Properties

Label 990.2.a.m.1.2
Level $990$
Weight $2$
Character 990.1
Self dual yes
Analytic conductor $7.905$
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] = [990,2,Mod(1,990)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(990, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("990.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 990.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: \(7.90518980011\)
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: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\chi\) \(=\) 990.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.37228 q^{7} +1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} +2.00000 q^{13} +3.37228 q^{14} +1.00000 q^{16} -1.37228 q^{17} +0.627719 q^{19} -1.00000 q^{20} +1.00000 q^{22} -2.74456 q^{23} +1.00000 q^{25} +2.00000 q^{26} +3.37228 q^{28} -1.37228 q^{29} +3.37228 q^{31} +1.00000 q^{32} -1.37228 q^{34} -3.37228 q^{35} +9.37228 q^{37} +0.627719 q^{38} -1.00000 q^{40} +11.4891 q^{41} -4.00000 q^{43} +1.00000 q^{44} -2.74456 q^{46} -2.74456 q^{47} +4.37228 q^{49} +1.00000 q^{50} +2.00000 q^{52} +4.11684 q^{53} -1.00000 q^{55} +3.37228 q^{56} -1.37228 q^{58} +2.74456 q^{59} -5.37228 q^{61} +3.37228 q^{62} +1.00000 q^{64} -2.00000 q^{65} +8.00000 q^{67} -1.37228 q^{68} -3.37228 q^{70} -10.1168 q^{71} -15.4891 q^{73} +9.37228 q^{74} +0.627719 q^{76} +3.37228 q^{77} -1.25544 q^{79} -1.00000 q^{80} +11.4891 q^{82} +2.74456 q^{83} +1.37228 q^{85} -4.00000 q^{86} +1.00000 q^{88} +1.37228 q^{89} +6.74456 q^{91} -2.74456 q^{92} -2.74456 q^{94} -0.627719 q^{95} -12.7446 q^{97} +4.37228 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} + 2 q^{8} - 2 q^{10} + 2 q^{11} + 4 q^{13} + q^{14} + 2 q^{16} + 3 q^{17} + 7 q^{19} - 2 q^{20} + 2 q^{22} + 6 q^{23} + 2 q^{25} + 4 q^{26} + q^{28} + 3 q^{29}+ \cdots + 3 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 3.37228 1.27460 0.637301 0.770615i \(-0.280051\pi\)
0.637301 + 0.770615i \(0.280051\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 3.37228 0.901280
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −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 \(0.477060\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −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
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 3.37228 0.637301
\(29\) −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 \(-0.402065\pi\)
0.302840 + 0.953041i \(0.402065\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −1.37228 −0.235344
\(35\) −3.37228 −0.570020
\(36\) 0 0
\(37\) 9.37228 1.54079 0.770397 0.637565i \(-0.220058\pi\)
0.770397 + 0.637565i \(0.220058\pi\)
\(38\) 0.627719 0.101829
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 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 \(0.598656\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −2.74456 −0.404664
\(47\) −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
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) 4.11684 0.565492 0.282746 0.959195i \(-0.408755\pi\)
0.282746 + 0.959195i \(0.408755\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 3.37228 0.450640
\(57\) 0 0
\(58\) −1.37228 −0.180189
\(59\) 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 \(-0.611757\pi\)
−0.343925 + 0.938997i \(0.611757\pi\)
\(62\) 3.37228 0.428280
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −1.37228 −0.166414
\(69\) 0 0
\(70\) −3.37228 −0.403065
\(71\) −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 \(-0.861205\pi\)
−0.906432 + 0.422351i \(0.861205\pi\)
\(74\) 9.37228 1.08951
\(75\) 0 0
\(76\) 0.627719 0.0720043
\(77\) 3.37228 0.384307
\(78\) 0 0
\(79\) −1.25544 −0.141248 −0.0706239 0.997503i \(-0.522499\pi\)
−0.0706239 + 0.997503i \(0.522499\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 11.4891 1.26876
\(83\) 2.74456 0.301255 0.150627 0.988591i \(-0.451871\pi\)
0.150627 + 0.988591i \(0.451871\pi\)
\(84\) 0 0
\(85\) 1.37228 0.148845
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 1.37228 0.145462 0.0727308 0.997352i \(-0.476829\pi\)
0.0727308 + 0.997352i \(0.476829\pi\)
\(90\) 0 0
\(91\) 6.74456 0.707022
\(92\) −2.74456 −0.286140
\(93\) 0 0
\(94\) −2.74456 −0.283080
\(95\) −0.627719 −0.0644026
\(96\) 0 0
\(97\) −12.7446 −1.29401 −0.647007 0.762484i \(-0.723980\pi\)
−0.647007 + 0.762484i \(0.723980\pi\)
\(98\) 4.37228 0.441667
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −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 \(-0.654844\pi\)
−0.467496 + 0.883995i \(0.654844\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 4.11684 0.399863
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −15.4891 −1.48359 −0.741795 0.670627i \(-0.766025\pi\)
−0.741795 + 0.670627i \(0.766025\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 0 0
\(112\) 3.37228 0.318651
\(113\) −3.25544 −0.306246 −0.153123 0.988207i \(-0.548933\pi\)
−0.153123 + 0.988207i \(0.548933\pi\)
\(114\) 0 0
\(115\) 2.74456 0.255932
\(116\) −1.37228 −0.127413
\(117\) 0 0
\(118\) 2.74456 0.252657
\(119\) −4.62772 −0.424222
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −5.37228 −0.486383
\(123\) 0 0
\(124\) 3.37228 0.302840
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −2.00000 −0.175412
\(131\) −22.1168 −1.93236 −0.966179 0.257873i \(-0.916978\pi\)
−0.966179 + 0.257873i \(0.916978\pi\)
\(132\) 0 0
\(133\) 2.11684 0.183554
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) −1.37228 −0.117672
\(137\) −8.74456 −0.747098 −0.373549 0.927610i \(-0.621859\pi\)
−0.373549 + 0.927610i \(0.621859\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −3.37228 −0.285010
\(141\) 0 0
\(142\) −10.1168 −0.848987
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) 1.37228 0.113962
\(146\) −15.4891 −1.28189
\(147\) 0 0
\(148\) 9.37228 0.770397
\(149\) −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 \(0.665852\pi\)
\(152\) 0.627719 0.0509147
\(153\) 0 0
\(154\) 3.37228 0.271746
\(155\) −3.37228 −0.270868
\(156\) 0 0
\(157\) 9.37228 0.747989 0.373995 0.927431i \(-0.377988\pi\)
0.373995 + 0.927431i \(0.377988\pi\)
\(158\) −1.25544 −0.0998772
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −9.25544 −0.729431
\(162\) 0 0
\(163\) −5.88316 −0.460804 −0.230402 0.973095i \(-0.574004\pi\)
−0.230402 + 0.973095i \(0.574004\pi\)
\(164\) 11.4891 0.897150
\(165\) 0 0
\(166\) 2.74456 0.213019
\(167\) 4.62772 0.358104 0.179052 0.983840i \(-0.442697\pi\)
0.179052 + 0.983840i \(0.442697\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 1.37228 0.105249
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 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
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 1.37228 0.102857
\(179\) 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 \(0.621207\pi\)
\(182\) 6.74456 0.499940
\(183\) 0 0
\(184\) −2.74456 −0.202332
\(185\) −9.37228 −0.689064
\(186\) 0 0
\(187\) −1.37228 −0.100351
\(188\) −2.74456 −0.200168
\(189\) 0 0
\(190\) −0.627719 −0.0455395
\(191\) 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 \(-0.320359\pi\)
0.534874 + 0.844932i \(0.320359\pi\)
\(194\) −12.7446 −0.915006
\(195\) 0 0
\(196\) 4.37228 0.312306
\(197\) 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 \(-0.278049\pi\)
0.642135 + 0.766592i \(0.278049\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −6.00000 −0.422159
\(203\) −4.62772 −0.324802
\(204\) 0 0
\(205\) −11.4891 −0.802435
\(206\) −9.48913 −0.661139
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 0.627719 0.0434202
\(210\) 0 0
\(211\) 6.11684 0.421101 0.210550 0.977583i \(-0.432474\pi\)
0.210550 + 0.977583i \(0.432474\pi\)
\(212\) 4.11684 0.282746
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) 11.3723 0.772001
\(218\) −15.4891 −1.04906
\(219\) 0 0
\(220\) −1.00000 −0.0674200
\(221\) −2.74456 −0.184619
\(222\) 0 0
\(223\) −18.7446 −1.25523 −0.627614 0.778524i \(-0.715969\pi\)
−0.627614 + 0.778524i \(0.715969\pi\)
\(224\) 3.37228 0.225320
\(225\) 0 0
\(226\) −3.25544 −0.216548
\(227\) 2.74456 0.182163 0.0910815 0.995843i \(-0.470968\pi\)
0.0910815 + 0.995843i \(0.470968\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 2.74456 0.180971
\(231\) 0 0
\(232\) −1.37228 −0.0900947
\(233\) −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
\(236\) 2.74456 0.178656
\(237\) 0 0
\(238\) −4.62772 −0.299970
\(239\) 14.7446 0.953746 0.476873 0.878972i \(-0.341770\pi\)
0.476873 + 0.878972i \(0.341770\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) −5.37228 −0.343925
\(245\) −4.37228 −0.279335
\(246\) 0 0
\(247\) 1.25544 0.0798816
\(248\) 3.37228 0.214140
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −2.74456 −0.173235 −0.0866176 0.996242i \(-0.527606\pi\)
−0.0866176 + 0.996242i \(0.527606\pi\)
\(252\) 0 0
\(253\) −2.74456 −0.172549
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 31.6060 1.96390
\(260\) −2.00000 −0.124035
\(261\) 0 0
\(262\) −22.1168 −1.36638
\(263\) 24.8614 1.53302 0.766510 0.642232i \(-0.221992\pi\)
0.766510 + 0.642232i \(0.221992\pi\)
\(264\) 0 0
\(265\) −4.11684 −0.252896
\(266\) 2.11684 0.129792
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) 8.74456 0.533165 0.266583 0.963812i \(-0.414105\pi\)
0.266583 + 0.963812i \(0.414105\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −1.37228 −0.0832068
\(273\) 0 0
\(274\) −8.74456 −0.528278
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −12.7446 −0.765747 −0.382873 0.923801i \(-0.625065\pi\)
−0.382873 + 0.923801i \(0.625065\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) −3.37228 −0.201532
\(281\) −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 \(0.450075\pi\)
\(284\) −10.1168 −0.600324
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 38.7446 2.28702
\(288\) 0 0
\(289\) −15.1168 −0.889226
\(290\) 1.37228 0.0805831
\(291\) 0 0
\(292\) −15.4891 −0.906432
\(293\) −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
\(296\) 9.37228 0.544753
\(297\) 0 0
\(298\) −21.6060 −1.25160
\(299\) −5.48913 −0.317444
\(300\) 0 0
\(301\) −13.4891 −0.777500
\(302\) −12.2337 −0.703970
\(303\) 0 0
\(304\) 0.627719 0.0360021
\(305\) 5.37228 0.307616
\(306\) 0 0
\(307\) 5.25544 0.299944 0.149972 0.988690i \(-0.452082\pi\)
0.149972 + 0.988690i \(0.452082\pi\)
\(308\) 3.37228 0.192154
\(309\) 0 0
\(310\) −3.37228 −0.191533
\(311\) −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 \(0.713581\pi\)
\(314\) 9.37228 0.528908
\(315\) 0 0
\(316\) −1.25544 −0.0706239
\(317\) 24.3505 1.36766 0.683831 0.729640i \(-0.260313\pi\)
0.683831 + 0.729640i \(0.260313\pi\)
\(318\) 0 0
\(319\) −1.37228 −0.0768330
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −9.25544 −0.515785
\(323\) −0.861407 −0.0479299
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) −5.88316 −0.325838
\(327\) 0 0
\(328\) 11.4891 0.634381
\(329\) −9.25544 −0.510269
\(330\) 0 0
\(331\) 30.9783 1.70272 0.851359 0.524583i \(-0.175779\pi\)
0.851359 + 0.524583i \(0.175779\pi\)
\(332\) 2.74456 0.150627
\(333\) 0 0
\(334\) 4.62772 0.253217
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) 24.1168 1.31373 0.656864 0.754009i \(-0.271883\pi\)
0.656864 + 0.754009i \(0.271883\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) 1.37228 0.0744224
\(341\) 3.37228 0.182619
\(342\) 0 0
\(343\) −8.86141 −0.478471
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 32.2337 1.73040 0.865198 0.501431i \(-0.167193\pi\)
0.865198 + 0.501431i \(0.167193\pi\)
\(348\) 0 0
\(349\) 19.4891 1.04323 0.521614 0.853181i \(-0.325330\pi\)
0.521614 + 0.853181i \(0.325330\pi\)
\(350\) 3.37228 0.180256
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 0.510875 0.0271911 0.0135956 0.999908i \(-0.495672\pi\)
0.0135956 + 0.999908i \(0.495672\pi\)
\(354\) 0 0
\(355\) 10.1168 0.536946
\(356\) 1.37228 0.0727308
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −18.6060 −0.979262
\(362\) −10.0000 −0.525588
\(363\) 0 0
\(364\) 6.74456 0.353511
\(365\) 15.4891 0.810738
\(366\) 0 0
\(367\) −4.00000 −0.208798 −0.104399 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) −2.74456 −0.143070
\(369\) 0 0
\(370\) −9.37228 −0.487242
\(371\) 13.8832 0.720778
\(372\) 0 0
\(373\) 31.4891 1.63045 0.815223 0.579148i \(-0.196615\pi\)
0.815223 + 0.579148i \(0.196615\pi\)
\(374\) −1.37228 −0.0709590
\(375\) 0 0
\(376\) −2.74456 −0.141540
\(377\) −2.74456 −0.141352
\(378\) 0 0
\(379\) −0.233688 −0.0120037 −0.00600187 0.999982i \(-0.501910\pi\)
−0.00600187 + 0.999982i \(0.501910\pi\)
\(380\) −0.627719 −0.0322013
\(381\) 0 0
\(382\) 5.48913 0.280848
\(383\) −32.2337 −1.64706 −0.823532 0.567269i \(-0.808000\pi\)
−0.823532 + 0.567269i \(0.808000\pi\)
\(384\) 0 0
\(385\) −3.37228 −0.171867
\(386\) 14.8614 0.756426
\(387\) 0 0
\(388\) −12.7446 −0.647007
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 3.76631 0.190471
\(392\) 4.37228 0.220834
\(393\) 0 0
\(394\) 20.7446 1.04510
\(395\) 1.25544 0.0631679
\(396\) 0 0
\(397\) 24.9783 1.25362 0.626811 0.779171i \(-0.284360\pi\)
0.626811 + 0.779171i \(0.284360\pi\)
\(398\) 18.1168 0.908115
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −13.3723 −0.667780 −0.333890 0.942612i \(-0.608361\pi\)
−0.333890 + 0.942612i \(0.608361\pi\)
\(402\) 0 0
\(403\) 6.74456 0.335971
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) −4.62772 −0.229670
\(407\) 9.37228 0.464567
\(408\) 0 0
\(409\) −1.76631 −0.0873385 −0.0436693 0.999046i \(-0.513905\pi\)
−0.0436693 + 0.999046i \(0.513905\pi\)
\(410\) −11.4891 −0.567407
\(411\) 0 0
\(412\) −9.48913 −0.467496
\(413\) 9.25544 0.455430
\(414\) 0 0
\(415\) −2.74456 −0.134725
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) 0.627719 0.0307027
\(419\) 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 \(0.419773\pi\)
\(422\) 6.11684 0.297763
\(423\) 0 0
\(424\) 4.11684 0.199932
\(425\) −1.37228 −0.0665654
\(426\) 0 0
\(427\) −18.1168 −0.876736
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) 34.9783 1.68484 0.842422 0.538819i \(-0.181129\pi\)
0.842422 + 0.538819i \(0.181129\pi\)
\(432\) 0 0
\(433\) 27.7228 1.33227 0.666137 0.745830i \(-0.267947\pi\)
0.666137 + 0.745830i \(0.267947\pi\)
\(434\) 11.3723 0.545887
\(435\) 0 0
\(436\) −15.4891 −0.741795
\(437\) −1.72281 −0.0824133
\(438\) 0 0
\(439\) 18.9783 0.905782 0.452891 0.891566i \(-0.350393\pi\)
0.452891 + 0.891566i \(0.350393\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 0 0
\(442\) −2.74456 −0.130546
\(443\) −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
\(446\) −18.7446 −0.887581
\(447\) 0 0
\(448\) 3.37228 0.159325
\(449\) −28.9783 −1.36757 −0.683784 0.729684i \(-0.739667\pi\)
−0.683784 + 0.729684i \(0.739667\pi\)
\(450\) 0 0
\(451\) 11.4891 0.541002
\(452\) −3.25544 −0.153123
\(453\) 0 0
\(454\) 2.74456 0.128809
\(455\) −6.74456 −0.316190
\(456\) 0 0
\(457\) −16.3505 −0.764846 −0.382423 0.923987i \(-0.624910\pi\)
−0.382423 + 0.923987i \(0.624910\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) 2.74456 0.127966
\(461\) −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 \(-0.501728\pi\)
−0.00543020 + 0.999985i \(0.501728\pi\)
\(464\) −1.37228 −0.0637066
\(465\) 0 0
\(466\) −1.37228 −0.0635697
\(467\) 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
\(470\) 2.74456 0.126597
\(471\) 0 0
\(472\) 2.74456 0.126329
\(473\) −4.00000 −0.183920
\(474\) 0 0
\(475\) 0.627719 0.0288017
\(476\) −4.62772 −0.212111
\(477\) 0 0
\(478\) 14.7446 0.674401
\(479\) −5.48913 −0.250805 −0.125402 0.992106i \(-0.540022\pi\)
−0.125402 + 0.992106i \(0.540022\pi\)
\(480\) 0 0
\(481\) 18.7446 0.854678
\(482\) −22.0000 −1.00207
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 12.7446 0.578701
\(486\) 0 0
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) −5.37228 −0.243192
\(489\) 0 0
\(490\) −4.37228 −0.197520
\(491\) 7.37228 0.332706 0.166353 0.986066i \(-0.446801\pi\)
0.166353 + 0.986066i \(0.446801\pi\)
\(492\) 0 0
\(493\) 1.88316 0.0848131
\(494\) 1.25544 0.0564848
\(495\) 0 0
\(496\) 3.37228 0.151420
\(497\) −34.1168 −1.53035
\(498\) 0 0
\(499\) −33.4891 −1.49918 −0.749590 0.661903i \(-0.769749\pi\)
−0.749590 + 0.661903i \(0.769749\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −2.74456 −0.122496
\(503\) 34.9783 1.55960 0.779802 0.626027i \(-0.215320\pi\)
0.779802 + 0.626027i \(0.215320\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) −2.74456 −0.122011
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) −9.76631 −0.432884 −0.216442 0.976295i \(-0.569445\pi\)
−0.216442 + 0.976295i \(0.569445\pi\)
\(510\) 0 0
\(511\) −52.2337 −2.31068
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −18.0000 −0.793946
\(515\) 9.48913 0.418141
\(516\) 0 0
\(517\) −2.74456 −0.120706
\(518\) 31.6060 1.38869
\(519\) 0 0
\(520\) −2.00000 −0.0877058
\(521\) −12.5109 −0.548111 −0.274056 0.961714i \(-0.588365\pi\)
−0.274056 + 0.961714i \(0.588365\pi\)
\(522\) 0 0
\(523\) 30.9783 1.35458 0.677292 0.735714i \(-0.263153\pi\)
0.677292 + 0.735714i \(0.263153\pi\)
\(524\) −22.1168 −0.966179
\(525\) 0 0
\(526\) 24.8614 1.08401
\(527\) −4.62772 −0.201587
\(528\) 0 0
\(529\) −15.4674 −0.672495
\(530\) −4.11684 −0.178824
\(531\) 0 0
\(532\) 2.11684 0.0917768
\(533\) 22.9783 0.995299
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) 8.00000 0.345547
\(537\) 0 0
\(538\) 8.74456 0.377005
\(539\) 4.37228 0.188327
\(540\) 0 0
\(541\) −20.1168 −0.864891 −0.432445 0.901660i \(-0.642349\pi\)
−0.432445 + 0.901660i \(0.642349\pi\)
\(542\) −16.0000 −0.687259
\(543\) 0 0
\(544\) −1.37228 −0.0588361
\(545\) 15.4891 0.663481
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) −8.74456 −0.373549
\(549\) 0 0
\(550\) 1.00000 0.0426401
\(551\) −0.861407 −0.0366972
\(552\) 0 0
\(553\) −4.23369 −0.180035
\(554\) −12.7446 −0.541465
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) −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
\(560\) −3.37228 −0.142505
\(561\) 0 0
\(562\) −18.0000 −0.759284
\(563\) 8.23369 0.347009 0.173504 0.984833i \(-0.444491\pi\)
0.173504 + 0.984833i \(0.444491\pi\)
\(564\) 0 0
\(565\) 3.25544 0.136957
\(566\) 5.25544 0.220903
\(567\) 0 0
\(568\) −10.1168 −0.424493
\(569\) 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 \(0.395761\pi\)
\(572\) 2.00000 0.0836242
\(573\) 0 0
\(574\) 38.7446 1.61717
\(575\) −2.74456 −0.114456
\(576\) 0 0
\(577\) 36.9783 1.53942 0.769712 0.638391i \(-0.220400\pi\)
0.769712 + 0.638391i \(0.220400\pi\)
\(578\) −15.1168 −0.628778
\(579\) 0 0
\(580\) 1.37228 0.0569809
\(581\) 9.25544 0.383980
\(582\) 0 0
\(583\) 4.11684 0.170502
\(584\) −15.4891 −0.640945
\(585\) 0 0
\(586\) −23.4891 −0.970327
\(587\) −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
\(590\) −2.74456 −0.112992
\(591\) 0 0
\(592\) 9.37228 0.385198
\(593\) 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
\(596\) −21.6060 −0.885015
\(597\) 0 0
\(598\) −5.48913 −0.224467
\(599\) 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 \(-0.609326\pi\)
−0.336746 + 0.941595i \(0.609326\pi\)
\(602\) −13.4891 −0.549776
\(603\) 0 0
\(604\) −12.2337 −0.497782
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) −5.88316 −0.238790 −0.119395 0.992847i \(-0.538095\pi\)
−0.119395 + 0.992847i \(0.538095\pi\)
\(608\) 0.627719 0.0254574
\(609\) 0 0
\(610\) 5.37228 0.217517
\(611\) −5.48913 −0.222066
\(612\) 0 0
\(613\) 20.5109 0.828426 0.414213 0.910180i \(-0.364057\pi\)
0.414213 + 0.910180i \(0.364057\pi\)
\(614\) 5.25544 0.212092
\(615\) 0 0
\(616\) 3.37228 0.135873
\(617\) 2.23369 0.0899249 0.0449624 0.998989i \(-0.485683\pi\)
0.0449624 + 0.998989i \(0.485683\pi\)
\(618\) 0 0
\(619\) −44.4674 −1.78729 −0.893647 0.448770i \(-0.851862\pi\)
−0.893647 + 0.448770i \(0.851862\pi\)
\(620\) −3.37228 −0.135434
\(621\) 0 0
\(622\) −19.3723 −0.776758
\(623\) 4.62772 0.185406
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −22.0000 −0.879297
\(627\) 0 0
\(628\) 9.37228 0.373995
\(629\) −12.8614 −0.512818
\(630\) 0 0
\(631\) 42.1168 1.67665 0.838323 0.545175i \(-0.183537\pi\)
0.838323 + 0.545175i \(0.183537\pi\)
\(632\) −1.25544 −0.0499386
\(633\) 0 0
\(634\) 24.3505 0.967083
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) 8.74456 0.346472
\(638\) −1.37228 −0.0543291
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 27.0951 1.07019 0.535096 0.844791i \(-0.320275\pi\)
0.535096 + 0.844791i \(0.320275\pi\)
\(642\) 0 0
\(643\) −5.88316 −0.232009 −0.116005 0.993249i \(-0.537009\pi\)
−0.116005 + 0.993249i \(0.537009\pi\)
\(644\) −9.25544 −0.364715
\(645\) 0 0
\(646\) −0.861407 −0.0338916
\(647\) 37.7228 1.48304 0.741518 0.670933i \(-0.234106\pi\)
0.741518 + 0.670933i \(0.234106\pi\)
\(648\) 0 0
\(649\) 2.74456 0.107734
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −5.88316 −0.230402
\(653\) −10.6277 −0.415895 −0.207947 0.978140i \(-0.566678\pi\)
−0.207947 + 0.978140i \(0.566678\pi\)
\(654\) 0 0
\(655\) 22.1168 0.864177
\(656\) 11.4891 0.448575
\(657\) 0 0
\(658\) −9.25544 −0.360815
\(659\) −12.8614 −0.501009 −0.250505 0.968115i \(-0.580597\pi\)
−0.250505 + 0.968115i \(0.580597\pi\)
\(660\) 0 0
\(661\) 8.51087 0.331035 0.165517 0.986207i \(-0.447071\pi\)
0.165517 + 0.986207i \(0.447071\pi\)
\(662\) 30.9783 1.20400
\(663\) 0 0
\(664\) 2.74456 0.106510
\(665\) −2.11684 −0.0820877
\(666\) 0 0
\(667\) 3.76631 0.145832
\(668\) 4.62772 0.179052
\(669\) 0 0
\(670\) −8.00000 −0.309067
\(671\) −5.37228 −0.207395
\(672\) 0 0
\(673\) 14.8614 0.572865 0.286433 0.958100i \(-0.407531\pi\)
0.286433 + 0.958100i \(0.407531\pi\)
\(674\) 24.1168 0.928946
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −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
\(680\) 1.37228 0.0526246
\(681\) 0 0
\(682\) 3.37228 0.129131
\(683\) 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
\(686\) −8.86141 −0.338330
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) 8.23369 0.313679
\(690\) 0 0
\(691\) 40.2337 1.53056 0.765281 0.643697i \(-0.222600\pi\)
0.765281 + 0.643697i \(0.222600\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) 32.2337 1.22357
\(695\) 4.00000 0.151729
\(696\) 0 0
\(697\) −15.7663 −0.597192
\(698\) 19.4891 0.737674
\(699\) 0 0
\(700\) 3.37228 0.127460
\(701\) 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
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 0.510875 0.0192270
\(707\) −20.2337 −0.760966
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 10.1168 0.379678
\(711\) 0 0
\(712\) 1.37228 0.0514284
\(713\) −9.25544 −0.346619
\(714\) 0 0
\(715\) −2.00000 −0.0747958
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) 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
\(722\) −18.6060 −0.692442
\(723\) 0 0
\(724\) −10.0000 −0.371647
\(725\) −1.37228 −0.0509652
\(726\) 0 0
\(727\) −24.2337 −0.898778 −0.449389 0.893336i \(-0.648358\pi\)
−0.449389 + 0.893336i \(0.648358\pi\)
\(728\) 6.74456 0.249970
\(729\) 0 0
\(730\) 15.4891 0.573278
\(731\) 5.48913 0.203023
\(732\) 0 0
\(733\) 46.2337 1.70768 0.853840 0.520535i \(-0.174268\pi\)
0.853840 + 0.520535i \(0.174268\pi\)
\(734\) −4.00000 −0.147643
\(735\) 0 0
\(736\) −2.74456 −0.101166
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) −20.4674 −0.752905 −0.376452 0.926436i \(-0.622856\pi\)
−0.376452 + 0.926436i \(0.622856\pi\)
\(740\) −9.37228 −0.344532
\(741\) 0 0
\(742\) 13.8832 0.509667
\(743\) 4.62772 0.169775 0.0848873 0.996391i \(-0.472947\pi\)
0.0848873 + 0.996391i \(0.472947\pi\)
\(744\) 0 0
\(745\) 21.6060 0.791581
\(746\) 31.4891 1.15290
\(747\) 0 0
\(748\) −1.37228 −0.0501756
\(749\) 40.4674 1.47865
\(750\) 0 0
\(751\) 8.86141 0.323357 0.161679 0.986843i \(-0.448309\pi\)
0.161679 + 0.986843i \(0.448309\pi\)
\(752\) −2.74456 −0.100084
\(753\) 0 0
\(754\) −2.74456 −0.0999511
\(755\) 12.2337 0.445229
\(756\) 0 0
\(757\) −20.9783 −0.762467 −0.381234 0.924479i \(-0.624501\pi\)
−0.381234 + 0.924479i \(0.624501\pi\)
\(758\) −0.233688 −0.00848793
\(759\) 0 0
\(760\) −0.627719 −0.0227697
\(761\) −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
\(764\) 5.48913 0.198590
\(765\) 0 0
\(766\) −32.2337 −1.16465
\(767\) 5.48913 0.198201
\(768\) 0 0
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) −3.37228 −0.121529
\(771\) 0 0
\(772\) 14.8614 0.534874
\(773\) 33.6060 1.20872 0.604361 0.796710i \(-0.293428\pi\)
0.604361 + 0.796710i \(0.293428\pi\)
\(774\) 0 0
\(775\) 3.37228 0.121136
\(776\) −12.7446 −0.457503
\(777\) 0 0
\(778\) −6.00000 −0.215110
\(779\) 7.21194 0.258395
\(780\) 0 0
\(781\) −10.1168 −0.362009
\(782\) 3.76631 0.134683
\(783\) 0 0
\(784\) 4.37228 0.156153
\(785\) −9.37228 −0.334511
\(786\) 0 0
\(787\) −44.4674 −1.58509 −0.792545 0.609813i \(-0.791245\pi\)
−0.792545 + 0.609813i \(0.791245\pi\)
\(788\) 20.7446 0.738994
\(789\) 0 0
\(790\) 1.25544 0.0446665
\(791\) −10.9783 −0.390342
\(792\) 0 0
\(793\) −10.7446 −0.381551
\(794\) 24.9783 0.886445
\(795\) 0 0
\(796\) 18.1168 0.642135
\(797\) −11.4891 −0.406966 −0.203483 0.979079i \(-0.565226\pi\)
−0.203483 + 0.979079i \(0.565226\pi\)
\(798\) 0 0
\(799\) 3.76631 0.133243
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −13.3723 −0.472192
\(803\) −15.4891 −0.546599
\(804\) 0 0
\(805\) 9.25544 0.326211
\(806\) 6.74456 0.237567
\(807\) 0 0
\(808\) −6.00000 −0.211079
\(809\) −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 \(0.211298\pi\)
\(812\) −4.62772 −0.162401
\(813\) 0 0
\(814\) 9.37228 0.328498
\(815\) 5.88316 0.206078
\(816\) 0 0
\(817\) −2.51087 −0.0878444
\(818\) −1.76631 −0.0617577
\(819\) 0 0
\(820\) −11.4891 −0.401218
\(821\) −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 \(-0.662277\pi\)
−0.488009 + 0.872838i \(0.662277\pi\)
\(824\) −9.48913 −0.330569
\(825\) 0 0
\(826\) 9.25544 0.322038
\(827\) −46.9783 −1.63359 −0.816797 0.576925i \(-0.804252\pi\)
−0.816797 + 0.576925i \(0.804252\pi\)
\(828\) 0 0
\(829\) −24.7446 −0.859414 −0.429707 0.902968i \(-0.641383\pi\)
−0.429707 + 0.902968i \(0.641383\pi\)
\(830\) −2.74456 −0.0952652
\(831\) 0 0
\(832\) 2.00000 0.0693375
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) −4.62772 −0.160149
\(836\) 0.627719 0.0217101
\(837\) 0 0
\(838\) 12.0000 0.414533
\(839\) 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
\(842\) 10.2337 0.352676
\(843\) 0 0
\(844\) 6.11684 0.210550
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) 3.37228 0.115873
\(848\) 4.11684 0.141373
\(849\) 0 0
\(850\) −1.37228 −0.0470689
\(851\) −25.7228 −0.881767
\(852\) 0 0
\(853\) −38.4674 −1.31710 −0.658549 0.752538i \(-0.728829\pi\)
−0.658549 + 0.752538i \(0.728829\pi\)
\(854\) −18.1168 −0.619946
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −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 \(-0.760114\pi\)
−0.729213 + 0.684287i \(0.760114\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) 34.9783 1.19136
\(863\) −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
\(866\) 27.7228 0.942060
\(867\) 0 0
\(868\) 11.3723 0.386000
\(869\) −1.25544 −0.0425878
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) −15.4891 −0.524528
\(873\) 0 0
\(874\) −1.72281 −0.0582750
\(875\) −3.37228 −0.114004
\(876\) 0 0
\(877\) 36.9783 1.24867 0.624333 0.781158i \(-0.285371\pi\)
0.624333 + 0.781158i \(0.285371\pi\)
\(878\) 18.9783 0.640485
\(879\) 0 0
\(880\) −1.00000 −0.0337100
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) 3.37228 0.113486 0.0567432 0.998389i \(-0.481928\pi\)
0.0567432 + 0.998389i \(0.481928\pi\)
\(884\) −2.74456 −0.0923096
\(885\) 0 0
\(886\) −29.4891 −0.990707
\(887\) 10.9783 0.368614 0.184307 0.982869i \(-0.440996\pi\)
0.184307 + 0.982869i \(0.440996\pi\)
\(888\) 0 0
\(889\) 26.9783 0.904821
\(890\) −1.37228 −0.0459990
\(891\) 0 0
\(892\) −18.7446 −0.627614
\(893\) −1.72281 −0.0576517
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 3.37228 0.112660
\(897\) 0 0
\(898\) −28.9783 −0.967017
\(899\) −4.62772 −0.154343
\(900\) 0 0
\(901\) −5.64947 −0.188211
\(902\) 11.4891 0.382546
\(903\) 0 0
\(904\) −3.25544 −0.108274
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) −0.394031 −0.0130836 −0.00654179 0.999979i \(-0.502082\pi\)
−0.00654179 + 0.999979i \(0.502082\pi\)
\(908\) 2.74456 0.0910815
\(909\) 0 0
\(910\) −6.74456 −0.223580
\(911\) 8.39403 0.278107 0.139053 0.990285i \(-0.455594\pi\)
0.139053 + 0.990285i \(0.455594\pi\)
\(912\) 0 0
\(913\) 2.74456 0.0908318
\(914\) −16.3505 −0.540828
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) −74.5842 −2.46299
\(918\) 0 0
\(919\) 18.9783 0.626035 0.313017 0.949747i \(-0.398660\pi\)
0.313017 + 0.949747i \(0.398660\pi\)
\(920\) 2.74456 0.0904856
\(921\) 0 0
\(922\) −16.1168 −0.530780
\(923\) −20.2337 −0.666000
\(924\) 0 0
\(925\) 9.37228 0.308159
\(926\) −0.233688 −0.00767946
\(927\) 0 0
\(928\) −1.37228 −0.0450473
\(929\) −24.3505 −0.798915 −0.399458 0.916752i \(-0.630801\pi\)
−0.399458 + 0.916752i \(0.630801\pi\)
\(930\) 0 0
\(931\) 2.74456 0.0899494
\(932\) −1.37228 −0.0449506
\(933\) 0 0
\(934\) 19.3723 0.633880
\(935\) 1.37228 0.0448784
\(936\) 0 0
\(937\) −28.5109 −0.931410 −0.465705 0.884940i \(-0.654199\pi\)
−0.465705 + 0.884940i \(0.654199\pi\)
\(938\) 26.9783 0.880871
\(939\) 0 0
\(940\) 2.74456 0.0895178
\(941\) 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
\(944\) 2.74456 0.0893279
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) −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
\(950\) 0.627719 0.0203659
\(951\) 0 0
\(952\) −4.62772 −0.149985
\(953\) −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
\(956\) 14.7446 0.476873
\(957\) 0 0
\(958\) −5.48913 −0.177346
\(959\) −29.4891 −0.952254
\(960\) 0 0
\(961\) −19.6277 −0.633152
\(962\) 18.7446 0.604349
\(963\) 0 0
\(964\) −22.0000 −0.708572
\(965\) −14.8614 −0.478406
\(966\) 0 0
\(967\) 47.6060 1.53090 0.765452 0.643493i \(-0.222515\pi\)
0.765452 + 0.643493i \(0.222515\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 12.7446 0.409203
\(971\) 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
\(974\) 20.0000 0.640841
\(975\) 0 0
\(976\) −5.37228 −0.171963
\(977\) −14.2337 −0.455376 −0.227688 0.973734i \(-0.573117\pi\)
−0.227688 + 0.973734i \(0.573117\pi\)
\(978\) 0 0
\(979\) 1.37228 0.0438583
\(980\) −4.37228 −0.139667
\(981\) 0 0
\(982\) 7.37228 0.235259
\(983\) 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
\(986\) 1.88316 0.0599719
\(987\) 0 0
\(988\) 1.25544 0.0399408
\(989\) 10.9783 0.349088
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 3.37228 0.107070
\(993\) 0 0
\(994\) −34.1168 −1.08212
\(995\) −18.1168 −0.574343
\(996\) 0 0
\(997\) 22.2337 0.704148 0.352074 0.935972i \(-0.385477\pi\)
0.352074 + 0.935972i \(0.385477\pi\)
\(998\) −33.4891 −1.06008
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 990.2.a.m.1.2 2
3.2 odd 2 110.2.a.d.1.1 2
4.3 odd 2 7920.2.a.bq.1.1 2
5.2 odd 4 4950.2.c.bc.199.4 4
5.3 odd 4 4950.2.c.bc.199.1 4
5.4 even 2 4950.2.a.bw.1.1 2
12.11 even 2 880.2.a.n.1.2 2
15.2 even 4 550.2.b.f.199.2 4
15.8 even 4 550.2.b.f.199.3 4
15.14 odd 2 550.2.a.n.1.2 2
21.20 even 2 5390.2.a.bp.1.2 2
24.5 odd 2 3520.2.a.bq.1.2 2
24.11 even 2 3520.2.a.bj.1.1 2
33.32 even 2 1210.2.a.r.1.1 2
60.23 odd 4 4400.2.b.p.4049.4 4
60.47 odd 4 4400.2.b.p.4049.1 4
60.59 even 2 4400.2.a.bl.1.1 2
132.131 odd 2 9680.2.a.bt.1.2 2
165.164 even 2 6050.2.a.cb.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.2.a.d.1.1 2 3.2 odd 2
550.2.a.n.1.2 2 15.14 odd 2
550.2.b.f.199.2 4 15.2 even 4
550.2.b.f.199.3 4 15.8 even 4
880.2.a.n.1.2 2 12.11 even 2
990.2.a.m.1.2 2 1.1 even 1 trivial
1210.2.a.r.1.1 2 33.32 even 2
3520.2.a.bj.1.1 2 24.11 even 2
3520.2.a.bq.1.2 2 24.5 odd 2
4400.2.a.bl.1.1 2 60.59 even 2
4400.2.b.p.4049.1 4 60.47 odd 4
4400.2.b.p.4049.4 4 60.23 odd 4
4950.2.a.bw.1.1 2 5.4 even 2
4950.2.c.bc.199.1 4 5.3 odd 4
4950.2.c.bc.199.4 4 5.2 odd 4
5390.2.a.bp.1.2 2 21.20 even 2
6050.2.a.cb.1.2 2 165.164 even 2
7920.2.a.bq.1.1 2 4.3 odd 2
9680.2.a.bt.1.2 2 132.131 odd 2