Properties

Label 9680.2.a.bt.1.2
Level $9680$
Weight $2$
Character 9680.1
Self dual yes
Analytic conductor $77.295$
Analytic rank $0$
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] = [9680,2,Mod(1,9680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9680, 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("9680.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9680.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: \(77.2951891566\)
Analytic rank: \(0\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 110)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 9680.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.37228 q^{3} +1.00000 q^{5} +3.37228 q^{7} +8.37228 q^{9} -2.00000 q^{13} +3.37228 q^{15} -1.37228 q^{17} +0.627719 q^{19} +11.3723 q^{21} -2.74456 q^{23} +1.00000 q^{25} +18.1168 q^{27} -1.37228 q^{29} -3.37228 q^{31} +3.37228 q^{35} +9.37228 q^{37} -6.74456 q^{39} +11.4891 q^{41} -4.00000 q^{43} +8.37228 q^{45} -2.74456 q^{47} +4.37228 q^{49} -4.62772 q^{51} -4.11684 q^{53} +2.11684 q^{57} +2.74456 q^{59} +5.37228 q^{61} +28.2337 q^{63} -2.00000 q^{65} -8.00000 q^{67} -9.25544 q^{69} -10.1168 q^{71} +15.4891 q^{73} +3.37228 q^{75} -1.25544 q^{79} +35.9783 q^{81} -2.74456 q^{83} -1.37228 q^{85} -4.62772 q^{87} -1.37228 q^{89} -6.74456 q^{91} -11.3723 q^{93} +0.627719 q^{95} -12.7446 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 2 q^{5} + q^{7} + 11 q^{9} - 4 q^{13} + q^{15} + 3 q^{17} + 7 q^{19} + 17 q^{21} + 6 q^{23} + 2 q^{25} + 19 q^{27} + 3 q^{29} - q^{31} + q^{35} + 13 q^{37} - 2 q^{39} - 8 q^{43} + 11 q^{45}+ \cdots - 14 q^{97}+O(q^{100}) \) 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
\(3\) 3.37228 1.94699 0.973494 0.228714i \(-0.0734519\pi\)
0.973494 + 0.228714i \(0.0734519\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 3.37228 1.27460 0.637301 0.770615i \(-0.280051\pi\)
0.637301 + 0.770615i \(0.280051\pi\)
\(8\) 0 0
\(9\) 8.37228 2.79076
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 3.37228 0.870719
\(16\) 0 0
\(17\) −1.37228 −0.332827 −0.166414 0.986056i \(-0.553219\pi\)
−0.166414 + 0.986056i \(0.553219\pi\)
\(18\) 0 0
\(19\) 0.627719 0.144009 0.0720043 0.997404i \(-0.477060\pi\)
0.0720043 + 0.997404i \(0.477060\pi\)
\(20\) 0 0
\(21\) 11.3723 2.48164
\(22\) 0 0
\(23\) −2.74456 −0.572281 −0.286140 0.958188i \(-0.592372\pi\)
−0.286140 + 0.958188i \(0.592372\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 18.1168 3.48659
\(28\) 0 0
\(29\) −1.37228 −0.254826 −0.127413 0.991850i \(-0.540667\pi\)
−0.127413 + 0.991850i \(0.540667\pi\)
\(30\) 0 0
\(31\) −3.37228 −0.605680 −0.302840 0.953041i \(-0.597935\pi\)
−0.302840 + 0.953041i \(0.597935\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.37228 0.570020
\(36\) 0 0
\(37\) 9.37228 1.54079 0.770397 0.637565i \(-0.220058\pi\)
0.770397 + 0.637565i \(0.220058\pi\)
\(38\) 0 0
\(39\) −6.74456 −1.07999
\(40\) 0 0
\(41\) 11.4891 1.79430 0.897150 0.441726i \(-0.145634\pi\)
0.897150 + 0.441726i \(0.145634\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 8.37228 1.24807
\(46\) 0 0
\(47\) −2.74456 −0.400336 −0.200168 0.979762i \(-0.564149\pi\)
−0.200168 + 0.979762i \(0.564149\pi\)
\(48\) 0 0
\(49\) 4.37228 0.624612
\(50\) 0 0
\(51\) −4.62772 −0.648010
\(52\) 0 0
\(53\) −4.11684 −0.565492 −0.282746 0.959195i \(-0.591245\pi\)
−0.282746 + 0.959195i \(0.591245\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.11684 0.280383
\(58\) 0 0
\(59\) 2.74456 0.357312 0.178656 0.983912i \(-0.442825\pi\)
0.178656 + 0.983912i \(0.442825\pi\)
\(60\) 0 0
\(61\) 5.37228 0.687850 0.343925 0.938997i \(-0.388243\pi\)
0.343925 + 0.938997i \(0.388243\pi\)
\(62\) 0 0
\(63\) 28.2337 3.55711
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) −9.25544 −1.11422
\(70\) 0 0
\(71\) −10.1168 −1.20065 −0.600324 0.799757i \(-0.704962\pi\)
−0.600324 + 0.799757i \(0.704962\pi\)
\(72\) 0 0
\(73\) 15.4891 1.81286 0.906432 0.422351i \(-0.138795\pi\)
0.906432 + 0.422351i \(0.138795\pi\)
\(74\) 0 0
\(75\) 3.37228 0.389398
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.25544 −0.141248 −0.0706239 0.997503i \(-0.522499\pi\)
−0.0706239 + 0.997503i \(0.522499\pi\)
\(80\) 0 0
\(81\) 35.9783 3.99758
\(82\) 0 0
\(83\) −2.74456 −0.301255 −0.150627 0.988591i \(-0.548129\pi\)
−0.150627 + 0.988591i \(0.548129\pi\)
\(84\) 0 0
\(85\) −1.37228 −0.148845
\(86\) 0 0
\(87\) −4.62772 −0.496144
\(88\) 0 0
\(89\) −1.37228 −0.145462 −0.0727308 0.997352i \(-0.523171\pi\)
−0.0727308 + 0.997352i \(0.523171\pi\)
\(90\) 0 0
\(91\) −6.74456 −0.707022
\(92\) 0 0
\(93\) −11.3723 −1.17925
\(94\) 0 0
\(95\) 0.627719 0.0644026
\(96\) 0 0
\(97\) −12.7446 −1.29401 −0.647007 0.762484i \(-0.723980\pi\)
−0.647007 + 0.762484i \(0.723980\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 9.48913 0.934991 0.467496 0.883995i \(-0.345156\pi\)
0.467496 + 0.883995i \(0.345156\pi\)
\(104\) 0 0
\(105\) 11.3723 1.10982
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 15.4891 1.48359 0.741795 0.670627i \(-0.233975\pi\)
0.741795 + 0.670627i \(0.233975\pi\)
\(110\) 0 0
\(111\) 31.6060 2.99991
\(112\) 0 0
\(113\) 3.25544 0.306246 0.153123 0.988207i \(-0.451067\pi\)
0.153123 + 0.988207i \(0.451067\pi\)
\(114\) 0 0
\(115\) −2.74456 −0.255932
\(116\) 0 0
\(117\) −16.7446 −1.54804
\(118\) 0 0
\(119\) −4.62772 −0.424222
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 38.7446 3.49348
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) −13.4891 −1.18765
\(130\) 0 0
\(131\) 22.1168 1.93236 0.966179 0.257873i \(-0.0830216\pi\)
0.966179 + 0.257873i \(0.0830216\pi\)
\(132\) 0 0
\(133\) 2.11684 0.183554
\(134\) 0 0
\(135\) 18.1168 1.55925
\(136\) 0 0
\(137\) 8.74456 0.747098 0.373549 0.927610i \(-0.378141\pi\)
0.373549 + 0.927610i \(0.378141\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −9.25544 −0.779448
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −1.37228 −0.113962
\(146\) 0 0
\(147\) 14.7446 1.21611
\(148\) 0 0
\(149\) −21.6060 −1.77003 −0.885015 0.465563i \(-0.845852\pi\)
−0.885015 + 0.465563i \(0.845852\pi\)
\(150\) 0 0
\(151\) −12.2337 −0.995563 −0.497782 0.867302i \(-0.665852\pi\)
−0.497782 + 0.867302i \(0.665852\pi\)
\(152\) 0 0
\(153\) −11.4891 −0.928841
\(154\) 0 0
\(155\) −3.37228 −0.270868
\(156\) 0 0
\(157\) 9.37228 0.747989 0.373995 0.927431i \(-0.377988\pi\)
0.373995 + 0.927431i \(0.377988\pi\)
\(158\) 0 0
\(159\) −13.8832 −1.10101
\(160\) 0 0
\(161\) −9.25544 −0.729431
\(162\) 0 0
\(163\) 5.88316 0.460804 0.230402 0.973095i \(-0.425996\pi\)
0.230402 + 0.973095i \(0.425996\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4.62772 −0.358104 −0.179052 0.983840i \(-0.557303\pi\)
−0.179052 + 0.983840i \(0.557303\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 5.25544 0.401893
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 3.37228 0.254921
\(176\) 0 0
\(177\) 9.25544 0.695681
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 18.1168 1.33924
\(184\) 0 0
\(185\) 9.37228 0.689064
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 61.0951 4.44401
\(190\) 0 0
\(191\) 5.48913 0.397179 0.198590 0.980083i \(-0.436364\pi\)
0.198590 + 0.980083i \(0.436364\pi\)
\(192\) 0 0
\(193\) −14.8614 −1.06975 −0.534874 0.844932i \(-0.679641\pi\)
−0.534874 + 0.844932i \(0.679641\pi\)
\(194\) 0 0
\(195\) −6.74456 −0.482988
\(196\) 0 0
\(197\) 20.7446 1.47799 0.738994 0.673712i \(-0.235301\pi\)
0.738994 + 0.673712i \(0.235301\pi\)
\(198\) 0 0
\(199\) −18.1168 −1.28427 −0.642135 0.766592i \(-0.721951\pi\)
−0.642135 + 0.766592i \(0.721951\pi\)
\(200\) 0 0
\(201\) −26.9783 −1.90290
\(202\) 0 0
\(203\) −4.62772 −0.324802
\(204\) 0 0
\(205\) 11.4891 0.802435
\(206\) 0 0
\(207\) −22.9783 −1.59710
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 6.11684 0.421101 0.210550 0.977583i \(-0.432474\pi\)
0.210550 + 0.977583i \(0.432474\pi\)
\(212\) 0 0
\(213\) −34.1168 −2.33765
\(214\) 0 0
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) −11.3723 −0.772001
\(218\) 0 0
\(219\) 52.2337 3.52963
\(220\) 0 0
\(221\) 2.74456 0.184619
\(222\) 0 0
\(223\) 18.7446 1.25523 0.627614 0.778524i \(-0.284031\pi\)
0.627614 + 0.778524i \(0.284031\pi\)
\(224\) 0 0
\(225\) 8.37228 0.558152
\(226\) 0 0
\(227\) −2.74456 −0.182163 −0.0910815 0.995843i \(-0.529032\pi\)
−0.0910815 + 0.995843i \(0.529032\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.37228 −0.0899011 −0.0449506 0.998989i \(-0.514313\pi\)
−0.0449506 + 0.998989i \(0.514313\pi\)
\(234\) 0 0
\(235\) −2.74456 −0.179036
\(236\) 0 0
\(237\) −4.23369 −0.275008
\(238\) 0 0
\(239\) −14.7446 −0.953746 −0.476873 0.878972i \(-0.658230\pi\)
−0.476873 + 0.878972i \(0.658230\pi\)
\(240\) 0 0
\(241\) 22.0000 1.41714 0.708572 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) 0 0
\(243\) 66.9783 4.29666
\(244\) 0 0
\(245\) 4.37228 0.279335
\(246\) 0 0
\(247\) −1.25544 −0.0798816
\(248\) 0 0
\(249\) −9.25544 −0.586540
\(250\) 0 0
\(251\) −2.74456 −0.173235 −0.0866176 0.996242i \(-0.527606\pi\)
−0.0866176 + 0.996242i \(0.527606\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −4.62772 −0.289799
\(256\) 0 0
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) 31.6060 1.96390
\(260\) 0 0
\(261\) −11.4891 −0.711159
\(262\) 0 0
\(263\) −24.8614 −1.53302 −0.766510 0.642232i \(-0.778008\pi\)
−0.766510 + 0.642232i \(0.778008\pi\)
\(264\) 0 0
\(265\) −4.11684 −0.252896
\(266\) 0 0
\(267\) −4.62772 −0.283212
\(268\) 0 0
\(269\) −8.74456 −0.533165 −0.266583 0.963812i \(-0.585895\pi\)
−0.266583 + 0.963812i \(0.585895\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) −22.7446 −1.37656
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 12.7446 0.765747 0.382873 0.923801i \(-0.374935\pi\)
0.382873 + 0.923801i \(0.374935\pi\)
\(278\) 0 0
\(279\) −28.2337 −1.69031
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) 5.25544 0.312403 0.156202 0.987725i \(-0.450075\pi\)
0.156202 + 0.987725i \(0.450075\pi\)
\(284\) 0 0
\(285\) 2.11684 0.125391
\(286\) 0 0
\(287\) 38.7446 2.28702
\(288\) 0 0
\(289\) −15.1168 −0.889226
\(290\) 0 0
\(291\) −42.9783 −2.51943
\(292\) 0 0
\(293\) −23.4891 −1.37225 −0.686125 0.727484i \(-0.740690\pi\)
−0.686125 + 0.727484i \(0.740690\pi\)
\(294\) 0 0
\(295\) 2.74456 0.159795
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5.48913 0.317444
\(300\) 0 0
\(301\) −13.4891 −0.777500
\(302\) 0 0
\(303\) −20.2337 −1.16240
\(304\) 0 0
\(305\) 5.37228 0.307616
\(306\) 0 0
\(307\) 5.25544 0.299944 0.149972 0.988690i \(-0.452082\pi\)
0.149972 + 0.988690i \(0.452082\pi\)
\(308\) 0 0
\(309\) 32.0000 1.82042
\(310\) 0 0
\(311\) −19.3723 −1.09850 −0.549251 0.835658i \(-0.685087\pi\)
−0.549251 + 0.835658i \(0.685087\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 0 0
\(315\) 28.2337 1.59079
\(316\) 0 0
\(317\) −24.3505 −1.36766 −0.683831 0.729640i \(-0.739687\pi\)
−0.683831 + 0.729640i \(0.739687\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −40.4674 −2.25867
\(322\) 0 0
\(323\) −0.861407 −0.0479299
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 0 0
\(327\) 52.2337 2.88853
\(328\) 0 0
\(329\) −9.25544 −0.510269
\(330\) 0 0
\(331\) −30.9783 −1.70272 −0.851359 0.524583i \(-0.824221\pi\)
−0.851359 + 0.524583i \(0.824221\pi\)
\(332\) 0 0
\(333\) 78.4674 4.29999
\(334\) 0 0
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) −24.1168 −1.31373 −0.656864 0.754009i \(-0.728117\pi\)
−0.656864 + 0.754009i \(0.728117\pi\)
\(338\) 0 0
\(339\) 10.9783 0.596257
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −8.86141 −0.478471
\(344\) 0 0
\(345\) −9.25544 −0.498296
\(346\) 0 0
\(347\) −32.2337 −1.73040 −0.865198 0.501431i \(-0.832807\pi\)
−0.865198 + 0.501431i \(0.832807\pi\)
\(348\) 0 0
\(349\) −19.4891 −1.04323 −0.521614 0.853181i \(-0.674670\pi\)
−0.521614 + 0.853181i \(0.674670\pi\)
\(350\) 0 0
\(351\) −36.2337 −1.93401
\(352\) 0 0
\(353\) −0.510875 −0.0271911 −0.0135956 0.999908i \(-0.504328\pi\)
−0.0135956 + 0.999908i \(0.504328\pi\)
\(354\) 0 0
\(355\) −10.1168 −0.536946
\(356\) 0 0
\(357\) −15.6060 −0.825955
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −18.6060 −0.979262
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 15.4891 0.810738
\(366\) 0 0
\(367\) 4.00000 0.208798 0.104399 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) 0 0
\(369\) 96.1902 5.00746
\(370\) 0 0
\(371\) −13.8832 −0.720778
\(372\) 0 0
\(373\) −31.4891 −1.63045 −0.815223 0.579148i \(-0.803385\pi\)
−0.815223 + 0.579148i \(0.803385\pi\)
\(374\) 0 0
\(375\) 3.37228 0.174144
\(376\) 0 0
\(377\) 2.74456 0.141352
\(378\) 0 0
\(379\) 0.233688 0.0120037 0.00600187 0.999982i \(-0.498090\pi\)
0.00600187 + 0.999982i \(0.498090\pi\)
\(380\) 0 0
\(381\) 26.9783 1.38214
\(382\) 0 0
\(383\) −32.2337 −1.64706 −0.823532 0.567269i \(-0.808000\pi\)
−0.823532 + 0.567269i \(0.808000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −33.4891 −1.70235
\(388\) 0 0
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 3.76631 0.190471
\(392\) 0 0
\(393\) 74.5842 3.76228
\(394\) 0 0
\(395\) −1.25544 −0.0631679
\(396\) 0 0
\(397\) 24.9783 1.25362 0.626811 0.779171i \(-0.284360\pi\)
0.626811 + 0.779171i \(0.284360\pi\)
\(398\) 0 0
\(399\) 7.13859 0.357377
\(400\) 0 0
\(401\) 13.3723 0.667780 0.333890 0.942612i \(-0.391639\pi\)
0.333890 + 0.942612i \(0.391639\pi\)
\(402\) 0 0
\(403\) 6.74456 0.335971
\(404\) 0 0
\(405\) 35.9783 1.78777
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.76631 0.0873385 0.0436693 0.999046i \(-0.486095\pi\)
0.0436693 + 0.999046i \(0.486095\pi\)
\(410\) 0 0
\(411\) 29.4891 1.45459
\(412\) 0 0
\(413\) 9.25544 0.455430
\(414\) 0 0
\(415\) −2.74456 −0.134725
\(416\) 0 0
\(417\) −13.4891 −0.660565
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 10.2337 0.498759 0.249380 0.968406i \(-0.419773\pi\)
0.249380 + 0.968406i \(0.419773\pi\)
\(422\) 0 0
\(423\) −22.9783 −1.11724
\(424\) 0 0
\(425\) −1.37228 −0.0665654
\(426\) 0 0
\(427\) 18.1168 0.876736
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −34.9783 −1.68484 −0.842422 0.538819i \(-0.818871\pi\)
−0.842422 + 0.538819i \(0.818871\pi\)
\(432\) 0 0
\(433\) 27.7228 1.33227 0.666137 0.745830i \(-0.267947\pi\)
0.666137 + 0.745830i \(0.267947\pi\)
\(434\) 0 0
\(435\) −4.62772 −0.221882
\(436\) 0 0
\(437\) −1.72281 −0.0824133
\(438\) 0 0
\(439\) 18.9783 0.905782 0.452891 0.891566i \(-0.350393\pi\)
0.452891 + 0.891566i \(0.350393\pi\)
\(440\) 0 0
\(441\) 36.6060 1.74314
\(442\) 0 0
\(443\) −29.4891 −1.40107 −0.700535 0.713618i \(-0.747055\pi\)
−0.700535 + 0.713618i \(0.747055\pi\)
\(444\) 0 0
\(445\) −1.37228 −0.0650524
\(446\) 0 0
\(447\) −72.8614 −3.44623
\(448\) 0 0
\(449\) 28.9783 1.36757 0.683784 0.729684i \(-0.260333\pi\)
0.683784 + 0.729684i \(0.260333\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −41.2554 −1.93835
\(454\) 0 0
\(455\) −6.74456 −0.316190
\(456\) 0 0
\(457\) 16.3505 0.764846 0.382423 0.923987i \(-0.375090\pi\)
0.382423 + 0.923987i \(0.375090\pi\)
\(458\) 0 0
\(459\) −24.8614 −1.16043
\(460\) 0 0
\(461\) −16.1168 −0.750636 −0.375318 0.926896i \(-0.622467\pi\)
−0.375318 + 0.926896i \(0.622467\pi\)
\(462\) 0 0
\(463\) 0.233688 0.0108604 0.00543020 0.999985i \(-0.498272\pi\)
0.00543020 + 0.999985i \(0.498272\pi\)
\(464\) 0 0
\(465\) −11.3723 −0.527377
\(466\) 0 0
\(467\) 19.3723 0.896442 0.448221 0.893923i \(-0.352058\pi\)
0.448221 + 0.893923i \(0.352058\pi\)
\(468\) 0 0
\(469\) −26.9783 −1.24574
\(470\) 0 0
\(471\) 31.6060 1.45633
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0.627719 0.0288017
\(476\) 0 0
\(477\) −34.4674 −1.57815
\(478\) 0 0
\(479\) 5.48913 0.250805 0.125402 0.992106i \(-0.459978\pi\)
0.125402 + 0.992106i \(0.459978\pi\)
\(480\) 0 0
\(481\) −18.7446 −0.854678
\(482\) 0 0
\(483\) −31.2119 −1.42019
\(484\) 0 0
\(485\) −12.7446 −0.578701
\(486\) 0 0
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 0 0
\(489\) 19.8397 0.897180
\(490\) 0 0
\(491\) −7.37228 −0.332706 −0.166353 0.986066i \(-0.553199\pi\)
−0.166353 + 0.986066i \(0.553199\pi\)
\(492\) 0 0
\(493\) 1.88316 0.0848131
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −34.1168 −1.53035
\(498\) 0 0
\(499\) 33.4891 1.49918 0.749590 0.661903i \(-0.230251\pi\)
0.749590 + 0.661903i \(0.230251\pi\)
\(500\) 0 0
\(501\) −15.6060 −0.697223
\(502\) 0 0
\(503\) −34.9783 −1.55960 −0.779802 0.626027i \(-0.784680\pi\)
−0.779802 + 0.626027i \(0.784680\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 0 0
\(507\) −30.3505 −1.34791
\(508\) 0 0
\(509\) 9.76631 0.432884 0.216442 0.976295i \(-0.430555\pi\)
0.216442 + 0.976295i \(0.430555\pi\)
\(510\) 0 0
\(511\) 52.2337 2.31068
\(512\) 0 0
\(513\) 11.3723 0.502098
\(514\) 0 0
\(515\) 9.48913 0.418141
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 20.2337 0.888160
\(520\) 0 0
\(521\) 12.5109 0.548111 0.274056 0.961714i \(-0.411635\pi\)
0.274056 + 0.961714i \(0.411635\pi\)
\(522\) 0 0
\(523\) 30.9783 1.35458 0.677292 0.735714i \(-0.263153\pi\)
0.677292 + 0.735714i \(0.263153\pi\)
\(524\) 0 0
\(525\) 11.3723 0.496327
\(526\) 0 0
\(527\) 4.62772 0.201587
\(528\) 0 0
\(529\) −15.4674 −0.672495
\(530\) 0 0
\(531\) 22.9783 0.997171
\(532\) 0 0
\(533\) −22.9783 −0.995299
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) 0 0
\(537\) 40.4674 1.74630
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 20.1168 0.864891 0.432445 0.901660i \(-0.357651\pi\)
0.432445 + 0.901660i \(0.357651\pi\)
\(542\) 0 0
\(543\) −33.7228 −1.44718
\(544\) 0 0
\(545\) 15.4891 0.663481
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 0 0
\(549\) 44.9783 1.91962
\(550\) 0 0
\(551\) −0.861407 −0.0366972
\(552\) 0 0
\(553\) −4.23369 −0.180035
\(554\) 0 0
\(555\) 31.6060 1.34160
\(556\) 0 0
\(557\) −4.97825 −0.210935 −0.105468 0.994423i \(-0.533634\pi\)
−0.105468 + 0.994423i \(0.533634\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −8.23369 −0.347009 −0.173504 0.984833i \(-0.555509\pi\)
−0.173504 + 0.984833i \(0.555509\pi\)
\(564\) 0 0
\(565\) 3.25544 0.136957
\(566\) 0 0
\(567\) 121.329 5.09533
\(568\) 0 0
\(569\) 15.2554 0.639541 0.319771 0.947495i \(-0.396394\pi\)
0.319771 + 0.947495i \(0.396394\pi\)
\(570\) 0 0
\(571\) 15.3723 0.643310 0.321655 0.946857i \(-0.395761\pi\)
0.321655 + 0.946857i \(0.395761\pi\)
\(572\) 0 0
\(573\) 18.5109 0.773303
\(574\) 0 0
\(575\) −2.74456 −0.114456
\(576\) 0 0
\(577\) 36.9783 1.53942 0.769712 0.638391i \(-0.220400\pi\)
0.769712 + 0.638391i \(0.220400\pi\)
\(578\) 0 0
\(579\) −50.1168 −2.08278
\(580\) 0 0
\(581\) −9.25544 −0.383980
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −16.7446 −0.692302
\(586\) 0 0
\(587\) −24.8614 −1.02614 −0.513070 0.858347i \(-0.671492\pi\)
−0.513070 + 0.858347i \(0.671492\pi\)
\(588\) 0 0
\(589\) −2.11684 −0.0872230
\(590\) 0 0
\(591\) 69.9565 2.87763
\(592\) 0 0
\(593\) 12.5109 0.513760 0.256880 0.966443i \(-0.417305\pi\)
0.256880 + 0.966443i \(0.417305\pi\)
\(594\) 0 0
\(595\) −4.62772 −0.189718
\(596\) 0 0
\(597\) −61.0951 −2.50046
\(598\) 0 0
\(599\) 39.6060 1.61826 0.809128 0.587632i \(-0.199940\pi\)
0.809128 + 0.587632i \(0.199940\pi\)
\(600\) 0 0
\(601\) 16.5109 0.673493 0.336746 0.941595i \(-0.390674\pi\)
0.336746 + 0.941595i \(0.390674\pi\)
\(602\) 0 0
\(603\) −66.9783 −2.72757
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −5.88316 −0.238790 −0.119395 0.992847i \(-0.538095\pi\)
−0.119395 + 0.992847i \(0.538095\pi\)
\(608\) 0 0
\(609\) −15.6060 −0.632386
\(610\) 0 0
\(611\) 5.48913 0.222066
\(612\) 0 0
\(613\) −20.5109 −0.828426 −0.414213 0.910180i \(-0.635943\pi\)
−0.414213 + 0.910180i \(0.635943\pi\)
\(614\) 0 0
\(615\) 38.7446 1.56233
\(616\) 0 0
\(617\) −2.23369 −0.0899249 −0.0449624 0.998989i \(-0.514317\pi\)
−0.0449624 + 0.998989i \(0.514317\pi\)
\(618\) 0 0
\(619\) 44.4674 1.78729 0.893647 0.448770i \(-0.148138\pi\)
0.893647 + 0.448770i \(0.148138\pi\)
\(620\) 0 0
\(621\) −49.7228 −1.99531
\(622\) 0 0
\(623\) −4.62772 −0.185406
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −12.8614 −0.512818
\(630\) 0 0
\(631\) −42.1168 −1.67665 −0.838323 0.545175i \(-0.816463\pi\)
−0.838323 + 0.545175i \(0.816463\pi\)
\(632\) 0 0
\(633\) 20.6277 0.819878
\(634\) 0 0
\(635\) 8.00000 0.317470
\(636\) 0 0
\(637\) −8.74456 −0.346472
\(638\) 0 0
\(639\) −84.7011 −3.35072
\(640\) 0 0
\(641\) −27.0951 −1.07019 −0.535096 0.844791i \(-0.679725\pi\)
−0.535096 + 0.844791i \(0.679725\pi\)
\(642\) 0 0
\(643\) 5.88316 0.232009 0.116005 0.993249i \(-0.462991\pi\)
0.116005 + 0.993249i \(0.462991\pi\)
\(644\) 0 0
\(645\) −13.4891 −0.531134
\(646\) 0 0
\(647\) 37.7228 1.48304 0.741518 0.670933i \(-0.234106\pi\)
0.741518 + 0.670933i \(0.234106\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −38.3505 −1.50308
\(652\) 0 0
\(653\) 10.6277 0.415895 0.207947 0.978140i \(-0.433322\pi\)
0.207947 + 0.978140i \(0.433322\pi\)
\(654\) 0 0
\(655\) 22.1168 0.864177
\(656\) 0 0
\(657\) 129.679 5.05927
\(658\) 0 0
\(659\) 12.8614 0.501009 0.250505 0.968115i \(-0.419403\pi\)
0.250505 + 0.968115i \(0.419403\pi\)
\(660\) 0 0
\(661\) 8.51087 0.331035 0.165517 0.986207i \(-0.447071\pi\)
0.165517 + 0.986207i \(0.447071\pi\)
\(662\) 0 0
\(663\) 9.25544 0.359451
\(664\) 0 0
\(665\) 2.11684 0.0820877
\(666\) 0 0
\(667\) 3.76631 0.145832
\(668\) 0 0
\(669\) 63.2119 2.44391
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −14.8614 −0.572865 −0.286433 0.958100i \(-0.592469\pi\)
−0.286433 + 0.958100i \(0.592469\pi\)
\(674\) 0 0
\(675\) 18.1168 0.697318
\(676\) 0 0
\(677\) −3.25544 −0.125117 −0.0625583 0.998041i \(-0.519926\pi\)
−0.0625583 + 0.998041i \(0.519926\pi\)
\(678\) 0 0
\(679\) −42.9783 −1.64935
\(680\) 0 0
\(681\) −9.25544 −0.354669
\(682\) 0 0
\(683\) 28.6277 1.09541 0.547705 0.836672i \(-0.315502\pi\)
0.547705 + 0.836672i \(0.315502\pi\)
\(684\) 0 0
\(685\) 8.74456 0.334113
\(686\) 0 0
\(687\) −33.7228 −1.28661
\(688\) 0 0
\(689\) 8.23369 0.313679
\(690\) 0 0
\(691\) −40.2337 −1.53056 −0.765281 0.643697i \(-0.777400\pi\)
−0.765281 + 0.643697i \(0.777400\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) −15.7663 −0.597192
\(698\) 0 0
\(699\) −4.62772 −0.175036
\(700\) 0 0
\(701\) 37.3723 1.41153 0.705766 0.708445i \(-0.250603\pi\)
0.705766 + 0.708445i \(0.250603\pi\)
\(702\) 0 0
\(703\) 5.88316 0.221887
\(704\) 0 0
\(705\) −9.25544 −0.348580
\(706\) 0 0
\(707\) −20.2337 −0.760966
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) −10.5109 −0.394189
\(712\) 0 0
\(713\) 9.25544 0.346619
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −49.7228 −1.85693
\(718\) 0 0
\(719\) −13.8832 −0.517754 −0.258877 0.965910i \(-0.583352\pi\)
−0.258877 + 0.965910i \(0.583352\pi\)
\(720\) 0 0
\(721\) 32.0000 1.19174
\(722\) 0 0
\(723\) 74.1902 2.75916
\(724\) 0 0
\(725\) −1.37228 −0.0509652
\(726\) 0 0
\(727\) 24.2337 0.898778 0.449389 0.893336i \(-0.351642\pi\)
0.449389 + 0.893336i \(0.351642\pi\)
\(728\) 0 0
\(729\) 117.935 4.36795
\(730\) 0 0
\(731\) 5.48913 0.203023
\(732\) 0 0
\(733\) −46.2337 −1.70768 −0.853840 0.520535i \(-0.825732\pi\)
−0.853840 + 0.520535i \(0.825732\pi\)
\(734\) 0 0
\(735\) 14.7446 0.543861
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −20.4674 −0.752905 −0.376452 0.926436i \(-0.622856\pi\)
−0.376452 + 0.926436i \(0.622856\pi\)
\(740\) 0 0
\(741\) −4.23369 −0.155528
\(742\) 0 0
\(743\) −4.62772 −0.169775 −0.0848873 0.996391i \(-0.527053\pi\)
−0.0848873 + 0.996391i \(0.527053\pi\)
\(744\) 0 0
\(745\) −21.6060 −0.791581
\(746\) 0 0
\(747\) −22.9783 −0.840730
\(748\) 0 0
\(749\) −40.4674 −1.47865
\(750\) 0 0
\(751\) −8.86141 −0.323357 −0.161679 0.986843i \(-0.551691\pi\)
−0.161679 + 0.986843i \(0.551691\pi\)
\(752\) 0 0
\(753\) −9.25544 −0.337287
\(754\) 0 0
\(755\) −12.2337 −0.445229
\(756\) 0 0
\(757\) −20.9783 −0.762467 −0.381234 0.924479i \(-0.624501\pi\)
−0.381234 + 0.924479i \(0.624501\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −4.97825 −0.180461 −0.0902307 0.995921i \(-0.528760\pi\)
−0.0902307 + 0.995921i \(0.528760\pi\)
\(762\) 0 0
\(763\) 52.2337 1.89099
\(764\) 0 0
\(765\) −11.4891 −0.415390
\(766\) 0 0
\(767\) −5.48913 −0.198201
\(768\) 0 0
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) 60.7011 2.18610
\(772\) 0 0
\(773\) −33.6060 −1.20872 −0.604361 0.796710i \(-0.706572\pi\)
−0.604361 + 0.796710i \(0.706572\pi\)
\(774\) 0 0
\(775\) −3.37228 −0.121136
\(776\) 0 0
\(777\) 106.584 3.82369
\(778\) 0 0
\(779\) 7.21194 0.258395
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −24.8614 −0.888474
\(784\) 0 0
\(785\) 9.37228 0.334511
\(786\) 0 0
\(787\) −44.4674 −1.58509 −0.792545 0.609813i \(-0.791245\pi\)
−0.792545 + 0.609813i \(0.791245\pi\)
\(788\) 0 0
\(789\) −83.8397 −2.98477
\(790\) 0 0
\(791\) 10.9783 0.390342
\(792\) 0 0
\(793\) −10.7446 −0.381551
\(794\) 0 0
\(795\) −13.8832 −0.492385
\(796\) 0 0
\(797\) 11.4891 0.406966 0.203483 0.979079i \(-0.434774\pi\)
0.203483 + 0.979079i \(0.434774\pi\)
\(798\) 0 0
\(799\) 3.76631 0.133243
\(800\) 0 0
\(801\) −11.4891 −0.405948
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −9.25544 −0.326211
\(806\) 0 0
\(807\) −29.4891 −1.03807
\(808\) 0 0
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) 44.8614 1.57530 0.787649 0.616125i \(-0.211298\pi\)
0.787649 + 0.616125i \(0.211298\pi\)
\(812\) 0 0
\(813\) −53.9565 −1.89234
\(814\) 0 0
\(815\) 5.88316 0.206078
\(816\) 0 0
\(817\) −2.51087 −0.0878444
\(818\) 0 0
\(819\) −56.4674 −1.97313
\(820\) 0 0
\(821\) −11.4891 −0.400973 −0.200487 0.979696i \(-0.564252\pi\)
−0.200487 + 0.979696i \(0.564252\pi\)
\(822\) 0 0
\(823\) 28.0000 0.976019 0.488009 0.872838i \(-0.337723\pi\)
0.488009 + 0.872838i \(0.337723\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 46.9783 1.63359 0.816797 0.576925i \(-0.195748\pi\)
0.816797 + 0.576925i \(0.195748\pi\)
\(828\) 0 0
\(829\) −24.7446 −0.859414 −0.429707 0.902968i \(-0.641383\pi\)
−0.429707 + 0.902968i \(0.641383\pi\)
\(830\) 0 0
\(831\) 42.9783 1.49090
\(832\) 0 0
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) −4.62772 −0.160149
\(836\) 0 0
\(837\) −61.0951 −2.11176
\(838\) 0 0
\(839\) 10.9783 0.379011 0.189506 0.981880i \(-0.439311\pi\)
0.189506 + 0.981880i \(0.439311\pi\)
\(840\) 0 0
\(841\) −27.1168 −0.935064
\(842\) 0 0
\(843\) −60.7011 −2.09066
\(844\) 0 0
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 17.7228 0.608245
\(850\) 0 0
\(851\) −25.7228 −0.881767
\(852\) 0 0
\(853\) 38.4674 1.31710 0.658549 0.752538i \(-0.271171\pi\)
0.658549 + 0.752538i \(0.271171\pi\)
\(854\) 0 0
\(855\) 5.25544 0.179732
\(856\) 0 0
\(857\) −36.3505 −1.24171 −0.620855 0.783925i \(-0.713215\pi\)
−0.620855 + 0.783925i \(0.713215\pi\)
\(858\) 0 0
\(859\) 42.7446 1.45843 0.729213 0.684287i \(-0.239886\pi\)
0.729213 + 0.684287i \(0.239886\pi\)
\(860\) 0 0
\(861\) 130.658 4.45280
\(862\) 0 0
\(863\) −21.2554 −0.723544 −0.361772 0.932267i \(-0.617828\pi\)
−0.361772 + 0.932267i \(0.617828\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) 0 0
\(867\) −50.9783 −1.73131
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 0 0
\(873\) −106.701 −3.61128
\(874\) 0 0
\(875\) 3.37228 0.114004
\(876\) 0 0
\(877\) −36.9783 −1.24867 −0.624333 0.781158i \(-0.714629\pi\)
−0.624333 + 0.781158i \(0.714629\pi\)
\(878\) 0 0
\(879\) −79.2119 −2.67175
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) −3.37228 −0.113486 −0.0567432 0.998389i \(-0.518072\pi\)
−0.0567432 + 0.998389i \(0.518072\pi\)
\(884\) 0 0
\(885\) 9.25544 0.311118
\(886\) 0 0
\(887\) −10.9783 −0.368614 −0.184307 0.982869i \(-0.559004\pi\)
−0.184307 + 0.982869i \(0.559004\pi\)
\(888\) 0 0
\(889\) 26.9783 0.904821
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1.72281 −0.0576517
\(894\) 0 0
\(895\) 12.0000 0.401116
\(896\) 0 0
\(897\) 18.5109 0.618060
\(898\) 0 0
\(899\) 4.62772 0.154343
\(900\) 0 0
\(901\) 5.64947 0.188211
\(902\) 0 0
\(903\) −45.4891 −1.51378
\(904\) 0 0
\(905\) −10.0000 −0.332411
\(906\) 0 0
\(907\) 0.394031 0.0130836 0.00654179 0.999979i \(-0.497918\pi\)
0.00654179 + 0.999979i \(0.497918\pi\)
\(908\) 0 0
\(909\) −50.2337 −1.66615
\(910\) 0 0
\(911\) 8.39403 0.278107 0.139053 0.990285i \(-0.455594\pi\)
0.139053 + 0.990285i \(0.455594\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 18.1168 0.598924
\(916\) 0 0
\(917\) 74.5842 2.46299
\(918\) 0 0
\(919\) 18.9783 0.626035 0.313017 0.949747i \(-0.398660\pi\)
0.313017 + 0.949747i \(0.398660\pi\)
\(920\) 0 0
\(921\) 17.7228 0.583987
\(922\) 0 0
\(923\) 20.2337 0.666000
\(924\) 0 0
\(925\) 9.37228 0.308159
\(926\) 0 0
\(927\) 79.4456 2.60934
\(928\) 0 0
\(929\) 24.3505 0.798915 0.399458 0.916752i \(-0.369199\pi\)
0.399458 + 0.916752i \(0.369199\pi\)
\(930\) 0 0
\(931\) 2.74456 0.0899494
\(932\) 0 0
\(933\) −65.3288 −2.13877
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 28.5109 0.931410 0.465705 0.884940i \(-0.345801\pi\)
0.465705 + 0.884940i \(0.345801\pi\)
\(938\) 0 0
\(939\) −74.1902 −2.42111
\(940\) 0 0
\(941\) 15.0951 0.492086 0.246043 0.969259i \(-0.420870\pi\)
0.246043 + 0.969259i \(0.420870\pi\)
\(942\) 0 0
\(943\) −31.5326 −1.02684
\(944\) 0 0
\(945\) 61.0951 1.98742
\(946\) 0 0
\(947\) −48.8614 −1.58778 −0.793891 0.608060i \(-0.791948\pi\)
−0.793891 + 0.608060i \(0.791948\pi\)
\(948\) 0 0
\(949\) −30.9783 −1.00560
\(950\) 0 0
\(951\) −82.1168 −2.66282
\(952\) 0 0
\(953\) −40.1168 −1.29951 −0.649756 0.760143i \(-0.725129\pi\)
−0.649756 + 0.760143i \(0.725129\pi\)
\(954\) 0 0
\(955\) 5.48913 0.177624
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 29.4891 0.952254
\(960\) 0 0
\(961\) −19.6277 −0.633152
\(962\) 0 0
\(963\) −100.467 −3.23752
\(964\) 0 0
\(965\) −14.8614 −0.478406
\(966\) 0 0
\(967\) 47.6060 1.53090 0.765452 0.643493i \(-0.222515\pi\)
0.765452 + 0.643493i \(0.222515\pi\)
\(968\) 0 0
\(969\) −2.90491 −0.0933190
\(970\) 0 0
\(971\) 1.02175 0.0327895 0.0163947 0.999866i \(-0.494781\pi\)
0.0163947 + 0.999866i \(0.494781\pi\)
\(972\) 0 0
\(973\) −13.4891 −0.432442
\(974\) 0 0
\(975\) −6.74456 −0.215999
\(976\) 0 0
\(977\) 14.2337 0.455376 0.227688 0.973734i \(-0.426883\pi\)
0.227688 + 0.973734i \(0.426883\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 129.679 4.14034
\(982\) 0 0
\(983\) 13.7228 0.437690 0.218845 0.975760i \(-0.429771\pi\)
0.218845 + 0.975760i \(0.429771\pi\)
\(984\) 0 0
\(985\) 20.7446 0.660977
\(986\) 0 0
\(987\) −31.2119 −0.993487
\(988\) 0 0
\(989\) 10.9783 0.349088
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 0 0
\(993\) −104.467 −3.31517
\(994\) 0 0
\(995\) −18.1168 −0.574343
\(996\) 0 0
\(997\) −22.2337 −0.704148 −0.352074 0.935972i \(-0.614523\pi\)
−0.352074 + 0.935972i \(0.614523\pi\)
\(998\) 0 0
\(999\) 169.796 5.37211
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 9680.2.a.bt.1.2 2
4.3 odd 2 1210.2.a.r.1.1 2
11.10 odd 2 880.2.a.n.1.2 2
20.19 odd 2 6050.2.a.cb.1.2 2
33.32 even 2 7920.2.a.bq.1.1 2
44.43 even 2 110.2.a.d.1.1 2
55.32 even 4 4400.2.b.p.4049.1 4
55.43 even 4 4400.2.b.p.4049.4 4
55.54 odd 2 4400.2.a.bl.1.1 2
88.21 odd 2 3520.2.a.bj.1.1 2
88.43 even 2 3520.2.a.bq.1.2 2
132.131 odd 2 990.2.a.m.1.2 2
220.43 odd 4 550.2.b.f.199.3 4
220.87 odd 4 550.2.b.f.199.2 4
220.219 even 2 550.2.a.n.1.2 2
308.307 odd 2 5390.2.a.bp.1.2 2
660.263 even 4 4950.2.c.bc.199.1 4
660.527 even 4 4950.2.c.bc.199.4 4
660.659 odd 2 4950.2.a.bw.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.2.a.d.1.1 2 44.43 even 2
550.2.a.n.1.2 2 220.219 even 2
550.2.b.f.199.2 4 220.87 odd 4
550.2.b.f.199.3 4 220.43 odd 4
880.2.a.n.1.2 2 11.10 odd 2
990.2.a.m.1.2 2 132.131 odd 2
1210.2.a.r.1.1 2 4.3 odd 2
3520.2.a.bj.1.1 2 88.21 odd 2
3520.2.a.bq.1.2 2 88.43 even 2
4400.2.a.bl.1.1 2 55.54 odd 2
4400.2.b.p.4049.1 4 55.32 even 4
4400.2.b.p.4049.4 4 55.43 even 4
4950.2.a.bw.1.1 2 660.659 odd 2
4950.2.c.bc.199.1 4 660.263 even 4
4950.2.c.bc.199.4 4 660.527 even 4
5390.2.a.bp.1.2 2 308.307 odd 2
6050.2.a.cb.1.2 2 20.19 odd 2
7920.2.a.bq.1.1 2 33.32 even 2
9680.2.a.bt.1.2 2 1.1 even 1 trivial