Properties

Label 1300.2.f.f.701.6
Level $1300$
Weight $2$
Character 1300.701
Analytic conductor $10.381$
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] = [1300,2,Mod(701,1300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1300, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1300.701");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.f (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: \(10.3805522628\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.796594176.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 5x^{6} - 2x^{5} + 63x^{4} - 64x^{3} + 46x^{2} - 16x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 701.6
Root \(2.62039 + 0.935532i\) of defining polynomial
Character \(\chi\) \(=\) 1300.701
Dual form 1300.2.f.f.701.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.646084 q^{3} +0.913701i q^{7} -2.58258 q^{9} +O(q^{10})\) \(q+0.646084 q^{3} +0.913701i q^{7} -2.58258 q^{9} +3.94748i q^{11} +(2.44949 + 2.64575i) q^{13} -6.19115 q^{17} -1.11905i q^{19} +0.590327i q^{21} -5.54506 q^{23} -3.60681 q^{27} -1.58258 q^{29} -9.60433i q^{31} +2.55040i q^{33} +7.84190i q^{37} +(1.58258 + 1.70938i) q^{39} +5.06653i q^{41} -6.83723 q^{43} +9.66930i q^{47} +6.16515 q^{49} -4.00000 q^{51} +1.29217 q^{53} -0.723000i q^{57} +9.01400i q^{59} -5.58258 q^{61} -2.35970i q^{63} +7.84190i q^{67} -3.58258 q^{69} -6.77590i q^{71} -7.84190i q^{73} -3.60681 q^{77} +7.16515 q^{79} +5.41742 q^{81} +16.5975i q^{83} -1.02248 q^{87} -11.3137i q^{89} +(-2.41742 + 2.23810i) q^{91} -6.20520i q^{93} +1.63670i q^{97} -10.1947i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{9} + 24 q^{29} - 24 q^{39} - 24 q^{49} - 32 q^{51} - 8 q^{61} + 8 q^{69} - 16 q^{79} + 80 q^{81} - 56 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(651\) \(677\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 0.646084 0.373017 0.186508 0.982453i \(-0.440283\pi\)
0.186508 + 0.982453i \(0.440283\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.913701i 0.345346i 0.984979 + 0.172673i \(0.0552404\pi\)
−0.984979 + 0.172673i \(0.944760\pi\)
\(8\) 0 0
\(9\) −2.58258 −0.860859
\(10\) 0 0
\(11\) 3.94748i 1.19021i 0.803648 + 0.595105i \(0.202889\pi\)
−0.803648 + 0.595105i \(0.797111\pi\)
\(12\) 0 0
\(13\) 2.44949 + 2.64575i 0.679366 + 0.733799i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.19115 −1.50157 −0.750787 0.660545i \(-0.770325\pi\)
−0.750787 + 0.660545i \(0.770325\pi\)
\(18\) 0 0
\(19\) 1.11905i 0.256728i −0.991727 0.128364i \(-0.959027\pi\)
0.991727 0.128364i \(-0.0409725\pi\)
\(20\) 0 0
\(21\) 0.590327i 0.128820i
\(22\) 0 0
\(23\) −5.54506 −1.15623 −0.578113 0.815957i \(-0.696211\pi\)
−0.578113 + 0.815957i \(0.696211\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −3.60681 −0.694131
\(28\) 0 0
\(29\) −1.58258 −0.293877 −0.146938 0.989146i \(-0.546942\pi\)
−0.146938 + 0.989146i \(0.546942\pi\)
\(30\) 0 0
\(31\) 9.60433i 1.72499i −0.506067 0.862494i \(-0.668901\pi\)
0.506067 0.862494i \(-0.331099\pi\)
\(32\) 0 0
\(33\) 2.55040i 0.443968i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.84190i 1.28920i 0.764520 + 0.644601i \(0.222976\pi\)
−0.764520 + 0.644601i \(0.777024\pi\)
\(38\) 0 0
\(39\) 1.58258 + 1.70938i 0.253415 + 0.273719i
\(40\) 0 0
\(41\) 5.06653i 0.791259i 0.918410 + 0.395629i \(0.129473\pi\)
−0.918410 + 0.395629i \(0.870527\pi\)
\(42\) 0 0
\(43\) −6.83723 −1.04267 −0.521334 0.853353i \(-0.674565\pi\)
−0.521334 + 0.853353i \(0.674565\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.66930i 1.41041i 0.709002 + 0.705207i \(0.249146\pi\)
−0.709002 + 0.705207i \(0.750854\pi\)
\(48\) 0 0
\(49\) 6.16515 0.880736
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) 1.29217 0.177493 0.0887464 0.996054i \(-0.471714\pi\)
0.0887464 + 0.996054i \(0.471714\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.723000i 0.0957637i
\(58\) 0 0
\(59\) 9.01400i 1.17352i 0.809760 + 0.586762i \(0.199597\pi\)
−0.809760 + 0.586762i \(0.800403\pi\)
\(60\) 0 0
\(61\) −5.58258 −0.714776 −0.357388 0.933956i \(-0.616333\pi\)
−0.357388 + 0.933956i \(0.616333\pi\)
\(62\) 0 0
\(63\) 2.35970i 0.297294i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.84190i 0.958041i 0.877804 + 0.479021i \(0.159008\pi\)
−0.877804 + 0.479021i \(0.840992\pi\)
\(68\) 0 0
\(69\) −3.58258 −0.431291
\(70\) 0 0
\(71\) 6.77590i 0.804152i −0.915606 0.402076i \(-0.868289\pi\)
0.915606 0.402076i \(-0.131711\pi\)
\(72\) 0 0
\(73\) 7.84190i 0.917825i −0.888481 0.458913i \(-0.848239\pi\)
0.888481 0.458913i \(-0.151761\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.60681 −0.411034
\(78\) 0 0
\(79\) 7.16515 0.806143 0.403071 0.915169i \(-0.367943\pi\)
0.403071 + 0.915169i \(0.367943\pi\)
\(80\) 0 0
\(81\) 5.41742 0.601936
\(82\) 0 0
\(83\) 16.5975i 1.82181i 0.412613 + 0.910907i \(0.364616\pi\)
−0.412613 + 0.910907i \(0.635384\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1.02248 −0.109621
\(88\) 0 0
\(89\) 11.3137i 1.19925i −0.800281 0.599625i \(-0.795316\pi\)
0.800281 0.599625i \(-0.204684\pi\)
\(90\) 0 0
\(91\) −2.41742 + 2.23810i −0.253415 + 0.234617i
\(92\) 0 0
\(93\) 6.20520i 0.643450i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.63670i 0.166182i 0.996542 + 0.0830909i \(0.0264792\pi\)
−0.996542 + 0.0830909i \(0.973521\pi\)
\(98\) 0 0
\(99\) 10.1947i 1.02460i
\(100\) 0 0
\(101\) 9.16515 0.911967 0.455983 0.889988i \(-0.349288\pi\)
0.455983 + 0.889988i \(0.349288\pi\)
\(102\) 0 0
\(103\) 1.93825 0.190982 0.0954908 0.995430i \(-0.469558\pi\)
0.0954908 + 0.995430i \(0.469558\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −8.12940 −0.785899 −0.392949 0.919560i \(-0.628545\pi\)
−0.392949 + 0.919560i \(0.628545\pi\)
\(108\) 0 0
\(109\) 8.48528i 0.812743i 0.913708 + 0.406371i \(0.133206\pi\)
−0.913708 + 0.406371i \(0.866794\pi\)
\(110\) 0 0
\(111\) 5.06653i 0.480893i
\(112\) 0 0
\(113\) −3.60681 −0.339300 −0.169650 0.985504i \(-0.554264\pi\)
−0.169650 + 0.985504i \(0.554264\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −6.32599 6.83285i −0.584838 0.631697i
\(118\) 0 0
\(119\) 5.65685i 0.518563i
\(120\) 0 0
\(121\) −4.58258 −0.416598
\(122\) 0 0
\(123\) 3.27340i 0.295153i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −11.7362 −1.04142 −0.520710 0.853734i \(-0.674333\pi\)
−0.520710 + 0.853734i \(0.674333\pi\)
\(128\) 0 0
\(129\) −4.41742 −0.388933
\(130\) 0 0
\(131\) 15.1652 1.32499 0.662493 0.749068i \(-0.269499\pi\)
0.662493 + 0.749068i \(0.269499\pi\)
\(132\) 0 0
\(133\) 1.02248 0.0886600
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.3923i 0.887875i −0.896058 0.443937i \(-0.853581\pi\)
0.896058 0.443937i \(-0.146419\pi\)
\(138\) 0 0
\(139\) −11.1652 −0.947016 −0.473508 0.880790i \(-0.657012\pi\)
−0.473508 + 0.880790i \(0.657012\pi\)
\(140\) 0 0
\(141\) 6.24718i 0.526108i
\(142\) 0 0
\(143\) −10.4440 + 9.66930i −0.873375 + 0.808588i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 3.98320 0.328529
\(148\) 0 0
\(149\) 2.82843i 0.231714i −0.993266 0.115857i \(-0.963039\pi\)
0.993266 0.115857i \(-0.0369614\pi\)
\(150\) 0 0
\(151\) 3.35715i 0.273201i 0.990626 + 0.136600i \(0.0436177\pi\)
−0.990626 + 0.136600i \(0.956382\pi\)
\(152\) 0 0
\(153\) 15.9891 1.29264
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 22.4499 1.79170 0.895850 0.444356i \(-0.146567\pi\)
0.895850 + 0.444356i \(0.146567\pi\)
\(158\) 0 0
\(159\) 0.834849 0.0662078
\(160\) 0 0
\(161\) 5.06653i 0.399298i
\(162\) 0 0
\(163\) 6.01450i 0.471092i 0.971863 + 0.235546i \(0.0756879\pi\)
−0.971863 + 0.235546i \(0.924312\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 16.5975i 1.28435i 0.766557 + 0.642177i \(0.221969\pi\)
−0.766557 + 0.642177i \(0.778031\pi\)
\(168\) 0 0
\(169\) −1.00000 + 12.9615i −0.0769231 + 0.997037i
\(170\) 0 0
\(171\) 2.89003i 0.221006i
\(172\) 0 0
\(173\) 11.0901 0.843167 0.421583 0.906790i \(-0.361474\pi\)
0.421583 + 0.906790i \(0.361474\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.82380i 0.437744i
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −14.4174 −1.07164 −0.535819 0.844333i \(-0.679997\pi\)
−0.535819 + 0.844333i \(0.679997\pi\)
\(182\) 0 0
\(183\) −3.60681 −0.266623
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 24.4394i 1.78719i
\(188\) 0 0
\(189\) 3.29555i 0.239716i
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) 20.9753i 1.50984i −0.655819 0.754918i \(-0.727677\pi\)
0.655819 0.754918i \(-0.272323\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.94630i 0.637398i −0.947856 0.318699i \(-0.896754\pi\)
0.947856 0.318699i \(-0.103246\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 5.06653i 0.357365i
\(202\) 0 0
\(203\) 1.44600i 0.101489i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 14.3205 0.995347
\(208\) 0 0
\(209\) 4.41742 0.305560
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 0 0
\(213\) 4.37780i 0.299962i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 8.77548 0.595719
\(218\) 0 0
\(219\) 5.06653i 0.342364i
\(220\) 0 0
\(221\) −15.1652 16.3802i −1.02012 1.10185i
\(222\) 0 0
\(223\) 7.84190i 0.525133i −0.964914 0.262566i \(-0.915431\pi\)
0.964914 0.262566i \(-0.0845689\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 11.1153i 0.737749i −0.929479 0.368874i \(-0.879743\pi\)
0.929479 0.368874i \(-0.120257\pi\)
\(228\) 0 0
\(229\) 2.23810i 0.147898i −0.997262 0.0739489i \(-0.976440\pi\)
0.997262 0.0739489i \(-0.0235602\pi\)
\(230\) 0 0
\(231\) −2.33030 −0.153323
\(232\) 0 0
\(233\) −2.58434 −0.169305 −0.0846527 0.996411i \(-0.526978\pi\)
−0.0846527 + 0.996411i \(0.526978\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 4.62929 0.300705
\(238\) 0 0
\(239\) 10.7850i 0.697623i 0.937193 + 0.348811i \(0.113415\pi\)
−0.937193 + 0.348811i \(0.886585\pi\)
\(240\) 0 0
\(241\) 27.6939i 1.78392i −0.452111 0.891962i \(-0.649329\pi\)
0.452111 0.891962i \(-0.350671\pi\)
\(242\) 0 0
\(243\) 14.3205 0.918663
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2.96073 2.74110i 0.188387 0.174412i
\(248\) 0 0
\(249\) 10.7234i 0.679567i
\(250\) 0 0
\(251\) 3.16515 0.199783 0.0998913 0.994998i \(-0.468150\pi\)
0.0998913 + 0.994998i \(0.468150\pi\)
\(252\) 0 0
\(253\) 21.8890i 1.37615i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 22.1803 1.38357 0.691783 0.722105i \(-0.256825\pi\)
0.691783 + 0.722105i \(0.256825\pi\)
\(258\) 0 0
\(259\) −7.16515 −0.445221
\(260\) 0 0
\(261\) 4.08712 0.252986
\(262\) 0 0
\(263\) −20.2420 −1.24818 −0.624088 0.781354i \(-0.714529\pi\)
−0.624088 + 0.781354i \(0.714529\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 7.30960i 0.447341i
\(268\) 0 0
\(269\) 9.16515 0.558809 0.279405 0.960173i \(-0.409863\pi\)
0.279405 + 0.960173i \(0.409863\pi\)
\(270\) 0 0
\(271\) 15.8515i 0.962911i 0.876471 + 0.481455i \(0.159892\pi\)
−0.876471 + 0.481455i \(0.840108\pi\)
\(272\) 0 0
\(273\) −1.56186 + 1.44600i −0.0945280 + 0.0875159i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5.92146 0.355786 0.177893 0.984050i \(-0.443072\pi\)
0.177893 + 0.984050i \(0.443072\pi\)
\(278\) 0 0
\(279\) 24.8039i 1.48497i
\(280\) 0 0
\(281\) 26.5133i 1.58165i 0.612042 + 0.790825i \(0.290348\pi\)
−0.612042 + 0.790825i \(0.709652\pi\)
\(282\) 0 0
\(283\) −6.83723 −0.406431 −0.203216 0.979134i \(-0.565139\pi\)
−0.203216 + 0.979134i \(0.565139\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −4.62929 −0.273258
\(288\) 0 0
\(289\) 21.3303 1.25472
\(290\) 0 0
\(291\) 1.05745i 0.0619886i
\(292\) 0 0
\(293\) 9.66930i 0.564887i 0.959284 + 0.282443i \(0.0911449\pi\)
−0.959284 + 0.282443i \(0.908855\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 14.2378i 0.826161i
\(298\) 0 0
\(299\) −13.5826 14.6709i −0.785501 0.848438i
\(300\) 0 0
\(301\) 6.24718i 0.360082i
\(302\) 0 0
\(303\) 5.92146 0.340179
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 18.4249i 1.05157i −0.850619 0.525783i \(-0.823773\pi\)
0.850619 0.525783i \(-0.176227\pi\)
\(308\) 0 0
\(309\) 1.25227 0.0712393
\(310\) 0 0
\(311\) −30.3303 −1.71987 −0.859937 0.510400i \(-0.829497\pi\)
−0.859937 + 0.510400i \(0.829497\pi\)
\(312\) 0 0
\(313\) 18.5734 1.04983 0.524916 0.851154i \(-0.324097\pi\)
0.524916 + 0.851154i \(0.324097\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4.18710i 0.235171i 0.993063 + 0.117586i \(0.0375154\pi\)
−0.993063 + 0.117586i \(0.962485\pi\)
\(318\) 0 0
\(319\) 6.24718i 0.349775i
\(320\) 0 0
\(321\) −5.25227 −0.293153
\(322\) 0 0
\(323\) 6.92820i 0.385496i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 5.48220i 0.303167i
\(328\) 0 0
\(329\) −8.83485 −0.487081
\(330\) 0 0
\(331\) 5.12813i 0.281868i 0.990019 + 0.140934i \(0.0450105\pi\)
−0.990019 + 0.140934i \(0.954990\pi\)
\(332\) 0 0
\(333\) 20.2523i 1.10982i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.02248 0.0556978 0.0278489 0.999612i \(-0.491134\pi\)
0.0278489 + 0.999612i \(0.491134\pi\)
\(338\) 0 0
\(339\) −2.33030 −0.126565
\(340\) 0 0
\(341\) 37.9129 2.05310
\(342\) 0 0
\(343\) 12.0290i 0.649505i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 24.1185 1.29475 0.647375 0.762172i \(-0.275867\pi\)
0.647375 + 0.762172i \(0.275867\pi\)
\(348\) 0 0
\(349\) 6.71430i 0.359408i 0.983721 + 0.179704i \(0.0575140\pi\)
−0.983721 + 0.179704i \(0.942486\pi\)
\(350\) 0 0
\(351\) −8.83485 9.54273i −0.471569 0.509353i
\(352\) 0 0
\(353\) 9.66930i 0.514645i 0.966326 + 0.257323i \(0.0828403\pi\)
−0.966326 + 0.257323i \(0.917160\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 3.65480i 0.193433i
\(358\) 0 0
\(359\) 2.29970i 0.121374i 0.998157 + 0.0606869i \(0.0193291\pi\)
−0.998157 + 0.0606869i \(0.980671\pi\)
\(360\) 0 0
\(361\) 17.7477 0.934091
\(362\) 0 0
\(363\) −2.96073 −0.155398
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.93825 −0.101176 −0.0505880 0.998720i \(-0.516110\pi\)
−0.0505880 + 0.998720i \(0.516110\pi\)
\(368\) 0 0
\(369\) 13.0847i 0.681162i
\(370\) 0 0
\(371\) 1.18065i 0.0612965i
\(372\) 0 0
\(373\) 14.6969 0.760979 0.380489 0.924785i \(-0.375756\pi\)
0.380489 + 0.924785i \(0.375756\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −3.87650 4.18710i −0.199650 0.215647i
\(378\) 0 0
\(379\) 31.0511i 1.59499i −0.603327 0.797494i \(-0.706159\pi\)
0.603327 0.797494i \(-0.293841\pi\)
\(380\) 0 0
\(381\) −7.58258 −0.388467
\(382\) 0 0
\(383\) 29.0079i 1.48224i 0.671375 + 0.741118i \(0.265704\pi\)
−0.671375 + 0.741118i \(0.734296\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 17.6577 0.897590
\(388\) 0 0
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 34.3303 1.73616
\(392\) 0 0
\(393\) 9.79796 0.494242
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 19.8709i 0.997292i −0.866806 0.498646i \(-0.833831\pi\)
0.866806 0.498646i \(-0.166169\pi\)
\(398\) 0 0
\(399\) 0.660606 0.0330716
\(400\) 0 0
\(401\) 22.6274i 1.12996i −0.825105 0.564980i \(-0.808884\pi\)
0.825105 0.564980i \(-0.191116\pi\)
\(402\) 0 0
\(403\) 25.4107 23.5257i 1.26580 1.17190i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −30.9557 −1.53442
\(408\) 0 0
\(409\) 32.1701i 1.59071i −0.606143 0.795356i \(-0.707284\pi\)
0.606143 0.795356i \(-0.292716\pi\)
\(410\) 0 0
\(411\) 6.71430i 0.331192i
\(412\) 0 0
\(413\) −8.23610 −0.405272
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −7.21362 −0.353253
\(418\) 0 0
\(419\) 21.4955 1.05012 0.525061 0.851065i \(-0.324043\pi\)
0.525061 + 0.851065i \(0.324043\pi\)
\(420\) 0 0
\(421\) 34.4082i 1.67696i 0.544936 + 0.838478i \(0.316554\pi\)
−0.544936 + 0.838478i \(0.683446\pi\)
\(422\) 0 0
\(423\) 24.9717i 1.21417i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 5.10080i 0.246845i
\(428\) 0 0
\(429\) −6.74773 + 6.24718i −0.325783 + 0.301617i
\(430\) 0 0
\(431\) 10.7850i 0.519494i −0.965677 0.259747i \(-0.916361\pi\)
0.965677 0.259747i \(-0.0836392\pi\)
\(432\) 0 0
\(433\) −9.79796 −0.470860 −0.235430 0.971891i \(-0.575650\pi\)
−0.235430 + 0.971891i \(0.575650\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 6.20520i 0.296835i
\(438\) 0 0
\(439\) −17.4955 −0.835012 −0.417506 0.908674i \(-0.637096\pi\)
−0.417506 + 0.908674i \(0.637096\pi\)
\(440\) 0 0
\(441\) −15.9220 −0.758189
\(442\) 0 0
\(443\) −6.56754 −0.312033 −0.156017 0.987754i \(-0.549865\pi\)
−0.156017 + 0.987754i \(0.549865\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 1.82740i 0.0864331i
\(448\) 0 0
\(449\) 3.88587i 0.183386i 0.995787 + 0.0916928i \(0.0292278\pi\)
−0.995787 + 0.0916928i \(0.970772\pi\)
\(450\) 0 0
\(451\) −20.0000 −0.941763
\(452\) 0 0
\(453\) 2.16900i 0.101909i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 12.2197i 0.571614i −0.958287 0.285807i \(-0.907738\pi\)
0.958287 0.285807i \(-0.0922616\pi\)
\(458\) 0 0
\(459\) 22.3303 1.04229
\(460\) 0 0
\(461\) 13.5518i 0.631171i 0.948897 + 0.315585i \(0.102201\pi\)
−0.948897 + 0.315585i \(0.897799\pi\)
\(462\) 0 0
\(463\) 33.7273i 1.56744i 0.621113 + 0.783721i \(0.286681\pi\)
−0.621113 + 0.783721i \(0.713319\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −5.27537 −0.244115 −0.122058 0.992523i \(-0.538949\pi\)
−0.122058 + 0.992523i \(0.538949\pi\)
\(468\) 0 0
\(469\) −7.16515 −0.330856
\(470\) 0 0
\(471\) 14.5045 0.668334
\(472\) 0 0
\(473\) 26.9898i 1.24099i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −3.33712 −0.152796
\(478\) 0 0
\(479\) 6.77590i 0.309599i 0.987946 + 0.154799i \(0.0494732\pi\)
−0.987946 + 0.154799i \(0.950527\pi\)
\(480\) 0 0
\(481\) −20.7477 + 19.2087i −0.946015 + 0.875840i
\(482\) 0 0
\(483\) 3.27340i 0.148945i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 14.7701i 0.669297i 0.942343 + 0.334649i \(0.108618\pi\)
−0.942343 + 0.334649i \(0.891382\pi\)
\(488\) 0 0
\(489\) 3.88587i 0.175725i
\(490\) 0 0
\(491\) −6.33030 −0.285683 −0.142841 0.989746i \(-0.545624\pi\)
−0.142841 + 0.989746i \(0.545624\pi\)
\(492\) 0 0
\(493\) 9.79796 0.441278
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.19115 0.277711
\(498\) 0 0
\(499\) 20.3277i 0.909993i 0.890493 + 0.454997i \(0.150360\pi\)
−0.890493 + 0.454997i \(0.849640\pi\)
\(500\) 0 0
\(501\) 10.7234i 0.479085i
\(502\) 0 0
\(503\) −42.6919 −1.90354 −0.951770 0.306813i \(-0.900737\pi\)
−0.951770 + 0.306813i \(0.900737\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −0.646084 + 8.37420i −0.0286936 + 0.371911i
\(508\) 0 0
\(509\) 24.8655i 1.10214i 0.834457 + 0.551072i \(0.185781\pi\)
−0.834457 + 0.551072i \(0.814219\pi\)
\(510\) 0 0
\(511\) 7.16515 0.316968
\(512\) 0 0
\(513\) 4.03620i 0.178203i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −38.1694 −1.67869
\(518\) 0 0
\(519\) 7.16515 0.314515
\(520\) 0 0
\(521\) 22.4174 0.982125 0.491063 0.871124i \(-0.336609\pi\)
0.491063 + 0.871124i \(0.336609\pi\)
\(522\) 0 0
\(523\) 35.2086 1.53957 0.769783 0.638306i \(-0.220364\pi\)
0.769783 + 0.638306i \(0.220364\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 59.4618i 2.59020i
\(528\) 0 0
\(529\) 7.74773 0.336858
\(530\) 0 0
\(531\) 23.2793i 1.01024i
\(532\) 0 0
\(533\) −13.4048 + 12.4104i −0.580625 + 0.537554i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 24.3368i 1.04826i
\(540\) 0 0
\(541\) 45.1316i 1.94036i −0.242385 0.970180i \(-0.577930\pi\)
0.242385 0.970180i \(-0.422070\pi\)
\(542\) 0 0
\(543\) −9.31486 −0.399739
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 26.4331 1.13020 0.565100 0.825022i \(-0.308838\pi\)
0.565100 + 0.825022i \(0.308838\pi\)
\(548\) 0 0
\(549\) 14.4174 0.615321
\(550\) 0 0
\(551\) 1.77098i 0.0754463i
\(552\) 0 0
\(553\) 6.54680i 0.278398i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 23.5257i 0.996816i −0.866942 0.498408i \(-0.833918\pi\)
0.866942 0.498408i \(-0.166082\pi\)
\(558\) 0 0
\(559\) −16.7477 18.0896i −0.708353 0.765109i
\(560\) 0 0
\(561\) 15.7899i 0.666650i
\(562\) 0 0
\(563\) −46.5684 −1.96263 −0.981313 0.192418i \(-0.938367\pi\)
−0.981313 + 0.192418i \(0.938367\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 4.94990i 0.207876i
\(568\) 0 0
\(569\) 10.4174 0.436721 0.218361 0.975868i \(-0.429929\pi\)
0.218361 + 0.975868i \(0.429929\pi\)
\(570\) 0 0
\(571\) −37.4955 −1.56914 −0.784568 0.620043i \(-0.787115\pi\)
−0.784568 + 0.620043i \(0.787115\pi\)
\(572\) 0 0
\(573\) −7.75301 −0.323886
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 46.5191i 1.93662i 0.249758 + 0.968308i \(0.419649\pi\)
−0.249758 + 0.968308i \(0.580351\pi\)
\(578\) 0 0
\(579\) 13.5518i 0.563194i
\(580\) 0 0
\(581\) −15.1652 −0.629156
\(582\) 0 0
\(583\) 5.10080i 0.211254i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 11.1153i 0.458778i 0.973335 + 0.229389i \(0.0736727\pi\)
−0.973335 + 0.229389i \(0.926327\pi\)
\(588\) 0 0
\(589\) −10.7477 −0.442852
\(590\) 0 0
\(591\) 5.78006i 0.237760i
\(592\) 0 0
\(593\) 15.8745i 0.651888i 0.945389 + 0.325944i \(0.105682\pi\)
−0.945389 + 0.325944i \(0.894318\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 2.58434 0.105770
\(598\) 0 0
\(599\) 42.3303 1.72957 0.864785 0.502143i \(-0.167455\pi\)
0.864785 + 0.502143i \(0.167455\pi\)
\(600\) 0 0
\(601\) 44.3303 1.80827 0.904135 0.427246i \(-0.140516\pi\)
0.904135 + 0.427246i \(0.140516\pi\)
\(602\) 0 0
\(603\) 20.2523i 0.824738i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −34.1862 −1.38757 −0.693787 0.720181i \(-0.744059\pi\)
−0.693787 + 0.720181i \(0.744059\pi\)
\(608\) 0 0
\(609\) 0.934237i 0.0378572i
\(610\) 0 0
\(611\) −25.5826 + 23.6849i −1.03496 + 0.958187i
\(612\) 0 0
\(613\) 24.6301i 0.994801i 0.867521 + 0.497400i \(0.165712\pi\)
−0.867521 + 0.497400i \(0.834288\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 28.2849i 1.13871i 0.822093 + 0.569354i \(0.192807\pi\)
−0.822093 + 0.569354i \(0.807193\pi\)
\(618\) 0 0
\(619\) 32.8221i 1.31923i 0.751603 + 0.659615i \(0.229281\pi\)
−0.751603 + 0.659615i \(0.770719\pi\)
\(620\) 0 0
\(621\) 20.0000 0.802572
\(622\) 0 0
\(623\) 10.3373 0.414157
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 2.85403 0.113979
\(628\) 0 0
\(629\) 48.5504i 1.93583i
\(630\) 0 0
\(631\) 24.3368i 0.968832i 0.874838 + 0.484416i \(0.160968\pi\)
−0.874838 + 0.484416i \(0.839032\pi\)
\(632\) 0 0
\(633\) 12.9217 0.513591
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 15.1015 + 16.3115i 0.598342 + 0.646283i
\(638\) 0 0
\(639\) 17.4993i 0.692261i
\(640\) 0 0
\(641\) −24.3303 −0.960989 −0.480495 0.876998i \(-0.659543\pi\)
−0.480495 + 0.876998i \(0.659543\pi\)
\(642\) 0 0
\(643\) 9.28790i 0.366279i −0.983087 0.183140i \(-0.941374\pi\)
0.983087 0.183140i \(-0.0586260\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −28.7478 −1.13019 −0.565096 0.825025i \(-0.691161\pi\)
−0.565096 + 0.825025i \(0.691161\pi\)
\(648\) 0 0
\(649\) −35.5826 −1.39674
\(650\) 0 0
\(651\) 5.66970 0.222213
\(652\) 0 0
\(653\) −15.2363 −0.596243 −0.298122 0.954528i \(-0.596360\pi\)
−0.298122 + 0.954528i \(0.596360\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 20.2523i 0.790118i
\(658\) 0 0
\(659\) −21.4955 −0.837344 −0.418672 0.908138i \(-0.637504\pi\)
−0.418672 + 0.908138i \(0.637504\pi\)
\(660\) 0 0
\(661\) 8.48528i 0.330039i 0.986290 + 0.165020i \(0.0527687\pi\)
−0.986290 + 0.165020i \(0.947231\pi\)
\(662\) 0 0
\(663\) −9.79796 10.5830i −0.380521 0.411010i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 8.77548 0.339788
\(668\) 0 0
\(669\) 5.06653i 0.195883i
\(670\) 0 0
\(671\) 22.0371i 0.850732i
\(672\) 0 0
\(673\) −1.02248 −0.0394136 −0.0197068 0.999806i \(-0.506273\pi\)
−0.0197068 + 0.999806i \(0.506273\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −28.6411 −1.10077 −0.550383 0.834912i \(-0.685518\pi\)
−0.550383 + 0.834912i \(0.685518\pi\)
\(678\) 0 0
\(679\) −1.49545 −0.0573903
\(680\) 0 0
\(681\) 7.18142i 0.275193i
\(682\) 0 0
\(683\) 11.1153i 0.425315i −0.977127 0.212658i \(-0.931788\pi\)
0.977127 0.212658i \(-0.0682119\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 1.44600i 0.0551683i
\(688\) 0 0
\(689\) 3.16515 + 3.41875i 0.120583 + 0.130244i
\(690\) 0 0
\(691\) 26.5749i 1.01096i 0.862839 + 0.505478i \(0.168684\pi\)
−0.862839 + 0.505478i \(0.831316\pi\)
\(692\) 0 0
\(693\) 9.31486 0.353842
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 31.3676i 1.18813i
\(698\) 0 0
\(699\) −1.66970 −0.0631537
\(700\) 0 0
\(701\) −33.1652 −1.25263 −0.626315 0.779570i \(-0.715438\pi\)
−0.626315 + 0.779570i \(0.715438\pi\)
\(702\) 0 0
\(703\) 8.77548 0.330974
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8.37420i 0.314944i
\(708\) 0 0
\(709\) 23.6849i 0.889504i 0.895654 + 0.444752i \(0.146708\pi\)
−0.895654 + 0.444752i \(0.853292\pi\)
\(710\) 0 0
\(711\) −18.5045 −0.693975
\(712\) 0 0
\(713\) 53.2566i 1.99448i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 6.96800i 0.260225i
\(718\) 0 0
\(719\) −21.4955 −0.801645 −0.400823 0.916156i \(-0.631276\pi\)
−0.400823 + 0.916156i \(0.631276\pi\)
\(720\) 0 0
\(721\) 1.77098i 0.0659548i
\(722\) 0 0
\(723\) 17.8926i 0.665433i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −26.4331 −0.980351 −0.490176 0.871624i \(-0.663067\pi\)
−0.490176 + 0.871624i \(0.663067\pi\)
\(728\) 0 0
\(729\) −7.00000 −0.259259
\(730\) 0 0
\(731\) 42.3303 1.56564
\(732\) 0 0
\(733\) 22.4213i 0.828150i −0.910243 0.414075i \(-0.864105\pi\)
0.910243 0.414075i \(-0.135895\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −30.9557 −1.14027
\(738\) 0 0
\(739\) 37.7654i 1.38922i 0.719385 + 0.694611i \(0.244424\pi\)
−0.719385 + 0.694611i \(0.755576\pi\)
\(740\) 0 0
\(741\) 1.91288 1.77098i 0.0702713 0.0650586i
\(742\) 0 0
\(743\) 29.0079i 1.06420i 0.846682 + 0.532099i \(0.178596\pi\)
−0.846682 + 0.532099i \(0.821404\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 42.8643i 1.56832i
\(748\) 0 0
\(749\) 7.42784i 0.271407i
\(750\) 0 0
\(751\) 23.1652 0.845308 0.422654 0.906291i \(-0.361098\pi\)
0.422654 + 0.906291i \(0.361098\pi\)
\(752\) 0 0
\(753\) 2.04495 0.0745222
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −41.0234 −1.49102 −0.745510 0.666494i \(-0.767794\pi\)
−0.745510 + 0.666494i \(0.767794\pi\)
\(758\) 0 0
\(759\) 14.1421i 0.513327i
\(760\) 0 0
\(761\) 11.3137i 0.410122i 0.978749 + 0.205061i \(0.0657392\pi\)
−0.978749 + 0.205061i \(0.934261\pi\)
\(762\) 0 0
\(763\) −7.75301 −0.280678
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −23.8488 + 22.0797i −0.861131 + 0.797252i
\(768\) 0 0
\(769\) 8.95240i 0.322832i −0.986886 0.161416i \(-0.948394\pi\)
0.986886 0.161416i \(-0.0516061\pi\)
\(770\) 0 0
\(771\) 14.3303 0.516093
\(772\) 0 0
\(773\) 9.66930i 0.347781i 0.984765 + 0.173890i \(0.0556338\pi\)
−0.984765 + 0.173890i \(0.944366\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −4.62929 −0.166075
\(778\) 0 0
\(779\) 5.66970 0.203138
\(780\) 0 0
\(781\) 26.7477 0.957109
\(782\) 0 0
\(783\) 5.70805 0.203989
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 18.4249i 0.656777i −0.944543 0.328389i \(-0.893494\pi\)
0.944543 0.328389i \(-0.106506\pi\)
\(788\) 0 0
\(789\) −13.0780 −0.465590
\(790\) 0 0
\(791\) 3.29555i 0.117176i
\(792\) 0 0
\(793\) −13.6745 14.7701i −0.485594 0.524502i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 25.0343 0.886760 0.443380 0.896334i \(-0.353779\pi\)
0.443380 + 0.896334i \(0.353779\pi\)
\(798\) 0 0
\(799\) 59.8641i 2.11784i
\(800\) 0 0
\(801\) 29.2185i 1.03239i
\(802\) 0 0
\(803\) 30.9557 1.09240
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 5.92146 0.208445
\(808\) 0 0
\(809\) −4.74773 −0.166921 −0.0834606 0.996511i \(-0.526597\pi\)
−0.0834606 + 0.996511i \(0.526597\pi\)
\(810\) 0 0
\(811\) 33.2892i 1.16894i −0.811415 0.584471i \(-0.801302\pi\)
0.811415 0.584471i \(-0.198698\pi\)
\(812\) 0 0
\(813\) 10.2414i 0.359182i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 7.65120i 0.267682i
\(818\) 0 0
\(819\) 6.24318 5.78006i 0.218154 0.201972i
\(820\) 0 0
\(821\) 19.7990i 0.690990i 0.938421 + 0.345495i \(0.112289\pi\)
−0.938421 + 0.345495i \(0.887711\pi\)
\(822\) 0 0
\(823\) 50.9280 1.77524 0.887620 0.460576i \(-0.152357\pi\)
0.887620 + 0.460576i \(0.152357\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 15.1515i 0.526870i 0.964677 + 0.263435i \(0.0848554\pi\)
−0.964677 + 0.263435i \(0.915145\pi\)
\(828\) 0 0
\(829\) 42.2432 1.46717 0.733583 0.679600i \(-0.237846\pi\)
0.733583 + 0.679600i \(0.237846\pi\)
\(830\) 0 0
\(831\) 3.82576 0.132714
\(832\) 0 0
\(833\) −38.1694 −1.32249
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 34.6410i 1.19737i
\(838\) 0 0
\(839\) 44.6029i 1.53986i −0.638126 0.769932i \(-0.720290\pi\)
0.638126 0.769932i \(-0.279710\pi\)
\(840\) 0 0
\(841\) −26.4955 −0.913636
\(842\) 0 0
\(843\) 17.1298i 0.589982i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 4.18710i 0.143871i
\(848\) 0 0
\(849\) −4.41742 −0.151606
\(850\) 0 0
\(851\) 43.4839i 1.49061i
\(852\) 0 0
\(853\) 33.7273i 1.15480i 0.816461 + 0.577401i \(0.195933\pi\)
−0.816461 + 0.577401i \(0.804067\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 14.4272 0.492825 0.246413 0.969165i \(-0.420748\pi\)
0.246413 + 0.969165i \(0.420748\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) −2.99091 −0.101930
\(862\) 0 0
\(863\) 9.66930i 0.329147i 0.986365 + 0.164573i \(0.0526248\pi\)
−0.986365 + 0.164573i \(0.947375\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 13.7812 0.468033
\(868\) 0 0
\(869\) 28.2843i 0.959478i
\(870\) 0 0
\(871\) −20.7477 + 19.2087i −0.703010 + 0.650861i
\(872\) 0 0
\(873\) 4.22690i 0.143059i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 41.7599i 1.41013i 0.709142 + 0.705066i \(0.249083\pi\)
−0.709142 + 0.705066i \(0.750917\pi\)
\(878\) 0 0
\(879\) 6.24718i 0.210712i
\(880\) 0 0
\(881\) 13.5826 0.457609 0.228804 0.973472i \(-0.426518\pi\)
0.228804 + 0.973472i \(0.426518\pi\)
\(882\) 0 0
\(883\) −24.3882 −0.820728 −0.410364 0.911922i \(-0.634598\pi\)
−0.410364 + 0.911922i \(0.634598\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 8.39909 0.282014 0.141007 0.990009i \(-0.454966\pi\)
0.141007 + 0.990009i \(0.454966\pi\)
\(888\) 0 0
\(889\) 10.7234i 0.359651i
\(890\) 0 0
\(891\) 21.3852i 0.716430i
\(892\) 0 0
\(893\) 10.8204 0.362092
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −8.77548 9.47860i −0.293005 0.316481i
\(898\) 0 0
\(899\) 15.1996i 0.506934i
\(900\) 0 0
\(901\) −8.00000 −0.266519
\(902\) 0 0
\(903\) 4.03620i 0.134316i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 50.9280 1.69104 0.845519 0.533945i \(-0.179291\pi\)
0.845519 + 0.533945i \(0.179291\pi\)
\(908\) 0 0
\(909\) −23.6697 −0.785074
\(910\) 0 0
\(911\) 54.3303 1.80004 0.900022 0.435845i \(-0.143551\pi\)
0.900022 + 0.435845i \(0.143551\pi\)
\(912\) 0 0
\(913\) −65.5183 −2.16834
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 13.8564i 0.457579i
\(918\) 0 0
\(919\) −50.3303 −1.66024 −0.830122 0.557582i \(-0.811729\pi\)
−0.830122 + 0.557582i \(0.811729\pi\)
\(920\) 0 0
\(921\) 11.9040i 0.392251i
\(922\) 0 0
\(923\) 17.9274 16.5975i 0.590086 0.546314i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −5.00568 −0.164408
\(928\) 0 0
\(929\) 46.7794i 1.53478i 0.641179 + 0.767391i \(0.278445\pi\)
−0.641179 + 0.767391i \(0.721555\pi\)
\(930\) 0 0
\(931\) 6.89911i 0.226109i
\(932\) 0 0
\(933\) −19.5959 −0.641542
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −37.1469 −1.21354 −0.606768 0.794879i \(-0.707534\pi\)
−0.606768 + 0.794879i \(0.707534\pi\)
\(938\) 0 0
\(939\) 12.0000 0.391605
\(940\) 0 0
\(941\) 18.0280i 0.587696i 0.955852 + 0.293848i \(0.0949360\pi\)
−0.955852 + 0.293848i \(0.905064\pi\)
\(942\) 0 0
\(943\) 28.0942i 0.914873i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 51.2385i 1.66503i −0.554004 0.832514i \(-0.686901\pi\)
0.554004 0.832514i \(-0.313099\pi\)
\(948\) 0 0
\(949\) 20.7477 19.2087i 0.673500 0.623540i
\(950\) 0 0
\(951\) 2.70522i 0.0877227i
\(952\) 0 0
\(953\) 22.7196 0.735961 0.367981 0.929833i \(-0.380049\pi\)
0.367981 + 0.929833i \(0.380049\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 4.03620i 0.130472i
\(958\) 0 0
\(959\) 9.49545 0.306624
\(960\) 0 0
\(961\) −61.2432 −1.97559
\(962\) 0 0
\(963\) 20.9948 0.676548
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 7.46050i 0.239914i −0.992779 0.119957i \(-0.961724\pi\)
0.992779 0.119957i \(-0.0382756\pi\)
\(968\) 0 0
\(969\) 4.47620i 0.143796i
\(970\) 0 0
\(971\) −24.6606 −0.791396 −0.395698 0.918381i \(-0.629497\pi\)
−0.395698 + 0.918381i \(0.629497\pi\)
\(972\) 0 0
\(973\) 10.2016i 0.327048i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 20.6337i 0.660131i −0.943958 0.330065i \(-0.892929\pi\)
0.943958 0.330065i \(-0.107071\pi\)
\(978\) 0 0
\(979\) 44.6606 1.42736
\(980\) 0 0
\(981\) 21.9139i 0.699656i
\(982\) 0 0
\(983\) 20.6337i 0.658113i 0.944310 + 0.329057i \(0.106731\pi\)
−0.944310 + 0.329057i \(0.893269\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −5.70805 −0.181689
\(988\) 0 0
\(989\) 37.9129 1.20556
\(990\) 0 0
\(991\) 29.4955 0.936954 0.468477 0.883476i \(-0.344803\pi\)
0.468477 + 0.883476i \(0.344803\pi\)
\(992\) 0 0
\(993\) 3.31320i 0.105141i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 52.8663 1.67429 0.837146 0.546979i \(-0.184222\pi\)
0.837146 + 0.546979i \(0.184222\pi\)
\(998\) 0 0
\(999\) 28.2843i 0.894875i
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 1300.2.f.f.701.6 8
5.2 odd 4 260.2.d.a.129.4 yes 8
5.3 odd 4 260.2.d.a.129.5 yes 8
5.4 even 2 inner 1300.2.f.f.701.3 8
13.12 even 2 inner 1300.2.f.f.701.5 8
15.2 even 4 2340.2.j.d.649.1 8
15.8 even 4 2340.2.j.d.649.6 8
20.3 even 4 1040.2.f.e.129.3 8
20.7 even 4 1040.2.f.e.129.6 8
65.8 even 4 3380.2.c.d.2029.5 8
65.12 odd 4 260.2.d.a.129.3 8
65.18 even 4 3380.2.c.d.2029.6 8
65.38 odd 4 260.2.d.a.129.6 yes 8
65.47 even 4 3380.2.c.d.2029.3 8
65.57 even 4 3380.2.c.d.2029.4 8
65.64 even 2 inner 1300.2.f.f.701.4 8
195.38 even 4 2340.2.j.d.649.3 8
195.77 even 4 2340.2.j.d.649.8 8
260.103 even 4 1040.2.f.e.129.4 8
260.207 even 4 1040.2.f.e.129.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.d.a.129.3 8 65.12 odd 4
260.2.d.a.129.4 yes 8 5.2 odd 4
260.2.d.a.129.5 yes 8 5.3 odd 4
260.2.d.a.129.6 yes 8 65.38 odd 4
1040.2.f.e.129.3 8 20.3 even 4
1040.2.f.e.129.4 8 260.103 even 4
1040.2.f.e.129.5 8 260.207 even 4
1040.2.f.e.129.6 8 20.7 even 4
1300.2.f.f.701.3 8 5.4 even 2 inner
1300.2.f.f.701.4 8 65.64 even 2 inner
1300.2.f.f.701.5 8 13.12 even 2 inner
1300.2.f.f.701.6 8 1.1 even 1 trivial
2340.2.j.d.649.1 8 15.2 even 4
2340.2.j.d.649.3 8 195.38 even 4
2340.2.j.d.649.6 8 15.8 even 4
2340.2.j.d.649.8 8 195.77 even 4
3380.2.c.d.2029.3 8 65.47 even 4
3380.2.c.d.2029.4 8 65.57 even 4
3380.2.c.d.2029.5 8 65.8 even 4
3380.2.c.d.2029.6 8 65.18 even 4