Properties

Label 8020.2.a.d.1.15
Level $8020$
Weight $2$
Character 8020.1
Self dual yes
Analytic conductor $64.040$
Analytic rank $1$
Dimension $29$
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: \(1\)
Dimension: \(29\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
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)
 
\(f(q)\) \(=\) \(q-0.107530 q^{3} +1.00000 q^{5} +2.68164 q^{7} -2.98844 q^{9} +O(q^{10})\) \(q-0.107530 q^{3} +1.00000 q^{5} +2.68164 q^{7} -2.98844 q^{9} +4.61949 q^{11} -2.28176 q^{13} -0.107530 q^{15} +1.13181 q^{17} -2.01120 q^{19} -0.288359 q^{21} -6.89522 q^{23} +1.00000 q^{25} +0.643940 q^{27} -3.21723 q^{29} +3.50595 q^{31} -0.496736 q^{33} +2.68164 q^{35} -2.91993 q^{37} +0.245358 q^{39} -9.50402 q^{41} -5.26560 q^{43} -2.98844 q^{45} -11.2358 q^{47} +0.191216 q^{49} -0.121704 q^{51} +2.54571 q^{53} +4.61949 q^{55} +0.216265 q^{57} -13.4719 q^{59} +8.98971 q^{61} -8.01393 q^{63} -2.28176 q^{65} -10.4558 q^{67} +0.741446 q^{69} +0.519620 q^{71} -0.240208 q^{73} -0.107530 q^{75} +12.3878 q^{77} +13.3375 q^{79} +8.89607 q^{81} +0.442286 q^{83} +1.13181 q^{85} +0.345950 q^{87} -18.4216 q^{89} -6.11886 q^{91} -0.376997 q^{93} -2.01120 q^{95} +14.0002 q^{97} -13.8051 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9} + 2 q^{11} - 23 q^{13} - 3 q^{15} - 30 q^{17} - 6 q^{19} - 16 q^{21} - 21 q^{23} + 29 q^{25} - 15 q^{27} - 35 q^{29} - 7 q^{31} - 36 q^{33} - 8 q^{35} - 31 q^{37} - 11 q^{39} - 24 q^{41} - 17 q^{43} + 10 q^{45} - 17 q^{47} + q^{49} + 8 q^{51} - 57 q^{53} + 2 q^{55} - 46 q^{57} - 9 q^{59} - 27 q^{61} - 34 q^{63} - 23 q^{65} - 21 q^{67} - 28 q^{69} - 19 q^{71} - 81 q^{73} - 3 q^{75} - 66 q^{77} - 17 q^{79} - 39 q^{81} - 30 q^{83} - 30 q^{85} - 20 q^{87} - 38 q^{89} + q^{91} - 75 q^{93} - 6 q^{95} - 48 q^{97} - 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.107530 −0.0620828 −0.0310414 0.999518i \(-0.509882\pi\)
−0.0310414 + 0.999518i \(0.509882\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.68164 1.01357 0.506783 0.862074i \(-0.330834\pi\)
0.506783 + 0.862074i \(0.330834\pi\)
\(8\) 0 0
\(9\) −2.98844 −0.996146
\(10\) 0 0
\(11\) 4.61949 1.39283 0.696414 0.717640i \(-0.254778\pi\)
0.696414 + 0.717640i \(0.254778\pi\)
\(12\) 0 0
\(13\) −2.28176 −0.632845 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(14\) 0 0
\(15\) −0.107530 −0.0277643
\(16\) 0 0
\(17\) 1.13181 0.274504 0.137252 0.990536i \(-0.456173\pi\)
0.137252 + 0.990536i \(0.456173\pi\)
\(18\) 0 0
\(19\) −2.01120 −0.461401 −0.230700 0.973025i \(-0.574102\pi\)
−0.230700 + 0.973025i \(0.574102\pi\)
\(20\) 0 0
\(21\) −0.288359 −0.0629250
\(22\) 0 0
\(23\) −6.89522 −1.43775 −0.718876 0.695138i \(-0.755343\pi\)
−0.718876 + 0.695138i \(0.755343\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0.643940 0.123926
\(28\) 0 0
\(29\) −3.21723 −0.597424 −0.298712 0.954343i \(-0.596557\pi\)
−0.298712 + 0.954343i \(0.596557\pi\)
\(30\) 0 0
\(31\) 3.50595 0.629688 0.314844 0.949143i \(-0.398048\pi\)
0.314844 + 0.949143i \(0.398048\pi\)
\(32\) 0 0
\(33\) −0.496736 −0.0864707
\(34\) 0 0
\(35\) 2.68164 0.453281
\(36\) 0 0
\(37\) −2.91993 −0.480034 −0.240017 0.970769i \(-0.577153\pi\)
−0.240017 + 0.970769i \(0.577153\pi\)
\(38\) 0 0
\(39\) 0.245358 0.0392888
\(40\) 0 0
\(41\) −9.50402 −1.48428 −0.742139 0.670246i \(-0.766189\pi\)
−0.742139 + 0.670246i \(0.766189\pi\)
\(42\) 0 0
\(43\) −5.26560 −0.802996 −0.401498 0.915860i \(-0.631510\pi\)
−0.401498 + 0.915860i \(0.631510\pi\)
\(44\) 0 0
\(45\) −2.98844 −0.445490
\(46\) 0 0
\(47\) −11.2358 −1.63890 −0.819451 0.573149i \(-0.805722\pi\)
−0.819451 + 0.573149i \(0.805722\pi\)
\(48\) 0 0
\(49\) 0.191216 0.0273166
\(50\) 0 0
\(51\) −0.121704 −0.0170420
\(52\) 0 0
\(53\) 2.54571 0.349680 0.174840 0.984597i \(-0.444059\pi\)
0.174840 + 0.984597i \(0.444059\pi\)
\(54\) 0 0
\(55\) 4.61949 0.622892
\(56\) 0 0
\(57\) 0.216265 0.0286450
\(58\) 0 0
\(59\) −13.4719 −1.75389 −0.876946 0.480589i \(-0.840423\pi\)
−0.876946 + 0.480589i \(0.840423\pi\)
\(60\) 0 0
\(61\) 8.98971 1.15101 0.575507 0.817797i \(-0.304805\pi\)
0.575507 + 0.817797i \(0.304805\pi\)
\(62\) 0 0
\(63\) −8.01393 −1.00966
\(64\) 0 0
\(65\) −2.28176 −0.283017
\(66\) 0 0
\(67\) −10.4558 −1.27738 −0.638688 0.769466i \(-0.720522\pi\)
−0.638688 + 0.769466i \(0.720522\pi\)
\(68\) 0 0
\(69\) 0.741446 0.0892596
\(70\) 0 0
\(71\) 0.519620 0.0616676 0.0308338 0.999525i \(-0.490184\pi\)
0.0308338 + 0.999525i \(0.490184\pi\)
\(72\) 0 0
\(73\) −0.240208 −0.0281142 −0.0140571 0.999901i \(-0.504475\pi\)
−0.0140571 + 0.999901i \(0.504475\pi\)
\(74\) 0 0
\(75\) −0.107530 −0.0124166
\(76\) 0 0
\(77\) 12.3878 1.41172
\(78\) 0 0
\(79\) 13.3375 1.50059 0.750294 0.661105i \(-0.229912\pi\)
0.750294 + 0.661105i \(0.229912\pi\)
\(80\) 0 0
\(81\) 8.89607 0.988452
\(82\) 0 0
\(83\) 0.442286 0.0485472 0.0242736 0.999705i \(-0.492273\pi\)
0.0242736 + 0.999705i \(0.492273\pi\)
\(84\) 0 0
\(85\) 1.13181 0.122762
\(86\) 0 0
\(87\) 0.345950 0.0370897
\(88\) 0 0
\(89\) −18.4216 −1.95268 −0.976341 0.216236i \(-0.930622\pi\)
−0.976341 + 0.216236i \(0.930622\pi\)
\(90\) 0 0
\(91\) −6.11886 −0.641431
\(92\) 0 0
\(93\) −0.376997 −0.0390927
\(94\) 0 0
\(95\) −2.01120 −0.206345
\(96\) 0 0
\(97\) 14.0002 1.42151 0.710754 0.703441i \(-0.248354\pi\)
0.710754 + 0.703441i \(0.248354\pi\)
\(98\) 0 0
\(99\) −13.8051 −1.38746
\(100\) 0 0
\(101\) −4.39922 −0.437739 −0.218869 0.975754i \(-0.570237\pi\)
−0.218869 + 0.975754i \(0.570237\pi\)
\(102\) 0 0
\(103\) −16.2889 −1.60499 −0.802495 0.596659i \(-0.796495\pi\)
−0.802495 + 0.596659i \(0.796495\pi\)
\(104\) 0 0
\(105\) −0.288359 −0.0281409
\(106\) 0 0
\(107\) −2.52153 −0.243765 −0.121883 0.992545i \(-0.538893\pi\)
−0.121883 + 0.992545i \(0.538893\pi\)
\(108\) 0 0
\(109\) −0.284536 −0.0272536 −0.0136268 0.999907i \(-0.504338\pi\)
−0.0136268 + 0.999907i \(0.504338\pi\)
\(110\) 0 0
\(111\) 0.313982 0.0298018
\(112\) 0 0
\(113\) 14.6911 1.38203 0.691013 0.722842i \(-0.257165\pi\)
0.691013 + 0.722842i \(0.257165\pi\)
\(114\) 0 0
\(115\) −6.89522 −0.642983
\(116\) 0 0
\(117\) 6.81888 0.630406
\(118\) 0 0
\(119\) 3.03511 0.278228
\(120\) 0 0
\(121\) 10.3397 0.939973
\(122\) 0 0
\(123\) 1.02197 0.0921481
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −16.4713 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(128\) 0 0
\(129\) 0.566212 0.0498522
\(130\) 0 0
\(131\) 12.9557 1.13194 0.565971 0.824425i \(-0.308501\pi\)
0.565971 + 0.824425i \(0.308501\pi\)
\(132\) 0 0
\(133\) −5.39332 −0.467660
\(134\) 0 0
\(135\) 0.643940 0.0554215
\(136\) 0 0
\(137\) 2.17677 0.185974 0.0929871 0.995667i \(-0.470358\pi\)
0.0929871 + 0.995667i \(0.470358\pi\)
\(138\) 0 0
\(139\) 2.77707 0.235548 0.117774 0.993040i \(-0.462424\pi\)
0.117774 + 0.993040i \(0.462424\pi\)
\(140\) 0 0
\(141\) 1.20819 0.101748
\(142\) 0 0
\(143\) −10.5406 −0.881445
\(144\) 0 0
\(145\) −3.21723 −0.267176
\(146\) 0 0
\(147\) −0.0205616 −0.00169589
\(148\) 0 0
\(149\) −20.9646 −1.71749 −0.858743 0.512407i \(-0.828754\pi\)
−0.858743 + 0.512407i \(0.828754\pi\)
\(150\) 0 0
\(151\) 11.4288 0.930061 0.465030 0.885295i \(-0.346043\pi\)
0.465030 + 0.885295i \(0.346043\pi\)
\(152\) 0 0
\(153\) −3.38234 −0.273446
\(154\) 0 0
\(155\) 3.50595 0.281605
\(156\) 0 0
\(157\) −17.9659 −1.43384 −0.716918 0.697158i \(-0.754448\pi\)
−0.716918 + 0.697158i \(0.754448\pi\)
\(158\) 0 0
\(159\) −0.273741 −0.0217091
\(160\) 0 0
\(161\) −18.4905 −1.45726
\(162\) 0 0
\(163\) −3.45117 −0.270316 −0.135158 0.990824i \(-0.543154\pi\)
−0.135158 + 0.990824i \(0.543154\pi\)
\(164\) 0 0
\(165\) −0.496736 −0.0386709
\(166\) 0 0
\(167\) 3.50747 0.271416 0.135708 0.990749i \(-0.456669\pi\)
0.135708 + 0.990749i \(0.456669\pi\)
\(168\) 0 0
\(169\) −7.79359 −0.599507
\(170\) 0 0
\(171\) 6.01034 0.459622
\(172\) 0 0
\(173\) 8.46228 0.643375 0.321688 0.946846i \(-0.395750\pi\)
0.321688 + 0.946846i \(0.395750\pi\)
\(174\) 0 0
\(175\) 2.68164 0.202713
\(176\) 0 0
\(177\) 1.44864 0.108886
\(178\) 0 0
\(179\) 0.447045 0.0334137 0.0167068 0.999860i \(-0.494682\pi\)
0.0167068 + 0.999860i \(0.494682\pi\)
\(180\) 0 0
\(181\) 12.8317 0.953770 0.476885 0.878966i \(-0.341766\pi\)
0.476885 + 0.878966i \(0.341766\pi\)
\(182\) 0 0
\(183\) −0.966667 −0.0714581
\(184\) 0 0
\(185\) −2.91993 −0.214678
\(186\) 0 0
\(187\) 5.22838 0.382337
\(188\) 0 0
\(189\) 1.72682 0.125607
\(190\) 0 0
\(191\) −0.116074 −0.00839880 −0.00419940 0.999991i \(-0.501337\pi\)
−0.00419940 + 0.999991i \(0.501337\pi\)
\(192\) 0 0
\(193\) −18.7963 −1.35299 −0.676495 0.736447i \(-0.736502\pi\)
−0.676495 + 0.736447i \(0.736502\pi\)
\(194\) 0 0
\(195\) 0.245358 0.0175705
\(196\) 0 0
\(197\) 9.96109 0.709699 0.354849 0.934924i \(-0.384532\pi\)
0.354849 + 0.934924i \(0.384532\pi\)
\(198\) 0 0
\(199\) −21.4699 −1.52196 −0.760981 0.648774i \(-0.775282\pi\)
−0.760981 + 0.648774i \(0.775282\pi\)
\(200\) 0 0
\(201\) 1.12431 0.0793030
\(202\) 0 0
\(203\) −8.62745 −0.605529
\(204\) 0 0
\(205\) −9.50402 −0.663790
\(206\) 0 0
\(207\) 20.6059 1.43221
\(208\) 0 0
\(209\) −9.29071 −0.642652
\(210\) 0 0
\(211\) −4.57433 −0.314910 −0.157455 0.987526i \(-0.550329\pi\)
−0.157455 + 0.987526i \(0.550329\pi\)
\(212\) 0 0
\(213\) −0.0558750 −0.00382849
\(214\) 0 0
\(215\) −5.26560 −0.359111
\(216\) 0 0
\(217\) 9.40172 0.638230
\(218\) 0 0
\(219\) 0.0258297 0.00174541
\(220\) 0 0
\(221\) −2.58251 −0.173719
\(222\) 0 0
\(223\) 25.9396 1.73705 0.868523 0.495649i \(-0.165070\pi\)
0.868523 + 0.495649i \(0.165070\pi\)
\(224\) 0 0
\(225\) −2.98844 −0.199229
\(226\) 0 0
\(227\) 12.9032 0.856414 0.428207 0.903681i \(-0.359145\pi\)
0.428207 + 0.903681i \(0.359145\pi\)
\(228\) 0 0
\(229\) 17.8607 1.18027 0.590134 0.807305i \(-0.299075\pi\)
0.590134 + 0.807305i \(0.299075\pi\)
\(230\) 0 0
\(231\) −1.33207 −0.0876437
\(232\) 0 0
\(233\) 15.9492 1.04487 0.522433 0.852680i \(-0.325024\pi\)
0.522433 + 0.852680i \(0.325024\pi\)
\(234\) 0 0
\(235\) −11.2358 −0.732940
\(236\) 0 0
\(237\) −1.43419 −0.0931606
\(238\) 0 0
\(239\) −2.47618 −0.160171 −0.0800855 0.996788i \(-0.525519\pi\)
−0.0800855 + 0.996788i \(0.525519\pi\)
\(240\) 0 0
\(241\) 18.8558 1.21461 0.607304 0.794470i \(-0.292251\pi\)
0.607304 + 0.794470i \(0.292251\pi\)
\(242\) 0 0
\(243\) −2.88842 −0.185292
\(244\) 0 0
\(245\) 0.191216 0.0122164
\(246\) 0 0
\(247\) 4.58906 0.291995
\(248\) 0 0
\(249\) −0.0475592 −0.00301394
\(250\) 0 0
\(251\) 2.57469 0.162513 0.0812567 0.996693i \(-0.474107\pi\)
0.0812567 + 0.996693i \(0.474107\pi\)
\(252\) 0 0
\(253\) −31.8524 −2.00254
\(254\) 0 0
\(255\) −0.121704 −0.00762140
\(256\) 0 0
\(257\) 18.2155 1.13625 0.568126 0.822941i \(-0.307668\pi\)
0.568126 + 0.822941i \(0.307668\pi\)
\(258\) 0 0
\(259\) −7.83022 −0.486546
\(260\) 0 0
\(261\) 9.61448 0.595121
\(262\) 0 0
\(263\) 6.91213 0.426220 0.213110 0.977028i \(-0.431641\pi\)
0.213110 + 0.977028i \(0.431641\pi\)
\(264\) 0 0
\(265\) 2.54571 0.156382
\(266\) 0 0
\(267\) 1.98088 0.121228
\(268\) 0 0
\(269\) −7.08625 −0.432056 −0.216028 0.976387i \(-0.569310\pi\)
−0.216028 + 0.976387i \(0.569310\pi\)
\(270\) 0 0
\(271\) 19.2813 1.17125 0.585627 0.810581i \(-0.300848\pi\)
0.585627 + 0.810581i \(0.300848\pi\)
\(272\) 0 0
\(273\) 0.657964 0.0398218
\(274\) 0 0
\(275\) 4.61949 0.278566
\(276\) 0 0
\(277\) −21.9048 −1.31613 −0.658067 0.752959i \(-0.728626\pi\)
−0.658067 + 0.752959i \(0.728626\pi\)
\(278\) 0 0
\(279\) −10.4773 −0.627261
\(280\) 0 0
\(281\) 5.29023 0.315589 0.157794 0.987472i \(-0.449562\pi\)
0.157794 + 0.987472i \(0.449562\pi\)
\(282\) 0 0
\(283\) −0.738913 −0.0439238 −0.0219619 0.999759i \(-0.506991\pi\)
−0.0219619 + 0.999759i \(0.506991\pi\)
\(284\) 0 0
\(285\) 0.216265 0.0128104
\(286\) 0 0
\(287\) −25.4864 −1.50441
\(288\) 0 0
\(289\) −15.7190 −0.924648
\(290\) 0 0
\(291\) −1.50545 −0.0882511
\(292\) 0 0
\(293\) 21.8949 1.27911 0.639556 0.768745i \(-0.279118\pi\)
0.639556 + 0.768745i \(0.279118\pi\)
\(294\) 0 0
\(295\) −13.4719 −0.784364
\(296\) 0 0
\(297\) 2.97467 0.172608
\(298\) 0 0
\(299\) 15.7332 0.909875
\(300\) 0 0
\(301\) −14.1205 −0.813890
\(302\) 0 0
\(303\) 0.473050 0.0271760
\(304\) 0 0
\(305\) 8.98971 0.514749
\(306\) 0 0
\(307\) 9.42666 0.538008 0.269004 0.963139i \(-0.413306\pi\)
0.269004 + 0.963139i \(0.413306\pi\)
\(308\) 0 0
\(309\) 1.75155 0.0996422
\(310\) 0 0
\(311\) −6.26994 −0.355536 −0.177768 0.984072i \(-0.556888\pi\)
−0.177768 + 0.984072i \(0.556888\pi\)
\(312\) 0 0
\(313\) −8.85701 −0.500628 −0.250314 0.968165i \(-0.580534\pi\)
−0.250314 + 0.968165i \(0.580534\pi\)
\(314\) 0 0
\(315\) −8.01393 −0.451534
\(316\) 0 0
\(317\) −12.1420 −0.681962 −0.340981 0.940070i \(-0.610759\pi\)
−0.340981 + 0.940070i \(0.610759\pi\)
\(318\) 0 0
\(319\) −14.8619 −0.832109
\(320\) 0 0
\(321\) 0.271141 0.0151336
\(322\) 0 0
\(323\) −2.27629 −0.126656
\(324\) 0 0
\(325\) −2.28176 −0.126569
\(326\) 0 0
\(327\) 0.0305963 0.00169198
\(328\) 0 0
\(329\) −30.1303 −1.66114
\(330\) 0 0
\(331\) −34.4670 −1.89448 −0.947238 0.320532i \(-0.896138\pi\)
−0.947238 + 0.320532i \(0.896138\pi\)
\(332\) 0 0
\(333\) 8.72603 0.478184
\(334\) 0 0
\(335\) −10.4558 −0.571260
\(336\) 0 0
\(337\) 22.1900 1.20877 0.604385 0.796693i \(-0.293419\pi\)
0.604385 + 0.796693i \(0.293419\pi\)
\(338\) 0 0
\(339\) −1.57975 −0.0858000
\(340\) 0 0
\(341\) 16.1957 0.877047
\(342\) 0 0
\(343\) −18.2587 −0.985879
\(344\) 0 0
\(345\) 0.741446 0.0399181
\(346\) 0 0
\(347\) −13.3192 −0.715011 −0.357506 0.933911i \(-0.616373\pi\)
−0.357506 + 0.933911i \(0.616373\pi\)
\(348\) 0 0
\(349\) −21.0432 −1.12641 −0.563207 0.826316i \(-0.690433\pi\)
−0.563207 + 0.826316i \(0.690433\pi\)
\(350\) 0 0
\(351\) −1.46931 −0.0784261
\(352\) 0 0
\(353\) −31.0319 −1.65166 −0.825831 0.563917i \(-0.809294\pi\)
−0.825831 + 0.563917i \(0.809294\pi\)
\(354\) 0 0
\(355\) 0.519620 0.0275786
\(356\) 0 0
\(357\) −0.326367 −0.0172732
\(358\) 0 0
\(359\) 34.8907 1.84146 0.920729 0.390202i \(-0.127595\pi\)
0.920729 + 0.390202i \(0.127595\pi\)
\(360\) 0 0
\(361\) −14.9551 −0.787110
\(362\) 0 0
\(363\) −1.11183 −0.0583561
\(364\) 0 0
\(365\) −0.240208 −0.0125731
\(366\) 0 0
\(367\) 2.86494 0.149549 0.0747744 0.997200i \(-0.476176\pi\)
0.0747744 + 0.997200i \(0.476176\pi\)
\(368\) 0 0
\(369\) 28.4022 1.47856
\(370\) 0 0
\(371\) 6.82668 0.354424
\(372\) 0 0
\(373\) −16.4919 −0.853917 −0.426958 0.904271i \(-0.640415\pi\)
−0.426958 + 0.904271i \(0.640415\pi\)
\(374\) 0 0
\(375\) −0.107530 −0.00555285
\(376\) 0 0
\(377\) 7.34092 0.378077
\(378\) 0 0
\(379\) 8.14872 0.418572 0.209286 0.977855i \(-0.432886\pi\)
0.209286 + 0.977855i \(0.432886\pi\)
\(380\) 0 0
\(381\) 1.77116 0.0907395
\(382\) 0 0
\(383\) −17.9974 −0.919625 −0.459813 0.888016i \(-0.652083\pi\)
−0.459813 + 0.888016i \(0.652083\pi\)
\(384\) 0 0
\(385\) 12.3878 0.631342
\(386\) 0 0
\(387\) 15.7359 0.799901
\(388\) 0 0
\(389\) 3.93065 0.199292 0.0996460 0.995023i \(-0.468229\pi\)
0.0996460 + 0.995023i \(0.468229\pi\)
\(390\) 0 0
\(391\) −7.80407 −0.394669
\(392\) 0 0
\(393\) −1.39313 −0.0702741
\(394\) 0 0
\(395\) 13.3375 0.671083
\(396\) 0 0
\(397\) −32.4096 −1.62659 −0.813295 0.581852i \(-0.802328\pi\)
−0.813295 + 0.581852i \(0.802328\pi\)
\(398\) 0 0
\(399\) 0.579946 0.0290336
\(400\) 0 0
\(401\) −1.00000 −0.0499376
\(402\) 0 0
\(403\) −7.99973 −0.398495
\(404\) 0 0
\(405\) 8.89607 0.442049
\(406\) 0 0
\(407\) −13.4886 −0.668605
\(408\) 0 0
\(409\) 19.3544 0.957012 0.478506 0.878084i \(-0.341179\pi\)
0.478506 + 0.878084i \(0.341179\pi\)
\(410\) 0 0
\(411\) −0.234069 −0.0115458
\(412\) 0 0
\(413\) −36.1268 −1.77769
\(414\) 0 0
\(415\) 0.442286 0.0217110
\(416\) 0 0
\(417\) −0.298620 −0.0146235
\(418\) 0 0
\(419\) 27.5338 1.34512 0.672558 0.740045i \(-0.265196\pi\)
0.672558 + 0.740045i \(0.265196\pi\)
\(420\) 0 0
\(421\) 17.1646 0.836549 0.418275 0.908321i \(-0.362635\pi\)
0.418275 + 0.908321i \(0.362635\pi\)
\(422\) 0 0
\(423\) 33.5773 1.63259
\(424\) 0 0
\(425\) 1.13181 0.0549008
\(426\) 0 0
\(427\) 24.1072 1.16663
\(428\) 0 0
\(429\) 1.13343 0.0547225
\(430\) 0 0
\(431\) −16.6915 −0.804002 −0.402001 0.915639i \(-0.631685\pi\)
−0.402001 + 0.915639i \(0.631685\pi\)
\(432\) 0 0
\(433\) −20.9644 −1.00748 −0.503742 0.863854i \(-0.668044\pi\)
−0.503742 + 0.863854i \(0.668044\pi\)
\(434\) 0 0
\(435\) 0.345950 0.0165870
\(436\) 0 0
\(437\) 13.8677 0.663380
\(438\) 0 0
\(439\) 15.9033 0.759023 0.379511 0.925187i \(-0.376092\pi\)
0.379511 + 0.925187i \(0.376092\pi\)
\(440\) 0 0
\(441\) −0.571438 −0.0272114
\(442\) 0 0
\(443\) 4.62494 0.219737 0.109869 0.993946i \(-0.464957\pi\)
0.109869 + 0.993946i \(0.464957\pi\)
\(444\) 0 0
\(445\) −18.4216 −0.873266
\(446\) 0 0
\(447\) 2.25433 0.106626
\(448\) 0 0
\(449\) −25.4521 −1.20116 −0.600579 0.799566i \(-0.705063\pi\)
−0.600579 + 0.799566i \(0.705063\pi\)
\(450\) 0 0
\(451\) −43.9037 −2.06735
\(452\) 0 0
\(453\) −1.22894 −0.0577407
\(454\) 0 0
\(455\) −6.11886 −0.286857
\(456\) 0 0
\(457\) 3.36936 0.157612 0.0788060 0.996890i \(-0.474889\pi\)
0.0788060 + 0.996890i \(0.474889\pi\)
\(458\) 0 0
\(459\) 0.728817 0.0340182
\(460\) 0 0
\(461\) −16.5242 −0.769608 −0.384804 0.922998i \(-0.625731\pi\)
−0.384804 + 0.922998i \(0.625731\pi\)
\(462\) 0 0
\(463\) 2.64568 0.122955 0.0614775 0.998108i \(-0.480419\pi\)
0.0614775 + 0.998108i \(0.480419\pi\)
\(464\) 0 0
\(465\) −0.376997 −0.0174828
\(466\) 0 0
\(467\) −26.7230 −1.23659 −0.618297 0.785945i \(-0.712177\pi\)
−0.618297 + 0.785945i \(0.712177\pi\)
\(468\) 0 0
\(469\) −28.0387 −1.29471
\(470\) 0 0
\(471\) 1.93188 0.0890165
\(472\) 0 0
\(473\) −24.3244 −1.11844
\(474\) 0 0
\(475\) −2.01120 −0.0922801
\(476\) 0 0
\(477\) −7.60769 −0.348332
\(478\) 0 0
\(479\) 0.974638 0.0445323 0.0222662 0.999752i \(-0.492912\pi\)
0.0222662 + 0.999752i \(0.492912\pi\)
\(480\) 0 0
\(481\) 6.66257 0.303787
\(482\) 0 0
\(483\) 1.98830 0.0904706
\(484\) 0 0
\(485\) 14.0002 0.635718
\(486\) 0 0
\(487\) −4.20800 −0.190683 −0.0953414 0.995445i \(-0.530394\pi\)
−0.0953414 + 0.995445i \(0.530394\pi\)
\(488\) 0 0
\(489\) 0.371106 0.0167820
\(490\) 0 0
\(491\) −8.07946 −0.364621 −0.182310 0.983241i \(-0.558358\pi\)
−0.182310 + 0.983241i \(0.558358\pi\)
\(492\) 0 0
\(493\) −3.64129 −0.163995
\(494\) 0 0
\(495\) −13.8051 −0.620491
\(496\) 0 0
\(497\) 1.39344 0.0625042
\(498\) 0 0
\(499\) 2.57209 0.115142 0.0575712 0.998341i \(-0.481664\pi\)
0.0575712 + 0.998341i \(0.481664\pi\)
\(500\) 0 0
\(501\) −0.377160 −0.0168503
\(502\) 0 0
\(503\) −14.3804 −0.641190 −0.320595 0.947216i \(-0.603883\pi\)
−0.320595 + 0.947216i \(0.603883\pi\)
\(504\) 0 0
\(505\) −4.39922 −0.195763
\(506\) 0 0
\(507\) 0.838048 0.0372190
\(508\) 0 0
\(509\) 19.1946 0.850783 0.425392 0.905009i \(-0.360136\pi\)
0.425392 + 0.905009i \(0.360136\pi\)
\(510\) 0 0
\(511\) −0.644152 −0.0284956
\(512\) 0 0
\(513\) −1.29509 −0.0571796
\(514\) 0 0
\(515\) −16.2889 −0.717774
\(516\) 0 0
\(517\) −51.9035 −2.28271
\(518\) 0 0
\(519\) −0.909953 −0.0399425
\(520\) 0 0
\(521\) −13.1825 −0.577536 −0.288768 0.957399i \(-0.593246\pi\)
−0.288768 + 0.957399i \(0.593246\pi\)
\(522\) 0 0
\(523\) 13.7109 0.599536 0.299768 0.954012i \(-0.403091\pi\)
0.299768 + 0.954012i \(0.403091\pi\)
\(524\) 0 0
\(525\) −0.288359 −0.0125850
\(526\) 0 0
\(527\) 3.96807 0.172852
\(528\) 0 0
\(529\) 24.5440 1.06713
\(530\) 0 0
\(531\) 40.2599 1.74713
\(532\) 0 0
\(533\) 21.6859 0.939319
\(534\) 0 0
\(535\) −2.52153 −0.109015
\(536\) 0 0
\(537\) −0.0480709 −0.00207441
\(538\) 0 0
\(539\) 0.883323 0.0380474
\(540\) 0 0
\(541\) 13.1710 0.566267 0.283134 0.959080i \(-0.408626\pi\)
0.283134 + 0.959080i \(0.408626\pi\)
\(542\) 0 0
\(543\) −1.37979 −0.0592126
\(544\) 0 0
\(545\) −0.284536 −0.0121882
\(546\) 0 0
\(547\) 44.9432 1.92163 0.960816 0.277187i \(-0.0894021\pi\)
0.960816 + 0.277187i \(0.0894021\pi\)
\(548\) 0 0
\(549\) −26.8652 −1.14658
\(550\) 0 0
\(551\) 6.47048 0.275652
\(552\) 0 0
\(553\) 35.7665 1.52094
\(554\) 0 0
\(555\) 0.313982 0.0133278
\(556\) 0 0
\(557\) −27.6421 −1.17123 −0.585616 0.810589i \(-0.699147\pi\)
−0.585616 + 0.810589i \(0.699147\pi\)
\(558\) 0 0
\(559\) 12.0148 0.508172
\(560\) 0 0
\(561\) −0.562210 −0.0237365
\(562\) 0 0
\(563\) −2.80614 −0.118265 −0.0591324 0.998250i \(-0.518833\pi\)
−0.0591324 + 0.998250i \(0.518833\pi\)
\(564\) 0 0
\(565\) 14.6911 0.618061
\(566\) 0 0
\(567\) 23.8561 1.00186
\(568\) 0 0
\(569\) 12.9648 0.543515 0.271757 0.962366i \(-0.412395\pi\)
0.271757 + 0.962366i \(0.412395\pi\)
\(570\) 0 0
\(571\) −9.21349 −0.385572 −0.192786 0.981241i \(-0.561752\pi\)
−0.192786 + 0.981241i \(0.561752\pi\)
\(572\) 0 0
\(573\) 0.0124815 0.000521421 0
\(574\) 0 0
\(575\) −6.89522 −0.287551
\(576\) 0 0
\(577\) 31.3358 1.30453 0.652264 0.757992i \(-0.273819\pi\)
0.652264 + 0.757992i \(0.273819\pi\)
\(578\) 0 0
\(579\) 2.02118 0.0839973
\(580\) 0 0
\(581\) 1.18605 0.0492058
\(582\) 0 0
\(583\) 11.7599 0.487044
\(584\) 0 0
\(585\) 6.81888 0.281926
\(586\) 0 0
\(587\) −26.3550 −1.08779 −0.543894 0.839154i \(-0.683051\pi\)
−0.543894 + 0.839154i \(0.683051\pi\)
\(588\) 0 0
\(589\) −7.05117 −0.290538
\(590\) 0 0
\(591\) −1.07112 −0.0440600
\(592\) 0 0
\(593\) 20.6956 0.849866 0.424933 0.905225i \(-0.360298\pi\)
0.424933 + 0.905225i \(0.360298\pi\)
\(594\) 0 0
\(595\) 3.03511 0.124427
\(596\) 0 0
\(597\) 2.30867 0.0944876
\(598\) 0 0
\(599\) −19.8701 −0.811871 −0.405936 0.913902i \(-0.633054\pi\)
−0.405936 + 0.913902i \(0.633054\pi\)
\(600\) 0 0
\(601\) −29.4627 −1.20181 −0.600905 0.799320i \(-0.705193\pi\)
−0.600905 + 0.799320i \(0.705193\pi\)
\(602\) 0 0
\(603\) 31.2464 1.27245
\(604\) 0 0
\(605\) 10.3397 0.420369
\(606\) 0 0
\(607\) −23.3291 −0.946897 −0.473448 0.880822i \(-0.656991\pi\)
−0.473448 + 0.880822i \(0.656991\pi\)
\(608\) 0 0
\(609\) 0.927714 0.0375929
\(610\) 0 0
\(611\) 25.6372 1.03717
\(612\) 0 0
\(613\) 27.8120 1.12332 0.561659 0.827369i \(-0.310163\pi\)
0.561659 + 0.827369i \(0.310163\pi\)
\(614\) 0 0
\(615\) 1.02197 0.0412099
\(616\) 0 0
\(617\) 14.7170 0.592482 0.296241 0.955113i \(-0.404267\pi\)
0.296241 + 0.955113i \(0.404267\pi\)
\(618\) 0 0
\(619\) −10.3808 −0.417240 −0.208620 0.977997i \(-0.566897\pi\)
−0.208620 + 0.977997i \(0.566897\pi\)
\(620\) 0 0
\(621\) −4.44010 −0.178175
\(622\) 0 0
\(623\) −49.4001 −1.97917
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0.999035 0.0398976
\(628\) 0 0
\(629\) −3.30480 −0.131771
\(630\) 0 0
\(631\) −27.0178 −1.07556 −0.537780 0.843085i \(-0.680737\pi\)
−0.537780 + 0.843085i \(0.680737\pi\)
\(632\) 0 0
\(633\) 0.491880 0.0195505
\(634\) 0 0
\(635\) −16.4713 −0.653642
\(636\) 0 0
\(637\) −0.436309 −0.0172872
\(638\) 0 0
\(639\) −1.55285 −0.0614299
\(640\) 0 0
\(641\) −14.0534 −0.555078 −0.277539 0.960714i \(-0.589519\pi\)
−0.277539 + 0.960714i \(0.589519\pi\)
\(642\) 0 0
\(643\) 22.7023 0.895291 0.447646 0.894211i \(-0.352263\pi\)
0.447646 + 0.894211i \(0.352263\pi\)
\(644\) 0 0
\(645\) 0.566212 0.0222946
\(646\) 0 0
\(647\) 27.4245 1.07817 0.539084 0.842252i \(-0.318771\pi\)
0.539084 + 0.842252i \(0.318771\pi\)
\(648\) 0 0
\(649\) −62.2333 −2.44287
\(650\) 0 0
\(651\) −1.01097 −0.0396231
\(652\) 0 0
\(653\) 39.5648 1.54829 0.774146 0.633008i \(-0.218180\pi\)
0.774146 + 0.633008i \(0.218180\pi\)
\(654\) 0 0
\(655\) 12.9557 0.506220
\(656\) 0 0
\(657\) 0.717846 0.0280058
\(658\) 0 0
\(659\) −15.7371 −0.613032 −0.306516 0.951865i \(-0.599163\pi\)
−0.306516 + 0.951865i \(0.599163\pi\)
\(660\) 0 0
\(661\) −22.1069 −0.859859 −0.429929 0.902863i \(-0.641462\pi\)
−0.429929 + 0.902863i \(0.641462\pi\)
\(662\) 0 0
\(663\) 0.277699 0.0107849
\(664\) 0 0
\(665\) −5.39332 −0.209144
\(666\) 0 0
\(667\) 22.1835 0.858948
\(668\) 0 0
\(669\) −2.78930 −0.107841
\(670\) 0 0
\(671\) 41.5279 1.60317
\(672\) 0 0
\(673\) −29.2498 −1.12750 −0.563749 0.825946i \(-0.690641\pi\)
−0.563749 + 0.825946i \(0.690641\pi\)
\(674\) 0 0
\(675\) 0.643940 0.0247852
\(676\) 0 0
\(677\) −27.5421 −1.05853 −0.529264 0.848457i \(-0.677532\pi\)
−0.529264 + 0.848457i \(0.677532\pi\)
\(678\) 0 0
\(679\) 37.5436 1.44079
\(680\) 0 0
\(681\) −1.38749 −0.0531686
\(682\) 0 0
\(683\) −19.6018 −0.750040 −0.375020 0.927017i \(-0.622364\pi\)
−0.375020 + 0.927017i \(0.622364\pi\)
\(684\) 0 0
\(685\) 2.17677 0.0831702
\(686\) 0 0
\(687\) −1.92057 −0.0732743
\(688\) 0 0
\(689\) −5.80868 −0.221293
\(690\) 0 0
\(691\) 34.4657 1.31114 0.655568 0.755136i \(-0.272429\pi\)
0.655568 + 0.755136i \(0.272429\pi\)
\(692\) 0 0
\(693\) −37.0203 −1.40628
\(694\) 0 0
\(695\) 2.77707 0.105340
\(696\) 0 0
\(697\) −10.7567 −0.407440
\(698\) 0 0
\(699\) −1.71502 −0.0648682
\(700\) 0 0
\(701\) −13.7043 −0.517604 −0.258802 0.965930i \(-0.583328\pi\)
−0.258802 + 0.965930i \(0.583328\pi\)
\(702\) 0 0
\(703\) 5.87256 0.221488
\(704\) 0 0
\(705\) 1.20819 0.0455029
\(706\) 0 0
\(707\) −11.7971 −0.443677
\(708\) 0 0
\(709\) 16.8716 0.633627 0.316813 0.948488i \(-0.397387\pi\)
0.316813 + 0.948488i \(0.397387\pi\)
\(710\) 0 0
\(711\) −39.8583 −1.49480
\(712\) 0 0
\(713\) −24.1743 −0.905335
\(714\) 0 0
\(715\) −10.5406 −0.394194
\(716\) 0 0
\(717\) 0.266265 0.00994385
\(718\) 0 0
\(719\) −21.6476 −0.807318 −0.403659 0.914909i \(-0.632262\pi\)
−0.403659 + 0.914909i \(0.632262\pi\)
\(720\) 0 0
\(721\) −43.6810 −1.62676
\(722\) 0 0
\(723\) −2.02757 −0.0754062
\(724\) 0 0
\(725\) −3.21723 −0.119485
\(726\) 0 0
\(727\) −9.41647 −0.349237 −0.174619 0.984636i \(-0.555869\pi\)
−0.174619 + 0.984636i \(0.555869\pi\)
\(728\) 0 0
\(729\) −26.3776 −0.976949
\(730\) 0 0
\(731\) −5.95965 −0.220426
\(732\) 0 0
\(733\) −13.1814 −0.486867 −0.243433 0.969918i \(-0.578274\pi\)
−0.243433 + 0.969918i \(0.578274\pi\)
\(734\) 0 0
\(735\) −0.0205616 −0.000758426 0
\(736\) 0 0
\(737\) −48.3003 −1.77917
\(738\) 0 0
\(739\) 47.8664 1.76079 0.880396 0.474239i \(-0.157277\pi\)
0.880396 + 0.474239i \(0.157277\pi\)
\(740\) 0 0
\(741\) −0.493464 −0.0181279
\(742\) 0 0
\(743\) −8.05550 −0.295528 −0.147764 0.989023i \(-0.547208\pi\)
−0.147764 + 0.989023i \(0.547208\pi\)
\(744\) 0 0
\(745\) −20.9646 −0.768083
\(746\) 0 0
\(747\) −1.32174 −0.0483601
\(748\) 0 0
\(749\) −6.76184 −0.247072
\(750\) 0 0
\(751\) 45.8484 1.67303 0.836515 0.547943i \(-0.184589\pi\)
0.836515 + 0.547943i \(0.184589\pi\)
\(752\) 0 0
\(753\) −0.276858 −0.0100893
\(754\) 0 0
\(755\) 11.4288 0.415936
\(756\) 0 0
\(757\) −36.0645 −1.31079 −0.655394 0.755287i \(-0.727497\pi\)
−0.655394 + 0.755287i \(0.727497\pi\)
\(758\) 0 0
\(759\) 3.42510 0.124323
\(760\) 0 0
\(761\) −20.6287 −0.747789 −0.373895 0.927471i \(-0.621978\pi\)
−0.373895 + 0.927471i \(0.621978\pi\)
\(762\) 0 0
\(763\) −0.763025 −0.0276234
\(764\) 0 0
\(765\) −3.38234 −0.122289
\(766\) 0 0
\(767\) 30.7396 1.10994
\(768\) 0 0
\(769\) −13.0496 −0.470580 −0.235290 0.971925i \(-0.575604\pi\)
−0.235290 + 0.971925i \(0.575604\pi\)
\(770\) 0 0
\(771\) −1.95872 −0.0705417
\(772\) 0 0
\(773\) −47.3700 −1.70378 −0.851891 0.523720i \(-0.824544\pi\)
−0.851891 + 0.523720i \(0.824544\pi\)
\(774\) 0 0
\(775\) 3.50595 0.125938
\(776\) 0 0
\(777\) 0.841987 0.0302061
\(778\) 0 0
\(779\) 19.1145 0.684847
\(780\) 0 0
\(781\) 2.40038 0.0858924
\(782\) 0 0
\(783\) −2.07170 −0.0740365
\(784\) 0 0
\(785\) −17.9659 −0.641231
\(786\) 0 0
\(787\) −42.4249 −1.51228 −0.756141 0.654408i \(-0.772918\pi\)
−0.756141 + 0.654408i \(0.772918\pi\)
\(788\) 0 0
\(789\) −0.743265 −0.0264609
\(790\) 0 0
\(791\) 39.3964 1.40077
\(792\) 0 0
\(793\) −20.5123 −0.728414
\(794\) 0 0
\(795\) −0.273741 −0.00970860
\(796\) 0 0
\(797\) −40.7859 −1.44471 −0.722355 0.691523i \(-0.756940\pi\)
−0.722355 + 0.691523i \(0.756940\pi\)
\(798\) 0 0
\(799\) −12.7167 −0.449885
\(800\) 0 0
\(801\) 55.0517 1.94516
\(802\) 0 0
\(803\) −1.10964 −0.0391583
\(804\) 0 0
\(805\) −18.4905 −0.651705
\(806\) 0 0
\(807\) 0.761988 0.0268232
\(808\) 0 0
\(809\) −29.5501 −1.03893 −0.519463 0.854493i \(-0.673868\pi\)
−0.519463 + 0.854493i \(0.673868\pi\)
\(810\) 0 0
\(811\) 2.77441 0.0974228 0.0487114 0.998813i \(-0.484489\pi\)
0.0487114 + 0.998813i \(0.484489\pi\)
\(812\) 0 0
\(813\) −2.07332 −0.0727146
\(814\) 0 0
\(815\) −3.45117 −0.120889
\(816\) 0 0
\(817\) 10.5902 0.370503
\(818\) 0 0
\(819\) 18.2858 0.638958
\(820\) 0 0
\(821\) −2.07217 −0.0723191 −0.0361595 0.999346i \(-0.511512\pi\)
−0.0361595 + 0.999346i \(0.511512\pi\)
\(822\) 0 0
\(823\) −34.5578 −1.20461 −0.602304 0.798267i \(-0.705751\pi\)
−0.602304 + 0.798267i \(0.705751\pi\)
\(824\) 0 0
\(825\) −0.496736 −0.0172941
\(826\) 0 0
\(827\) 13.4854 0.468935 0.234467 0.972124i \(-0.424665\pi\)
0.234467 + 0.972124i \(0.424665\pi\)
\(828\) 0 0
\(829\) −18.1037 −0.628768 −0.314384 0.949296i \(-0.601798\pi\)
−0.314384 + 0.949296i \(0.601798\pi\)
\(830\) 0 0
\(831\) 2.35544 0.0817092
\(832\) 0 0
\(833\) 0.216421 0.00749853
\(834\) 0 0
\(835\) 3.50747 0.121381
\(836\) 0 0
\(837\) 2.25762 0.0780348
\(838\) 0 0
\(839\) 13.4443 0.464149 0.232074 0.972698i \(-0.425449\pi\)
0.232074 + 0.972698i \(0.425449\pi\)
\(840\) 0 0
\(841\) −18.6495 −0.643085
\(842\) 0 0
\(843\) −0.568861 −0.0195926
\(844\) 0 0
\(845\) −7.79359 −0.268108
\(846\) 0 0
\(847\) 27.7274 0.952725
\(848\) 0 0
\(849\) 0.0794557 0.00272691
\(850\) 0 0
\(851\) 20.1336 0.690170
\(852\) 0 0
\(853\) 37.0538 1.26870 0.634348 0.773047i \(-0.281268\pi\)
0.634348 + 0.773047i \(0.281268\pi\)
\(854\) 0 0
\(855\) 6.01034 0.205549
\(856\) 0 0
\(857\) −57.7654 −1.97323 −0.986615 0.163068i \(-0.947861\pi\)
−0.986615 + 0.163068i \(0.947861\pi\)
\(858\) 0 0
\(859\) 55.2127 1.88383 0.941916 0.335848i \(-0.109023\pi\)
0.941916 + 0.335848i \(0.109023\pi\)
\(860\) 0 0
\(861\) 2.74057 0.0933982
\(862\) 0 0
\(863\) 23.7506 0.808480 0.404240 0.914653i \(-0.367536\pi\)
0.404240 + 0.914653i \(0.367536\pi\)
\(864\) 0 0
\(865\) 8.46228 0.287726
\(866\) 0 0
\(867\) 1.69027 0.0574047
\(868\) 0 0
\(869\) 61.6125 2.09006
\(870\) 0 0
\(871\) 23.8575 0.808381
\(872\) 0 0
\(873\) −41.8388 −1.41603
\(874\) 0 0
\(875\) 2.68164 0.0906561
\(876\) 0 0
\(877\) 24.7407 0.835434 0.417717 0.908577i \(-0.362830\pi\)
0.417717 + 0.908577i \(0.362830\pi\)
\(878\) 0 0
\(879\) −2.35436 −0.0794107
\(880\) 0 0
\(881\) −12.0942 −0.407463 −0.203731 0.979027i \(-0.565307\pi\)
−0.203731 + 0.979027i \(0.565307\pi\)
\(882\) 0 0
\(883\) 3.44520 0.115940 0.0579702 0.998318i \(-0.481537\pi\)
0.0579702 + 0.998318i \(0.481537\pi\)
\(884\) 0 0
\(885\) 1.44864 0.0486955
\(886\) 0 0
\(887\) 16.3037 0.547425 0.273712 0.961812i \(-0.411748\pi\)
0.273712 + 0.961812i \(0.411748\pi\)
\(888\) 0 0
\(889\) −44.1701 −1.48142
\(890\) 0 0
\(891\) 41.0953 1.37674
\(892\) 0 0
\(893\) 22.5973 0.756191
\(894\) 0 0
\(895\) 0.447045 0.0149431
\(896\) 0 0
\(897\) −1.69180 −0.0564875
\(898\) 0 0
\(899\) −11.2794 −0.376190
\(900\) 0 0
\(901\) 2.88126 0.0959885
\(902\) 0 0
\(903\) 1.51838 0.0505285
\(904\) 0 0
\(905\) 12.8317 0.426539
\(906\) 0 0
\(907\) 21.6352 0.718384 0.359192 0.933264i \(-0.383052\pi\)
0.359192 + 0.933264i \(0.383052\pi\)
\(908\) 0 0
\(909\) 13.1468 0.436051
\(910\) 0 0
\(911\) 29.4463 0.975600 0.487800 0.872956i \(-0.337800\pi\)
0.487800 + 0.872956i \(0.337800\pi\)
\(912\) 0 0
\(913\) 2.04314 0.0676179
\(914\) 0 0
\(915\) −0.966667 −0.0319570
\(916\) 0 0
\(917\) 34.7425 1.14730
\(918\) 0 0
\(919\) 38.7772 1.27914 0.639570 0.768733i \(-0.279112\pi\)
0.639570 + 0.768733i \(0.279112\pi\)
\(920\) 0 0
\(921\) −1.01365 −0.0334010
\(922\) 0 0
\(923\) −1.18565 −0.0390260
\(924\) 0 0
\(925\) −2.91993 −0.0960067
\(926\) 0 0
\(927\) 48.6783 1.59880
\(928\) 0 0
\(929\) 11.0566 0.362754 0.181377 0.983414i \(-0.441945\pi\)
0.181377 + 0.983414i \(0.441945\pi\)
\(930\) 0 0
\(931\) −0.384574 −0.0126039
\(932\) 0 0
\(933\) 0.674210 0.0220726
\(934\) 0 0
\(935\) 5.22838 0.170986
\(936\) 0 0
\(937\) 13.5811 0.443674 0.221837 0.975084i \(-0.428795\pi\)
0.221837 + 0.975084i \(0.428795\pi\)
\(938\) 0 0
\(939\) 0.952399 0.0310804
\(940\) 0 0
\(941\) 1.55187 0.0505895 0.0252947 0.999680i \(-0.491948\pi\)
0.0252947 + 0.999680i \(0.491948\pi\)
\(942\) 0 0
\(943\) 65.5323 2.13403
\(944\) 0 0
\(945\) 1.72682 0.0561734
\(946\) 0 0
\(947\) 27.8176 0.903951 0.451975 0.892030i \(-0.350719\pi\)
0.451975 + 0.892030i \(0.350719\pi\)
\(948\) 0 0
\(949\) 0.548096 0.0177919
\(950\) 0 0
\(951\) 1.30563 0.0423381
\(952\) 0 0
\(953\) −41.8854 −1.35680 −0.678401 0.734692i \(-0.737327\pi\)
−0.678401 + 0.734692i \(0.737327\pi\)
\(954\) 0 0
\(955\) −0.116074 −0.00375606
\(956\) 0 0
\(957\) 1.59811 0.0516596
\(958\) 0 0
\(959\) 5.83733 0.188497
\(960\) 0 0
\(961\) −18.7083 −0.603493
\(962\) 0 0
\(963\) 7.53543 0.242826
\(964\) 0 0
\(965\) −18.7963 −0.605076
\(966\) 0 0
\(967\) 0.112018 0.00360225 0.00180112 0.999998i \(-0.499427\pi\)
0.00180112 + 0.999998i \(0.499427\pi\)
\(968\) 0 0
\(969\) 0.244771 0.00786317
\(970\) 0 0
\(971\) −45.4862 −1.45972 −0.729860 0.683596i \(-0.760415\pi\)
−0.729860 + 0.683596i \(0.760415\pi\)
\(972\) 0 0
\(973\) 7.44713 0.238744
\(974\) 0 0
\(975\) 0.245358 0.00785776
\(976\) 0 0
\(977\) −40.5369 −1.29689 −0.648446 0.761261i \(-0.724581\pi\)
−0.648446 + 0.761261i \(0.724581\pi\)
\(978\) 0 0
\(979\) −85.0983 −2.71975
\(980\) 0 0
\(981\) 0.850319 0.0271486
\(982\) 0 0
\(983\) −37.3299 −1.19064 −0.595319 0.803489i \(-0.702975\pi\)
−0.595319 + 0.803489i \(0.702975\pi\)
\(984\) 0 0
\(985\) 9.96109 0.317387
\(986\) 0 0
\(987\) 3.23992 0.103128
\(988\) 0 0
\(989\) 36.3074 1.15451
\(990\) 0 0
\(991\) 37.6416 1.19573 0.597863 0.801598i \(-0.296017\pi\)
0.597863 + 0.801598i \(0.296017\pi\)
\(992\) 0 0
\(993\) 3.70625 0.117614
\(994\) 0 0
\(995\) −21.4699 −0.680642
\(996\) 0 0
\(997\) −32.0529 −1.01512 −0.507562 0.861615i \(-0.669453\pi\)
−0.507562 + 0.861615i \(0.669453\pi\)
\(998\) 0 0
\(999\) −1.88026 −0.0594888
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.d.1.15 29
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8020.2.a.d.1.15 29 1.1 even 1 trivial