Properties

Label 2736.3.o.r.721.16
Level $2736$
Weight $3$
Character 2736.721
Analytic conductor $74.551$
Analytic rank $0$
Dimension $20$
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] = [2736,3,Mod(721,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.721");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2736.o (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: \(74.5506003290\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{19} + 80 x^{18} - 152 x^{17} + 4326 x^{16} - 10096 x^{15} + 70116 x^{14} - 93436 x^{13} + 597327 x^{12} - 837564 x^{11} + 2838686 x^{10} - 2898276 x^{9} + \cdots + 36100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{37} \)
Twist minimal: no (minimal twist has level 456)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 721.16
Root \(0.855886 - 1.48244i\) of defining polynomial
Character \(\chi\) \(=\) 2736.721
Dual form 2736.3.o.r.721.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.78754 q^{5} -0.363991 q^{7} +O(q^{10})\) \(q+3.78754 q^{5} -0.363991 q^{7} -8.03560 q^{11} +17.0897i q^{13} -31.1440 q^{17} +(11.3197 - 15.2599i) q^{19} +34.4232 q^{23} -10.6546 q^{25} -5.21270i q^{29} +4.14913i q^{31} -1.37863 q^{35} +37.7395i q^{37} -36.6108i q^{41} -53.5899 q^{43} +23.4910 q^{47} -48.8675 q^{49} -79.9131i q^{53} -30.4351 q^{55} -31.2858i q^{59} +59.5399 q^{61} +64.7279i q^{65} -108.865i q^{67} -100.436i q^{71} -132.046 q^{73} +2.92489 q^{77} -14.6681i q^{79} +69.7261 q^{83} -117.959 q^{85} -150.974i q^{89} -6.22051i q^{91} +(42.8738 - 57.7973i) q^{95} -51.4768i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 16 q^{7} + 16 q^{11} - 32 q^{17} - 40 q^{19} + 64 q^{23} + 68 q^{25} - 208 q^{35} - 64 q^{43} + 48 q^{47} + 20 q^{49} + 336 q^{55} + 184 q^{61} + 104 q^{73} - 88 q^{77} + 224 q^{83} - 136 q^{85} - 320 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 0
\(4\) 0 0
\(5\) 3.78754 0.757507 0.378754 0.925498i \(-0.376353\pi\)
0.378754 + 0.925498i \(0.376353\pi\)
\(6\) 0 0
\(7\) −0.363991 −0.0519988 −0.0259994 0.999662i \(-0.508277\pi\)
−0.0259994 + 0.999662i \(0.508277\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −8.03560 −0.730509 −0.365255 0.930908i \(-0.619018\pi\)
−0.365255 + 0.930908i \(0.619018\pi\)
\(12\) 0 0
\(13\) 17.0897i 1.31459i 0.753632 + 0.657296i \(0.228300\pi\)
−0.753632 + 0.657296i \(0.771700\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −31.1440 −1.83200 −0.916000 0.401179i \(-0.868600\pi\)
−0.916000 + 0.401179i \(0.868600\pi\)
\(18\) 0 0
\(19\) 11.3197 15.2599i 0.595775 0.803152i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 34.4232 1.49666 0.748330 0.663327i \(-0.230856\pi\)
0.748330 + 0.663327i \(0.230856\pi\)
\(24\) 0 0
\(25\) −10.6546 −0.426183
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.21270i 0.179748i −0.995953 0.0898741i \(-0.971354\pi\)
0.995953 0.0898741i \(-0.0286465\pi\)
\(30\) 0 0
\(31\) 4.14913i 0.133843i 0.997758 + 0.0669215i \(0.0213177\pi\)
−0.997758 + 0.0669215i \(0.978682\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.37863 −0.0393894
\(36\) 0 0
\(37\) 37.7395i 1.01999i 0.860178 + 0.509994i \(0.170352\pi\)
−0.860178 + 0.509994i \(0.829648\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 36.6108i 0.892947i −0.894797 0.446473i \(-0.852680\pi\)
0.894797 0.446473i \(-0.147320\pi\)
\(42\) 0 0
\(43\) −53.5899 −1.24628 −0.623138 0.782112i \(-0.714142\pi\)
−0.623138 + 0.782112i \(0.714142\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 23.4910 0.499809 0.249905 0.968270i \(-0.419601\pi\)
0.249905 + 0.968270i \(0.419601\pi\)
\(48\) 0 0
\(49\) −48.8675 −0.997296
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 79.9131i 1.50779i −0.656992 0.753897i \(-0.728172\pi\)
0.656992 0.753897i \(-0.271828\pi\)
\(54\) 0 0
\(55\) −30.4351 −0.553366
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 31.2858i 0.530268i −0.964212 0.265134i \(-0.914584\pi\)
0.964212 0.265134i \(-0.0854161\pi\)
\(60\) 0 0
\(61\) 59.5399 0.976064 0.488032 0.872826i \(-0.337715\pi\)
0.488032 + 0.872826i \(0.337715\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 64.7279i 0.995813i
\(66\) 0 0
\(67\) 108.865i 1.62485i −0.583068 0.812423i \(-0.698148\pi\)
0.583068 0.812423i \(-0.301852\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 100.436i 1.41459i −0.706916 0.707297i \(-0.749914\pi\)
0.706916 0.707297i \(-0.250086\pi\)
\(72\) 0 0
\(73\) −132.046 −1.80885 −0.904427 0.426628i \(-0.859701\pi\)
−0.904427 + 0.426628i \(0.859701\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.92489 0.0379856
\(78\) 0 0
\(79\) 14.6681i 0.185673i −0.995681 0.0928364i \(-0.970407\pi\)
0.995681 0.0928364i \(-0.0295933\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 69.7261 0.840074 0.420037 0.907507i \(-0.362017\pi\)
0.420037 + 0.907507i \(0.362017\pi\)
\(84\) 0 0
\(85\) −117.959 −1.38775
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 150.974i 1.69634i −0.529723 0.848171i \(-0.677704\pi\)
0.529723 0.848171i \(-0.322296\pi\)
\(90\) 0 0
\(91\) 6.22051i 0.0683572i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 42.8738 57.7973i 0.451304 0.608393i
\(96\) 0 0
\(97\) 51.4768i 0.530689i −0.964154 0.265345i \(-0.914514\pi\)
0.964154 0.265345i \(-0.0854857\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 91.1672 0.902646 0.451323 0.892361i \(-0.350952\pi\)
0.451323 + 0.892361i \(0.350952\pi\)
\(102\) 0 0
\(103\) 60.0393i 0.582906i −0.956585 0.291453i \(-0.905861\pi\)
0.956585 0.291453i \(-0.0941386\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 31.8715i 0.297864i 0.988847 + 0.148932i \(0.0475836\pi\)
−0.988847 + 0.148932i \(0.952416\pi\)
\(108\) 0 0
\(109\) 119.535i 1.09665i 0.836266 + 0.548324i \(0.184734\pi\)
−0.836266 + 0.548324i \(0.815266\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 126.867i 1.12272i 0.827573 + 0.561358i \(0.189721\pi\)
−0.827573 + 0.561358i \(0.810279\pi\)
\(114\) 0 0
\(115\) 130.379 1.13373
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 11.3361 0.0952617
\(120\) 0 0
\(121\) −56.4291 −0.466356
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −135.043 −1.08034
\(126\) 0 0
\(127\) 212.297i 1.67163i 0.549012 + 0.835814i \(0.315004\pi\)
−0.549012 + 0.835814i \(0.684996\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −254.455 −1.94241 −0.971203 0.238254i \(-0.923425\pi\)
−0.971203 + 0.238254i \(0.923425\pi\)
\(132\) 0 0
\(133\) −4.12028 + 5.55446i −0.0309796 + 0.0417629i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 7.84460 0.0572599 0.0286299 0.999590i \(-0.490886\pi\)
0.0286299 + 0.999590i \(0.490886\pi\)
\(138\) 0 0
\(139\) 123.658 0.889625 0.444813 0.895624i \(-0.353270\pi\)
0.444813 + 0.895624i \(0.353270\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 137.326i 0.960322i
\(144\) 0 0
\(145\) 19.7433i 0.136161i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3.13822 −0.0210619 −0.0105309 0.999945i \(-0.503352\pi\)
−0.0105309 + 0.999945i \(0.503352\pi\)
\(150\) 0 0
\(151\) 9.31415i 0.0616831i −0.999524 0.0308415i \(-0.990181\pi\)
0.999524 0.0308415i \(-0.00981873\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 15.7150i 0.101387i
\(156\) 0 0
\(157\) −162.867 −1.03737 −0.518684 0.854966i \(-0.673578\pi\)
−0.518684 + 0.854966i \(0.673578\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −12.5297 −0.0778245
\(162\) 0 0
\(163\) −67.5305 −0.414297 −0.207149 0.978309i \(-0.566418\pi\)
−0.207149 + 0.978309i \(0.566418\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 41.7421i 0.249953i −0.992160 0.124976i \(-0.960114\pi\)
0.992160 0.124976i \(-0.0398855\pi\)
\(168\) 0 0
\(169\) −123.058 −0.728154
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 106.215i 0.613957i 0.951716 + 0.306978i \(0.0993179\pi\)
−0.951716 + 0.306978i \(0.900682\pi\)
\(174\) 0 0
\(175\) 3.87817 0.0221610
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 292.085i 1.63176i −0.578220 0.815881i \(-0.696252\pi\)
0.578220 0.815881i \(-0.303748\pi\)
\(180\) 0 0
\(181\) 55.2420i 0.305204i −0.988288 0.152602i \(-0.951235\pi\)
0.988288 0.152602i \(-0.0487653\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 142.940i 0.772647i
\(186\) 0 0
\(187\) 250.261 1.33829
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 183.468 0.960563 0.480282 0.877114i \(-0.340534\pi\)
0.480282 + 0.877114i \(0.340534\pi\)
\(192\) 0 0
\(193\) 122.921i 0.636896i 0.947940 + 0.318448i \(0.103162\pi\)
−0.947940 + 0.318448i \(0.896838\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −270.998 −1.37563 −0.687813 0.725888i \(-0.741429\pi\)
−0.687813 + 0.725888i \(0.741429\pi\)
\(198\) 0 0
\(199\) −170.801 −0.858295 −0.429148 0.903234i \(-0.641186\pi\)
−0.429148 + 0.903234i \(0.641186\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.89738i 0.00934669i
\(204\) 0 0
\(205\) 138.665i 0.676413i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −90.9608 + 122.622i −0.435219 + 0.586710i
\(210\) 0 0
\(211\) 150.250i 0.712083i −0.934470 0.356042i \(-0.884126\pi\)
0.934470 0.356042i \(-0.115874\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −202.973 −0.944063
\(216\) 0 0
\(217\) 1.51025i 0.00695967i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 532.242i 2.40833i
\(222\) 0 0
\(223\) 394.139i 1.76744i −0.468015 0.883721i \(-0.655031\pi\)
0.468015 0.883721i \(-0.344969\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 187.168i 0.824528i −0.911064 0.412264i \(-0.864738\pi\)
0.911064 0.412264i \(-0.135262\pi\)
\(228\) 0 0
\(229\) 386.050 1.68581 0.842905 0.538062i \(-0.180844\pi\)
0.842905 + 0.538062i \(0.180844\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −28.9836 −0.124393 −0.0621965 0.998064i \(-0.519811\pi\)
−0.0621965 + 0.998064i \(0.519811\pi\)
\(234\) 0 0
\(235\) 88.9731 0.378609
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 23.3544 0.0977174 0.0488587 0.998806i \(-0.484442\pi\)
0.0488587 + 0.998806i \(0.484442\pi\)
\(240\) 0 0
\(241\) 152.708i 0.633643i 0.948485 + 0.316821i \(0.102616\pi\)
−0.948485 + 0.316821i \(0.897384\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −185.087 −0.755459
\(246\) 0 0
\(247\) 260.787 + 193.451i 1.05582 + 0.783201i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −122.289 −0.487208 −0.243604 0.969875i \(-0.578330\pi\)
−0.243604 + 0.969875i \(0.578330\pi\)
\(252\) 0 0
\(253\) −276.611 −1.09332
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 184.338i 0.717268i −0.933478 0.358634i \(-0.883243\pi\)
0.933478 0.358634i \(-0.116757\pi\)
\(258\) 0 0
\(259\) 13.7369i 0.0530381i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −130.697 −0.496945 −0.248473 0.968639i \(-0.579929\pi\)
−0.248473 + 0.968639i \(0.579929\pi\)
\(264\) 0 0
\(265\) 302.674i 1.14217i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 91.2275i 0.339136i 0.985519 + 0.169568i \(0.0542372\pi\)
−0.985519 + 0.169568i \(0.945763\pi\)
\(270\) 0 0
\(271\) −211.416 −0.780134 −0.390067 0.920786i \(-0.627548\pi\)
−0.390067 + 0.920786i \(0.627548\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 85.6160 0.311331
\(276\) 0 0
\(277\) −205.340 −0.741298 −0.370649 0.928773i \(-0.620865\pi\)
−0.370649 + 0.928773i \(0.620865\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 488.604i 1.73881i −0.494104 0.869403i \(-0.664504\pi\)
0.494104 0.869403i \(-0.335496\pi\)
\(282\) 0 0
\(283\) −216.924 −0.766515 −0.383258 0.923641i \(-0.625198\pi\)
−0.383258 + 0.923641i \(0.625198\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 13.3260i 0.0464321i
\(288\) 0 0
\(289\) 680.948 2.35622
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 153.381i 0.523485i −0.965138 0.261743i \(-0.915703\pi\)
0.965138 0.261743i \(-0.0842972\pi\)
\(294\) 0 0
\(295\) 118.496i 0.401681i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 588.282i 1.96750i
\(300\) 0 0
\(301\) 19.5062 0.0648048
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 225.510 0.739376
\(306\) 0 0
\(307\) 165.972i 0.540624i −0.962773 0.270312i \(-0.912873\pi\)
0.962773 0.270312i \(-0.0871269\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −183.105 −0.588762 −0.294381 0.955688i \(-0.595114\pi\)
−0.294381 + 0.955688i \(0.595114\pi\)
\(312\) 0 0
\(313\) 243.119 0.776738 0.388369 0.921504i \(-0.373039\pi\)
0.388369 + 0.921504i \(0.373039\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 381.617i 1.20384i 0.798556 + 0.601920i \(0.205597\pi\)
−0.798556 + 0.601920i \(0.794403\pi\)
\(318\) 0 0
\(319\) 41.8872i 0.131308i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −352.541 + 475.253i −1.09146 + 1.47137i
\(324\) 0 0
\(325\) 182.084i 0.560257i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −8.55053 −0.0259895
\(330\) 0 0
\(331\) 116.607i 0.352287i −0.984365 0.176143i \(-0.943638\pi\)
0.984365 0.176143i \(-0.0563622\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 412.329i 1.23083i
\(336\) 0 0
\(337\) 274.008i 0.813081i −0.913633 0.406541i \(-0.866735\pi\)
0.913633 0.406541i \(-0.133265\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 33.3408i 0.0977735i
\(342\) 0 0
\(343\) 35.6229 0.103857
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −531.768 −1.53247 −0.766236 0.642560i \(-0.777872\pi\)
−0.766236 + 0.642560i \(0.777872\pi\)
\(348\) 0 0
\(349\) 190.284 0.545227 0.272614 0.962124i \(-0.412112\pi\)
0.272614 + 0.962124i \(0.412112\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −534.492 −1.51414 −0.757071 0.653333i \(-0.773370\pi\)
−0.757071 + 0.653333i \(0.773370\pi\)
\(354\) 0 0
\(355\) 380.406i 1.07157i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 234.606 0.653500 0.326750 0.945111i \(-0.394046\pi\)
0.326750 + 0.945111i \(0.394046\pi\)
\(360\) 0 0
\(361\) −104.728 345.475i −0.290105 0.956995i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −500.130 −1.37022
\(366\) 0 0
\(367\) −386.010 −1.05180 −0.525899 0.850547i \(-0.676271\pi\)
−0.525899 + 0.850547i \(0.676271\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 29.0877i 0.0784035i
\(372\) 0 0
\(373\) 196.177i 0.525944i 0.964804 + 0.262972i \(0.0847026\pi\)
−0.964804 + 0.262972i \(0.915297\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 89.0835 0.236296
\(378\) 0 0
\(379\) 208.596i 0.550384i −0.961389 0.275192i \(-0.911259\pi\)
0.961389 0.275192i \(-0.0887415\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 481.127i 1.25621i 0.778130 + 0.628104i \(0.216169\pi\)
−0.778130 + 0.628104i \(0.783831\pi\)
\(384\) 0 0
\(385\) 11.0781 0.0287744
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −361.111 −0.928306 −0.464153 0.885755i \(-0.653641\pi\)
−0.464153 + 0.885755i \(0.653641\pi\)
\(390\) 0 0
\(391\) −1072.08 −2.74188
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 55.5561i 0.140648i
\(396\) 0 0
\(397\) 317.576 0.799941 0.399970 0.916528i \(-0.369020\pi\)
0.399970 + 0.916528i \(0.369020\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 115.524i 0.288089i −0.989571 0.144045i \(-0.953989\pi\)
0.989571 0.144045i \(-0.0460109\pi\)
\(402\) 0 0
\(403\) −70.9074 −0.175949
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 303.260i 0.745110i
\(408\) 0 0
\(409\) 363.245i 0.888130i 0.895995 + 0.444065i \(0.146464\pi\)
−0.895995 + 0.444065i \(0.853536\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 11.3878i 0.0275733i
\(414\) 0 0
\(415\) 264.090 0.636362
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −104.376 −0.249108 −0.124554 0.992213i \(-0.539750\pi\)
−0.124554 + 0.992213i \(0.539750\pi\)
\(420\) 0 0
\(421\) 720.332i 1.71100i −0.517800 0.855502i \(-0.673249\pi\)
0.517800 0.855502i \(-0.326751\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 331.826 0.780767
\(426\) 0 0
\(427\) −21.6720 −0.0507541
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 515.127i 1.19519i −0.801798 0.597595i \(-0.796123\pi\)
0.801798 0.597595i \(-0.203877\pi\)
\(432\) 0 0
\(433\) 4.37332i 0.0101000i −0.999987 0.00505002i \(-0.998393\pi\)
0.999987 0.00505002i \(-0.00160748\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 389.661 525.294i 0.891672 1.20204i
\(438\) 0 0
\(439\) 492.579i 1.12205i 0.827800 + 0.561024i \(0.189592\pi\)
−0.827800 + 0.561024i \(0.810408\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −314.485 −0.709898 −0.354949 0.934886i \(-0.615502\pi\)
−0.354949 + 0.934886i \(0.615502\pi\)
\(444\) 0 0
\(445\) 571.821i 1.28499i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 169.326i 0.377118i 0.982062 + 0.188559i \(0.0603817\pi\)
−0.982062 + 0.188559i \(0.939618\pi\)
\(450\) 0 0
\(451\) 294.190i 0.652306i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 23.5604i 0.0517811i
\(456\) 0 0
\(457\) −489.329 −1.07074 −0.535371 0.844617i \(-0.679828\pi\)
−0.535371 + 0.844617i \(0.679828\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 225.978 0.490191 0.245096 0.969499i \(-0.421181\pi\)
0.245096 + 0.969499i \(0.421181\pi\)
\(462\) 0 0
\(463\) −198.625 −0.428995 −0.214498 0.976725i \(-0.568811\pi\)
−0.214498 + 0.976725i \(0.568811\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 125.888 0.269568 0.134784 0.990875i \(-0.456966\pi\)
0.134784 + 0.990875i \(0.456966\pi\)
\(468\) 0 0
\(469\) 39.6258i 0.0844900i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 430.627 0.910416
\(474\) 0 0
\(475\) −120.607 + 162.588i −0.253909 + 0.342290i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 576.515 1.20358 0.601790 0.798654i \(-0.294454\pi\)
0.601790 + 0.798654i \(0.294454\pi\)
\(480\) 0 0
\(481\) −644.957 −1.34087
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 194.970i 0.402001i
\(486\) 0 0
\(487\) 546.813i 1.12282i 0.827538 + 0.561410i \(0.189741\pi\)
−0.827538 + 0.561410i \(0.810259\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −520.490 −1.06006 −0.530030 0.847979i \(-0.677819\pi\)
−0.530030 + 0.847979i \(0.677819\pi\)
\(492\) 0 0
\(493\) 162.344i 0.329299i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 36.5579i 0.0735572i
\(498\) 0 0
\(499\) 796.126 1.59544 0.797722 0.603026i \(-0.206038\pi\)
0.797722 + 0.603026i \(0.206038\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 285.969 0.568528 0.284264 0.958746i \(-0.408251\pi\)
0.284264 + 0.958746i \(0.408251\pi\)
\(504\) 0 0
\(505\) 345.299 0.683761
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 430.356i 0.845493i −0.906248 0.422746i \(-0.861066\pi\)
0.906248 0.422746i \(-0.138934\pi\)
\(510\) 0 0
\(511\) 48.0637 0.0940582
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 227.401i 0.441555i
\(516\) 0 0
\(517\) −188.765 −0.365115
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 467.790i 0.897869i 0.893565 + 0.448934i \(0.148196\pi\)
−0.893565 + 0.448934i \(0.851804\pi\)
\(522\) 0 0
\(523\) 637.179i 1.21831i 0.793049 + 0.609157i \(0.208492\pi\)
−0.793049 + 0.609157i \(0.791508\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 129.220i 0.245200i
\(528\) 0 0
\(529\) 655.956 1.23999
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 625.668 1.17386
\(534\) 0 0
\(535\) 120.714i 0.225634i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 392.680 0.728534
\(540\) 0 0
\(541\) 390.249 0.721348 0.360674 0.932692i \(-0.382547\pi\)
0.360674 + 0.932692i \(0.382547\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 452.742i 0.830719i
\(546\) 0 0
\(547\) 438.152i 0.801009i 0.916295 + 0.400504i \(0.131165\pi\)
−0.916295 + 0.400504i \(0.868835\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −79.5452 59.0063i −0.144365 0.107090i
\(552\) 0 0
\(553\) 5.33908i 0.00965475i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 708.962 1.27282 0.636411 0.771350i \(-0.280418\pi\)
0.636411 + 0.771350i \(0.280418\pi\)
\(558\) 0 0
\(559\) 915.835i 1.63835i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 752.460i 1.33652i −0.743929 0.668259i \(-0.767040\pi\)
0.743929 0.668259i \(-0.232960\pi\)
\(564\) 0 0
\(565\) 480.513i 0.850465i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 481.551i 0.846311i 0.906057 + 0.423155i \(0.139078\pi\)
−0.906057 + 0.423155i \(0.860922\pi\)
\(570\) 0 0
\(571\) 686.556 1.20237 0.601187 0.799108i \(-0.294695\pi\)
0.601187 + 0.799108i \(0.294695\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −366.764 −0.637851
\(576\) 0 0
\(577\) 780.175 1.35212 0.676062 0.736845i \(-0.263685\pi\)
0.676062 + 0.736845i \(0.263685\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −25.3797 −0.0436828
\(582\) 0 0
\(583\) 642.150i 1.10146i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 162.138 0.276215 0.138108 0.990417i \(-0.455898\pi\)
0.138108 + 0.990417i \(0.455898\pi\)
\(588\) 0 0
\(589\) 63.3152 + 46.9670i 0.107496 + 0.0797402i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −265.476 −0.447683 −0.223841 0.974626i \(-0.571860\pi\)
−0.223841 + 0.974626i \(0.571860\pi\)
\(594\) 0 0
\(595\) 42.9360 0.0721614
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 73.4493i 0.122620i 0.998119 + 0.0613099i \(0.0195278\pi\)
−0.998119 + 0.0613099i \(0.980472\pi\)
\(600\) 0 0
\(601\) 1056.53i 1.75795i 0.476869 + 0.878974i \(0.341771\pi\)
−0.476869 + 0.878974i \(0.658229\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −213.727 −0.353268
\(606\) 0 0
\(607\) 743.206i 1.22439i 0.790706 + 0.612196i \(0.209714\pi\)
−0.790706 + 0.612196i \(0.790286\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 401.455i 0.657045i
\(612\) 0 0
\(613\) −1037.93 −1.69319 −0.846597 0.532235i \(-0.821352\pi\)
−0.846597 + 0.532235i \(0.821352\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 709.227 1.14948 0.574738 0.818337i \(-0.305104\pi\)
0.574738 + 0.818337i \(0.305104\pi\)
\(618\) 0 0
\(619\) −551.078 −0.890272 −0.445136 0.895463i \(-0.646845\pi\)
−0.445136 + 0.895463i \(0.646845\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 54.9534i 0.0882077i
\(624\) 0 0
\(625\) −245.116 −0.392185
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1175.36i 1.86862i
\(630\) 0 0
\(631\) −63.8795 −0.101235 −0.0506177 0.998718i \(-0.516119\pi\)
−0.0506177 + 0.998718i \(0.516119\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 804.082i 1.26627i
\(636\) 0 0
\(637\) 835.131i 1.31104i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 579.947i 0.904753i 0.891827 + 0.452377i \(0.149424\pi\)
−0.891827 + 0.452377i \(0.850576\pi\)
\(642\) 0 0
\(643\) −948.558 −1.47521 −0.737603 0.675234i \(-0.764043\pi\)
−0.737603 + 0.675234i \(0.764043\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 244.438 0.377803 0.188901 0.981996i \(-0.439507\pi\)
0.188901 + 0.981996i \(0.439507\pi\)
\(648\) 0 0
\(649\) 251.400i 0.387365i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −247.358 −0.378803 −0.189401 0.981900i \(-0.560655\pi\)
−0.189401 + 0.981900i \(0.560655\pi\)
\(654\) 0 0
\(655\) −963.758 −1.47139
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 179.151i 0.271853i −0.990719 0.135926i \(-0.956599\pi\)
0.990719 0.135926i \(-0.0434011\pi\)
\(660\) 0 0
\(661\) 779.968i 1.17998i −0.807410 0.589991i \(-0.799131\pi\)
0.807410 0.589991i \(-0.200869\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −15.6057 + 21.0377i −0.0234672 + 0.0316357i
\(666\) 0 0
\(667\) 179.438i 0.269022i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −478.439 −0.713024
\(672\) 0 0
\(673\) 485.801i 0.721843i −0.932596 0.360922i \(-0.882462\pi\)
0.932596 0.360922i \(-0.117538\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 270.632i 0.399751i 0.979821 + 0.199876i \(0.0640538\pi\)
−0.979821 + 0.199876i \(0.935946\pi\)
\(678\) 0 0
\(679\) 18.7371i 0.0275952i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 418.415i 0.612614i 0.951933 + 0.306307i \(0.0990934\pi\)
−0.951933 + 0.306307i \(0.900907\pi\)
\(684\) 0 0
\(685\) 29.7117 0.0433748
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1365.69 1.98214
\(690\) 0 0
\(691\) −639.926 −0.926086 −0.463043 0.886336i \(-0.653242\pi\)
−0.463043 + 0.886336i \(0.653242\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 468.359 0.673898
\(696\) 0 0
\(697\) 1140.21i 1.63588i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1137.14 −1.62217 −0.811085 0.584928i \(-0.801123\pi\)
−0.811085 + 0.584928i \(0.801123\pi\)
\(702\) 0 0
\(703\) 575.901 + 427.201i 0.819204 + 0.607683i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −33.1841 −0.0469365
\(708\) 0 0
\(709\) −1297.36 −1.82984 −0.914921 0.403634i \(-0.867747\pi\)
−0.914921 + 0.403634i \(0.867747\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 142.826i 0.200317i
\(714\) 0 0
\(715\) 520.128i 0.727451i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 232.620 0.323532 0.161766 0.986829i \(-0.448281\pi\)
0.161766 + 0.986829i \(0.448281\pi\)
\(720\) 0 0
\(721\) 21.8538i 0.0303104i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 55.5391i 0.0766057i
\(726\) 0 0
\(727\) 1275.80 1.75489 0.877445 0.479678i \(-0.159247\pi\)
0.877445 + 0.479678i \(0.159247\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1669.00 2.28318
\(732\) 0 0
\(733\) 543.697 0.741742 0.370871 0.928684i \(-0.379059\pi\)
0.370871 + 0.928684i \(0.379059\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 874.794i 1.18697i
\(738\) 0 0
\(739\) 829.948 1.12307 0.561535 0.827453i \(-0.310211\pi\)
0.561535 + 0.827453i \(0.310211\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 988.314i 1.33017i 0.746769 + 0.665084i \(0.231604\pi\)
−0.746769 + 0.665084i \(0.768396\pi\)
\(744\) 0 0
\(745\) −11.8861 −0.0159545
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 11.6009i 0.0154886i
\(750\) 0 0
\(751\) 271.579i 0.361624i −0.983518 0.180812i \(-0.942127\pi\)
0.983518 0.180812i \(-0.0578725\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 35.2777i 0.0467254i
\(756\) 0 0
\(757\) −621.388 −0.820856 −0.410428 0.911893i \(-0.634621\pi\)
−0.410428 + 0.911893i \(0.634621\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 639.560 0.840420 0.420210 0.907427i \(-0.361956\pi\)
0.420210 + 0.907427i \(0.361956\pi\)
\(762\) 0 0
\(763\) 43.5096i 0.0570244i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 534.665 0.697086
\(768\) 0 0
\(769\) 839.975 1.09229 0.546147 0.837689i \(-0.316094\pi\)
0.546147 + 0.837689i \(0.316094\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 811.652i 1.05000i −0.851101 0.525001i \(-0.824065\pi\)
0.851101 0.525001i \(-0.175935\pi\)
\(774\) 0 0
\(775\) 44.2072i 0.0570416i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −558.677 414.424i −0.717171 0.531995i
\(780\) 0 0
\(781\) 807.066i 1.03337i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −616.864 −0.785814
\(786\) 0 0
\(787\) 463.393i 0.588810i −0.955681 0.294405i \(-0.904879\pi\)
0.955681 0.294405i \(-0.0951215\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 46.1785i 0.0583799i
\(792\) 0 0
\(793\) 1017.52i 1.28313i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 191.296i 0.240021i −0.992773 0.120010i \(-0.961707\pi\)
0.992773 0.120010i \(-0.0382928\pi\)
\(798\) 0 0
\(799\) −731.604 −0.915650
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1061.07 1.32139
\(804\) 0 0
\(805\) −47.4568 −0.0589526
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 65.3407 0.0807672 0.0403836 0.999184i \(-0.487142\pi\)
0.0403836 + 0.999184i \(0.487142\pi\)
\(810\) 0 0
\(811\) 469.404i 0.578796i 0.957209 + 0.289398i \(0.0934551\pi\)
−0.957209 + 0.289398i \(0.906545\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −255.774 −0.313833
\(816\) 0 0
\(817\) −606.622 + 817.775i −0.742500 + 1.00095i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −854.605 −1.04093 −0.520466 0.853883i \(-0.674242\pi\)
−0.520466 + 0.853883i \(0.674242\pi\)
\(822\) 0 0
\(823\) 24.7419 0.0300630 0.0150315 0.999887i \(-0.495215\pi\)
0.0150315 + 0.999887i \(0.495215\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1516.75i 1.83404i −0.398841 0.917020i \(-0.630587\pi\)
0.398841 0.917020i \(-0.369413\pi\)
\(828\) 0 0
\(829\) 922.491i 1.11278i 0.830923 + 0.556388i \(0.187813\pi\)
−0.830923 + 0.556388i \(0.812187\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1521.93 1.82705
\(834\) 0 0
\(835\) 158.100i 0.189341i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1051.69i 1.25351i 0.779218 + 0.626753i \(0.215617\pi\)
−0.779218 + 0.626753i \(0.784383\pi\)
\(840\) 0 0
\(841\) 813.828 0.967691
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −466.087 −0.551582
\(846\) 0 0
\(847\) 20.5397 0.0242499
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1299.11i 1.52657i
\(852\) 0 0
\(853\) 937.436 1.09899 0.549494 0.835498i \(-0.314821\pi\)
0.549494 + 0.835498i \(0.314821\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1508.90i 1.76067i 0.474348 + 0.880337i \(0.342684\pi\)
−0.474348 + 0.880337i \(0.657316\pi\)
\(858\) 0 0
\(859\) −1441.65 −1.67829 −0.839144 0.543909i \(-0.816944\pi\)
−0.839144 + 0.543909i \(0.816944\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1143.05i 1.32450i 0.749281 + 0.662252i \(0.230399\pi\)
−0.749281 + 0.662252i \(0.769601\pi\)
\(864\) 0 0
\(865\) 402.291i 0.465077i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 117.867i 0.135636i
\(870\) 0 0
\(871\) 1860.47 2.13601
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 49.1545 0.0561765
\(876\) 0 0
\(877\) 809.431i 0.922954i −0.887152 0.461477i \(-0.847320\pi\)
0.887152 0.461477i \(-0.152680\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 356.009 0.404096 0.202048 0.979376i \(-0.435240\pi\)
0.202048 + 0.979376i \(0.435240\pi\)
\(882\) 0 0
\(883\) 584.439 0.661879 0.330940 0.943652i \(-0.392634\pi\)
0.330940 + 0.943652i \(0.392634\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1056.68i 1.19129i 0.803247 + 0.595646i \(0.203104\pi\)
−0.803247 + 0.595646i \(0.796896\pi\)
\(888\) 0 0
\(889\) 77.2742i 0.0869226i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 265.912 358.470i 0.297774 0.401422i
\(894\) 0 0
\(895\) 1106.28i 1.23607i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 21.6282 0.0240580
\(900\) 0 0
\(901\) 2488.81i 2.76228i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 209.231i 0.231194i
\(906\) 0 0
\(907\) 1768.14i 1.94944i −0.223426 0.974721i \(-0.571724\pi\)
0.223426 0.974721i \(-0.428276\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 277.297i 0.304388i 0.988351 + 0.152194i \(0.0486338\pi\)
−0.988351 + 0.152194i \(0.951366\pi\)
\(912\) 0 0
\(913\) −560.291 −0.613682
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 92.6195 0.101003
\(918\) 0 0
\(919\) 1438.17 1.56493 0.782463 0.622697i \(-0.213963\pi\)
0.782463 + 0.622697i \(0.213963\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1716.43 1.85962
\(924\) 0 0
\(925\) 402.099i 0.434701i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −501.787 −0.540136 −0.270068 0.962841i \(-0.587046\pi\)
−0.270068 + 0.962841i \(0.587046\pi\)
\(930\) 0 0
\(931\) −553.167 + 745.712i −0.594164 + 0.800980i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 947.871 1.01377
\(936\) 0 0
\(937\) −516.414 −0.551136 −0.275568 0.961282i \(-0.588866\pi\)
−0.275568 + 0.961282i \(0.588866\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 14.8505i 0.0157816i −0.999969 0.00789079i \(-0.997488\pi\)
0.999969 0.00789079i \(-0.00251174\pi\)
\(942\) 0 0
\(943\) 1260.26i 1.33644i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −121.697 −0.128508 −0.0642541 0.997934i \(-0.520467\pi\)
−0.0642541 + 0.997934i \(0.520467\pi\)
\(948\) 0 0
\(949\) 2256.63i 2.37791i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 294.229i 0.308740i 0.988013 + 0.154370i \(0.0493347\pi\)
−0.988013 + 0.154370i \(0.950665\pi\)
\(954\) 0 0
\(955\) 694.890 0.727633
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −2.85537 −0.00297744
\(960\) 0 0
\(961\) 943.785 0.982086
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 465.567i 0.482453i
\(966\) 0 0
\(967\) 1443.55 1.49282 0.746408 0.665489i \(-0.231777\pi\)
0.746408 + 0.665489i \(0.231777\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 595.844i 0.613639i −0.951768 0.306820i \(-0.900735\pi\)
0.951768 0.306820i \(-0.0992649\pi\)
\(972\) 0 0
\(973\) −45.0104 −0.0462594
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1755.69i 1.79702i 0.438956 + 0.898509i \(0.355349\pi\)
−0.438956 + 0.898509i \(0.644651\pi\)
\(978\) 0 0
\(979\) 1213.17i 1.23919i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1591.20i 1.61872i −0.587312 0.809361i \(-0.699814\pi\)
0.587312 0.809361i \(-0.300186\pi\)
\(984\) 0 0
\(985\) −1026.42 −1.04205
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1844.73 −1.86525
\(990\) 0 0
\(991\) 781.379i 0.788475i 0.919009 + 0.394237i \(0.128991\pi\)
−0.919009 + 0.394237i \(0.871009\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −646.914 −0.650165
\(996\) 0 0
\(997\) −907.423 −0.910154 −0.455077 0.890452i \(-0.650388\pi\)
−0.455077 + 0.890452i \(0.650388\pi\)
\(998\) 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 2736.3.o.r.721.16 20
3.2 odd 2 912.3.o.e.721.13 20
4.3 odd 2 1368.3.o.c.721.16 20
12.11 even 2 456.3.o.a.265.3 20
19.18 odd 2 inner 2736.3.o.r.721.15 20
57.56 even 2 912.3.o.e.721.3 20
76.75 even 2 1368.3.o.c.721.15 20
228.227 odd 2 456.3.o.a.265.13 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
456.3.o.a.265.3 20 12.11 even 2
456.3.o.a.265.13 yes 20 228.227 odd 2
912.3.o.e.721.3 20 57.56 even 2
912.3.o.e.721.13 20 3.2 odd 2
1368.3.o.c.721.15 20 76.75 even 2
1368.3.o.c.721.16 20 4.3 odd 2
2736.3.o.r.721.15 20 19.18 odd 2 inner
2736.3.o.r.721.16 20 1.1 even 1 trivial