Properties

Label 2052.3.be.a.125.8
Level $2052$
Weight $3$
Character 2052.125
Analytic conductor $55.913$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2052,3,Mod(125,2052)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2052, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2052.125");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2052 = 2^{2} \cdot 3^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2052.be (of order \(6\), degree \(2\), 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: \(55.9129502467\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 684)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 125.8
Character \(\chi\) \(=\) 2052.125
Dual form 2052.3.be.a.197.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-5.40523 + 3.12071i) q^{5} +(4.14502 + 7.17938i) q^{7} +O(q^{10})\) \(q+(-5.40523 + 3.12071i) q^{5} +(4.14502 + 7.17938i) q^{7} +(-16.6134 + 9.59174i) q^{11} +5.01910 q^{13} +(25.7470 + 14.8650i) q^{17} +(17.0105 + 8.46421i) q^{19} +10.5502i q^{23} +(6.97770 - 12.0857i) q^{25} +(25.7875 + 14.8884i) q^{29} +(-2.96277 + 5.13166i) q^{31} +(-44.8096 - 25.8708i) q^{35} +52.1430 q^{37} +(40.6831 - 23.4884i) q^{41} -60.4290 q^{43} +(25.9940 + 15.0076i) q^{47} +(-9.86235 + 17.0821i) q^{49} +(-42.0135 + 24.2565i) q^{53} +(59.8661 - 103.691i) q^{55} +(-83.0645 + 47.9573i) q^{59} +(-46.3314 + 80.2483i) q^{61} +(-27.1294 + 15.6632i) q^{65} +24.0870 q^{67} +(1.03023 + 0.594802i) q^{71} +(-20.6395 + 35.7487i) q^{73} +(-137.725 - 79.5158i) q^{77} -12.1846 q^{79} +(124.538 - 71.9023i) q^{83} -185.558 q^{85} +(-31.0467 + 17.9248i) q^{89} +(20.8043 + 36.0340i) q^{91} +(-118.360 + 7.33387i) q^{95} +86.6880 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 80 q + q^{7} - 18 q^{11} + 10 q^{13} - 9 q^{17} + 20 q^{19} + 200 q^{25} + 27 q^{29} - 8 q^{31} + 22 q^{37} + 54 q^{41} + 88 q^{43} - 198 q^{47} - 267 q^{49} - 36 q^{53} - 171 q^{59} + 7 q^{61} + 144 q^{65} + 154 q^{67} - 135 q^{71} + 43 q^{73} - 216 q^{77} + 34 q^{79} + 171 q^{83} + 216 q^{89} + 122 q^{91} + 216 q^{95} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2052\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1027\) \(1217\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\)

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 0
\(4\) 0 0
\(5\) −5.40523 + 3.12071i −1.08105 + 0.624143i −0.931179 0.364563i \(-0.881218\pi\)
−0.149868 + 0.988706i \(0.547885\pi\)
\(6\) 0 0
\(7\) 4.14502 + 7.17938i 0.592145 + 1.02563i 0.993943 + 0.109897i \(0.0350522\pi\)
−0.401798 + 0.915729i \(0.631614\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −16.6134 + 9.59174i −1.51031 + 0.871976i −0.510379 + 0.859950i \(0.670495\pi\)
−0.999928 + 0.0120264i \(0.996172\pi\)
\(12\) 0 0
\(13\) 5.01910 0.386085 0.193042 0.981190i \(-0.438165\pi\)
0.193042 + 0.981190i \(0.438165\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 25.7470 + 14.8650i 1.51453 + 0.874414i 0.999855 + 0.0170290i \(0.00542075\pi\)
0.514675 + 0.857385i \(0.327913\pi\)
\(18\) 0 0
\(19\) 17.0105 + 8.46421i 0.895290 + 0.445485i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 10.5502i 0.458703i 0.973344 + 0.229351i \(0.0736605\pi\)
−0.973344 + 0.229351i \(0.926339\pi\)
\(24\) 0 0
\(25\) 6.97770 12.0857i 0.279108 0.483429i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 25.7875 + 14.8884i 0.889225 + 0.513394i 0.873689 0.486485i \(-0.161721\pi\)
0.0155362 + 0.999879i \(0.495054\pi\)
\(30\) 0 0
\(31\) −2.96277 + 5.13166i −0.0955732 + 0.165538i −0.909848 0.414942i \(-0.863802\pi\)
0.814275 + 0.580480i \(0.197135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −44.8096 25.8708i −1.28027 0.739166i
\(36\) 0 0
\(37\) 52.1430 1.40927 0.704635 0.709570i \(-0.251111\pi\)
0.704635 + 0.709570i \(0.251111\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 40.6831 23.4884i 0.992271 0.572888i 0.0863184 0.996268i \(-0.472490\pi\)
0.905952 + 0.423380i \(0.139156\pi\)
\(42\) 0 0
\(43\) −60.4290 −1.40533 −0.702663 0.711523i \(-0.748006\pi\)
−0.702663 + 0.711523i \(0.748006\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 25.9940 + 15.0076i 0.553064 + 0.319311i 0.750357 0.661033i \(-0.229882\pi\)
−0.197293 + 0.980345i \(0.563215\pi\)
\(48\) 0 0
\(49\) −9.86235 + 17.0821i −0.201272 + 0.348614i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −42.0135 + 24.2565i −0.792707 + 0.457670i −0.840915 0.541167i \(-0.817983\pi\)
0.0482074 + 0.998837i \(0.484649\pi\)
\(54\) 0 0
\(55\) 59.8661 103.691i 1.08847 1.88529i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −83.0645 + 47.9573i −1.40787 + 0.812835i −0.995183 0.0980364i \(-0.968744\pi\)
−0.412689 + 0.910872i \(0.635411\pi\)
\(60\) 0 0
\(61\) −46.3314 + 80.2483i −0.759530 + 1.31555i 0.183560 + 0.983009i \(0.441238\pi\)
−0.943090 + 0.332537i \(0.892095\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −27.1294 + 15.6632i −0.417375 + 0.240972i
\(66\) 0 0
\(67\) 24.0870 0.359507 0.179753 0.983712i \(-0.442470\pi\)
0.179753 + 0.983712i \(0.442470\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.03023 + 0.594802i 0.0145103 + 0.00837750i 0.507238 0.861806i \(-0.330667\pi\)
−0.492727 + 0.870184i \(0.664000\pi\)
\(72\) 0 0
\(73\) −20.6395 + 35.7487i −0.282733 + 0.489709i −0.972057 0.234745i \(-0.924574\pi\)
0.689324 + 0.724454i \(0.257908\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −137.725 79.5158i −1.78864 1.03267i
\(78\) 0 0
\(79\) −12.1846 −0.154235 −0.0771177 0.997022i \(-0.524572\pi\)
−0.0771177 + 0.997022i \(0.524572\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 124.538 71.9023i 1.50046 0.866293i 0.500463 0.865758i \(-0.333163\pi\)
1.00000 0.000535193i \(-0.000170357\pi\)
\(84\) 0 0
\(85\) −185.558 −2.18304
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −31.0467 + 17.9248i −0.348839 + 0.201402i −0.664174 0.747578i \(-0.731217\pi\)
0.315335 + 0.948981i \(0.397883\pi\)
\(90\) 0 0
\(91\) 20.8043 + 36.0340i 0.228618 + 0.395978i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −118.360 + 7.33387i −1.24590 + 0.0771986i
\(96\) 0 0
\(97\) 86.6880 0.893691 0.446845 0.894611i \(-0.352547\pi\)
0.446845 + 0.894611i \(0.352547\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −120.418 69.5234i −1.19226 0.688350i −0.233440 0.972371i \(-0.574998\pi\)
−0.958818 + 0.284021i \(0.908331\pi\)
\(102\) 0 0
\(103\) −49.7978 + 86.2523i −0.483474 + 0.837401i −0.999820 0.0189791i \(-0.993958\pi\)
0.516346 + 0.856380i \(0.327292\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 160.156i 1.49679i −0.663254 0.748394i \(-0.730825\pi\)
0.663254 0.748394i \(-0.269175\pi\)
\(108\) 0 0
\(109\) −25.5188 + 44.1999i −0.234117 + 0.405503i −0.959016 0.283352i \(-0.908553\pi\)
0.724898 + 0.688856i \(0.241887\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 137.055 + 79.1287i 1.21288 + 0.700254i 0.963385 0.268123i \(-0.0864033\pi\)
0.249491 + 0.968377i \(0.419737\pi\)
\(114\) 0 0
\(115\) −32.9240 57.0261i −0.286296 0.495879i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 246.463i 2.07112i
\(120\) 0 0
\(121\) 123.503 213.913i 1.02068 1.76788i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 68.9341i 0.551473i
\(126\) 0 0
\(127\) −102.738 177.947i −0.808958 1.40116i −0.913586 0.406646i \(-0.866698\pi\)
0.104628 0.994511i \(-0.466635\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −80.9155 + 46.7166i −0.617676 + 0.356615i −0.775963 0.630778i \(-0.782736\pi\)
0.158288 + 0.987393i \(0.449403\pi\)
\(132\) 0 0
\(133\) 9.74105 + 157.209i 0.0732410 + 1.18202i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.71952 3.87952i −0.0490476 0.0283176i 0.475276 0.879837i \(-0.342348\pi\)
−0.524323 + 0.851519i \(0.675682\pi\)
\(138\) 0 0
\(139\) 68.3785 0.491932 0.245966 0.969278i \(-0.420895\pi\)
0.245966 + 0.969278i \(0.420895\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −83.3842 + 48.1419i −0.583106 + 0.336656i
\(144\) 0 0
\(145\) −185.850 −1.28173
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 43.0650 24.8636i 0.289027 0.166870i −0.348476 0.937318i \(-0.613301\pi\)
0.637503 + 0.770448i \(0.279967\pi\)
\(150\) 0 0
\(151\) −105.562 182.839i −0.699088 1.21086i −0.968783 0.247910i \(-0.920256\pi\)
0.269695 0.962946i \(-0.413077\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 36.9838i 0.238605i
\(156\) 0 0
\(157\) 6.28248 + 10.8816i 0.0400158 + 0.0693094i 0.885340 0.464945i \(-0.153926\pi\)
−0.845324 + 0.534254i \(0.820593\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −75.7437 + 43.7306i −0.470458 + 0.271619i
\(162\) 0 0
\(163\) 56.3032 0.345418 0.172709 0.984973i \(-0.444748\pi\)
0.172709 + 0.984973i \(0.444748\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 184.027i 1.10196i 0.834520 + 0.550978i \(0.185745\pi\)
−0.834520 + 0.550978i \(0.814255\pi\)
\(168\) 0 0
\(169\) −143.809 −0.850939
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 245.490i 1.41902i 0.704697 + 0.709508i \(0.251083\pi\)
−0.704697 + 0.709508i \(0.748917\pi\)
\(174\) 0 0
\(175\) 115.691 0.661090
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 135.043i 0.754431i 0.926126 + 0.377215i \(0.123118\pi\)
−0.926126 + 0.377215i \(0.876882\pi\)
\(180\) 0 0
\(181\) 4.36155 + 7.55442i 0.0240970 + 0.0417372i 0.877822 0.478986i \(-0.158996\pi\)
−0.853725 + 0.520723i \(0.825662\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −281.845 + 162.723i −1.52349 + 0.879586i
\(186\) 0 0
\(187\) −570.326 −3.04987
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 190.729 110.118i 0.998582 0.576531i 0.0907535 0.995873i \(-0.471072\pi\)
0.907828 + 0.419342i \(0.137739\pi\)
\(192\) 0 0
\(193\) −128.964 223.372i −0.668207 1.15737i −0.978405 0.206696i \(-0.933729\pi\)
0.310199 0.950672i \(-0.399604\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 193.499i 0.982229i −0.871095 0.491114i \(-0.836590\pi\)
0.871095 0.491114i \(-0.163410\pi\)
\(198\) 0 0
\(199\) 34.0099 + 58.9068i 0.170904 + 0.296014i 0.938736 0.344637i \(-0.111998\pi\)
−0.767832 + 0.640651i \(0.778665\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 246.851i 1.21602i
\(204\) 0 0
\(205\) −146.601 + 253.921i −0.715127 + 1.23864i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −363.788 + 22.5412i −1.74061 + 0.107853i
\(210\) 0 0
\(211\) 16.4900 + 28.5616i 0.0781518 + 0.135363i 0.902453 0.430789i \(-0.141765\pi\)
−0.824301 + 0.566152i \(0.808431\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 326.633 188.582i 1.51922 0.877123i
\(216\) 0 0
\(217\) −49.1229 −0.226373
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 129.227 + 74.6091i 0.584737 + 0.337598i
\(222\) 0 0
\(223\) −301.114 −1.35029 −0.675143 0.737687i \(-0.735918\pi\)
−0.675143 + 0.737687i \(0.735918\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −250.357 + 144.544i −1.10289 + 0.636756i −0.936979 0.349385i \(-0.886391\pi\)
−0.165914 + 0.986140i \(0.553057\pi\)
\(228\) 0 0
\(229\) −135.468 + 234.637i −0.591563 + 1.02462i 0.402459 + 0.915438i \(0.368156\pi\)
−0.994022 + 0.109179i \(0.965178\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −37.9517 21.9115i −0.162883 0.0940406i 0.416343 0.909208i \(-0.363312\pi\)
−0.579226 + 0.815167i \(0.696645\pi\)
\(234\) 0 0
\(235\) −187.338 −0.797183
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 290.122 + 167.502i 1.21390 + 0.700845i 0.963606 0.267326i \(-0.0861400\pi\)
0.250292 + 0.968170i \(0.419473\pi\)
\(240\) 0 0
\(241\) −84.7677 + 146.822i −0.351733 + 0.609220i −0.986553 0.163440i \(-0.947741\pi\)
0.634820 + 0.772660i \(0.281074\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 123.110i 0.502491i
\(246\) 0 0
\(247\) 85.3774 + 42.4827i 0.345657 + 0.171995i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −140.235 + 80.9650i −0.558707 + 0.322570i −0.752626 0.658448i \(-0.771213\pi\)
0.193919 + 0.981017i \(0.437880\pi\)
\(252\) 0 0
\(253\) −101.194 175.274i −0.399978 0.692782i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 59.0684i 0.229838i 0.993375 + 0.114919i \(0.0366609\pi\)
−0.993375 + 0.114919i \(0.963339\pi\)
\(258\) 0 0
\(259\) 216.134 + 374.355i 0.834493 + 1.44538i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 177.086i 0.673329i −0.941625 0.336665i \(-0.890701\pi\)
0.941625 0.336665i \(-0.109299\pi\)
\(264\) 0 0
\(265\) 151.395 262.224i 0.571302 0.989525i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 111.617 + 64.4419i 0.414932 + 0.239561i 0.692907 0.721027i \(-0.256330\pi\)
−0.277975 + 0.960588i \(0.589663\pi\)
\(270\) 0 0
\(271\) 43.8719 75.9883i 0.161889 0.280400i −0.773657 0.633604i \(-0.781575\pi\)
0.935546 + 0.353205i \(0.114908\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 267.713i 0.973501i
\(276\) 0 0
\(277\) −120.322 208.404i −0.434375 0.752360i 0.562869 0.826546i \(-0.309698\pi\)
−0.997244 + 0.0741858i \(0.976364\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −72.5547 41.8895i −0.258202 0.149073i 0.365312 0.930885i \(-0.380962\pi\)
−0.623514 + 0.781812i \(0.714296\pi\)
\(282\) 0 0
\(283\) −185.400 321.122i −0.655123 1.13471i −0.981863 0.189592i \(-0.939284\pi\)
0.326740 0.945114i \(-0.394050\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 337.264 + 194.720i 1.17514 + 0.678466i
\(288\) 0 0
\(289\) 297.439 + 515.180i 1.02920 + 1.78263i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 119.643 + 69.0758i 0.408337 + 0.235754i 0.690075 0.723738i \(-0.257577\pi\)
−0.281738 + 0.959491i \(0.590911\pi\)
\(294\) 0 0
\(295\) 299.322 518.441i 1.01465 1.75743i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 52.9523i 0.177098i
\(300\) 0 0
\(301\) −250.479 433.843i −0.832157 1.44134i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 578.347i 1.89622i
\(306\) 0 0
\(307\) 231.124 400.319i 0.752848 1.30397i −0.193589 0.981083i \(-0.562013\pi\)
0.946437 0.322889i \(-0.104654\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 336.644 + 194.362i 1.08246 + 0.624957i 0.931558 0.363592i \(-0.118450\pi\)
0.150899 + 0.988549i \(0.451783\pi\)
\(312\) 0 0
\(313\) 282.706 489.661i 0.903213 1.56441i 0.0799157 0.996802i \(-0.474535\pi\)
0.823298 0.567610i \(-0.192132\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.1334 + 5.85051i 0.0319665 + 0.0184559i 0.515898 0.856650i \(-0.327458\pi\)
−0.483932 + 0.875106i \(0.660792\pi\)
\(318\) 0 0
\(319\) −571.224 −1.79067
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 312.149 + 470.790i 0.966405 + 1.45755i
\(324\) 0 0
\(325\) 35.0217 60.6594i 0.107759 0.186644i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 248.828i 0.756315i
\(330\) 0 0
\(331\) −36.1568 62.6255i −0.109235 0.189201i 0.806225 0.591608i \(-0.201507\pi\)
−0.915461 + 0.402408i \(0.868173\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −130.196 + 75.1685i −0.388644 + 0.224383i
\(336\) 0 0
\(337\) 265.547 + 459.941i 0.787973 + 1.36481i 0.927207 + 0.374550i \(0.122203\pi\)
−0.139233 + 0.990260i \(0.544464\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 113.672i 0.333350i
\(342\) 0 0
\(343\) 242.693 0.707561
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −274.157 + 158.284i −0.790077 + 0.456151i −0.839990 0.542602i \(-0.817439\pi\)
0.0499127 + 0.998754i \(0.484106\pi\)
\(348\) 0 0
\(349\) 331.633 + 574.406i 0.950239 + 1.64586i 0.744906 + 0.667169i \(0.232494\pi\)
0.205332 + 0.978692i \(0.434172\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 396.651 229.006i 1.12366 0.648743i 0.181325 0.983423i \(-0.441962\pi\)
0.942332 + 0.334680i \(0.108628\pi\)
\(354\) 0 0
\(355\) −7.42483 −0.0209150
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −116.361 67.1811i −0.324125 0.187134i 0.329104 0.944294i \(-0.393253\pi\)
−0.653230 + 0.757160i \(0.726587\pi\)
\(360\) 0 0
\(361\) 217.714 + 287.961i 0.603087 + 0.797676i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 257.640i 0.705864i
\(366\) 0 0
\(367\) 49.6894 86.0646i 0.135393 0.234508i −0.790354 0.612650i \(-0.790103\pi\)
0.925748 + 0.378142i \(0.123437\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −348.293 201.087i −0.938796 0.542014i
\(372\) 0 0
\(373\) 185.179 320.739i 0.496457 0.859889i −0.503534 0.863975i \(-0.667967\pi\)
0.999992 + 0.00408598i \(0.00130061\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 129.430 + 74.7265i 0.343316 + 0.198214i
\(378\) 0 0
\(379\) −480.220 −1.26707 −0.633536 0.773713i \(-0.718397\pi\)
−0.633536 + 0.773713i \(0.718397\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −47.8137 + 27.6053i −0.124840 + 0.0720764i −0.561120 0.827735i \(-0.689629\pi\)
0.436280 + 0.899811i \(0.356296\pi\)
\(384\) 0 0
\(385\) 992.584 2.57814
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −247.457 142.869i −0.636136 0.367273i 0.146988 0.989138i \(-0.453042\pi\)
−0.783125 + 0.621865i \(0.786375\pi\)
\(390\) 0 0
\(391\) −156.829 + 271.635i −0.401096 + 0.694719i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 65.8605 38.0246i 0.166736 0.0962648i
\(396\) 0 0
\(397\) 189.291 327.861i 0.476802 0.825846i −0.522844 0.852428i \(-0.675129\pi\)
0.999647 + 0.0265823i \(0.00846239\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −142.320 + 82.1683i −0.354912 + 0.204909i −0.666847 0.745195i \(-0.732356\pi\)
0.311935 + 0.950104i \(0.399023\pi\)
\(402\) 0 0
\(403\) −14.8704 + 25.7563i −0.0368993 + 0.0639115i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −866.271 + 500.142i −2.12843 + 1.22885i
\(408\) 0 0
\(409\) 211.156 0.516273 0.258136 0.966108i \(-0.416892\pi\)
0.258136 + 0.966108i \(0.416892\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −688.607 397.568i −1.66733 0.962634i
\(414\) 0 0
\(415\) −448.773 + 777.297i −1.08138 + 1.87301i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −539.656 311.571i −1.28796 0.743606i −0.309672 0.950843i \(-0.600219\pi\)
−0.978291 + 0.207238i \(0.933553\pi\)
\(420\) 0 0
\(421\) 224.827 0.534032 0.267016 0.963692i \(-0.413962\pi\)
0.267016 + 0.963692i \(0.413962\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 359.310 207.447i 0.845434 0.488112i
\(426\) 0 0
\(427\) −768.177 −1.79901
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 475.176 274.343i 1.10250 0.636527i 0.165621 0.986189i \(-0.447037\pi\)
0.936876 + 0.349663i \(0.113704\pi\)
\(432\) 0 0
\(433\) 78.5854 + 136.114i 0.181491 + 0.314351i 0.942388 0.334521i \(-0.108574\pi\)
−0.760898 + 0.648872i \(0.775241\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −89.2988 + 179.464i −0.204345 + 0.410672i
\(438\) 0 0
\(439\) −210.141 −0.478682 −0.239341 0.970936i \(-0.576931\pi\)
−0.239341 + 0.970936i \(0.576931\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −221.312 127.774i −0.499575 0.288430i 0.228963 0.973435i \(-0.426467\pi\)
−0.728538 + 0.685005i \(0.759800\pi\)
\(444\) 0 0
\(445\) 111.876 193.776i 0.251408 0.435451i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 499.573i 1.11264i −0.830970 0.556318i \(-0.812214\pi\)
0.830970 0.556318i \(-0.187786\pi\)
\(450\) 0 0
\(451\) −450.589 + 780.443i −0.999089 + 1.73047i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −224.904 129.848i −0.494294 0.285381i
\(456\) 0 0
\(457\) −398.855 690.836i −0.872767 1.51168i −0.859123 0.511770i \(-0.828990\pi\)
−0.0136444 0.999907i \(-0.504343\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 52.9853i 0.114936i 0.998347 + 0.0574678i \(0.0183026\pi\)
−0.998347 + 0.0574678i \(0.981697\pi\)
\(462\) 0 0
\(463\) −95.1208 + 164.754i −0.205445 + 0.355840i −0.950274 0.311414i \(-0.899197\pi\)
0.744830 + 0.667255i \(0.232531\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 316.369i 0.677449i −0.940886 0.338725i \(-0.890004\pi\)
0.940886 0.338725i \(-0.109996\pi\)
\(468\) 0 0
\(469\) 99.8409 + 172.929i 0.212880 + 0.368719i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1003.93 579.619i 2.12247 1.22541i
\(474\) 0 0
\(475\) 220.990 146.524i 0.465242 0.308471i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −380.884 219.904i −0.795166 0.459089i 0.0466123 0.998913i \(-0.485157\pi\)
−0.841778 + 0.539824i \(0.818491\pi\)
\(480\) 0 0
\(481\) 261.711 0.544098
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −468.569 + 270.528i −0.966121 + 0.557790i
\(486\) 0 0
\(487\) 182.917 0.375600 0.187800 0.982207i \(-0.439864\pi\)
0.187800 + 0.982207i \(0.439864\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −399.817 + 230.834i −0.814291 + 0.470131i −0.848444 0.529286i \(-0.822460\pi\)
0.0341528 + 0.999417i \(0.489127\pi\)
\(492\) 0 0
\(493\) 442.635 + 766.665i 0.897839 + 1.55510i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 9.86186i 0.0198428i
\(498\) 0 0
\(499\) 138.657 + 240.161i 0.277870 + 0.481285i 0.970855 0.239667i \(-0.0770383\pi\)
−0.692985 + 0.720952i \(0.743705\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 735.341 424.549i 1.46191 0.844034i 0.462810 0.886458i \(-0.346841\pi\)
0.999100 + 0.0424236i \(0.0135079\pi\)
\(504\) 0 0
\(505\) 867.850 1.71851
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 66.3025i 0.130260i 0.997877 + 0.0651301i \(0.0207462\pi\)
−0.997877 + 0.0651301i \(0.979254\pi\)
\(510\) 0 0
\(511\) −342.205 −0.669677
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 621.618i 1.20703i
\(516\) 0 0
\(517\) −575.797 −1.11373
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 92.5065i 0.177556i 0.996051 + 0.0887778i \(0.0282961\pi\)
−0.996051 + 0.0887778i \(0.971704\pi\)
\(522\) 0 0
\(523\) 52.7222 + 91.3175i 0.100807 + 0.174603i 0.912017 0.410151i \(-0.134524\pi\)
−0.811210 + 0.584755i \(0.801191\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −152.565 + 88.0833i −0.289497 + 0.167141i
\(528\) 0 0
\(529\) 417.694 0.789592
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 204.193 117.891i 0.383100 0.221183i
\(534\) 0 0
\(535\) 499.802 + 865.683i 0.934209 + 1.61810i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 378.388i 0.702019i
\(540\) 0 0
\(541\) −6.28202 10.8808i −0.0116119 0.0201124i 0.860161 0.510022i \(-0.170363\pi\)
−0.871773 + 0.489910i \(0.837030\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 318.547i 0.584491i
\(546\) 0 0
\(547\) 48.3736 83.7856i 0.0884344 0.153173i −0.818415 0.574627i \(-0.805147\pi\)
0.906850 + 0.421455i \(0.138480\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 312.640 + 471.531i 0.567405 + 0.855773i
\(552\) 0 0
\(553\) −50.5053 87.4778i −0.0913297 0.158188i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 429.405 247.917i 0.770925 0.445094i −0.0622792 0.998059i \(-0.519837\pi\)
0.833205 + 0.552965i \(0.186504\pi\)
\(558\) 0 0
\(559\) −303.299 −0.542574
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −707.636 408.554i −1.25690 0.725673i −0.284431 0.958697i \(-0.591805\pi\)
−0.972471 + 0.233024i \(0.925138\pi\)
\(564\) 0 0
\(565\) −987.752 −1.74823
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −369.359 + 213.249i −0.649137 + 0.374779i −0.788126 0.615515i \(-0.788948\pi\)
0.138988 + 0.990294i \(0.455615\pi\)
\(570\) 0 0
\(571\) 86.9476 150.598i 0.152273 0.263744i −0.779790 0.626041i \(-0.784674\pi\)
0.932063 + 0.362297i \(0.118008\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 127.506 + 73.6158i 0.221750 + 0.128028i
\(576\) 0 0
\(577\) −291.789 −0.505700 −0.252850 0.967505i \(-0.581368\pi\)
−0.252850 + 0.967505i \(0.581368\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1032.43 + 596.073i 1.77698 + 1.02594i
\(582\) 0 0
\(583\) 465.324 805.965i 0.798154 1.38244i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 203.125i 0.346040i −0.984918 0.173020i \(-0.944647\pi\)
0.984918 0.173020i \(-0.0553525\pi\)
\(588\) 0 0
\(589\) −93.8336 + 62.2147i −0.159310 + 0.105628i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 256.281 147.964i 0.432177 0.249517i −0.268097 0.963392i \(-0.586395\pi\)
0.700274 + 0.713875i \(0.253061\pi\)
\(594\) 0 0
\(595\) −769.142 1332.19i −1.29268 2.23898i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1082.64i 1.80741i 0.428160 + 0.903703i \(0.359162\pi\)
−0.428160 + 0.903703i \(0.640838\pi\)
\(600\) 0 0
\(601\) 524.803 + 908.986i 0.873217 + 1.51246i 0.858650 + 0.512563i \(0.171304\pi\)
0.0145672 + 0.999894i \(0.495363\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1541.67i 2.54821i
\(606\) 0 0
\(607\) 177.821 307.995i 0.292951 0.507405i −0.681556 0.731766i \(-0.738696\pi\)
0.974506 + 0.224361i \(0.0720295\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 130.466 + 75.3248i 0.213529 + 0.123281i
\(612\) 0 0
\(613\) −279.541 + 484.179i −0.456021 + 0.789852i −0.998746 0.0500583i \(-0.984059\pi\)
0.542725 + 0.839911i \(0.317393\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 393.300i 0.637439i −0.947849 0.318719i \(-0.896747\pi\)
0.947849 0.318719i \(-0.103253\pi\)
\(618\) 0 0
\(619\) 315.129 + 545.820i 0.509094 + 0.881777i 0.999945 + 0.0105328i \(0.00335277\pi\)
−0.490851 + 0.871244i \(0.663314\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −257.378 148.597i −0.413127 0.238519i
\(624\) 0 0
\(625\) 389.566 + 674.748i 0.623305 + 1.07960i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1342.53 + 775.108i 2.13438 + 1.23229i
\(630\) 0 0
\(631\) −436.567 756.155i −0.691865 1.19834i −0.971226 0.238159i \(-0.923456\pi\)
0.279362 0.960186i \(-0.409877\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1110.64 + 641.230i 1.74904 + 1.00981i
\(636\) 0 0
\(637\) −49.5001 + 85.7367i −0.0777082 + 0.134594i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1027.74i 1.60335i −0.597763 0.801673i \(-0.703943\pi\)
0.597763 0.801673i \(-0.296057\pi\)
\(642\) 0 0
\(643\) 237.714 + 411.733i 0.369695 + 0.640331i 0.989518 0.144411i \(-0.0461288\pi\)
−0.619823 + 0.784742i \(0.712795\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 80.3000i 0.124111i 0.998073 + 0.0620557i \(0.0197656\pi\)
−0.998073 + 0.0620557i \(0.980234\pi\)
\(648\) 0 0
\(649\) 919.987 1593.46i 1.41755 2.45526i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 666.603 + 384.863i 1.02083 + 0.589377i 0.914345 0.404937i \(-0.132706\pi\)
0.106487 + 0.994314i \(0.466040\pi\)
\(654\) 0 0
\(655\) 291.578 505.028i 0.445157 0.771035i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −370.850 214.111i −0.562747 0.324902i 0.191500 0.981493i \(-0.438665\pi\)
−0.754247 + 0.656590i \(0.771998\pi\)
\(660\) 0 0
\(661\) 996.915 1.50819 0.754096 0.656764i \(-0.228075\pi\)
0.754096 + 0.656764i \(0.228075\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −543.257 819.353i −0.816928 1.23211i
\(666\) 0 0
\(667\) −157.075 + 272.063i −0.235495 + 0.407890i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1777.59i 2.64917i
\(672\) 0 0
\(673\) 431.710 + 747.743i 0.641470 + 1.11106i 0.985105 + 0.171956i \(0.0550086\pi\)
−0.343634 + 0.939104i \(0.611658\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 914.184 527.804i 1.35035 0.779622i 0.362048 0.932160i \(-0.382078\pi\)
0.988298 + 0.152537i \(0.0487445\pi\)
\(678\) 0 0
\(679\) 359.323 + 622.366i 0.529195 + 0.916592i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 828.737i 1.21338i 0.794939 + 0.606689i \(0.207503\pi\)
−0.794939 + 0.606689i \(0.792497\pi\)
\(684\) 0 0
\(685\) 48.4274 0.0706970
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −210.870 + 121.746i −0.306052 + 0.176699i
\(690\) 0 0
\(691\) 118.397 + 205.070i 0.171342 + 0.296773i 0.938889 0.344219i \(-0.111856\pi\)
−0.767547 + 0.640993i \(0.778523\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −369.602 + 213.390i −0.531801 + 0.307036i
\(696\) 0 0
\(697\) 1396.62 2.00377
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 664.672 + 383.748i 0.948177 + 0.547430i 0.892514 0.451020i \(-0.148940\pi\)
0.0556627 + 0.998450i \(0.482273\pi\)
\(702\) 0 0
\(703\) 886.979 + 441.349i 1.26171 + 0.627808i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1152.70i 1.63041i
\(708\) 0 0
\(709\) 498.446 863.333i 0.703026 1.21768i −0.264373 0.964421i \(-0.585165\pi\)
0.967399 0.253257i \(-0.0815017\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −54.1399 31.2577i −0.0759326 0.0438397i
\(714\) 0 0
\(715\) 300.474 520.436i 0.420243 0.727883i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −858.980 495.932i −1.19469 0.689753i −0.235321 0.971918i \(-0.575614\pi\)
−0.959366 + 0.282165i \(0.908947\pi\)
\(720\) 0 0
\(721\) −825.651 −1.14515
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 359.875 207.774i 0.496379 0.286585i
\(726\) 0 0
\(727\) −226.650 −0.311761 −0.155881 0.987776i \(-0.549822\pi\)
−0.155881 + 0.987776i \(0.549822\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1555.87 898.280i −2.12841 1.22884i
\(732\) 0 0
\(733\) −394.329 + 682.998i −0.537966 + 0.931785i 0.461047 + 0.887376i \(0.347474\pi\)
−0.999013 + 0.0444091i \(0.985859\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −400.166 + 231.036i −0.542965 + 0.313481i
\(738\) 0 0
\(739\) 83.0957 143.926i 0.112443 0.194758i −0.804311 0.594208i \(-0.797466\pi\)
0.916755 + 0.399450i \(0.130799\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 838.921 484.351i 1.12910 0.651886i 0.185391 0.982665i \(-0.440645\pi\)
0.943708 + 0.330779i \(0.107311\pi\)
\(744\) 0 0
\(745\) −155.184 + 268.787i −0.208301 + 0.360788i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1149.82 663.851i 1.53515 0.886316i
\(750\) 0 0
\(751\) 1252.30 1.66750 0.833752 0.552139i \(-0.186188\pi\)
0.833752 + 0.552139i \(0.186188\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1141.18 + 658.859i 1.51149 + 0.872661i
\(756\) 0 0
\(757\) 592.866 1026.87i 0.783179 1.35651i −0.146902 0.989151i \(-0.546930\pi\)
0.930081 0.367355i \(-0.119736\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 643.588 + 371.576i 0.845714 + 0.488273i 0.859202 0.511636i \(-0.170960\pi\)
−0.0134887 + 0.999909i \(0.504294\pi\)
\(762\) 0 0
\(763\) −423.104 −0.554526
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −416.909 + 240.702i −0.543558 + 0.313823i
\(768\) 0 0
\(769\) −962.692 −1.25187 −0.625937 0.779873i \(-0.715283\pi\)
−0.625937 + 0.779873i \(0.715283\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −34.6745 + 20.0193i −0.0448570 + 0.0258982i −0.522261 0.852786i \(-0.674911\pi\)
0.477404 + 0.878684i \(0.341578\pi\)
\(774\) 0 0
\(775\) 41.3466 + 71.6144i 0.0533504 + 0.0924057i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 890.851 55.1992i 1.14358 0.0708590i
\(780\) 0 0
\(781\) −22.8207 −0.0292199
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −67.9166 39.2117i −0.0865179 0.0499511i
\(786\) 0 0
\(787\) 718.501 1244.48i 0.912961 1.58130i 0.103102 0.994671i \(-0.467123\pi\)
0.809859 0.586625i \(-0.199544\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1311.96i 1.65861i
\(792\) 0 0
\(793\) −232.542 + 402.774i −0.293243 + 0.507912i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 772.318 + 445.898i 0.969031 + 0.559470i 0.898941 0.438070i \(-0.144338\pi\)
0.0700902 + 0.997541i \(0.477671\pi\)
\(798\) 0 0
\(799\) 446.178 + 772.804i 0.558421 + 0.967214i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 791.876i 0.986147i
\(804\) 0 0
\(805\) 272.941 472.748i 0.339058 0.587265i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 14.9006i 0.0184186i −0.999958 0.00920928i \(-0.997069\pi\)
0.999958 0.00920928i \(-0.00293145\pi\)
\(810\) 0 0
\(811\) 741.642 + 1284.56i 0.914478 + 1.58392i 0.807663 + 0.589644i \(0.200732\pi\)
0.106815 + 0.994279i \(0.465935\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −304.332 + 175.706i −0.373413 + 0.215590i
\(816\) 0 0
\(817\) −1027.93 511.484i −1.25817 0.626051i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1155.51 667.132i −1.40744 0.812585i −0.412298 0.911049i \(-0.635274\pi\)
−0.995141 + 0.0984639i \(0.968607\pi\)
\(822\) 0 0
\(823\) −940.461 −1.14272 −0.571362 0.820698i \(-0.693585\pi\)
−0.571362 + 0.820698i \(0.693585\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1348.77 + 778.714i −1.63092 + 0.941613i −0.647112 + 0.762395i \(0.724024\pi\)
−0.983809 + 0.179218i \(0.942643\pi\)
\(828\) 0 0
\(829\) −215.811 −0.260327 −0.130163 0.991493i \(-0.541550\pi\)
−0.130163 + 0.991493i \(0.541550\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −507.852 + 293.208i −0.609666 + 0.351991i
\(834\) 0 0
\(835\) −574.294 994.707i −0.687778 1.19127i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 304.380i 0.362789i 0.983410 + 0.181394i \(0.0580610\pi\)
−0.983410 + 0.181394i \(0.941939\pi\)
\(840\) 0 0
\(841\) 22.8311 + 39.5446i 0.0271476 + 0.0470209i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 777.319 448.785i 0.919904 0.531107i
\(846\) 0 0
\(847\) 2047.68 2.41757
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 550.117i 0.646437i
\(852\) 0 0
\(853\) −1090.96 −1.27897 −0.639487 0.768802i \(-0.720853\pi\)
−0.639487 + 0.768802i \(0.720853\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1057.91i 1.23443i −0.786793 0.617217i \(-0.788260\pi\)
0.786793 0.617217i \(-0.211740\pi\)
\(858\) 0 0
\(859\) 658.540 0.766635 0.383318 0.923617i \(-0.374781\pi\)
0.383318 + 0.923617i \(0.374781\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 611.208i 0.708236i −0.935201 0.354118i \(-0.884781\pi\)
0.935201 0.354118i \(-0.115219\pi\)
\(864\) 0 0
\(865\) −766.103 1326.93i −0.885668 1.53402i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 202.427 116.871i 0.232943 0.134489i
\(870\) 0 0
\(871\) 120.895 0.138800
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 494.904 285.733i 0.565605 0.326552i
\(876\) 0 0
\(877\) 66.2995 + 114.834i 0.0755980 + 0.130940i 0.901346 0.433099i \(-0.142580\pi\)
−0.825748 + 0.564039i \(0.809247\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 75.4585i 0.0856510i −0.999083 0.0428255i \(-0.986364\pi\)
0.999083 0.0428255i \(-0.0136360\pi\)
\(882\) 0 0
\(883\) −374.686 648.975i −0.424333 0.734966i 0.572025 0.820236i \(-0.306158\pi\)
−0.996358 + 0.0852698i \(0.972825\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1377.75i 1.55327i 0.629949 + 0.776636i \(0.283076\pi\)
−0.629949 + 0.776636i \(0.716924\pi\)
\(888\) 0 0
\(889\) 851.699 1475.19i 0.958042 1.65938i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 315.143 + 475.306i 0.352904 + 0.532258i
\(894\) 0 0
\(895\) −421.431 729.940i −0.470872 0.815575i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −152.805 + 88.2220i −0.169972 + 0.0981334i
\(900\) 0 0
\(901\) −1442.30 −1.60077
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −47.1504 27.2223i −0.0520999 0.0300799i
\(906\) 0 0
\(907\) −564.480 −0.622359 −0.311179 0.950351i \(-0.600724\pi\)
−0.311179 + 0.950351i \(0.600724\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −100.741 + 58.1627i −0.110583 + 0.0638450i −0.554271 0.832336i \(-0.687003\pi\)
0.443689 + 0.896181i \(0.353670\pi\)
\(912\) 0 0
\(913\) −1379.34 + 2389.08i −1.51077 + 2.61674i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −670.792 387.282i −0.731507 0.422336i
\(918\) 0 0
\(919\) −806.769 −0.877877 −0.438939 0.898517i \(-0.644645\pi\)
−0.438939 + 0.898517i \(0.644645\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 5.17082 + 2.98537i 0.00560218 + 0.00323442i
\(924\) 0 0
\(925\) 363.838 630.186i 0.393338 0.681282i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 403.812i 0.434674i −0.976097 0.217337i \(-0.930263\pi\)
0.976097 0.217337i \(-0.0697371\pi\)
\(930\) 0 0
\(931\) −312.350 + 207.098i −0.335499 + 0.222447i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 3082.75 1779.82i 3.29705 1.90356i
\(936\) 0 0
\(937\) 119.261 + 206.566i 0.127279 + 0.220454i 0.922622 0.385706i \(-0.126042\pi\)
−0.795342 + 0.606161i \(0.792709\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 773.528i 0.822027i 0.911629 + 0.411014i \(0.134825\pi\)
−0.911629 + 0.411014i \(0.865175\pi\)
\(942\) 0 0
\(943\) 247.807 + 429.214i 0.262785 + 0.455157i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 801.869i 0.846747i 0.905955 + 0.423373i \(0.139154\pi\)
−0.905955 + 0.423373i \(0.860846\pi\)
\(948\) 0 0
\(949\) −103.592 + 179.426i −0.109159 + 0.189069i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1027.59 + 593.282i 1.07827 + 0.622542i 0.930430 0.366469i \(-0.119433\pi\)
0.147843 + 0.989011i \(0.452767\pi\)
\(954\) 0 0
\(955\) −687.290 + 1190.42i −0.719676 + 1.24651i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 64.3227i 0.0670726i
\(960\) 0 0
\(961\) 462.944 + 801.843i 0.481732 + 0.834384i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1394.16 + 804.918i 1.44472 + 0.834112i
\(966\) 0 0
\(967\) 546.727 + 946.959i 0.565385 + 0.979275i 0.997014 + 0.0772241i \(0.0246057\pi\)
−0.431629 + 0.902051i \(0.642061\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −184.215 106.357i −0.189717 0.109533i 0.402133 0.915581i \(-0.368269\pi\)
−0.591850 + 0.806048i \(0.701602\pi\)
\(972\) 0 0
\(973\) 283.430 + 490.915i 0.291295 + 0.504538i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −639.024 368.941i −0.654068 0.377626i 0.135945 0.990716i \(-0.456593\pi\)
−0.790013 + 0.613090i \(0.789926\pi\)
\(978\) 0 0
\(979\) 343.860 595.583i 0.351236 0.608359i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1508.76i 1.53485i 0.641136 + 0.767427i \(0.278464\pi\)
−0.641136 + 0.767427i \(0.721536\pi\)
\(984\) 0 0
\(985\) 603.855 + 1045.91i 0.613051 + 1.06183i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 637.536i 0.644627i
\(990\) 0 0
\(991\) −751.118 + 1300.97i −0.757939 + 1.31279i 0.185960 + 0.982557i \(0.440460\pi\)
−0.943900 + 0.330232i \(0.892873\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −367.662 212.270i −0.369510 0.213337i
\(996\) 0 0
\(997\) −269.965 + 467.594i −0.270778 + 0.469001i −0.969061 0.246821i \(-0.920614\pi\)
0.698283 + 0.715821i \(0.253948\pi\)
\(998\) 0 0
\(999\) 0 0
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 2052.3.be.a.125.8 80
3.2 odd 2 684.3.be.a.581.32 yes 80
9.2 odd 6 2052.3.m.a.1493.33 80
9.7 even 3 684.3.m.a.353.4 80
19.7 even 3 2052.3.m.a.881.8 80
57.26 odd 6 684.3.m.a.653.4 yes 80
171.7 even 3 684.3.be.a.425.32 yes 80
171.83 odd 6 inner 2052.3.be.a.197.8 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
684.3.m.a.353.4 80 9.7 even 3
684.3.m.a.653.4 yes 80 57.26 odd 6
684.3.be.a.425.32 yes 80 171.7 even 3
684.3.be.a.581.32 yes 80 3.2 odd 2
2052.3.m.a.881.8 80 19.7 even 3
2052.3.m.a.1493.33 80 9.2 odd 6
2052.3.be.a.125.8 80 1.1 even 1 trivial
2052.3.be.a.197.8 80 171.83 odd 6 inner