Properties

Label 6020.2.a.k.1.2
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: \( x^{13} - 27 x^{11} - 2 x^{10} + 268 x^{9} + 37 x^{8} - 1201 x^{7} - 189 x^{6} + 2384 x^{5} + 231 x^{4} - 1729 x^{3} + 20 x^{2} + 105 x - 4 \) 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 1.2
Root \(-2.54322\) 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)
 
\(f(q)\) \(=\) \(q-2.54322 q^{3} +1.00000 q^{5} -1.00000 q^{7} +3.46796 q^{9} +O(q^{10})\) \(q-2.54322 q^{3} +1.00000 q^{5} -1.00000 q^{7} +3.46796 q^{9} -3.23106 q^{11} -4.70161 q^{13} -2.54322 q^{15} -0.826062 q^{17} -5.65222 q^{19} +2.54322 q^{21} -7.98232 q^{23} +1.00000 q^{25} -1.19012 q^{27} -2.88745 q^{29} -8.99139 q^{31} +8.21730 q^{33} -1.00000 q^{35} +4.79116 q^{37} +11.9572 q^{39} -2.32201 q^{41} +1.00000 q^{43} +3.46796 q^{45} +0.524113 q^{47} +1.00000 q^{49} +2.10085 q^{51} -6.07174 q^{53} -3.23106 q^{55} +14.3748 q^{57} -4.71720 q^{59} -4.31640 q^{61} -3.46796 q^{63} -4.70161 q^{65} +11.6915 q^{67} +20.3008 q^{69} +2.06811 q^{71} +6.28496 q^{73} -2.54322 q^{75} +3.23106 q^{77} -13.6842 q^{79} -7.37714 q^{81} +11.4005 q^{83} -0.826062 q^{85} +7.34343 q^{87} -11.7618 q^{89} +4.70161 q^{91} +22.8671 q^{93} -5.65222 q^{95} +12.9020 q^{97} -11.2052 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 13 q^{5} - 13 q^{7} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 13 q^{5} - 13 q^{7} + 15 q^{9} + 11 q^{13} + 16 q^{17} + 3 q^{23} + 13 q^{25} + 6 q^{27} + 10 q^{29} - q^{31} + 14 q^{33} - 13 q^{35} + 16 q^{37} - 14 q^{39} + 23 q^{41} + 13 q^{43} + 15 q^{45} + 2 q^{47} + 13 q^{49} + 4 q^{51} + 20 q^{53} + 22 q^{57} + 2 q^{59} + 5 q^{61} - 15 q^{63} + 11 q^{65} + 19 q^{67} + 16 q^{69} + 4 q^{71} + 34 q^{73} - 15 q^{79} + 17 q^{81} + 27 q^{83} + 16 q^{85} - 5 q^{87} + 3 q^{89} - 11 q^{91} + 35 q^{93} + 45 q^{97} + q^{99}+O(q^{100}) \) 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
\(3\) −2.54322 −1.46833 −0.734164 0.678972i \(-0.762426\pi\)
−0.734164 + 0.678972i \(0.762426\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 3.46796 1.15599
\(10\) 0 0
\(11\) −3.23106 −0.974202 −0.487101 0.873346i \(-0.661946\pi\)
−0.487101 + 0.873346i \(0.661946\pi\)
\(12\) 0 0
\(13\) −4.70161 −1.30399 −0.651996 0.758222i \(-0.726068\pi\)
−0.651996 + 0.758222i \(0.726068\pi\)
\(14\) 0 0
\(15\) −2.54322 −0.656656
\(16\) 0 0
\(17\) −0.826062 −0.200349 −0.100175 0.994970i \(-0.531940\pi\)
−0.100175 + 0.994970i \(0.531940\pi\)
\(18\) 0 0
\(19\) −5.65222 −1.29671 −0.648354 0.761339i \(-0.724542\pi\)
−0.648354 + 0.761339i \(0.724542\pi\)
\(20\) 0 0
\(21\) 2.54322 0.554976
\(22\) 0 0
\(23\) −7.98232 −1.66443 −0.832215 0.554453i \(-0.812927\pi\)
−0.832215 + 0.554453i \(0.812927\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.19012 −0.229038
\(28\) 0 0
\(29\) −2.88745 −0.536187 −0.268093 0.963393i \(-0.586394\pi\)
−0.268093 + 0.963393i \(0.586394\pi\)
\(30\) 0 0
\(31\) −8.99139 −1.61490 −0.807451 0.589935i \(-0.799153\pi\)
−0.807451 + 0.589935i \(0.799153\pi\)
\(32\) 0 0
\(33\) 8.21730 1.43045
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 4.79116 0.787662 0.393831 0.919183i \(-0.371149\pi\)
0.393831 + 0.919183i \(0.371149\pi\)
\(38\) 0 0
\(39\) 11.9572 1.91469
\(40\) 0 0
\(41\) −2.32201 −0.362637 −0.181319 0.983424i \(-0.558037\pi\)
−0.181319 + 0.983424i \(0.558037\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499
\(44\) 0 0
\(45\) 3.46796 0.516973
\(46\) 0 0
\(47\) 0.524113 0.0764497 0.0382248 0.999269i \(-0.487830\pi\)
0.0382248 + 0.999269i \(0.487830\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.10085 0.294179
\(52\) 0 0
\(53\) −6.07174 −0.834018 −0.417009 0.908902i \(-0.636922\pi\)
−0.417009 + 0.908902i \(0.636922\pi\)
\(54\) 0 0
\(55\) −3.23106 −0.435676
\(56\) 0 0
\(57\) 14.3748 1.90399
\(58\) 0 0
\(59\) −4.71720 −0.614127 −0.307063 0.951689i \(-0.599346\pi\)
−0.307063 + 0.951689i \(0.599346\pi\)
\(60\) 0 0
\(61\) −4.31640 −0.552658 −0.276329 0.961063i \(-0.589118\pi\)
−0.276329 + 0.961063i \(0.589118\pi\)
\(62\) 0 0
\(63\) −3.46796 −0.436922
\(64\) 0 0
\(65\) −4.70161 −0.583163
\(66\) 0 0
\(67\) 11.6915 1.42835 0.714173 0.699969i \(-0.246803\pi\)
0.714173 + 0.699969i \(0.246803\pi\)
\(68\) 0 0
\(69\) 20.3008 2.44393
\(70\) 0 0
\(71\) 2.06811 0.245440 0.122720 0.992441i \(-0.460838\pi\)
0.122720 + 0.992441i \(0.460838\pi\)
\(72\) 0 0
\(73\) 6.28496 0.735598 0.367799 0.929905i \(-0.380111\pi\)
0.367799 + 0.929905i \(0.380111\pi\)
\(74\) 0 0
\(75\) −2.54322 −0.293666
\(76\) 0 0
\(77\) 3.23106 0.368214
\(78\) 0 0
\(79\) −13.6842 −1.53959 −0.769796 0.638290i \(-0.779642\pi\)
−0.769796 + 0.638290i \(0.779642\pi\)
\(80\) 0 0
\(81\) −7.37714 −0.819682
\(82\) 0 0
\(83\) 11.4005 1.25137 0.625683 0.780078i \(-0.284820\pi\)
0.625683 + 0.780078i \(0.284820\pi\)
\(84\) 0 0
\(85\) −0.826062 −0.0895990
\(86\) 0 0
\(87\) 7.34343 0.787298
\(88\) 0 0
\(89\) −11.7618 −1.24675 −0.623376 0.781922i \(-0.714239\pi\)
−0.623376 + 0.781922i \(0.714239\pi\)
\(90\) 0 0
\(91\) 4.70161 0.492863
\(92\) 0 0
\(93\) 22.8671 2.37120
\(94\) 0 0
\(95\) −5.65222 −0.579905
\(96\) 0 0
\(97\) 12.9020 1.31000 0.654998 0.755631i \(-0.272670\pi\)
0.654998 + 0.755631i \(0.272670\pi\)
\(98\) 0 0
\(99\) −11.2052 −1.12616
\(100\) 0 0
\(101\) 2.64856 0.263542 0.131771 0.991280i \(-0.457934\pi\)
0.131771 + 0.991280i \(0.457934\pi\)
\(102\) 0 0
\(103\) −13.2293 −1.30353 −0.651763 0.758423i \(-0.725970\pi\)
−0.651763 + 0.758423i \(0.725970\pi\)
\(104\) 0 0
\(105\) 2.54322 0.248193
\(106\) 0 0
\(107\) −3.56234 −0.344384 −0.172192 0.985063i \(-0.555085\pi\)
−0.172192 + 0.985063i \(0.555085\pi\)
\(108\) 0 0
\(109\) 17.8358 1.70836 0.854179 0.519979i \(-0.174060\pi\)
0.854179 + 0.519979i \(0.174060\pi\)
\(110\) 0 0
\(111\) −12.1850 −1.15655
\(112\) 0 0
\(113\) 1.17570 0.110600 0.0553001 0.998470i \(-0.482388\pi\)
0.0553001 + 0.998470i \(0.482388\pi\)
\(114\) 0 0
\(115\) −7.98232 −0.744356
\(116\) 0 0
\(117\) −16.3050 −1.50740
\(118\) 0 0
\(119\) 0.826062 0.0757250
\(120\) 0 0
\(121\) −0.560233 −0.0509303
\(122\) 0 0
\(123\) 5.90538 0.532470
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 2.59158 0.229966 0.114983 0.993367i \(-0.463319\pi\)
0.114983 + 0.993367i \(0.463319\pi\)
\(128\) 0 0
\(129\) −2.54322 −0.223918
\(130\) 0 0
\(131\) 2.53621 0.221589 0.110795 0.993843i \(-0.464660\pi\)
0.110795 + 0.993843i \(0.464660\pi\)
\(132\) 0 0
\(133\) 5.65222 0.490110
\(134\) 0 0
\(135\) −1.19012 −0.102429
\(136\) 0 0
\(137\) 18.6291 1.59160 0.795798 0.605563i \(-0.207052\pi\)
0.795798 + 0.605563i \(0.207052\pi\)
\(138\) 0 0
\(139\) −18.7700 −1.59205 −0.796027 0.605261i \(-0.793069\pi\)
−0.796027 + 0.605261i \(0.793069\pi\)
\(140\) 0 0
\(141\) −1.33293 −0.112253
\(142\) 0 0
\(143\) 15.1912 1.27035
\(144\) 0 0
\(145\) −2.88745 −0.239790
\(146\) 0 0
\(147\) −2.54322 −0.209761
\(148\) 0 0
\(149\) −19.2059 −1.57341 −0.786704 0.617330i \(-0.788214\pi\)
−0.786704 + 0.617330i \(0.788214\pi\)
\(150\) 0 0
\(151\) −14.1547 −1.15190 −0.575948 0.817486i \(-0.695367\pi\)
−0.575948 + 0.817486i \(0.695367\pi\)
\(152\) 0 0
\(153\) −2.86475 −0.231601
\(154\) 0 0
\(155\) −8.99139 −0.722206
\(156\) 0 0
\(157\) −15.8703 −1.26659 −0.633293 0.773912i \(-0.718297\pi\)
−0.633293 + 0.773912i \(0.718297\pi\)
\(158\) 0 0
\(159\) 15.4418 1.22461
\(160\) 0 0
\(161\) 7.98232 0.629095
\(162\) 0 0
\(163\) −10.0290 −0.785536 −0.392768 0.919638i \(-0.628482\pi\)
−0.392768 + 0.919638i \(0.628482\pi\)
\(164\) 0 0
\(165\) 8.21730 0.639716
\(166\) 0 0
\(167\) −20.6275 −1.59621 −0.798103 0.602521i \(-0.794163\pi\)
−0.798103 + 0.602521i \(0.794163\pi\)
\(168\) 0 0
\(169\) 9.10515 0.700396
\(170\) 0 0
\(171\) −19.6017 −1.49898
\(172\) 0 0
\(173\) −12.1560 −0.924202 −0.462101 0.886827i \(-0.652904\pi\)
−0.462101 + 0.886827i \(0.652904\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 11.9969 0.901739
\(178\) 0 0
\(179\) 25.4334 1.90098 0.950492 0.310749i \(-0.100580\pi\)
0.950492 + 0.310749i \(0.100580\pi\)
\(180\) 0 0
\(181\) −6.06167 −0.450560 −0.225280 0.974294i \(-0.572330\pi\)
−0.225280 + 0.974294i \(0.572330\pi\)
\(182\) 0 0
\(183\) 10.9775 0.811483
\(184\) 0 0
\(185\) 4.79116 0.352253
\(186\) 0 0
\(187\) 2.66906 0.195181
\(188\) 0 0
\(189\) 1.19012 0.0865684
\(190\) 0 0
\(191\) 6.94351 0.502415 0.251207 0.967933i \(-0.419172\pi\)
0.251207 + 0.967933i \(0.419172\pi\)
\(192\) 0 0
\(193\) −10.7134 −0.771171 −0.385585 0.922672i \(-0.626000\pi\)
−0.385585 + 0.922672i \(0.626000\pi\)
\(194\) 0 0
\(195\) 11.9572 0.856274
\(196\) 0 0
\(197\) 4.86251 0.346440 0.173220 0.984883i \(-0.444583\pi\)
0.173220 + 0.984883i \(0.444583\pi\)
\(198\) 0 0
\(199\) 9.92150 0.703316 0.351658 0.936128i \(-0.385618\pi\)
0.351658 + 0.936128i \(0.385618\pi\)
\(200\) 0 0
\(201\) −29.7341 −2.09728
\(202\) 0 0
\(203\) 2.88745 0.202660
\(204\) 0 0
\(205\) −2.32201 −0.162176
\(206\) 0 0
\(207\) −27.6824 −1.92406
\(208\) 0 0
\(209\) 18.2627 1.26326
\(210\) 0 0
\(211\) 10.5109 0.723601 0.361800 0.932256i \(-0.382162\pi\)
0.361800 + 0.932256i \(0.382162\pi\)
\(212\) 0 0
\(213\) −5.25966 −0.360386
\(214\) 0 0
\(215\) 1.00000 0.0681994
\(216\) 0 0
\(217\) 8.99139 0.610375
\(218\) 0 0
\(219\) −15.9840 −1.08010
\(220\) 0 0
\(221\) 3.88382 0.261254
\(222\) 0 0
\(223\) −9.08545 −0.608407 −0.304203 0.952607i \(-0.598390\pi\)
−0.304203 + 0.952607i \(0.598390\pi\)
\(224\) 0 0
\(225\) 3.46796 0.231197
\(226\) 0 0
\(227\) 22.8035 1.51352 0.756760 0.653693i \(-0.226781\pi\)
0.756760 + 0.653693i \(0.226781\pi\)
\(228\) 0 0
\(229\) 2.21632 0.146458 0.0732291 0.997315i \(-0.476670\pi\)
0.0732291 + 0.997315i \(0.476670\pi\)
\(230\) 0 0
\(231\) −8.21730 −0.540658
\(232\) 0 0
\(233\) 12.1345 0.794957 0.397478 0.917612i \(-0.369885\pi\)
0.397478 + 0.917612i \(0.369885\pi\)
\(234\) 0 0
\(235\) 0.524113 0.0341893
\(236\) 0 0
\(237\) 34.8019 2.26062
\(238\) 0 0
\(239\) 0.846447 0.0547521 0.0273760 0.999625i \(-0.491285\pi\)
0.0273760 + 0.999625i \(0.491285\pi\)
\(240\) 0 0
\(241\) 11.8902 0.765917 0.382958 0.923766i \(-0.374905\pi\)
0.382958 + 0.923766i \(0.374905\pi\)
\(242\) 0 0
\(243\) 22.3320 1.43260
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 26.5745 1.69090
\(248\) 0 0
\(249\) −28.9939 −1.83741
\(250\) 0 0
\(251\) −1.32084 −0.0833709 −0.0416855 0.999131i \(-0.513273\pi\)
−0.0416855 + 0.999131i \(0.513273\pi\)
\(252\) 0 0
\(253\) 25.7914 1.62149
\(254\) 0 0
\(255\) 2.10085 0.131561
\(256\) 0 0
\(257\) 1.58738 0.0990178 0.0495089 0.998774i \(-0.484234\pi\)
0.0495089 + 0.998774i \(0.484234\pi\)
\(258\) 0 0
\(259\) −4.79116 −0.297708
\(260\) 0 0
\(261\) −10.0136 −0.619824
\(262\) 0 0
\(263\) −20.3740 −1.25632 −0.628158 0.778086i \(-0.716191\pi\)
−0.628158 + 0.778086i \(0.716191\pi\)
\(264\) 0 0
\(265\) −6.07174 −0.372984
\(266\) 0 0
\(267\) 29.9129 1.83064
\(268\) 0 0
\(269\) −2.01643 −0.122944 −0.0614718 0.998109i \(-0.519579\pi\)
−0.0614718 + 0.998109i \(0.519579\pi\)
\(270\) 0 0
\(271\) 5.44996 0.331062 0.165531 0.986205i \(-0.447066\pi\)
0.165531 + 0.986205i \(0.447066\pi\)
\(272\) 0 0
\(273\) −11.9572 −0.723684
\(274\) 0 0
\(275\) −3.23106 −0.194840
\(276\) 0 0
\(277\) 9.91936 0.595997 0.297998 0.954566i \(-0.403681\pi\)
0.297998 + 0.954566i \(0.403681\pi\)
\(278\) 0 0
\(279\) −31.1818 −1.86680
\(280\) 0 0
\(281\) −6.29128 −0.375306 −0.187653 0.982235i \(-0.560088\pi\)
−0.187653 + 0.982235i \(0.560088\pi\)
\(282\) 0 0
\(283\) 20.3795 1.21144 0.605719 0.795679i \(-0.292886\pi\)
0.605719 + 0.795679i \(0.292886\pi\)
\(284\) 0 0
\(285\) 14.3748 0.851491
\(286\) 0 0
\(287\) 2.32201 0.137064
\(288\) 0 0
\(289\) −16.3176 −0.959860
\(290\) 0 0
\(291\) −32.8125 −1.92350
\(292\) 0 0
\(293\) 20.3636 1.18966 0.594828 0.803853i \(-0.297220\pi\)
0.594828 + 0.803853i \(0.297220\pi\)
\(294\) 0 0
\(295\) −4.71720 −0.274646
\(296\) 0 0
\(297\) 3.84535 0.223130
\(298\) 0 0
\(299\) 37.5298 2.17040
\(300\) 0 0
\(301\) −1.00000 −0.0576390
\(302\) 0 0
\(303\) −6.73588 −0.386966
\(304\) 0 0
\(305\) −4.31640 −0.247156
\(306\) 0 0
\(307\) 17.9777 1.02604 0.513021 0.858376i \(-0.328526\pi\)
0.513021 + 0.858376i \(0.328526\pi\)
\(308\) 0 0
\(309\) 33.6451 1.91400
\(310\) 0 0
\(311\) −22.8813 −1.29748 −0.648740 0.761010i \(-0.724704\pi\)
−0.648740 + 0.761010i \(0.724704\pi\)
\(312\) 0 0
\(313\) −33.2726 −1.88068 −0.940339 0.340238i \(-0.889492\pi\)
−0.940339 + 0.340238i \(0.889492\pi\)
\(314\) 0 0
\(315\) −3.46796 −0.195397
\(316\) 0 0
\(317\) 5.06981 0.284749 0.142374 0.989813i \(-0.454526\pi\)
0.142374 + 0.989813i \(0.454526\pi\)
\(318\) 0 0
\(319\) 9.32955 0.522354
\(320\) 0 0
\(321\) 9.05980 0.505669
\(322\) 0 0
\(323\) 4.66908 0.259795
\(324\) 0 0
\(325\) −4.70161 −0.260798
\(326\) 0 0
\(327\) −45.3603 −2.50843
\(328\) 0 0
\(329\) −0.524113 −0.0288953
\(330\) 0 0
\(331\) 13.1196 0.721116 0.360558 0.932737i \(-0.382586\pi\)
0.360558 + 0.932737i \(0.382586\pi\)
\(332\) 0 0
\(333\) 16.6155 0.910526
\(334\) 0 0
\(335\) 11.6915 0.638776
\(336\) 0 0
\(337\) 21.7963 1.18732 0.593661 0.804715i \(-0.297682\pi\)
0.593661 + 0.804715i \(0.297682\pi\)
\(338\) 0 0
\(339\) −2.99005 −0.162397
\(340\) 0 0
\(341\) 29.0517 1.57324
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 20.3008 1.09296
\(346\) 0 0
\(347\) −3.70002 −0.198628 −0.0993138 0.995056i \(-0.531665\pi\)
−0.0993138 + 0.995056i \(0.531665\pi\)
\(348\) 0 0
\(349\) 18.3883 0.984301 0.492151 0.870510i \(-0.336211\pi\)
0.492151 + 0.870510i \(0.336211\pi\)
\(350\) 0 0
\(351\) 5.59548 0.298664
\(352\) 0 0
\(353\) −19.2852 −1.02645 −0.513225 0.858254i \(-0.671549\pi\)
−0.513225 + 0.858254i \(0.671549\pi\)
\(354\) 0 0
\(355\) 2.06811 0.109764
\(356\) 0 0
\(357\) −2.10085 −0.111189
\(358\) 0 0
\(359\) 2.86451 0.151183 0.0755915 0.997139i \(-0.475916\pi\)
0.0755915 + 0.997139i \(0.475916\pi\)
\(360\) 0 0
\(361\) 12.9476 0.681452
\(362\) 0 0
\(363\) 1.42480 0.0747824
\(364\) 0 0
\(365\) 6.28496 0.328970
\(366\) 0 0
\(367\) −26.9686 −1.40775 −0.703873 0.710325i \(-0.748548\pi\)
−0.703873 + 0.710325i \(0.748548\pi\)
\(368\) 0 0
\(369\) −8.05263 −0.419203
\(370\) 0 0
\(371\) 6.07174 0.315229
\(372\) 0 0
\(373\) 28.0224 1.45095 0.725473 0.688251i \(-0.241621\pi\)
0.725473 + 0.688251i \(0.241621\pi\)
\(374\) 0 0
\(375\) −2.54322 −0.131331
\(376\) 0 0
\(377\) 13.5757 0.699184
\(378\) 0 0
\(379\) 11.1535 0.572915 0.286458 0.958093i \(-0.407522\pi\)
0.286458 + 0.958093i \(0.407522\pi\)
\(380\) 0 0
\(381\) −6.59096 −0.337665
\(382\) 0 0
\(383\) −3.99036 −0.203898 −0.101949 0.994790i \(-0.532508\pi\)
−0.101949 + 0.994790i \(0.532508\pi\)
\(384\) 0 0
\(385\) 3.23106 0.164670
\(386\) 0 0
\(387\) 3.46796 0.176286
\(388\) 0 0
\(389\) 5.89279 0.298776 0.149388 0.988779i \(-0.452270\pi\)
0.149388 + 0.988779i \(0.452270\pi\)
\(390\) 0 0
\(391\) 6.59389 0.333467
\(392\) 0 0
\(393\) −6.45012 −0.325366
\(394\) 0 0
\(395\) −13.6842 −0.688526
\(396\) 0 0
\(397\) −17.3012 −0.868320 −0.434160 0.900836i \(-0.642955\pi\)
−0.434160 + 0.900836i \(0.642955\pi\)
\(398\) 0 0
\(399\) −14.3748 −0.719641
\(400\) 0 0
\(401\) −2.48895 −0.124292 −0.0621462 0.998067i \(-0.519795\pi\)
−0.0621462 + 0.998067i \(0.519795\pi\)
\(402\) 0 0
\(403\) 42.2740 2.10582
\(404\) 0 0
\(405\) −7.37714 −0.366573
\(406\) 0 0
\(407\) −15.4805 −0.767342
\(408\) 0 0
\(409\) −5.50827 −0.272367 −0.136183 0.990684i \(-0.543484\pi\)
−0.136183 + 0.990684i \(0.543484\pi\)
\(410\) 0 0
\(411\) −47.3780 −2.33698
\(412\) 0 0
\(413\) 4.71720 0.232118
\(414\) 0 0
\(415\) 11.4005 0.559628
\(416\) 0 0
\(417\) 47.7363 2.33766
\(418\) 0 0
\(419\) 14.2976 0.698485 0.349242 0.937032i \(-0.386439\pi\)
0.349242 + 0.937032i \(0.386439\pi\)
\(420\) 0 0
\(421\) −4.46188 −0.217459 −0.108729 0.994071i \(-0.534678\pi\)
−0.108729 + 0.994071i \(0.534678\pi\)
\(422\) 0 0
\(423\) 1.81760 0.0883748
\(424\) 0 0
\(425\) −0.826062 −0.0400699
\(426\) 0 0
\(427\) 4.31640 0.208885
\(428\) 0 0
\(429\) −38.6345 −1.86529
\(430\) 0 0
\(431\) 30.3253 1.46072 0.730359 0.683063i \(-0.239353\pi\)
0.730359 + 0.683063i \(0.239353\pi\)
\(432\) 0 0
\(433\) 38.7065 1.86011 0.930057 0.367414i \(-0.119757\pi\)
0.930057 + 0.367414i \(0.119757\pi\)
\(434\) 0 0
\(435\) 7.34343 0.352090
\(436\) 0 0
\(437\) 45.1178 2.15828
\(438\) 0 0
\(439\) 8.29330 0.395818 0.197909 0.980220i \(-0.436585\pi\)
0.197909 + 0.980220i \(0.436585\pi\)
\(440\) 0 0
\(441\) 3.46796 0.165141
\(442\) 0 0
\(443\) 7.64506 0.363228 0.181614 0.983370i \(-0.441868\pi\)
0.181614 + 0.983370i \(0.441868\pi\)
\(444\) 0 0
\(445\) −11.7618 −0.557565
\(446\) 0 0
\(447\) 48.8448 2.31028
\(448\) 0 0
\(449\) 14.5289 0.685659 0.342829 0.939398i \(-0.388615\pi\)
0.342829 + 0.939398i \(0.388615\pi\)
\(450\) 0 0
\(451\) 7.50256 0.353282
\(452\) 0 0
\(453\) 35.9986 1.69136
\(454\) 0 0
\(455\) 4.70161 0.220415
\(456\) 0 0
\(457\) 28.8478 1.34945 0.674723 0.738071i \(-0.264263\pi\)
0.674723 + 0.738071i \(0.264263\pi\)
\(458\) 0 0
\(459\) 0.983112 0.0458877
\(460\) 0 0
\(461\) 21.7276 1.01196 0.505978 0.862546i \(-0.331132\pi\)
0.505978 + 0.862546i \(0.331132\pi\)
\(462\) 0 0
\(463\) 27.1384 1.26123 0.630615 0.776096i \(-0.282803\pi\)
0.630615 + 0.776096i \(0.282803\pi\)
\(464\) 0 0
\(465\) 22.8671 1.06043
\(466\) 0 0
\(467\) 5.07991 0.235070 0.117535 0.993069i \(-0.462501\pi\)
0.117535 + 0.993069i \(0.462501\pi\)
\(468\) 0 0
\(469\) −11.6915 −0.539864
\(470\) 0 0
\(471\) 40.3616 1.85976
\(472\) 0 0
\(473\) −3.23106 −0.148564
\(474\) 0 0
\(475\) −5.65222 −0.259342
\(476\) 0 0
\(477\) −21.0565 −0.964113
\(478\) 0 0
\(479\) −14.4541 −0.660423 −0.330211 0.943907i \(-0.607120\pi\)
−0.330211 + 0.943907i \(0.607120\pi\)
\(480\) 0 0
\(481\) −22.5262 −1.02711
\(482\) 0 0
\(483\) −20.3008 −0.923718
\(484\) 0 0
\(485\) 12.9020 0.585848
\(486\) 0 0
\(487\) −17.1656 −0.777848 −0.388924 0.921270i \(-0.627153\pi\)
−0.388924 + 0.921270i \(0.627153\pi\)
\(488\) 0 0
\(489\) 25.5061 1.15342
\(490\) 0 0
\(491\) −16.4552 −0.742613 −0.371307 0.928510i \(-0.621090\pi\)
−0.371307 + 0.928510i \(0.621090\pi\)
\(492\) 0 0
\(493\) 2.38522 0.107425
\(494\) 0 0
\(495\) −11.2052 −0.503636
\(496\) 0 0
\(497\) −2.06811 −0.0927674
\(498\) 0 0
\(499\) −19.1019 −0.855120 −0.427560 0.903987i \(-0.640627\pi\)
−0.427560 + 0.903987i \(0.640627\pi\)
\(500\) 0 0
\(501\) 52.4603 2.34375
\(502\) 0 0
\(503\) −19.3764 −0.863952 −0.431976 0.901885i \(-0.642183\pi\)
−0.431976 + 0.901885i \(0.642183\pi\)
\(504\) 0 0
\(505\) 2.64856 0.117860
\(506\) 0 0
\(507\) −23.1564 −1.02841
\(508\) 0 0
\(509\) 6.30352 0.279399 0.139699 0.990194i \(-0.455386\pi\)
0.139699 + 0.990194i \(0.455386\pi\)
\(510\) 0 0
\(511\) −6.28496 −0.278030
\(512\) 0 0
\(513\) 6.72681 0.296996
\(514\) 0 0
\(515\) −13.2293 −0.582954
\(516\) 0 0
\(517\) −1.69344 −0.0744775
\(518\) 0 0
\(519\) 30.9153 1.35703
\(520\) 0 0
\(521\) −6.52354 −0.285801 −0.142901 0.989737i \(-0.545643\pi\)
−0.142901 + 0.989737i \(0.545643\pi\)
\(522\) 0 0
\(523\) −22.4101 −0.979925 −0.489963 0.871743i \(-0.662990\pi\)
−0.489963 + 0.871743i \(0.662990\pi\)
\(524\) 0 0
\(525\) 2.54322 0.110995
\(526\) 0 0
\(527\) 7.42744 0.323544
\(528\) 0 0
\(529\) 40.7175 1.77033
\(530\) 0 0
\(531\) −16.3590 −0.709922
\(532\) 0 0
\(533\) 10.9172 0.472876
\(534\) 0 0
\(535\) −3.56234 −0.154013
\(536\) 0 0
\(537\) −64.6828 −2.79127
\(538\) 0 0
\(539\) −3.23106 −0.139172
\(540\) 0 0
\(541\) −34.1446 −1.46799 −0.733996 0.679154i \(-0.762347\pi\)
−0.733996 + 0.679154i \(0.762347\pi\)
\(542\) 0 0
\(543\) 15.4161 0.661570
\(544\) 0 0
\(545\) 17.8358 0.764001
\(546\) 0 0
\(547\) −24.1558 −1.03283 −0.516413 0.856339i \(-0.672733\pi\)
−0.516413 + 0.856339i \(0.672733\pi\)
\(548\) 0 0
\(549\) −14.9691 −0.638865
\(550\) 0 0
\(551\) 16.3205 0.695278
\(552\) 0 0
\(553\) 13.6842 0.581911
\(554\) 0 0
\(555\) −12.1850 −0.517223
\(556\) 0 0
\(557\) 11.1381 0.471938 0.235969 0.971761i \(-0.424174\pi\)
0.235969 + 0.971761i \(0.424174\pi\)
\(558\) 0 0
\(559\) −4.70161 −0.198857
\(560\) 0 0
\(561\) −6.78799 −0.286589
\(562\) 0 0
\(563\) −42.4073 −1.78726 −0.893628 0.448808i \(-0.851849\pi\)
−0.893628 + 0.448808i \(0.851849\pi\)
\(564\) 0 0
\(565\) 1.17570 0.0494619
\(566\) 0 0
\(567\) 7.37714 0.309811
\(568\) 0 0
\(569\) −28.5083 −1.19513 −0.597566 0.801820i \(-0.703865\pi\)
−0.597566 + 0.801820i \(0.703865\pi\)
\(570\) 0 0
\(571\) 27.3986 1.14659 0.573297 0.819347i \(-0.305664\pi\)
0.573297 + 0.819347i \(0.305664\pi\)
\(572\) 0 0
\(573\) −17.6589 −0.737709
\(574\) 0 0
\(575\) −7.98232 −0.332886
\(576\) 0 0
\(577\) −5.16264 −0.214924 −0.107462 0.994209i \(-0.534272\pi\)
−0.107462 + 0.994209i \(0.534272\pi\)
\(578\) 0 0
\(579\) 27.2466 1.13233
\(580\) 0 0
\(581\) −11.4005 −0.472972
\(582\) 0 0
\(583\) 19.6182 0.812502
\(584\) 0 0
\(585\) −16.3050 −0.674128
\(586\) 0 0
\(587\) −29.4702 −1.21637 −0.608183 0.793796i \(-0.708102\pi\)
−0.608183 + 0.793796i \(0.708102\pi\)
\(588\) 0 0
\(589\) 50.8213 2.09406
\(590\) 0 0
\(591\) −12.3664 −0.508687
\(592\) 0 0
\(593\) 40.8935 1.67930 0.839648 0.543131i \(-0.182761\pi\)
0.839648 + 0.543131i \(0.182761\pi\)
\(594\) 0 0
\(595\) 0.826062 0.0338652
\(596\) 0 0
\(597\) −25.2325 −1.03270
\(598\) 0 0
\(599\) −1.90595 −0.0778751 −0.0389376 0.999242i \(-0.512397\pi\)
−0.0389376 + 0.999242i \(0.512397\pi\)
\(600\) 0 0
\(601\) −26.6359 −1.08650 −0.543251 0.839570i \(-0.682807\pi\)
−0.543251 + 0.839570i \(0.682807\pi\)
\(602\) 0 0
\(603\) 40.5457 1.65115
\(604\) 0 0
\(605\) −0.560233 −0.0227767
\(606\) 0 0
\(607\) −37.6611 −1.52862 −0.764309 0.644850i \(-0.776920\pi\)
−0.764309 + 0.644850i \(0.776920\pi\)
\(608\) 0 0
\(609\) −7.34343 −0.297571
\(610\) 0 0
\(611\) −2.46417 −0.0996898
\(612\) 0 0
\(613\) 24.5251 0.990558 0.495279 0.868734i \(-0.335066\pi\)
0.495279 + 0.868734i \(0.335066\pi\)
\(614\) 0 0
\(615\) 5.90538 0.238128
\(616\) 0 0
\(617\) 26.5372 1.06835 0.534174 0.845374i \(-0.320623\pi\)
0.534174 + 0.845374i \(0.320623\pi\)
\(618\) 0 0
\(619\) 12.4489 0.500363 0.250181 0.968199i \(-0.419510\pi\)
0.250181 + 0.968199i \(0.419510\pi\)
\(620\) 0 0
\(621\) 9.49991 0.381218
\(622\) 0 0
\(623\) 11.7618 0.471228
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −46.4460 −1.85487
\(628\) 0 0
\(629\) −3.95780 −0.157808
\(630\) 0 0
\(631\) −10.0114 −0.398546 −0.199273 0.979944i \(-0.563858\pi\)
−0.199273 + 0.979944i \(0.563858\pi\)
\(632\) 0 0
\(633\) −26.7315 −1.06248
\(634\) 0 0
\(635\) 2.59158 0.102844
\(636\) 0 0
\(637\) −4.70161 −0.186285
\(638\) 0 0
\(639\) 7.17212 0.283725
\(640\) 0 0
\(641\) −35.0043 −1.38259 −0.691294 0.722574i \(-0.742959\pi\)
−0.691294 + 0.722574i \(0.742959\pi\)
\(642\) 0 0
\(643\) 9.41630 0.371342 0.185671 0.982612i \(-0.440554\pi\)
0.185671 + 0.982612i \(0.440554\pi\)
\(644\) 0 0
\(645\) −2.54322 −0.100139
\(646\) 0 0
\(647\) −38.2706 −1.50457 −0.752286 0.658836i \(-0.771049\pi\)
−0.752286 + 0.658836i \(0.771049\pi\)
\(648\) 0 0
\(649\) 15.2416 0.598283
\(650\) 0 0
\(651\) −22.8671 −0.896231
\(652\) 0 0
\(653\) 1.55237 0.0607489 0.0303745 0.999539i \(-0.490330\pi\)
0.0303745 + 0.999539i \(0.490330\pi\)
\(654\) 0 0
\(655\) 2.53621 0.0990978
\(656\) 0 0
\(657\) 21.7960 0.850341
\(658\) 0 0
\(659\) 15.3540 0.598107 0.299054 0.954236i \(-0.403329\pi\)
0.299054 + 0.954236i \(0.403329\pi\)
\(660\) 0 0
\(661\) −19.7984 −0.770069 −0.385035 0.922902i \(-0.625810\pi\)
−0.385035 + 0.922902i \(0.625810\pi\)
\(662\) 0 0
\(663\) −9.87740 −0.383607
\(664\) 0 0
\(665\) 5.65222 0.219184
\(666\) 0 0
\(667\) 23.0486 0.892445
\(668\) 0 0
\(669\) 23.1063 0.893340
\(670\) 0 0
\(671\) 13.9466 0.538401
\(672\) 0 0
\(673\) 27.5922 1.06360 0.531801 0.846869i \(-0.321515\pi\)
0.531801 + 0.846869i \(0.321515\pi\)
\(674\) 0 0
\(675\) −1.19012 −0.0458077
\(676\) 0 0
\(677\) −1.31356 −0.0504841 −0.0252421 0.999681i \(-0.508036\pi\)
−0.0252421 + 0.999681i \(0.508036\pi\)
\(678\) 0 0
\(679\) −12.9020 −0.495132
\(680\) 0 0
\(681\) −57.9942 −2.22234
\(682\) 0 0
\(683\) 32.0362 1.22583 0.612916 0.790148i \(-0.289996\pi\)
0.612916 + 0.790148i \(0.289996\pi\)
\(684\) 0 0
\(685\) 18.6291 0.711783
\(686\) 0 0
\(687\) −5.63657 −0.215049
\(688\) 0 0
\(689\) 28.5470 1.08755
\(690\) 0 0
\(691\) 16.0286 0.609755 0.304878 0.952392i \(-0.401384\pi\)
0.304878 + 0.952392i \(0.401384\pi\)
\(692\) 0 0
\(693\) 11.2052 0.425650
\(694\) 0 0
\(695\) −18.7700 −0.711988
\(696\) 0 0
\(697\) 1.91812 0.0726541
\(698\) 0 0
\(699\) −30.8606 −1.16726
\(700\) 0 0
\(701\) −41.8591 −1.58100 −0.790498 0.612464i \(-0.790178\pi\)
−0.790498 + 0.612464i \(0.790178\pi\)
\(702\) 0 0
\(703\) −27.0807 −1.02137
\(704\) 0 0
\(705\) −1.33293 −0.0502012
\(706\) 0 0
\(707\) −2.64856 −0.0996095
\(708\) 0 0
\(709\) 23.1658 0.870010 0.435005 0.900428i \(-0.356747\pi\)
0.435005 + 0.900428i \(0.356747\pi\)
\(710\) 0 0
\(711\) −47.4562 −1.77975
\(712\) 0 0
\(713\) 71.7722 2.68789
\(714\) 0 0
\(715\) 15.1912 0.568119
\(716\) 0 0
\(717\) −2.15270 −0.0803940
\(718\) 0 0
\(719\) −6.82345 −0.254472 −0.127236 0.991872i \(-0.540611\pi\)
−0.127236 + 0.991872i \(0.540611\pi\)
\(720\) 0 0
\(721\) 13.2293 0.492686
\(722\) 0 0
\(723\) −30.2394 −1.12462
\(724\) 0 0
\(725\) −2.88745 −0.107237
\(726\) 0 0
\(727\) 36.5855 1.35688 0.678440 0.734656i \(-0.262656\pi\)
0.678440 + 0.734656i \(0.262656\pi\)
\(728\) 0 0
\(729\) −34.6638 −1.28384
\(730\) 0 0
\(731\) −0.826062 −0.0305530
\(732\) 0 0
\(733\) −6.20968 −0.229360 −0.114680 0.993402i \(-0.536584\pi\)
−0.114680 + 0.993402i \(0.536584\pi\)
\(734\) 0 0
\(735\) −2.54322 −0.0938080
\(736\) 0 0
\(737\) −37.7760 −1.39150
\(738\) 0 0
\(739\) 3.70123 0.136152 0.0680760 0.997680i \(-0.478314\pi\)
0.0680760 + 0.997680i \(0.478314\pi\)
\(740\) 0 0
\(741\) −67.5848 −2.48279
\(742\) 0 0
\(743\) −35.9570 −1.31913 −0.659566 0.751646i \(-0.729260\pi\)
−0.659566 + 0.751646i \(0.729260\pi\)
\(744\) 0 0
\(745\) −19.2059 −0.703649
\(746\) 0 0
\(747\) 39.5364 1.44656
\(748\) 0 0
\(749\) 3.56234 0.130165
\(750\) 0 0
\(751\) 35.3333 1.28933 0.644666 0.764464i \(-0.276996\pi\)
0.644666 + 0.764464i \(0.276996\pi\)
\(752\) 0 0
\(753\) 3.35919 0.122416
\(754\) 0 0
\(755\) −14.1547 −0.515143
\(756\) 0 0
\(757\) −30.5795 −1.11143 −0.555716 0.831373i \(-0.687556\pi\)
−0.555716 + 0.831373i \(0.687556\pi\)
\(758\) 0 0
\(759\) −65.5931 −2.38088
\(760\) 0 0
\(761\) −24.7467 −0.897067 −0.448533 0.893766i \(-0.648054\pi\)
−0.448533 + 0.893766i \(0.648054\pi\)
\(762\) 0 0
\(763\) −17.8358 −0.645699
\(764\) 0 0
\(765\) −2.86475 −0.103575
\(766\) 0 0
\(767\) 22.1784 0.800816
\(768\) 0 0
\(769\) 39.6176 1.42865 0.714324 0.699815i \(-0.246734\pi\)
0.714324 + 0.699815i \(0.246734\pi\)
\(770\) 0 0
\(771\) −4.03704 −0.145391
\(772\) 0 0
\(773\) −4.70111 −0.169087 −0.0845435 0.996420i \(-0.526943\pi\)
−0.0845435 + 0.996420i \(0.526943\pi\)
\(774\) 0 0
\(775\) −8.99139 −0.322980
\(776\) 0 0
\(777\) 12.1850 0.437133
\(778\) 0 0
\(779\) 13.1245 0.470235
\(780\) 0 0
\(781\) −6.68219 −0.239108
\(782\) 0 0
\(783\) 3.43641 0.122807
\(784\) 0 0
\(785\) −15.8703 −0.566434
\(786\) 0 0
\(787\) −4.24502 −0.151319 −0.0756594 0.997134i \(-0.524106\pi\)
−0.0756594 + 0.997134i \(0.524106\pi\)
\(788\) 0 0
\(789\) 51.8156 1.84468
\(790\) 0 0
\(791\) −1.17570 −0.0418029
\(792\) 0 0
\(793\) 20.2940 0.720662
\(794\) 0 0
\(795\) 15.4418 0.547663
\(796\) 0 0
\(797\) 16.3478 0.579069 0.289535 0.957168i \(-0.406500\pi\)
0.289535 + 0.957168i \(0.406500\pi\)
\(798\) 0 0
\(799\) −0.432949 −0.0153166
\(800\) 0 0
\(801\) −40.7896 −1.44123
\(802\) 0 0
\(803\) −20.3071 −0.716621
\(804\) 0 0
\(805\) 7.98232 0.281340
\(806\) 0 0
\(807\) 5.12821 0.180522
\(808\) 0 0
\(809\) −23.6565 −0.831719 −0.415860 0.909429i \(-0.636519\pi\)
−0.415860 + 0.909429i \(0.636519\pi\)
\(810\) 0 0
\(811\) −8.78363 −0.308435 −0.154217 0.988037i \(-0.549286\pi\)
−0.154217 + 0.988037i \(0.549286\pi\)
\(812\) 0 0
\(813\) −13.8604 −0.486107
\(814\) 0 0
\(815\) −10.0290 −0.351302
\(816\) 0 0
\(817\) −5.65222 −0.197746
\(818\) 0 0
\(819\) 16.3050 0.569742
\(820\) 0 0
\(821\) −1.55306 −0.0542023 −0.0271011 0.999633i \(-0.508628\pi\)
−0.0271011 + 0.999633i \(0.508628\pi\)
\(822\) 0 0
\(823\) 50.7299 1.76833 0.884166 0.467173i \(-0.154727\pi\)
0.884166 + 0.467173i \(0.154727\pi\)
\(824\) 0 0
\(825\) 8.21730 0.286090
\(826\) 0 0
\(827\) 8.92543 0.310368 0.155184 0.987886i \(-0.450403\pi\)
0.155184 + 0.987886i \(0.450403\pi\)
\(828\) 0 0
\(829\) −30.0578 −1.04395 −0.521975 0.852961i \(-0.674805\pi\)
−0.521975 + 0.852961i \(0.674805\pi\)
\(830\) 0 0
\(831\) −25.2271 −0.875118
\(832\) 0 0
\(833\) −0.826062 −0.0286213
\(834\) 0 0
\(835\) −20.6275 −0.713845
\(836\) 0 0
\(837\) 10.7008 0.369875
\(838\) 0 0
\(839\) 33.6236 1.16082 0.580408 0.814326i \(-0.302893\pi\)
0.580408 + 0.814326i \(0.302893\pi\)
\(840\) 0 0
\(841\) −20.6626 −0.712504
\(842\) 0 0
\(843\) 16.0001 0.551072
\(844\) 0 0
\(845\) 9.10515 0.313227
\(846\) 0 0
\(847\) 0.560233 0.0192498
\(848\) 0 0
\(849\) −51.8296 −1.77879
\(850\) 0 0
\(851\) −38.2446 −1.31101
\(852\) 0 0
\(853\) 23.3386 0.799098 0.399549 0.916712i \(-0.369167\pi\)
0.399549 + 0.916712i \(0.369167\pi\)
\(854\) 0 0
\(855\) −19.6017 −0.670363
\(856\) 0 0
\(857\) 2.38382 0.0814298 0.0407149 0.999171i \(-0.487036\pi\)
0.0407149 + 0.999171i \(0.487036\pi\)
\(858\) 0 0
\(859\) −8.69240 −0.296581 −0.148290 0.988944i \(-0.547377\pi\)
−0.148290 + 0.988944i \(0.547377\pi\)
\(860\) 0 0
\(861\) −5.90538 −0.201255
\(862\) 0 0
\(863\) 53.6653 1.82679 0.913393 0.407078i \(-0.133452\pi\)
0.913393 + 0.407078i \(0.133452\pi\)
\(864\) 0 0
\(865\) −12.1560 −0.413316
\(866\) 0 0
\(867\) 41.4993 1.40939
\(868\) 0 0
\(869\) 44.2145 1.49987
\(870\) 0 0
\(871\) −54.9690 −1.86255
\(872\) 0 0
\(873\) 44.7435 1.51434
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −46.1380 −1.55797 −0.778985 0.627043i \(-0.784265\pi\)
−0.778985 + 0.627043i \(0.784265\pi\)
\(878\) 0 0
\(879\) −51.7892 −1.74681
\(880\) 0 0
\(881\) −22.1977 −0.747858 −0.373929 0.927457i \(-0.621990\pi\)
−0.373929 + 0.927457i \(0.621990\pi\)
\(882\) 0 0
\(883\) 28.6248 0.963301 0.481650 0.876363i \(-0.340038\pi\)
0.481650 + 0.876363i \(0.340038\pi\)
\(884\) 0 0
\(885\) 11.9969 0.403270
\(886\) 0 0
\(887\) 39.9140 1.34018 0.670091 0.742279i \(-0.266255\pi\)
0.670091 + 0.742279i \(0.266255\pi\)
\(888\) 0 0
\(889\) −2.59158 −0.0869189
\(890\) 0 0
\(891\) 23.8360 0.798536
\(892\) 0 0
\(893\) −2.96240 −0.0991329
\(894\) 0 0
\(895\) 25.4334 0.850146
\(896\) 0 0
\(897\) −95.4464 −3.18686
\(898\) 0 0
\(899\) 25.9622 0.865889
\(900\) 0 0
\(901\) 5.01563 0.167095
\(902\) 0 0
\(903\) 2.54322 0.0846330
\(904\) 0 0
\(905\) −6.06167 −0.201497
\(906\) 0 0
\(907\) 27.8337 0.924203 0.462101 0.886827i \(-0.347096\pi\)
0.462101 + 0.886827i \(0.347096\pi\)
\(908\) 0 0
\(909\) 9.18511 0.304651
\(910\) 0 0
\(911\) −2.81976 −0.0934228 −0.0467114 0.998908i \(-0.514874\pi\)
−0.0467114 + 0.998908i \(0.514874\pi\)
\(912\) 0 0
\(913\) −36.8357 −1.21908
\(914\) 0 0
\(915\) 10.9775 0.362906
\(916\) 0 0
\(917\) −2.53621 −0.0837529
\(918\) 0 0
\(919\) 44.3219 1.46204 0.731022 0.682354i \(-0.239044\pi\)
0.731022 + 0.682354i \(0.239044\pi\)
\(920\) 0 0
\(921\) −45.7212 −1.50656
\(922\) 0 0
\(923\) −9.72345 −0.320051
\(924\) 0 0
\(925\) 4.79116 0.157532
\(926\) 0 0
\(927\) −45.8788 −1.50686
\(928\) 0 0
\(929\) 8.54170 0.280244 0.140122 0.990134i \(-0.455251\pi\)
0.140122 + 0.990134i \(0.455251\pi\)
\(930\) 0 0
\(931\) −5.65222 −0.185244
\(932\) 0 0
\(933\) 58.1922 1.90513
\(934\) 0 0
\(935\) 2.66906 0.0872875
\(936\) 0 0
\(937\) −1.35442 −0.0442469 −0.0221234 0.999755i \(-0.507043\pi\)
−0.0221234 + 0.999755i \(0.507043\pi\)
\(938\) 0 0
\(939\) 84.6195 2.76145
\(940\) 0 0
\(941\) −36.3210 −1.18403 −0.592016 0.805926i \(-0.701668\pi\)
−0.592016 + 0.805926i \(0.701668\pi\)
\(942\) 0 0
\(943\) 18.5350 0.603584
\(944\) 0 0
\(945\) 1.19012 0.0387146
\(946\) 0 0
\(947\) −57.4358 −1.86641 −0.933207 0.359339i \(-0.883002\pi\)
−0.933207 + 0.359339i \(0.883002\pi\)
\(948\) 0 0
\(949\) −29.5494 −0.959215
\(950\) 0 0
\(951\) −12.8936 −0.418105
\(952\) 0 0
\(953\) −39.6811 −1.28540 −0.642698 0.766120i \(-0.722185\pi\)
−0.642698 + 0.766120i \(0.722185\pi\)
\(954\) 0 0
\(955\) 6.94351 0.224687
\(956\) 0 0
\(957\) −23.7271 −0.766987
\(958\) 0 0
\(959\) −18.6291 −0.601567
\(960\) 0 0
\(961\) 49.8451 1.60791
\(962\) 0 0
\(963\) −12.3540 −0.398103
\(964\) 0 0
\(965\) −10.7134 −0.344878
\(966\) 0 0
\(967\) −42.4196 −1.36412 −0.682061 0.731295i \(-0.738916\pi\)
−0.682061 + 0.731295i \(0.738916\pi\)
\(968\) 0 0
\(969\) −11.8745 −0.381464
\(970\) 0 0
\(971\) −37.5379 −1.20465 −0.602324 0.798252i \(-0.705758\pi\)
−0.602324 + 0.798252i \(0.705758\pi\)
\(972\) 0 0
\(973\) 18.7700 0.601740
\(974\) 0 0
\(975\) 11.9572 0.382938
\(976\) 0 0
\(977\) −37.3123 −1.19373 −0.596864 0.802343i \(-0.703587\pi\)
−0.596864 + 0.802343i \(0.703587\pi\)
\(978\) 0 0
\(979\) 38.0032 1.21459
\(980\) 0 0
\(981\) 61.8537 1.97484
\(982\) 0 0
\(983\) −11.6499 −0.371574 −0.185787 0.982590i \(-0.559484\pi\)
−0.185787 + 0.982590i \(0.559484\pi\)
\(984\) 0 0
\(985\) 4.86251 0.154933
\(986\) 0 0
\(987\) 1.33293 0.0424277
\(988\) 0 0
\(989\) −7.98232 −0.253823
\(990\) 0 0
\(991\) −20.3456 −0.646298 −0.323149 0.946348i \(-0.604742\pi\)
−0.323149 + 0.946348i \(0.604742\pi\)
\(992\) 0 0
\(993\) −33.3659 −1.05883
\(994\) 0 0
\(995\) 9.92150 0.314533
\(996\) 0 0
\(997\) −42.2181 −1.33706 −0.668530 0.743685i \(-0.733076\pi\)
−0.668530 + 0.743685i \(0.733076\pi\)
\(998\) 0 0
\(999\) −5.70205 −0.180405
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.k.1.2 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6020.2.a.k.1.2 13 1.1 even 1 trivial