Properties

Label 3648.2.a.bk.1.1
Level $3648$
Weight $2$
Character 3648.1
Self dual yes
Analytic conductor $29.129$
Analytic rank $1$
Dimension $2$
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] = [3648,2,Mod(1,3648)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3648, 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("3648.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3648 = 2^{6} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3648.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: \(29.1294266574\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 228)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 3648.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -4.37228 q^{5} -2.37228 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -4.37228 q^{5} -2.37228 q^{7} +1.00000 q^{9} +4.37228 q^{11} -2.00000 q^{13} +4.37228 q^{15} -4.37228 q^{17} -1.00000 q^{19} +2.37228 q^{21} +2.74456 q^{23} +14.1168 q^{25} -1.00000 q^{27} +8.74456 q^{29} +4.74456 q^{31} -4.37228 q^{33} +10.3723 q^{35} +6.74456 q^{37} +2.00000 q^{39} +2.37228 q^{43} -4.37228 q^{45} -7.62772 q^{47} -1.37228 q^{49} +4.37228 q^{51} -8.74456 q^{53} -19.1168 q^{55} +1.00000 q^{57} +8.37228 q^{61} -2.37228 q^{63} +8.74456 q^{65} -13.4891 q^{67} -2.74456 q^{69} -12.0000 q^{71} +0.372281 q^{73} -14.1168 q^{75} -10.3723 q^{77} +8.00000 q^{79} +1.00000 q^{81} +2.74456 q^{83} +19.1168 q^{85} -8.74456 q^{87} +3.25544 q^{89} +4.74456 q^{91} -4.74456 q^{93} +4.37228 q^{95} +14.0000 q^{97} +4.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 3 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 3 q^{5} + q^{7} + 2 q^{9} + 3 q^{11} - 4 q^{13} + 3 q^{15} - 3 q^{17} - 2 q^{19} - q^{21} - 6 q^{23} + 11 q^{25} - 2 q^{27} + 6 q^{29} - 2 q^{31} - 3 q^{33} + 15 q^{35} + 2 q^{37} + 4 q^{39} - q^{43} - 3 q^{45} - 21 q^{47} + 3 q^{49} + 3 q^{51} - 6 q^{53} - 21 q^{55} + 2 q^{57} + 11 q^{61} + q^{63} + 6 q^{65} - 4 q^{67} + 6 q^{69} - 24 q^{71} - 5 q^{73} - 11 q^{75} - 15 q^{77} + 16 q^{79} + 2 q^{81} - 6 q^{83} + 21 q^{85} - 6 q^{87} + 18 q^{89} - 2 q^{91} + 2 q^{93} + 3 q^{95} + 28 q^{97} + 3 q^{99}+O(q^{100}) \) 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
\(5\) −4.37228 −1.95534 −0.977672 0.210138i \(-0.932609\pi\)
−0.977672 + 0.210138i \(0.932609\pi\)
\(6\) 0 0
\(7\) −2.37228 −0.896638 −0.448319 0.893874i \(-0.647977\pi\)
−0.448319 + 0.893874i \(0.647977\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.37228 1.31829 0.659146 0.752015i \(-0.270918\pi\)
0.659146 + 0.752015i \(0.270918\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 4.37228 1.12892
\(16\) 0 0
\(17\) −4.37228 −1.06043 −0.530217 0.847862i \(-0.677890\pi\)
−0.530217 + 0.847862i \(0.677890\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 2.37228 0.517674
\(22\) 0 0
\(23\) 2.74456 0.572281 0.286140 0.958188i \(-0.407628\pi\)
0.286140 + 0.958188i \(0.407628\pi\)
\(24\) 0 0
\(25\) 14.1168 2.82337
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 8.74456 1.62382 0.811912 0.583779i \(-0.198427\pi\)
0.811912 + 0.583779i \(0.198427\pi\)
\(30\) 0 0
\(31\) 4.74456 0.852149 0.426074 0.904688i \(-0.359896\pi\)
0.426074 + 0.904688i \(0.359896\pi\)
\(32\) 0 0
\(33\) −4.37228 −0.761116
\(34\) 0 0
\(35\) 10.3723 1.75324
\(36\) 0 0
\(37\) 6.74456 1.10880 0.554400 0.832251i \(-0.312948\pi\)
0.554400 + 0.832251i \(0.312948\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 2.37228 0.361770 0.180885 0.983504i \(-0.442104\pi\)
0.180885 + 0.983504i \(0.442104\pi\)
\(44\) 0 0
\(45\) −4.37228 −0.651781
\(46\) 0 0
\(47\) −7.62772 −1.11262 −0.556309 0.830976i \(-0.687783\pi\)
−0.556309 + 0.830976i \(0.687783\pi\)
\(48\) 0 0
\(49\) −1.37228 −0.196040
\(50\) 0 0
\(51\) 4.37228 0.612242
\(52\) 0 0
\(53\) −8.74456 −1.20116 −0.600579 0.799565i \(-0.705063\pi\)
−0.600579 + 0.799565i \(0.705063\pi\)
\(54\) 0 0
\(55\) −19.1168 −2.57771
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 8.37228 1.07196 0.535980 0.844230i \(-0.319942\pi\)
0.535980 + 0.844230i \(0.319942\pi\)
\(62\) 0 0
\(63\) −2.37228 −0.298879
\(64\) 0 0
\(65\) 8.74456 1.08463
\(66\) 0 0
\(67\) −13.4891 −1.64796 −0.823979 0.566620i \(-0.808251\pi\)
−0.823979 + 0.566620i \(0.808251\pi\)
\(68\) 0 0
\(69\) −2.74456 −0.330407
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 0.372281 0.0435722 0.0217861 0.999763i \(-0.493065\pi\)
0.0217861 + 0.999763i \(0.493065\pi\)
\(74\) 0 0
\(75\) −14.1168 −1.63007
\(76\) 0 0
\(77\) −10.3723 −1.18203
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 2.74456 0.301255 0.150627 0.988591i \(-0.451871\pi\)
0.150627 + 0.988591i \(0.451871\pi\)
\(84\) 0 0
\(85\) 19.1168 2.07351
\(86\) 0 0
\(87\) −8.74456 −0.937516
\(88\) 0 0
\(89\) 3.25544 0.345076 0.172538 0.985003i \(-0.444803\pi\)
0.172538 + 0.985003i \(0.444803\pi\)
\(90\) 0 0
\(91\) 4.74456 0.497365
\(92\) 0 0
\(93\) −4.74456 −0.491988
\(94\) 0 0
\(95\) 4.37228 0.448587
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) 4.37228 0.439431
\(100\) 0 0
\(101\) 11.4891 1.14321 0.571605 0.820529i \(-0.306321\pi\)
0.571605 + 0.820529i \(0.306321\pi\)
\(102\) 0 0
\(103\) −12.7446 −1.25576 −0.627880 0.778311i \(-0.716077\pi\)
−0.627880 + 0.778311i \(0.716077\pi\)
\(104\) 0 0
\(105\) −10.3723 −1.01223
\(106\) 0 0
\(107\) −3.25544 −0.314715 −0.157358 0.987542i \(-0.550297\pi\)
−0.157358 + 0.987542i \(0.550297\pi\)
\(108\) 0 0
\(109\) −16.2337 −1.55491 −0.777453 0.628941i \(-0.783489\pi\)
−0.777453 + 0.628941i \(0.783489\pi\)
\(110\) 0 0
\(111\) −6.74456 −0.640166
\(112\) 0 0
\(113\) 3.25544 0.306246 0.153123 0.988207i \(-0.451067\pi\)
0.153123 + 0.988207i \(0.451067\pi\)
\(114\) 0 0
\(115\) −12.0000 −1.11901
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 10.3723 0.950825
\(120\) 0 0
\(121\) 8.11684 0.737895
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −39.8614 −3.56531
\(126\) 0 0
\(127\) 22.2337 1.97292 0.986460 0.164000i \(-0.0524396\pi\)
0.986460 + 0.164000i \(0.0524396\pi\)
\(128\) 0 0
\(129\) −2.37228 −0.208868
\(130\) 0 0
\(131\) −9.86141 −0.861595 −0.430798 0.902449i \(-0.641768\pi\)
−0.430798 + 0.902449i \(0.641768\pi\)
\(132\) 0 0
\(133\) 2.37228 0.205703
\(134\) 0 0
\(135\) 4.37228 0.376306
\(136\) 0 0
\(137\) −4.37228 −0.373549 −0.186775 0.982403i \(-0.559803\pi\)
−0.186775 + 0.982403i \(0.559803\pi\)
\(138\) 0 0
\(139\) −9.62772 −0.816612 −0.408306 0.912845i \(-0.633880\pi\)
−0.408306 + 0.912845i \(0.633880\pi\)
\(140\) 0 0
\(141\) 7.62772 0.642370
\(142\) 0 0
\(143\) −8.74456 −0.731257
\(144\) 0 0
\(145\) −38.2337 −3.17513
\(146\) 0 0
\(147\) 1.37228 0.113184
\(148\) 0 0
\(149\) 4.37228 0.358191 0.179096 0.983832i \(-0.442683\pi\)
0.179096 + 0.983832i \(0.442683\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) −4.37228 −0.353478
\(154\) 0 0
\(155\) −20.7446 −1.66624
\(156\) 0 0
\(157\) 15.4891 1.23617 0.618083 0.786113i \(-0.287909\pi\)
0.618083 + 0.786113i \(0.287909\pi\)
\(158\) 0 0
\(159\) 8.74456 0.693489
\(160\) 0 0
\(161\) −6.51087 −0.513129
\(162\) 0 0
\(163\) −13.4891 −1.05655 −0.528275 0.849073i \(-0.677161\pi\)
−0.528275 + 0.849073i \(0.677161\pi\)
\(164\) 0 0
\(165\) 19.1168 1.48824
\(166\) 0 0
\(167\) −14.2337 −1.10144 −0.550718 0.834691i \(-0.685646\pi\)
−0.550718 + 0.834691i \(0.685646\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) −20.7446 −1.57718 −0.788590 0.614919i \(-0.789189\pi\)
−0.788590 + 0.614919i \(0.789189\pi\)
\(174\) 0 0
\(175\) −33.4891 −2.53154
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −3.25544 −0.243323 −0.121661 0.992572i \(-0.538822\pi\)
−0.121661 + 0.992572i \(0.538822\pi\)
\(180\) 0 0
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) −8.37228 −0.618897
\(184\) 0 0
\(185\) −29.4891 −2.16808
\(186\) 0 0
\(187\) −19.1168 −1.39796
\(188\) 0 0
\(189\) 2.37228 0.172558
\(190\) 0 0
\(191\) −16.3723 −1.18466 −0.592328 0.805697i \(-0.701791\pi\)
−0.592328 + 0.805697i \(0.701791\pi\)
\(192\) 0 0
\(193\) 5.25544 0.378295 0.189147 0.981949i \(-0.439428\pi\)
0.189147 + 0.981949i \(0.439428\pi\)
\(194\) 0 0
\(195\) −8.74456 −0.626211
\(196\) 0 0
\(197\) −11.4891 −0.818566 −0.409283 0.912407i \(-0.634221\pi\)
−0.409283 + 0.912407i \(0.634221\pi\)
\(198\) 0 0
\(199\) 3.11684 0.220947 0.110474 0.993879i \(-0.464763\pi\)
0.110474 + 0.993879i \(0.464763\pi\)
\(200\) 0 0
\(201\) 13.4891 0.951450
\(202\) 0 0
\(203\) −20.7446 −1.45598
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.74456 0.190760
\(208\) 0 0
\(209\) −4.37228 −0.302437
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) 12.0000 0.822226
\(214\) 0 0
\(215\) −10.3723 −0.707384
\(216\) 0 0
\(217\) −11.2554 −0.764069
\(218\) 0 0
\(219\) −0.372281 −0.0251564
\(220\) 0 0
\(221\) 8.74456 0.588223
\(222\) 0 0
\(223\) −9.48913 −0.635439 −0.317719 0.948185i \(-0.602917\pi\)
−0.317719 + 0.948185i \(0.602917\pi\)
\(224\) 0 0
\(225\) 14.1168 0.941123
\(226\) 0 0
\(227\) −26.2337 −1.74119 −0.870596 0.491999i \(-0.836266\pi\)
−0.870596 + 0.491999i \(0.836266\pi\)
\(228\) 0 0
\(229\) −0.372281 −0.0246010 −0.0123005 0.999924i \(-0.503915\pi\)
−0.0123005 + 0.999924i \(0.503915\pi\)
\(230\) 0 0
\(231\) 10.3723 0.682446
\(232\) 0 0
\(233\) 4.37228 0.286438 0.143219 0.989691i \(-0.454255\pi\)
0.143219 + 0.989691i \(0.454255\pi\)
\(234\) 0 0
\(235\) 33.3505 2.17555
\(236\) 0 0
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) −9.86141 −0.637881 −0.318941 0.947775i \(-0.603327\pi\)
−0.318941 + 0.947775i \(0.603327\pi\)
\(240\) 0 0
\(241\) 10.7446 0.692118 0.346059 0.938213i \(-0.387520\pi\)
0.346059 + 0.938213i \(0.387520\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 6.00000 0.383326
\(246\) 0 0
\(247\) 2.00000 0.127257
\(248\) 0 0
\(249\) −2.74456 −0.173930
\(250\) 0 0
\(251\) −4.37228 −0.275976 −0.137988 0.990434i \(-0.544064\pi\)
−0.137988 + 0.990434i \(0.544064\pi\)
\(252\) 0 0
\(253\) 12.0000 0.754434
\(254\) 0 0
\(255\) −19.1168 −1.19714
\(256\) 0 0
\(257\) 5.48913 0.342402 0.171201 0.985236i \(-0.445235\pi\)
0.171201 + 0.985236i \(0.445235\pi\)
\(258\) 0 0
\(259\) −16.0000 −0.994192
\(260\) 0 0
\(261\) 8.74456 0.541275
\(262\) 0 0
\(263\) −2.13859 −0.131871 −0.0659357 0.997824i \(-0.521003\pi\)
−0.0659357 + 0.997824i \(0.521003\pi\)
\(264\) 0 0
\(265\) 38.2337 2.34868
\(266\) 0 0
\(267\) −3.25544 −0.199230
\(268\) 0 0
\(269\) −3.25544 −0.198488 −0.0992438 0.995063i \(-0.531642\pi\)
−0.0992438 + 0.995063i \(0.531642\pi\)
\(270\) 0 0
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) 0 0
\(273\) −4.74456 −0.287154
\(274\) 0 0
\(275\) 61.7228 3.72203
\(276\) 0 0
\(277\) −6.88316 −0.413569 −0.206784 0.978387i \(-0.566300\pi\)
−0.206784 + 0.978387i \(0.566300\pi\)
\(278\) 0 0
\(279\) 4.74456 0.284050
\(280\) 0 0
\(281\) −15.2554 −0.910063 −0.455032 0.890475i \(-0.650372\pi\)
−0.455032 + 0.890475i \(0.650372\pi\)
\(282\) 0 0
\(283\) 8.88316 0.528049 0.264024 0.964516i \(-0.414950\pi\)
0.264024 + 0.964516i \(0.414950\pi\)
\(284\) 0 0
\(285\) −4.37228 −0.258992
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2.11684 0.124520
\(290\) 0 0
\(291\) −14.0000 −0.820695
\(292\) 0 0
\(293\) −6.51087 −0.380369 −0.190185 0.981748i \(-0.560909\pi\)
−0.190185 + 0.981748i \(0.560909\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −4.37228 −0.253705
\(298\) 0 0
\(299\) −5.48913 −0.317444
\(300\) 0 0
\(301\) −5.62772 −0.324376
\(302\) 0 0
\(303\) −11.4891 −0.660033
\(304\) 0 0
\(305\) −36.6060 −2.09605
\(306\) 0 0
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) 12.7446 0.725013
\(310\) 0 0
\(311\) 9.86141 0.559189 0.279595 0.960118i \(-0.409800\pi\)
0.279595 + 0.960118i \(0.409800\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 0 0
\(315\) 10.3723 0.584412
\(316\) 0 0
\(317\) −6.51087 −0.365687 −0.182844 0.983142i \(-0.558530\pi\)
−0.182844 + 0.983142i \(0.558530\pi\)
\(318\) 0 0
\(319\) 38.2337 2.14068
\(320\) 0 0
\(321\) 3.25544 0.181701
\(322\) 0 0
\(323\) 4.37228 0.243280
\(324\) 0 0
\(325\) −28.2337 −1.56612
\(326\) 0 0
\(327\) 16.2337 0.897725
\(328\) 0 0
\(329\) 18.0951 0.997615
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 0 0
\(333\) 6.74456 0.369600
\(334\) 0 0
\(335\) 58.9783 3.22233
\(336\) 0 0
\(337\) −3.48913 −0.190065 −0.0950324 0.995474i \(-0.530295\pi\)
−0.0950324 + 0.995474i \(0.530295\pi\)
\(338\) 0 0
\(339\) −3.25544 −0.176811
\(340\) 0 0
\(341\) 20.7446 1.12338
\(342\) 0 0
\(343\) 19.8614 1.07242
\(344\) 0 0
\(345\) 12.0000 0.646058
\(346\) 0 0
\(347\) 1.11684 0.0599553 0.0299777 0.999551i \(-0.490456\pi\)
0.0299777 + 0.999551i \(0.490456\pi\)
\(348\) 0 0
\(349\) −24.3723 −1.30462 −0.652309 0.757953i \(-0.726200\pi\)
−0.652309 + 0.757953i \(0.726200\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) 30.0000 1.59674 0.798369 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) 0 0
\(355\) 52.4674 2.78468
\(356\) 0 0
\(357\) −10.3723 −0.548959
\(358\) 0 0
\(359\) −33.8614 −1.78714 −0.893568 0.448927i \(-0.851806\pi\)
−0.893568 + 0.448927i \(0.851806\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −8.11684 −0.426024
\(364\) 0 0
\(365\) −1.62772 −0.0851987
\(366\) 0 0
\(367\) 2.51087 0.131067 0.0655333 0.997850i \(-0.479125\pi\)
0.0655333 + 0.997850i \(0.479125\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 20.7446 1.07700
\(372\) 0 0
\(373\) −4.23369 −0.219212 −0.109606 0.993975i \(-0.534959\pi\)
−0.109606 + 0.993975i \(0.534959\pi\)
\(374\) 0 0
\(375\) 39.8614 2.05843
\(376\) 0 0
\(377\) −17.4891 −0.900736
\(378\) 0 0
\(379\) 35.7228 1.83496 0.917479 0.397785i \(-0.130221\pi\)
0.917479 + 0.397785i \(0.130221\pi\)
\(380\) 0 0
\(381\) −22.2337 −1.13907
\(382\) 0 0
\(383\) −26.2337 −1.34048 −0.670239 0.742145i \(-0.733809\pi\)
−0.670239 + 0.742145i \(0.733809\pi\)
\(384\) 0 0
\(385\) 45.3505 2.31128
\(386\) 0 0
\(387\) 2.37228 0.120590
\(388\) 0 0
\(389\) 19.6277 0.995165 0.497582 0.867417i \(-0.334221\pi\)
0.497582 + 0.867417i \(0.334221\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) 0 0
\(393\) 9.86141 0.497442
\(394\) 0 0
\(395\) −34.9783 −1.75995
\(396\) 0 0
\(397\) 10.6060 0.532298 0.266149 0.963932i \(-0.414249\pi\)
0.266149 + 0.963932i \(0.414249\pi\)
\(398\) 0 0
\(399\) −2.37228 −0.118763
\(400\) 0 0
\(401\) −17.4891 −0.873365 −0.436683 0.899616i \(-0.643847\pi\)
−0.436683 + 0.899616i \(0.643847\pi\)
\(402\) 0 0
\(403\) −9.48913 −0.472687
\(404\) 0 0
\(405\) −4.37228 −0.217260
\(406\) 0 0
\(407\) 29.4891 1.46172
\(408\) 0 0
\(409\) −38.4674 −1.90209 −0.951045 0.309053i \(-0.899988\pi\)
−0.951045 + 0.309053i \(0.899988\pi\)
\(410\) 0 0
\(411\) 4.37228 0.215669
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) 9.62772 0.471471
\(418\) 0 0
\(419\) −37.7228 −1.84288 −0.921440 0.388521i \(-0.872986\pi\)
−0.921440 + 0.388521i \(0.872986\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 0 0
\(423\) −7.62772 −0.370872
\(424\) 0 0
\(425\) −61.7228 −2.99400
\(426\) 0 0
\(427\) −19.8614 −0.961161
\(428\) 0 0
\(429\) 8.74456 0.422191
\(430\) 0 0
\(431\) 8.74456 0.421211 0.210605 0.977571i \(-0.432456\pi\)
0.210605 + 0.977571i \(0.432456\pi\)
\(432\) 0 0
\(433\) −15.4891 −0.744360 −0.372180 0.928161i \(-0.621390\pi\)
−0.372180 + 0.928161i \(0.621390\pi\)
\(434\) 0 0
\(435\) 38.2337 1.83317
\(436\) 0 0
\(437\) −2.74456 −0.131290
\(438\) 0 0
\(439\) −30.2337 −1.44298 −0.721488 0.692427i \(-0.756541\pi\)
−0.721488 + 0.692427i \(0.756541\pi\)
\(440\) 0 0
\(441\) −1.37228 −0.0653467
\(442\) 0 0
\(443\) 7.62772 0.362404 0.181202 0.983446i \(-0.442001\pi\)
0.181202 + 0.983446i \(0.442001\pi\)
\(444\) 0 0
\(445\) −14.2337 −0.674742
\(446\) 0 0
\(447\) −4.37228 −0.206802
\(448\) 0 0
\(449\) 22.9783 1.08441 0.542205 0.840246i \(-0.317589\pi\)
0.542205 + 0.840246i \(0.317589\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −8.00000 −0.375873
\(454\) 0 0
\(455\) −20.7446 −0.972520
\(456\) 0 0
\(457\) −13.8614 −0.648409 −0.324205 0.945987i \(-0.605097\pi\)
−0.324205 + 0.945987i \(0.605097\pi\)
\(458\) 0 0
\(459\) 4.37228 0.204081
\(460\) 0 0
\(461\) 39.3505 1.83274 0.916368 0.400336i \(-0.131107\pi\)
0.916368 + 0.400336i \(0.131107\pi\)
\(462\) 0 0
\(463\) 17.3505 0.806348 0.403174 0.915123i \(-0.367907\pi\)
0.403174 + 0.915123i \(0.367907\pi\)
\(464\) 0 0
\(465\) 20.7446 0.962006
\(466\) 0 0
\(467\) 13.1168 0.606975 0.303488 0.952835i \(-0.401849\pi\)
0.303488 + 0.952835i \(0.401849\pi\)
\(468\) 0 0
\(469\) 32.0000 1.47762
\(470\) 0 0
\(471\) −15.4891 −0.713701
\(472\) 0 0
\(473\) 10.3723 0.476918
\(474\) 0 0
\(475\) −14.1168 −0.647725
\(476\) 0 0
\(477\) −8.74456 −0.400386
\(478\) 0 0
\(479\) 3.76631 0.172087 0.0860436 0.996291i \(-0.472578\pi\)
0.0860436 + 0.996291i \(0.472578\pi\)
\(480\) 0 0
\(481\) −13.4891 −0.615051
\(482\) 0 0
\(483\) 6.51087 0.296255
\(484\) 0 0
\(485\) −61.2119 −2.77949
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 0 0
\(489\) 13.4891 0.609999
\(490\) 0 0
\(491\) 38.7446 1.74852 0.874259 0.485460i \(-0.161348\pi\)
0.874259 + 0.485460i \(0.161348\pi\)
\(492\) 0 0
\(493\) −38.2337 −1.72196
\(494\) 0 0
\(495\) −19.1168 −0.859238
\(496\) 0 0
\(497\) 28.4674 1.27694
\(498\) 0 0
\(499\) −17.3505 −0.776716 −0.388358 0.921508i \(-0.626958\pi\)
−0.388358 + 0.921508i \(0.626958\pi\)
\(500\) 0 0
\(501\) 14.2337 0.635914
\(502\) 0 0
\(503\) 14.7446 0.657428 0.328714 0.944430i \(-0.393385\pi\)
0.328714 + 0.944430i \(0.393385\pi\)
\(504\) 0 0
\(505\) −50.2337 −2.23537
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 0 0
\(509\) 3.25544 0.144295 0.0721474 0.997394i \(-0.477015\pi\)
0.0721474 + 0.997394i \(0.477015\pi\)
\(510\) 0 0
\(511\) −0.883156 −0.0390685
\(512\) 0 0
\(513\) 1.00000 0.0441511
\(514\) 0 0
\(515\) 55.7228 2.45544
\(516\) 0 0
\(517\) −33.3505 −1.46675
\(518\) 0 0
\(519\) 20.7446 0.910585
\(520\) 0 0
\(521\) −17.4891 −0.766212 −0.383106 0.923704i \(-0.625146\pi\)
−0.383106 + 0.923704i \(0.625146\pi\)
\(522\) 0 0
\(523\) 7.25544 0.317258 0.158629 0.987338i \(-0.449293\pi\)
0.158629 + 0.987338i \(0.449293\pi\)
\(524\) 0 0
\(525\) 33.4891 1.46159
\(526\) 0 0
\(527\) −20.7446 −0.903647
\(528\) 0 0
\(529\) −15.4674 −0.672495
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 14.2337 0.615376
\(536\) 0 0
\(537\) 3.25544 0.140482
\(538\) 0 0
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) 1.86141 0.0800281 0.0400141 0.999199i \(-0.487260\pi\)
0.0400141 + 0.999199i \(0.487260\pi\)
\(542\) 0 0
\(543\) −22.0000 −0.944110
\(544\) 0 0
\(545\) 70.9783 3.04037
\(546\) 0 0
\(547\) 7.25544 0.310220 0.155110 0.987897i \(-0.450427\pi\)
0.155110 + 0.987897i \(0.450427\pi\)
\(548\) 0 0
\(549\) 8.37228 0.357320
\(550\) 0 0
\(551\) −8.74456 −0.372531
\(552\) 0 0
\(553\) −18.9783 −0.807037
\(554\) 0 0
\(555\) 29.4891 1.25174
\(556\) 0 0
\(557\) 21.8614 0.926298 0.463149 0.886281i \(-0.346720\pi\)
0.463149 + 0.886281i \(0.346720\pi\)
\(558\) 0 0
\(559\) −4.74456 −0.200674
\(560\) 0 0
\(561\) 19.1168 0.807114
\(562\) 0 0
\(563\) −1.02175 −0.0430616 −0.0215308 0.999768i \(-0.506854\pi\)
−0.0215308 + 0.999768i \(0.506854\pi\)
\(564\) 0 0
\(565\) −14.2337 −0.598816
\(566\) 0 0
\(567\) −2.37228 −0.0996265
\(568\) 0 0
\(569\) −32.7446 −1.37272 −0.686362 0.727260i \(-0.740793\pi\)
−0.686362 + 0.727260i \(0.740793\pi\)
\(570\) 0 0
\(571\) 26.9783 1.12900 0.564502 0.825431i \(-0.309068\pi\)
0.564502 + 0.825431i \(0.309068\pi\)
\(572\) 0 0
\(573\) 16.3723 0.683962
\(574\) 0 0
\(575\) 38.7446 1.61576
\(576\) 0 0
\(577\) 21.1168 0.879106 0.439553 0.898217i \(-0.355137\pi\)
0.439553 + 0.898217i \(0.355137\pi\)
\(578\) 0 0
\(579\) −5.25544 −0.218409
\(580\) 0 0
\(581\) −6.51087 −0.270117
\(582\) 0 0
\(583\) −38.2337 −1.58348
\(584\) 0 0
\(585\) 8.74456 0.361543
\(586\) 0 0
\(587\) 6.60597 0.272658 0.136329 0.990664i \(-0.456470\pi\)
0.136329 + 0.990664i \(0.456470\pi\)
\(588\) 0 0
\(589\) −4.74456 −0.195496
\(590\) 0 0
\(591\) 11.4891 0.472599
\(592\) 0 0
\(593\) −16.9783 −0.697213 −0.348607 0.937269i \(-0.613345\pi\)
−0.348607 + 0.937269i \(0.613345\pi\)
\(594\) 0 0
\(595\) −45.3505 −1.85919
\(596\) 0 0
\(597\) −3.11684 −0.127564
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −1.25544 −0.0512104 −0.0256052 0.999672i \(-0.508151\pi\)
−0.0256052 + 0.999672i \(0.508151\pi\)
\(602\) 0 0
\(603\) −13.4891 −0.549320
\(604\) 0 0
\(605\) −35.4891 −1.44284
\(606\) 0 0
\(607\) −19.2554 −0.781554 −0.390777 0.920485i \(-0.627794\pi\)
−0.390777 + 0.920485i \(0.627794\pi\)
\(608\) 0 0
\(609\) 20.7446 0.840612
\(610\) 0 0
\(611\) 15.2554 0.617169
\(612\) 0 0
\(613\) 29.1168 1.17602 0.588009 0.808854i \(-0.299912\pi\)
0.588009 + 0.808854i \(0.299912\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −10.8832 −0.438139 −0.219070 0.975709i \(-0.570302\pi\)
−0.219070 + 0.975709i \(0.570302\pi\)
\(618\) 0 0
\(619\) −6.97825 −0.280480 −0.140240 0.990118i \(-0.544787\pi\)
−0.140240 + 0.990118i \(0.544787\pi\)
\(620\) 0 0
\(621\) −2.74456 −0.110136
\(622\) 0 0
\(623\) −7.72281 −0.309408
\(624\) 0 0
\(625\) 103.701 4.14804
\(626\) 0 0
\(627\) 4.37228 0.174612
\(628\) 0 0
\(629\) −29.4891 −1.17581
\(630\) 0 0
\(631\) 4.13859 0.164755 0.0823774 0.996601i \(-0.473749\pi\)
0.0823774 + 0.996601i \(0.473749\pi\)
\(632\) 0 0
\(633\) −4.00000 −0.158986
\(634\) 0 0
\(635\) −97.2119 −3.85774
\(636\) 0 0
\(637\) 2.74456 0.108744
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) 14.2337 0.562197 0.281098 0.959679i \(-0.409301\pi\)
0.281098 + 0.959679i \(0.409301\pi\)
\(642\) 0 0
\(643\) 37.3505 1.47296 0.736481 0.676459i \(-0.236486\pi\)
0.736481 + 0.676459i \(0.236486\pi\)
\(644\) 0 0
\(645\) 10.3723 0.408408
\(646\) 0 0
\(647\) −21.8614 −0.859461 −0.429730 0.902957i \(-0.641391\pi\)
−0.429730 + 0.902957i \(0.641391\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 11.2554 0.441135
\(652\) 0 0
\(653\) −30.6060 −1.19770 −0.598852 0.800860i \(-0.704376\pi\)
−0.598852 + 0.800860i \(0.704376\pi\)
\(654\) 0 0
\(655\) 43.1168 1.68471
\(656\) 0 0
\(657\) 0.372281 0.0145241
\(658\) 0 0
\(659\) 8.74456 0.340640 0.170320 0.985389i \(-0.445520\pi\)
0.170320 + 0.985389i \(0.445520\pi\)
\(660\) 0 0
\(661\) 24.2337 0.942581 0.471291 0.881978i \(-0.343788\pi\)
0.471291 + 0.881978i \(0.343788\pi\)
\(662\) 0 0
\(663\) −8.74456 −0.339611
\(664\) 0 0
\(665\) −10.3723 −0.402220
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) 0 0
\(669\) 9.48913 0.366871
\(670\) 0 0
\(671\) 36.6060 1.41316
\(672\) 0 0
\(673\) −36.2337 −1.39671 −0.698353 0.715753i \(-0.746083\pi\)
−0.698353 + 0.715753i \(0.746083\pi\)
\(674\) 0 0
\(675\) −14.1168 −0.543358
\(676\) 0 0
\(677\) −8.74456 −0.336081 −0.168040 0.985780i \(-0.553744\pi\)
−0.168040 + 0.985780i \(0.553744\pi\)
\(678\) 0 0
\(679\) −33.2119 −1.27456
\(680\) 0 0
\(681\) 26.2337 1.00528
\(682\) 0 0
\(683\) −20.7446 −0.793769 −0.396884 0.917869i \(-0.629909\pi\)
−0.396884 + 0.917869i \(0.629909\pi\)
\(684\) 0 0
\(685\) 19.1168 0.730417
\(686\) 0 0
\(687\) 0.372281 0.0142034
\(688\) 0 0
\(689\) 17.4891 0.666283
\(690\) 0 0
\(691\) −29.3505 −1.11655 −0.558273 0.829657i \(-0.688536\pi\)
−0.558273 + 0.829657i \(0.688536\pi\)
\(692\) 0 0
\(693\) −10.3723 −0.394010
\(694\) 0 0
\(695\) 42.0951 1.59676
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −4.37228 −0.165375
\(700\) 0 0
\(701\) −12.5109 −0.472529 −0.236265 0.971689i \(-0.575923\pi\)
−0.236265 + 0.971689i \(0.575923\pi\)
\(702\) 0 0
\(703\) −6.74456 −0.254376
\(704\) 0 0
\(705\) −33.3505 −1.25605
\(706\) 0 0
\(707\) −27.2554 −1.02505
\(708\) 0 0
\(709\) 50.4674 1.89534 0.947671 0.319248i \(-0.103430\pi\)
0.947671 + 0.319248i \(0.103430\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 13.0217 0.487668
\(714\) 0 0
\(715\) 38.2337 1.42986
\(716\) 0 0
\(717\) 9.86141 0.368281
\(718\) 0 0
\(719\) −10.8832 −0.405873 −0.202937 0.979192i \(-0.565049\pi\)
−0.202937 + 0.979192i \(0.565049\pi\)
\(720\) 0 0
\(721\) 30.2337 1.12596
\(722\) 0 0
\(723\) −10.7446 −0.399595
\(724\) 0 0
\(725\) 123.446 4.58466
\(726\) 0 0
\(727\) −39.5842 −1.46810 −0.734049 0.679097i \(-0.762372\pi\)
−0.734049 + 0.679097i \(0.762372\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −10.3723 −0.383633
\(732\) 0 0
\(733\) −2.00000 −0.0738717 −0.0369358 0.999318i \(-0.511760\pi\)
−0.0369358 + 0.999318i \(0.511760\pi\)
\(734\) 0 0
\(735\) −6.00000 −0.221313
\(736\) 0 0
\(737\) −58.9783 −2.17249
\(738\) 0 0
\(739\) −44.6060 −1.64086 −0.820429 0.571749i \(-0.806265\pi\)
−0.820429 + 0.571749i \(0.806265\pi\)
\(740\) 0 0
\(741\) −2.00000 −0.0734718
\(742\) 0 0
\(743\) 40.4674 1.48460 0.742302 0.670065i \(-0.233734\pi\)
0.742302 + 0.670065i \(0.233734\pi\)
\(744\) 0 0
\(745\) −19.1168 −0.700387
\(746\) 0 0
\(747\) 2.74456 0.100418
\(748\) 0 0
\(749\) 7.72281 0.282185
\(750\) 0 0
\(751\) −2.97825 −0.108678 −0.0543390 0.998523i \(-0.517305\pi\)
−0.0543390 + 0.998523i \(0.517305\pi\)
\(752\) 0 0
\(753\) 4.37228 0.159335
\(754\) 0 0
\(755\) −34.9783 −1.27299
\(756\) 0 0
\(757\) −32.0951 −1.16652 −0.583258 0.812287i \(-0.698222\pi\)
−0.583258 + 0.812287i \(0.698222\pi\)
\(758\) 0 0
\(759\) −12.0000 −0.435572
\(760\) 0 0
\(761\) 41.5842 1.50743 0.753713 0.657203i \(-0.228261\pi\)
0.753713 + 0.657203i \(0.228261\pi\)
\(762\) 0 0
\(763\) 38.5109 1.39419
\(764\) 0 0
\(765\) 19.1168 0.691171
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 21.1168 0.761493 0.380746 0.924679i \(-0.375667\pi\)
0.380746 + 0.924679i \(0.375667\pi\)
\(770\) 0 0
\(771\) −5.48913 −0.197686
\(772\) 0 0
\(773\) −41.4891 −1.49226 −0.746130 0.665800i \(-0.768090\pi\)
−0.746130 + 0.665800i \(0.768090\pi\)
\(774\) 0 0
\(775\) 66.9783 2.40593
\(776\) 0 0
\(777\) 16.0000 0.573997
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −52.4674 −1.87743
\(782\) 0 0
\(783\) −8.74456 −0.312505
\(784\) 0 0
\(785\) −67.7228 −2.41713
\(786\) 0 0
\(787\) 21.4891 0.766005 0.383002 0.923747i \(-0.374890\pi\)
0.383002 + 0.923747i \(0.374890\pi\)
\(788\) 0 0
\(789\) 2.13859 0.0761359
\(790\) 0 0
\(791\) −7.72281 −0.274592
\(792\) 0 0
\(793\) −16.7446 −0.594617
\(794\) 0 0
\(795\) −38.2337 −1.35601
\(796\) 0 0
\(797\) 6.51087 0.230627 0.115314 0.993329i \(-0.463213\pi\)
0.115314 + 0.993329i \(0.463213\pi\)
\(798\) 0 0
\(799\) 33.3505 1.17986
\(800\) 0 0
\(801\) 3.25544 0.115025
\(802\) 0 0
\(803\) 1.62772 0.0574409
\(804\) 0 0
\(805\) 28.4674 1.00334
\(806\) 0 0
\(807\) 3.25544 0.114597
\(808\) 0 0
\(809\) −19.6277 −0.690074 −0.345037 0.938589i \(-0.612134\pi\)
−0.345037 + 0.938589i \(0.612134\pi\)
\(810\) 0 0
\(811\) −34.2337 −1.20211 −0.601054 0.799209i \(-0.705252\pi\)
−0.601054 + 0.799209i \(0.705252\pi\)
\(812\) 0 0
\(813\) 4.00000 0.140286
\(814\) 0 0
\(815\) 58.9783 2.06592
\(816\) 0 0
\(817\) −2.37228 −0.0829956
\(818\) 0 0
\(819\) 4.74456 0.165788
\(820\) 0 0
\(821\) −41.5842 −1.45130 −0.725650 0.688064i \(-0.758461\pi\)
−0.725650 + 0.688064i \(0.758461\pi\)
\(822\) 0 0
\(823\) −37.3505 −1.30196 −0.650979 0.759096i \(-0.725641\pi\)
−0.650979 + 0.759096i \(0.725641\pi\)
\(824\) 0 0
\(825\) −61.7228 −2.14891
\(826\) 0 0
\(827\) 17.4891 0.608156 0.304078 0.952647i \(-0.401652\pi\)
0.304078 + 0.952647i \(0.401652\pi\)
\(828\) 0 0
\(829\) 7.76631 0.269735 0.134868 0.990864i \(-0.456939\pi\)
0.134868 + 0.990864i \(0.456939\pi\)
\(830\) 0 0
\(831\) 6.88316 0.238774
\(832\) 0 0
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) 62.2337 2.15369
\(836\) 0 0
\(837\) −4.74456 −0.163996
\(838\) 0 0
\(839\) 8.74456 0.301896 0.150948 0.988542i \(-0.451767\pi\)
0.150948 + 0.988542i \(0.451767\pi\)
\(840\) 0 0
\(841\) 47.4674 1.63681
\(842\) 0 0
\(843\) 15.2554 0.525425
\(844\) 0 0
\(845\) 39.3505 1.35370
\(846\) 0 0
\(847\) −19.2554 −0.661625
\(848\) 0 0
\(849\) −8.88316 −0.304869
\(850\) 0 0
\(851\) 18.5109 0.634545
\(852\) 0 0
\(853\) −31.4891 −1.07817 −0.539084 0.842252i \(-0.681229\pi\)
−0.539084 + 0.842252i \(0.681229\pi\)
\(854\) 0 0
\(855\) 4.37228 0.149529
\(856\) 0 0
\(857\) 10.9783 0.375010 0.187505 0.982264i \(-0.439960\pi\)
0.187505 + 0.982264i \(0.439960\pi\)
\(858\) 0 0
\(859\) −44.6060 −1.52194 −0.760968 0.648789i \(-0.775276\pi\)
−0.760968 + 0.648789i \(0.775276\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.02175 0.0347808 0.0173904 0.999849i \(-0.494464\pi\)
0.0173904 + 0.999849i \(0.494464\pi\)
\(864\) 0 0
\(865\) 90.7011 3.08393
\(866\) 0 0
\(867\) −2.11684 −0.0718918
\(868\) 0 0
\(869\) 34.9783 1.18656
\(870\) 0 0
\(871\) 26.9783 0.914123
\(872\) 0 0
\(873\) 14.0000 0.473828
\(874\) 0 0
\(875\) 94.5625 3.19679
\(876\) 0 0
\(877\) 16.5109 0.557533 0.278766 0.960359i \(-0.410075\pi\)
0.278766 + 0.960359i \(0.410075\pi\)
\(878\) 0 0
\(879\) 6.51087 0.219606
\(880\) 0 0
\(881\) −56.8397 −1.91498 −0.957488 0.288472i \(-0.906853\pi\)
−0.957488 + 0.288472i \(0.906853\pi\)
\(882\) 0 0
\(883\) 1.35053 0.0454490 0.0227245 0.999742i \(-0.492766\pi\)
0.0227245 + 0.999742i \(0.492766\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −41.4891 −1.39307 −0.696534 0.717524i \(-0.745276\pi\)
−0.696534 + 0.717524i \(0.745276\pi\)
\(888\) 0 0
\(889\) −52.7446 −1.76900
\(890\) 0 0
\(891\) 4.37228 0.146477
\(892\) 0 0
\(893\) 7.62772 0.255252
\(894\) 0 0
\(895\) 14.2337 0.475780
\(896\) 0 0
\(897\) 5.48913 0.183277
\(898\) 0 0
\(899\) 41.4891 1.38374
\(900\) 0 0
\(901\) 38.2337 1.27375
\(902\) 0 0
\(903\) 5.62772 0.187279
\(904\) 0 0
\(905\) −96.1902 −3.19747
\(906\) 0 0
\(907\) −5.76631 −0.191467 −0.0957336 0.995407i \(-0.530520\pi\)
−0.0957336 + 0.995407i \(0.530520\pi\)
\(908\) 0 0
\(909\) 11.4891 0.381070
\(910\) 0 0
\(911\) −38.2337 −1.26674 −0.633369 0.773850i \(-0.718329\pi\)
−0.633369 + 0.773850i \(0.718329\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 0 0
\(915\) 36.6060 1.21016
\(916\) 0 0
\(917\) 23.3940 0.772539
\(918\) 0 0
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) 0 0
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) 95.2119 3.13055
\(926\) 0 0
\(927\) −12.7446 −0.418586
\(928\) 0 0
\(929\) 11.4891 0.376946 0.188473 0.982078i \(-0.439646\pi\)
0.188473 + 0.982078i \(0.439646\pi\)
\(930\) 0 0
\(931\) 1.37228 0.0449747
\(932\) 0 0
\(933\) −9.86141 −0.322848
\(934\) 0 0
\(935\) 83.5842 2.73350
\(936\) 0 0
\(937\) 16.8397 0.550128 0.275064 0.961426i \(-0.411301\pi\)
0.275064 + 0.961426i \(0.411301\pi\)
\(938\) 0 0
\(939\) −14.0000 −0.456873
\(940\) 0 0
\(941\) −27.2554 −0.888502 −0.444251 0.895902i \(-0.646530\pi\)
−0.444251 + 0.895902i \(0.646530\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −10.3723 −0.337410
\(946\) 0 0
\(947\) −13.7228 −0.445932 −0.222966 0.974826i \(-0.571574\pi\)
−0.222966 + 0.974826i \(0.571574\pi\)
\(948\) 0 0
\(949\) −0.744563 −0.0241695
\(950\) 0 0
\(951\) 6.51087 0.211130
\(952\) 0 0
\(953\) 16.4674 0.533431 0.266715 0.963775i \(-0.414062\pi\)
0.266715 + 0.963775i \(0.414062\pi\)
\(954\) 0 0
\(955\) 71.5842 2.31641
\(956\) 0 0
\(957\) −38.2337 −1.23592
\(958\) 0 0
\(959\) 10.3723 0.334938
\(960\) 0 0
\(961\) −8.48913 −0.273843
\(962\) 0 0
\(963\) −3.25544 −0.104905
\(964\) 0 0
\(965\) −22.9783 −0.739696
\(966\) 0 0
\(967\) −50.9783 −1.63935 −0.819675 0.572829i \(-0.805846\pi\)
−0.819675 + 0.572829i \(0.805846\pi\)
\(968\) 0 0
\(969\) −4.37228 −0.140458
\(970\) 0 0
\(971\) 1.02175 0.0327895 0.0163947 0.999866i \(-0.494781\pi\)
0.0163947 + 0.999866i \(0.494781\pi\)
\(972\) 0 0
\(973\) 22.8397 0.732206
\(974\) 0 0
\(975\) 28.2337 0.904202
\(976\) 0 0
\(977\) 7.72281 0.247075 0.123537 0.992340i \(-0.460576\pi\)
0.123537 + 0.992340i \(0.460576\pi\)
\(978\) 0 0
\(979\) 14.2337 0.454911
\(980\) 0 0
\(981\) −16.2337 −0.518302
\(982\) 0 0
\(983\) −6.51087 −0.207665 −0.103832 0.994595i \(-0.533111\pi\)
−0.103832 + 0.994595i \(0.533111\pi\)
\(984\) 0 0
\(985\) 50.2337 1.60058
\(986\) 0 0
\(987\) −18.0951 −0.575973
\(988\) 0 0
\(989\) 6.51087 0.207034
\(990\) 0 0
\(991\) −37.9565 −1.20573 −0.602864 0.797844i \(-0.705974\pi\)
−0.602864 + 0.797844i \(0.705974\pi\)
\(992\) 0 0
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) −13.6277 −0.432028
\(996\) 0 0
\(997\) −18.8832 −0.598036 −0.299018 0.954248i \(-0.596659\pi\)
−0.299018 + 0.954248i \(0.596659\pi\)
\(998\) 0 0
\(999\) −6.74456 −0.213389
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 3648.2.a.bk.1.1 2
4.3 odd 2 3648.2.a.bq.1.1 2
8.3 odd 2 912.2.a.n.1.2 2
8.5 even 2 228.2.a.c.1.2 2
24.5 odd 2 684.2.a.d.1.1 2
24.11 even 2 2736.2.a.y.1.1 2
40.13 odd 4 5700.2.f.m.3649.4 4
40.29 even 2 5700.2.a.t.1.2 2
40.37 odd 4 5700.2.f.m.3649.1 4
152.37 odd 2 4332.2.a.i.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
228.2.a.c.1.2 2 8.5 even 2
684.2.a.d.1.1 2 24.5 odd 2
912.2.a.n.1.2 2 8.3 odd 2
2736.2.a.y.1.1 2 24.11 even 2
3648.2.a.bk.1.1 2 1.1 even 1 trivial
3648.2.a.bq.1.1 2 4.3 odd 2
4332.2.a.i.1.2 2 152.37 odd 2
5700.2.a.t.1.2 2 40.29 even 2
5700.2.f.m.3649.1 4 40.37 odd 4
5700.2.f.m.3649.4 4 40.13 odd 4