Properties

Label 588.3.c.j.197.4
Level $588$
Weight $3$
Character 588.197
Analytic conductor $16.022$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [588,3,Mod(197,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.197");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 588.c (of order \(2\), degree \(1\), 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: \(16.0218395444\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.6018425749504.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 18x^{6} + 123x^{4} + 304x^{2} + 441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 197.4
Root \(0.707107 + 2.83573i\) of defining polynomial
Character \(\chi\) \(=\) 588.197
Dual form 588.3.c.j.197.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.79703 + 2.40223i) q^{3} -0.867013i q^{5} +(-2.54138 - 8.63374i) q^{9} +O(q^{10})\) \(q+(-1.79703 + 2.40223i) q^{3} -0.867013i q^{5} +(-2.54138 - 8.63374i) q^{9} -2.45228i q^{11} +6.53953 q^{13} +(2.08276 + 1.55805i) q^{15} +18.7484i q^{17} +13.6106 q^{19} -41.9933i q^{23} +24.2483 q^{25} +(25.3071 + 9.41009i) q^{27} +22.8358i q^{29} +8.48528 q^{31} +(5.89094 + 4.40682i) q^{33} -41.7449 q^{37} +(-11.7517 + 15.7094i) q^{39} +71.1998i q^{41} +59.7449 q^{43} +(-7.48556 + 2.20341i) q^{45} +69.7916i q^{47} +(-45.0380 - 33.6914i) q^{51} +62.3768i q^{53} -2.12616 q^{55} +(-24.4586 + 32.6957i) q^{57} -57.1123i q^{59} +59.2166 q^{61} -5.66986i q^{65} +66.0000 q^{67} +(100.877 + 75.4631i) q^{69} +53.7938i q^{71} +95.4546 q^{73} +(-43.5748 + 58.2499i) q^{75} -40.9932 q^{79} +(-68.0828 + 43.8832i) q^{81} +23.0835i q^{83} +16.2551 q^{85} +(-54.8569 - 41.0367i) q^{87} -5.52795i q^{89} +(-15.2483 + 20.3836i) q^{93} -11.8006i q^{95} -85.5648 q^{97} +(-21.1724 + 6.23219i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{9} - 32 q^{15} + 48 q^{25} + 104 q^{37} - 240 q^{39} + 40 q^{43} - 44 q^{51} - 220 q^{57} + 528 q^{67} + 256 q^{79} - 496 q^{81} + 568 q^{85} + 24 q^{93} - 656 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) −1.79703 + 2.40223i −0.599009 + 0.800742i
\(4\) 0 0
\(5\) 0.867013i 0.173403i −0.996234 0.0867013i \(-0.972367\pi\)
0.996234 0.0867013i \(-0.0276326\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −2.54138 8.63374i −0.282376 0.959304i
\(10\) 0 0
\(11\) 2.45228i 0.222935i −0.993768 0.111467i \(-0.964445\pi\)
0.993768 0.111467i \(-0.0355551\pi\)
\(12\) 0 0
\(13\) 6.53953 0.503041 0.251520 0.967852i \(-0.419069\pi\)
0.251520 + 0.967852i \(0.419069\pi\)
\(14\) 0 0
\(15\) 2.08276 + 1.55805i 0.138851 + 0.103870i
\(16\) 0 0
\(17\) 18.7484i 1.10285i 0.834225 + 0.551424i \(0.185915\pi\)
−0.834225 + 0.551424i \(0.814085\pi\)
\(18\) 0 0
\(19\) 13.6106 0.716347 0.358174 0.933655i \(-0.383400\pi\)
0.358174 + 0.933655i \(0.383400\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 41.9933i 1.82579i −0.408189 0.912897i \(-0.633840\pi\)
0.408189 0.912897i \(-0.366160\pi\)
\(24\) 0 0
\(25\) 24.2483 0.969932
\(26\) 0 0
\(27\) 25.3071 + 9.41009i 0.937301 + 0.348522i
\(28\) 0 0
\(29\) 22.8358i 0.787443i 0.919230 + 0.393722i \(0.128813\pi\)
−0.919230 + 0.393722i \(0.871187\pi\)
\(30\) 0 0
\(31\) 8.48528 0.273719 0.136859 0.990590i \(-0.456299\pi\)
0.136859 + 0.990590i \(0.456299\pi\)
\(32\) 0 0
\(33\) 5.89094 + 4.40682i 0.178513 + 0.133540i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −41.7449 −1.12824 −0.564120 0.825693i \(-0.690784\pi\)
−0.564120 + 0.825693i \(0.690784\pi\)
\(38\) 0 0
\(39\) −11.7517 + 15.7094i −0.301326 + 0.402806i
\(40\) 0 0
\(41\) 71.1998i 1.73658i 0.496057 + 0.868290i \(0.334781\pi\)
−0.496057 + 0.868290i \(0.665219\pi\)
\(42\) 0 0
\(43\) 59.7449 1.38942 0.694708 0.719292i \(-0.255534\pi\)
0.694708 + 0.719292i \(0.255534\pi\)
\(44\) 0 0
\(45\) −7.48556 + 2.20341i −0.166346 + 0.0489647i
\(46\) 0 0
\(47\) 69.7916i 1.48493i 0.669886 + 0.742464i \(0.266343\pi\)
−0.669886 + 0.742464i \(0.733657\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −45.0380 33.6914i −0.883097 0.660617i
\(52\) 0 0
\(53\) 62.3768i 1.17692i 0.808526 + 0.588461i \(0.200266\pi\)
−0.808526 + 0.588461i \(0.799734\pi\)
\(54\) 0 0
\(55\) −2.12616 −0.0386575
\(56\) 0 0
\(57\) −24.4586 + 32.6957i −0.429099 + 0.573609i
\(58\) 0 0
\(59\) 57.1123i 0.968005i −0.875067 0.484002i \(-0.839183\pi\)
0.875067 0.484002i \(-0.160817\pi\)
\(60\) 0 0
\(61\) 59.2166 0.970763 0.485382 0.874302i \(-0.338681\pi\)
0.485382 + 0.874302i \(0.338681\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.66986i 0.0872286i
\(66\) 0 0
\(67\) 66.0000 0.985075 0.492537 0.870291i \(-0.336070\pi\)
0.492537 + 0.870291i \(0.336070\pi\)
\(68\) 0 0
\(69\) 100.877 + 75.4631i 1.46199 + 1.09367i
\(70\) 0 0
\(71\) 53.7938i 0.757660i 0.925466 + 0.378830i \(0.123673\pi\)
−0.925466 + 0.378830i \(0.876327\pi\)
\(72\) 0 0
\(73\) 95.4546 1.30760 0.653798 0.756669i \(-0.273174\pi\)
0.653798 + 0.756669i \(0.273174\pi\)
\(74\) 0 0
\(75\) −43.5748 + 58.2499i −0.580998 + 0.776665i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −40.9932 −0.518901 −0.259450 0.965756i \(-0.583541\pi\)
−0.259450 + 0.965756i \(0.583541\pi\)
\(80\) 0 0
\(81\) −68.0828 + 43.8832i −0.840528 + 0.541768i
\(82\) 0 0
\(83\) 23.0835i 0.278114i 0.990284 + 0.139057i \(0.0444072\pi\)
−0.990284 + 0.139057i \(0.955593\pi\)
\(84\) 0 0
\(85\) 16.2551 0.191237
\(86\) 0 0
\(87\) −54.8569 41.0367i −0.630539 0.471686i
\(88\) 0 0
\(89\) 5.52795i 0.0621118i −0.999518 0.0310559i \(-0.990113\pi\)
0.999518 0.0310559i \(-0.00988700\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −15.2483 + 20.3836i −0.163960 + 0.219178i
\(94\) 0 0
\(95\) 11.8006i 0.124217i
\(96\) 0 0
\(97\) −85.5648 −0.882111 −0.441055 0.897480i \(-0.645396\pi\)
−0.441055 + 0.897480i \(0.645396\pi\)
\(98\) 0 0
\(99\) −21.1724 + 6.23219i −0.213862 + 0.0629514i
\(100\) 0 0
\(101\) 89.0812i 0.881992i 0.897509 + 0.440996i \(0.145375\pi\)
−0.897509 + 0.440996i \(0.854625\pi\)
\(102\) 0 0
\(103\) 35.0042 0.339847 0.169923 0.985457i \(-0.445648\pi\)
0.169923 + 0.985457i \(0.445648\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 88.1258i 0.823606i −0.911273 0.411803i \(-0.864899\pi\)
0.911273 0.411803i \(-0.135101\pi\)
\(108\) 0 0
\(109\) 133.745 1.22702 0.613509 0.789688i \(-0.289758\pi\)
0.613509 + 0.789688i \(0.289758\pi\)
\(110\) 0 0
\(111\) 75.0167 100.281i 0.675826 0.903429i
\(112\) 0 0
\(113\) 70.4990i 0.623885i −0.950101 0.311942i \(-0.899020\pi\)
0.950101 0.311942i \(-0.100980\pi\)
\(114\) 0 0
\(115\) −36.4087 −0.316598
\(116\) 0 0
\(117\) −16.6194 56.4605i −0.142046 0.482569i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 114.986 0.950300
\(122\) 0 0
\(123\) −171.038 127.948i −1.39055 1.04023i
\(124\) 0 0
\(125\) 42.6989i 0.341591i
\(126\) 0 0
\(127\) 142.993 1.12593 0.562965 0.826481i \(-0.309661\pi\)
0.562965 + 0.826481i \(0.309661\pi\)
\(128\) 0 0
\(129\) −107.363 + 143.521i −0.832273 + 1.11256i
\(130\) 0 0
\(131\) 116.500i 0.889311i −0.895702 0.444656i \(-0.853326\pi\)
0.895702 0.444656i \(-0.146674\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 8.15868 21.9416i 0.0604346 0.162530i
\(136\) 0 0
\(137\) 57.4723i 0.419506i −0.977754 0.209753i \(-0.932734\pi\)
0.977754 0.209753i \(-0.0672659\pi\)
\(138\) 0 0
\(139\) −252.242 −1.81469 −0.907346 0.420385i \(-0.861895\pi\)
−0.907346 + 0.420385i \(0.861895\pi\)
\(140\) 0 0
\(141\) −167.655 125.417i −1.18904 0.889486i
\(142\) 0 0
\(143\) 16.0368i 0.112145i
\(144\) 0 0
\(145\) 19.7990 0.136545
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 129.197i 0.867097i 0.901130 + 0.433548i \(0.142739\pi\)
−0.901130 + 0.433548i \(0.857261\pi\)
\(150\) 0 0
\(151\) 54.5103 0.360995 0.180498 0.983575i \(-0.442229\pi\)
0.180498 + 0.983575i \(0.442229\pi\)
\(152\) 0 0
\(153\) 161.869 47.6469i 1.05797 0.311418i
\(154\) 0 0
\(155\) 7.35685i 0.0474636i
\(156\) 0 0
\(157\) 170.588 1.08655 0.543275 0.839555i \(-0.317184\pi\)
0.543275 + 0.839555i \(0.317184\pi\)
\(158\) 0 0
\(159\) −149.843 112.093i −0.942411 0.704987i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 182.241 1.11805 0.559023 0.829152i \(-0.311176\pi\)
0.559023 + 0.829152i \(0.311176\pi\)
\(164\) 0 0
\(165\) 3.82078 5.10753i 0.0231562 0.0309547i
\(166\) 0 0
\(167\) 305.607i 1.82998i −0.403471 0.914992i \(-0.632197\pi\)
0.403471 0.914992i \(-0.367803\pi\)
\(168\) 0 0
\(169\) −126.235 −0.746950
\(170\) 0 0
\(171\) −34.5897 117.510i −0.202279 0.687195i
\(172\) 0 0
\(173\) 102.302i 0.591339i −0.955290 0.295670i \(-0.904457\pi\)
0.955290 0.295670i \(-0.0955427\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 137.197 + 102.632i 0.775122 + 0.579844i
\(178\) 0 0
\(179\) 190.809i 1.06597i −0.846124 0.532986i \(-0.821070\pi\)
0.846124 0.532986i \(-0.178930\pi\)
\(180\) 0 0
\(181\) −227.859 −1.25889 −0.629445 0.777045i \(-0.716718\pi\)
−0.629445 + 0.777045i \(0.716718\pi\)
\(182\) 0 0
\(183\) −106.414 + 142.252i −0.581496 + 0.777331i
\(184\) 0 0
\(185\) 36.1934i 0.195640i
\(186\) 0 0
\(187\) 45.9765 0.245863
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 248.742i 1.30231i 0.758943 + 0.651157i \(0.225716\pi\)
−0.758943 + 0.651157i \(0.774284\pi\)
\(192\) 0 0
\(193\) 27.7586 0.143827 0.0719134 0.997411i \(-0.477089\pi\)
0.0719134 + 0.997411i \(0.477089\pi\)
\(194\) 0 0
\(195\) 13.6203 + 10.1889i 0.0698476 + 0.0522507i
\(196\) 0 0
\(197\) 351.121i 1.78234i −0.453671 0.891169i \(-0.649886\pi\)
0.453671 0.891169i \(-0.350114\pi\)
\(198\) 0 0
\(199\) −129.386 −0.650181 −0.325090 0.945683i \(-0.605395\pi\)
−0.325090 + 0.945683i \(0.605395\pi\)
\(200\) 0 0
\(201\) −118.604 + 158.547i −0.590069 + 0.788791i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 61.7312 0.301128
\(206\) 0 0
\(207\) −362.559 + 106.721i −1.75149 + 0.515560i
\(208\) 0 0
\(209\) 33.3771i 0.159699i
\(210\) 0 0
\(211\) 143.503 0.680111 0.340056 0.940405i \(-0.389554\pi\)
0.340056 + 0.940405i \(0.389554\pi\)
\(212\) 0 0
\(213\) −129.225 96.6690i −0.606690 0.453845i
\(214\) 0 0
\(215\) 51.7996i 0.240928i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −171.535 + 229.303i −0.783263 + 1.04705i
\(220\) 0 0
\(221\) 122.606i 0.554777i
\(222\) 0 0
\(223\) −303.685 −1.36182 −0.680909 0.732368i \(-0.738415\pi\)
−0.680909 + 0.732368i \(0.738415\pi\)
\(224\) 0 0
\(225\) −61.6241 209.353i −0.273885 0.930459i
\(226\) 0 0
\(227\) 357.518i 1.57497i −0.616335 0.787484i \(-0.711383\pi\)
0.616335 0.787484i \(-0.288617\pi\)
\(228\) 0 0
\(229\) −207.699 −0.906984 −0.453492 0.891260i \(-0.649822\pi\)
−0.453492 + 0.891260i \(0.649822\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 346.216i 1.48591i 0.669343 + 0.742953i \(0.266576\pi\)
−0.669343 + 0.742953i \(0.733424\pi\)
\(234\) 0 0
\(235\) 60.5103 0.257491
\(236\) 0 0
\(237\) 73.6658 98.4748i 0.310826 0.415506i
\(238\) 0 0
\(239\) 377.331i 1.57879i 0.613886 + 0.789395i \(0.289605\pi\)
−0.613886 + 0.789395i \(0.710395\pi\)
\(240\) 0 0
\(241\) 356.353 1.47864 0.739321 0.673353i \(-0.235147\pi\)
0.739321 + 0.673353i \(0.235147\pi\)
\(242\) 0 0
\(243\) 16.9292 242.410i 0.0696675 0.997570i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 89.0068 0.360352
\(248\) 0 0
\(249\) −55.4518 41.4817i −0.222698 0.166593i
\(250\) 0 0
\(251\) 98.7289i 0.393342i 0.980470 + 0.196671i \(0.0630131\pi\)
−0.980470 + 0.196671i \(0.936987\pi\)
\(252\) 0 0
\(253\) −102.979 −0.407033
\(254\) 0 0
\(255\) −29.2109 + 39.0485i −0.114553 + 0.153131i
\(256\) 0 0
\(257\) 323.274i 1.25787i 0.777456 + 0.628937i \(0.216510\pi\)
−0.777456 + 0.628937i \(0.783490\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 197.159 58.0346i 0.755397 0.222355i
\(262\) 0 0
\(263\) 71.8815i 0.273314i 0.990618 + 0.136657i \(0.0436358\pi\)
−0.990618 + 0.136657i \(0.956364\pi\)
\(264\) 0 0
\(265\) 54.0816 0.204081
\(266\) 0 0
\(267\) 13.2794 + 9.93389i 0.0497356 + 0.0372056i
\(268\) 0 0
\(269\) 42.4837i 0.157932i 0.996877 + 0.0789659i \(0.0251618\pi\)
−0.996877 + 0.0789659i \(0.974838\pi\)
\(270\) 0 0
\(271\) 88.3835 0.326138 0.163069 0.986615i \(-0.447861\pi\)
0.163069 + 0.986615i \(0.447861\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 59.4637i 0.216232i
\(276\) 0 0
\(277\) 84.9795 0.306785 0.153393 0.988165i \(-0.450980\pi\)
0.153393 + 0.988165i \(0.450980\pi\)
\(278\) 0 0
\(279\) −21.5643 73.2597i −0.0772915 0.262579i
\(280\) 0 0
\(281\) 237.707i 0.845932i −0.906146 0.422966i \(-0.860989\pi\)
0.906146 0.422966i \(-0.139011\pi\)
\(282\) 0 0
\(283\) −184.721 −0.652723 −0.326362 0.945245i \(-0.605823\pi\)
−0.326362 + 0.945245i \(0.605823\pi\)
\(284\) 0 0
\(285\) 28.3476 + 21.2060i 0.0994654 + 0.0744069i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −62.5034 −0.216275
\(290\) 0 0
\(291\) 153.762 205.546i 0.528393 0.706343i
\(292\) 0 0
\(293\) 103.384i 0.352846i 0.984314 + 0.176423i \(0.0564527\pi\)
−0.984314 + 0.176423i \(0.943547\pi\)
\(294\) 0 0
\(295\) −49.5171 −0.167855
\(296\) 0 0
\(297\) 23.0762 62.0603i 0.0776977 0.208957i
\(298\) 0 0
\(299\) 274.616i 0.918449i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −213.993 160.081i −0.706248 0.528321i
\(304\) 0 0
\(305\) 51.3416i 0.168333i
\(306\) 0 0
\(307\) 138.422 0.450887 0.225443 0.974256i \(-0.427617\pi\)
0.225443 + 0.974256i \(0.427617\pi\)
\(308\) 0 0
\(309\) −62.9035 + 84.0880i −0.203571 + 0.272130i
\(310\) 0 0
\(311\) 231.265i 0.743619i 0.928309 + 0.371809i \(0.121263\pi\)
−0.928309 + 0.371809i \(0.878737\pi\)
\(312\) 0 0
\(313\) 312.522 0.998472 0.499236 0.866466i \(-0.333614\pi\)
0.499236 + 0.866466i \(0.333614\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 541.164i 1.70714i −0.520976 0.853572i \(-0.674432\pi\)
0.520976 0.853572i \(-0.325568\pi\)
\(318\) 0 0
\(319\) 56.0000 0.175549
\(320\) 0 0
\(321\) 211.698 + 158.365i 0.659496 + 0.493348i
\(322\) 0 0
\(323\) 255.177i 0.790022i
\(324\) 0 0
\(325\) 158.572 0.487915
\(326\) 0 0
\(327\) −240.343 + 321.285i −0.734995 + 0.982524i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −108.738 −0.328514 −0.164257 0.986418i \(-0.552523\pi\)
−0.164257 + 0.986418i \(0.552523\pi\)
\(332\) 0 0
\(333\) 106.090 + 360.414i 0.318587 + 1.08232i
\(334\) 0 0
\(335\) 57.2229i 0.170815i
\(336\) 0 0
\(337\) 327.717 0.972455 0.486228 0.873832i \(-0.338373\pi\)
0.486228 + 0.873832i \(0.338373\pi\)
\(338\) 0 0
\(339\) 169.354 + 126.689i 0.499571 + 0.373713i
\(340\) 0 0
\(341\) 20.8083i 0.0610215i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 65.4275 87.4620i 0.189645 0.253513i
\(346\) 0 0
\(347\) 306.832i 0.884241i −0.896956 0.442120i \(-0.854226\pi\)
0.896956 0.442120i \(-0.145774\pi\)
\(348\) 0 0
\(349\) 85.7161 0.245605 0.122802 0.992431i \(-0.460812\pi\)
0.122802 + 0.992431i \(0.460812\pi\)
\(350\) 0 0
\(351\) 165.497 + 61.5375i 0.471500 + 0.175321i
\(352\) 0 0
\(353\) 450.719i 1.27682i 0.769695 + 0.638412i \(0.220408\pi\)
−0.769695 + 0.638412i \(0.779592\pi\)
\(354\) 0 0
\(355\) 46.6400 0.131380
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 432.499i 1.20473i −0.798220 0.602366i \(-0.794225\pi\)
0.798220 0.602366i \(-0.205775\pi\)
\(360\) 0 0
\(361\) −175.752 −0.486847
\(362\) 0 0
\(363\) −206.634 + 276.223i −0.569239 + 0.760945i
\(364\) 0 0
\(365\) 82.7604i 0.226741i
\(366\) 0 0
\(367\) −595.716 −1.62320 −0.811602 0.584211i \(-0.801404\pi\)
−0.811602 + 0.584211i \(0.801404\pi\)
\(368\) 0 0
\(369\) 614.720 180.946i 1.66591 0.490368i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −428.952 −1.15001 −0.575003 0.818151i \(-0.694999\pi\)
−0.575003 + 0.818151i \(0.694999\pi\)
\(374\) 0 0
\(375\) 102.572 + 76.7312i 0.273527 + 0.204616i
\(376\) 0 0
\(377\) 149.336i 0.396116i
\(378\) 0 0
\(379\) 304.228 0.802712 0.401356 0.915922i \(-0.368539\pi\)
0.401356 + 0.915922i \(0.368539\pi\)
\(380\) 0 0
\(381\) −256.963 + 343.502i −0.674443 + 0.901580i
\(382\) 0 0
\(383\) 100.783i 0.263141i 0.991307 + 0.131570i \(0.0420019\pi\)
−0.991307 + 0.131570i \(0.957998\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −151.834 515.821i −0.392337 1.33287i
\(388\) 0 0
\(389\) 411.811i 1.05864i 0.848423 + 0.529320i \(0.177553\pi\)
−0.848423 + 0.529320i \(0.822447\pi\)
\(390\) 0 0
\(391\) 787.308 2.01357
\(392\) 0 0
\(393\) 279.859 + 209.353i 0.712109 + 0.532706i
\(394\) 0 0
\(395\) 35.5416i 0.0899788i
\(396\) 0 0
\(397\) −481.696 −1.21334 −0.606670 0.794954i \(-0.707495\pi\)
−0.606670 + 0.794954i \(0.707495\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 245.681i 0.612671i −0.951924 0.306335i \(-0.900897\pi\)
0.951924 0.306335i \(-0.0991029\pi\)
\(402\) 0 0
\(403\) 55.4897 0.137692
\(404\) 0 0
\(405\) 38.0473 + 59.0287i 0.0939441 + 0.145750i
\(406\) 0 0
\(407\) 102.370i 0.251524i
\(408\) 0 0
\(409\) 272.582 0.666461 0.333230 0.942845i \(-0.391861\pi\)
0.333230 + 0.942845i \(0.391861\pi\)
\(410\) 0 0
\(411\) 138.061 + 103.279i 0.335916 + 0.251288i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 20.0137 0.0482258
\(416\) 0 0
\(417\) 453.286 605.943i 1.08702 1.45310i
\(418\) 0 0
\(419\) 477.375i 1.13932i 0.821881 + 0.569660i \(0.192925\pi\)
−0.821881 + 0.569660i \(0.807075\pi\)
\(420\) 0 0
\(421\) −711.476 −1.68997 −0.844983 0.534793i \(-0.820390\pi\)
−0.844983 + 0.534793i \(0.820390\pi\)
\(422\) 0 0
\(423\) 602.562 177.367i 1.42450 0.419308i
\(424\) 0 0
\(425\) 454.617i 1.06969i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 38.5240 + 28.8185i 0.0897995 + 0.0671761i
\(430\) 0 0
\(431\) 581.166i 1.34841i 0.738543 + 0.674207i \(0.235514\pi\)
−0.738543 + 0.674207i \(0.764486\pi\)
\(432\) 0 0
\(433\) 358.137 0.827107 0.413554 0.910480i \(-0.364287\pi\)
0.413554 + 0.910480i \(0.364287\pi\)
\(434\) 0 0
\(435\) −35.5793 + 47.5617i −0.0817916 + 0.109337i
\(436\) 0 0
\(437\) 571.553i 1.30790i
\(438\) 0 0
\(439\) 620.830 1.41419 0.707096 0.707118i \(-0.250005\pi\)
0.707096 + 0.707118i \(0.250005\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 162.303i 0.366373i −0.983078 0.183186i \(-0.941359\pi\)
0.983078 0.183186i \(-0.0586412\pi\)
\(444\) 0 0
\(445\) −4.79281 −0.0107704
\(446\) 0 0
\(447\) −310.361 232.171i −0.694321 0.519399i
\(448\) 0 0
\(449\) 800.020i 1.78178i 0.454217 + 0.890891i \(0.349919\pi\)
−0.454217 + 0.890891i \(0.650081\pi\)
\(450\) 0 0
\(451\) 174.602 0.387144
\(452\) 0 0
\(453\) −97.9565 + 130.946i −0.216239 + 0.289064i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −682.738 −1.49396 −0.746978 0.664849i \(-0.768496\pi\)
−0.746978 + 0.664849i \(0.768496\pi\)
\(458\) 0 0
\(459\) −176.424 + 474.469i −0.384367 + 1.03370i
\(460\) 0 0
\(461\) 443.241i 0.961478i 0.876864 + 0.480739i \(0.159632\pi\)
−0.876864 + 0.480739i \(0.840368\pi\)
\(462\) 0 0
\(463\) 21.5034 0.0464437 0.0232218 0.999730i \(-0.492608\pi\)
0.0232218 + 0.999730i \(0.492608\pi\)
\(464\) 0 0
\(465\) 17.6728 + 13.2205i 0.0380061 + 0.0284311i
\(466\) 0 0
\(467\) 610.243i 1.30673i 0.757043 + 0.653365i \(0.226643\pi\)
−0.757043 + 0.653365i \(0.773357\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −306.552 + 409.792i −0.650853 + 0.870046i
\(472\) 0 0
\(473\) 146.511i 0.309749i
\(474\) 0 0
\(475\) 330.034 0.694808
\(476\) 0 0
\(477\) 538.545 158.523i 1.12903 0.332334i
\(478\) 0 0
\(479\) 126.793i 0.264704i −0.991203 0.132352i \(-0.957747\pi\)
0.991203 0.132352i \(-0.0422530\pi\)
\(480\) 0 0
\(481\) −272.992 −0.567550
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 74.1858i 0.152960i
\(486\) 0 0
\(487\) −240.497 −0.493833 −0.246916 0.969037i \(-0.579417\pi\)
−0.246916 + 0.969037i \(0.579417\pi\)
\(488\) 0 0
\(489\) −327.493 + 437.785i −0.669720 + 0.895266i
\(490\) 0 0
\(491\) 602.463i 1.22701i −0.789690 0.613506i \(-0.789759\pi\)
0.789690 0.613506i \(-0.210241\pi\)
\(492\) 0 0
\(493\) −428.136 −0.868430
\(494\) 0 0
\(495\) 5.40339 + 18.3567i 0.0109159 + 0.0370843i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 341.517 0.684403 0.342202 0.939627i \(-0.388827\pi\)
0.342202 + 0.939627i \(0.388827\pi\)
\(500\) 0 0
\(501\) 734.138 + 549.185i 1.46535 + 1.09618i
\(502\) 0 0
\(503\) 42.6989i 0.0848885i 0.999099 + 0.0424443i \(0.0135145\pi\)
−0.999099 + 0.0424443i \(0.986486\pi\)
\(504\) 0 0
\(505\) 77.2346 0.152940
\(506\) 0 0
\(507\) 226.847 303.244i 0.447430 0.598114i
\(508\) 0 0
\(509\) 777.024i 1.52657i −0.646062 0.763285i \(-0.723585\pi\)
0.646062 0.763285i \(-0.276415\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 344.445 + 128.077i 0.671433 + 0.249663i
\(514\) 0 0
\(515\) 30.3491i 0.0589303i
\(516\) 0 0
\(517\) 171.149 0.331042
\(518\) 0 0
\(519\) 245.752 + 183.839i 0.473510 + 0.354218i
\(520\) 0 0
\(521\) 105.659i 0.202801i 0.994846 + 0.101400i \(0.0323323\pi\)
−0.994846 + 0.101400i \(0.967668\pi\)
\(522\) 0 0
\(523\) −415.589 −0.794625 −0.397312 0.917683i \(-0.630057\pi\)
−0.397312 + 0.917683i \(0.630057\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 159.086i 0.301870i
\(528\) 0 0
\(529\) −1234.43 −2.33353
\(530\) 0 0
\(531\) −493.092 + 145.144i −0.928611 + 0.273341i
\(532\) 0 0
\(533\) 465.613i 0.873570i
\(534\) 0 0
\(535\) −76.4063 −0.142815
\(536\) 0 0
\(537\) 458.366 + 342.889i 0.853568 + 0.638527i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −283.503 −0.524036 −0.262018 0.965063i \(-0.584388\pi\)
−0.262018 + 0.965063i \(0.584388\pi\)
\(542\) 0 0
\(543\) 409.469 547.369i 0.754087 1.00805i
\(544\) 0 0
\(545\) 115.959i 0.212768i
\(546\) 0 0
\(547\) −398.269 −0.728097 −0.364048 0.931380i \(-0.618606\pi\)
−0.364048 + 0.931380i \(0.618606\pi\)
\(548\) 0 0
\(549\) −150.492 511.260i −0.274120 0.931257i
\(550\) 0 0
\(551\) 310.809i 0.564083i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −86.9446 65.0405i −0.156657 0.117190i
\(556\) 0 0
\(557\) 60.8463i 0.109239i 0.998507 + 0.0546196i \(0.0173946\pi\)
−0.998507 + 0.0546196i \(0.982605\pi\)
\(558\) 0 0
\(559\) 390.703 0.698932
\(560\) 0 0
\(561\) −82.6210 + 110.446i −0.147275 + 0.196873i
\(562\) 0 0
\(563\) 2.92691i 0.00519878i 0.999997 + 0.00259939i \(0.000827412\pi\)
−0.999997 + 0.00259939i \(0.999173\pi\)
\(564\) 0 0
\(565\) −61.1236 −0.108183
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 751.131i 1.32009i 0.751226 + 0.660045i \(0.229463\pi\)
−0.751226 + 0.660045i \(0.770537\pi\)
\(570\) 0 0
\(571\) 166.724 0.291987 0.145993 0.989286i \(-0.453362\pi\)
0.145993 + 0.989286i \(0.453362\pi\)
\(572\) 0 0
\(573\) −597.535 446.996i −1.04282 0.780098i
\(574\) 0 0
\(575\) 1018.26i 1.77090i
\(576\) 0 0
\(577\) −117.682 −0.203956 −0.101978 0.994787i \(-0.532517\pi\)
−0.101978 + 0.994787i \(0.532517\pi\)
\(578\) 0 0
\(579\) −49.8829 + 66.6823i −0.0861536 + 0.115168i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 152.966 0.262377
\(584\) 0 0
\(585\) −48.9521 + 14.4093i −0.0836787 + 0.0246312i
\(586\) 0 0
\(587\) 650.556i 1.10827i −0.832426 0.554137i \(-0.813049\pi\)
0.832426 0.554137i \(-0.186951\pi\)
\(588\) 0 0
\(589\) 115.490 0.196078
\(590\) 0 0
\(591\) 843.471 + 630.974i 1.42719 + 1.06764i
\(592\) 0 0
\(593\) 377.459i 0.636524i −0.948003 0.318262i \(-0.896901\pi\)
0.948003 0.318262i \(-0.103099\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 232.510 310.814i 0.389464 0.520627i
\(598\) 0 0
\(599\) 76.7861i 0.128190i −0.997944 0.0640952i \(-0.979584\pi\)
0.997944 0.0640952i \(-0.0204161\pi\)
\(600\) 0 0
\(601\) −436.612 −0.726476 −0.363238 0.931696i \(-0.618329\pi\)
−0.363238 + 0.931696i \(0.618329\pi\)
\(602\) 0 0
\(603\) −167.731 569.827i −0.278161 0.944986i
\(604\) 0 0
\(605\) 99.6947i 0.164785i
\(606\) 0 0
\(607\) −751.240 −1.23763 −0.618814 0.785537i \(-0.712387\pi\)
−0.618814 + 0.785537i \(0.712387\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 456.404i 0.746979i
\(612\) 0 0
\(613\) −188.752 −0.307915 −0.153957 0.988077i \(-0.549202\pi\)
−0.153957 + 0.988077i \(0.549202\pi\)
\(614\) 0 0
\(615\) −110.933 + 148.292i −0.180378 + 0.241126i
\(616\) 0 0
\(617\) 548.513i 0.889000i 0.895779 + 0.444500i \(0.146619\pi\)
−0.895779 + 0.444500i \(0.853381\pi\)
\(618\) 0 0
\(619\) −907.716 −1.46642 −0.733211 0.680001i \(-0.761979\pi\)
−0.733211 + 0.680001i \(0.761979\pi\)
\(620\) 0 0
\(621\) 395.160 1062.73i 0.636329 1.71132i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 569.187 0.910699
\(626\) 0 0
\(627\) 80.1792 + 59.9795i 0.127878 + 0.0956611i
\(628\) 0 0
\(629\) 782.650i 1.24428i
\(630\) 0 0
\(631\) −1032.40 −1.63613 −0.818067 0.575123i \(-0.804954\pi\)
−0.818067 + 0.575123i \(0.804954\pi\)
\(632\) 0 0
\(633\) −257.880 + 344.728i −0.407393 + 0.544593i
\(634\) 0 0
\(635\) 123.977i 0.195239i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 464.442 136.711i 0.726826 0.213945i
\(640\) 0 0
\(641\) 1183.64i 1.84655i −0.384141 0.923274i \(-0.625502\pi\)
0.384141 0.923274i \(-0.374498\pi\)
\(642\) 0 0
\(643\) 810.525 1.26054 0.630268 0.776378i \(-0.282945\pi\)
0.630268 + 0.776378i \(0.282945\pi\)
\(644\) 0 0
\(645\) 124.434 + 93.0853i 0.192921 + 0.144318i
\(646\) 0 0
\(647\) 122.895i 0.189945i 0.995480 + 0.0949727i \(0.0302764\pi\)
−0.995480 + 0.0949727i \(0.969724\pi\)
\(648\) 0 0
\(649\) −140.056 −0.215802
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 442.160i 0.677121i −0.940945 0.338560i \(-0.890060\pi\)
0.940945 0.338560i \(-0.109940\pi\)
\(654\) 0 0
\(655\) −101.007 −0.154209
\(656\) 0 0
\(657\) −242.586 824.129i −0.369234 1.25438i
\(658\) 0 0
\(659\) 219.158i 0.332562i −0.986078 0.166281i \(-0.946824\pi\)
0.986078 0.166281i \(-0.0531758\pi\)
\(660\) 0 0
\(661\) 135.984 0.205724 0.102862 0.994696i \(-0.467200\pi\)
0.102862 + 0.994696i \(0.467200\pi\)
\(662\) 0 0
\(663\) −294.527 220.326i −0.444234 0.332317i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 958.952 1.43771
\(668\) 0 0
\(669\) 545.731 729.521i 0.815742 1.09046i
\(670\) 0 0
\(671\) 145.216i 0.216417i
\(672\) 0 0
\(673\) −553.531 −0.822483 −0.411241 0.911527i \(-0.634905\pi\)
−0.411241 + 0.911527i \(0.634905\pi\)
\(674\) 0 0
\(675\) 613.654 + 228.179i 0.909117 + 0.338042i
\(676\) 0 0
\(677\) 676.881i 0.999824i −0.866076 0.499912i \(-0.833366\pi\)
0.866076 0.499912i \(-0.166634\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 858.838 + 642.469i 1.26114 + 0.943420i
\(682\) 0 0
\(683\) 257.021i 0.376311i −0.982139 0.188156i \(-0.939749\pi\)
0.982139 0.188156i \(-0.0602509\pi\)
\(684\) 0 0
\(685\) −49.8292 −0.0727434
\(686\) 0 0
\(687\) 373.241 498.941i 0.543292 0.726260i
\(688\) 0 0
\(689\) 407.915i 0.592039i
\(690\) 0 0
\(691\) 493.039 0.713515 0.356757 0.934197i \(-0.383882\pi\)
0.356757 + 0.934197i \(0.383882\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 218.697i 0.314672i
\(696\) 0 0
\(697\) −1334.88 −1.91518
\(698\) 0 0
\(699\) −831.690 622.160i −1.18983 0.890072i
\(700\) 0 0
\(701\) 891.207i 1.27134i −0.771962 0.635668i \(-0.780725\pi\)
0.771962 0.635668i \(-0.219275\pi\)
\(702\) 0 0
\(703\) −568.172 −0.808211
\(704\) 0 0
\(705\) −108.739 + 145.359i −0.154239 + 0.206183i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −972.241 −1.37129 −0.685643 0.727938i \(-0.740479\pi\)
−0.685643 + 0.727938i \(0.740479\pi\)
\(710\) 0 0
\(711\) 104.179 + 353.924i 0.146525 + 0.497783i
\(712\) 0 0
\(713\) 356.325i 0.499754i
\(714\) 0 0
\(715\) −13.9041 −0.0194463
\(716\) 0 0
\(717\) −906.433 678.074i −1.26420 0.945709i
\(718\) 0 0
\(719\) 708.530i 0.985438i −0.870189 0.492719i \(-0.836003\pi\)
0.870189 0.492719i \(-0.163997\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −640.376 + 856.040i −0.885720 + 1.18401i
\(724\) 0 0
\(725\) 553.730i 0.763766i
\(726\) 0 0
\(727\) −855.912 −1.17732 −0.588660 0.808381i \(-0.700344\pi\)
−0.588660 + 0.808381i \(0.700344\pi\)
\(728\) 0 0
\(729\) 551.900 + 476.285i 0.757065 + 0.653340i
\(730\) 0 0
\(731\) 1120.12i 1.53231i
\(732\) 0 0
\(733\) −1272.19 −1.73560 −0.867799 0.496916i \(-0.834466\pi\)
−0.867799 + 0.496916i \(0.834466\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 161.851i 0.219608i
\(738\) 0 0
\(739\) 84.2277 0.113975 0.0569876 0.998375i \(-0.481850\pi\)
0.0569876 + 0.998375i \(0.481850\pi\)
\(740\) 0 0
\(741\) −159.948 + 213.815i −0.215854 + 0.288549i
\(742\) 0 0
\(743\) 236.020i 0.317658i −0.987306 0.158829i \(-0.949228\pi\)
0.987306 0.158829i \(-0.0507718\pi\)
\(744\) 0 0
\(745\) 112.016 0.150357
\(746\) 0 0
\(747\) 199.297 58.6640i 0.266796 0.0785327i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 1068.39 1.42262 0.711310 0.702879i \(-0.248102\pi\)
0.711310 + 0.702879i \(0.248102\pi\)
\(752\) 0 0
\(753\) −237.169 177.419i −0.314966 0.235616i
\(754\) 0 0
\(755\) 47.2611i 0.0625975i
\(756\) 0 0
\(757\) 800.187 1.05705 0.528525 0.848918i \(-0.322745\pi\)
0.528525 + 0.848918i \(0.322745\pi\)
\(758\) 0 0
\(759\) 185.057 247.380i 0.243817 0.325929i
\(760\) 0 0
\(761\) 801.190i 1.05281i 0.850234 + 0.526406i \(0.176461\pi\)
−0.850234 + 0.526406i \(0.823539\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −41.3105 140.343i −0.0540007 0.183454i
\(766\) 0 0
\(767\) 373.487i 0.486946i
\(768\) 0 0
\(769\) −236.096 −0.307017 −0.153509 0.988147i \(-0.549057\pi\)
−0.153509 + 0.988147i \(0.549057\pi\)
\(770\) 0 0
\(771\) −776.576 580.932i −1.00723 0.753478i
\(772\) 0 0
\(773\) 1387.16i 1.79451i −0.441513 0.897255i \(-0.645558\pi\)
0.441513 0.897255i \(-0.354442\pi\)
\(774\) 0 0
\(775\) 205.754 0.265488
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 969.071i 1.24399i
\(780\) 0 0
\(781\) 131.918 0.168909
\(782\) 0 0
\(783\) −214.887 + 577.910i −0.274441 + 0.738071i
\(784\) 0 0
\(785\) 147.902i 0.188411i
\(786\) 0 0
\(787\) 374.225 0.475509 0.237754 0.971325i \(-0.423589\pi\)
0.237754 + 0.971325i \(0.423589\pi\)
\(788\) 0 0
\(789\) −172.676 129.173i −0.218854 0.163718i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 387.248 0.488333
\(794\) 0 0
\(795\) −97.1861 + 129.916i −0.122247 + 0.163417i
\(796\) 0 0
\(797\) 1150.47i 1.44350i −0.692155 0.721749i \(-0.743339\pi\)
0.692155 0.721749i \(-0.256661\pi\)
\(798\) 0 0
\(799\) −1308.48 −1.63765
\(800\) 0 0
\(801\) −47.7269 + 14.0486i −0.0595841 + 0.0175389i
\(802\) 0 0
\(803\) 234.082i 0.291509i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −102.055 76.3443i −0.126463 0.0946026i
\(808\) 0 0
\(809\) 983.620i 1.21585i −0.793996 0.607923i \(-0.792003\pi\)
0.793996 0.607923i \(-0.207997\pi\)
\(810\) 0 0
\(811\) 1159.05 1.42916 0.714578 0.699555i \(-0.246619\pi\)
0.714578 + 0.699555i \(0.246619\pi\)
\(812\) 0 0
\(813\) −158.828 + 212.317i −0.195360 + 0.261153i
\(814\) 0 0
\(815\) 158.006i 0.193872i
\(816\) 0 0
\(817\) 813.163 0.995304
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 563.548i 0.686416i 0.939259 + 0.343208i \(0.111514\pi\)
−0.939259 + 0.343208i \(0.888486\pi\)
\(822\) 0 0
\(823\) 429.503 0.521875 0.260938 0.965356i \(-0.415968\pi\)
0.260938 + 0.965356i \(0.415968\pi\)
\(824\) 0 0
\(825\) 142.845 + 106.858i 0.173146 + 0.129525i
\(826\) 0 0
\(827\) 911.451i 1.10212i 0.834467 + 0.551059i \(0.185776\pi\)
−0.834467 + 0.551059i \(0.814224\pi\)
\(828\) 0 0
\(829\) 59.9769 0.0723485 0.0361743 0.999345i \(-0.488483\pi\)
0.0361743 + 0.999345i \(0.488483\pi\)
\(830\) 0 0
\(831\) −152.710 + 204.140i −0.183767 + 0.245656i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −264.966 −0.317324
\(836\) 0 0
\(837\) 214.738 + 79.8473i 0.256557 + 0.0953970i
\(838\) 0 0
\(839\) 1124.25i 1.33999i 0.742364 + 0.669996i \(0.233704\pi\)
−0.742364 + 0.669996i \(0.766296\pi\)
\(840\) 0 0
\(841\) 319.524 0.379933
\(842\) 0 0
\(843\) 571.025 + 427.166i 0.677373 + 0.506721i
\(844\) 0 0
\(845\) 109.447i 0.129523i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 331.948 443.741i 0.390987 0.522663i
\(850\) 0 0
\(851\) 1753.00i 2.05993i
\(852\) 0 0
\(853\) −776.915 −0.910804 −0.455402 0.890286i \(-0.650504\pi\)
−0.455402 + 0.890286i \(0.650504\pi\)
\(854\) 0 0
\(855\) −101.883 + 29.9897i −0.119161 + 0.0350757i
\(856\) 0 0
\(857\) 1691.79i 1.97409i −0.160455 0.987043i \(-0.551296\pi\)
0.160455 0.987043i \(-0.448704\pi\)
\(858\) 0 0
\(859\) −671.951 −0.782248 −0.391124 0.920338i \(-0.627914\pi\)
−0.391124 + 0.920338i \(0.627914\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 674.501i 0.781577i 0.920480 + 0.390789i \(0.127798\pi\)
−0.920480 + 0.390789i \(0.872202\pi\)
\(864\) 0 0
\(865\) −88.6969 −0.102540
\(866\) 0 0
\(867\) 112.320 150.147i 0.129551 0.173180i
\(868\) 0 0
\(869\) 100.527i 0.115681i
\(870\) 0 0
\(871\) 431.609 0.495532
\(872\) 0 0
\(873\) 217.453 + 738.744i 0.249087 + 0.846212i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −824.159 −0.939748 −0.469874 0.882733i \(-0.655701\pi\)
−0.469874 + 0.882733i \(0.655701\pi\)
\(878\) 0 0
\(879\) −248.352 185.784i −0.282539 0.211358i
\(880\) 0 0
\(881\) 911.504i 1.03462i 0.855797 + 0.517312i \(0.173067\pi\)
−0.855797 + 0.517312i \(0.826933\pi\)
\(882\) 0 0
\(883\) 917.973 1.03961 0.519803 0.854286i \(-0.326005\pi\)
0.519803 + 0.854286i \(0.326005\pi\)
\(884\) 0 0
\(885\) 88.9837 118.951i 0.100547 0.134408i
\(886\) 0 0
\(887\) 95.8020i 0.108007i −0.998541 0.0540034i \(-0.982802\pi\)
0.998541 0.0540034i \(-0.0171982\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 107.614 + 166.958i 0.120779 + 0.187383i
\(892\) 0 0
\(893\) 949.905i 1.06372i
\(894\) 0 0
\(895\) −165.434 −0.184842
\(896\) 0 0
\(897\) 659.690 + 493.493i 0.735440 + 0.550159i
\(898\) 0 0
\(899\) 193.769i 0.215538i
\(900\) 0 0
\(901\) −1169.47 −1.29797
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 197.557i 0.218295i
\(906\) 0 0
\(907\) −26.4555 −0.0291681 −0.0145841 0.999894i \(-0.504642\pi\)
−0.0145841 + 0.999894i \(0.504642\pi\)
\(908\) 0 0
\(909\) 769.103 226.389i 0.846098 0.249053i
\(910\) 0 0
\(911\) 713.573i 0.783285i −0.920117 0.391643i \(-0.871907\pi\)
0.920117 0.391643i \(-0.128093\pi\)
\(912\) 0 0
\(913\) 56.6073 0.0620014
\(914\) 0 0
\(915\) 123.334 + 92.2622i 0.134791 + 0.100833i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −675.408 −0.734937 −0.367469 0.930036i \(-0.619776\pi\)
−0.367469 + 0.930036i \(0.619776\pi\)
\(920\) 0 0
\(921\) −248.749 + 332.521i −0.270085 + 0.361044i
\(922\) 0 0
\(923\) 351.786i 0.381134i
\(924\) 0 0
\(925\) −1012.24 −1.09432
\(926\) 0 0
\(927\) −88.9590 302.217i −0.0959644 0.326016i
\(928\) 0 0
\(929\) 1151.01i 1.23898i 0.785006 + 0.619488i \(0.212660\pi\)
−0.785006 + 0.619488i \(0.787340\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −555.552 415.590i −0.595447 0.445435i
\(934\) 0 0
\(935\) 39.8622i 0.0426334i
\(936\) 0 0
\(937\) −344.395 −0.367551 −0.183775 0.982968i \(-0.558832\pi\)
−0.183775 + 0.982968i \(0.558832\pi\)
\(938\) 0 0
\(939\) −561.610 + 750.748i −0.598094 + 0.799519i
\(940\) 0 0
\(941\) 667.780i 0.709649i 0.934933 + 0.354825i \(0.115459\pi\)
−0.934933 + 0.354825i \(0.884541\pi\)
\(942\) 0 0
\(943\) 2989.91 3.17064
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1436.97i 1.51739i −0.651444 0.758697i \(-0.725836\pi\)
0.651444 0.758697i \(-0.274164\pi\)
\(948\) 0 0
\(949\) 624.228 0.657774
\(950\) 0 0
\(951\) 1300.00 + 972.488i 1.36698 + 1.02259i
\(952\) 0 0
\(953\) 1005.39i 1.05497i 0.849564 + 0.527485i \(0.176865\pi\)
−0.849564 + 0.527485i \(0.823135\pi\)
\(954\) 0 0
\(955\) 215.663 0.225825
\(956\) 0 0
\(957\) −100.634 + 134.525i −0.105155 + 0.140569i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −889.000 −0.925078
\(962\) 0 0
\(963\) −760.855 + 223.961i −0.790088 + 0.232566i
\(964\) 0 0
\(965\) 24.0670i 0.0249399i
\(966\) 0 0
\(967\) 1695.39 1.75325 0.876626 0.481173i \(-0.159789\pi\)
0.876626 + 0.481173i \(0.159789\pi\)
\(968\) 0 0
\(969\) −612.993 458.561i −0.632604 0.473231i
\(970\) 0 0
\(971\) 268.430i 0.276447i 0.990401 + 0.138224i \(0.0441393\pi\)
−0.990401 + 0.138224i \(0.955861\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −284.959 + 380.927i −0.292266 + 0.390694i
\(976\) 0 0
\(977\) 1321.25i 1.35236i −0.736738 0.676179i \(-0.763635\pi\)
0.736738 0.676179i \(-0.236365\pi\)
\(978\) 0 0
\(979\) −13.5561 −0.0138469
\(980\) 0 0
\(981\) −339.897 1154.72i −0.346480 1.17708i
\(982\) 0 0
\(983\) 628.543i 0.639413i 0.947517 + 0.319707i \(0.103584\pi\)
−0.947517 + 0.319707i \(0.896416\pi\)
\(984\) 0 0
\(985\) −304.426 −0.309062
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2508.88i 2.53679i
\(990\) 0 0
\(991\) 908.911 0.917165 0.458583 0.888652i \(-0.348357\pi\)
0.458583 + 0.888652i \(0.348357\pi\)
\(992\) 0 0
\(993\) 195.405 261.213i 0.196783 0.263055i
\(994\) 0 0
\(995\) 112.179i 0.112743i
\(996\) 0 0
\(997\) −962.792 −0.965690 −0.482845 0.875706i \(-0.660396\pi\)
−0.482845 + 0.875706i \(0.660396\pi\)
\(998\) 0 0
\(999\) −1056.44 392.823i −1.05750 0.393216i
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 588.3.c.j.197.4 yes 8
3.2 odd 2 inner 588.3.c.j.197.3 8
7.2 even 3 588.3.p.i.557.3 16
7.3 odd 6 588.3.p.i.569.1 16
7.4 even 3 588.3.p.i.569.8 16
7.5 odd 6 588.3.p.i.557.6 16
7.6 odd 2 inner 588.3.c.j.197.5 yes 8
21.2 odd 6 588.3.p.i.557.8 16
21.5 even 6 588.3.p.i.557.1 16
21.11 odd 6 588.3.p.i.569.3 16
21.17 even 6 588.3.p.i.569.6 16
21.20 even 2 inner 588.3.c.j.197.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
588.3.c.j.197.3 8 3.2 odd 2 inner
588.3.c.j.197.4 yes 8 1.1 even 1 trivial
588.3.c.j.197.5 yes 8 7.6 odd 2 inner
588.3.c.j.197.6 yes 8 21.20 even 2 inner
588.3.p.i.557.1 16 21.5 even 6
588.3.p.i.557.3 16 7.2 even 3
588.3.p.i.557.6 16 7.5 odd 6
588.3.p.i.557.8 16 21.2 odd 6
588.3.p.i.569.1 16 7.3 odd 6
588.3.p.i.569.3 16 21.11 odd 6
588.3.p.i.569.6 16 21.17 even 6
588.3.p.i.569.8 16 7.4 even 3