Properties

Label 8112.2.a.bz.1.3
Level $8112$
Weight $2$
Character 8112.1
Self dual yes
Analytic conductor $64.775$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8112,2,Mod(1,8112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8112, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8112.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8112.a (trivial)

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: yes
Analytic conductor: \(64.7746461197\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1014)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 8112.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +0.692021 q^{5} +0.356896 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +0.692021 q^{5} +0.356896 q^{7} +1.00000 q^{9} +2.93900 q^{11} -0.692021 q^{15} +6.71379 q^{17} -7.20775 q^{19} -0.356896 q^{21} -2.39612 q^{23} -4.52111 q^{25} -1.00000 q^{27} +7.82908 q^{29} -2.76271 q^{31} -2.93900 q^{33} +0.246980 q^{35} -10.0978 q^{37} -4.89008 q^{41} -6.59179 q^{43} +0.692021 q^{45} +4.98792 q^{47} -6.87263 q^{49} -6.71379 q^{51} -8.88769 q^{53} +2.03385 q^{55} +7.20775 q^{57} +1.64310 q^{59} -6.49396 q^{61} +0.356896 q^{63} +13.5254 q^{67} +2.39612 q^{69} +6.81163 q^{71} -3.18598 q^{73} +4.52111 q^{75} +1.04892 q^{77} -15.0465 q^{79} +1.00000 q^{81} -14.8267 q^{83} +4.64609 q^{85} -7.82908 q^{87} -0.396125 q^{89} +2.76271 q^{93} -4.98792 q^{95} +0.417895 q^{97} +2.93900 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} - q^{11} + 3 q^{15} + 12 q^{17} - 4 q^{19} + 3 q^{21} - 16 q^{23} + 2 q^{25} - 3 q^{27} + 13 q^{29} + 9 q^{31} + q^{33} - 4 q^{35} - 12 q^{37} - 14 q^{41} + 8 q^{43} - 3 q^{45} - 4 q^{47} - 4 q^{49} - 12 q^{51} + 15 q^{53} + 22 q^{55} + 4 q^{57} + 9 q^{59} - 10 q^{61} - 3 q^{63} + 6 q^{67} + 16 q^{69} - 6 q^{71} + 5 q^{73} - 2 q^{75} - 6 q^{77} + 5 q^{79} + 3 q^{81} + 7 q^{83} - 26 q^{85} - 13 q^{87} - 10 q^{89} - 9 q^{93} + 4 q^{95} + 7 q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0.692021 0.309481 0.154741 0.987955i \(-0.450546\pi\)
0.154741 + 0.987955i \(0.450546\pi\)
\(6\) 0 0
\(7\) 0.356896 0.134894 0.0674470 0.997723i \(-0.478515\pi\)
0.0674470 + 0.997723i \(0.478515\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.93900 0.886142 0.443071 0.896486i \(-0.353889\pi\)
0.443071 + 0.896486i \(0.353889\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) −0.692021 −0.178679
\(16\) 0 0
\(17\) 6.71379 1.62833 0.814167 0.580631i \(-0.197194\pi\)
0.814167 + 0.580631i \(0.197194\pi\)
\(18\) 0 0
\(19\) −7.20775 −1.65357 −0.826786 0.562517i \(-0.809833\pi\)
−0.826786 + 0.562517i \(0.809833\pi\)
\(20\) 0 0
\(21\) −0.356896 −0.0778811
\(22\) 0 0
\(23\) −2.39612 −0.499627 −0.249813 0.968294i \(-0.580369\pi\)
−0.249813 + 0.968294i \(0.580369\pi\)
\(24\) 0 0
\(25\) −4.52111 −0.904221
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 7.82908 1.45382 0.726912 0.686730i \(-0.240955\pi\)
0.726912 + 0.686730i \(0.240955\pi\)
\(30\) 0 0
\(31\) −2.76271 −0.496197 −0.248099 0.968735i \(-0.579806\pi\)
−0.248099 + 0.968735i \(0.579806\pi\)
\(32\) 0 0
\(33\) −2.93900 −0.511614
\(34\) 0 0
\(35\) 0.246980 0.0417472
\(36\) 0 0
\(37\) −10.0978 −1.66007 −0.830037 0.557708i \(-0.811681\pi\)
−0.830037 + 0.557708i \(0.811681\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.89008 −0.763703 −0.381851 0.924224i \(-0.624713\pi\)
−0.381851 + 0.924224i \(0.624713\pi\)
\(42\) 0 0
\(43\) −6.59179 −1.00524 −0.502620 0.864508i \(-0.667630\pi\)
−0.502620 + 0.864508i \(0.667630\pi\)
\(44\) 0 0
\(45\) 0.692021 0.103160
\(46\) 0 0
\(47\) 4.98792 0.727563 0.363781 0.931484i \(-0.381486\pi\)
0.363781 + 0.931484i \(0.381486\pi\)
\(48\) 0 0
\(49\) −6.87263 −0.981804
\(50\) 0 0
\(51\) −6.71379 −0.940119
\(52\) 0 0
\(53\) −8.88769 −1.22082 −0.610409 0.792086i \(-0.708995\pi\)
−0.610409 + 0.792086i \(0.708995\pi\)
\(54\) 0 0
\(55\) 2.03385 0.274245
\(56\) 0 0
\(57\) 7.20775 0.954690
\(58\) 0 0
\(59\) 1.64310 0.213914 0.106957 0.994264i \(-0.465889\pi\)
0.106957 + 0.994264i \(0.465889\pi\)
\(60\) 0 0
\(61\) −6.49396 −0.831466 −0.415733 0.909487i \(-0.636475\pi\)
−0.415733 + 0.909487i \(0.636475\pi\)
\(62\) 0 0
\(63\) 0.356896 0.0449647
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 13.5254 1.65239 0.826196 0.563382i \(-0.190500\pi\)
0.826196 + 0.563382i \(0.190500\pi\)
\(68\) 0 0
\(69\) 2.39612 0.288459
\(70\) 0 0
\(71\) 6.81163 0.808391 0.404196 0.914673i \(-0.367551\pi\)
0.404196 + 0.914673i \(0.367551\pi\)
\(72\) 0 0
\(73\) −3.18598 −0.372891 −0.186445 0.982465i \(-0.559697\pi\)
−0.186445 + 0.982465i \(0.559697\pi\)
\(74\) 0 0
\(75\) 4.52111 0.522052
\(76\) 0 0
\(77\) 1.04892 0.119535
\(78\) 0 0
\(79\) −15.0465 −1.69287 −0.846433 0.532495i \(-0.821255\pi\)
−0.846433 + 0.532495i \(0.821255\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −14.8267 −1.62744 −0.813720 0.581256i \(-0.802561\pi\)
−0.813720 + 0.581256i \(0.802561\pi\)
\(84\) 0 0
\(85\) 4.64609 0.503939
\(86\) 0 0
\(87\) −7.82908 −0.839366
\(88\) 0 0
\(89\) −0.396125 −0.0419891 −0.0209946 0.999780i \(-0.506683\pi\)
−0.0209946 + 0.999780i \(0.506683\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 2.76271 0.286480
\(94\) 0 0
\(95\) −4.98792 −0.511750
\(96\) 0 0
\(97\) 0.417895 0.0424308 0.0212154 0.999775i \(-0.493246\pi\)
0.0212154 + 0.999775i \(0.493246\pi\)
\(98\) 0 0
\(99\) 2.93900 0.295381
\(100\) 0 0
\(101\) 10.0151 0.996536 0.498268 0.867023i \(-0.333970\pi\)
0.498268 + 0.867023i \(0.333970\pi\)
\(102\) 0 0
\(103\) 9.62565 0.948443 0.474222 0.880406i \(-0.342730\pi\)
0.474222 + 0.880406i \(0.342730\pi\)
\(104\) 0 0
\(105\) −0.246980 −0.0241027
\(106\) 0 0
\(107\) 6.63102 0.641045 0.320523 0.947241i \(-0.396141\pi\)
0.320523 + 0.947241i \(0.396141\pi\)
\(108\) 0 0
\(109\) −12.9879 −1.24402 −0.622008 0.783011i \(-0.713683\pi\)
−0.622008 + 0.783011i \(0.713683\pi\)
\(110\) 0 0
\(111\) 10.0978 0.958444
\(112\) 0 0
\(113\) 0.792249 0.0745285 0.0372643 0.999305i \(-0.488136\pi\)
0.0372643 + 0.999305i \(0.488136\pi\)
\(114\) 0 0
\(115\) −1.65817 −0.154625
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.39612 0.219652
\(120\) 0 0
\(121\) −2.36227 −0.214752
\(122\) 0 0
\(123\) 4.89008 0.440924
\(124\) 0 0
\(125\) −6.58881 −0.589321
\(126\) 0 0
\(127\) 18.2174 1.61654 0.808268 0.588815i \(-0.200405\pi\)
0.808268 + 0.588815i \(0.200405\pi\)
\(128\) 0 0
\(129\) 6.59179 0.580375
\(130\) 0 0
\(131\) −2.73556 −0.239007 −0.119504 0.992834i \(-0.538130\pi\)
−0.119504 + 0.992834i \(0.538130\pi\)
\(132\) 0 0
\(133\) −2.57242 −0.223057
\(134\) 0 0
\(135\) −0.692021 −0.0595597
\(136\) 0 0
\(137\) 7.64742 0.653363 0.326681 0.945135i \(-0.394070\pi\)
0.326681 + 0.945135i \(0.394070\pi\)
\(138\) 0 0
\(139\) −3.38404 −0.287031 −0.143515 0.989648i \(-0.545841\pi\)
−0.143515 + 0.989648i \(0.545841\pi\)
\(140\) 0 0
\(141\) −4.98792 −0.420059
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 5.41789 0.449932
\(146\) 0 0
\(147\) 6.87263 0.566845
\(148\) 0 0
\(149\) −20.8170 −1.70540 −0.852698 0.522405i \(-0.825035\pi\)
−0.852698 + 0.522405i \(0.825035\pi\)
\(150\) 0 0
\(151\) 0.895461 0.0728715 0.0364358 0.999336i \(-0.488400\pi\)
0.0364358 + 0.999336i \(0.488400\pi\)
\(152\) 0 0
\(153\) 6.71379 0.542778
\(154\) 0 0
\(155\) −1.91185 −0.153564
\(156\) 0 0
\(157\) 8.59179 0.685700 0.342850 0.939390i \(-0.388608\pi\)
0.342850 + 0.939390i \(0.388608\pi\)
\(158\) 0 0
\(159\) 8.88769 0.704840
\(160\) 0 0
\(161\) −0.855167 −0.0673966
\(162\) 0 0
\(163\) −1.72587 −0.135181 −0.0675904 0.997713i \(-0.521531\pi\)
−0.0675904 + 0.997713i \(0.521531\pi\)
\(164\) 0 0
\(165\) −2.03385 −0.158335
\(166\) 0 0
\(167\) −21.1400 −1.63587 −0.817933 0.575314i \(-0.804880\pi\)
−0.817933 + 0.575314i \(0.804880\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) −7.20775 −0.551190
\(172\) 0 0
\(173\) 9.35450 0.711210 0.355605 0.934636i \(-0.384275\pi\)
0.355605 + 0.934636i \(0.384275\pi\)
\(174\) 0 0
\(175\) −1.61356 −0.121974
\(176\) 0 0
\(177\) −1.64310 −0.123503
\(178\) 0 0
\(179\) −3.17523 −0.237328 −0.118664 0.992934i \(-0.537861\pi\)
−0.118664 + 0.992934i \(0.537861\pi\)
\(180\) 0 0
\(181\) −19.7995 −1.47169 −0.735844 0.677151i \(-0.763214\pi\)
−0.735844 + 0.677151i \(0.763214\pi\)
\(182\) 0 0
\(183\) 6.49396 0.480047
\(184\) 0 0
\(185\) −6.98792 −0.513762
\(186\) 0 0
\(187\) 19.7318 1.44294
\(188\) 0 0
\(189\) −0.356896 −0.0259604
\(190\) 0 0
\(191\) −15.2620 −1.10432 −0.552161 0.833737i \(-0.686197\pi\)
−0.552161 + 0.833737i \(0.686197\pi\)
\(192\) 0 0
\(193\) 4.76809 0.343214 0.171607 0.985165i \(-0.445104\pi\)
0.171607 + 0.985165i \(0.445104\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.2349 0.871700 0.435850 0.900019i \(-0.356448\pi\)
0.435850 + 0.900019i \(0.356448\pi\)
\(198\) 0 0
\(199\) 11.8485 0.839915 0.419958 0.907544i \(-0.362045\pi\)
0.419958 + 0.907544i \(0.362045\pi\)
\(200\) 0 0
\(201\) −13.5254 −0.954009
\(202\) 0 0
\(203\) 2.79417 0.196112
\(204\) 0 0
\(205\) −3.38404 −0.236352
\(206\) 0 0
\(207\) −2.39612 −0.166542
\(208\) 0 0
\(209\) −21.1836 −1.46530
\(210\) 0 0
\(211\) 17.2620 1.18837 0.594184 0.804329i \(-0.297475\pi\)
0.594184 + 0.804329i \(0.297475\pi\)
\(212\) 0 0
\(213\) −6.81163 −0.466725
\(214\) 0 0
\(215\) −4.56166 −0.311103
\(216\) 0 0
\(217\) −0.985999 −0.0669340
\(218\) 0 0
\(219\) 3.18598 0.215289
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −6.76809 −0.453225 −0.226612 0.973985i \(-0.572765\pi\)
−0.226612 + 0.973985i \(0.572765\pi\)
\(224\) 0 0
\(225\) −4.52111 −0.301407
\(226\) 0 0
\(227\) −23.6799 −1.57169 −0.785846 0.618422i \(-0.787772\pi\)
−0.785846 + 0.618422i \(0.787772\pi\)
\(228\) 0 0
\(229\) −8.29829 −0.548366 −0.274183 0.961677i \(-0.588407\pi\)
−0.274183 + 0.961677i \(0.588407\pi\)
\(230\) 0 0
\(231\) −1.04892 −0.0690137
\(232\) 0 0
\(233\) −23.9651 −1.57000 −0.785002 0.619493i \(-0.787338\pi\)
−0.785002 + 0.619493i \(0.787338\pi\)
\(234\) 0 0
\(235\) 3.45175 0.225167
\(236\) 0 0
\(237\) 15.0465 0.977377
\(238\) 0 0
\(239\) −12.6160 −0.816058 −0.408029 0.912969i \(-0.633784\pi\)
−0.408029 + 0.912969i \(0.633784\pi\)
\(240\) 0 0
\(241\) 26.3937 1.70017 0.850085 0.526646i \(-0.176551\pi\)
0.850085 + 0.526646i \(0.176551\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −4.75600 −0.303850
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 14.8267 0.939603
\(250\) 0 0
\(251\) 30.0344 1.89576 0.947879 0.318632i \(-0.103223\pi\)
0.947879 + 0.318632i \(0.103223\pi\)
\(252\) 0 0
\(253\) −7.04221 −0.442740
\(254\) 0 0
\(255\) −4.64609 −0.290949
\(256\) 0 0
\(257\) 11.2620 0.702507 0.351254 0.936280i \(-0.385756\pi\)
0.351254 + 0.936280i \(0.385756\pi\)
\(258\) 0 0
\(259\) −3.60388 −0.223934
\(260\) 0 0
\(261\) 7.82908 0.484608
\(262\) 0 0
\(263\) 5.54958 0.342202 0.171101 0.985254i \(-0.445268\pi\)
0.171101 + 0.985254i \(0.445268\pi\)
\(264\) 0 0
\(265\) −6.15047 −0.377821
\(266\) 0 0
\(267\) 0.396125 0.0242424
\(268\) 0 0
\(269\) 16.6872 1.01744 0.508719 0.860932i \(-0.330119\pi\)
0.508719 + 0.860932i \(0.330119\pi\)
\(270\) 0 0
\(271\) 6.61356 0.401745 0.200873 0.979617i \(-0.435622\pi\)
0.200873 + 0.979617i \(0.435622\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −13.2875 −0.801269
\(276\) 0 0
\(277\) −21.7995 −1.30981 −0.654904 0.755712i \(-0.727291\pi\)
−0.654904 + 0.755712i \(0.727291\pi\)
\(278\) 0 0
\(279\) −2.76271 −0.165399
\(280\) 0 0
\(281\) 20.5918 1.22840 0.614202 0.789149i \(-0.289478\pi\)
0.614202 + 0.789149i \(0.289478\pi\)
\(282\) 0 0
\(283\) 13.0121 0.773488 0.386744 0.922187i \(-0.373600\pi\)
0.386744 + 0.922187i \(0.373600\pi\)
\(284\) 0 0
\(285\) 4.98792 0.295459
\(286\) 0 0
\(287\) −1.74525 −0.103019
\(288\) 0 0
\(289\) 28.0750 1.65147
\(290\) 0 0
\(291\) −0.417895 −0.0244974
\(292\) 0 0
\(293\) −14.9390 −0.872746 −0.436373 0.899766i \(-0.643737\pi\)
−0.436373 + 0.899766i \(0.643737\pi\)
\(294\) 0 0
\(295\) 1.13706 0.0662024
\(296\) 0 0
\(297\) −2.93900 −0.170538
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −2.35258 −0.135601
\(302\) 0 0
\(303\) −10.0151 −0.575350
\(304\) 0 0
\(305\) −4.49396 −0.257323
\(306\) 0 0
\(307\) −26.0301 −1.48562 −0.742809 0.669503i \(-0.766507\pi\)
−0.742809 + 0.669503i \(0.766507\pi\)
\(308\) 0 0
\(309\) −9.62565 −0.547584
\(310\) 0 0
\(311\) 4.81163 0.272842 0.136421 0.990651i \(-0.456440\pi\)
0.136421 + 0.990651i \(0.456440\pi\)
\(312\) 0 0
\(313\) −26.0411 −1.47193 −0.735966 0.677018i \(-0.763272\pi\)
−0.735966 + 0.677018i \(0.763272\pi\)
\(314\) 0 0
\(315\) 0.246980 0.0139157
\(316\) 0 0
\(317\) −11.5211 −0.647090 −0.323545 0.946213i \(-0.604875\pi\)
−0.323545 + 0.946213i \(0.604875\pi\)
\(318\) 0 0
\(319\) 23.0097 1.28830
\(320\) 0 0
\(321\) −6.63102 −0.370108
\(322\) 0 0
\(323\) −48.3913 −2.69257
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 12.9879 0.718234
\(328\) 0 0
\(329\) 1.78017 0.0981438
\(330\) 0 0
\(331\) −3.43834 −0.188988 −0.0944940 0.995525i \(-0.530123\pi\)
−0.0944940 + 0.995525i \(0.530123\pi\)
\(332\) 0 0
\(333\) −10.0978 −0.553358
\(334\) 0 0
\(335\) 9.35988 0.511385
\(336\) 0 0
\(337\) 8.20105 0.446739 0.223370 0.974734i \(-0.428294\pi\)
0.223370 + 0.974734i \(0.428294\pi\)
\(338\) 0 0
\(339\) −0.792249 −0.0430291
\(340\) 0 0
\(341\) −8.11960 −0.439701
\(342\) 0 0
\(343\) −4.95108 −0.267333
\(344\) 0 0
\(345\) 1.65817 0.0892729
\(346\) 0 0
\(347\) −7.86294 −0.422105 −0.211052 0.977475i \(-0.567689\pi\)
−0.211052 + 0.977475i \(0.567689\pi\)
\(348\) 0 0
\(349\) −18.7245 −1.00230 −0.501151 0.865360i \(-0.667090\pi\)
−0.501151 + 0.865360i \(0.667090\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −31.5448 −1.67896 −0.839480 0.543391i \(-0.817140\pi\)
−0.839480 + 0.543391i \(0.817140\pi\)
\(354\) 0 0
\(355\) 4.71379 0.250182
\(356\) 0 0
\(357\) −2.39612 −0.126816
\(358\) 0 0
\(359\) −2.39612 −0.126463 −0.0632313 0.997999i \(-0.520141\pi\)
−0.0632313 + 0.997999i \(0.520141\pi\)
\(360\) 0 0
\(361\) 32.9517 1.73430
\(362\) 0 0
\(363\) 2.36227 0.123987
\(364\) 0 0
\(365\) −2.20477 −0.115403
\(366\) 0 0
\(367\) 0.00431187 0.000225078 0 0.000112539 1.00000i \(-0.499964\pi\)
0.000112539 1.00000i \(0.499964\pi\)
\(368\) 0 0
\(369\) −4.89008 −0.254568
\(370\) 0 0
\(371\) −3.17198 −0.164681
\(372\) 0 0
\(373\) −32.3129 −1.67310 −0.836549 0.547892i \(-0.815430\pi\)
−0.836549 + 0.547892i \(0.815430\pi\)
\(374\) 0 0
\(375\) 6.58881 0.340245
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −19.7560 −1.01480 −0.507399 0.861711i \(-0.669393\pi\)
−0.507399 + 0.861711i \(0.669393\pi\)
\(380\) 0 0
\(381\) −18.2174 −0.933308
\(382\) 0 0
\(383\) −28.8116 −1.47221 −0.736103 0.676870i \(-0.763336\pi\)
−0.736103 + 0.676870i \(0.763336\pi\)
\(384\) 0 0
\(385\) 0.725873 0.0369939
\(386\) 0 0
\(387\) −6.59179 −0.335080
\(388\) 0 0
\(389\) 34.7821 1.76352 0.881761 0.471697i \(-0.156358\pi\)
0.881761 + 0.471697i \(0.156358\pi\)
\(390\) 0 0
\(391\) −16.0871 −0.813559
\(392\) 0 0
\(393\) 2.73556 0.137991
\(394\) 0 0
\(395\) −10.4125 −0.523911
\(396\) 0 0
\(397\) −5.15346 −0.258645 −0.129322 0.991603i \(-0.541280\pi\)
−0.129322 + 0.991603i \(0.541280\pi\)
\(398\) 0 0
\(399\) 2.57242 0.128782
\(400\) 0 0
\(401\) 13.3250 0.665417 0.332708 0.943030i \(-0.392038\pi\)
0.332708 + 0.943030i \(0.392038\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0.692021 0.0343868
\(406\) 0 0
\(407\) −29.6775 −1.47106
\(408\) 0 0
\(409\) −24.0237 −1.18789 −0.593947 0.804504i \(-0.702431\pi\)
−0.593947 + 0.804504i \(0.702431\pi\)
\(410\) 0 0
\(411\) −7.64742 −0.377219
\(412\) 0 0
\(413\) 0.586417 0.0288557
\(414\) 0 0
\(415\) −10.2604 −0.503663
\(416\) 0 0
\(417\) 3.38404 0.165717
\(418\) 0 0
\(419\) −13.8049 −0.674415 −0.337207 0.941430i \(-0.609482\pi\)
−0.337207 + 0.941430i \(0.609482\pi\)
\(420\) 0 0
\(421\) 7.72587 0.376536 0.188268 0.982118i \(-0.439713\pi\)
0.188268 + 0.982118i \(0.439713\pi\)
\(422\) 0 0
\(423\) 4.98792 0.242521
\(424\) 0 0
\(425\) −30.3538 −1.47237
\(426\) 0 0
\(427\) −2.31767 −0.112160
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −0.640120 −0.0308335 −0.0154168 0.999881i \(-0.504907\pi\)
−0.0154168 + 0.999881i \(0.504907\pi\)
\(432\) 0 0
\(433\) 21.2760 1.02246 0.511231 0.859443i \(-0.329190\pi\)
0.511231 + 0.859443i \(0.329190\pi\)
\(434\) 0 0
\(435\) −5.41789 −0.259768
\(436\) 0 0
\(437\) 17.2707 0.826168
\(438\) 0 0
\(439\) −12.7181 −0.607002 −0.303501 0.952831i \(-0.598156\pi\)
−0.303501 + 0.952831i \(0.598156\pi\)
\(440\) 0 0
\(441\) −6.87263 −0.327268
\(442\) 0 0
\(443\) −22.5972 −1.07362 −0.536812 0.843702i \(-0.680372\pi\)
−0.536812 + 0.843702i \(0.680372\pi\)
\(444\) 0 0
\(445\) −0.274127 −0.0129949
\(446\) 0 0
\(447\) 20.8170 0.984610
\(448\) 0 0
\(449\) 11.6474 0.549676 0.274838 0.961491i \(-0.411376\pi\)
0.274838 + 0.961491i \(0.411376\pi\)
\(450\) 0 0
\(451\) −14.3720 −0.676749
\(452\) 0 0
\(453\) −0.895461 −0.0420724
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −21.1890 −0.991178 −0.495589 0.868557i \(-0.665048\pi\)
−0.495589 + 0.868557i \(0.665048\pi\)
\(458\) 0 0
\(459\) −6.71379 −0.313373
\(460\) 0 0
\(461\) 24.0694 1.12102 0.560511 0.828147i \(-0.310605\pi\)
0.560511 + 0.828147i \(0.310605\pi\)
\(462\) 0 0
\(463\) 18.1715 0.844502 0.422251 0.906479i \(-0.361240\pi\)
0.422251 + 0.906479i \(0.361240\pi\)
\(464\) 0 0
\(465\) 1.91185 0.0886601
\(466\) 0 0
\(467\) 2.93123 0.135641 0.0678206 0.997698i \(-0.478395\pi\)
0.0678206 + 0.997698i \(0.478395\pi\)
\(468\) 0 0
\(469\) 4.82717 0.222898
\(470\) 0 0
\(471\) −8.59179 −0.395889
\(472\) 0 0
\(473\) −19.3733 −0.890785
\(474\) 0 0
\(475\) 32.5870 1.49519
\(476\) 0 0
\(477\) −8.88769 −0.406939
\(478\) 0 0
\(479\) 30.7090 1.40313 0.701565 0.712605i \(-0.252485\pi\)
0.701565 + 0.712605i \(0.252485\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0.855167 0.0389114
\(484\) 0 0
\(485\) 0.289192 0.0131315
\(486\) 0 0
\(487\) −24.1497 −1.09433 −0.547164 0.837025i \(-0.684293\pi\)
−0.547164 + 0.837025i \(0.684293\pi\)
\(488\) 0 0
\(489\) 1.72587 0.0780467
\(490\) 0 0
\(491\) −14.5972 −0.658761 −0.329381 0.944197i \(-0.606840\pi\)
−0.329381 + 0.944197i \(0.606840\pi\)
\(492\) 0 0
\(493\) 52.5628 2.36731
\(494\) 0 0
\(495\) 2.03385 0.0914148
\(496\) 0 0
\(497\) 2.43104 0.109047
\(498\) 0 0
\(499\) −6.85517 −0.306879 −0.153440 0.988158i \(-0.549035\pi\)
−0.153440 + 0.988158i \(0.549035\pi\)
\(500\) 0 0
\(501\) 21.1400 0.944468
\(502\) 0 0
\(503\) 26.7332 1.19197 0.595987 0.802994i \(-0.296761\pi\)
0.595987 + 0.802994i \(0.296761\pi\)
\(504\) 0 0
\(505\) 6.93064 0.308409
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −20.6595 −0.915716 −0.457858 0.889025i \(-0.651383\pi\)
−0.457858 + 0.889025i \(0.651383\pi\)
\(510\) 0 0
\(511\) −1.13706 −0.0503007
\(512\) 0 0
\(513\) 7.20775 0.318230
\(514\) 0 0
\(515\) 6.66115 0.293525
\(516\) 0 0
\(517\) 14.6595 0.644724
\(518\) 0 0
\(519\) −9.35450 −0.410617
\(520\) 0 0
\(521\) −15.0965 −0.661390 −0.330695 0.943738i \(-0.607283\pi\)
−0.330695 + 0.943738i \(0.607283\pi\)
\(522\) 0 0
\(523\) −0.0349168 −0.00152680 −0.000763402 1.00000i \(-0.500243\pi\)
−0.000763402 1.00000i \(0.500243\pi\)
\(524\) 0 0
\(525\) 1.61356 0.0704217
\(526\) 0 0
\(527\) −18.5483 −0.807975
\(528\) 0 0
\(529\) −17.2586 −0.750373
\(530\) 0 0
\(531\) 1.64310 0.0713046
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 4.58881 0.198392
\(536\) 0 0
\(537\) 3.17523 0.137021
\(538\) 0 0
\(539\) −20.1987 −0.870018
\(540\) 0 0
\(541\) 13.0858 0.562600 0.281300 0.959620i \(-0.409234\pi\)
0.281300 + 0.959620i \(0.409234\pi\)
\(542\) 0 0
\(543\) 19.7995 0.849680
\(544\) 0 0
\(545\) −8.98792 −0.385000
\(546\) 0 0
\(547\) 5.97584 0.255508 0.127754 0.991806i \(-0.459223\pi\)
0.127754 + 0.991806i \(0.459223\pi\)
\(548\) 0 0
\(549\) −6.49396 −0.277155
\(550\) 0 0
\(551\) −56.4301 −2.40400
\(552\) 0 0
\(553\) −5.37004 −0.228357
\(554\) 0 0
\(555\) 6.98792 0.296621
\(556\) 0 0
\(557\) −10.4397 −0.442343 −0.221171 0.975235i \(-0.570988\pi\)
−0.221171 + 0.975235i \(0.570988\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −19.7318 −0.833079
\(562\) 0 0
\(563\) 5.66056 0.238564 0.119282 0.992860i \(-0.461941\pi\)
0.119282 + 0.992860i \(0.461941\pi\)
\(564\) 0 0
\(565\) 0.548253 0.0230652
\(566\) 0 0
\(567\) 0.356896 0.0149882
\(568\) 0 0
\(569\) 20.6568 0.865980 0.432990 0.901399i \(-0.357459\pi\)
0.432990 + 0.901399i \(0.357459\pi\)
\(570\) 0 0
\(571\) −44.3672 −1.85671 −0.928354 0.371697i \(-0.878776\pi\)
−0.928354 + 0.371697i \(0.878776\pi\)
\(572\) 0 0
\(573\) 15.2620 0.637581
\(574\) 0 0
\(575\) 10.8331 0.451773
\(576\) 0 0
\(577\) 29.4426 1.22571 0.612857 0.790194i \(-0.290020\pi\)
0.612857 + 0.790194i \(0.290020\pi\)
\(578\) 0 0
\(579\) −4.76809 −0.198155
\(580\) 0 0
\(581\) −5.29159 −0.219532
\(582\) 0 0
\(583\) −26.1209 −1.08182
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 19.5636 0.807475 0.403738 0.914875i \(-0.367711\pi\)
0.403738 + 0.914875i \(0.367711\pi\)
\(588\) 0 0
\(589\) 19.9129 0.820498
\(590\) 0 0
\(591\) −12.2349 −0.503276
\(592\) 0 0
\(593\) −10.8793 −0.446761 −0.223380 0.974731i \(-0.571709\pi\)
−0.223380 + 0.974731i \(0.571709\pi\)
\(594\) 0 0
\(595\) 1.65817 0.0679783
\(596\) 0 0
\(597\) −11.8485 −0.484925
\(598\) 0 0
\(599\) −16.0543 −0.655961 −0.327980 0.944685i \(-0.606368\pi\)
−0.327980 + 0.944685i \(0.606368\pi\)
\(600\) 0 0
\(601\) 12.8955 0.526017 0.263008 0.964794i \(-0.415285\pi\)
0.263008 + 0.964794i \(0.415285\pi\)
\(602\) 0 0
\(603\) 13.5254 0.550798
\(604\) 0 0
\(605\) −1.63474 −0.0664618
\(606\) 0 0
\(607\) −44.4741 −1.80515 −0.902574 0.430534i \(-0.858325\pi\)
−0.902574 + 0.430534i \(0.858325\pi\)
\(608\) 0 0
\(609\) −2.79417 −0.113225
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 42.0253 1.69739 0.848694 0.528884i \(-0.177390\pi\)
0.848694 + 0.528884i \(0.177390\pi\)
\(614\) 0 0
\(615\) 3.38404 0.136458
\(616\) 0 0
\(617\) −16.6655 −0.670926 −0.335463 0.942053i \(-0.608893\pi\)
−0.335463 + 0.942053i \(0.608893\pi\)
\(618\) 0 0
\(619\) −39.7512 −1.59774 −0.798868 0.601506i \(-0.794568\pi\)
−0.798868 + 0.601506i \(0.794568\pi\)
\(620\) 0 0
\(621\) 2.39612 0.0961532
\(622\) 0 0
\(623\) −0.141375 −0.00566408
\(624\) 0 0
\(625\) 18.0459 0.721837
\(626\) 0 0
\(627\) 21.1836 0.845991
\(628\) 0 0
\(629\) −67.7948 −2.70315
\(630\) 0 0
\(631\) −14.5767 −0.580290 −0.290145 0.956983i \(-0.593704\pi\)
−0.290145 + 0.956983i \(0.593704\pi\)
\(632\) 0 0
\(633\) −17.2620 −0.686105
\(634\) 0 0
\(635\) 12.6069 0.500288
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 6.81163 0.269464
\(640\) 0 0
\(641\) 38.4349 1.51809 0.759043 0.651040i \(-0.225667\pi\)
0.759043 + 0.651040i \(0.225667\pi\)
\(642\) 0 0
\(643\) 1.04221 0.0411009 0.0205504 0.999789i \(-0.493458\pi\)
0.0205504 + 0.999789i \(0.493458\pi\)
\(644\) 0 0
\(645\) 4.56166 0.179615
\(646\) 0 0
\(647\) 28.1608 1.10711 0.553557 0.832811i \(-0.313270\pi\)
0.553557 + 0.832811i \(0.313270\pi\)
\(648\) 0 0
\(649\) 4.82908 0.189558
\(650\) 0 0
\(651\) 0.985999 0.0386444
\(652\) 0 0
\(653\) 10.6203 0.415603 0.207802 0.978171i \(-0.433369\pi\)
0.207802 + 0.978171i \(0.433369\pi\)
\(654\) 0 0
\(655\) −1.89307 −0.0739683
\(656\) 0 0
\(657\) −3.18598 −0.124297
\(658\) 0 0
\(659\) 40.3629 1.57231 0.786157 0.618027i \(-0.212068\pi\)
0.786157 + 0.618027i \(0.212068\pi\)
\(660\) 0 0
\(661\) −31.9168 −1.24142 −0.620709 0.784041i \(-0.713155\pi\)
−0.620709 + 0.784041i \(0.713155\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.78017 −0.0690319
\(666\) 0 0
\(667\) −18.7595 −0.726369
\(668\) 0 0
\(669\) 6.76809 0.261669
\(670\) 0 0
\(671\) −19.0858 −0.736797
\(672\) 0 0
\(673\) 3.82802 0.147559 0.0737797 0.997275i \(-0.476494\pi\)
0.0737797 + 0.997275i \(0.476494\pi\)
\(674\) 0 0
\(675\) 4.52111 0.174017
\(676\) 0 0
\(677\) 1.78927 0.0687670 0.0343835 0.999409i \(-0.489053\pi\)
0.0343835 + 0.999409i \(0.489053\pi\)
\(678\) 0 0
\(679\) 0.149145 0.00572366
\(680\) 0 0
\(681\) 23.6799 0.907417
\(682\) 0 0
\(683\) −0.759725 −0.0290701 −0.0145350 0.999894i \(-0.504627\pi\)
−0.0145350 + 0.999894i \(0.504627\pi\)
\(684\) 0 0
\(685\) 5.29218 0.202204
\(686\) 0 0
\(687\) 8.29829 0.316600
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −4.65950 −0.177256 −0.0886278 0.996065i \(-0.528248\pi\)
−0.0886278 + 0.996065i \(0.528248\pi\)
\(692\) 0 0
\(693\) 1.04892 0.0398451
\(694\) 0 0
\(695\) −2.34183 −0.0888307
\(696\) 0 0
\(697\) −32.8310 −1.24356
\(698\) 0 0
\(699\) 23.9651 0.906443
\(700\) 0 0
\(701\) −24.3284 −0.918872 −0.459436 0.888211i \(-0.651948\pi\)
−0.459436 + 0.888211i \(0.651948\pi\)
\(702\) 0 0
\(703\) 72.7827 2.74505
\(704\) 0 0
\(705\) −3.45175 −0.130000
\(706\) 0 0
\(707\) 3.57434 0.134427
\(708\) 0 0
\(709\) −29.3927 −1.10386 −0.551932 0.833889i \(-0.686109\pi\)
−0.551932 + 0.833889i \(0.686109\pi\)
\(710\) 0 0
\(711\) −15.0465 −0.564289
\(712\) 0 0
\(713\) 6.61979 0.247913
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 12.6160 0.471152
\(718\) 0 0
\(719\) 51.4878 1.92017 0.960086 0.279704i \(-0.0902363\pi\)
0.960086 + 0.279704i \(0.0902363\pi\)
\(720\) 0 0
\(721\) 3.43535 0.127939
\(722\) 0 0
\(723\) −26.3937 −0.981593
\(724\) 0 0
\(725\) −35.3961 −1.31458
\(726\) 0 0
\(727\) 3.67324 0.136233 0.0681164 0.997677i \(-0.478301\pi\)
0.0681164 + 0.997677i \(0.478301\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −44.2559 −1.63686
\(732\) 0 0
\(733\) 18.3612 0.678187 0.339093 0.940753i \(-0.389880\pi\)
0.339093 + 0.940753i \(0.389880\pi\)
\(734\) 0 0
\(735\) 4.75600 0.175428
\(736\) 0 0
\(737\) 39.7512 1.46425
\(738\) 0 0
\(739\) −1.68233 −0.0618856 −0.0309428 0.999521i \(-0.509851\pi\)
−0.0309428 + 0.999521i \(0.509851\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −48.2935 −1.77172 −0.885858 0.463956i \(-0.846430\pi\)
−0.885858 + 0.463956i \(0.846430\pi\)
\(744\) 0 0
\(745\) −14.4058 −0.527788
\(746\) 0 0
\(747\) −14.8267 −0.542480
\(748\) 0 0
\(749\) 2.36658 0.0864731
\(750\) 0 0
\(751\) −11.7011 −0.426980 −0.213490 0.976945i \(-0.568483\pi\)
−0.213490 + 0.976945i \(0.568483\pi\)
\(752\) 0 0
\(753\) −30.0344 −1.09452
\(754\) 0 0
\(755\) 0.619678 0.0225524
\(756\) 0 0
\(757\) 25.1836 0.915313 0.457657 0.889129i \(-0.348689\pi\)
0.457657 + 0.889129i \(0.348689\pi\)
\(758\) 0 0
\(759\) 7.04221 0.255616
\(760\) 0 0
\(761\) 22.4155 0.812561 0.406281 0.913748i \(-0.366826\pi\)
0.406281 + 0.913748i \(0.366826\pi\)
\(762\) 0 0
\(763\) −4.63533 −0.167810
\(764\) 0 0
\(765\) 4.64609 0.167980
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −0.132751 −0.00478714 −0.00239357 0.999997i \(-0.500762\pi\)
−0.00239357 + 0.999997i \(0.500762\pi\)
\(770\) 0 0
\(771\) −11.2620 −0.405593
\(772\) 0 0
\(773\) −48.0694 −1.72893 −0.864467 0.502689i \(-0.832344\pi\)
−0.864467 + 0.502689i \(0.832344\pi\)
\(774\) 0 0
\(775\) 12.4905 0.448672
\(776\) 0 0
\(777\) 3.60388 0.129288
\(778\) 0 0
\(779\) 35.2465 1.26284
\(780\) 0 0
\(781\) 20.0194 0.716350
\(782\) 0 0
\(783\) −7.82908 −0.279789
\(784\) 0 0
\(785\) 5.94571 0.212211
\(786\) 0 0
\(787\) −6.20908 −0.221330 −0.110665 0.993858i \(-0.535298\pi\)
−0.110665 + 0.993858i \(0.535298\pi\)
\(788\) 0 0
\(789\) −5.54958 −0.197570
\(790\) 0 0
\(791\) 0.282750 0.0100534
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 6.15047 0.218135
\(796\) 0 0
\(797\) 0.327830 0.0116123 0.00580616 0.999983i \(-0.498152\pi\)
0.00580616 + 0.999983i \(0.498152\pi\)
\(798\) 0 0
\(799\) 33.4878 1.18471
\(800\) 0 0
\(801\) −0.396125 −0.0139964
\(802\) 0 0
\(803\) −9.36360 −0.330434
\(804\) 0 0
\(805\) −0.591794 −0.0208580
\(806\) 0 0
\(807\) −16.6872 −0.587419
\(808\) 0 0
\(809\) −37.4383 −1.31626 −0.658131 0.752904i \(-0.728653\pi\)
−0.658131 + 0.752904i \(0.728653\pi\)
\(810\) 0 0
\(811\) −17.1448 −0.602037 −0.301018 0.953618i \(-0.597327\pi\)
−0.301018 + 0.953618i \(0.597327\pi\)
\(812\) 0 0
\(813\) −6.61356 −0.231948
\(814\) 0 0
\(815\) −1.19434 −0.0418360
\(816\) 0 0
\(817\) 47.5120 1.66223
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −17.7885 −0.620824 −0.310412 0.950602i \(-0.600467\pi\)
−0.310412 + 0.950602i \(0.600467\pi\)
\(822\) 0 0
\(823\) −12.2301 −0.426315 −0.213157 0.977018i \(-0.568375\pi\)
−0.213157 + 0.977018i \(0.568375\pi\)
\(824\) 0 0
\(825\) 13.2875 0.462613
\(826\) 0 0
\(827\) −20.5623 −0.715020 −0.357510 0.933909i \(-0.616374\pi\)
−0.357510 + 0.933909i \(0.616374\pi\)
\(828\) 0 0
\(829\) −25.4470 −0.883809 −0.441905 0.897062i \(-0.645697\pi\)
−0.441905 + 0.897062i \(0.645697\pi\)
\(830\) 0 0
\(831\) 21.7995 0.756218
\(832\) 0 0
\(833\) −46.1414 −1.59870
\(834\) 0 0
\(835\) −14.6294 −0.506270
\(836\) 0 0
\(837\) 2.76271 0.0954932
\(838\) 0 0
\(839\) −5.76676 −0.199091 −0.0995453 0.995033i \(-0.531739\pi\)
−0.0995453 + 0.995033i \(0.531739\pi\)
\(840\) 0 0
\(841\) 32.2946 1.11361
\(842\) 0 0
\(843\) −20.5918 −0.709219
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −0.843085 −0.0289688
\(848\) 0 0
\(849\) −13.0121 −0.446573
\(850\) 0 0
\(851\) 24.1957 0.829417
\(852\) 0 0
\(853\) −21.1728 −0.724944 −0.362472 0.931995i \(-0.618067\pi\)
−0.362472 + 0.931995i \(0.618067\pi\)
\(854\) 0 0
\(855\) −4.98792 −0.170583
\(856\) 0 0
\(857\) 12.0086 0.410207 0.205103 0.978740i \(-0.434247\pi\)
0.205103 + 0.978740i \(0.434247\pi\)
\(858\) 0 0
\(859\) 1.66296 0.0567393 0.0283697 0.999598i \(-0.490968\pi\)
0.0283697 + 0.999598i \(0.490968\pi\)
\(860\) 0 0
\(861\) 1.74525 0.0594780
\(862\) 0 0
\(863\) 23.8323 0.811262 0.405631 0.914037i \(-0.367052\pi\)
0.405631 + 0.914037i \(0.367052\pi\)
\(864\) 0 0
\(865\) 6.47352 0.220106
\(866\) 0 0
\(867\) −28.0750 −0.953477
\(868\) 0 0
\(869\) −44.2218 −1.50012
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0.417895 0.0141436
\(874\) 0 0
\(875\) −2.35152 −0.0794959
\(876\) 0 0
\(877\) −42.3177 −1.42897 −0.714483 0.699653i \(-0.753338\pi\)
−0.714483 + 0.699653i \(0.753338\pi\)
\(878\) 0 0
\(879\) 14.9390 0.503880
\(880\) 0 0
\(881\) 22.2741 0.750434 0.375217 0.926937i \(-0.377568\pi\)
0.375217 + 0.926937i \(0.377568\pi\)
\(882\) 0 0
\(883\) 8.54229 0.287471 0.143735 0.989616i \(-0.454089\pi\)
0.143735 + 0.989616i \(0.454089\pi\)
\(884\) 0 0
\(885\) −1.13706 −0.0382220
\(886\) 0 0
\(887\) 18.9142 0.635078 0.317539 0.948245i \(-0.397144\pi\)
0.317539 + 0.948245i \(0.397144\pi\)
\(888\) 0 0
\(889\) 6.50173 0.218061
\(890\) 0 0
\(891\) 2.93900 0.0984602
\(892\) 0 0
\(893\) −35.9517 −1.20308
\(894\) 0 0
\(895\) −2.19733 −0.0734485
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −21.6295 −0.721384
\(900\) 0 0
\(901\) −59.6701 −1.98790
\(902\) 0 0
\(903\) 2.35258 0.0782891
\(904\) 0 0
\(905\) −13.7017 −0.455460
\(906\) 0 0
\(907\) −13.9517 −0.463258 −0.231629 0.972804i \(-0.574405\pi\)
−0.231629 + 0.972804i \(0.574405\pi\)
\(908\) 0 0
\(909\) 10.0151 0.332179
\(910\) 0 0
\(911\) −45.0422 −1.49232 −0.746158 0.665769i \(-0.768103\pi\)
−0.746158 + 0.665769i \(0.768103\pi\)
\(912\) 0 0
\(913\) −43.5757 −1.44214
\(914\) 0 0
\(915\) 4.49396 0.148566
\(916\) 0 0
\(917\) −0.976311 −0.0322406
\(918\) 0 0
\(919\) −39.9976 −1.31940 −0.659700 0.751529i \(-0.729317\pi\)
−0.659700 + 0.751529i \(0.729317\pi\)
\(920\) 0 0
\(921\) 26.0301 0.857722
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 45.6534 1.50107
\(926\) 0 0
\(927\) 9.62565 0.316148
\(928\) 0 0
\(929\) 34.6848 1.13797 0.568986 0.822347i \(-0.307336\pi\)
0.568986 + 0.822347i \(0.307336\pi\)
\(930\) 0 0
\(931\) 49.5362 1.62348
\(932\) 0 0
\(933\) −4.81163 −0.157526
\(934\) 0 0
\(935\) 13.6549 0.446562
\(936\) 0 0
\(937\) −19.1260 −0.624821 −0.312410 0.949947i \(-0.601136\pi\)
−0.312410 + 0.949947i \(0.601136\pi\)
\(938\) 0 0
\(939\) 26.0411 0.849821
\(940\) 0 0
\(941\) −22.5972 −0.736647 −0.368323 0.929698i \(-0.620068\pi\)
−0.368323 + 0.929698i \(0.620068\pi\)
\(942\) 0 0
\(943\) 11.7172 0.381566
\(944\) 0 0
\(945\) −0.246980 −0.00803425
\(946\) 0 0
\(947\) 27.4359 0.891548 0.445774 0.895145i \(-0.352928\pi\)
0.445774 + 0.895145i \(0.352928\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 11.5211 0.373597
\(952\) 0 0
\(953\) −1.84787 −0.0598584 −0.0299292 0.999552i \(-0.509528\pi\)
−0.0299292 + 0.999552i \(0.509528\pi\)
\(954\) 0 0
\(955\) −10.5617 −0.341767
\(956\) 0 0
\(957\) −23.0097 −0.743798
\(958\) 0 0
\(959\) 2.72933 0.0881347
\(960\) 0 0
\(961\) −23.3674 −0.753788
\(962\) 0 0
\(963\) 6.63102 0.213682
\(964\) 0 0
\(965\) 3.29962 0.106218
\(966\) 0 0
\(967\) 8.88471 0.285713 0.142856 0.989743i \(-0.454371\pi\)
0.142856 + 0.989743i \(0.454371\pi\)
\(968\) 0 0
\(969\) 48.3913 1.55455
\(970\) 0 0
\(971\) −35.0863 −1.12597 −0.562987 0.826466i \(-0.690348\pi\)
−0.562987 + 0.826466i \(0.690348\pi\)
\(972\) 0 0
\(973\) −1.20775 −0.0387187
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −8.33704 −0.266726 −0.133363 0.991067i \(-0.542578\pi\)
−0.133363 + 0.991067i \(0.542578\pi\)
\(978\) 0 0
\(979\) −1.16421 −0.0372083
\(980\) 0 0
\(981\) −12.9879 −0.414672
\(982\) 0 0
\(983\) 55.6051 1.77353 0.886763 0.462224i \(-0.152949\pi\)
0.886763 + 0.462224i \(0.152949\pi\)
\(984\) 0 0
\(985\) 8.46681 0.269775
\(986\) 0 0
\(987\) −1.78017 −0.0566634
\(988\) 0 0
\(989\) 15.7948 0.502244
\(990\) 0 0
\(991\) 43.5967 1.38489 0.692447 0.721468i \(-0.256532\pi\)
0.692447 + 0.721468i \(0.256532\pi\)
\(992\) 0 0
\(993\) 3.43834 0.109112
\(994\) 0 0
\(995\) 8.19939 0.259938
\(996\) 0 0
\(997\) 22.4590 0.711285 0.355643 0.934622i \(-0.384262\pi\)
0.355643 + 0.934622i \(0.384262\pi\)
\(998\) 0 0
\(999\) 10.0978 0.319481
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 8112.2.a.bz.1.3 3
4.3 odd 2 1014.2.a.o.1.3 yes 3
12.11 even 2 3042.2.a.bd.1.1 3
13.12 even 2 8112.2.a.ce.1.1 3
52.3 odd 6 1014.2.e.k.529.3 6
52.7 even 12 1014.2.i.g.361.6 12
52.11 even 12 1014.2.i.g.823.3 12
52.15 even 12 1014.2.i.g.823.4 12
52.19 even 12 1014.2.i.g.361.1 12
52.23 odd 6 1014.2.e.m.529.1 6
52.31 even 4 1014.2.b.g.337.1 6
52.35 odd 6 1014.2.e.k.991.3 6
52.43 odd 6 1014.2.e.m.991.1 6
52.47 even 4 1014.2.b.g.337.6 6
52.51 odd 2 1014.2.a.m.1.1 3
156.47 odd 4 3042.2.b.r.1351.1 6
156.83 odd 4 3042.2.b.r.1351.6 6
156.155 even 2 3042.2.a.be.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1014.2.a.m.1.1 3 52.51 odd 2
1014.2.a.o.1.3 yes 3 4.3 odd 2
1014.2.b.g.337.1 6 52.31 even 4
1014.2.b.g.337.6 6 52.47 even 4
1014.2.e.k.529.3 6 52.3 odd 6
1014.2.e.k.991.3 6 52.35 odd 6
1014.2.e.m.529.1 6 52.23 odd 6
1014.2.e.m.991.1 6 52.43 odd 6
1014.2.i.g.361.1 12 52.19 even 12
1014.2.i.g.361.6 12 52.7 even 12
1014.2.i.g.823.3 12 52.11 even 12
1014.2.i.g.823.4 12 52.15 even 12
3042.2.a.bd.1.1 3 12.11 even 2
3042.2.a.be.1.3 3 156.155 even 2
3042.2.b.r.1351.1 6 156.47 odd 4
3042.2.b.r.1351.6 6 156.83 odd 4
8112.2.a.bz.1.3 3 1.1 even 1 trivial
8112.2.a.ce.1.1 3 13.12 even 2