Properties

Label 4028.2.a.f.1.19
Level $4028$
Weight $2$
Character 4028.1
Self dual yes
Analytic conductor $32.164$
Analytic rank $0$
Dimension $19$
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] = [4028,2,Mod(1,4028)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4028, 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("4028.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4028.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: \(32.1637419342\)
Analytic rank: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 4 x^{18} - 30 x^{17} + 124 x^{16} + 364 x^{15} - 1554 x^{14} - 2310 x^{13} + 10113 x^{12} + 8368 x^{11} - 36567 x^{10} - 18074 x^{9} + 72868 x^{8} + 23819 x^{7} + \cdots + 139 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.19
Root \(3.30281\) of defining polynomial
Character \(\chi\) \(=\) 4028.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.30281 q^{3} -1.36124 q^{5} +2.46762 q^{7} +7.90858 q^{9} +O(q^{10})\) \(q+3.30281 q^{3} -1.36124 q^{5} +2.46762 q^{7} +7.90858 q^{9} -1.61612 q^{11} +3.79921 q^{13} -4.49592 q^{15} +5.80568 q^{17} -1.00000 q^{19} +8.15008 q^{21} +0.190450 q^{23} -3.14703 q^{25} +16.2121 q^{27} -0.449717 q^{29} +3.15951 q^{31} -5.33773 q^{33} -3.35902 q^{35} +3.75573 q^{37} +12.5481 q^{39} -5.76176 q^{41} -11.1067 q^{43} -10.7655 q^{45} -3.59038 q^{47} -0.910860 q^{49} +19.1751 q^{51} -1.00000 q^{53} +2.19992 q^{55} -3.30281 q^{57} +7.88021 q^{59} -2.92689 q^{61} +19.5153 q^{63} -5.17164 q^{65} +8.20479 q^{67} +0.629019 q^{69} -2.88321 q^{71} -2.34889 q^{73} -10.3940 q^{75} -3.98796 q^{77} -4.56433 q^{79} +29.8198 q^{81} -3.53645 q^{83} -7.90292 q^{85} -1.48533 q^{87} +8.05543 q^{89} +9.37501 q^{91} +10.4353 q^{93} +1.36124 q^{95} -3.69548 q^{97} -12.7812 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9} + q^{11} - q^{13} + 8 q^{15} + 3 q^{17} - 19 q^{19} + 8 q^{21} + 10 q^{23} + 21 q^{25} + 28 q^{27} + 2 q^{29} + 25 q^{31} + q^{33} + 20 q^{35} + 19 q^{37} + 37 q^{39} - 9 q^{41} + 35 q^{43} + 37 q^{45} + 23 q^{47} + 30 q^{49} + 34 q^{51} - 19 q^{53} + 40 q^{55} - 4 q^{57} + 16 q^{59} + 21 q^{61} + 3 q^{63} - 10 q^{65} + 67 q^{67} + 23 q^{69} + 18 q^{71} - 20 q^{73} + 33 q^{75} + 37 q^{77} + 2 q^{79} + 23 q^{81} + 38 q^{83} + 8 q^{85} + 18 q^{87} - q^{89} - 9 q^{91} + 14 q^{93} - 4 q^{95} - 21 q^{97} + 33 q^{99}+O(q^{100}) \) 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
\(3\) 3.30281 1.90688 0.953440 0.301582i \(-0.0975149\pi\)
0.953440 + 0.301582i \(0.0975149\pi\)
\(4\) 0 0
\(5\) −1.36124 −0.608765 −0.304382 0.952550i \(-0.598450\pi\)
−0.304382 + 0.952550i \(0.598450\pi\)
\(6\) 0 0
\(7\) 2.46762 0.932672 0.466336 0.884608i \(-0.345574\pi\)
0.466336 + 0.884608i \(0.345574\pi\)
\(8\) 0 0
\(9\) 7.90858 2.63619
\(10\) 0 0
\(11\) −1.61612 −0.487278 −0.243639 0.969866i \(-0.578341\pi\)
−0.243639 + 0.969866i \(0.578341\pi\)
\(12\) 0 0
\(13\) 3.79921 1.05371 0.526856 0.849954i \(-0.323371\pi\)
0.526856 + 0.849954i \(0.323371\pi\)
\(14\) 0 0
\(15\) −4.49592 −1.16084
\(16\) 0 0
\(17\) 5.80568 1.40808 0.704042 0.710158i \(-0.251377\pi\)
0.704042 + 0.710158i \(0.251377\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 8.15008 1.77849
\(22\) 0 0
\(23\) 0.190450 0.0397115 0.0198557 0.999803i \(-0.493679\pi\)
0.0198557 + 0.999803i \(0.493679\pi\)
\(24\) 0 0
\(25\) −3.14703 −0.629405
\(26\) 0 0
\(27\) 16.2121 3.12002
\(28\) 0 0
\(29\) −0.449717 −0.0835104 −0.0417552 0.999128i \(-0.513295\pi\)
−0.0417552 + 0.999128i \(0.513295\pi\)
\(30\) 0 0
\(31\) 3.15951 0.567465 0.283733 0.958903i \(-0.408427\pi\)
0.283733 + 0.958903i \(0.408427\pi\)
\(32\) 0 0
\(33\) −5.33773 −0.929180
\(34\) 0 0
\(35\) −3.35902 −0.567778
\(36\) 0 0
\(37\) 3.75573 0.617439 0.308719 0.951153i \(-0.400100\pi\)
0.308719 + 0.951153i \(0.400100\pi\)
\(38\) 0 0
\(39\) 12.5481 2.00930
\(40\) 0 0
\(41\) −5.76176 −0.899836 −0.449918 0.893070i \(-0.648547\pi\)
−0.449918 + 0.893070i \(0.648547\pi\)
\(42\) 0 0
\(43\) −11.1067 −1.69376 −0.846878 0.531788i \(-0.821520\pi\)
−0.846878 + 0.531788i \(0.821520\pi\)
\(44\) 0 0
\(45\) −10.7655 −1.60482
\(46\) 0 0
\(47\) −3.59038 −0.523710 −0.261855 0.965107i \(-0.584334\pi\)
−0.261855 + 0.965107i \(0.584334\pi\)
\(48\) 0 0
\(49\) −0.910860 −0.130123
\(50\) 0 0
\(51\) 19.1751 2.68505
\(52\) 0 0
\(53\) −1.00000 −0.137361
\(54\) 0 0
\(55\) 2.19992 0.296638
\(56\) 0 0
\(57\) −3.30281 −0.437468
\(58\) 0 0
\(59\) 7.88021 1.02592 0.512958 0.858414i \(-0.328550\pi\)
0.512958 + 0.858414i \(0.328550\pi\)
\(60\) 0 0
\(61\) −2.92689 −0.374750 −0.187375 0.982288i \(-0.559998\pi\)
−0.187375 + 0.982288i \(0.559998\pi\)
\(62\) 0 0
\(63\) 19.5153 2.45870
\(64\) 0 0
\(65\) −5.17164 −0.641463
\(66\) 0 0
\(67\) 8.20479 1.00237 0.501187 0.865339i \(-0.332897\pi\)
0.501187 + 0.865339i \(0.332897\pi\)
\(68\) 0 0
\(69\) 0.629019 0.0757250
\(70\) 0 0
\(71\) −2.88321 −0.342174 −0.171087 0.985256i \(-0.554728\pi\)
−0.171087 + 0.985256i \(0.554728\pi\)
\(72\) 0 0
\(73\) −2.34889 −0.274917 −0.137458 0.990508i \(-0.543893\pi\)
−0.137458 + 0.990508i \(0.543893\pi\)
\(74\) 0 0
\(75\) −10.3940 −1.20020
\(76\) 0 0
\(77\) −3.98796 −0.454470
\(78\) 0 0
\(79\) −4.56433 −0.513527 −0.256764 0.966474i \(-0.582656\pi\)
−0.256764 + 0.966474i \(0.582656\pi\)
\(80\) 0 0
\(81\) 29.8198 3.31332
\(82\) 0 0
\(83\) −3.53645 −0.388176 −0.194088 0.980984i \(-0.562175\pi\)
−0.194088 + 0.980984i \(0.562175\pi\)
\(84\) 0 0
\(85\) −7.90292 −0.857192
\(86\) 0 0
\(87\) −1.48533 −0.159244
\(88\) 0 0
\(89\) 8.05543 0.853874 0.426937 0.904281i \(-0.359593\pi\)
0.426937 + 0.904281i \(0.359593\pi\)
\(90\) 0 0
\(91\) 9.37501 0.982768
\(92\) 0 0
\(93\) 10.4353 1.08209
\(94\) 0 0
\(95\) 1.36124 0.139660
\(96\) 0 0
\(97\) −3.69548 −0.375219 −0.187610 0.982244i \(-0.560074\pi\)
−0.187610 + 0.982244i \(0.560074\pi\)
\(98\) 0 0
\(99\) −12.7812 −1.28456
\(100\) 0 0
\(101\) −11.2888 −1.12328 −0.561641 0.827381i \(-0.689830\pi\)
−0.561641 + 0.827381i \(0.689830\pi\)
\(102\) 0 0
\(103\) 14.4463 1.42343 0.711716 0.702468i \(-0.247918\pi\)
0.711716 + 0.702468i \(0.247918\pi\)
\(104\) 0 0
\(105\) −11.0942 −1.08268
\(106\) 0 0
\(107\) 17.4687 1.68877 0.844384 0.535739i \(-0.179967\pi\)
0.844384 + 0.535739i \(0.179967\pi\)
\(108\) 0 0
\(109\) 13.2491 1.26903 0.634517 0.772909i \(-0.281199\pi\)
0.634517 + 0.772909i \(0.281199\pi\)
\(110\) 0 0
\(111\) 12.4045 1.17738
\(112\) 0 0
\(113\) −11.9483 −1.12400 −0.562002 0.827136i \(-0.689969\pi\)
−0.562002 + 0.827136i \(0.689969\pi\)
\(114\) 0 0
\(115\) −0.259248 −0.0241750
\(116\) 0 0
\(117\) 30.0464 2.77779
\(118\) 0 0
\(119\) 14.3262 1.31328
\(120\) 0 0
\(121\) −8.38816 −0.762560
\(122\) 0 0
\(123\) −19.0300 −1.71588
\(124\) 0 0
\(125\) 11.0901 0.991925
\(126\) 0 0
\(127\) 20.1382 1.78698 0.893488 0.449086i \(-0.148250\pi\)
0.893488 + 0.449086i \(0.148250\pi\)
\(128\) 0 0
\(129\) −36.6833 −3.22979
\(130\) 0 0
\(131\) −17.3360 −1.51465 −0.757326 0.653037i \(-0.773495\pi\)
−0.757326 + 0.653037i \(0.773495\pi\)
\(132\) 0 0
\(133\) −2.46762 −0.213970
\(134\) 0 0
\(135\) −22.0686 −1.89936
\(136\) 0 0
\(137\) −4.76509 −0.407109 −0.203555 0.979064i \(-0.565249\pi\)
−0.203555 + 0.979064i \(0.565249\pi\)
\(138\) 0 0
\(139\) −0.585771 −0.0496844 −0.0248422 0.999691i \(-0.507908\pi\)
−0.0248422 + 0.999691i \(0.507908\pi\)
\(140\) 0 0
\(141\) −11.8583 −0.998653
\(142\) 0 0
\(143\) −6.13997 −0.513450
\(144\) 0 0
\(145\) 0.612173 0.0508382
\(146\) 0 0
\(147\) −3.00840 −0.248129
\(148\) 0 0
\(149\) −18.8537 −1.54455 −0.772277 0.635286i \(-0.780882\pi\)
−0.772277 + 0.635286i \(0.780882\pi\)
\(150\) 0 0
\(151\) −0.325422 −0.0264824 −0.0132412 0.999912i \(-0.504215\pi\)
−0.0132412 + 0.999912i \(0.504215\pi\)
\(152\) 0 0
\(153\) 45.9147 3.71198
\(154\) 0 0
\(155\) −4.30085 −0.345453
\(156\) 0 0
\(157\) 2.24083 0.178837 0.0894187 0.995994i \(-0.471499\pi\)
0.0894187 + 0.995994i \(0.471499\pi\)
\(158\) 0 0
\(159\) −3.30281 −0.261930
\(160\) 0 0
\(161\) 0.469957 0.0370378
\(162\) 0 0
\(163\) −2.83476 −0.222036 −0.111018 0.993818i \(-0.535411\pi\)
−0.111018 + 0.993818i \(0.535411\pi\)
\(164\) 0 0
\(165\) 7.26593 0.565652
\(166\) 0 0
\(167\) −4.36625 −0.337871 −0.168935 0.985627i \(-0.554033\pi\)
−0.168935 + 0.985627i \(0.554033\pi\)
\(168\) 0 0
\(169\) 1.43402 0.110309
\(170\) 0 0
\(171\) −7.90858 −0.604784
\(172\) 0 0
\(173\) 4.20871 0.319982 0.159991 0.987118i \(-0.448853\pi\)
0.159991 + 0.987118i \(0.448853\pi\)
\(174\) 0 0
\(175\) −7.76566 −0.587029
\(176\) 0 0
\(177\) 26.0269 1.95630
\(178\) 0 0
\(179\) −13.8986 −1.03883 −0.519416 0.854521i \(-0.673851\pi\)
−0.519416 + 0.854521i \(0.673851\pi\)
\(180\) 0 0
\(181\) 15.9446 1.18515 0.592577 0.805514i \(-0.298110\pi\)
0.592577 + 0.805514i \(0.298110\pi\)
\(182\) 0 0
\(183\) −9.66699 −0.714604
\(184\) 0 0
\(185\) −5.11245 −0.375875
\(186\) 0 0
\(187\) −9.38266 −0.686128
\(188\) 0 0
\(189\) 40.0053 2.90996
\(190\) 0 0
\(191\) 14.5324 1.05152 0.525762 0.850631i \(-0.323780\pi\)
0.525762 + 0.850631i \(0.323780\pi\)
\(192\) 0 0
\(193\) 20.1303 1.44901 0.724506 0.689268i \(-0.242068\pi\)
0.724506 + 0.689268i \(0.242068\pi\)
\(194\) 0 0
\(195\) −17.0810 −1.22319
\(196\) 0 0
\(197\) 20.3371 1.44896 0.724479 0.689297i \(-0.242081\pi\)
0.724479 + 0.689297i \(0.242081\pi\)
\(198\) 0 0
\(199\) −23.3629 −1.65615 −0.828075 0.560617i \(-0.810564\pi\)
−0.828075 + 0.560617i \(0.810564\pi\)
\(200\) 0 0
\(201\) 27.0989 1.91141
\(202\) 0 0
\(203\) −1.10973 −0.0778878
\(204\) 0 0
\(205\) 7.84314 0.547788
\(206\) 0 0
\(207\) 1.50619 0.104687
\(208\) 0 0
\(209\) 1.61612 0.111789
\(210\) 0 0
\(211\) 10.3865 0.715039 0.357519 0.933906i \(-0.383623\pi\)
0.357519 + 0.933906i \(0.383623\pi\)
\(212\) 0 0
\(213\) −9.52272 −0.652486
\(214\) 0 0
\(215\) 15.1189 1.03110
\(216\) 0 0
\(217\) 7.79647 0.529259
\(218\) 0 0
\(219\) −7.75795 −0.524234
\(220\) 0 0
\(221\) 22.0570 1.48372
\(222\) 0 0
\(223\) 3.50818 0.234925 0.117462 0.993077i \(-0.462524\pi\)
0.117462 + 0.993077i \(0.462524\pi\)
\(224\) 0 0
\(225\) −24.8885 −1.65923
\(226\) 0 0
\(227\) −27.3856 −1.81764 −0.908821 0.417185i \(-0.863017\pi\)
−0.908821 + 0.417185i \(0.863017\pi\)
\(228\) 0 0
\(229\) 20.5269 1.35646 0.678228 0.734852i \(-0.262748\pi\)
0.678228 + 0.734852i \(0.262748\pi\)
\(230\) 0 0
\(231\) −13.1715 −0.866620
\(232\) 0 0
\(233\) −7.24612 −0.474709 −0.237355 0.971423i \(-0.576280\pi\)
−0.237355 + 0.971423i \(0.576280\pi\)
\(234\) 0 0
\(235\) 4.88736 0.318816
\(236\) 0 0
\(237\) −15.0751 −0.979235
\(238\) 0 0
\(239\) 0.604707 0.0391152 0.0195576 0.999809i \(-0.493774\pi\)
0.0195576 + 0.999809i \(0.493774\pi\)
\(240\) 0 0
\(241\) 23.0652 1.48576 0.742880 0.669425i \(-0.233459\pi\)
0.742880 + 0.669425i \(0.233459\pi\)
\(242\) 0 0
\(243\) 49.8531 3.19808
\(244\) 0 0
\(245\) 1.23990 0.0792142
\(246\) 0 0
\(247\) −3.79921 −0.241738
\(248\) 0 0
\(249\) −11.6802 −0.740204
\(250\) 0 0
\(251\) 2.72188 0.171803 0.0859017 0.996304i \(-0.472623\pi\)
0.0859017 + 0.996304i \(0.472623\pi\)
\(252\) 0 0
\(253\) −0.307789 −0.0193505
\(254\) 0 0
\(255\) −26.1019 −1.63456
\(256\) 0 0
\(257\) −14.9536 −0.932780 −0.466390 0.884579i \(-0.654446\pi\)
−0.466390 + 0.884579i \(0.654446\pi\)
\(258\) 0 0
\(259\) 9.26771 0.575868
\(260\) 0 0
\(261\) −3.55662 −0.220150
\(262\) 0 0
\(263\) −7.81332 −0.481790 −0.240895 0.970551i \(-0.577441\pi\)
−0.240895 + 0.970551i \(0.577441\pi\)
\(264\) 0 0
\(265\) 1.36124 0.0836203
\(266\) 0 0
\(267\) 26.6056 1.62824
\(268\) 0 0
\(269\) 6.73144 0.410423 0.205212 0.978718i \(-0.434212\pi\)
0.205212 + 0.978718i \(0.434212\pi\)
\(270\) 0 0
\(271\) −0.275159 −0.0167147 −0.00835736 0.999965i \(-0.502660\pi\)
−0.00835736 + 0.999965i \(0.502660\pi\)
\(272\) 0 0
\(273\) 30.9639 1.87402
\(274\) 0 0
\(275\) 5.08596 0.306695
\(276\) 0 0
\(277\) −21.8188 −1.31096 −0.655482 0.755211i \(-0.727535\pi\)
−0.655482 + 0.755211i \(0.727535\pi\)
\(278\) 0 0
\(279\) 24.9872 1.49595
\(280\) 0 0
\(281\) −5.59747 −0.333917 −0.166959 0.985964i \(-0.553395\pi\)
−0.166959 + 0.985964i \(0.553395\pi\)
\(282\) 0 0
\(283\) −1.62668 −0.0966963 −0.0483481 0.998831i \(-0.515396\pi\)
−0.0483481 + 0.998831i \(0.515396\pi\)
\(284\) 0 0
\(285\) 4.49592 0.266315
\(286\) 0 0
\(287\) −14.2178 −0.839252
\(288\) 0 0
\(289\) 16.7059 0.982702
\(290\) 0 0
\(291\) −12.2055 −0.715498
\(292\) 0 0
\(293\) −19.2205 −1.12287 −0.561436 0.827520i \(-0.689751\pi\)
−0.561436 + 0.827520i \(0.689751\pi\)
\(294\) 0 0
\(295\) −10.7269 −0.624541
\(296\) 0 0
\(297\) −26.2007 −1.52032
\(298\) 0 0
\(299\) 0.723559 0.0418445
\(300\) 0 0
\(301\) −27.4071 −1.57972
\(302\) 0 0
\(303\) −37.2849 −2.14196
\(304\) 0 0
\(305\) 3.98421 0.228135
\(306\) 0 0
\(307\) −2.00548 −0.114459 −0.0572294 0.998361i \(-0.518227\pi\)
−0.0572294 + 0.998361i \(0.518227\pi\)
\(308\) 0 0
\(309\) 47.7133 2.71431
\(310\) 0 0
\(311\) −25.6226 −1.45293 −0.726463 0.687205i \(-0.758837\pi\)
−0.726463 + 0.687205i \(0.758837\pi\)
\(312\) 0 0
\(313\) −27.1232 −1.53310 −0.766548 0.642187i \(-0.778027\pi\)
−0.766548 + 0.642187i \(0.778027\pi\)
\(314\) 0 0
\(315\) −26.5651 −1.49677
\(316\) 0 0
\(317\) −12.2414 −0.687543 −0.343772 0.939053i \(-0.611705\pi\)
−0.343772 + 0.939053i \(0.611705\pi\)
\(318\) 0 0
\(319\) 0.726796 0.0406928
\(320\) 0 0
\(321\) 57.6960 3.22028
\(322\) 0 0
\(323\) −5.80568 −0.323037
\(324\) 0 0
\(325\) −11.9562 −0.663212
\(326\) 0 0
\(327\) 43.7593 2.41990
\(328\) 0 0
\(329\) −8.85968 −0.488450
\(330\) 0 0
\(331\) 12.7150 0.698881 0.349441 0.936959i \(-0.386372\pi\)
0.349441 + 0.936959i \(0.386372\pi\)
\(332\) 0 0
\(333\) 29.7025 1.62769
\(334\) 0 0
\(335\) −11.1687 −0.610210
\(336\) 0 0
\(337\) −19.2102 −1.04645 −0.523223 0.852196i \(-0.675271\pi\)
−0.523223 + 0.852196i \(0.675271\pi\)
\(338\) 0 0
\(339\) −39.4631 −2.14334
\(340\) 0 0
\(341\) −5.10614 −0.276513
\(342\) 0 0
\(343\) −19.5210 −1.05403
\(344\) 0 0
\(345\) −0.856246 −0.0460987
\(346\) 0 0
\(347\) 3.63724 0.195257 0.0976286 0.995223i \(-0.468874\pi\)
0.0976286 + 0.995223i \(0.468874\pi\)
\(348\) 0 0
\(349\) 14.4674 0.774421 0.387210 0.921991i \(-0.373439\pi\)
0.387210 + 0.921991i \(0.373439\pi\)
\(350\) 0 0
\(351\) 61.5933 3.28761
\(352\) 0 0
\(353\) −18.8466 −1.00310 −0.501552 0.865127i \(-0.667237\pi\)
−0.501552 + 0.865127i \(0.667237\pi\)
\(354\) 0 0
\(355\) 3.92474 0.208304
\(356\) 0 0
\(357\) 47.3168 2.50427
\(358\) 0 0
\(359\) −10.0552 −0.530695 −0.265348 0.964153i \(-0.585487\pi\)
−0.265348 + 0.964153i \(0.585487\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −27.7045 −1.45411
\(364\) 0 0
\(365\) 3.19740 0.167360
\(366\) 0 0
\(367\) 20.2269 1.05584 0.527918 0.849295i \(-0.322973\pi\)
0.527918 + 0.849295i \(0.322973\pi\)
\(368\) 0 0
\(369\) −45.5673 −2.37214
\(370\) 0 0
\(371\) −2.46762 −0.128112
\(372\) 0 0
\(373\) −6.58343 −0.340877 −0.170438 0.985368i \(-0.554518\pi\)
−0.170438 + 0.985368i \(0.554518\pi\)
\(374\) 0 0
\(375\) 36.6284 1.89148
\(376\) 0 0
\(377\) −1.70857 −0.0879960
\(378\) 0 0
\(379\) −9.72104 −0.499336 −0.249668 0.968331i \(-0.580322\pi\)
−0.249668 + 0.968331i \(0.580322\pi\)
\(380\) 0 0
\(381\) 66.5127 3.40755
\(382\) 0 0
\(383\) −15.3123 −0.782421 −0.391210 0.920301i \(-0.627943\pi\)
−0.391210 + 0.920301i \(0.627943\pi\)
\(384\) 0 0
\(385\) 5.42857 0.276666
\(386\) 0 0
\(387\) −87.8382 −4.46506
\(388\) 0 0
\(389\) −4.61393 −0.233935 −0.116968 0.993136i \(-0.537317\pi\)
−0.116968 + 0.993136i \(0.537317\pi\)
\(390\) 0 0
\(391\) 1.10569 0.0559171
\(392\) 0 0
\(393\) −57.2575 −2.88826
\(394\) 0 0
\(395\) 6.21315 0.312617
\(396\) 0 0
\(397\) 20.0394 1.00575 0.502875 0.864359i \(-0.332275\pi\)
0.502875 + 0.864359i \(0.332275\pi\)
\(398\) 0 0
\(399\) −8.15008 −0.408014
\(400\) 0 0
\(401\) −28.5669 −1.42656 −0.713282 0.700877i \(-0.752792\pi\)
−0.713282 + 0.700877i \(0.752792\pi\)
\(402\) 0 0
\(403\) 12.0037 0.597945
\(404\) 0 0
\(405\) −40.5920 −2.01703
\(406\) 0 0
\(407\) −6.06970 −0.300864
\(408\) 0 0
\(409\) −1.36323 −0.0674073 −0.0337037 0.999432i \(-0.510730\pi\)
−0.0337037 + 0.999432i \(0.510730\pi\)
\(410\) 0 0
\(411\) −15.7382 −0.776309
\(412\) 0 0
\(413\) 19.4453 0.956843
\(414\) 0 0
\(415\) 4.81395 0.236308
\(416\) 0 0
\(417\) −1.93469 −0.0947422
\(418\) 0 0
\(419\) −27.8248 −1.35933 −0.679666 0.733521i \(-0.737875\pi\)
−0.679666 + 0.733521i \(0.737875\pi\)
\(420\) 0 0
\(421\) −24.6880 −1.20322 −0.601610 0.798790i \(-0.705474\pi\)
−0.601610 + 0.798790i \(0.705474\pi\)
\(422\) 0 0
\(423\) −28.3948 −1.38060
\(424\) 0 0
\(425\) −18.2706 −0.886256
\(426\) 0 0
\(427\) −7.22246 −0.349519
\(428\) 0 0
\(429\) −20.2792 −0.979089
\(430\) 0 0
\(431\) −13.6653 −0.658234 −0.329117 0.944289i \(-0.606751\pi\)
−0.329117 + 0.944289i \(0.606751\pi\)
\(432\) 0 0
\(433\) −0.497570 −0.0239117 −0.0119558 0.999929i \(-0.503806\pi\)
−0.0119558 + 0.999929i \(0.503806\pi\)
\(434\) 0 0
\(435\) 2.02189 0.0969424
\(436\) 0 0
\(437\) −0.190450 −0.00911044
\(438\) 0 0
\(439\) 30.9878 1.47897 0.739484 0.673174i \(-0.235069\pi\)
0.739484 + 0.673174i \(0.235069\pi\)
\(440\) 0 0
\(441\) −7.20361 −0.343029
\(442\) 0 0
\(443\) −9.62366 −0.457234 −0.228617 0.973516i \(-0.573420\pi\)
−0.228617 + 0.973516i \(0.573420\pi\)
\(444\) 0 0
\(445\) −10.9654 −0.519809
\(446\) 0 0
\(447\) −62.2702 −2.94528
\(448\) 0 0
\(449\) −33.3689 −1.57477 −0.787387 0.616459i \(-0.788567\pi\)
−0.787387 + 0.616459i \(0.788567\pi\)
\(450\) 0 0
\(451\) 9.31168 0.438470
\(452\) 0 0
\(453\) −1.07481 −0.0504988
\(454\) 0 0
\(455\) −12.7616 −0.598275
\(456\) 0 0
\(457\) −7.12651 −0.333364 −0.166682 0.986011i \(-0.553305\pi\)
−0.166682 + 0.986011i \(0.553305\pi\)
\(458\) 0 0
\(459\) 94.1223 4.39325
\(460\) 0 0
\(461\) 21.9427 1.02197 0.510987 0.859589i \(-0.329280\pi\)
0.510987 + 0.859589i \(0.329280\pi\)
\(462\) 0 0
\(463\) 12.3472 0.573823 0.286911 0.957957i \(-0.407371\pi\)
0.286911 + 0.957957i \(0.407371\pi\)
\(464\) 0 0
\(465\) −14.2049 −0.658737
\(466\) 0 0
\(467\) 10.4938 0.485593 0.242797 0.970077i \(-0.421935\pi\)
0.242797 + 0.970077i \(0.421935\pi\)
\(468\) 0 0
\(469\) 20.2463 0.934887
\(470\) 0 0
\(471\) 7.40103 0.341022
\(472\) 0 0
\(473\) 17.9497 0.825329
\(474\) 0 0
\(475\) 3.14703 0.144395
\(476\) 0 0
\(477\) −7.90858 −0.362109
\(478\) 0 0
\(479\) 38.0827 1.74004 0.870021 0.493014i \(-0.164105\pi\)
0.870021 + 0.493014i \(0.164105\pi\)
\(480\) 0 0
\(481\) 14.2688 0.650603
\(482\) 0 0
\(483\) 1.55218 0.0706266
\(484\) 0 0
\(485\) 5.03043 0.228420
\(486\) 0 0
\(487\) −9.50810 −0.430853 −0.215427 0.976520i \(-0.569114\pi\)
−0.215427 + 0.976520i \(0.569114\pi\)
\(488\) 0 0
\(489\) −9.36269 −0.423396
\(490\) 0 0
\(491\) −25.0069 −1.12854 −0.564272 0.825589i \(-0.690843\pi\)
−0.564272 + 0.825589i \(0.690843\pi\)
\(492\) 0 0
\(493\) −2.61092 −0.117590
\(494\) 0 0
\(495\) 17.3983 0.781994
\(496\) 0 0
\(497\) −7.11467 −0.319137
\(498\) 0 0
\(499\) −24.7494 −1.10794 −0.553968 0.832538i \(-0.686887\pi\)
−0.553968 + 0.832538i \(0.686887\pi\)
\(500\) 0 0
\(501\) −14.4209 −0.644279
\(502\) 0 0
\(503\) −14.8402 −0.661690 −0.330845 0.943685i \(-0.607334\pi\)
−0.330845 + 0.943685i \(0.607334\pi\)
\(504\) 0 0
\(505\) 15.3668 0.683814
\(506\) 0 0
\(507\) 4.73631 0.210347
\(508\) 0 0
\(509\) 33.3200 1.47688 0.738441 0.674319i \(-0.235563\pi\)
0.738441 + 0.674319i \(0.235563\pi\)
\(510\) 0 0
\(511\) −5.79617 −0.256407
\(512\) 0 0
\(513\) −16.2121 −0.715782
\(514\) 0 0
\(515\) −19.6648 −0.866535
\(516\) 0 0
\(517\) 5.80247 0.255192
\(518\) 0 0
\(519\) 13.9006 0.610168
\(520\) 0 0
\(521\) −21.5146 −0.942572 −0.471286 0.881980i \(-0.656210\pi\)
−0.471286 + 0.881980i \(0.656210\pi\)
\(522\) 0 0
\(523\) 38.5273 1.68468 0.842340 0.538946i \(-0.181177\pi\)
0.842340 + 0.538946i \(0.181177\pi\)
\(524\) 0 0
\(525\) −25.6485 −1.11939
\(526\) 0 0
\(527\) 18.3431 0.799039
\(528\) 0 0
\(529\) −22.9637 −0.998423
\(530\) 0 0
\(531\) 62.3212 2.70451
\(532\) 0 0
\(533\) −21.8902 −0.948168
\(534\) 0 0
\(535\) −23.7792 −1.02806
\(536\) 0 0
\(537\) −45.9046 −1.98093
\(538\) 0 0
\(539\) 1.47206 0.0634060
\(540\) 0 0
\(541\) 10.5751 0.454658 0.227329 0.973818i \(-0.427001\pi\)
0.227329 + 0.973818i \(0.427001\pi\)
\(542\) 0 0
\(543\) 52.6621 2.25995
\(544\) 0 0
\(545\) −18.0352 −0.772544
\(546\) 0 0
\(547\) −28.8990 −1.23563 −0.617816 0.786323i \(-0.711982\pi\)
−0.617816 + 0.786323i \(0.711982\pi\)
\(548\) 0 0
\(549\) −23.1476 −0.987914
\(550\) 0 0
\(551\) 0.449717 0.0191586
\(552\) 0 0
\(553\) −11.2630 −0.478952
\(554\) 0 0
\(555\) −16.8855 −0.716749
\(556\) 0 0
\(557\) −21.8756 −0.926900 −0.463450 0.886123i \(-0.653389\pi\)
−0.463450 + 0.886123i \(0.653389\pi\)
\(558\) 0 0
\(559\) −42.1967 −1.78473
\(560\) 0 0
\(561\) −30.9892 −1.30836
\(562\) 0 0
\(563\) 27.0135 1.13848 0.569242 0.822170i \(-0.307237\pi\)
0.569242 + 0.822170i \(0.307237\pi\)
\(564\) 0 0
\(565\) 16.2645 0.684254
\(566\) 0 0
\(567\) 73.5840 3.09024
\(568\) 0 0
\(569\) 5.30868 0.222551 0.111276 0.993790i \(-0.464506\pi\)
0.111276 + 0.993790i \(0.464506\pi\)
\(570\) 0 0
\(571\) 17.7065 0.740992 0.370496 0.928834i \(-0.379188\pi\)
0.370496 + 0.928834i \(0.379188\pi\)
\(572\) 0 0
\(573\) 47.9977 2.00513
\(574\) 0 0
\(575\) −0.599350 −0.0249946
\(576\) 0 0
\(577\) 9.79362 0.407714 0.203857 0.979001i \(-0.434652\pi\)
0.203857 + 0.979001i \(0.434652\pi\)
\(578\) 0 0
\(579\) 66.4867 2.76309
\(580\) 0 0
\(581\) −8.72660 −0.362041
\(582\) 0 0
\(583\) 1.61612 0.0669327
\(584\) 0 0
\(585\) −40.9003 −1.69102
\(586\) 0 0
\(587\) 46.0667 1.90137 0.950687 0.310150i \(-0.100379\pi\)
0.950687 + 0.310150i \(0.100379\pi\)
\(588\) 0 0
\(589\) −3.15951 −0.130185
\(590\) 0 0
\(591\) 67.1696 2.76299
\(592\) 0 0
\(593\) −39.2147 −1.61036 −0.805178 0.593033i \(-0.797930\pi\)
−0.805178 + 0.593033i \(0.797930\pi\)
\(594\) 0 0
\(595\) −19.5014 −0.799479
\(596\) 0 0
\(597\) −77.1632 −3.15808
\(598\) 0 0
\(599\) −11.4553 −0.468051 −0.234025 0.972230i \(-0.575190\pi\)
−0.234025 + 0.972230i \(0.575190\pi\)
\(600\) 0 0
\(601\) −11.4147 −0.465614 −0.232807 0.972523i \(-0.574791\pi\)
−0.232807 + 0.972523i \(0.574791\pi\)
\(602\) 0 0
\(603\) 64.8882 2.64245
\(604\) 0 0
\(605\) 11.4183 0.464220
\(606\) 0 0
\(607\) −9.50158 −0.385657 −0.192829 0.981232i \(-0.561766\pi\)
−0.192829 + 0.981232i \(0.561766\pi\)
\(608\) 0 0
\(609\) −3.66523 −0.148523
\(610\) 0 0
\(611\) −13.6406 −0.551840
\(612\) 0 0
\(613\) 36.1349 1.45947 0.729737 0.683727i \(-0.239642\pi\)
0.729737 + 0.683727i \(0.239642\pi\)
\(614\) 0 0
\(615\) 25.9044 1.04457
\(616\) 0 0
\(617\) −43.1518 −1.73722 −0.868612 0.495493i \(-0.834988\pi\)
−0.868612 + 0.495493i \(0.834988\pi\)
\(618\) 0 0
\(619\) −42.7775 −1.71937 −0.859687 0.510821i \(-0.829341\pi\)
−0.859687 + 0.510821i \(0.829341\pi\)
\(620\) 0 0
\(621\) 3.08759 0.123901
\(622\) 0 0
\(623\) 19.8777 0.796385
\(624\) 0 0
\(625\) 0.638911 0.0255564
\(626\) 0 0
\(627\) 5.33773 0.213169
\(628\) 0 0
\(629\) 21.8046 0.869406
\(630\) 0 0
\(631\) −19.6862 −0.783693 −0.391847 0.920031i \(-0.628164\pi\)
−0.391847 + 0.920031i \(0.628164\pi\)
\(632\) 0 0
\(633\) 34.3048 1.36349
\(634\) 0 0
\(635\) −27.4129 −1.08785
\(636\) 0 0
\(637\) −3.46055 −0.137112
\(638\) 0 0
\(639\) −22.8021 −0.902038
\(640\) 0 0
\(641\) 4.89125 0.193193 0.0965963 0.995324i \(-0.469204\pi\)
0.0965963 + 0.995324i \(0.469204\pi\)
\(642\) 0 0
\(643\) 47.4838 1.87258 0.936290 0.351228i \(-0.114236\pi\)
0.936290 + 0.351228i \(0.114236\pi\)
\(644\) 0 0
\(645\) 49.9348 1.96618
\(646\) 0 0
\(647\) 9.60803 0.377731 0.188865 0.982003i \(-0.439519\pi\)
0.188865 + 0.982003i \(0.439519\pi\)
\(648\) 0 0
\(649\) −12.7353 −0.499906
\(650\) 0 0
\(651\) 25.7503 1.00923
\(652\) 0 0
\(653\) 40.3300 1.57823 0.789117 0.614243i \(-0.210539\pi\)
0.789117 + 0.614243i \(0.210539\pi\)
\(654\) 0 0
\(655\) 23.5984 0.922067
\(656\) 0 0
\(657\) −18.5764 −0.724734
\(658\) 0 0
\(659\) 15.6104 0.608095 0.304047 0.952657i \(-0.401662\pi\)
0.304047 + 0.952657i \(0.401662\pi\)
\(660\) 0 0
\(661\) 2.80323 0.109033 0.0545164 0.998513i \(-0.482638\pi\)
0.0545164 + 0.998513i \(0.482638\pi\)
\(662\) 0 0
\(663\) 72.8502 2.82927
\(664\) 0 0
\(665\) 3.35902 0.130257
\(666\) 0 0
\(667\) −0.0856485 −0.00331632
\(668\) 0 0
\(669\) 11.5869 0.447974
\(670\) 0 0
\(671\) 4.73021 0.182608
\(672\) 0 0
\(673\) −9.27073 −0.357360 −0.178680 0.983907i \(-0.557183\pi\)
−0.178680 + 0.983907i \(0.557183\pi\)
\(674\) 0 0
\(675\) −51.0199 −1.96376
\(676\) 0 0
\(677\) −2.03024 −0.0780285 −0.0390143 0.999239i \(-0.512422\pi\)
−0.0390143 + 0.999239i \(0.512422\pi\)
\(678\) 0 0
\(679\) −9.11903 −0.349956
\(680\) 0 0
\(681\) −90.4494 −3.46603
\(682\) 0 0
\(683\) −24.0043 −0.918498 −0.459249 0.888308i \(-0.651881\pi\)
−0.459249 + 0.888308i \(0.651881\pi\)
\(684\) 0 0
\(685\) 6.48643 0.247834
\(686\) 0 0
\(687\) 67.7965 2.58660
\(688\) 0 0
\(689\) −3.79921 −0.144739
\(690\) 0 0
\(691\) 36.7232 1.39702 0.698508 0.715602i \(-0.253847\pi\)
0.698508 + 0.715602i \(0.253847\pi\)
\(692\) 0 0
\(693\) −31.5391 −1.19807
\(694\) 0 0
\(695\) 0.797374 0.0302461
\(696\) 0 0
\(697\) −33.4509 −1.26704
\(698\) 0 0
\(699\) −23.9326 −0.905213
\(700\) 0 0
\(701\) −43.5117 −1.64341 −0.821707 0.569910i \(-0.806978\pi\)
−0.821707 + 0.569910i \(0.806978\pi\)
\(702\) 0 0
\(703\) −3.75573 −0.141650
\(704\) 0 0
\(705\) 16.1420 0.607945
\(706\) 0 0
\(707\) −27.8566 −1.04765
\(708\) 0 0
\(709\) 25.1065 0.942894 0.471447 0.881894i \(-0.343732\pi\)
0.471447 + 0.881894i \(0.343732\pi\)
\(710\) 0 0
\(711\) −36.0973 −1.35376
\(712\) 0 0
\(713\) 0.601728 0.0225349
\(714\) 0 0
\(715\) 8.35798 0.312571
\(716\) 0 0
\(717\) 1.99723 0.0745881
\(718\) 0 0
\(719\) 38.3607 1.43061 0.715307 0.698811i \(-0.246287\pi\)
0.715307 + 0.698811i \(0.246287\pi\)
\(720\) 0 0
\(721\) 35.6478 1.32759
\(722\) 0 0
\(723\) 76.1800 2.83316
\(724\) 0 0
\(725\) 1.41527 0.0525619
\(726\) 0 0
\(727\) −19.2956 −0.715632 −0.357816 0.933792i \(-0.616479\pi\)
−0.357816 + 0.933792i \(0.616479\pi\)
\(728\) 0 0
\(729\) 75.1958 2.78503
\(730\) 0 0
\(731\) −64.4819 −2.38495
\(732\) 0 0
\(733\) 7.02369 0.259426 0.129713 0.991552i \(-0.458594\pi\)
0.129713 + 0.991552i \(0.458594\pi\)
\(734\) 0 0
\(735\) 4.09515 0.151052
\(736\) 0 0
\(737\) −13.2599 −0.488435
\(738\) 0 0
\(739\) 37.5531 1.38141 0.690706 0.723136i \(-0.257300\pi\)
0.690706 + 0.723136i \(0.257300\pi\)
\(740\) 0 0
\(741\) −12.5481 −0.460966
\(742\) 0 0
\(743\) 16.1947 0.594127 0.297063 0.954858i \(-0.403993\pi\)
0.297063 + 0.954858i \(0.403993\pi\)
\(744\) 0 0
\(745\) 25.6644 0.940270
\(746\) 0 0
\(747\) −27.9683 −1.02331
\(748\) 0 0
\(749\) 43.1062 1.57507
\(750\) 0 0
\(751\) 5.79622 0.211507 0.105754 0.994392i \(-0.466275\pi\)
0.105754 + 0.994392i \(0.466275\pi\)
\(752\) 0 0
\(753\) 8.98986 0.327609
\(754\) 0 0
\(755\) 0.442977 0.0161216
\(756\) 0 0
\(757\) −20.6846 −0.751796 −0.375898 0.926661i \(-0.622666\pi\)
−0.375898 + 0.926661i \(0.622666\pi\)
\(758\) 0 0
\(759\) −1.01657 −0.0368991
\(760\) 0 0
\(761\) 53.1447 1.92649 0.963247 0.268617i \(-0.0865664\pi\)
0.963247 + 0.268617i \(0.0865664\pi\)
\(762\) 0 0
\(763\) 32.6937 1.18359
\(764\) 0 0
\(765\) −62.5009 −2.25972
\(766\) 0 0
\(767\) 29.9386 1.08102
\(768\) 0 0
\(769\) 34.8739 1.25758 0.628792 0.777574i \(-0.283550\pi\)
0.628792 + 0.777574i \(0.283550\pi\)
\(770\) 0 0
\(771\) −49.3890 −1.77870
\(772\) 0 0
\(773\) 35.6436 1.28201 0.641006 0.767536i \(-0.278518\pi\)
0.641006 + 0.767536i \(0.278518\pi\)
\(774\) 0 0
\(775\) −9.94307 −0.357166
\(776\) 0 0
\(777\) 30.6095 1.09811
\(778\) 0 0
\(779\) 5.76176 0.206436
\(780\) 0 0
\(781\) 4.65961 0.166734
\(782\) 0 0
\(783\) −7.29087 −0.260554
\(784\) 0 0
\(785\) −3.05030 −0.108870
\(786\) 0 0
\(787\) −19.5134 −0.695579 −0.347789 0.937573i \(-0.613068\pi\)
−0.347789 + 0.937573i \(0.613068\pi\)
\(788\) 0 0
\(789\) −25.8059 −0.918716
\(790\) 0 0
\(791\) −29.4839 −1.04833
\(792\) 0 0
\(793\) −11.1199 −0.394879
\(794\) 0 0
\(795\) 4.49592 0.159454
\(796\) 0 0
\(797\) −27.2396 −0.964878 −0.482439 0.875930i \(-0.660249\pi\)
−0.482439 + 0.875930i \(0.660249\pi\)
\(798\) 0 0
\(799\) −20.8446 −0.737428
\(800\) 0 0
\(801\) 63.7070 2.25098
\(802\) 0 0
\(803\) 3.79608 0.133961
\(804\) 0 0
\(805\) −0.639724 −0.0225473
\(806\) 0 0
\(807\) 22.2327 0.782628
\(808\) 0 0
\(809\) 23.1297 0.813198 0.406599 0.913607i \(-0.366715\pi\)
0.406599 + 0.913607i \(0.366715\pi\)
\(810\) 0 0
\(811\) −16.0645 −0.564102 −0.282051 0.959399i \(-0.591015\pi\)
−0.282051 + 0.959399i \(0.591015\pi\)
\(812\) 0 0
\(813\) −0.908799 −0.0318730
\(814\) 0 0
\(815\) 3.85879 0.135168
\(816\) 0 0
\(817\) 11.1067 0.388574
\(818\) 0 0
\(819\) 74.1430 2.59076
\(820\) 0 0
\(821\) 0.205065 0.00715682 0.00357841 0.999994i \(-0.498861\pi\)
0.00357841 + 0.999994i \(0.498861\pi\)
\(822\) 0 0
\(823\) −20.6080 −0.718351 −0.359175 0.933270i \(-0.616942\pi\)
−0.359175 + 0.933270i \(0.616942\pi\)
\(824\) 0 0
\(825\) 16.7980 0.584831
\(826\) 0 0
\(827\) 0.897926 0.0312239 0.0156120 0.999878i \(-0.495030\pi\)
0.0156120 + 0.999878i \(0.495030\pi\)
\(828\) 0 0
\(829\) −18.5387 −0.643874 −0.321937 0.946761i \(-0.604334\pi\)
−0.321937 + 0.946761i \(0.604334\pi\)
\(830\) 0 0
\(831\) −72.0634 −2.49985
\(832\) 0 0
\(833\) −5.28816 −0.183224
\(834\) 0 0
\(835\) 5.94351 0.205684
\(836\) 0 0
\(837\) 51.2224 1.77050
\(838\) 0 0
\(839\) 42.3721 1.46285 0.731423 0.681924i \(-0.238857\pi\)
0.731423 + 0.681924i \(0.238857\pi\)
\(840\) 0 0
\(841\) −28.7978 −0.993026
\(842\) 0 0
\(843\) −18.4874 −0.636740
\(844\) 0 0
\(845\) −1.95205 −0.0671525
\(846\) 0 0
\(847\) −20.6988 −0.711219
\(848\) 0 0
\(849\) −5.37263 −0.184388
\(850\) 0 0
\(851\) 0.715278 0.0245194
\(852\) 0 0
\(853\) −19.0764 −0.653165 −0.326582 0.945169i \(-0.605897\pi\)
−0.326582 + 0.945169i \(0.605897\pi\)
\(854\) 0 0
\(855\) 10.7655 0.368171
\(856\) 0 0
\(857\) −3.07869 −0.105166 −0.0525831 0.998617i \(-0.516745\pi\)
−0.0525831 + 0.998617i \(0.516745\pi\)
\(858\) 0 0
\(859\) 27.2804 0.930794 0.465397 0.885102i \(-0.345911\pi\)
0.465397 + 0.885102i \(0.345911\pi\)
\(860\) 0 0
\(861\) −46.9588 −1.60035
\(862\) 0 0
\(863\) 17.4081 0.592579 0.296290 0.955098i \(-0.404251\pi\)
0.296290 + 0.955098i \(0.404251\pi\)
\(864\) 0 0
\(865\) −5.72906 −0.194794
\(866\) 0 0
\(867\) 55.1766 1.87389
\(868\) 0 0
\(869\) 7.37649 0.250230
\(870\) 0 0
\(871\) 31.1717 1.05621
\(872\) 0 0
\(873\) −29.2260 −0.989150
\(874\) 0 0
\(875\) 27.3660 0.925140
\(876\) 0 0
\(877\) −23.2110 −0.783781 −0.391890 0.920012i \(-0.628179\pi\)
−0.391890 + 0.920012i \(0.628179\pi\)
\(878\) 0 0
\(879\) −63.4817 −2.14118
\(880\) 0 0
\(881\) 29.0262 0.977918 0.488959 0.872307i \(-0.337377\pi\)
0.488959 + 0.872307i \(0.337377\pi\)
\(882\) 0 0
\(883\) 33.6123 1.13114 0.565572 0.824699i \(-0.308655\pi\)
0.565572 + 0.824699i \(0.308655\pi\)
\(884\) 0 0
\(885\) −35.4288 −1.19093
\(886\) 0 0
\(887\) −49.0131 −1.64570 −0.822849 0.568259i \(-0.807617\pi\)
−0.822849 + 0.568259i \(0.807617\pi\)
\(888\) 0 0
\(889\) 49.6934 1.66666
\(890\) 0 0
\(891\) −48.1924 −1.61451
\(892\) 0 0
\(893\) 3.59038 0.120147
\(894\) 0 0
\(895\) 18.9194 0.632405
\(896\) 0 0
\(897\) 2.38978 0.0797924
\(898\) 0 0
\(899\) −1.42089 −0.0473893
\(900\) 0 0
\(901\) −5.80568 −0.193415
\(902\) 0 0
\(903\) −90.5205 −3.01233
\(904\) 0 0
\(905\) −21.7045 −0.721481
\(906\) 0 0
\(907\) −57.7290 −1.91686 −0.958431 0.285326i \(-0.907898\pi\)
−0.958431 + 0.285326i \(0.907898\pi\)
\(908\) 0 0
\(909\) −89.2787 −2.96119
\(910\) 0 0
\(911\) 17.3471 0.574734 0.287367 0.957821i \(-0.407220\pi\)
0.287367 + 0.957821i \(0.407220\pi\)
\(912\) 0 0
\(913\) 5.71532 0.189149
\(914\) 0 0
\(915\) 13.1591 0.435026
\(916\) 0 0
\(917\) −42.7786 −1.41267
\(918\) 0 0
\(919\) −28.3392 −0.934825 −0.467412 0.884039i \(-0.654814\pi\)
−0.467412 + 0.884039i \(0.654814\pi\)
\(920\) 0 0
\(921\) −6.62373 −0.218259
\(922\) 0 0
\(923\) −10.9539 −0.360553
\(924\) 0 0
\(925\) −11.8194 −0.388619
\(926\) 0 0
\(927\) 114.249 3.75244
\(928\) 0 0
\(929\) −28.1079 −0.922191 −0.461095 0.887351i \(-0.652543\pi\)
−0.461095 + 0.887351i \(0.652543\pi\)
\(930\) 0 0
\(931\) 0.910860 0.0298522
\(932\) 0 0
\(933\) −84.6268 −2.77056
\(934\) 0 0
\(935\) 12.7720 0.417691
\(936\) 0 0
\(937\) −27.6376 −0.902881 −0.451440 0.892301i \(-0.649090\pi\)
−0.451440 + 0.892301i \(0.649090\pi\)
\(938\) 0 0
\(939\) −89.5830 −2.92343
\(940\) 0 0
\(941\) −3.41271 −0.111251 −0.0556256 0.998452i \(-0.517715\pi\)
−0.0556256 + 0.998452i \(0.517715\pi\)
\(942\) 0 0
\(943\) −1.09732 −0.0357338
\(944\) 0 0
\(945\) −54.4568 −1.77148
\(946\) 0 0
\(947\) 5.42927 0.176428 0.0882138 0.996102i \(-0.471884\pi\)
0.0882138 + 0.996102i \(0.471884\pi\)
\(948\) 0 0
\(949\) −8.92394 −0.289683
\(950\) 0 0
\(951\) −40.4309 −1.31106
\(952\) 0 0
\(953\) 33.4941 1.08498 0.542491 0.840062i \(-0.317481\pi\)
0.542491 + 0.840062i \(0.317481\pi\)
\(954\) 0 0
\(955\) −19.7820 −0.640131
\(956\) 0 0
\(957\) 2.40047 0.0775962
\(958\) 0 0
\(959\) −11.7584 −0.379700
\(960\) 0 0
\(961\) −21.0175 −0.677983
\(962\) 0 0
\(963\) 138.153 4.45192
\(964\) 0 0
\(965\) −27.4022 −0.882108
\(966\) 0 0
\(967\) −56.4322 −1.81474 −0.907369 0.420335i \(-0.861912\pi\)
−0.907369 + 0.420335i \(0.861912\pi\)
\(968\) 0 0
\(969\) −19.1751 −0.615992
\(970\) 0 0
\(971\) 34.6433 1.11176 0.555878 0.831264i \(-0.312382\pi\)
0.555878 + 0.831264i \(0.312382\pi\)
\(972\) 0 0
\(973\) −1.44546 −0.0463393
\(974\) 0 0
\(975\) −39.4892 −1.26467
\(976\) 0 0
\(977\) −34.9373 −1.11774 −0.558871 0.829255i \(-0.688765\pi\)
−0.558871 + 0.829255i \(0.688765\pi\)
\(978\) 0 0
\(979\) −13.0185 −0.416074
\(980\) 0 0
\(981\) 104.782 3.34542
\(982\) 0 0
\(983\) 2.37952 0.0758948 0.0379474 0.999280i \(-0.487918\pi\)
0.0379474 + 0.999280i \(0.487918\pi\)
\(984\) 0 0
\(985\) −27.6836 −0.882074
\(986\) 0 0
\(987\) −29.2619 −0.931415
\(988\) 0 0
\(989\) −2.11527 −0.0672615
\(990\) 0 0
\(991\) −5.64207 −0.179226 −0.0896132 0.995977i \(-0.528563\pi\)
−0.0896132 + 0.995977i \(0.528563\pi\)
\(992\) 0 0
\(993\) 41.9954 1.33268
\(994\) 0 0
\(995\) 31.8025 1.00821
\(996\) 0 0
\(997\) −54.2183 −1.71711 −0.858556 0.512720i \(-0.828638\pi\)
−0.858556 + 0.512720i \(0.828638\pi\)
\(998\) 0 0
\(999\) 60.8884 1.92642
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 4028.2.a.f.1.19 19
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4028.2.a.f.1.19 19 1.1 even 1 trivial