Properties

Label 3800.2.d.l.3649.3
Level $3800$
Weight $2$
Character 3800.3649
Analytic conductor $30.343$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3800,2,Mod(3649,3800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3800.3649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.d (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(30.3431527681\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.3356224.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 8x^{4} + 16x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 760)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3649.3
Root \(-2.11491i\) of defining polynomial
Character \(\chi\) \(=\) 3800.3649
Dual form 3800.2.d.l.3649.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.642074i q^{3} -3.58774i q^{7} +2.58774 q^{9} +O(q^{10})\) \(q-0.642074i q^{3} -3.58774i q^{7} +2.58774 q^{9} +1.35793i q^{13} -5.58774i q^{17} -1.00000 q^{19} -2.30359 q^{21} -4.87189i q^{23} -3.58774i q^{27} -9.58774 q^{29} -7.17548 q^{31} +0.945668i q^{37} +0.871889 q^{39} +10.4596 q^{41} -2.71585i q^{43} +5.89134i q^{47} -5.87189 q^{49} -3.58774 q^{51} +9.81756i q^{53} +0.642074i q^{57} -10.1560 q^{59} +3.28415 q^{61} -9.28415i q^{63} -10.3859i q^{67} -3.12811 q^{69} +14.3510 q^{71} -4.15604i q^{73} -1.28415 q^{79} +5.45963 q^{81} -11.1755i q^{83} +6.15604i q^{87} +6.45963 q^{89} +4.87189 q^{91} +4.60719i q^{93} +13.4053i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 8 q^{9} - 6 q^{19} + 6 q^{21} - 34 q^{29} + 4 q^{31} - 22 q^{39} + 12 q^{41} - 8 q^{49} + 2 q^{51} - 30 q^{59} + 16 q^{61} - 46 q^{69} - 8 q^{71} - 4 q^{79} - 18 q^{81} - 12 q^{89} + 2 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) − 0.642074i − 0.370701i −0.982672 0.185351i \(-0.940658\pi\)
0.982672 0.185351i \(-0.0593421\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 3.58774i − 1.35604i −0.735044 0.678019i \(-0.762839\pi\)
0.735044 0.678019i \(-0.237161\pi\)
\(8\) 0 0
\(9\) 2.58774 0.862580
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 1.35793i 0.376621i 0.982110 + 0.188311i \(0.0603011\pi\)
−0.982110 + 0.188311i \(0.939699\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 5.58774i − 1.35523i −0.735419 0.677613i \(-0.763014\pi\)
0.735419 0.677613i \(-0.236986\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −2.30359 −0.502685
\(22\) 0 0
\(23\) − 4.87189i − 1.01586i −0.861399 0.507930i \(-0.830411\pi\)
0.861399 0.507930i \(-0.169589\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 3.58774i − 0.690461i
\(28\) 0 0
\(29\) −9.58774 −1.78040 −0.890199 0.455571i \(-0.849435\pi\)
−0.890199 + 0.455571i \(0.849435\pi\)
\(30\) 0 0
\(31\) −7.17548 −1.28875 −0.644377 0.764708i \(-0.722883\pi\)
−0.644377 + 0.764708i \(0.722883\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.945668i 0.155467i 0.996974 + 0.0777334i \(0.0247683\pi\)
−0.996974 + 0.0777334i \(0.975232\pi\)
\(38\) 0 0
\(39\) 0.871889 0.139614
\(40\) 0 0
\(41\) 10.4596 1.63352 0.816760 0.576978i \(-0.195768\pi\)
0.816760 + 0.576978i \(0.195768\pi\)
\(42\) 0 0
\(43\) − 2.71585i − 0.414164i −0.978324 0.207082i \(-0.933603\pi\)
0.978324 0.207082i \(-0.0663966\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.89134i 0.859340i 0.902986 + 0.429670i \(0.141370\pi\)
−0.902986 + 0.429670i \(0.858630\pi\)
\(48\) 0 0
\(49\) −5.87189 −0.838841
\(50\) 0 0
\(51\) −3.58774 −0.502384
\(52\) 0 0
\(53\) 9.81756i 1.34855i 0.738483 + 0.674273i \(0.235543\pi\)
−0.738483 + 0.674273i \(0.764457\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.642074i 0.0850447i
\(58\) 0 0
\(59\) −10.1560 −1.32220 −0.661102 0.750296i \(-0.729911\pi\)
−0.661102 + 0.750296i \(0.729911\pi\)
\(60\) 0 0
\(61\) 3.28415 0.420492 0.210246 0.977649i \(-0.432574\pi\)
0.210246 + 0.977649i \(0.432574\pi\)
\(62\) 0 0
\(63\) − 9.28415i − 1.16969i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 10.3859i − 1.26883i −0.772991 0.634417i \(-0.781240\pi\)
0.772991 0.634417i \(-0.218760\pi\)
\(68\) 0 0
\(69\) −3.12811 −0.376580
\(70\) 0 0
\(71\) 14.3510 1.70315 0.851573 0.524236i \(-0.175649\pi\)
0.851573 + 0.524236i \(0.175649\pi\)
\(72\) 0 0
\(73\) − 4.15604i − 0.486427i −0.969973 0.243214i \(-0.921798\pi\)
0.969973 0.243214i \(-0.0782016\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.28415 −0.144478 −0.0722389 0.997387i \(-0.523014\pi\)
−0.0722389 + 0.997387i \(0.523014\pi\)
\(80\) 0 0
\(81\) 5.45963 0.606626
\(82\) 0 0
\(83\) − 11.1755i − 1.22667i −0.789823 0.613334i \(-0.789828\pi\)
0.789823 0.613334i \(-0.210172\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.15604i 0.659996i
\(88\) 0 0
\(89\) 6.45963 0.684719 0.342360 0.939569i \(-0.388774\pi\)
0.342360 + 0.939569i \(0.388774\pi\)
\(90\) 0 0
\(91\) 4.87189 0.510713
\(92\) 0 0
\(93\) 4.60719i 0.477743i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.4053i 1.36110i 0.732701 + 0.680551i \(0.238259\pi\)
−0.732701 + 0.680551i \(0.761741\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −19.0668 −1.89722 −0.948610 0.316449i \(-0.897510\pi\)
−0.948610 + 0.316449i \(0.897510\pi\)
\(102\) 0 0
\(103\) − 9.05433i − 0.892150i −0.894996 0.446075i \(-0.852821\pi\)
0.894996 0.446075i \(-0.147179\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.24926i 0.507465i 0.967274 + 0.253733i \(0.0816583\pi\)
−0.967274 + 0.253733i \(0.918342\pi\)
\(108\) 0 0
\(109\) 6.04737 0.579233 0.289617 0.957143i \(-0.406472\pi\)
0.289617 + 0.957143i \(0.406472\pi\)
\(110\) 0 0
\(111\) 0.607188 0.0576318
\(112\) 0 0
\(113\) − 6.22982i − 0.586052i −0.956105 0.293026i \(-0.905338\pi\)
0.956105 0.293026i \(-0.0946622\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 3.51396i 0.324866i
\(118\) 0 0
\(119\) −20.0474 −1.83774
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) − 6.71585i − 0.605548i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 14.1212i 1.25305i 0.779402 + 0.626525i \(0.215523\pi\)
−0.779402 + 0.626525i \(0.784477\pi\)
\(128\) 0 0
\(129\) −1.74378 −0.153531
\(130\) 0 0
\(131\) −18.3510 −1.60333 −0.801666 0.597773i \(-0.796053\pi\)
−0.801666 + 0.597773i \(0.796053\pi\)
\(132\) 0 0
\(133\) 3.58774i 0.311097i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 3.01945i − 0.257969i −0.991647 0.128984i \(-0.958828\pi\)
0.991647 0.128984i \(-0.0411717\pi\)
\(138\) 0 0
\(139\) −11.7827 −0.999393 −0.499697 0.866201i \(-0.666555\pi\)
−0.499697 + 0.866201i \(0.666555\pi\)
\(140\) 0 0
\(141\) 3.78267 0.318558
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 3.77018i 0.310960i
\(148\) 0 0
\(149\) −20.4985 −1.67930 −0.839652 0.543124i \(-0.817241\pi\)
−0.839652 + 0.543124i \(0.817241\pi\)
\(150\) 0 0
\(151\) −3.85244 −0.313507 −0.156754 0.987638i \(-0.550103\pi\)
−0.156754 + 0.987638i \(0.550103\pi\)
\(152\) 0 0
\(153\) − 14.4596i − 1.16899i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 3.89134i − 0.310562i −0.987870 0.155281i \(-0.950372\pi\)
0.987870 0.155281i \(-0.0496284\pi\)
\(158\) 0 0
\(159\) 6.30359 0.499908
\(160\) 0 0
\(161\) −17.4791 −1.37754
\(162\) 0 0
\(163\) − 18.7159i − 1.46594i −0.680262 0.732969i \(-0.738134\pi\)
0.680262 0.732969i \(-0.261866\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.09323i 0.239361i 0.992812 + 0.119681i \(0.0381870\pi\)
−0.992812 + 0.119681i \(0.961813\pi\)
\(168\) 0 0
\(169\) 11.1560 0.858157
\(170\) 0 0
\(171\) −2.58774 −0.197890
\(172\) 0 0
\(173\) 7.51396i 0.571276i 0.958338 + 0.285638i \(0.0922055\pi\)
−0.958338 + 0.285638i \(0.907795\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 6.52092i 0.490143i
\(178\) 0 0
\(179\) 4.45963 0.333328 0.166664 0.986014i \(-0.446700\pi\)
0.166664 + 0.986014i \(0.446700\pi\)
\(180\) 0 0
\(181\) 8.56829 0.636876 0.318438 0.947944i \(-0.396842\pi\)
0.318438 + 0.947944i \(0.396842\pi\)
\(182\) 0 0
\(183\) − 2.10866i − 0.155877i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −12.8719 −0.936292
\(190\) 0 0
\(191\) 7.73530 0.559707 0.279853 0.960043i \(-0.409714\pi\)
0.279853 + 0.960043i \(0.409714\pi\)
\(192\) 0 0
\(193\) 6.08226i 0.437810i 0.975746 + 0.218905i \(0.0702486\pi\)
−0.975746 + 0.218905i \(0.929751\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.00000i 0.142494i 0.997459 + 0.0712470i \(0.0226979\pi\)
−0.997459 + 0.0712470i \(0.977302\pi\)
\(198\) 0 0
\(199\) 6.91078 0.489892 0.244946 0.969537i \(-0.421230\pi\)
0.244946 + 0.969537i \(0.421230\pi\)
\(200\) 0 0
\(201\) −6.66848 −0.470358
\(202\) 0 0
\(203\) 34.3983i 2.41429i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 12.6072i − 0.876260i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −12.1949 −0.839534 −0.419767 0.907632i \(-0.637888\pi\)
−0.419767 + 0.907632i \(0.637888\pi\)
\(212\) 0 0
\(213\) − 9.21438i − 0.631359i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 25.7438i 1.74760i
\(218\) 0 0
\(219\) −2.66848 −0.180319
\(220\) 0 0
\(221\) 7.58774 0.510407
\(222\) 0 0
\(223\) − 1.87885i − 0.125817i −0.998019 0.0629085i \(-0.979962\pi\)
0.998019 0.0629085i \(-0.0200376\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 7.35793i − 0.488363i −0.969730 0.244181i \(-0.921481\pi\)
0.969730 0.244181i \(-0.0785192\pi\)
\(228\) 0 0
\(229\) −13.0279 −0.860909 −0.430455 0.902612i \(-0.641647\pi\)
−0.430455 + 0.902612i \(0.641647\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 26.7019i − 1.74930i −0.484753 0.874651i \(-0.661091\pi\)
0.484753 0.874651i \(-0.338909\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0.824517i 0.0535581i
\(238\) 0 0
\(239\) 9.47908 0.613151 0.306575 0.951846i \(-0.400817\pi\)
0.306575 + 0.951846i \(0.400817\pi\)
\(240\) 0 0
\(241\) −24.2034 −1.55908 −0.779539 0.626353i \(-0.784547\pi\)
−0.779539 + 0.626353i \(0.784547\pi\)
\(242\) 0 0
\(243\) − 14.2687i − 0.915338i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 1.35793i − 0.0864028i
\(248\) 0 0
\(249\) −7.17548 −0.454728
\(250\) 0 0
\(251\) 9.43171 0.595324 0.297662 0.954671i \(-0.403793\pi\)
0.297662 + 0.954671i \(0.403793\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 25.1102i − 1.56633i −0.621814 0.783165i \(-0.713604\pi\)
0.621814 0.783165i \(-0.286396\pi\)
\(258\) 0 0
\(259\) 3.39281 0.210819
\(260\) 0 0
\(261\) −24.8106 −1.53574
\(262\) 0 0
\(263\) − 28.7019i − 1.76984i −0.465746 0.884918i \(-0.654214\pi\)
0.465746 0.884918i \(-0.345786\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 4.14756i − 0.253826i
\(268\) 0 0
\(269\) 0.350966 0.0213988 0.0106994 0.999943i \(-0.496594\pi\)
0.0106994 + 0.999943i \(0.496594\pi\)
\(270\) 0 0
\(271\) 16.8719 1.02489 0.512447 0.858719i \(-0.328739\pi\)
0.512447 + 0.858719i \(0.328739\pi\)
\(272\) 0 0
\(273\) − 3.12811i − 0.189322i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 6.14756i 0.369371i 0.982798 + 0.184685i \(0.0591266\pi\)
−0.982798 + 0.184685i \(0.940873\pi\)
\(278\) 0 0
\(279\) −18.5683 −1.11165
\(280\) 0 0
\(281\) −17.0279 −1.01580 −0.507900 0.861416i \(-0.669578\pi\)
−0.507900 + 0.861416i \(0.669578\pi\)
\(282\) 0 0
\(283\) 5.74378i 0.341432i 0.985320 + 0.170716i \(0.0546081\pi\)
−0.985320 + 0.170716i \(0.945392\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 37.5264i − 2.21512i
\(288\) 0 0
\(289\) −14.2229 −0.836639
\(290\) 0 0
\(291\) 8.60719 0.504562
\(292\) 0 0
\(293\) − 14.2772i − 0.834082i −0.908888 0.417041i \(-0.863067\pi\)
0.908888 0.417041i \(-0.136933\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.61567 0.382594
\(300\) 0 0
\(301\) −9.74378 −0.561622
\(302\) 0 0
\(303\) 12.2423i 0.703302i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 8.83700i − 0.504354i −0.967681 0.252177i \(-0.918853\pi\)
0.967681 0.252177i \(-0.0811466\pi\)
\(308\) 0 0
\(309\) −5.81355 −0.330721
\(310\) 0 0
\(311\) 23.2229 1.31685 0.658424 0.752648i \(-0.271224\pi\)
0.658424 + 0.752648i \(0.271224\pi\)
\(312\) 0 0
\(313\) 30.1421i 1.70373i 0.523759 + 0.851867i \(0.324529\pi\)
−0.523759 + 0.851867i \(0.675471\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 5.96511i − 0.335034i −0.985869 0.167517i \(-0.946425\pi\)
0.985869 0.167517i \(-0.0535749\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 3.37041 0.188118
\(322\) 0 0
\(323\) 5.58774i 0.310910i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 3.88286i − 0.214723i
\(328\) 0 0
\(329\) 21.1366 1.16530
\(330\) 0 0
\(331\) −2.15604 −0.118506 −0.0592532 0.998243i \(-0.518872\pi\)
−0.0592532 + 0.998243i \(0.518872\pi\)
\(332\) 0 0
\(333\) 2.44714i 0.134103i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 0.945668i − 0.0515138i −0.999668 0.0257569i \(-0.991800\pi\)
0.999668 0.0257569i \(-0.00819958\pi\)
\(338\) 0 0
\(339\) −4.00000 −0.217250
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 4.04737i − 0.218538i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 10.2562i − 0.550583i −0.961361 0.275291i \(-0.911226\pi\)
0.961361 0.275291i \(-0.0887743\pi\)
\(348\) 0 0
\(349\) 24.0558 1.28768 0.643840 0.765160i \(-0.277340\pi\)
0.643840 + 0.765160i \(0.277340\pi\)
\(350\) 0 0
\(351\) 4.87189 0.260042
\(352\) 0 0
\(353\) 15.3315i 0.816014i 0.912979 + 0.408007i \(0.133776\pi\)
−0.912979 + 0.408007i \(0.866224\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 12.8719i 0.681253i
\(358\) 0 0
\(359\) −15.1281 −0.798431 −0.399216 0.916857i \(-0.630718\pi\)
−0.399216 + 0.916857i \(0.630718\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 7.06281i 0.370701i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 33.6740i 1.75777i 0.477035 + 0.878884i \(0.341712\pi\)
−0.477035 + 0.878884i \(0.658288\pi\)
\(368\) 0 0
\(369\) 27.0668 1.40904
\(370\) 0 0
\(371\) 35.2229 1.82868
\(372\) 0 0
\(373\) 16.0738i 0.832269i 0.909303 + 0.416134i \(0.136615\pi\)
−0.909303 + 0.416134i \(0.863385\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 13.0194i − 0.670536i
\(378\) 0 0
\(379\) 6.00848 0.308635 0.154317 0.988021i \(-0.450682\pi\)
0.154317 + 0.988021i \(0.450682\pi\)
\(380\) 0 0
\(381\) 9.06682 0.464507
\(382\) 0 0
\(383\) − 15.8649i − 0.810660i −0.914170 0.405330i \(-0.867157\pi\)
0.914170 0.405330i \(-0.132843\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 7.02792i − 0.357249i
\(388\) 0 0
\(389\) −15.7438 −0.798241 −0.399121 0.916898i \(-0.630685\pi\)
−0.399121 + 0.916898i \(0.630685\pi\)
\(390\) 0 0
\(391\) −27.2229 −1.37672
\(392\) 0 0
\(393\) 11.7827i 0.594357i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 23.9861i 1.20383i 0.798561 + 0.601913i \(0.205595\pi\)
−0.798561 + 0.601913i \(0.794405\pi\)
\(398\) 0 0
\(399\) 2.30359 0.115324
\(400\) 0 0
\(401\) −24.5683 −1.22688 −0.613441 0.789741i \(-0.710215\pi\)
−0.613441 + 0.789741i \(0.710215\pi\)
\(402\) 0 0
\(403\) − 9.74378i − 0.485372i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 14.9193 0.737710 0.368855 0.929487i \(-0.379750\pi\)
0.368855 + 0.929487i \(0.379750\pi\)
\(410\) 0 0
\(411\) −1.93871 −0.0956294
\(412\) 0 0
\(413\) 36.4372i 1.79296i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 7.56534i 0.370476i
\(418\) 0 0
\(419\) −14.0389 −0.685845 −0.342922 0.939364i \(-0.611417\pi\)
−0.342922 + 0.939364i \(0.611417\pi\)
\(420\) 0 0
\(421\) 39.9387 1.94649 0.973247 0.229762i \(-0.0737949\pi\)
0.973247 + 0.229762i \(0.0737949\pi\)
\(422\) 0 0
\(423\) 15.2453i 0.741250i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 11.7827i − 0.570203i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 27.0279 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(432\) 0 0
\(433\) − 40.0683i − 1.92556i −0.270284 0.962781i \(-0.587118\pi\)
0.270284 0.962781i \(-0.412882\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.87189i 0.233054i
\(438\) 0 0
\(439\) −6.42074 −0.306445 −0.153223 0.988192i \(-0.548965\pi\)
−0.153223 + 0.988192i \(0.548965\pi\)
\(440\) 0 0
\(441\) −15.1949 −0.723568
\(442\) 0 0
\(443\) − 35.2702i − 1.67574i −0.545871 0.837870i \(-0.683801\pi\)
0.545871 0.837870i \(-0.316199\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 13.1616i 0.622521i
\(448\) 0 0
\(449\) −5.78267 −0.272901 −0.136451 0.990647i \(-0.543569\pi\)
−0.136451 + 0.990647i \(0.543569\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 2.47355i 0.116218i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 9.12811i 0.426995i 0.976944 + 0.213498i \(0.0684855\pi\)
−0.976944 + 0.213498i \(0.931514\pi\)
\(458\) 0 0
\(459\) −20.0474 −0.935731
\(460\) 0 0
\(461\) 8.86341 0.412810 0.206405 0.978467i \(-0.433824\pi\)
0.206405 + 0.978467i \(0.433824\pi\)
\(462\) 0 0
\(463\) − 21.1366i − 0.982301i −0.871075 0.491150i \(-0.836577\pi\)
0.871075 0.491150i \(-0.163423\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 7.78267i − 0.360139i −0.983654 0.180070i \(-0.942368\pi\)
0.983654 0.180070i \(-0.0576323\pi\)
\(468\) 0 0
\(469\) −37.2617 −1.72059
\(470\) 0 0
\(471\) −2.49852 −0.115126
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 25.4053i 1.16323i
\(478\) 0 0
\(479\) −40.4068 −1.84623 −0.923117 0.384519i \(-0.874367\pi\)
−0.923117 + 0.384519i \(0.874367\pi\)
\(480\) 0 0
\(481\) −1.28415 −0.0585521
\(482\) 0 0
\(483\) 11.2229i 0.510658i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 5.82603i − 0.264003i −0.991250 0.132001i \(-0.957860\pi\)
0.991250 0.132001i \(-0.0421403\pi\)
\(488\) 0 0
\(489\) −12.0170 −0.543426
\(490\) 0 0
\(491\) 22.6630 1.02277 0.511384 0.859352i \(-0.329133\pi\)
0.511384 + 0.859352i \(0.329133\pi\)
\(492\) 0 0
\(493\) 53.5738i 2.41284i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 51.4876i − 2.30953i
\(498\) 0 0
\(499\) 25.7438 1.15245 0.576225 0.817291i \(-0.304525\pi\)
0.576225 + 0.817291i \(0.304525\pi\)
\(500\) 0 0
\(501\) 1.98608 0.0887315
\(502\) 0 0
\(503\) 25.4791i 1.13606i 0.823009 + 0.568028i \(0.192293\pi\)
−0.823009 + 0.568028i \(0.807707\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 7.16300i − 0.318120i
\(508\) 0 0
\(509\) 9.54037 0.422869 0.211435 0.977392i \(-0.432186\pi\)
0.211435 + 0.977392i \(0.432186\pi\)
\(510\) 0 0
\(511\) −14.9108 −0.659614
\(512\) 0 0
\(513\) 3.58774i 0.158403i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 4.82452 0.211773
\(520\) 0 0
\(521\) 6.31207 0.276537 0.138268 0.990395i \(-0.455846\pi\)
0.138268 + 0.990395i \(0.455846\pi\)
\(522\) 0 0
\(523\) 8.93719i 0.390796i 0.980724 + 0.195398i \(0.0625999\pi\)
−0.980724 + 0.195398i \(0.937400\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 40.0947i 1.74655i
\(528\) 0 0
\(529\) −0.735300 −0.0319696
\(530\) 0 0
\(531\) −26.2812 −1.14051
\(532\) 0 0
\(533\) 14.2034i 0.615218i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 2.86341i − 0.123565i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 8.71585 0.374724 0.187362 0.982291i \(-0.440006\pi\)
0.187362 + 0.982291i \(0.440006\pi\)
\(542\) 0 0
\(543\) − 5.50148i − 0.236091i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 26.9457i 1.15211i 0.817410 + 0.576057i \(0.195409\pi\)
−0.817410 + 0.576057i \(0.804591\pi\)
\(548\) 0 0
\(549\) 8.49852 0.362708
\(550\) 0 0
\(551\) 9.58774 0.408452
\(552\) 0 0
\(553\) 4.60719i 0.195918i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 23.3619i 0.989877i 0.868928 + 0.494938i \(0.164809\pi\)
−0.868928 + 0.494938i \(0.835191\pi\)
\(558\) 0 0
\(559\) 3.68793 0.155983
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 31.6476i 1.33379i 0.745153 + 0.666894i \(0.232376\pi\)
−0.745153 + 0.666894i \(0.767624\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 19.5877i − 0.822608i
\(568\) 0 0
\(569\) 20.8804 0.875350 0.437675 0.899133i \(-0.355802\pi\)
0.437675 + 0.899133i \(0.355802\pi\)
\(570\) 0 0
\(571\) 42.6630 1.78539 0.892696 0.450659i \(-0.148811\pi\)
0.892696 + 0.450659i \(0.148811\pi\)
\(572\) 0 0
\(573\) − 4.96663i − 0.207484i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 8.59871i 0.357969i 0.983852 + 0.178985i \(0.0572812\pi\)
−0.983852 + 0.178985i \(0.942719\pi\)
\(578\) 0 0
\(579\) 3.90526 0.162297
\(580\) 0 0
\(581\) −40.0947 −1.66341
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 29.7438i − 1.22766i −0.789439 0.613829i \(-0.789629\pi\)
0.789439 0.613829i \(-0.210371\pi\)
\(588\) 0 0
\(589\) 7.17548 0.295661
\(590\) 0 0
\(591\) 1.28415 0.0528228
\(592\) 0 0
\(593\) 4.56829i 0.187597i 0.995591 + 0.0937987i \(0.0299010\pi\)
−0.995591 + 0.0937987i \(0.970099\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 4.43723i − 0.181604i
\(598\) 0 0
\(599\) −14.7159 −0.601273 −0.300637 0.953739i \(-0.597199\pi\)
−0.300637 + 0.953739i \(0.597199\pi\)
\(600\) 0 0
\(601\) 37.3230 1.52244 0.761219 0.648495i \(-0.224601\pi\)
0.761219 + 0.648495i \(0.224601\pi\)
\(602\) 0 0
\(603\) − 26.8759i − 1.09447i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 23.0404i − 0.935181i −0.883945 0.467591i \(-0.845122\pi\)
0.883945 0.467591i \(-0.154878\pi\)
\(608\) 0 0
\(609\) 22.0863 0.894981
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) 32.4985i 1.31260i 0.754499 + 0.656302i \(0.227880\pi\)
−0.754499 + 0.656302i \(0.772120\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 32.2982i 1.30027i 0.759817 + 0.650137i \(0.225289\pi\)
−0.759817 + 0.650137i \(0.774711\pi\)
\(618\) 0 0
\(619\) 34.5683 1.38942 0.694709 0.719291i \(-0.255533\pi\)
0.694709 + 0.719291i \(0.255533\pi\)
\(620\) 0 0
\(621\) −17.4791 −0.701411
\(622\) 0 0
\(623\) − 23.1755i − 0.928506i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 5.28415 0.210693
\(630\) 0 0
\(631\) −19.2702 −0.767136 −0.383568 0.923513i \(-0.625305\pi\)
−0.383568 + 0.923513i \(0.625305\pi\)
\(632\) 0 0
\(633\) 7.83004i 0.311216i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 7.97359i − 0.315925i
\(638\) 0 0
\(639\) 37.1366 1.46910
\(640\) 0 0
\(641\) −3.06682 −0.121132 −0.0605660 0.998164i \(-0.519291\pi\)
−0.0605660 + 0.998164i \(0.519291\pi\)
\(642\) 0 0
\(643\) − 25.5264i − 1.00666i −0.864093 0.503332i \(-0.832107\pi\)
0.864093 0.503332i \(-0.167893\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 25.4791i − 1.00169i −0.865538 0.500843i \(-0.833023\pi\)
0.865538 0.500843i \(-0.166977\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 16.5294 0.647838
\(652\) 0 0
\(653\) − 36.2034i − 1.41675i −0.705837 0.708374i \(-0.749429\pi\)
0.705837 0.708374i \(-0.250571\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 10.7547i − 0.419583i
\(658\) 0 0
\(659\) 42.6406 1.66104 0.830522 0.556986i \(-0.188042\pi\)
0.830522 + 0.556986i \(0.188042\pi\)
\(660\) 0 0
\(661\) 24.4681 0.951699 0.475850 0.879527i \(-0.342141\pi\)
0.475850 + 0.879527i \(0.342141\pi\)
\(662\) 0 0
\(663\) − 4.87189i − 0.189208i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 46.7104i 1.80863i
\(668\) 0 0
\(669\) −1.20636 −0.0466406
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) − 26.7672i − 1.03180i −0.856649 0.515901i \(-0.827457\pi\)
0.856649 0.515901i \(-0.172543\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 5.65304i 0.217264i 0.994082 + 0.108632i \(0.0346470\pi\)
−0.994082 + 0.108632i \(0.965353\pi\)
\(678\) 0 0
\(679\) 48.0947 1.84571
\(680\) 0 0
\(681\) −4.72433 −0.181037
\(682\) 0 0
\(683\) − 11.1102i − 0.425119i −0.977148 0.212560i \(-0.931820\pi\)
0.977148 0.212560i \(-0.0681800\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 8.36489i 0.319140i
\(688\) 0 0
\(689\) −13.3315 −0.507890
\(690\) 0 0
\(691\) −5.96111 −0.226771 −0.113386 0.993551i \(-0.536170\pi\)
−0.113386 + 0.993551i \(0.536170\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 58.4457i − 2.21379i
\(698\) 0 0
\(699\) −17.1446 −0.648469
\(700\) 0 0
\(701\) −25.8524 −0.976433 −0.488217 0.872722i \(-0.662352\pi\)
−0.488217 + 0.872722i \(0.662352\pi\)
\(702\) 0 0
\(703\) − 0.945668i − 0.0356665i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 68.4068i 2.57270i
\(708\) 0 0
\(709\) 23.4317 0.879996 0.439998 0.897999i \(-0.354979\pi\)
0.439998 + 0.897999i \(0.354979\pi\)
\(710\) 0 0
\(711\) −3.32304 −0.124624
\(712\) 0 0
\(713\) 34.9582i 1.30919i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 6.08627i − 0.227296i
\(718\) 0 0
\(719\) 6.20885 0.231551 0.115776 0.993275i \(-0.463065\pi\)
0.115776 + 0.993275i \(0.463065\pi\)
\(720\) 0 0
\(721\) −32.4846 −1.20979
\(722\) 0 0
\(723\) 15.5404i 0.577953i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 29.7214i − 1.10230i −0.834405 0.551152i \(-0.814188\pi\)
0.834405 0.551152i \(-0.185812\pi\)
\(728\) 0 0
\(729\) 7.21733 0.267308
\(730\) 0 0
\(731\) −15.1755 −0.561286
\(732\) 0 0
\(733\) − 7.52645i − 0.277996i −0.990293 0.138998i \(-0.955612\pi\)
0.990293 0.138998i \(-0.0443881\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −22.9582 −0.844529 −0.422265 0.906473i \(-0.638765\pi\)
−0.422265 + 0.906473i \(0.638765\pi\)
\(740\) 0 0
\(741\) −0.871889 −0.0320296
\(742\) 0 0
\(743\) 0.524931i 0.0192579i 0.999954 + 0.00962893i \(0.00306503\pi\)
−0.999954 + 0.00962893i \(0.996935\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 28.9193i − 1.05810i
\(748\) 0 0
\(749\) 18.8330 0.688143
\(750\) 0 0
\(751\) 21.3619 0.779508 0.389754 0.920919i \(-0.372560\pi\)
0.389754 + 0.920919i \(0.372560\pi\)
\(752\) 0 0
\(753\) − 6.05585i − 0.220687i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 50.4068i − 1.83207i −0.401102 0.916033i \(-0.631373\pi\)
0.401102 0.916033i \(-0.368627\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −20.2338 −0.733476 −0.366738 0.930324i \(-0.619525\pi\)
−0.366738 + 0.930324i \(0.619525\pi\)
\(762\) 0 0
\(763\) − 21.6964i − 0.785463i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 13.7911i − 0.497970i
\(768\) 0 0
\(769\) 27.1142 0.977763 0.488881 0.872350i \(-0.337405\pi\)
0.488881 + 0.872350i \(0.337405\pi\)
\(770\) 0 0
\(771\) −16.1226 −0.580641
\(772\) 0 0
\(773\) − 12.1685i − 0.437671i −0.975762 0.218836i \(-0.929774\pi\)
0.975762 0.218836i \(-0.0702259\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 2.17843i − 0.0781509i
\(778\) 0 0
\(779\) −10.4596 −0.374755
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 34.3983i 1.22930i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 36.1296i 1.28788i 0.765075 + 0.643941i \(0.222702\pi\)
−0.765075 + 0.643941i \(0.777298\pi\)
\(788\) 0 0
\(789\) −18.4288 −0.656081
\(790\) 0 0
\(791\) −22.3510 −0.794709
\(792\) 0 0
\(793\) 4.45963i 0.158366i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 42.4417i − 1.50336i −0.659527 0.751681i \(-0.729243\pi\)
0.659527 0.751681i \(-0.270757\pi\)
\(798\) 0 0
\(799\) 32.9193 1.16460
\(800\) 0 0
\(801\) 16.7159 0.590626
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 0.225346i − 0.00793255i
\(808\) 0 0
\(809\) 37.3006 1.31142 0.655710 0.755013i \(-0.272369\pi\)
0.655710 + 0.755013i \(0.272369\pi\)
\(810\) 0 0
\(811\) 27.4402 0.963555 0.481778 0.876294i \(-0.339991\pi\)
0.481778 + 0.876294i \(0.339991\pi\)
\(812\) 0 0
\(813\) − 10.8330i − 0.379930i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 2.71585i 0.0950157i
\(818\) 0 0
\(819\) 12.6072 0.440531
\(820\) 0 0
\(821\) 47.0140 1.64080 0.820400 0.571790i \(-0.193751\pi\)
0.820400 + 0.571790i \(0.193751\pi\)
\(822\) 0 0
\(823\) 35.9776i 1.25410i 0.778979 + 0.627050i \(0.215738\pi\)
−0.778979 + 0.627050i \(0.784262\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 14.6032i − 0.507802i −0.967230 0.253901i \(-0.918286\pi\)
0.967230 0.253901i \(-0.0817138\pi\)
\(828\) 0 0
\(829\) 6.96663 0.241961 0.120981 0.992655i \(-0.461396\pi\)
0.120981 + 0.992655i \(0.461396\pi\)
\(830\) 0 0
\(831\) 3.94719 0.136926
\(832\) 0 0
\(833\) 32.8106i 1.13682i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 25.7438i 0.889835i
\(838\) 0 0
\(839\) 0.459630 0.0158682 0.00793410 0.999969i \(-0.497474\pi\)
0.00793410 + 0.999969i \(0.497474\pi\)
\(840\) 0 0
\(841\) 62.9248 2.16982
\(842\) 0 0
\(843\) 10.9332i 0.376559i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 39.4652i 1.35604i
\(848\) 0 0
\(849\) 3.68793 0.126569
\(850\) 0 0
\(851\) 4.60719 0.157932
\(852\) 0 0
\(853\) − 5.48755i − 0.187890i −0.995577 0.0939451i \(-0.970052\pi\)
0.995577 0.0939451i \(-0.0299478\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 52.6586i − 1.79878i −0.437145 0.899391i \(-0.644010\pi\)
0.437145 0.899391i \(-0.355990\pi\)
\(858\) 0 0
\(859\) 4.29512 0.146547 0.0732737 0.997312i \(-0.476655\pi\)
0.0732737 + 0.997312i \(0.476655\pi\)
\(860\) 0 0
\(861\) −24.0947 −0.821147
\(862\) 0 0
\(863\) − 14.6506i − 0.498711i −0.968412 0.249355i \(-0.919781\pi\)
0.968412 0.249355i \(-0.0802187\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 9.13212i 0.310143i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 14.1032 0.477869
\(872\) 0 0
\(873\) 34.6894i 1.17406i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 19.7089i 0.665522i 0.943011 + 0.332761i \(0.107980\pi\)
−0.943011 + 0.332761i \(0.892020\pi\)
\(878\) 0 0
\(879\) −9.16701 −0.309195
\(880\) 0 0
\(881\) 1.32304 0.0445744 0.0222872 0.999752i \(-0.492905\pi\)
0.0222872 + 0.999752i \(0.492905\pi\)
\(882\) 0 0
\(883\) − 25.0668i − 0.843566i −0.906697 0.421783i \(-0.861404\pi\)
0.906697 0.421783i \(-0.138596\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 36.5669i 1.22779i 0.789386 + 0.613897i \(0.210399\pi\)
−0.789386 + 0.613897i \(0.789601\pi\)
\(888\) 0 0
\(889\) 50.6630 1.69918
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 5.89134i − 0.197146i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 4.24774i − 0.141828i
\(898\) 0 0
\(899\) 68.7967 2.29450
\(900\) 0 0
\(901\) 54.8580 1.82758
\(902\) 0 0
\(903\) 6.25622i 0.208194i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 40.7368i 1.35264i 0.736606 + 0.676322i \(0.236427\pi\)
−0.736606 + 0.676322i \(0.763573\pi\)
\(908\) 0 0
\(909\) −49.3400 −1.63650
\(910\) 0 0
\(911\) −58.2982 −1.93150 −0.965752 0.259467i \(-0.916453\pi\)
−0.965752 + 0.259467i \(0.916453\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 65.8385i 2.17418i
\(918\) 0 0
\(919\) 14.5209 0.479001 0.239501 0.970896i \(-0.423016\pi\)
0.239501 + 0.970896i \(0.423016\pi\)
\(920\) 0 0
\(921\) −5.67401 −0.186965
\(922\) 0 0
\(923\) 19.4876i 0.641441i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 23.4303i − 0.769551i
\(928\) 0 0
\(929\) −13.7523 −0.451197 −0.225598 0.974220i \(-0.572434\pi\)
−0.225598 + 0.974220i \(0.572434\pi\)
\(930\) 0 0
\(931\) 5.87189 0.192443
\(932\) 0 0
\(933\) − 14.9108i − 0.488157i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 12.4512i 0.406761i 0.979100 + 0.203381i \(0.0651929\pi\)
−0.979100 + 0.203381i \(0.934807\pi\)
\(938\) 0 0
\(939\) 19.3535 0.631576
\(940\) 0 0
\(941\) −34.3594 −1.12009 −0.560043 0.828464i \(-0.689215\pi\)
−0.560043 + 0.828464i \(0.689215\pi\)
\(942\) 0 0
\(943\) − 50.9582i − 1.65943i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 23.4876i − 0.763243i −0.924319 0.381621i \(-0.875366\pi\)
0.924319 0.381621i \(-0.124634\pi\)
\(948\) 0 0
\(949\) 5.64359 0.183199
\(950\) 0 0
\(951\) −3.83004 −0.124198
\(952\) 0 0
\(953\) 48.5808i 1.57369i 0.617153 + 0.786843i \(0.288286\pi\)
−0.617153 + 0.786843i \(0.711714\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −10.8330 −0.349816
\(960\) 0 0
\(961\) 20.4876 0.660889
\(962\) 0 0
\(963\) 13.5837i 0.437730i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 53.3261i − 1.71485i −0.514608 0.857425i \(-0.672063\pi\)
0.514608 0.857425i \(-0.327937\pi\)
\(968\) 0 0
\(969\) 3.58774 0.115255
\(970\) 0 0
\(971\) −33.1366 −1.06340 −0.531702 0.846932i \(-0.678447\pi\)
−0.531702 + 0.846932i \(0.678447\pi\)
\(972\) 0 0
\(973\) 42.2732i 1.35522i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 5.62263i − 0.179884i −0.995947 0.0899419i \(-0.971332\pi\)
0.995947 0.0899419i \(-0.0286681\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 15.6490 0.499635
\(982\) 0 0
\(983\) 21.3664i 0.681482i 0.940157 + 0.340741i \(0.110678\pi\)
−0.940157 + 0.340741i \(0.889322\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 13.5712i − 0.431978i
\(988\) 0 0
\(989\) −13.2313 −0.420732
\(990\) 0 0
\(991\) 41.1446 1.30700 0.653501 0.756926i \(-0.273300\pi\)
0.653501 + 0.756926i \(0.273300\pi\)
\(992\) 0 0
\(993\) 1.38433i 0.0439305i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 48.5155i 1.53650i 0.640150 + 0.768250i \(0.278872\pi\)
−0.640150 + 0.768250i \(0.721128\pi\)
\(998\) 0 0
\(999\) 3.39281 0.107344
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 3800.2.d.l.3649.3 6
5.2 odd 4 760.2.a.j.1.2 3
5.3 odd 4 3800.2.a.x.1.2 3
5.4 even 2 inner 3800.2.d.l.3649.4 6
15.2 even 4 6840.2.a.bg.1.3 3
20.3 even 4 7600.2.a.bq.1.2 3
20.7 even 4 1520.2.a.s.1.2 3
40.27 even 4 6080.2.a.bq.1.2 3
40.37 odd 4 6080.2.a.bv.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.j.1.2 3 5.2 odd 4
1520.2.a.s.1.2 3 20.7 even 4
3800.2.a.x.1.2 3 5.3 odd 4
3800.2.d.l.3649.3 6 1.1 even 1 trivial
3800.2.d.l.3649.4 6 5.4 even 2 inner
6080.2.a.bq.1.2 3 40.27 even 4
6080.2.a.bv.1.2 3 40.37 odd 4
6840.2.a.bg.1.3 3 15.2 even 4
7600.2.a.bq.1.2 3 20.3 even 4