Properties

Label 6160.2.a.x.1.1
Level $6160$
Weight $2$
Character 6160.1
Self dual yes
Analytic conductor $49.188$
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] = [6160,2,Mod(1,6160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6160.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6160.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: \(49.1878476451\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3080)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 6160.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.82843 q^{3} -1.00000 q^{5} -1.00000 q^{7} +5.00000 q^{9} +O(q^{10})\) \(q-2.82843 q^{3} -1.00000 q^{5} -1.00000 q^{7} +5.00000 q^{9} +1.00000 q^{11} +4.82843 q^{13} +2.82843 q^{15} -0.828427 q^{17} +2.82843 q^{21} +4.00000 q^{23} +1.00000 q^{25} -5.65685 q^{27} -2.00000 q^{29} -6.82843 q^{31} -2.82843 q^{33} +1.00000 q^{35} +11.6569 q^{37} -13.6569 q^{39} +8.82843 q^{41} -1.65685 q^{43} -5.00000 q^{45} +2.82843 q^{47} +1.00000 q^{49} +2.34315 q^{51} +3.65685 q^{53} -1.00000 q^{55} +6.82843 q^{59} +3.17157 q^{61} -5.00000 q^{63} -4.82843 q^{65} -5.65685 q^{67} -11.3137 q^{69} -5.65685 q^{71} -3.17157 q^{73} -2.82843 q^{75} -1.00000 q^{77} +1.00000 q^{81} -5.65685 q^{83} +0.828427 q^{85} +5.65685 q^{87} -17.3137 q^{89} -4.82843 q^{91} +19.3137 q^{93} -9.31371 q^{97} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} - 2 q^{7} + 10 q^{9} + 2 q^{11} + 4 q^{13} + 4 q^{17} + 8 q^{23} + 2 q^{25} - 4 q^{29} - 8 q^{31} + 2 q^{35} + 12 q^{37} - 16 q^{39} + 12 q^{41} + 8 q^{43} - 10 q^{45} + 2 q^{49} + 16 q^{51} - 4 q^{53} - 2 q^{55} + 8 q^{59} + 12 q^{61} - 10 q^{63} - 4 q^{65} - 12 q^{73} - 2 q^{77} + 2 q^{81} - 4 q^{85} - 12 q^{89} - 4 q^{91} + 16 q^{93} + 4 q^{97} + 10 q^{99}+O(q^{100}) \) 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
\(3\) −2.82843 −1.63299 −0.816497 0.577350i \(-0.804087\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 5.00000 1.66667
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 4.82843 1.33916 0.669582 0.742738i \(-0.266473\pi\)
0.669582 + 0.742738i \(0.266473\pi\)
\(14\) 0 0
\(15\) 2.82843 0.730297
\(16\) 0 0
\(17\) −0.828427 −0.200923 −0.100462 0.994941i \(-0.532032\pi\)
−0.100462 + 0.994941i \(0.532032\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 2.82843 0.617213
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.65685 −1.08866
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −6.82843 −1.22642 −0.613211 0.789919i \(-0.710122\pi\)
−0.613211 + 0.789919i \(0.710122\pi\)
\(32\) 0 0
\(33\) −2.82843 −0.492366
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 11.6569 1.91638 0.958188 0.286141i \(-0.0923726\pi\)
0.958188 + 0.286141i \(0.0923726\pi\)
\(38\) 0 0
\(39\) −13.6569 −2.18685
\(40\) 0 0
\(41\) 8.82843 1.37877 0.689384 0.724396i \(-0.257881\pi\)
0.689384 + 0.724396i \(0.257881\pi\)
\(42\) 0 0
\(43\) −1.65685 −0.252668 −0.126334 0.991988i \(-0.540321\pi\)
−0.126334 + 0.991988i \(0.540321\pi\)
\(44\) 0 0
\(45\) −5.00000 −0.745356
\(46\) 0 0
\(47\) 2.82843 0.412568 0.206284 0.978492i \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.34315 0.328106
\(52\) 0 0
\(53\) 3.65685 0.502308 0.251154 0.967947i \(-0.419190\pi\)
0.251154 + 0.967947i \(0.419190\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.82843 0.888985 0.444493 0.895782i \(-0.353384\pi\)
0.444493 + 0.895782i \(0.353384\pi\)
\(60\) 0 0
\(61\) 3.17157 0.406078 0.203039 0.979171i \(-0.434918\pi\)
0.203039 + 0.979171i \(0.434918\pi\)
\(62\) 0 0
\(63\) −5.00000 −0.629941
\(64\) 0 0
\(65\) −4.82843 −0.598893
\(66\) 0 0
\(67\) −5.65685 −0.691095 −0.345547 0.938401i \(-0.612307\pi\)
−0.345547 + 0.938401i \(0.612307\pi\)
\(68\) 0 0
\(69\) −11.3137 −1.36201
\(70\) 0 0
\(71\) −5.65685 −0.671345 −0.335673 0.941979i \(-0.608964\pi\)
−0.335673 + 0.941979i \(0.608964\pi\)
\(72\) 0 0
\(73\) −3.17157 −0.371205 −0.185602 0.982625i \(-0.559424\pi\)
−0.185602 + 0.982625i \(0.559424\pi\)
\(74\) 0 0
\(75\) −2.82843 −0.326599
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −5.65685 −0.620920 −0.310460 0.950586i \(-0.600483\pi\)
−0.310460 + 0.950586i \(0.600483\pi\)
\(84\) 0 0
\(85\) 0.828427 0.0898555
\(86\) 0 0
\(87\) 5.65685 0.606478
\(88\) 0 0
\(89\) −17.3137 −1.83525 −0.917625 0.397448i \(-0.869896\pi\)
−0.917625 + 0.397448i \(0.869896\pi\)
\(90\) 0 0
\(91\) −4.82843 −0.506157
\(92\) 0 0
\(93\) 19.3137 2.00274
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −9.31371 −0.945664 −0.472832 0.881153i \(-0.656768\pi\)
−0.472832 + 0.881153i \(0.656768\pi\)
\(98\) 0 0
\(99\) 5.00000 0.502519
\(100\) 0 0
\(101\) −10.4853 −1.04332 −0.521662 0.853152i \(-0.674688\pi\)
−0.521662 + 0.853152i \(0.674688\pi\)
\(102\) 0 0
\(103\) −5.17157 −0.509570 −0.254785 0.966998i \(-0.582005\pi\)
−0.254785 + 0.966998i \(0.582005\pi\)
\(104\) 0 0
\(105\) −2.82843 −0.276026
\(106\) 0 0
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) −5.31371 −0.508961 −0.254480 0.967078i \(-0.581904\pi\)
−0.254480 + 0.967078i \(0.581904\pi\)
\(110\) 0 0
\(111\) −32.9706 −3.12943
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) 0 0
\(117\) 24.1421 2.23194
\(118\) 0 0
\(119\) 0.828427 0.0759418
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −24.9706 −2.25152
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 4.68629 0.412605
\(130\) 0 0
\(131\) −10.3431 −0.903685 −0.451842 0.892098i \(-0.649233\pi\)
−0.451842 + 0.892098i \(0.649233\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 5.65685 0.486864
\(136\) 0 0
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 4.82843 0.403773
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 0 0
\(147\) −2.82843 −0.233285
\(148\) 0 0
\(149\) 8.34315 0.683497 0.341749 0.939791i \(-0.388981\pi\)
0.341749 + 0.939791i \(0.388981\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) −4.14214 −0.334872
\(154\) 0 0
\(155\) 6.82843 0.548472
\(156\) 0 0
\(157\) −1.31371 −0.104845 −0.0524227 0.998625i \(-0.516694\pi\)
−0.0524227 + 0.998625i \(0.516694\pi\)
\(158\) 0 0
\(159\) −10.3431 −0.820265
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) 24.9706 1.95585 0.977923 0.208967i \(-0.0670101\pi\)
0.977923 + 0.208967i \(0.0670101\pi\)
\(164\) 0 0
\(165\) 2.82843 0.220193
\(166\) 0 0
\(167\) −3.31371 −0.256422 −0.128211 0.991747i \(-0.540924\pi\)
−0.128211 + 0.991747i \(0.540924\pi\)
\(168\) 0 0
\(169\) 10.3137 0.793362
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 8.14214 0.619035 0.309518 0.950894i \(-0.399832\pi\)
0.309518 + 0.950894i \(0.399832\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −19.3137 −1.45171
\(178\) 0 0
\(179\) 15.3137 1.14460 0.572300 0.820044i \(-0.306051\pi\)
0.572300 + 0.820044i \(0.306051\pi\)
\(180\) 0 0
\(181\) 15.6569 1.16376 0.581882 0.813273i \(-0.302316\pi\)
0.581882 + 0.813273i \(0.302316\pi\)
\(182\) 0 0
\(183\) −8.97056 −0.663123
\(184\) 0 0
\(185\) −11.6569 −0.857029
\(186\) 0 0
\(187\) −0.828427 −0.0605806
\(188\) 0 0
\(189\) 5.65685 0.411476
\(190\) 0 0
\(191\) 19.3137 1.39749 0.698745 0.715370i \(-0.253742\pi\)
0.698745 + 0.715370i \(0.253742\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 0 0
\(195\) 13.6569 0.977988
\(196\) 0 0
\(197\) 21.3137 1.51854 0.759269 0.650776i \(-0.225556\pi\)
0.759269 + 0.650776i \(0.225556\pi\)
\(198\) 0 0
\(199\) −17.1716 −1.21726 −0.608630 0.793454i \(-0.708281\pi\)
−0.608630 + 0.793454i \(0.708281\pi\)
\(200\) 0 0
\(201\) 16.0000 1.12855
\(202\) 0 0
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) −8.82843 −0.616604
\(206\) 0 0
\(207\) 20.0000 1.39010
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) 16.0000 1.09630
\(214\) 0 0
\(215\) 1.65685 0.112997
\(216\) 0 0
\(217\) 6.82843 0.463544
\(218\) 0 0
\(219\) 8.97056 0.606174
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) 2.82843 0.189405 0.0947027 0.995506i \(-0.469810\pi\)
0.0947027 + 0.995506i \(0.469810\pi\)
\(224\) 0 0
\(225\) 5.00000 0.333333
\(226\) 0 0
\(227\) 19.3137 1.28190 0.640948 0.767584i \(-0.278541\pi\)
0.640948 + 0.767584i \(0.278541\pi\)
\(228\) 0 0
\(229\) −22.9706 −1.51794 −0.758969 0.651127i \(-0.774296\pi\)
−0.758969 + 0.651127i \(0.774296\pi\)
\(230\) 0 0
\(231\) 2.82843 0.186097
\(232\) 0 0
\(233\) 27.6569 1.81186 0.905930 0.423427i \(-0.139173\pi\)
0.905930 + 0.423427i \(0.139173\pi\)
\(234\) 0 0
\(235\) −2.82843 −0.184506
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −2.34315 −0.151565 −0.0757827 0.997124i \(-0.524146\pi\)
−0.0757827 + 0.997124i \(0.524146\pi\)
\(240\) 0 0
\(241\) −21.7990 −1.40420 −0.702098 0.712080i \(-0.747753\pi\)
−0.702098 + 0.712080i \(0.747753\pi\)
\(242\) 0 0
\(243\) 14.1421 0.907218
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 16.0000 1.01396
\(250\) 0 0
\(251\) −18.1421 −1.14512 −0.572561 0.819862i \(-0.694050\pi\)
−0.572561 + 0.819862i \(0.694050\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 0 0
\(255\) −2.34315 −0.146733
\(256\) 0 0
\(257\) −0.343146 −0.0214048 −0.0107024 0.999943i \(-0.503407\pi\)
−0.0107024 + 0.999943i \(0.503407\pi\)
\(258\) 0 0
\(259\) −11.6569 −0.724322
\(260\) 0 0
\(261\) −10.0000 −0.618984
\(262\) 0 0
\(263\) 11.3137 0.697633 0.348817 0.937191i \(-0.386584\pi\)
0.348817 + 0.937191i \(0.386584\pi\)
\(264\) 0 0
\(265\) −3.65685 −0.224639
\(266\) 0 0
\(267\) 48.9706 2.99695
\(268\) 0 0
\(269\) −3.65685 −0.222962 −0.111481 0.993767i \(-0.535559\pi\)
−0.111481 + 0.993767i \(0.535559\pi\)
\(270\) 0 0
\(271\) −13.6569 −0.829595 −0.414797 0.909914i \(-0.636148\pi\)
−0.414797 + 0.909914i \(0.636148\pi\)
\(272\) 0 0
\(273\) 13.6569 0.826550
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) −34.1421 −2.04404
\(280\) 0 0
\(281\) −6.97056 −0.415829 −0.207914 0.978147i \(-0.566668\pi\)
−0.207914 + 0.978147i \(0.566668\pi\)
\(282\) 0 0
\(283\) 11.3137 0.672530 0.336265 0.941767i \(-0.390836\pi\)
0.336265 + 0.941767i \(0.390836\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8.82843 −0.521126
\(288\) 0 0
\(289\) −16.3137 −0.959630
\(290\) 0 0
\(291\) 26.3431 1.54426
\(292\) 0 0
\(293\) 2.48528 0.145192 0.0725958 0.997361i \(-0.476872\pi\)
0.0725958 + 0.997361i \(0.476872\pi\)
\(294\) 0 0
\(295\) −6.82843 −0.397566
\(296\) 0 0
\(297\) −5.65685 −0.328244
\(298\) 0 0
\(299\) 19.3137 1.11694
\(300\) 0 0
\(301\) 1.65685 0.0954995
\(302\) 0 0
\(303\) 29.6569 1.70374
\(304\) 0 0
\(305\) −3.17157 −0.181604
\(306\) 0 0
\(307\) 16.0000 0.913168 0.456584 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 0 0
\(309\) 14.6274 0.832125
\(310\) 0 0
\(311\) 26.1421 1.48238 0.741192 0.671293i \(-0.234261\pi\)
0.741192 + 0.671293i \(0.234261\pi\)
\(312\) 0 0
\(313\) 4.34315 0.245489 0.122745 0.992438i \(-0.460830\pi\)
0.122745 + 0.992438i \(0.460830\pi\)
\(314\) 0 0
\(315\) 5.00000 0.281718
\(316\) 0 0
\(317\) −26.9706 −1.51482 −0.757409 0.652941i \(-0.773535\pi\)
−0.757409 + 0.652941i \(0.773535\pi\)
\(318\) 0 0
\(319\) −2.00000 −0.111979
\(320\) 0 0
\(321\) −11.3137 −0.631470
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 4.82843 0.267833
\(326\) 0 0
\(327\) 15.0294 0.831130
\(328\) 0 0
\(329\) −2.82843 −0.155936
\(330\) 0 0
\(331\) 34.6274 1.90329 0.951647 0.307192i \(-0.0993895\pi\)
0.951647 + 0.307192i \(0.0993895\pi\)
\(332\) 0 0
\(333\) 58.2843 3.19396
\(334\) 0 0
\(335\) 5.65685 0.309067
\(336\) 0 0
\(337\) −15.6569 −0.852883 −0.426442 0.904515i \(-0.640233\pi\)
−0.426442 + 0.904515i \(0.640233\pi\)
\(338\) 0 0
\(339\) 16.9706 0.921714
\(340\) 0 0
\(341\) −6.82843 −0.369780
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 11.3137 0.609110
\(346\) 0 0
\(347\) −32.2843 −1.73311 −0.866555 0.499081i \(-0.833671\pi\)
−0.866555 + 0.499081i \(0.833671\pi\)
\(348\) 0 0
\(349\) 6.48528 0.347149 0.173575 0.984821i \(-0.444468\pi\)
0.173575 + 0.984821i \(0.444468\pi\)
\(350\) 0 0
\(351\) −27.3137 −1.45790
\(352\) 0 0
\(353\) 26.9706 1.43550 0.717749 0.696302i \(-0.245172\pi\)
0.717749 + 0.696302i \(0.245172\pi\)
\(354\) 0 0
\(355\) 5.65685 0.300235
\(356\) 0 0
\(357\) −2.34315 −0.124012
\(358\) 0 0
\(359\) 27.3137 1.44156 0.720781 0.693163i \(-0.243783\pi\)
0.720781 + 0.693163i \(0.243783\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) −2.82843 −0.148454
\(364\) 0 0
\(365\) 3.17157 0.166008
\(366\) 0 0
\(367\) 6.14214 0.320617 0.160308 0.987067i \(-0.448751\pi\)
0.160308 + 0.987067i \(0.448751\pi\)
\(368\) 0 0
\(369\) 44.1421 2.29795
\(370\) 0 0
\(371\) −3.65685 −0.189854
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 0 0
\(375\) 2.82843 0.146059
\(376\) 0 0
\(377\) −9.65685 −0.497353
\(378\) 0 0
\(379\) 9.65685 0.496039 0.248020 0.968755i \(-0.420220\pi\)
0.248020 + 0.968755i \(0.420220\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 16.4853 0.842359 0.421179 0.906977i \(-0.361616\pi\)
0.421179 + 0.906977i \(0.361616\pi\)
\(384\) 0 0
\(385\) 1.00000 0.0509647
\(386\) 0 0
\(387\) −8.28427 −0.421113
\(388\) 0 0
\(389\) 9.31371 0.472224 0.236112 0.971726i \(-0.424127\pi\)
0.236112 + 0.971726i \(0.424127\pi\)
\(390\) 0 0
\(391\) −3.31371 −0.167581
\(392\) 0 0
\(393\) 29.2548 1.47571
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2.68629 −0.134147 −0.0670735 0.997748i \(-0.521366\pi\)
−0.0670735 + 0.997748i \(0.521366\pi\)
\(402\) 0 0
\(403\) −32.9706 −1.64238
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 11.6569 0.577809
\(408\) 0 0
\(409\) 3.17157 0.156824 0.0784121 0.996921i \(-0.475015\pi\)
0.0784121 + 0.996921i \(0.475015\pi\)
\(410\) 0 0
\(411\) −28.2843 −1.39516
\(412\) 0 0
\(413\) −6.82843 −0.336005
\(414\) 0 0
\(415\) 5.65685 0.277684
\(416\) 0 0
\(417\) −22.6274 −1.10807
\(418\) 0 0
\(419\) −1.17157 −0.0572351 −0.0286175 0.999590i \(-0.509110\pi\)
−0.0286175 + 0.999590i \(0.509110\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) 0 0
\(423\) 14.1421 0.687614
\(424\) 0 0
\(425\) −0.828427 −0.0401846
\(426\) 0 0
\(427\) −3.17157 −0.153483
\(428\) 0 0
\(429\) −13.6569 −0.659359
\(430\) 0 0
\(431\) 12.6863 0.611077 0.305539 0.952180i \(-0.401164\pi\)
0.305539 + 0.952180i \(0.401164\pi\)
\(432\) 0 0
\(433\) 29.3137 1.40873 0.704363 0.709839i \(-0.251233\pi\)
0.704363 + 0.709839i \(0.251233\pi\)
\(434\) 0 0
\(435\) −5.65685 −0.271225
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0.970563 0.0463224 0.0231612 0.999732i \(-0.492627\pi\)
0.0231612 + 0.999732i \(0.492627\pi\)
\(440\) 0 0
\(441\) 5.00000 0.238095
\(442\) 0 0
\(443\) 38.6274 1.83524 0.917622 0.397454i \(-0.130106\pi\)
0.917622 + 0.397454i \(0.130106\pi\)
\(444\) 0 0
\(445\) 17.3137 0.820748
\(446\) 0 0
\(447\) −23.5980 −1.11615
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 8.82843 0.415714
\(452\) 0 0
\(453\) 22.6274 1.06313
\(454\) 0 0
\(455\) 4.82843 0.226360
\(456\) 0 0
\(457\) −40.6274 −1.90047 −0.950235 0.311533i \(-0.899157\pi\)
−0.950235 + 0.311533i \(0.899157\pi\)
\(458\) 0 0
\(459\) 4.68629 0.218737
\(460\) 0 0
\(461\) −1.51472 −0.0705475 −0.0352737 0.999378i \(-0.511230\pi\)
−0.0352737 + 0.999378i \(0.511230\pi\)
\(462\) 0 0
\(463\) −39.3137 −1.82706 −0.913531 0.406768i \(-0.866656\pi\)
−0.913531 + 0.406768i \(0.866656\pi\)
\(464\) 0 0
\(465\) −19.3137 −0.895652
\(466\) 0 0
\(467\) 36.7696 1.70149 0.850746 0.525577i \(-0.176151\pi\)
0.850746 + 0.525577i \(0.176151\pi\)
\(468\) 0 0
\(469\) 5.65685 0.261209
\(470\) 0 0
\(471\) 3.71573 0.171212
\(472\) 0 0
\(473\) −1.65685 −0.0761822
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 18.2843 0.837179
\(478\) 0 0
\(479\) −24.9706 −1.14093 −0.570467 0.821320i \(-0.693238\pi\)
−0.570467 + 0.821320i \(0.693238\pi\)
\(480\) 0 0
\(481\) 56.2843 2.56634
\(482\) 0 0
\(483\) 11.3137 0.514792
\(484\) 0 0
\(485\) 9.31371 0.422914
\(486\) 0 0
\(487\) 18.6274 0.844089 0.422044 0.906575i \(-0.361313\pi\)
0.422044 + 0.906575i \(0.361313\pi\)
\(488\) 0 0
\(489\) −70.6274 −3.19388
\(490\) 0 0
\(491\) −28.9706 −1.30742 −0.653712 0.756744i \(-0.726789\pi\)
−0.653712 + 0.756744i \(0.726789\pi\)
\(492\) 0 0
\(493\) 1.65685 0.0746210
\(494\) 0 0
\(495\) −5.00000 −0.224733
\(496\) 0 0
\(497\) 5.65685 0.253745
\(498\) 0 0
\(499\) 20.9706 0.938771 0.469386 0.882993i \(-0.344475\pi\)
0.469386 + 0.882993i \(0.344475\pi\)
\(500\) 0 0
\(501\) 9.37258 0.418736
\(502\) 0 0
\(503\) 32.9706 1.47008 0.735042 0.678021i \(-0.237162\pi\)
0.735042 + 0.678021i \(0.237162\pi\)
\(504\) 0 0
\(505\) 10.4853 0.466589
\(506\) 0 0
\(507\) −29.1716 −1.29556
\(508\) 0 0
\(509\) −3.65685 −0.162087 −0.0810436 0.996711i \(-0.525825\pi\)
−0.0810436 + 0.996711i \(0.525825\pi\)
\(510\) 0 0
\(511\) 3.17157 0.140302
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5.17157 0.227887
\(516\) 0 0
\(517\) 2.82843 0.124394
\(518\) 0 0
\(519\) −23.0294 −1.01088
\(520\) 0 0
\(521\) −0.343146 −0.0150335 −0.00751674 0.999972i \(-0.502393\pi\)
−0.00751674 + 0.999972i \(0.502393\pi\)
\(522\) 0 0
\(523\) 30.6274 1.33924 0.669622 0.742702i \(-0.266456\pi\)
0.669622 + 0.742702i \(0.266456\pi\)
\(524\) 0 0
\(525\) 2.82843 0.123443
\(526\) 0 0
\(527\) 5.65685 0.246416
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 34.1421 1.48164
\(532\) 0 0
\(533\) 42.6274 1.84640
\(534\) 0 0
\(535\) −4.00000 −0.172935
\(536\) 0 0
\(537\) −43.3137 −1.86912
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 18.2843 0.786102 0.393051 0.919517i \(-0.371420\pi\)
0.393051 + 0.919517i \(0.371420\pi\)
\(542\) 0 0
\(543\) −44.2843 −1.90042
\(544\) 0 0
\(545\) 5.31371 0.227614
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) 15.8579 0.676797
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 32.9706 1.39952
\(556\) 0 0
\(557\) 45.3137 1.92000 0.960002 0.279994i \(-0.0903325\pi\)
0.960002 + 0.279994i \(0.0903325\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 2.34315 0.0989277
\(562\) 0 0
\(563\) −10.3431 −0.435912 −0.217956 0.975959i \(-0.569939\pi\)
−0.217956 + 0.975959i \(0.569939\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 18.9706 0.795287 0.397644 0.917540i \(-0.369828\pi\)
0.397644 + 0.917540i \(0.369828\pi\)
\(570\) 0 0
\(571\) −31.3137 −1.31044 −0.655219 0.755439i \(-0.727424\pi\)
−0.655219 + 0.755439i \(0.727424\pi\)
\(572\) 0 0
\(573\) −54.6274 −2.28209
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 0 0
\(579\) −39.5980 −1.64564
\(580\) 0 0
\(581\) 5.65685 0.234686
\(582\) 0 0
\(583\) 3.65685 0.151451
\(584\) 0 0
\(585\) −24.1421 −0.998154
\(586\) 0 0
\(587\) −3.79899 −0.156801 −0.0784005 0.996922i \(-0.524981\pi\)
−0.0784005 + 0.996922i \(0.524981\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −60.2843 −2.47976
\(592\) 0 0
\(593\) 27.4558 1.12748 0.563738 0.825954i \(-0.309363\pi\)
0.563738 + 0.825954i \(0.309363\pi\)
\(594\) 0 0
\(595\) −0.828427 −0.0339622
\(596\) 0 0
\(597\) 48.5685 1.98778
\(598\) 0 0
\(599\) −21.6569 −0.884875 −0.442438 0.896799i \(-0.645886\pi\)
−0.442438 + 0.896799i \(0.645886\pi\)
\(600\) 0 0
\(601\) −3.85786 −0.157366 −0.0786828 0.996900i \(-0.525071\pi\)
−0.0786828 + 0.996900i \(0.525071\pi\)
\(602\) 0 0
\(603\) −28.2843 −1.15182
\(604\) 0 0
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 24.0000 0.974130 0.487065 0.873366i \(-0.338067\pi\)
0.487065 + 0.873366i \(0.338067\pi\)
\(608\) 0 0
\(609\) −5.65685 −0.229227
\(610\) 0 0
\(611\) 13.6569 0.552497
\(612\) 0 0
\(613\) −5.02944 −0.203137 −0.101569 0.994829i \(-0.532386\pi\)
−0.101569 + 0.994829i \(0.532386\pi\)
\(614\) 0 0
\(615\) 24.9706 1.00691
\(616\) 0 0
\(617\) −23.9411 −0.963833 −0.481917 0.876217i \(-0.660059\pi\)
−0.481917 + 0.876217i \(0.660059\pi\)
\(618\) 0 0
\(619\) 7.79899 0.313468 0.156734 0.987641i \(-0.449903\pi\)
0.156734 + 0.987641i \(0.449903\pi\)
\(620\) 0 0
\(621\) −22.6274 −0.908007
\(622\) 0 0
\(623\) 17.3137 0.693659
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −9.65685 −0.385044
\(630\) 0 0
\(631\) 25.9411 1.03270 0.516350 0.856378i \(-0.327290\pi\)
0.516350 + 0.856378i \(0.327290\pi\)
\(632\) 0 0
\(633\) −11.3137 −0.449680
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 4.82843 0.191309
\(638\) 0 0
\(639\) −28.2843 −1.11891
\(640\) 0 0
\(641\) −12.6274 −0.498753 −0.249376 0.968407i \(-0.580226\pi\)
−0.249376 + 0.968407i \(0.580226\pi\)
\(642\) 0 0
\(643\) −9.85786 −0.388756 −0.194378 0.980927i \(-0.562269\pi\)
−0.194378 + 0.980927i \(0.562269\pi\)
\(644\) 0 0
\(645\) −4.68629 −0.184523
\(646\) 0 0
\(647\) −33.4558 −1.31528 −0.657642 0.753330i \(-0.728446\pi\)
−0.657642 + 0.753330i \(0.728446\pi\)
\(648\) 0 0
\(649\) 6.82843 0.268039
\(650\) 0 0
\(651\) −19.3137 −0.756964
\(652\) 0 0
\(653\) −26.9706 −1.05544 −0.527720 0.849418i \(-0.676953\pi\)
−0.527720 + 0.849418i \(0.676953\pi\)
\(654\) 0 0
\(655\) 10.3431 0.404140
\(656\) 0 0
\(657\) −15.8579 −0.618674
\(658\) 0 0
\(659\) 28.9706 1.12853 0.564266 0.825593i \(-0.309159\pi\)
0.564266 + 0.825593i \(0.309159\pi\)
\(660\) 0 0
\(661\) 14.6863 0.571231 0.285615 0.958344i \(-0.407802\pi\)
0.285615 + 0.958344i \(0.407802\pi\)
\(662\) 0 0
\(663\) 11.3137 0.439388
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −8.00000 −0.309761
\(668\) 0 0
\(669\) −8.00000 −0.309298
\(670\) 0 0
\(671\) 3.17157 0.122437
\(672\) 0 0
\(673\) 34.2843 1.32156 0.660781 0.750579i \(-0.270225\pi\)
0.660781 + 0.750579i \(0.270225\pi\)
\(674\) 0 0
\(675\) −5.65685 −0.217732
\(676\) 0 0
\(677\) −46.4853 −1.78657 −0.893287 0.449486i \(-0.851607\pi\)
−0.893287 + 0.449486i \(0.851607\pi\)
\(678\) 0 0
\(679\) 9.31371 0.357427
\(680\) 0 0
\(681\) −54.6274 −2.09333
\(682\) 0 0
\(683\) 19.3137 0.739019 0.369509 0.929227i \(-0.379526\pi\)
0.369509 + 0.929227i \(0.379526\pi\)
\(684\) 0 0
\(685\) −10.0000 −0.382080
\(686\) 0 0
\(687\) 64.9706 2.47878
\(688\) 0 0
\(689\) 17.6569 0.672673
\(690\) 0 0
\(691\) 18.1421 0.690159 0.345080 0.938573i \(-0.387852\pi\)
0.345080 + 0.938573i \(0.387852\pi\)
\(692\) 0 0
\(693\) −5.00000 −0.189934
\(694\) 0 0
\(695\) −8.00000 −0.303457
\(696\) 0 0
\(697\) −7.31371 −0.277026
\(698\) 0 0
\(699\) −78.2254 −2.95876
\(700\) 0 0
\(701\) −29.3137 −1.10716 −0.553582 0.832795i \(-0.686739\pi\)
−0.553582 + 0.832795i \(0.686739\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 8.00000 0.301297
\(706\) 0 0
\(707\) 10.4853 0.394340
\(708\) 0 0
\(709\) 20.6274 0.774679 0.387339 0.921937i \(-0.373394\pi\)
0.387339 + 0.921937i \(0.373394\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −27.3137 −1.02291
\(714\) 0 0
\(715\) −4.82843 −0.180573
\(716\) 0 0
\(717\) 6.62742 0.247505
\(718\) 0 0
\(719\) −21.4558 −0.800168 −0.400084 0.916478i \(-0.631019\pi\)
−0.400084 + 0.916478i \(0.631019\pi\)
\(720\) 0 0
\(721\) 5.17157 0.192599
\(722\) 0 0
\(723\) 61.6569 2.29304
\(724\) 0 0
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) −26.8284 −0.995011 −0.497506 0.867461i \(-0.665751\pi\)
−0.497506 + 0.867461i \(0.665751\pi\)
\(728\) 0 0
\(729\) −43.0000 −1.59259
\(730\) 0 0
\(731\) 1.37258 0.0507668
\(732\) 0 0
\(733\) 11.8579 0.437980 0.218990 0.975727i \(-0.429724\pi\)
0.218990 + 0.975727i \(0.429724\pi\)
\(734\) 0 0
\(735\) 2.82843 0.104328
\(736\) 0 0
\(737\) −5.65685 −0.208373
\(738\) 0 0
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 30.6274 1.12361 0.561805 0.827269i \(-0.310107\pi\)
0.561805 + 0.827269i \(0.310107\pi\)
\(744\) 0 0
\(745\) −8.34315 −0.305669
\(746\) 0 0
\(747\) −28.2843 −1.03487
\(748\) 0 0
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) 28.2843 1.03211 0.516054 0.856556i \(-0.327400\pi\)
0.516054 + 0.856556i \(0.327400\pi\)
\(752\) 0 0
\(753\) 51.3137 1.86998
\(754\) 0 0
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) −15.6569 −0.569058 −0.284529 0.958667i \(-0.591837\pi\)
−0.284529 + 0.958667i \(0.591837\pi\)
\(758\) 0 0
\(759\) −11.3137 −0.410662
\(760\) 0 0
\(761\) 46.4853 1.68509 0.842545 0.538626i \(-0.181056\pi\)
0.842545 + 0.538626i \(0.181056\pi\)
\(762\) 0 0
\(763\) 5.31371 0.192369
\(764\) 0 0
\(765\) 4.14214 0.149759
\(766\) 0 0
\(767\) 32.9706 1.19050
\(768\) 0 0
\(769\) −1.51472 −0.0546222 −0.0273111 0.999627i \(-0.508694\pi\)
−0.0273111 + 0.999627i \(0.508694\pi\)
\(770\) 0 0
\(771\) 0.970563 0.0349540
\(772\) 0 0
\(773\) −22.9706 −0.826194 −0.413097 0.910687i \(-0.635553\pi\)
−0.413097 + 0.910687i \(0.635553\pi\)
\(774\) 0 0
\(775\) −6.82843 −0.245284
\(776\) 0 0
\(777\) 32.9706 1.18281
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −5.65685 −0.202418
\(782\) 0 0
\(783\) 11.3137 0.404319
\(784\) 0 0
\(785\) 1.31371 0.0468883
\(786\) 0 0
\(787\) 8.97056 0.319766 0.159883 0.987136i \(-0.448888\pi\)
0.159883 + 0.987136i \(0.448888\pi\)
\(788\) 0 0
\(789\) −32.0000 −1.13923
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 15.3137 0.543806
\(794\) 0 0
\(795\) 10.3431 0.366834
\(796\) 0 0
\(797\) −26.6863 −0.945277 −0.472638 0.881256i \(-0.656698\pi\)
−0.472638 + 0.881256i \(0.656698\pi\)
\(798\) 0 0
\(799\) −2.34315 −0.0828945
\(800\) 0 0
\(801\) −86.5685 −3.05875
\(802\) 0 0
\(803\) −3.17157 −0.111922
\(804\) 0 0
\(805\) 4.00000 0.140981
\(806\) 0 0
\(807\) 10.3431 0.364096
\(808\) 0 0
\(809\) −2.68629 −0.0944450 −0.0472225 0.998884i \(-0.515037\pi\)
−0.0472225 + 0.998884i \(0.515037\pi\)
\(810\) 0 0
\(811\) −28.2843 −0.993195 −0.496598 0.867981i \(-0.665417\pi\)
−0.496598 + 0.867981i \(0.665417\pi\)
\(812\) 0 0
\(813\) 38.6274 1.35472
\(814\) 0 0
\(815\) −24.9706 −0.874681
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −24.1421 −0.843594
\(820\) 0 0
\(821\) 54.9706 1.91849 0.959243 0.282583i \(-0.0911912\pi\)
0.959243 + 0.282583i \(0.0911912\pi\)
\(822\) 0 0
\(823\) 28.9706 1.00985 0.504925 0.863163i \(-0.331520\pi\)
0.504925 + 0.863163i \(0.331520\pi\)
\(824\) 0 0
\(825\) −2.82843 −0.0984732
\(826\) 0 0
\(827\) −1.65685 −0.0576145 −0.0288072 0.999585i \(-0.509171\pi\)
−0.0288072 + 0.999585i \(0.509171\pi\)
\(828\) 0 0
\(829\) 45.3137 1.57381 0.786905 0.617074i \(-0.211682\pi\)
0.786905 + 0.617074i \(0.211682\pi\)
\(830\) 0 0
\(831\) −5.65685 −0.196234
\(832\) 0 0
\(833\) −0.828427 −0.0287033
\(834\) 0 0
\(835\) 3.31371 0.114676
\(836\) 0 0
\(837\) 38.6274 1.33516
\(838\) 0 0
\(839\) 19.1127 0.659844 0.329922 0.944008i \(-0.392978\pi\)
0.329922 + 0.944008i \(0.392978\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 19.7157 0.679046
\(844\) 0 0
\(845\) −10.3137 −0.354802
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 0 0
\(849\) −32.0000 −1.09824
\(850\) 0 0
\(851\) 46.6274 1.59837
\(852\) 0 0
\(853\) −40.4264 −1.38417 −0.692087 0.721814i \(-0.743309\pi\)
−0.692087 + 0.721814i \(0.743309\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 20.8284 0.711486 0.355743 0.934584i \(-0.384228\pi\)
0.355743 + 0.934584i \(0.384228\pi\)
\(858\) 0 0
\(859\) 31.7990 1.08497 0.542484 0.840066i \(-0.317484\pi\)
0.542484 + 0.840066i \(0.317484\pi\)
\(860\) 0 0
\(861\) 24.9706 0.850995
\(862\) 0 0
\(863\) −24.2843 −0.826646 −0.413323 0.910584i \(-0.635632\pi\)
−0.413323 + 0.910584i \(0.635632\pi\)
\(864\) 0 0
\(865\) −8.14214 −0.276841
\(866\) 0 0
\(867\) 46.1421 1.56707
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −27.3137 −0.925490
\(872\) 0 0
\(873\) −46.5685 −1.57611
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −58.2843 −1.96812 −0.984060 0.177837i \(-0.943090\pi\)
−0.984060 + 0.177837i \(0.943090\pi\)
\(878\) 0 0
\(879\) −7.02944 −0.237097
\(880\) 0 0
\(881\) 36.3431 1.22443 0.612216 0.790691i \(-0.290278\pi\)
0.612216 + 0.790691i \(0.290278\pi\)
\(882\) 0 0
\(883\) 8.00000 0.269221 0.134611 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) 0 0
\(885\) 19.3137 0.649223
\(886\) 0 0
\(887\) 16.9706 0.569816 0.284908 0.958555i \(-0.408037\pi\)
0.284908 + 0.958555i \(0.408037\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −15.3137 −0.511881
\(896\) 0 0
\(897\) −54.6274 −1.82396
\(898\) 0 0
\(899\) 13.6569 0.455482
\(900\) 0 0
\(901\) −3.02944 −0.100925
\(902\) 0 0
\(903\) −4.68629 −0.155950
\(904\) 0 0
\(905\) −15.6569 −0.520451
\(906\) 0 0
\(907\) −8.97056 −0.297863 −0.148931 0.988848i \(-0.547583\pi\)
−0.148931 + 0.988848i \(0.547583\pi\)
\(908\) 0 0
\(909\) −52.4264 −1.73887
\(910\) 0 0
\(911\) 36.2843 1.20215 0.601076 0.799192i \(-0.294739\pi\)
0.601076 + 0.799192i \(0.294739\pi\)
\(912\) 0 0
\(913\) −5.65685 −0.187215
\(914\) 0 0
\(915\) 8.97056 0.296558
\(916\) 0 0
\(917\) 10.3431 0.341561
\(918\) 0 0
\(919\) −48.9706 −1.61539 −0.807695 0.589601i \(-0.799285\pi\)
−0.807695 + 0.589601i \(0.799285\pi\)
\(920\) 0 0
\(921\) −45.2548 −1.49120
\(922\) 0 0
\(923\) −27.3137 −0.899042
\(924\) 0 0
\(925\) 11.6569 0.383275
\(926\) 0 0
\(927\) −25.8579 −0.849284
\(928\) 0 0
\(929\) 39.6569 1.30110 0.650550 0.759464i \(-0.274539\pi\)
0.650550 + 0.759464i \(0.274539\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −73.9411 −2.42072
\(934\) 0 0
\(935\) 0.828427 0.0270925
\(936\) 0 0
\(937\) 14.2010 0.463927 0.231963 0.972725i \(-0.425485\pi\)
0.231963 + 0.972725i \(0.425485\pi\)
\(938\) 0 0
\(939\) −12.2843 −0.400882
\(940\) 0 0
\(941\) −38.7696 −1.26385 −0.631926 0.775029i \(-0.717735\pi\)
−0.631926 + 0.775029i \(0.717735\pi\)
\(942\) 0 0
\(943\) 35.3137 1.14997
\(944\) 0 0
\(945\) −5.65685 −0.184017
\(946\) 0 0
\(947\) 17.9411 0.583008 0.291504 0.956570i \(-0.405844\pi\)
0.291504 + 0.956570i \(0.405844\pi\)
\(948\) 0 0
\(949\) −15.3137 −0.497104
\(950\) 0 0
\(951\) 76.2843 2.47369
\(952\) 0 0
\(953\) −22.6863 −0.734881 −0.367441 0.930047i \(-0.619766\pi\)
−0.367441 + 0.930047i \(0.619766\pi\)
\(954\) 0 0
\(955\) −19.3137 −0.624977
\(956\) 0 0
\(957\) 5.65685 0.182860
\(958\) 0 0
\(959\) −10.0000 −0.322917
\(960\) 0 0
\(961\) 15.6274 0.504110
\(962\) 0 0
\(963\) 20.0000 0.644491
\(964\) 0 0
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) 40.9706 1.31752 0.658762 0.752351i \(-0.271080\pi\)
0.658762 + 0.752351i \(0.271080\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −22.8284 −0.732599 −0.366300 0.930497i \(-0.619375\pi\)
−0.366300 + 0.930497i \(0.619375\pi\)
\(972\) 0 0
\(973\) −8.00000 −0.256468
\(974\) 0 0
\(975\) −13.6569 −0.437369
\(976\) 0 0
\(977\) −17.3137 −0.553915 −0.276957 0.960882i \(-0.589326\pi\)
−0.276957 + 0.960882i \(0.589326\pi\)
\(978\) 0 0
\(979\) −17.3137 −0.553349
\(980\) 0 0
\(981\) −26.5685 −0.848268
\(982\) 0 0
\(983\) 26.8284 0.855694 0.427847 0.903851i \(-0.359272\pi\)
0.427847 + 0.903851i \(0.359272\pi\)
\(984\) 0 0
\(985\) −21.3137 −0.679111
\(986\) 0 0
\(987\) 8.00000 0.254643
\(988\) 0 0
\(989\) −6.62742 −0.210740
\(990\) 0 0
\(991\) −12.2843 −0.390223 −0.195111 0.980781i \(-0.562507\pi\)
−0.195111 + 0.980781i \(0.562507\pi\)
\(992\) 0 0
\(993\) −97.9411 −3.10807
\(994\) 0 0
\(995\) 17.1716 0.544375
\(996\) 0 0
\(997\) −46.4853 −1.47220 −0.736102 0.676871i \(-0.763336\pi\)
−0.736102 + 0.676871i \(0.763336\pi\)
\(998\) 0 0
\(999\) −65.9411 −2.08628
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 6160.2.a.x.1.1 2
4.3 odd 2 3080.2.a.g.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3080.2.a.g.1.2 2 4.3 odd 2
6160.2.a.x.1.1 2 1.1 even 1 trivial