\)
−0.0114392 + 0.999935i \(0.503641\pi\)
\(542\) 0 0
\(543\) −3.39460 −0.145676
\(544\) 0 0
\(545\) −13.5363 −0.579831
\(546\) 0 0
\(547\) 18.1102 0.774335 0.387167 0.922009i \(-0.373454\pi\)
0.387167 + 0.922009i \(0.373454\pi\)
\(548\) 0 0
\(549\) 36.1606 1.54330
\(550\) 0 0
\(551\) −0.00633591 −0.000269919 0
\(552\) 0 0
\(553\) 12.8850 0.547926
\(554\) 0 0
\(555\) −4.43215 −0.188134
\(556\) 0 0
\(557\) −17.4203 −0.738124 −0.369062 0.929405i \(-0.620321\pi\)
−0.369062 + 0.929405i \(0.620321\pi\)
\(558\) 0 0
\(559\) 19.4134 0.821100
\(560\) 0 0
\(561\) −10.1056 −0.426658
\(562\) 0 0
\(563\) 36.1757 1.52463 0.762313 0.647209i \(-0.224064\pi\)
0.762313 + 0.647209i \(0.224064\pi\)
\(564\) 0 0
\(565\) −7.38594 −0.310729
\(566\) 0 0
\(567\) 11.1233 0.467136
\(568\) 0 0
\(569\) 10.3021 0.431886 0.215943 0.976406i \(-0.430718\pi\)
0.215943 + 0.976406i \(0.430718\pi\)
\(570\) 0 0
\(571\) −7.62701 −0.319180 −0.159590 0.987183i \(-0.551017\pi\)
−0.159590 + 0.987183i \(0.551017\pi\)
\(572\) 0 0
\(573\) −8.11084 −0.338835
\(574\) 0 0
\(575\) 8.46543 0.353033
\(576\) 0 0
\(577\) −28.9812 −1.20650 −0.603252 0.797551i \(-0.706129\pi\)
−0.603252 + 0.797551i \(0.706129\pi\)
\(578\) 0 0
\(579\) −2.12474 −0.0883012
\(580\) 0 0
\(581\) −3.15470 −0.130879
\(582\) 0 0
\(583\) −19.8861 −0.823596
\(584\) 0 0
\(585\) −9.35145 −0.386635
\(586\) 0 0
\(587\) 28.7378 1.18613 0.593067 0.805153i \(-0.297917\pi\)
0.593067 + 0.805153i \(0.297917\pi\)
\(588\) 0 0
\(589\) 0.0206769 0.000851977 0
\(590\) 0 0
\(591\) 9.60167 0.394960
\(592\) 0 0
\(593\) −34.6436 −1.42264 −0.711321 0.702868i \(-0.751903\pi\)
−0.711321 + 0.702868i \(0.751903\pi\)
\(594\) 0 0
\(595\) 5.15283 0.211245
\(596\) 0 0
\(597\) −11.5192 −0.471451
\(598\) 0 0
\(599\) 0.821286 0.0335568 0.0167784 0.999859i \(-0.494659\pi\)
0.0167784 + 0.999859i \(0.494659\pi\)
\(600\) 0 0
\(601\) 3.98090 0.162384 0.0811921 0.996698i \(-0.474127\pi\)
0.0811921 + 0.996698i \(0.474127\pi\)
\(602\) 0 0
\(603\) −5.88782 −0.239771
\(604\) 0 0
\(605\) −31.4905 −1.28027
\(606\) 0 0
\(607\) 17.0648 0.692638 0.346319 0.938117i \(-0.387432\pi\)
0.346319 + 0.938117i \(0.387432\pi\)
\(608\) 0 0
\(609\) −0.508938 −0.0206232
\(610\) 0 0
\(611\) −13.0729 −0.528872
\(612\) 0 0
\(613\) −23.0452 −0.930785 −0.465393 0.885104i \(-0.654087\pi\)
−0.465393 + 0.885104i \(0.654087\pi\)
\(614\) 0 0
\(615\) −2.41719 −0.0974707
\(616\) 0 0
\(617\) −8.88644 −0.357755 −0.178877 0.983871i \(-0.557247\pi\)
−0.178877 + 0.983871i \(0.557247\pi\)
\(618\) 0 0
\(619\) −11.1230 −0.447072 −0.223536 0.974696i \(-0.571760\pi\)
−0.223536 + 0.974696i \(0.571760\pi\)
\(620\) 0 0
\(621\) 23.3921 0.938691
\(622\) 0 0
\(623\) −15.7424 −0.630704
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −0.0296135 −0.00118265
\(628\) 0 0
\(629\) 29.9675 1.19488
\(630\) 0 0
\(631\) −44.7216 −1.78034 −0.890169 0.455631i \(-0.849414\pi\)
−0.890169 + 0.455631i \(0.849414\pi\)
\(632\) 0 0
\(633\) −12.7503 −0.506778
\(634\) 0 0
\(635\) −1.93979 −0.0769782
\(636\) 0 0
\(637\) 15.0765 0.597352
\(638\) 0 0
\(639\) −8.42964 −0.333471
\(640\) 0 0
\(641\) 20.3811 0.805006 0.402503 0.915419i \(-0.368140\pi\)
0.402503 + 0.915419i \(0.368140\pi\)
\(642\) 0 0
\(643\) −29.3474 −1.15735 −0.578674 0.815559i \(-0.696430\pi\)
−0.578674 + 0.815559i \(0.696430\pi\)
\(644\) 0 0
\(645\) −2.75426 −0.108449
\(646\) 0 0
\(647\) 1.25165 0.0492073 0.0246037 0.999697i \(-0.492168\pi\)
0.0246037 + 0.999697i \(0.492168\pi\)
\(648\) 0 0
\(649\) 66.9451 2.62783
\(650\) 0 0
\(651\) 1.66089 0.0650955
\(652\) 0 0
\(653\) 32.3003 1.26401 0.632004 0.774965i \(-0.282232\pi\)
0.632004 + 0.774965i \(0.282232\pi\)
\(654\) 0 0
\(655\) 12.2579 0.478957
\(656\) 0 0
\(657\) −5.43027 −0.211855
\(658\) 0 0
\(659\) −33.2366 −1.29471 −0.647357 0.762187i \(-0.724126\pi\)
−0.647357 + 0.762187i \(0.724126\pi\)
\(660\) 0 0
\(661\) −10.6990 −0.416143 −0.208072 0.978114i \(-0.566719\pi\)
−0.208072 + 0.978114i \(0.566719\pi\)
\(662\) 0 0
\(663\) −5.23242 −0.203210
\(664\) 0 0
\(665\) 0.0150999 0.000585550 0
\(666\) 0 0
\(667\) 5.65332 0.218897
\(668\) 0 0
\(669\) −6.00408 −0.232131
\(670\) 0 0
\(671\) 85.0726 3.28419
\(672\) 0 0
\(673\) 2.79966 0.107919 0.0539596 0.998543i \(-0.482816\pi\)
0.0539596 + 0.998543i \(0.482816\pi\)
\(674\) 0 0
\(675\) 2.76324 0.106357
\(676\) 0 0
\(677\) −11.7940 −0.453279 −0.226639 0.973979i \(-0.572774\pi\)
−0.226639 + 0.973979i \(0.572774\pi\)
\(678\) 0 0
\(679\) 4.81420 0.184752
\(680\) 0 0
\(681\) −6.21349 −0.238101
\(682\) 0 0
\(683\) −45.2908 −1.73300 −0.866502 0.499174i \(-0.833637\pi\)
−0.866502 + 0.499174i \(0.833637\pi\)
\(684\) 0 0
\(685\) −12.3710 −0.472672
\(686\) 0 0
\(687\) 12.3829 0.472438
\(688\) 0 0
\(689\) −10.2965 −0.392266
\(690\) 0 0
\(691\) −17.5846 −0.668951 −0.334476 0.942404i \(-0.608559\pi\)
−0.334476 + 0.942404i \(0.608559\pi\)
\(692\) 0 0
\(693\) 28.7447 1.09192
\(694\) 0 0
\(695\) −2.21075 −0.0838584
\(696\) 0 0
\(697\) 16.3436 0.619057
\(698\) 0 0
\(699\) −8.91339 −0.337136
\(700\) 0 0
\(701\) −17.5026 −0.661062 −0.330531 0.943795i \(-0.607228\pi\)
−0.330531 + 0.943795i \(0.607228\pi\)
\(702\) 0 0
\(703\) 0.0878172 0.00331209
\(704\) 0 0
\(705\) 1.85470 0.0698521
\(706\) 0 0
\(707\) −18.5800 −0.698775
\(708\) 0 0
\(709\) 25.7252 0.966128 0.483064 0.875585i \(-0.339524\pi\)
0.483064 + 0.875585i \(0.339524\pi\)
\(710\) 0 0
\(711\) −22.4314 −0.841241
\(712\) 0 0
\(713\) −18.4493 −0.690932
\(714\) 0 0
\(715\) −22.0005 −0.822774
\(716\) 0 0
\(717\) 6.00921 0.224418
\(718\) 0 0
\(719\) 13.7240 0.511818 0.255909 0.966701i \(-0.417625\pi\)
0.255909 + 0.966701i \(0.417625\pi\)
\(720\) 0 0
\(721\) −12.7933 −0.476446
\(722\) 0 0
\(723\) −7.60269 −0.282747
\(724\) 0 0
\(725\) 0.667812 0.0248019
\(726\) 0 0
\(727\) −46.8546 −1.73774 −0.868870 0.495041i \(-0.835153\pi\)
−0.868870 + 0.495041i \(0.835153\pi\)
\(728\) 0 0
\(729\) −15.3950 −0.570186
\(730\) 0 0
\(731\) 18.6226 0.688781
\(732\) 0 0
\(733\) 16.3702 0.604647 0.302324 0.953205i \(-0.402238\pi\)
0.302324 + 0.953205i \(0.402238\pi\)
\(734\) 0 0
\(735\) −2.13896 −0.0788967
\(736\) 0 0
\(737\) −13.8519 −0.510241
\(738\) 0 0
\(739\) −4.25605 −0.156561 −0.0782806 0.996931i \(-0.524943\pi\)
−0.0782806 + 0.996931i \(0.524943\pi\)
\(740\) 0 0
\(741\) −0.0153332 −0.000563278 0
\(742\) 0 0
\(743\) 18.9608 0.695605 0.347802 0.937568i \(-0.386928\pi\)
0.347802 + 0.937568i \(0.386928\pi\)
\(744\) 0 0
\(745\) −2.91615 −0.106839
\(746\) 0 0
\(747\) 5.49199 0.200941
\(748\) 0 0
\(749\) 31.6590 1.15679
\(750\) 0 0
\(751\) 29.7461 1.08545 0.542724 0.839911i \(-0.317393\pi\)
0.542724 + 0.839911i \(0.317393\pi\)
\(752\) 0 0
\(753\) −11.4788 −0.418311
\(754\) 0 0
\(755\) 21.1186 0.768584
\(756\) 0 0
\(757\) 44.3794 1.61300 0.806499 0.591235i \(-0.201360\pi\)
0.806499 + 0.591235i \(0.201360\pi\)
\(758\) 0 0
\(759\) 26.4232 0.959100
\(760\) 0 0
\(761\) −7.37209 −0.267238 −0.133619 0.991033i \(-0.542660\pi\)
−0.133619 + 0.991033i \(0.542660\pi\)
\(762\) 0 0
\(763\) 21.5437 0.779934
\(764\) 0 0
\(765\) −8.97050 −0.324329
\(766\) 0 0
\(767\) 34.6626 1.25159
\(768\) 0 0
\(769\) 47.3210 1.70644 0.853219 0.521553i \(-0.174647\pi\)
0.853219 + 0.521553i \(0.174647\pi\)
\(770\) 0 0
\(771\) −2.42691 −0.0874031
\(772\) 0 0
\(773\) −34.2867 −1.23321 −0.616604 0.787274i \(-0.711492\pi\)
−0.616604 + 0.787274i \(0.711492\pi\)
\(774\) 0 0
\(775\) −2.17937 −0.0782853
\(776\) 0 0
\(777\) 7.05400 0.253061
\(778\) 0 0
\(779\) 0.0478935 0.00171596
\(780\) 0 0
\(781\) −19.8319 −0.709640
\(782\) 0 0
\(783\) 1.84533 0.0659467
\(784\) 0 0
\(785\) 11.3435 0.404869
\(786\) 0 0
\(787\) 26.4729 0.943656 0.471828 0.881691i \(-0.343594\pi\)
0.471828 + 0.881691i \(0.343594\pi\)
\(788\) 0 0
\(789\) 5.37383 0.191314
\(790\) 0 0
\(791\) 11.7551 0.417963
\(792\) 0 0
\(793\) 44.0485 1.56421
\(794\) 0 0
\(795\) 1.46081 0.0518095
\(796\) 0 0
\(797\) 13.2108 0.467950 0.233975 0.972243i \(-0.424827\pi\)
0.233975 + 0.972243i \(0.424827\pi\)
\(798\) 0 0
\(799\) −12.5403 −0.443645
\(800\) 0 0
\(801\) 27.4057 0.968333
\(802\) 0 0
\(803\) −12.7754 −0.450835
\(804\) 0 0
\(805\) −13.4732 −0.474867
\(806\) 0 0
\(807\) −5.27622 −0.185732
\(808\) 0 0
\(809\) 1.69076 0.0594441 0.0297220 0.999558i \(-0.490538\pi\)
0.0297220 + 0.999558i \(0.490538\pi\)
\(810\) 0 0
\(811\) 38.2707 1.34386 0.671932 0.740613i \(-0.265465\pi\)
0.671932 + 0.740613i \(0.265465\pi\)
\(812\) 0 0
\(813\) 1.25827 0.0441294
\(814\) 0 0
\(815\) −14.0049 −0.490569
\(816\) 0 0
\(817\) 0.0545719 0.00190923
\(818\) 0 0
\(819\) 14.8833 0.520065
\(820\) 0 0
\(821\) 26.0505 0.909170 0.454585 0.890703i \(-0.349788\pi\)
0.454585 + 0.890703i \(0.349788\pi\)
\(822\) 0 0
\(823\) −46.9913 −1.63801 −0.819007 0.573783i \(-0.805475\pi\)
−0.819007 + 0.573783i \(0.805475\pi\)
\(824\) 0 0
\(825\) 3.12130 0.108670
\(826\) 0 0
\(827\) 6.23256 0.216727 0.108364 0.994111i \(-0.465439\pi\)
0.108364 + 0.994111i \(0.465439\pi\)
\(828\) 0 0
\(829\) 17.9160 0.622247 0.311124 0.950369i \(-0.399295\pi\)
0.311124 + 0.950369i \(0.399295\pi\)
\(830\) 0 0
\(831\) −10.2354 −0.355062
\(832\) 0 0
\(833\) 14.4623 0.501090
\(834\) 0 0
\(835\) −0.315144 −0.0109060
\(836\) 0 0
\(837\) −6.02213 −0.208155
\(838\) 0 0
\(839\) −40.6949 −1.40495 −0.702473 0.711711i \(-0.747921\pi\)
−0.702473 + 0.711711i \(0.747921\pi\)
\(840\) 0 0
\(841\) −28.5540 −0.984622
\(842\) 0 0
\(843\) −4.64651 −0.160034
\(844\) 0 0
\(845\) 1.60865 0.0553394
\(846\) 0 0
\(847\) 50.1187 1.72210
\(848\) 0 0
\(849\) 10.1322 0.347737
\(850\) 0 0
\(851\) −78.3563 −2.68602
\(852\) 0 0
\(853\) −41.9318 −1.43572 −0.717859 0.696188i \(-0.754878\pi\)
−0.717859 + 0.696188i \(0.754878\pi\)
\(854\) 0 0
\(855\) −0.0262873 −0.000899007 0
\(856\) 0 0
\(857\) −14.5473 −0.496927 −0.248464 0.968641i \(-0.579926\pi\)
−0.248464 + 0.968641i \(0.579926\pi\)
\(858\) 0 0
\(859\) 26.4475 0.902378 0.451189 0.892428i \(-0.351000\pi\)
0.451189 + 0.892428i \(0.351000\pi\)
\(860\) 0 0
\(861\) 3.84709 0.131108
\(862\) 0 0
\(863\) −55.3001 −1.88244 −0.941218 0.337798i \(-0.890318\pi\)
−0.941218 + 0.337798i \(0.890318\pi\)
\(864\) 0 0
\(865\) −17.9627 −0.610751
\(866\) 0 0
\(867\) 3.12100 0.105995
\(868\) 0 0
\(869\) −52.7728 −1.79019
\(870\) 0 0
\(871\) −7.17217 −0.243020
\(872\) 0 0
\(873\) −8.38099 −0.283654
\(874\) 0 0
\(875\) −1.59155 −0.0538042
\(876\) 0 0
\(877\) 0.509653 0.0172098 0.00860489 0.999963i \(-0.497261\pi\)
0.00860489 + 0.999963i \(0.497261\pi\)
\(878\) 0 0
\(879\) 10.5913 0.357237
\(880\) 0 0
\(881\) 46.6582 1.57195 0.785977 0.618256i \(-0.212161\pi\)
0.785977 + 0.618256i \(0.212161\pi\)
\(882\) 0 0
\(883\) −33.1322 −1.11499 −0.557494 0.830181i \(-0.688237\pi\)
−0.557494 + 0.830181i \(0.688237\pi\)
\(884\) 0 0
\(885\) −4.91771 −0.165307
\(886\) 0 0
\(887\) 31.4172 1.05489 0.527443 0.849590i \(-0.323151\pi\)
0.527443 + 0.849590i \(0.323151\pi\)
\(888\) 0 0
\(889\) 3.08727 0.103544
\(890\) 0 0
\(891\) −45.5575 −1.52623
\(892\) 0 0
\(893\) −0.0367484 −0.00122974
\(894\) 0 0
\(895\) 12.5698 0.420164
\(896\) 0 0
\(897\) 13.6813 0.456804
\(898\) 0 0
\(899\) −1.45541 −0.0485406
\(900\) 0 0
\(901\) −9.87707 −0.329053
\(902\) 0 0
\(903\) 4.38354 0.145875
\(904\) 0 0
\(905\) −7.08922 −0.235654
\(906\) 0 0
\(907\) −9.89124 −0.328433 −0.164217 0.986424i \(-0.552510\pi\)
−0.164217 + 0.986424i \(0.552510\pi\)
\(908\) 0 0
\(909\) 32.3458 1.07284
\(910\) 0 0
\(911\) 33.8943 1.12297 0.561485 0.827487i \(-0.310230\pi\)
0.561485 + 0.827487i \(0.310230\pi\)
\(912\) 0 0
\(913\) 12.9206 0.427611
\(914\) 0 0
\(915\) −6.24934 −0.206597
\(916\) 0 0
\(917\) −19.5091 −0.644248
\(918\) 0 0
\(919\) −20.3342 −0.670762 −0.335381 0.942083i \(-0.608865\pi\)
−0.335381 + 0.942083i \(0.608865\pi\)
\(920\) 0 0
\(921\) 6.95812 0.229278
\(922\) 0 0
\(923\) −10.2685 −0.337990
\(924\) 0 0
\(925\) −9.25603 −0.304336
\(926\) 0 0
\(927\) 22.2716 0.731496
\(928\) 0 0
\(929\) −8.07264 −0.264855 −0.132427 0.991193i \(-0.542277\pi\)
−0.132427 + 0.991193i \(0.542277\pi\)
\(930\) 0 0
\(931\) 0.0423806 0.00138897
\(932\) 0 0
\(933\) 8.48875 0.277909
\(934\) 0 0
\(935\) −21.1043 −0.690185
\(936\) 0 0
\(937\) 32.9023 1.07487 0.537436 0.843304i \(-0.319393\pi\)
0.537436 + 0.843304i \(0.319393\pi\)
\(938\) 0 0
\(939\) −12.2559 −0.399957
\(940\) 0 0
\(941\) 14.8664 0.484629 0.242315 0.970198i \(-0.422093\pi\)
0.242315 + 0.970198i \(0.422093\pi\)
\(942\) 0 0
\(943\) −42.7337 −1.39160
\(944\) 0 0
\(945\) −4.39784 −0.143062
\(946\) 0 0
\(947\) 13.5736 0.441082 0.220541 0.975378i \(-0.429218\pi\)
0.220541 + 0.975378i \(0.429218\pi\)
\(948\) 0 0
\(949\) −6.61481 −0.214726
\(950\) 0 0
\(951\) −7.87369 −0.255322
\(952\) 0 0
\(953\) −41.3361 −1.33901 −0.669504 0.742808i \(-0.733493\pi\)
−0.669504 + 0.742808i \(0.733493\pi\)
\(954\) 0 0
\(955\) −16.9385 −0.548118
\(956\) 0 0
\(957\) 2.08444 0.0673805
\(958\) 0 0
\(959\) 19.6891 0.635794
\(960\) 0 0
\(961\) −26.2503 −0.846785
\(962\) 0 0
\(963\) −55.1147 −1.77605
\(964\) 0 0
\(965\) −4.43727 −0.142841
\(966\) 0 0
\(967\) −30.5352 −0.981946 −0.490973 0.871175i \(-0.663359\pi\)
−0.490973 + 0.871175i \(0.663359\pi\)
\(968\) 0 0
\(969\) −0.0147085 −0.000472507 0
\(970\) 0 0
\(971\) 37.0974 1.19051 0.595256 0.803536i \(-0.297051\pi\)
0.595256 + 0.803536i \(0.297051\pi\)
\(972\) 0 0
\(973\) 3.51851 0.112798
\(974\) 0 0
\(975\) 1.61613 0.0517577
\(976\) 0 0
\(977\) −4.96666 −0.158898 −0.0794488 0.996839i \(-0.525316\pi\)
−0.0794488 + 0.996839i \(0.525316\pi\)
\(978\) 0 0
\(979\) 64.4756 2.06065
\(980\) 0 0
\(981\) −37.5052 −1.19745
\(982\) 0 0
\(983\) 2.46334 0.0785684 0.0392842 0.999228i \(-0.487492\pi\)
0.0392842 + 0.999228i \(0.487492\pi\)
\(984\) 0 0
\(985\) 20.0520 0.638909
\(986\) 0 0
\(987\) −2.95185 −0.0939584
\(988\) 0 0
\(989\) −48.6927 −1.54834
\(990\) 0 0
\(991\) 28.5590 0.907206 0.453603 0.891204i \(-0.350138\pi\)
0.453603 + 0.891204i \(0.350138\pi\)
\(992\) 0 0
\(993\) 3.38453 0.107405
\(994\) 0 0
\(995\) −24.0566 −0.762644
\(996\) 0 0
\(997\) −31.5746 −0.999977 −0.499989 0.866032i \(-0.666662\pi\)
−0.499989 + 0.866032i \(0.666662\pi\)
\(998\) 0 0
\(999\) −25.5767 −0.809210
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.e.1.17 35
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8020.2.a.e.1.17 35 1.1 even 1 trivial