Properties

Label 6020.2.a.j.1.5
Level $6020$
Weight $2$
Character 6020.1
Self dual yes
Analytic conductor $48.070$
Analytic rank $0$
Dimension $13$
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] = [6020,2,Mod(1,6020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6020, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6020.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6020.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(48.0699420168\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - x^{12} - 28 x^{11} + 26 x^{10} + 286 x^{9} - 235 x^{8} - 1298 x^{7} + 895 x^{6} + 2571 x^{5} - 1548 x^{4} - 1833 x^{3} + 1119 x^{2} + 342 x - 216 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.700922\) of defining polynomial
Character \(\chi\) \(=\) 6020.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.700922 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.50871 q^{9} +O(q^{10})\) \(q-0.700922 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.50871 q^{9} -5.50079 q^{11} +4.90479 q^{13} +0.700922 q^{15} -4.01768 q^{17} +7.12968 q^{19} +0.700922 q^{21} -4.84223 q^{23} +1.00000 q^{25} +3.86117 q^{27} -0.850146 q^{29} -9.51909 q^{31} +3.85563 q^{33} +1.00000 q^{35} -8.77907 q^{37} -3.43787 q^{39} +7.93687 q^{41} -1.00000 q^{43} +2.50871 q^{45} +8.69012 q^{47} +1.00000 q^{49} +2.81608 q^{51} -12.3811 q^{53} +5.50079 q^{55} -4.99735 q^{57} -12.2176 q^{59} -12.3673 q^{61} +2.50871 q^{63} -4.90479 q^{65} +2.41256 q^{67} +3.39402 q^{69} -9.30283 q^{71} +4.44040 q^{73} -0.700922 q^{75} +5.50079 q^{77} -9.91718 q^{79} +4.81974 q^{81} +14.9630 q^{83} +4.01768 q^{85} +0.595886 q^{87} +10.9581 q^{89} -4.90479 q^{91} +6.67214 q^{93} -7.12968 q^{95} +7.86462 q^{97} +13.7999 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - q^{3} - 13 q^{5} - 13 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - q^{3} - 13 q^{5} - 13 q^{7} + 18 q^{9} + 6 q^{11} + q^{13} + q^{15} - 6 q^{17} + 2 q^{19} + q^{21} - 6 q^{23} + 13 q^{25} - q^{27} + 19 q^{29} - 24 q^{31} + 17 q^{33} + 13 q^{35} + 15 q^{39} + 4 q^{41} - 13 q^{43} - 18 q^{45} - 3 q^{47} + 13 q^{49} + 7 q^{51} - 5 q^{53} - 6 q^{55} + 6 q^{57} - 6 q^{59} + 3 q^{61} - 18 q^{63} - q^{65} - 2 q^{67} + 20 q^{69} + 18 q^{71} + 14 q^{73} - q^{75} - 6 q^{77} + 12 q^{79} + 37 q^{81} + 2 q^{83} + 6 q^{85} - 2 q^{87} + 17 q^{89} - q^{91} + 15 q^{93} - 2 q^{95} + 17 q^{97} + 80 q^{99}+O(q^{100}) \) 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
\(3\) −0.700922 −0.404677 −0.202339 0.979316i \(-0.564854\pi\)
−0.202339 + 0.979316i \(0.564854\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −2.50871 −0.836236
\(10\) 0 0
\(11\) −5.50079 −1.65855 −0.829276 0.558840i \(-0.811247\pi\)
−0.829276 + 0.558840i \(0.811247\pi\)
\(12\) 0 0
\(13\) 4.90479 1.36034 0.680171 0.733053i \(-0.261905\pi\)
0.680171 + 0.733053i \(0.261905\pi\)
\(14\) 0 0
\(15\) 0.700922 0.180977
\(16\) 0 0
\(17\) −4.01768 −0.974431 −0.487215 0.873282i \(-0.661987\pi\)
−0.487215 + 0.873282i \(0.661987\pi\)
\(18\) 0 0
\(19\) 7.12968 1.63566 0.817830 0.575460i \(-0.195177\pi\)
0.817830 + 0.575460i \(0.195177\pi\)
\(20\) 0 0
\(21\) 0.700922 0.152954
\(22\) 0 0
\(23\) −4.84223 −1.00967 −0.504837 0.863215i \(-0.668447\pi\)
−0.504837 + 0.863215i \(0.668447\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 3.86117 0.743083
\(28\) 0 0
\(29\) −0.850146 −0.157868 −0.0789341 0.996880i \(-0.525152\pi\)
−0.0789341 + 0.996880i \(0.525152\pi\)
\(30\) 0 0
\(31\) −9.51909 −1.70968 −0.854840 0.518892i \(-0.826345\pi\)
−0.854840 + 0.518892i \(0.826345\pi\)
\(32\) 0 0
\(33\) 3.85563 0.671178
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −8.77907 −1.44327 −0.721635 0.692274i \(-0.756609\pi\)
−0.721635 + 0.692274i \(0.756609\pi\)
\(38\) 0 0
\(39\) −3.43787 −0.550500
\(40\) 0 0
\(41\) 7.93687 1.23953 0.619766 0.784787i \(-0.287228\pi\)
0.619766 + 0.784787i \(0.287228\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499
\(44\) 0 0
\(45\) 2.50871 0.373976
\(46\) 0 0
\(47\) 8.69012 1.26758 0.633792 0.773504i \(-0.281498\pi\)
0.633792 + 0.773504i \(0.281498\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.81608 0.394330
\(52\) 0 0
\(53\) −12.3811 −1.70068 −0.850341 0.526233i \(-0.823604\pi\)
−0.850341 + 0.526233i \(0.823604\pi\)
\(54\) 0 0
\(55\) 5.50079 0.741727
\(56\) 0 0
\(57\) −4.99735 −0.661915
\(58\) 0 0
\(59\) −12.2176 −1.59060 −0.795301 0.606215i \(-0.792687\pi\)
−0.795301 + 0.606215i \(0.792687\pi\)
\(60\) 0 0
\(61\) −12.3673 −1.58346 −0.791732 0.610868i \(-0.790820\pi\)
−0.791732 + 0.610868i \(0.790820\pi\)
\(62\) 0 0
\(63\) 2.50871 0.316068
\(64\) 0 0
\(65\) −4.90479 −0.608364
\(66\) 0 0
\(67\) 2.41256 0.294742 0.147371 0.989081i \(-0.452919\pi\)
0.147371 + 0.989081i \(0.452919\pi\)
\(68\) 0 0
\(69\) 3.39402 0.408593
\(70\) 0 0
\(71\) −9.30283 −1.10404 −0.552021 0.833830i \(-0.686143\pi\)
−0.552021 + 0.833830i \(0.686143\pi\)
\(72\) 0 0
\(73\) 4.44040 0.519709 0.259855 0.965648i \(-0.416325\pi\)
0.259855 + 0.965648i \(0.416325\pi\)
\(74\) 0 0
\(75\) −0.700922 −0.0809355
\(76\) 0 0
\(77\) 5.50079 0.626873
\(78\) 0 0
\(79\) −9.91718 −1.11577 −0.557885 0.829918i \(-0.688387\pi\)
−0.557885 + 0.829918i \(0.688387\pi\)
\(80\) 0 0
\(81\) 4.81974 0.535527
\(82\) 0 0
\(83\) 14.9630 1.64241 0.821203 0.570636i \(-0.193303\pi\)
0.821203 + 0.570636i \(0.193303\pi\)
\(84\) 0 0
\(85\) 4.01768 0.435779
\(86\) 0 0
\(87\) 0.595886 0.0638857
\(88\) 0 0
\(89\) 10.9581 1.16155 0.580776 0.814063i \(-0.302749\pi\)
0.580776 + 0.814063i \(0.302749\pi\)
\(90\) 0 0
\(91\) −4.90479 −0.514161
\(92\) 0 0
\(93\) 6.67214 0.691869
\(94\) 0 0
\(95\) −7.12968 −0.731489
\(96\) 0 0
\(97\) 7.86462 0.798531 0.399266 0.916835i \(-0.369265\pi\)
0.399266 + 0.916835i \(0.369265\pi\)
\(98\) 0 0
\(99\) 13.7999 1.38694
\(100\) 0 0
\(101\) 7.60449 0.756675 0.378338 0.925668i \(-0.376496\pi\)
0.378338 + 0.925668i \(0.376496\pi\)
\(102\) 0 0
\(103\) 0.00896932 0.000883773 0 0.000441887 1.00000i \(-0.499859\pi\)
0.000441887 1.00000i \(0.499859\pi\)
\(104\) 0 0
\(105\) −0.700922 −0.0684030
\(106\) 0 0
\(107\) −16.5317 −1.59818 −0.799088 0.601214i \(-0.794684\pi\)
−0.799088 + 0.601214i \(0.794684\pi\)
\(108\) 0 0
\(109\) 19.5225 1.86991 0.934957 0.354762i \(-0.115438\pi\)
0.934957 + 0.354762i \(0.115438\pi\)
\(110\) 0 0
\(111\) 6.15344 0.584059
\(112\) 0 0
\(113\) −0.964913 −0.0907714 −0.0453857 0.998970i \(-0.514452\pi\)
−0.0453857 + 0.998970i \(0.514452\pi\)
\(114\) 0 0
\(115\) 4.84223 0.451540
\(116\) 0 0
\(117\) −12.3047 −1.13757
\(118\) 0 0
\(119\) 4.01768 0.368300
\(120\) 0 0
\(121\) 19.2587 1.75079
\(122\) 0 0
\(123\) −5.56313 −0.501610
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 8.35361 0.741263 0.370632 0.928780i \(-0.379141\pi\)
0.370632 + 0.928780i \(0.379141\pi\)
\(128\) 0 0
\(129\) 0.700922 0.0617127
\(130\) 0 0
\(131\) −20.3261 −1.77590 −0.887948 0.459943i \(-0.847870\pi\)
−0.887948 + 0.459943i \(0.847870\pi\)
\(132\) 0 0
\(133\) −7.12968 −0.618221
\(134\) 0 0
\(135\) −3.86117 −0.332317
\(136\) 0 0
\(137\) −12.6583 −1.08147 −0.540737 0.841192i \(-0.681855\pi\)
−0.540737 + 0.841192i \(0.681855\pi\)
\(138\) 0 0
\(139\) 4.61985 0.391851 0.195925 0.980619i \(-0.437229\pi\)
0.195925 + 0.980619i \(0.437229\pi\)
\(140\) 0 0
\(141\) −6.09109 −0.512963
\(142\) 0 0
\(143\) −26.9802 −2.25620
\(144\) 0 0
\(145\) 0.850146 0.0706008
\(146\) 0 0
\(147\) −0.700922 −0.0578111
\(148\) 0 0
\(149\) 12.6885 1.03948 0.519740 0.854325i \(-0.326029\pi\)
0.519740 + 0.854325i \(0.326029\pi\)
\(150\) 0 0
\(151\) 8.65548 0.704373 0.352186 0.935930i \(-0.385438\pi\)
0.352186 + 0.935930i \(0.385438\pi\)
\(152\) 0 0
\(153\) 10.0792 0.814854
\(154\) 0 0
\(155\) 9.51909 0.764592
\(156\) 0 0
\(157\) 4.39864 0.351049 0.175525 0.984475i \(-0.443838\pi\)
0.175525 + 0.984475i \(0.443838\pi\)
\(158\) 0 0
\(159\) 8.67822 0.688227
\(160\) 0 0
\(161\) 4.84223 0.381621
\(162\) 0 0
\(163\) 11.8824 0.930698 0.465349 0.885127i \(-0.345929\pi\)
0.465349 + 0.885127i \(0.345929\pi\)
\(164\) 0 0
\(165\) −3.85563 −0.300160
\(166\) 0 0
\(167\) 24.4849 1.89470 0.947351 0.320197i \(-0.103749\pi\)
0.947351 + 0.320197i \(0.103749\pi\)
\(168\) 0 0
\(169\) 11.0569 0.850532
\(170\) 0 0
\(171\) −17.8863 −1.36780
\(172\) 0 0
\(173\) 23.2616 1.76854 0.884272 0.466972i \(-0.154655\pi\)
0.884272 + 0.466972i \(0.154655\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 8.56362 0.643681
\(178\) 0 0
\(179\) −0.497675 −0.0371979 −0.0185990 0.999827i \(-0.505921\pi\)
−0.0185990 + 0.999827i \(0.505921\pi\)
\(180\) 0 0
\(181\) 2.06830 0.153735 0.0768677 0.997041i \(-0.475508\pi\)
0.0768677 + 0.997041i \(0.475508\pi\)
\(182\) 0 0
\(183\) 8.66848 0.640792
\(184\) 0 0
\(185\) 8.77907 0.645450
\(186\) 0 0
\(187\) 22.1004 1.61614
\(188\) 0 0
\(189\) −3.86117 −0.280859
\(190\) 0 0
\(191\) 15.6610 1.13319 0.566597 0.823995i \(-0.308260\pi\)
0.566597 + 0.823995i \(0.308260\pi\)
\(192\) 0 0
\(193\) 14.3643 1.03396 0.516981 0.855997i \(-0.327056\pi\)
0.516981 + 0.855997i \(0.327056\pi\)
\(194\) 0 0
\(195\) 3.43787 0.246191
\(196\) 0 0
\(197\) −6.66662 −0.474977 −0.237488 0.971390i \(-0.576324\pi\)
−0.237488 + 0.971390i \(0.576324\pi\)
\(198\) 0 0
\(199\) 12.3085 0.872525 0.436263 0.899819i \(-0.356302\pi\)
0.436263 + 0.899819i \(0.356302\pi\)
\(200\) 0 0
\(201\) −1.69102 −0.119275
\(202\) 0 0
\(203\) 0.850146 0.0596685
\(204\) 0 0
\(205\) −7.93687 −0.554335
\(206\) 0 0
\(207\) 12.1477 0.844326
\(208\) 0 0
\(209\) −39.2189 −2.71283
\(210\) 0 0
\(211\) 24.7807 1.70597 0.852985 0.521935i \(-0.174790\pi\)
0.852985 + 0.521935i \(0.174790\pi\)
\(212\) 0 0
\(213\) 6.52055 0.446781
\(214\) 0 0
\(215\) 1.00000 0.0681994
\(216\) 0 0
\(217\) 9.51909 0.646198
\(218\) 0 0
\(219\) −3.11237 −0.210315
\(220\) 0 0
\(221\) −19.7059 −1.32556
\(222\) 0 0
\(223\) −22.5342 −1.50900 −0.754500 0.656300i \(-0.772121\pi\)
−0.754500 + 0.656300i \(0.772121\pi\)
\(224\) 0 0
\(225\) −2.50871 −0.167247
\(226\) 0 0
\(227\) 12.9115 0.856967 0.428484 0.903550i \(-0.359048\pi\)
0.428484 + 0.903550i \(0.359048\pi\)
\(228\) 0 0
\(229\) 11.1759 0.738527 0.369264 0.929325i \(-0.379610\pi\)
0.369264 + 0.929325i \(0.379610\pi\)
\(230\) 0 0
\(231\) −3.85563 −0.253682
\(232\) 0 0
\(233\) 11.6628 0.764057 0.382029 0.924150i \(-0.375226\pi\)
0.382029 + 0.924150i \(0.375226\pi\)
\(234\) 0 0
\(235\) −8.69012 −0.566881
\(236\) 0 0
\(237\) 6.95117 0.451527
\(238\) 0 0
\(239\) −21.9495 −1.41979 −0.709897 0.704305i \(-0.751259\pi\)
−0.709897 + 0.704305i \(0.751259\pi\)
\(240\) 0 0
\(241\) 20.2572 1.30488 0.652441 0.757840i \(-0.273745\pi\)
0.652441 + 0.757840i \(0.273745\pi\)
\(242\) 0 0
\(243\) −14.9618 −0.959799
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 34.9695 2.22506
\(248\) 0 0
\(249\) −10.4879 −0.664645
\(250\) 0 0
\(251\) 9.77732 0.617139 0.308570 0.951202i \(-0.400150\pi\)
0.308570 + 0.951202i \(0.400150\pi\)
\(252\) 0 0
\(253\) 26.6361 1.67460
\(254\) 0 0
\(255\) −2.81608 −0.176350
\(256\) 0 0
\(257\) 14.6053 0.911054 0.455527 0.890222i \(-0.349451\pi\)
0.455527 + 0.890222i \(0.349451\pi\)
\(258\) 0 0
\(259\) 8.77907 0.545505
\(260\) 0 0
\(261\) 2.13277 0.132015
\(262\) 0 0
\(263\) −18.1734 −1.12062 −0.560309 0.828284i \(-0.689318\pi\)
−0.560309 + 0.828284i \(0.689318\pi\)
\(264\) 0 0
\(265\) 12.3811 0.760568
\(266\) 0 0
\(267\) −7.68074 −0.470054
\(268\) 0 0
\(269\) −22.4154 −1.36669 −0.683345 0.730096i \(-0.739476\pi\)
−0.683345 + 0.730096i \(0.739476\pi\)
\(270\) 0 0
\(271\) −20.3825 −1.23815 −0.619076 0.785331i \(-0.712493\pi\)
−0.619076 + 0.785331i \(0.712493\pi\)
\(272\) 0 0
\(273\) 3.43787 0.208069
\(274\) 0 0
\(275\) −5.50079 −0.331710
\(276\) 0 0
\(277\) −21.3170 −1.28081 −0.640406 0.768037i \(-0.721234\pi\)
−0.640406 + 0.768037i \(0.721234\pi\)
\(278\) 0 0
\(279\) 23.8806 1.42970
\(280\) 0 0
\(281\) −12.1995 −0.727760 −0.363880 0.931446i \(-0.618548\pi\)
−0.363880 + 0.931446i \(0.618548\pi\)
\(282\) 0 0
\(283\) 0.577862 0.0343503 0.0171752 0.999852i \(-0.494533\pi\)
0.0171752 + 0.999852i \(0.494533\pi\)
\(284\) 0 0
\(285\) 4.99735 0.296017
\(286\) 0 0
\(287\) −7.93687 −0.468499
\(288\) 0 0
\(289\) −0.858242 −0.0504848
\(290\) 0 0
\(291\) −5.51248 −0.323148
\(292\) 0 0
\(293\) −31.8053 −1.85809 −0.929043 0.369973i \(-0.879367\pi\)
−0.929043 + 0.369973i \(0.879367\pi\)
\(294\) 0 0
\(295\) 12.2176 0.711339
\(296\) 0 0
\(297\) −21.2395 −1.23244
\(298\) 0 0
\(299\) −23.7501 −1.37350
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) 0 0
\(303\) −5.33016 −0.306209
\(304\) 0 0
\(305\) 12.3673 0.708147
\(306\) 0 0
\(307\) −2.79472 −0.159503 −0.0797516 0.996815i \(-0.525413\pi\)
−0.0797516 + 0.996815i \(0.525413\pi\)
\(308\) 0 0
\(309\) −0.00628679 −0.000357643 0
\(310\) 0 0
\(311\) 21.2066 1.20252 0.601258 0.799055i \(-0.294666\pi\)
0.601258 + 0.799055i \(0.294666\pi\)
\(312\) 0 0
\(313\) −14.4215 −0.815151 −0.407575 0.913172i \(-0.633626\pi\)
−0.407575 + 0.913172i \(0.633626\pi\)
\(314\) 0 0
\(315\) −2.50871 −0.141350
\(316\) 0 0
\(317\) 18.2521 1.02514 0.512571 0.858645i \(-0.328693\pi\)
0.512571 + 0.858645i \(0.328693\pi\)
\(318\) 0 0
\(319\) 4.67648 0.261832
\(320\) 0 0
\(321\) 11.5874 0.646746
\(322\) 0 0
\(323\) −28.6448 −1.59384
\(324\) 0 0
\(325\) 4.90479 0.272069
\(326\) 0 0
\(327\) −13.6837 −0.756712
\(328\) 0 0
\(329\) −8.69012 −0.479102
\(330\) 0 0
\(331\) −12.0934 −0.664716 −0.332358 0.943153i \(-0.607844\pi\)
−0.332358 + 0.943153i \(0.607844\pi\)
\(332\) 0 0
\(333\) 22.0241 1.20691
\(334\) 0 0
\(335\) −2.41256 −0.131812
\(336\) 0 0
\(337\) −15.1131 −0.823266 −0.411633 0.911350i \(-0.635041\pi\)
−0.411633 + 0.911350i \(0.635041\pi\)
\(338\) 0 0
\(339\) 0.676329 0.0367331
\(340\) 0 0
\(341\) 52.3625 2.83559
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −3.39402 −0.182728
\(346\) 0 0
\(347\) −12.5488 −0.673657 −0.336829 0.941566i \(-0.609354\pi\)
−0.336829 + 0.941566i \(0.609354\pi\)
\(348\) 0 0
\(349\) −7.90686 −0.423245 −0.211622 0.977352i \(-0.567875\pi\)
−0.211622 + 0.977352i \(0.567875\pi\)
\(350\) 0 0
\(351\) 18.9382 1.01085
\(352\) 0 0
\(353\) 8.61723 0.458649 0.229325 0.973350i \(-0.426348\pi\)
0.229325 + 0.973350i \(0.426348\pi\)
\(354\) 0 0
\(355\) 9.30283 0.493743
\(356\) 0 0
\(357\) −2.81608 −0.149043
\(358\) 0 0
\(359\) 3.39810 0.179345 0.0896724 0.995971i \(-0.471418\pi\)
0.0896724 + 0.995971i \(0.471418\pi\)
\(360\) 0 0
\(361\) 31.8323 1.67538
\(362\) 0 0
\(363\) −13.4988 −0.708506
\(364\) 0 0
\(365\) −4.44040 −0.232421
\(366\) 0 0
\(367\) 9.53561 0.497755 0.248877 0.968535i \(-0.419938\pi\)
0.248877 + 0.968535i \(0.419938\pi\)
\(368\) 0 0
\(369\) −19.9113 −1.03654
\(370\) 0 0
\(371\) 12.3811 0.642797
\(372\) 0 0
\(373\) 11.5512 0.598101 0.299050 0.954237i \(-0.403330\pi\)
0.299050 + 0.954237i \(0.403330\pi\)
\(374\) 0 0
\(375\) 0.700922 0.0361955
\(376\) 0 0
\(377\) −4.16978 −0.214755
\(378\) 0 0
\(379\) 22.0125 1.13071 0.565353 0.824849i \(-0.308740\pi\)
0.565353 + 0.824849i \(0.308740\pi\)
\(380\) 0 0
\(381\) −5.85523 −0.299973
\(382\) 0 0
\(383\) −26.6300 −1.36073 −0.680366 0.732873i \(-0.738179\pi\)
−0.680366 + 0.732873i \(0.738179\pi\)
\(384\) 0 0
\(385\) −5.50079 −0.280346
\(386\) 0 0
\(387\) 2.50871 0.127525
\(388\) 0 0
\(389\) −4.09908 −0.207831 −0.103916 0.994586i \(-0.533137\pi\)
−0.103916 + 0.994586i \(0.533137\pi\)
\(390\) 0 0
\(391\) 19.4545 0.983858
\(392\) 0 0
\(393\) 14.2470 0.718665
\(394\) 0 0
\(395\) 9.91718 0.498987
\(396\) 0 0
\(397\) 27.9487 1.40271 0.701353 0.712814i \(-0.252579\pi\)
0.701353 + 0.712814i \(0.252579\pi\)
\(398\) 0 0
\(399\) 4.99735 0.250180
\(400\) 0 0
\(401\) 4.04308 0.201902 0.100951 0.994891i \(-0.467811\pi\)
0.100951 + 0.994891i \(0.467811\pi\)
\(402\) 0 0
\(403\) −46.6891 −2.32575
\(404\) 0 0
\(405\) −4.81974 −0.239495
\(406\) 0 0
\(407\) 48.2918 2.39374
\(408\) 0 0
\(409\) 37.2768 1.84322 0.921609 0.388120i \(-0.126875\pi\)
0.921609 + 0.388120i \(0.126875\pi\)
\(410\) 0 0
\(411\) 8.87250 0.437648
\(412\) 0 0
\(413\) 12.2176 0.601191
\(414\) 0 0
\(415\) −14.9630 −0.734507
\(416\) 0 0
\(417\) −3.23816 −0.158573
\(418\) 0 0
\(419\) −20.5634 −1.00459 −0.502294 0.864697i \(-0.667510\pi\)
−0.502294 + 0.864697i \(0.667510\pi\)
\(420\) 0 0
\(421\) 26.8001 1.30616 0.653078 0.757290i \(-0.273477\pi\)
0.653078 + 0.757290i \(0.273477\pi\)
\(422\) 0 0
\(423\) −21.8010 −1.06000
\(424\) 0 0
\(425\) −4.01768 −0.194886
\(426\) 0 0
\(427\) 12.3673 0.598493
\(428\) 0 0
\(429\) 18.9110 0.913032
\(430\) 0 0
\(431\) 31.7229 1.52804 0.764018 0.645195i \(-0.223224\pi\)
0.764018 + 0.645195i \(0.223224\pi\)
\(432\) 0 0
\(433\) 0.937133 0.0450358 0.0225179 0.999746i \(-0.492832\pi\)
0.0225179 + 0.999746i \(0.492832\pi\)
\(434\) 0 0
\(435\) −0.595886 −0.0285705
\(436\) 0 0
\(437\) −34.5235 −1.65148
\(438\) 0 0
\(439\) 13.4961 0.644133 0.322066 0.946717i \(-0.395623\pi\)
0.322066 + 0.946717i \(0.395623\pi\)
\(440\) 0 0
\(441\) −2.50871 −0.119462
\(442\) 0 0
\(443\) −26.5597 −1.26189 −0.630945 0.775828i \(-0.717332\pi\)
−0.630945 + 0.775828i \(0.717332\pi\)
\(444\) 0 0
\(445\) −10.9581 −0.519462
\(446\) 0 0
\(447\) −8.89362 −0.420654
\(448\) 0 0
\(449\) −0.941531 −0.0444336 −0.0222168 0.999753i \(-0.507072\pi\)
−0.0222168 + 0.999753i \(0.507072\pi\)
\(450\) 0 0
\(451\) −43.6591 −2.05583
\(452\) 0 0
\(453\) −6.06682 −0.285044
\(454\) 0 0
\(455\) 4.90479 0.229940
\(456\) 0 0
\(457\) 12.4449 0.582149 0.291075 0.956700i \(-0.405987\pi\)
0.291075 + 0.956700i \(0.405987\pi\)
\(458\) 0 0
\(459\) −15.5130 −0.724083
\(460\) 0 0
\(461\) 31.2939 1.45750 0.728752 0.684778i \(-0.240101\pi\)
0.728752 + 0.684778i \(0.240101\pi\)
\(462\) 0 0
\(463\) −11.5292 −0.535809 −0.267904 0.963445i \(-0.586331\pi\)
−0.267904 + 0.963445i \(0.586331\pi\)
\(464\) 0 0
\(465\) −6.67214 −0.309413
\(466\) 0 0
\(467\) 0.0190768 0.000882768 0 0.000441384 1.00000i \(-0.499860\pi\)
0.000441384 1.00000i \(0.499860\pi\)
\(468\) 0 0
\(469\) −2.41256 −0.111402
\(470\) 0 0
\(471\) −3.08310 −0.142062
\(472\) 0 0
\(473\) 5.50079 0.252927
\(474\) 0 0
\(475\) 7.12968 0.327132
\(476\) 0 0
\(477\) 31.0607 1.42217
\(478\) 0 0
\(479\) 1.71186 0.0782169 0.0391084 0.999235i \(-0.487548\pi\)
0.0391084 + 0.999235i \(0.487548\pi\)
\(480\) 0 0
\(481\) −43.0595 −1.96334
\(482\) 0 0
\(483\) −3.39402 −0.154433
\(484\) 0 0
\(485\) −7.86462 −0.357114
\(486\) 0 0
\(487\) −20.7652 −0.940960 −0.470480 0.882411i \(-0.655919\pi\)
−0.470480 + 0.882411i \(0.655919\pi\)
\(488\) 0 0
\(489\) −8.32860 −0.376632
\(490\) 0 0
\(491\) 21.9787 0.991886 0.495943 0.868355i \(-0.334823\pi\)
0.495943 + 0.868355i \(0.334823\pi\)
\(492\) 0 0
\(493\) 3.41561 0.153832
\(494\) 0 0
\(495\) −13.7999 −0.620259
\(496\) 0 0
\(497\) 9.30283 0.417289
\(498\) 0 0
\(499\) 10.8772 0.486929 0.243464 0.969910i \(-0.421716\pi\)
0.243464 + 0.969910i \(0.421716\pi\)
\(500\) 0 0
\(501\) −17.1620 −0.766743
\(502\) 0 0
\(503\) −7.75037 −0.345572 −0.172786 0.984959i \(-0.555277\pi\)
−0.172786 + 0.984959i \(0.555277\pi\)
\(504\) 0 0
\(505\) −7.60449 −0.338395
\(506\) 0 0
\(507\) −7.75004 −0.344191
\(508\) 0 0
\(509\) 28.8415 1.27838 0.639189 0.769049i \(-0.279270\pi\)
0.639189 + 0.769049i \(0.279270\pi\)
\(510\) 0 0
\(511\) −4.44040 −0.196432
\(512\) 0 0
\(513\) 27.5289 1.21543
\(514\) 0 0
\(515\) −0.00896932 −0.000395235 0
\(516\) 0 0
\(517\) −47.8025 −2.10235
\(518\) 0 0
\(519\) −16.3045 −0.715690
\(520\) 0 0
\(521\) 7.79743 0.341612 0.170806 0.985305i \(-0.445363\pi\)
0.170806 + 0.985305i \(0.445363\pi\)
\(522\) 0 0
\(523\) −7.09397 −0.310198 −0.155099 0.987899i \(-0.549570\pi\)
−0.155099 + 0.987899i \(0.549570\pi\)
\(524\) 0 0
\(525\) 0.700922 0.0305907
\(526\) 0 0
\(527\) 38.2447 1.66596
\(528\) 0 0
\(529\) 0.447180 0.0194426
\(530\) 0 0
\(531\) 30.6505 1.33012
\(532\) 0 0
\(533\) 38.9287 1.68619
\(534\) 0 0
\(535\) 16.5317 0.714726
\(536\) 0 0
\(537\) 0.348831 0.0150532
\(538\) 0 0
\(539\) −5.50079 −0.236936
\(540\) 0 0
\(541\) 12.0751 0.519148 0.259574 0.965723i \(-0.416418\pi\)
0.259574 + 0.965723i \(0.416418\pi\)
\(542\) 0 0
\(543\) −1.44972 −0.0622132
\(544\) 0 0
\(545\) −19.5225 −0.836251
\(546\) 0 0
\(547\) −13.4066 −0.573226 −0.286613 0.958046i \(-0.592529\pi\)
−0.286613 + 0.958046i \(0.592529\pi\)
\(548\) 0 0
\(549\) 31.0258 1.32415
\(550\) 0 0
\(551\) −6.06126 −0.258219
\(552\) 0 0
\(553\) 9.91718 0.421721
\(554\) 0 0
\(555\) −6.15344 −0.261199
\(556\) 0 0
\(557\) −17.8946 −0.758219 −0.379109 0.925352i \(-0.623770\pi\)
−0.379109 + 0.925352i \(0.623770\pi\)
\(558\) 0 0
\(559\) −4.90479 −0.207450
\(560\) 0 0
\(561\) −15.4907 −0.654017
\(562\) 0 0
\(563\) 19.8359 0.835983 0.417992 0.908451i \(-0.362734\pi\)
0.417992 + 0.908451i \(0.362734\pi\)
\(564\) 0 0
\(565\) 0.964913 0.0405942
\(566\) 0 0
\(567\) −4.81974 −0.202410
\(568\) 0 0
\(569\) 3.40891 0.142909 0.0714544 0.997444i \(-0.477236\pi\)
0.0714544 + 0.997444i \(0.477236\pi\)
\(570\) 0 0
\(571\) 33.8121 1.41499 0.707497 0.706717i \(-0.249824\pi\)
0.707497 + 0.706717i \(0.249824\pi\)
\(572\) 0 0
\(573\) −10.9772 −0.458578
\(574\) 0 0
\(575\) −4.84223 −0.201935
\(576\) 0 0
\(577\) 40.2352 1.67501 0.837506 0.546427i \(-0.184013\pi\)
0.837506 + 0.546427i \(0.184013\pi\)
\(578\) 0 0
\(579\) −10.0682 −0.418421
\(580\) 0 0
\(581\) −14.9630 −0.620771
\(582\) 0 0
\(583\) 68.1061 2.82067
\(584\) 0 0
\(585\) 12.3047 0.508736
\(586\) 0 0
\(587\) 2.95467 0.121952 0.0609762 0.998139i \(-0.480579\pi\)
0.0609762 + 0.998139i \(0.480579\pi\)
\(588\) 0 0
\(589\) −67.8680 −2.79645
\(590\) 0 0
\(591\) 4.67278 0.192212
\(592\) 0 0
\(593\) −45.9000 −1.88489 −0.942444 0.334364i \(-0.891478\pi\)
−0.942444 + 0.334364i \(0.891478\pi\)
\(594\) 0 0
\(595\) −4.01768 −0.164709
\(596\) 0 0
\(597\) −8.62729 −0.353091
\(598\) 0 0
\(599\) 21.5193 0.879257 0.439628 0.898180i \(-0.355110\pi\)
0.439628 + 0.898180i \(0.355110\pi\)
\(600\) 0 0
\(601\) −20.2130 −0.824505 −0.412253 0.911070i \(-0.635258\pi\)
−0.412253 + 0.911070i \(0.635258\pi\)
\(602\) 0 0
\(603\) −6.05242 −0.246474
\(604\) 0 0
\(605\) −19.2587 −0.782978
\(606\) 0 0
\(607\) 11.3892 0.462273 0.231136 0.972921i \(-0.425756\pi\)
0.231136 + 0.972921i \(0.425756\pi\)
\(608\) 0 0
\(609\) −0.595886 −0.0241465
\(610\) 0 0
\(611\) 42.6232 1.72435
\(612\) 0 0
\(613\) −0.744736 −0.0300796 −0.0150398 0.999887i \(-0.504787\pi\)
−0.0150398 + 0.999887i \(0.504787\pi\)
\(614\) 0 0
\(615\) 5.56313 0.224327
\(616\) 0 0
\(617\) −19.2496 −0.774961 −0.387481 0.921878i \(-0.626655\pi\)
−0.387481 + 0.921878i \(0.626655\pi\)
\(618\) 0 0
\(619\) −36.2763 −1.45807 −0.729034 0.684478i \(-0.760030\pi\)
−0.729034 + 0.684478i \(0.760030\pi\)
\(620\) 0 0
\(621\) −18.6967 −0.750272
\(622\) 0 0
\(623\) −10.9581 −0.439025
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 27.4894 1.09782
\(628\) 0 0
\(629\) 35.2715 1.40637
\(630\) 0 0
\(631\) 4.85163 0.193140 0.0965702 0.995326i \(-0.469213\pi\)
0.0965702 + 0.995326i \(0.469213\pi\)
\(632\) 0 0
\(633\) −17.3693 −0.690368
\(634\) 0 0
\(635\) −8.35361 −0.331503
\(636\) 0 0
\(637\) 4.90479 0.194335
\(638\) 0 0
\(639\) 23.3381 0.923240
\(640\) 0 0
\(641\) 21.3651 0.843871 0.421935 0.906626i \(-0.361351\pi\)
0.421935 + 0.906626i \(0.361351\pi\)
\(642\) 0 0
\(643\) 5.44504 0.214732 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(644\) 0 0
\(645\) −0.700922 −0.0275988
\(646\) 0 0
\(647\) −26.5641 −1.04434 −0.522171 0.852841i \(-0.674878\pi\)
−0.522171 + 0.852841i \(0.674878\pi\)
\(648\) 0 0
\(649\) 67.2067 2.63809
\(650\) 0 0
\(651\) −6.67214 −0.261502
\(652\) 0 0
\(653\) −13.8253 −0.541026 −0.270513 0.962716i \(-0.587193\pi\)
−0.270513 + 0.962716i \(0.587193\pi\)
\(654\) 0 0
\(655\) 20.3261 0.794205
\(656\) 0 0
\(657\) −11.1397 −0.434600
\(658\) 0 0
\(659\) −35.5177 −1.38357 −0.691787 0.722101i \(-0.743176\pi\)
−0.691787 + 0.722101i \(0.743176\pi\)
\(660\) 0 0
\(661\) 32.7029 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(662\) 0 0
\(663\) 13.8123 0.536424
\(664\) 0 0
\(665\) 7.12968 0.276477
\(666\) 0 0
\(667\) 4.11660 0.159395
\(668\) 0 0
\(669\) 15.7947 0.610659
\(670\) 0 0
\(671\) 68.0297 2.62626
\(672\) 0 0
\(673\) 9.00976 0.347301 0.173650 0.984807i \(-0.444444\pi\)
0.173650 + 0.984807i \(0.444444\pi\)
\(674\) 0 0
\(675\) 3.86117 0.148617
\(676\) 0 0
\(677\) 16.4758 0.633218 0.316609 0.948556i \(-0.397456\pi\)
0.316609 + 0.948556i \(0.397456\pi\)
\(678\) 0 0
\(679\) −7.86462 −0.301816
\(680\) 0 0
\(681\) −9.04996 −0.346795
\(682\) 0 0
\(683\) 23.7149 0.907426 0.453713 0.891148i \(-0.350099\pi\)
0.453713 + 0.891148i \(0.350099\pi\)
\(684\) 0 0
\(685\) 12.6583 0.483650
\(686\) 0 0
\(687\) −7.83346 −0.298865
\(688\) 0 0
\(689\) −60.7269 −2.31351
\(690\) 0 0
\(691\) 17.6835 0.672711 0.336355 0.941735i \(-0.390806\pi\)
0.336355 + 0.941735i \(0.390806\pi\)
\(692\) 0 0
\(693\) −13.7999 −0.524214
\(694\) 0 0
\(695\) −4.61985 −0.175241
\(696\) 0 0
\(697\) −31.8878 −1.20784
\(698\) 0 0
\(699\) −8.17473 −0.309197
\(700\) 0 0
\(701\) 30.9125 1.16755 0.583774 0.811916i \(-0.301575\pi\)
0.583774 + 0.811916i \(0.301575\pi\)
\(702\) 0 0
\(703\) −62.5919 −2.36070
\(704\) 0 0
\(705\) 6.09109 0.229404
\(706\) 0 0
\(707\) −7.60449 −0.285996
\(708\) 0 0
\(709\) −2.07846 −0.0780584 −0.0390292 0.999238i \(-0.512427\pi\)
−0.0390292 + 0.999238i \(0.512427\pi\)
\(710\) 0 0
\(711\) 24.8793 0.933047
\(712\) 0 0
\(713\) 46.0936 1.72622
\(714\) 0 0
\(715\) 26.9802 1.00900
\(716\) 0 0
\(717\) 15.3849 0.574559
\(718\) 0 0
\(719\) −9.55907 −0.356493 −0.178247 0.983986i \(-0.557042\pi\)
−0.178247 + 0.983986i \(0.557042\pi\)
\(720\) 0 0
\(721\) −0.00896932 −0.000334035 0
\(722\) 0 0
\(723\) −14.1987 −0.528056
\(724\) 0 0
\(725\) −0.850146 −0.0315736
\(726\) 0 0
\(727\) −30.8862 −1.14551 −0.572753 0.819728i \(-0.694125\pi\)
−0.572753 + 0.819728i \(0.694125\pi\)
\(728\) 0 0
\(729\) −3.97219 −0.147118
\(730\) 0 0
\(731\) 4.01768 0.148599
\(732\) 0 0
\(733\) −33.4356 −1.23497 −0.617485 0.786582i \(-0.711849\pi\)
−0.617485 + 0.786582i \(0.711849\pi\)
\(734\) 0 0
\(735\) 0.700922 0.0258539
\(736\) 0 0
\(737\) −13.2710 −0.488844
\(738\) 0 0
\(739\) 30.1993 1.11090 0.555449 0.831550i \(-0.312546\pi\)
0.555449 + 0.831550i \(0.312546\pi\)
\(740\) 0 0
\(741\) −24.5109 −0.900431
\(742\) 0 0
\(743\) −25.9868 −0.953363 −0.476681 0.879076i \(-0.658160\pi\)
−0.476681 + 0.879076i \(0.658160\pi\)
\(744\) 0 0
\(745\) −12.6885 −0.464869
\(746\) 0 0
\(747\) −37.5379 −1.37344
\(748\) 0 0
\(749\) 16.5317 0.604054
\(750\) 0 0
\(751\) −19.0619 −0.695580 −0.347790 0.937572i \(-0.613068\pi\)
−0.347790 + 0.937572i \(0.613068\pi\)
\(752\) 0 0
\(753\) −6.85314 −0.249742
\(754\) 0 0
\(755\) −8.65548 −0.315005
\(756\) 0 0
\(757\) 34.7062 1.26142 0.630709 0.776019i \(-0.282764\pi\)
0.630709 + 0.776019i \(0.282764\pi\)
\(758\) 0 0
\(759\) −18.6698 −0.677672
\(760\) 0 0
\(761\) 11.6874 0.423670 0.211835 0.977305i \(-0.432056\pi\)
0.211835 + 0.977305i \(0.432056\pi\)
\(762\) 0 0
\(763\) −19.5225 −0.706761
\(764\) 0 0
\(765\) −10.0792 −0.364414
\(766\) 0 0
\(767\) −59.9249 −2.16376
\(768\) 0 0
\(769\) −0.361368 −0.0130313 −0.00651564 0.999979i \(-0.502074\pi\)
−0.00651564 + 0.999979i \(0.502074\pi\)
\(770\) 0 0
\(771\) −10.2372 −0.368683
\(772\) 0 0
\(773\) 8.22709 0.295908 0.147954 0.988994i \(-0.452731\pi\)
0.147954 + 0.988994i \(0.452731\pi\)
\(774\) 0 0
\(775\) −9.51909 −0.341936
\(776\) 0 0
\(777\) −6.15344 −0.220754
\(778\) 0 0
\(779\) 56.5873 2.02745
\(780\) 0 0
\(781\) 51.1729 1.83111
\(782\) 0 0
\(783\) −3.28256 −0.117309
\(784\) 0 0
\(785\) −4.39864 −0.156994
\(786\) 0 0
\(787\) −34.2262 −1.22003 −0.610015 0.792390i \(-0.708837\pi\)
−0.610015 + 0.792390i \(0.708837\pi\)
\(788\) 0 0
\(789\) 12.7381 0.453489
\(790\) 0 0
\(791\) 0.964913 0.0343084
\(792\) 0 0
\(793\) −60.6587 −2.15405
\(794\) 0 0
\(795\) −8.67822 −0.307785
\(796\) 0 0
\(797\) 17.5824 0.622800 0.311400 0.950279i \(-0.399202\pi\)
0.311400 + 0.950279i \(0.399202\pi\)
\(798\) 0 0
\(799\) −34.9141 −1.23517
\(800\) 0 0
\(801\) −27.4906 −0.971332
\(802\) 0 0
\(803\) −24.4257 −0.861964
\(804\) 0 0
\(805\) −4.84223 −0.170666
\(806\) 0 0
\(807\) 15.7114 0.553069
\(808\) 0 0
\(809\) −4.04653 −0.142269 −0.0711343 0.997467i \(-0.522662\pi\)
−0.0711343 + 0.997467i \(0.522662\pi\)
\(810\) 0 0
\(811\) −29.0783 −1.02108 −0.510538 0.859855i \(-0.670554\pi\)
−0.510538 + 0.859855i \(0.670554\pi\)
\(812\) 0 0
\(813\) 14.2866 0.501052
\(814\) 0 0
\(815\) −11.8824 −0.416221
\(816\) 0 0
\(817\) −7.12968 −0.249436
\(818\) 0 0
\(819\) 12.3047 0.429960
\(820\) 0 0
\(821\) 22.8888 0.798826 0.399413 0.916771i \(-0.369214\pi\)
0.399413 + 0.916771i \(0.369214\pi\)
\(822\) 0 0
\(823\) −48.1394 −1.67803 −0.839017 0.544105i \(-0.816869\pi\)
−0.839017 + 0.544105i \(0.816869\pi\)
\(824\) 0 0
\(825\) 3.85563 0.134236
\(826\) 0 0
\(827\) −10.0777 −0.350436 −0.175218 0.984530i \(-0.556063\pi\)
−0.175218 + 0.984530i \(0.556063\pi\)
\(828\) 0 0
\(829\) −38.0579 −1.32180 −0.660902 0.750472i \(-0.729826\pi\)
−0.660902 + 0.750472i \(0.729826\pi\)
\(830\) 0 0
\(831\) 14.9415 0.518316
\(832\) 0 0
\(833\) −4.01768 −0.139204
\(834\) 0 0
\(835\) −24.4849 −0.847336
\(836\) 0 0
\(837\) −36.7549 −1.27043
\(838\) 0 0
\(839\) 24.4667 0.844686 0.422343 0.906436i \(-0.361208\pi\)
0.422343 + 0.906436i \(0.361208\pi\)
\(840\) 0 0
\(841\) −28.2773 −0.975078
\(842\) 0 0
\(843\) 8.55088 0.294508
\(844\) 0 0
\(845\) −11.0569 −0.380370
\(846\) 0 0
\(847\) −19.2587 −0.661737
\(848\) 0 0
\(849\) −0.405036 −0.0139008
\(850\) 0 0
\(851\) 42.5103 1.45723
\(852\) 0 0
\(853\) 3.62748 0.124203 0.0621013 0.998070i \(-0.480220\pi\)
0.0621013 + 0.998070i \(0.480220\pi\)
\(854\) 0 0
\(855\) 17.8863 0.611698
\(856\) 0 0
\(857\) 28.6691 0.979319 0.489659 0.871914i \(-0.337121\pi\)
0.489659 + 0.871914i \(0.337121\pi\)
\(858\) 0 0
\(859\) −22.3605 −0.762932 −0.381466 0.924383i \(-0.624581\pi\)
−0.381466 + 0.924383i \(0.624581\pi\)
\(860\) 0 0
\(861\) 5.56313 0.189591
\(862\) 0 0
\(863\) 49.0247 1.66882 0.834411 0.551143i \(-0.185808\pi\)
0.834411 + 0.551143i \(0.185808\pi\)
\(864\) 0 0
\(865\) −23.2616 −0.790917
\(866\) 0 0
\(867\) 0.601561 0.0204301
\(868\) 0 0
\(869\) 54.5523 1.85056
\(870\) 0 0
\(871\) 11.8331 0.400950
\(872\) 0 0
\(873\) −19.7300 −0.667761
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −21.4759 −0.725188 −0.362594 0.931947i \(-0.618109\pi\)
−0.362594 + 0.931947i \(0.618109\pi\)
\(878\) 0 0
\(879\) 22.2930 0.751925
\(880\) 0 0
\(881\) −16.1537 −0.544231 −0.272116 0.962265i \(-0.587723\pi\)
−0.272116 + 0.962265i \(0.587723\pi\)
\(882\) 0 0
\(883\) 11.8838 0.399921 0.199960 0.979804i \(-0.435919\pi\)
0.199960 + 0.979804i \(0.435919\pi\)
\(884\) 0 0
\(885\) −8.56362 −0.287863
\(886\) 0 0
\(887\) −36.1485 −1.21375 −0.606875 0.794798i \(-0.707577\pi\)
−0.606875 + 0.794798i \(0.707577\pi\)
\(888\) 0 0
\(889\) −8.35361 −0.280171
\(890\) 0 0
\(891\) −26.5124 −0.888199
\(892\) 0 0
\(893\) 61.9577 2.07334
\(894\) 0 0
\(895\) 0.497675 0.0166354
\(896\) 0 0
\(897\) 16.6470 0.555826
\(898\) 0 0
\(899\) 8.09262 0.269904
\(900\) 0 0
\(901\) 49.7435 1.65720
\(902\) 0 0
\(903\) −0.700922 −0.0233252
\(904\) 0 0
\(905\) −2.06830 −0.0687526
\(906\) 0 0
\(907\) 19.0738 0.633337 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(908\) 0 0
\(909\) −19.0775 −0.632759
\(910\) 0 0
\(911\) −6.51196 −0.215751 −0.107875 0.994164i \(-0.534405\pi\)
−0.107875 + 0.994164i \(0.534405\pi\)
\(912\) 0 0
\(913\) −82.3086 −2.72402
\(914\) 0 0
\(915\) −8.66848 −0.286571
\(916\) 0 0
\(917\) 20.3261 0.671226
\(918\) 0 0
\(919\) −39.8181 −1.31348 −0.656740 0.754117i \(-0.728065\pi\)
−0.656740 + 0.754117i \(0.728065\pi\)
\(920\) 0 0
\(921\) 1.95888 0.0645473
\(922\) 0 0
\(923\) −45.6284 −1.50188
\(924\) 0 0
\(925\) −8.77907 −0.288654
\(926\) 0 0
\(927\) −0.0225014 −0.000739043 0
\(928\) 0 0
\(929\) 31.1467 1.02189 0.510946 0.859613i \(-0.329295\pi\)
0.510946 + 0.859613i \(0.329295\pi\)
\(930\) 0 0
\(931\) 7.12968 0.233666
\(932\) 0 0
\(933\) −14.8642 −0.486632
\(934\) 0 0
\(935\) −22.1004 −0.722761
\(936\) 0 0
\(937\) −25.4314 −0.830809 −0.415404 0.909637i \(-0.636360\pi\)
−0.415404 + 0.909637i \(0.636360\pi\)
\(938\) 0 0
\(939\) 10.1083 0.329873
\(940\) 0 0
\(941\) 10.3306 0.336769 0.168384 0.985721i \(-0.446145\pi\)
0.168384 + 0.985721i \(0.446145\pi\)
\(942\) 0 0
\(943\) −38.4321 −1.25152
\(944\) 0 0
\(945\) 3.86117 0.125604
\(946\) 0 0
\(947\) 25.2624 0.820916 0.410458 0.911879i \(-0.365369\pi\)
0.410458 + 0.911879i \(0.365369\pi\)
\(948\) 0 0
\(949\) 21.7792 0.706983
\(950\) 0 0
\(951\) −12.7933 −0.414851
\(952\) 0 0
\(953\) −46.8248 −1.51680 −0.758402 0.651787i \(-0.774020\pi\)
−0.758402 + 0.651787i \(0.774020\pi\)
\(954\) 0 0
\(955\) −15.6610 −0.506780
\(956\) 0 0
\(957\) −3.27784 −0.105958
\(958\) 0 0
\(959\) 12.6583 0.408759
\(960\) 0 0
\(961\) 59.6131 1.92300
\(962\) 0 0
\(963\) 41.4731 1.33645
\(964\) 0 0
\(965\) −14.3643 −0.462402
\(966\) 0 0
\(967\) −2.76684 −0.0889756 −0.0444878 0.999010i \(-0.514166\pi\)
−0.0444878 + 0.999010i \(0.514166\pi\)
\(968\) 0 0
\(969\) 20.0777 0.644990
\(970\) 0 0
\(971\) −4.43069 −0.142188 −0.0710939 0.997470i \(-0.522649\pi\)
−0.0710939 + 0.997470i \(0.522649\pi\)
\(972\) 0 0
\(973\) −4.61985 −0.148106
\(974\) 0 0
\(975\) −3.43787 −0.110100
\(976\) 0 0
\(977\) 24.0333 0.768893 0.384447 0.923147i \(-0.374392\pi\)
0.384447 + 0.923147i \(0.374392\pi\)
\(978\) 0 0
\(979\) −60.2780 −1.92649
\(980\) 0 0
\(981\) −48.9762 −1.56369
\(982\) 0 0
\(983\) −46.1819 −1.47297 −0.736487 0.676452i \(-0.763517\pi\)
−0.736487 + 0.676452i \(0.763517\pi\)
\(984\) 0 0
\(985\) 6.66662 0.212416
\(986\) 0 0
\(987\) 6.09109 0.193882
\(988\) 0 0
\(989\) 4.84223 0.153974
\(990\) 0 0
\(991\) −27.3610 −0.869152 −0.434576 0.900635i \(-0.643102\pi\)
−0.434576 + 0.900635i \(0.643102\pi\)
\(992\) 0 0
\(993\) 8.47656 0.268996
\(994\) 0 0
\(995\) −12.3085 −0.390205
\(996\) 0 0
\(997\) 43.0922 1.36474 0.682372 0.731005i \(-0.260948\pi\)
0.682372 + 0.731005i \(0.260948\pi\)
\(998\) 0 0
\(999\) −33.8975 −1.07247
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 6020.2.a.j.1.5 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6020.2.a.j.1.5 13 1.1 even 1 trivial