Properties

Label 810.2.a.k.1.2
Level $810$
Weight $2$
Character 810.1
Self dual yes
Analytic conductor $6.468$
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] = [810,2,Mod(1,810)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(810, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("810.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 810.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: \(6.46788256372\)
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 90)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 810.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +3.37228 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +3.37228 q^{7} +1.00000 q^{8} +1.00000 q^{10} +4.37228 q^{11} -6.74456 q^{13} +3.37228 q^{14} +1.00000 q^{16} +1.62772 q^{17} -2.37228 q^{19} +1.00000 q^{20} +4.37228 q^{22} -1.37228 q^{23} +1.00000 q^{25} -6.74456 q^{26} +3.37228 q^{28} -1.37228 q^{29} +4.74456 q^{31} +1.00000 q^{32} +1.62772 q^{34} +3.37228 q^{35} -4.00000 q^{37} -2.37228 q^{38} +1.00000 q^{40} +3.00000 q^{41} -5.62772 q^{43} +4.37228 q^{44} -1.37228 q^{46} +7.37228 q^{47} +4.37228 q^{49} +1.00000 q^{50} -6.74456 q^{52} +11.4891 q^{53} +4.37228 q^{55} +3.37228 q^{56} -1.37228 q^{58} -4.37228 q^{59} -8.11684 q^{61} +4.74456 q^{62} +1.00000 q^{64} -6.74456 q^{65} -7.00000 q^{67} +1.62772 q^{68} +3.37228 q^{70} +6.00000 q^{71} +3.11684 q^{73} -4.00000 q^{74} -2.37228 q^{76} +14.7446 q^{77} +2.00000 q^{79} +1.00000 q^{80} +3.00000 q^{82} -7.37228 q^{83} +1.62772 q^{85} -5.62772 q^{86} +4.37228 q^{88} -16.1168 q^{89} -22.7446 q^{91} -1.37228 q^{92} +7.37228 q^{94} -2.37228 q^{95} -8.37228 q^{97} +4.37228 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + q^{7} + 2 q^{8} + 2 q^{10} + 3 q^{11} - 2 q^{13} + q^{14} + 2 q^{16} + 9 q^{17} + q^{19} + 2 q^{20} + 3 q^{22} + 3 q^{23} + 2 q^{25} - 2 q^{26} + q^{28} + 3 q^{29} - 2 q^{31} + 2 q^{32} + 9 q^{34} + q^{35} - 8 q^{37} + q^{38} + 2 q^{40} + 6 q^{41} - 17 q^{43} + 3 q^{44} + 3 q^{46} + 9 q^{47} + 3 q^{49} + 2 q^{50} - 2 q^{52} + 3 q^{55} + q^{56} + 3 q^{58} - 3 q^{59} + q^{61} - 2 q^{62} + 2 q^{64} - 2 q^{65} - 14 q^{67} + 9 q^{68} + q^{70} + 12 q^{71} - 11 q^{73} - 8 q^{74} + q^{76} + 18 q^{77} + 4 q^{79} + 2 q^{80} + 6 q^{82} - 9 q^{83} + 9 q^{85} - 17 q^{86} + 3 q^{88} - 15 q^{89} - 34 q^{91} + 3 q^{92} + 9 q^{94} + q^{95} - 11 q^{97} + 3 q^{98}+O(q^{100}) \) 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
\(11\) 4.37228 1.31829 0.659146 0.752015i \(-0.270918\pi\)
0.659146 + 0.752015i \(0.270918\pi\)
\(12\) 0 0
\(13\) −6.74456 −1.87061 −0.935303 0.353849i \(-0.884873\pi\)
−0.935303 + 0.353849i \(0.884873\pi\)
\(14\) 3.37228 0.901280
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.62772 0.394780 0.197390 0.980325i \(-0.436754\pi\)
0.197390 + 0.980325i \(0.436754\pi\)
\(18\) 0 0
\(19\) −2.37228 −0.544239 −0.272119 0.962264i \(-0.587725\pi\)
−0.272119 + 0.962264i \(0.587725\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 4.37228 0.932174
\(23\) −1.37228 −0.286140 −0.143070 0.989713i \(-0.545697\pi\)
−0.143070 + 0.989713i \(0.545697\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −6.74456 −1.32272
\(27\) 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
\(31\) 4.74456 0.852149 0.426074 0.904688i \(-0.359896\pi\)
0.426074 + 0.904688i \(0.359896\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 1.62772 0.279151
\(35\) 3.37228 0.570020
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) −2.37228 −0.384835
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) −5.62772 −0.858219 −0.429110 0.903252i \(-0.641173\pi\)
−0.429110 + 0.903252i \(0.641173\pi\)
\(44\) 4.37228 0.659146
\(45\) 0 0
\(46\) −1.37228 −0.202332
\(47\) 7.37228 1.07536 0.537679 0.843150i \(-0.319301\pi\)
0.537679 + 0.843150i \(0.319301\pi\)
\(48\) 0 0
\(49\) 4.37228 0.624612
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −6.74456 −0.935303
\(53\) 11.4891 1.57815 0.789076 0.614295i \(-0.210560\pi\)
0.789076 + 0.614295i \(0.210560\pi\)
\(54\) 0 0
\(55\) 4.37228 0.589558
\(56\) 3.37228 0.450640
\(57\) 0 0
\(58\) −1.37228 −0.180189
\(59\) −4.37228 −0.569223 −0.284611 0.958643i \(-0.591865\pi\)
−0.284611 + 0.958643i \(0.591865\pi\)
\(60\) 0 0
\(61\) −8.11684 −1.03926 −0.519628 0.854393i \(-0.673929\pi\)
−0.519628 + 0.854393i \(0.673929\pi\)
\(62\) 4.74456 0.602560
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −6.74456 −0.836560
\(66\) 0 0
\(67\) −7.00000 −0.855186 −0.427593 0.903971i \(-0.640638\pi\)
−0.427593 + 0.903971i \(0.640638\pi\)
\(68\) 1.62772 0.197390
\(69\) 0 0
\(70\) 3.37228 0.403065
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 3.11684 0.364799 0.182399 0.983225i \(-0.441614\pi\)
0.182399 + 0.983225i \(0.441614\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) −2.37228 −0.272119
\(77\) 14.7446 1.68030
\(78\) 0 0
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 3.00000 0.331295
\(83\) −7.37228 −0.809213 −0.404607 0.914491i \(-0.632592\pi\)
−0.404607 + 0.914491i \(0.632592\pi\)
\(84\) 0 0
\(85\) 1.62772 0.176551
\(86\) −5.62772 −0.606853
\(87\) 0 0
\(88\) 4.37228 0.466087
\(89\) −16.1168 −1.70838 −0.854191 0.519959i \(-0.825947\pi\)
−0.854191 + 0.519959i \(0.825947\pi\)
\(90\) 0 0
\(91\) −22.7446 −2.38428
\(92\) −1.37228 −0.143070
\(93\) 0 0
\(94\) 7.37228 0.760393
\(95\) −2.37228 −0.243391
\(96\) 0 0
\(97\) −8.37228 −0.850076 −0.425038 0.905175i \(-0.639739\pi\)
−0.425038 + 0.905175i \(0.639739\pi\)
\(98\) 4.37228 0.441667
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −2.74456 −0.273094 −0.136547 0.990634i \(-0.543601\pi\)
−0.136547 + 0.990634i \(0.543601\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −6.74456 −0.661359
\(105\) 0 0
\(106\) 11.4891 1.11592
\(107\) −8.48913 −0.820675 −0.410337 0.911934i \(-0.634589\pi\)
−0.410337 + 0.911934i \(0.634589\pi\)
\(108\) 0 0
\(109\) 15.3723 1.47240 0.736199 0.676765i \(-0.236619\pi\)
0.736199 + 0.676765i \(0.236619\pi\)
\(110\) 4.37228 0.416881
\(111\) 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
\(115\) −1.37228 −0.127966
\(116\) −1.37228 −0.127413
\(117\) 0 0
\(118\) −4.37228 −0.402501
\(119\) 5.48913 0.503187
\(120\) 0 0
\(121\) 8.11684 0.737895
\(122\) −8.11684 −0.734865
\(123\) 0 0
\(124\) 4.74456 0.426074
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −8.11684 −0.720253 −0.360127 0.932903i \(-0.617267\pi\)
−0.360127 + 0.932903i \(0.617267\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −6.74456 −0.591537
\(131\) 2.74456 0.239794 0.119897 0.992786i \(-0.461744\pi\)
0.119897 + 0.992786i \(0.461744\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) −7.00000 −0.604708
\(135\) 0 0
\(136\) 1.62772 0.139576
\(137\) 19.1168 1.63326 0.816631 0.577160i \(-0.195839\pi\)
0.816631 + 0.577160i \(0.195839\pi\)
\(138\) 0 0
\(139\) 0.883156 0.0749083 0.0374542 0.999298i \(-0.488075\pi\)
0.0374542 + 0.999298i \(0.488075\pi\)
\(140\) 3.37228 0.285010
\(141\) 0 0
\(142\) 6.00000 0.503509
\(143\) −29.4891 −2.46600
\(144\) 0 0
\(145\) −1.37228 −0.113962
\(146\) 3.11684 0.257952
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) −1.88316 −0.154274 −0.0771371 0.997020i \(-0.524578\pi\)
−0.0771371 + 0.997020i \(0.524578\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) −2.37228 −0.192417
\(153\) 0 0
\(154\) 14.7446 1.18815
\(155\) 4.74456 0.381092
\(156\) 0 0
\(157\) −6.74456 −0.538275 −0.269137 0.963102i \(-0.586739\pi\)
−0.269137 + 0.963102i \(0.586739\pi\)
\(158\) 2.00000 0.159111
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) −4.62772 −0.364715
\(162\) 0 0
\(163\) −21.4891 −1.68316 −0.841579 0.540134i \(-0.818374\pi\)
−0.841579 + 0.540134i \(0.818374\pi\)
\(164\) 3.00000 0.234261
\(165\) 0 0
\(166\) −7.37228 −0.572200
\(167\) −13.3723 −1.03478 −0.517389 0.855750i \(-0.673096\pi\)
−0.517389 + 0.855750i \(0.673096\pi\)
\(168\) 0 0
\(169\) 32.4891 2.49916
\(170\) 1.62772 0.124840
\(171\) 0 0
\(172\) −5.62772 −0.429110
\(173\) −20.7446 −1.57718 −0.788590 0.614919i \(-0.789189\pi\)
−0.788590 + 0.614919i \(0.789189\pi\)
\(174\) 0 0
\(175\) 3.37228 0.254921
\(176\) 4.37228 0.329573
\(177\) 0 0
\(178\) −16.1168 −1.20801
\(179\) 14.7446 1.10206 0.551030 0.834485i \(-0.314235\pi\)
0.551030 + 0.834485i \(0.314235\pi\)
\(180\) 0 0
\(181\) 20.8614 1.55062 0.775308 0.631583i \(-0.217595\pi\)
0.775308 + 0.631583i \(0.217595\pi\)
\(182\) −22.7446 −1.68594
\(183\) 0 0
\(184\) −1.37228 −0.101166
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) 7.11684 0.520435
\(188\) 7.37228 0.537679
\(189\) 0 0
\(190\) −2.37228 −0.172103
\(191\) −17.4891 −1.26547 −0.632734 0.774369i \(-0.718067\pi\)
−0.632734 + 0.774369i \(0.718067\pi\)
\(192\) 0 0
\(193\) 21.1168 1.52002 0.760012 0.649909i \(-0.225193\pi\)
0.760012 + 0.649909i \(0.225193\pi\)
\(194\) −8.37228 −0.601095
\(195\) 0 0
\(196\) 4.37228 0.312306
\(197\) 5.48913 0.391084 0.195542 0.980695i \(-0.437353\pi\)
0.195542 + 0.980695i \(0.437353\pi\)
\(198\) 0 0
\(199\) 13.4891 0.956219 0.478109 0.878300i \(-0.341322\pi\)
0.478109 + 0.878300i \(0.341322\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −2.74456 −0.193107
\(203\) −4.62772 −0.324802
\(204\) 0 0
\(205\) 3.00000 0.209529
\(206\) −16.0000 −1.11477
\(207\) 0 0
\(208\) −6.74456 −0.467651
\(209\) −10.3723 −0.717466
\(210\) 0 0
\(211\) −18.7446 −1.29043 −0.645214 0.764002i \(-0.723232\pi\)
−0.645214 + 0.764002i \(0.723232\pi\)
\(212\) 11.4891 0.789076
\(213\) 0 0
\(214\) −8.48913 −0.580305
\(215\) −5.62772 −0.383807
\(216\) 0 0
\(217\) 16.0000 1.08615
\(218\) 15.3723 1.04114
\(219\) 0 0
\(220\) 4.37228 0.294779
\(221\) −10.9783 −0.738477
\(222\) 0 0
\(223\) 6.62772 0.443825 0.221912 0.975067i \(-0.428770\pi\)
0.221912 + 0.975067i \(0.428770\pi\)
\(224\) 3.37228 0.225320
\(225\) 0 0
\(226\) −3.25544 −0.216548
\(227\) −19.1168 −1.26883 −0.634415 0.772993i \(-0.718759\pi\)
−0.634415 + 0.772993i \(0.718759\pi\)
\(228\) 0 0
\(229\) 12.6277 0.834463 0.417232 0.908800i \(-0.363000\pi\)
0.417232 + 0.908800i \(0.363000\pi\)
\(230\) −1.37228 −0.0904856
\(231\) 0 0
\(232\) −1.37228 −0.0900947
\(233\) −7.11684 −0.466240 −0.233120 0.972448i \(-0.574893\pi\)
−0.233120 + 0.972448i \(0.574893\pi\)
\(234\) 0 0
\(235\) 7.37228 0.480915
\(236\) −4.37228 −0.284611
\(237\) 0 0
\(238\) 5.48913 0.355807
\(239\) 3.25544 0.210577 0.105288 0.994442i \(-0.466423\pi\)
0.105288 + 0.994442i \(0.466423\pi\)
\(240\) 0 0
\(241\) −12.4891 −0.804495 −0.402248 0.915531i \(-0.631771\pi\)
−0.402248 + 0.915531i \(0.631771\pi\)
\(242\) 8.11684 0.521770
\(243\) 0 0
\(244\) −8.11684 −0.519628
\(245\) 4.37228 0.279335
\(246\) 0 0
\(247\) 16.0000 1.01806
\(248\) 4.74456 0.301280
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 24.6060 1.55312 0.776558 0.630046i \(-0.216964\pi\)
0.776558 + 0.630046i \(0.216964\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) −8.11684 −0.509296
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −4.37228 −0.272735 −0.136368 0.990658i \(-0.543543\pi\)
−0.136368 + 0.990658i \(0.543543\pi\)
\(258\) 0 0
\(259\) −13.4891 −0.838173
\(260\) −6.74456 −0.418280
\(261\) 0 0
\(262\) 2.74456 0.169560
\(263\) 17.4891 1.07843 0.539213 0.842170i \(-0.318722\pi\)
0.539213 + 0.842170i \(0.318722\pi\)
\(264\) 0 0
\(265\) 11.4891 0.705771
\(266\) −8.00000 −0.490511
\(267\) 0 0
\(268\) −7.00000 −0.427593
\(269\) 1.37228 0.0836695 0.0418347 0.999125i \(-0.486680\pi\)
0.0418347 + 0.999125i \(0.486680\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 1.62772 0.0986949
\(273\) 0 0
\(274\) 19.1168 1.15489
\(275\) 4.37228 0.263658
\(276\) 0 0
\(277\) 16.7446 1.00608 0.503042 0.864262i \(-0.332214\pi\)
0.503042 + 0.864262i \(0.332214\pi\)
\(278\) 0.883156 0.0529682
\(279\) 0 0
\(280\) 3.37228 0.201532
\(281\) 1.37228 0.0818634 0.0409317 0.999162i \(-0.486967\pi\)
0.0409317 + 0.999162i \(0.486967\pi\)
\(282\) 0 0
\(283\) −3.13859 −0.186570 −0.0932850 0.995639i \(-0.529737\pi\)
−0.0932850 + 0.995639i \(0.529737\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) −29.4891 −1.74373
\(287\) 10.1168 0.597178
\(288\) 0 0
\(289\) −14.3505 −0.844149
\(290\) −1.37228 −0.0805831
\(291\) 0 0
\(292\) 3.11684 0.182399
\(293\) −26.2337 −1.53259 −0.766294 0.642490i \(-0.777901\pi\)
−0.766294 + 0.642490i \(0.777901\pi\)
\(294\) 0 0
\(295\) −4.37228 −0.254564
\(296\) −4.00000 −0.232495
\(297\) 0 0
\(298\) −1.88316 −0.109088
\(299\) 9.25544 0.535256
\(300\) 0 0
\(301\) −18.9783 −1.09389
\(302\) −10.0000 −0.575435
\(303\) 0 0
\(304\) −2.37228 −0.136060
\(305\) −8.11684 −0.464769
\(306\) 0 0
\(307\) 1.23369 0.0704103 0.0352051 0.999380i \(-0.488792\pi\)
0.0352051 + 0.999380i \(0.488792\pi\)
\(308\) 14.7446 0.840149
\(309\) 0 0
\(310\) 4.74456 0.269473
\(311\) 20.7446 1.17632 0.588158 0.808746i \(-0.299853\pi\)
0.588158 + 0.808746i \(0.299853\pi\)
\(312\) 0 0
\(313\) 3.62772 0.205051 0.102525 0.994730i \(-0.467308\pi\)
0.102525 + 0.994730i \(0.467308\pi\)
\(314\) −6.74456 −0.380618
\(315\) 0 0
\(316\) 2.00000 0.112509
\(317\) −2.74456 −0.154150 −0.0770750 0.997025i \(-0.524558\pi\)
−0.0770750 + 0.997025i \(0.524558\pi\)
\(318\) 0 0
\(319\) −6.00000 −0.335936
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −4.62772 −0.257893
\(323\) −3.86141 −0.214854
\(324\) 0 0
\(325\) −6.74456 −0.374121
\(326\) −21.4891 −1.19017
\(327\) 0 0
\(328\) 3.00000 0.165647
\(329\) 24.8614 1.37065
\(330\) 0 0
\(331\) 16.2337 0.892284 0.446142 0.894962i \(-0.352798\pi\)
0.446142 + 0.894962i \(0.352798\pi\)
\(332\) −7.37228 −0.404607
\(333\) 0 0
\(334\) −13.3723 −0.731699
\(335\) −7.00000 −0.382451
\(336\) 0 0
\(337\) −2.37228 −0.129226 −0.0646132 0.997910i \(-0.520581\pi\)
−0.0646132 + 0.997910i \(0.520581\pi\)
\(338\) 32.4891 1.76718
\(339\) 0 0
\(340\) 1.62772 0.0882754
\(341\) 20.7446 1.12338
\(342\) 0 0
\(343\) −8.86141 −0.478471
\(344\) −5.62772 −0.303426
\(345\) 0 0
\(346\) −20.7446 −1.11523
\(347\) 4.88316 0.262142 0.131071 0.991373i \(-0.458158\pi\)
0.131071 + 0.991373i \(0.458158\pi\)
\(348\) 0 0
\(349\) 18.1168 0.969772 0.484886 0.874577i \(-0.338861\pi\)
0.484886 + 0.874577i \(0.338861\pi\)
\(350\) 3.37228 0.180256
\(351\) 0 0
\(352\) 4.37228 0.233043
\(353\) 21.3505 1.13637 0.568187 0.822899i \(-0.307645\pi\)
0.568187 + 0.822899i \(0.307645\pi\)
\(354\) 0 0
\(355\) 6.00000 0.318447
\(356\) −16.1168 −0.854191
\(357\) 0 0
\(358\) 14.7446 0.779274
\(359\) 17.4891 0.923041 0.461520 0.887130i \(-0.347304\pi\)
0.461520 + 0.887130i \(0.347304\pi\)
\(360\) 0 0
\(361\) −13.3723 −0.703804
\(362\) 20.8614 1.09645
\(363\) 0 0
\(364\) −22.7446 −1.19214
\(365\) 3.11684 0.163143
\(366\) 0 0
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) −1.37228 −0.0715351
\(369\) 0 0
\(370\) −4.00000 −0.207950
\(371\) 38.7446 2.01152
\(372\) 0 0
\(373\) −15.4891 −0.801997 −0.400998 0.916079i \(-0.631337\pi\)
−0.400998 + 0.916079i \(0.631337\pi\)
\(374\) 7.11684 0.368003
\(375\) 0 0
\(376\) 7.37228 0.380196
\(377\) 9.25544 0.476679
\(378\) 0 0
\(379\) 17.8614 0.917479 0.458739 0.888571i \(-0.348301\pi\)
0.458739 + 0.888571i \(0.348301\pi\)
\(380\) −2.37228 −0.121695
\(381\) 0 0
\(382\) −17.4891 −0.894821
\(383\) 22.9783 1.17413 0.587067 0.809538i \(-0.300283\pi\)
0.587067 + 0.809538i \(0.300283\pi\)
\(384\) 0 0
\(385\) 14.7446 0.751452
\(386\) 21.1168 1.07482
\(387\) 0 0
\(388\) −8.37228 −0.425038
\(389\) 4.62772 0.234635 0.117317 0.993094i \(-0.462571\pi\)
0.117317 + 0.993094i \(0.462571\pi\)
\(390\) 0 0
\(391\) −2.23369 −0.112962
\(392\) 4.37228 0.220834
\(393\) 0 0
\(394\) 5.48913 0.276538
\(395\) 2.00000 0.100631
\(396\) 0 0
\(397\) 22.7446 1.14152 0.570758 0.821118i \(-0.306649\pi\)
0.570758 + 0.821118i \(0.306649\pi\)
\(398\) 13.4891 0.676149
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 1.11684 0.0557725 0.0278863 0.999611i \(-0.491122\pi\)
0.0278863 + 0.999611i \(0.491122\pi\)
\(402\) 0 0
\(403\) −32.0000 −1.59403
\(404\) −2.74456 −0.136547
\(405\) 0 0
\(406\) −4.62772 −0.229670
\(407\) −17.4891 −0.866904
\(408\) 0 0
\(409\) 17.8614 0.883190 0.441595 0.897215i \(-0.354413\pi\)
0.441595 + 0.897215i \(0.354413\pi\)
\(410\) 3.00000 0.148159
\(411\) 0 0
\(412\) −16.0000 −0.788263
\(413\) −14.7446 −0.725532
\(414\) 0 0
\(415\) −7.37228 −0.361891
\(416\) −6.74456 −0.330679
\(417\) 0 0
\(418\) −10.3723 −0.507325
\(419\) −25.7228 −1.25664 −0.628321 0.777954i \(-0.716257\pi\)
−0.628321 + 0.777954i \(0.716257\pi\)
\(420\) 0 0
\(421\) 30.4674 1.48489 0.742445 0.669908i \(-0.233666\pi\)
0.742445 + 0.669908i \(0.233666\pi\)
\(422\) −18.7446 −0.912471
\(423\) 0 0
\(424\) 11.4891 0.557961
\(425\) 1.62772 0.0789560
\(426\) 0 0
\(427\) −27.3723 −1.32464
\(428\) −8.48913 −0.410337
\(429\) 0 0
\(430\) −5.62772 −0.271393
\(431\) −8.23369 −0.396603 −0.198301 0.980141i \(-0.563542\pi\)
−0.198301 + 0.980141i \(0.563542\pi\)
\(432\) 0 0
\(433\) 6.37228 0.306232 0.153116 0.988208i \(-0.451069\pi\)
0.153116 + 0.988208i \(0.451069\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) 15.3723 0.736199
\(437\) 3.25544 0.155729
\(438\) 0 0
\(439\) −18.2337 −0.870246 −0.435123 0.900371i \(-0.643295\pi\)
−0.435123 + 0.900371i \(0.643295\pi\)
\(440\) 4.37228 0.208440
\(441\) 0 0
\(442\) −10.9783 −0.522182
\(443\) 3.51087 0.166807 0.0834033 0.996516i \(-0.473421\pi\)
0.0834033 + 0.996516i \(0.473421\pi\)
\(444\) 0 0
\(445\) −16.1168 −0.764012
\(446\) 6.62772 0.313832
\(447\) 0 0
\(448\) 3.37228 0.159325
\(449\) −9.86141 −0.465389 −0.232694 0.972550i \(-0.574754\pi\)
−0.232694 + 0.972550i \(0.574754\pi\)
\(450\) 0 0
\(451\) 13.1168 0.617648
\(452\) −3.25544 −0.153123
\(453\) 0 0
\(454\) −19.1168 −0.897198
\(455\) −22.7446 −1.06628
\(456\) 0 0
\(457\) 12.8832 0.602649 0.301324 0.953522i \(-0.402571\pi\)
0.301324 + 0.953522i \(0.402571\pi\)
\(458\) 12.6277 0.590055
\(459\) 0 0
\(460\) −1.37228 −0.0639829
\(461\) −1.88316 −0.0877073 −0.0438537 0.999038i \(-0.513964\pi\)
−0.0438537 + 0.999038i \(0.513964\pi\)
\(462\) 0 0
\(463\) 20.0000 0.929479 0.464739 0.885448i \(-0.346148\pi\)
0.464739 + 0.885448i \(0.346148\pi\)
\(464\) −1.37228 −0.0637066
\(465\) 0 0
\(466\) −7.11684 −0.329681
\(467\) −43.1168 −1.99521 −0.997605 0.0691713i \(-0.977964\pi\)
−0.997605 + 0.0691713i \(0.977964\pi\)
\(468\) 0 0
\(469\) −23.6060 −1.09002
\(470\) 7.37228 0.340058
\(471\) 0 0
\(472\) −4.37228 −0.201251
\(473\) −24.6060 −1.13138
\(474\) 0 0
\(475\) −2.37228 −0.108848
\(476\) 5.48913 0.251594
\(477\) 0 0
\(478\) 3.25544 0.148900
\(479\) 0.510875 0.0233425 0.0116712 0.999932i \(-0.496285\pi\)
0.0116712 + 0.999932i \(0.496285\pi\)
\(480\) 0 0
\(481\) 26.9783 1.23010
\(482\) −12.4891 −0.568864
\(483\) 0 0
\(484\) 8.11684 0.368947
\(485\) −8.37228 −0.380166
\(486\) 0 0
\(487\) −12.7446 −0.577511 −0.288756 0.957403i \(-0.593242\pi\)
−0.288756 + 0.957403i \(0.593242\pi\)
\(488\) −8.11684 −0.367432
\(489\) 0 0
\(490\) 4.37228 0.197520
\(491\) 36.6060 1.65200 0.826002 0.563667i \(-0.190610\pi\)
0.826002 + 0.563667i \(0.190610\pi\)
\(492\) 0 0
\(493\) −2.23369 −0.100600
\(494\) 16.0000 0.719874
\(495\) 0 0
\(496\) 4.74456 0.213037
\(497\) 20.2337 0.907605
\(498\) 0 0
\(499\) 15.1168 0.676723 0.338361 0.941016i \(-0.390127\pi\)
0.338361 + 0.941016i \(0.390127\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 24.6060 1.09822
\(503\) 6.86141 0.305935 0.152967 0.988231i \(-0.451117\pi\)
0.152967 + 0.988231i \(0.451117\pi\)
\(504\) 0 0
\(505\) −2.74456 −0.122131
\(506\) −6.00000 −0.266733
\(507\) 0 0
\(508\) −8.11684 −0.360127
\(509\) 42.3505 1.87715 0.938577 0.345069i \(-0.112145\pi\)
0.938577 + 0.345069i \(0.112145\pi\)
\(510\) 0 0
\(511\) 10.5109 0.464974
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −4.37228 −0.192853
\(515\) −16.0000 −0.705044
\(516\) 0 0
\(517\) 32.2337 1.41764
\(518\) −13.4891 −0.592678
\(519\) 0 0
\(520\) −6.74456 −0.295769
\(521\) 6.76631 0.296438 0.148219 0.988955i \(-0.452646\pi\)
0.148219 + 0.988955i \(0.452646\pi\)
\(522\) 0 0
\(523\) 6.11684 0.267471 0.133735 0.991017i \(-0.457303\pi\)
0.133735 + 0.991017i \(0.457303\pi\)
\(524\) 2.74456 0.119897
\(525\) 0 0
\(526\) 17.4891 0.762562
\(527\) 7.72281 0.336411
\(528\) 0 0
\(529\) −21.1168 −0.918124
\(530\) 11.4891 0.499056
\(531\) 0 0
\(532\) −8.00000 −0.346844
\(533\) −20.2337 −0.876418
\(534\) 0 0
\(535\) −8.48913 −0.367017
\(536\) −7.00000 −0.302354
\(537\) 0 0
\(538\) 1.37228 0.0591632
\(539\) 19.1168 0.823421
\(540\) 0 0
\(541\) 27.3723 1.17683 0.588413 0.808560i \(-0.299753\pi\)
0.588413 + 0.808560i \(0.299753\pi\)
\(542\) 8.00000 0.343629
\(543\) 0 0
\(544\) 1.62772 0.0697879
\(545\) 15.3723 0.658476
\(546\) 0 0
\(547\) −29.4674 −1.25993 −0.629967 0.776622i \(-0.716932\pi\)
−0.629967 + 0.776622i \(0.716932\pi\)
\(548\) 19.1168 0.816631
\(549\) 0 0
\(550\) 4.37228 0.186435
\(551\) 3.25544 0.138686
\(552\) 0 0
\(553\) 6.74456 0.286808
\(554\) 16.7446 0.711408
\(555\) 0 0
\(556\) 0.883156 0.0374542
\(557\) 44.2337 1.87424 0.937121 0.349005i \(-0.113480\pi\)
0.937121 + 0.349005i \(0.113480\pi\)
\(558\) 0 0
\(559\) 37.9565 1.60539
\(560\) 3.37228 0.142505
\(561\) 0 0
\(562\) 1.37228 0.0578862
\(563\) 40.7228 1.71626 0.858131 0.513431i \(-0.171626\pi\)
0.858131 + 0.513431i \(0.171626\pi\)
\(564\) 0 0
\(565\) −3.25544 −0.136957
\(566\) −3.13859 −0.131925
\(567\) 0 0
\(568\) 6.00000 0.251754
\(569\) 30.6060 1.28307 0.641534 0.767094i \(-0.278298\pi\)
0.641534 + 0.767094i \(0.278298\pi\)
\(570\) 0 0
\(571\) 8.60597 0.360149 0.180074 0.983653i \(-0.442366\pi\)
0.180074 + 0.983653i \(0.442366\pi\)
\(572\) −29.4891 −1.23300
\(573\) 0 0
\(574\) 10.1168 0.422269
\(575\) −1.37228 −0.0572281
\(576\) 0 0
\(577\) −41.1168 −1.71172 −0.855858 0.517210i \(-0.826970\pi\)
−0.855858 + 0.517210i \(0.826970\pi\)
\(578\) −14.3505 −0.596903
\(579\) 0 0
\(580\) −1.37228 −0.0569809
\(581\) −24.8614 −1.03142
\(582\) 0 0
\(583\) 50.2337 2.08047
\(584\) 3.11684 0.128976
\(585\) 0 0
\(586\) −26.2337 −1.08370
\(587\) −27.0000 −1.11441 −0.557205 0.830375i \(-0.688126\pi\)
−0.557205 + 0.830375i \(0.688126\pi\)
\(588\) 0 0
\(589\) −11.2554 −0.463772
\(590\) −4.37228 −0.180004
\(591\) 0 0
\(592\) −4.00000 −0.164399
\(593\) 19.7228 0.809919 0.404959 0.914335i \(-0.367286\pi\)
0.404959 + 0.914335i \(0.367286\pi\)
\(594\) 0 0
\(595\) 5.48913 0.225032
\(596\) −1.88316 −0.0771371
\(597\) 0 0
\(598\) 9.25544 0.378483
\(599\) 3.76631 0.153887 0.0769437 0.997035i \(-0.475484\pi\)
0.0769437 + 0.997035i \(0.475484\pi\)
\(600\) 0 0
\(601\) −1.86141 −0.0759284 −0.0379642 0.999279i \(-0.512087\pi\)
−0.0379642 + 0.999279i \(0.512087\pi\)
\(602\) −18.9783 −0.773496
\(603\) 0 0
\(604\) −10.0000 −0.406894
\(605\) 8.11684 0.329997
\(606\) 0 0
\(607\) 18.1168 0.735340 0.367670 0.929956i \(-0.380156\pi\)
0.367670 + 0.929956i \(0.380156\pi\)
\(608\) −2.37228 −0.0962087
\(609\) 0 0
\(610\) −8.11684 −0.328641
\(611\) −49.7228 −2.01157
\(612\) 0 0
\(613\) 34.2337 1.38269 0.691343 0.722527i \(-0.257019\pi\)
0.691343 + 0.722527i \(0.257019\pi\)
\(614\) 1.23369 0.0497876
\(615\) 0 0
\(616\) 14.7446 0.594075
\(617\) 4.88316 0.196588 0.0982942 0.995157i \(-0.468661\pi\)
0.0982942 + 0.995157i \(0.468661\pi\)
\(618\) 0 0
\(619\) −20.8832 −0.839365 −0.419682 0.907671i \(-0.637859\pi\)
−0.419682 + 0.907671i \(0.637859\pi\)
\(620\) 4.74456 0.190546
\(621\) 0 0
\(622\) 20.7446 0.831781
\(623\) −54.3505 −2.17751
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 3.62772 0.144993
\(627\) 0 0
\(628\) −6.74456 −0.269137
\(629\) −6.51087 −0.259606
\(630\) 0 0
\(631\) −23.7228 −0.944390 −0.472195 0.881494i \(-0.656538\pi\)
−0.472195 + 0.881494i \(0.656538\pi\)
\(632\) 2.00000 0.0795557
\(633\) 0 0
\(634\) −2.74456 −0.109001
\(635\) −8.11684 −0.322107
\(636\) 0 0
\(637\) −29.4891 −1.16840
\(638\) −6.00000 −0.237542
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 39.0000 1.54041 0.770204 0.637798i \(-0.220155\pi\)
0.770204 + 0.637798i \(0.220155\pi\)
\(642\) 0 0
\(643\) 11.0000 0.433798 0.216899 0.976194i \(-0.430406\pi\)
0.216899 + 0.976194i \(0.430406\pi\)
\(644\) −4.62772 −0.182358
\(645\) 0 0
\(646\) −3.86141 −0.151925
\(647\) 39.0951 1.53699 0.768493 0.639858i \(-0.221007\pi\)
0.768493 + 0.639858i \(0.221007\pi\)
\(648\) 0 0
\(649\) −19.1168 −0.750402
\(650\) −6.74456 −0.264544
\(651\) 0 0
\(652\) −21.4891 −0.841579
\(653\) −19.7228 −0.771813 −0.385907 0.922538i \(-0.626111\pi\)
−0.385907 + 0.922538i \(0.626111\pi\)
\(654\) 0 0
\(655\) 2.74456 0.107239
\(656\) 3.00000 0.117130
\(657\) 0 0
\(658\) 24.8614 0.969199
\(659\) −17.4891 −0.681280 −0.340640 0.940194i \(-0.610644\pi\)
−0.340640 + 0.940194i \(0.610644\pi\)
\(660\) 0 0
\(661\) −12.2337 −0.475835 −0.237918 0.971285i \(-0.576465\pi\)
−0.237918 + 0.971285i \(0.576465\pi\)
\(662\) 16.2337 0.630940
\(663\) 0 0
\(664\) −7.37228 −0.286100
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) 1.88316 0.0729161
\(668\) −13.3723 −0.517389
\(669\) 0 0
\(670\) −7.00000 −0.270434
\(671\) −35.4891 −1.37004
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) −2.37228 −0.0913769
\(675\) 0 0
\(676\) 32.4891 1.24958
\(677\) −13.7228 −0.527411 −0.263705 0.964603i \(-0.584945\pi\)
−0.263705 + 0.964603i \(0.584945\pi\)
\(678\) 0 0
\(679\) −28.2337 −1.08351
\(680\) 1.62772 0.0624202
\(681\) 0 0
\(682\) 20.7446 0.794350
\(683\) 30.0951 1.15156 0.575778 0.817606i \(-0.304699\pi\)
0.575778 + 0.817606i \(0.304699\pi\)
\(684\) 0 0
\(685\) 19.1168 0.730417
\(686\) −8.86141 −0.338330
\(687\) 0 0
\(688\) −5.62772 −0.214555
\(689\) −77.4891 −2.95210
\(690\) 0 0
\(691\) −36.2337 −1.37839 −0.689197 0.724574i \(-0.742037\pi\)
−0.689197 + 0.724574i \(0.742037\pi\)
\(692\) −20.7446 −0.788590
\(693\) 0 0
\(694\) 4.88316 0.185362
\(695\) 0.883156 0.0335000
\(696\) 0 0
\(697\) 4.88316 0.184963
\(698\) 18.1168 0.685733
\(699\) 0 0
\(700\) 3.37228 0.127460
\(701\) −42.8614 −1.61885 −0.809426 0.587221i \(-0.800222\pi\)
−0.809426 + 0.587221i \(0.800222\pi\)
\(702\) 0 0
\(703\) 9.48913 0.357889
\(704\) 4.37228 0.164787
\(705\) 0 0
\(706\) 21.3505 0.803538
\(707\) −9.25544 −0.348087
\(708\) 0 0
\(709\) 2.86141 0.107462 0.0537312 0.998555i \(-0.482889\pi\)
0.0537312 + 0.998555i \(0.482889\pi\)
\(710\) 6.00000 0.225176
\(711\) 0 0
\(712\) −16.1168 −0.604004
\(713\) −6.51087 −0.243834
\(714\) 0 0
\(715\) −29.4891 −1.10283
\(716\) 14.7446 0.551030
\(717\) 0 0
\(718\) 17.4891 0.652688
\(719\) 3.76631 0.140460 0.0702299 0.997531i \(-0.477627\pi\)
0.0702299 + 0.997531i \(0.477627\pi\)
\(720\) 0 0
\(721\) −53.9565 −2.00945
\(722\) −13.3723 −0.497665
\(723\) 0 0
\(724\) 20.8614 0.775308
\(725\) −1.37228 −0.0509652
\(726\) 0 0
\(727\) 18.1168 0.671917 0.335958 0.941877i \(-0.390940\pi\)
0.335958 + 0.941877i \(0.390940\pi\)
\(728\) −22.7446 −0.842970
\(729\) 0 0
\(730\) 3.11684 0.115360
\(731\) −9.16034 −0.338808
\(732\) 0 0
\(733\) −0.233688 −0.00863146 −0.00431573 0.999991i \(-0.501374\pi\)
−0.00431573 + 0.999991i \(0.501374\pi\)
\(734\) −16.0000 −0.590571
\(735\) 0 0
\(736\) −1.37228 −0.0505830
\(737\) −30.6060 −1.12739
\(738\) 0 0
\(739\) −41.1168 −1.51251 −0.756254 0.654278i \(-0.772972\pi\)
−0.756254 + 0.654278i \(0.772972\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) 38.7446 1.42236
\(743\) −7.88316 −0.289205 −0.144602 0.989490i \(-0.546190\pi\)
−0.144602 + 0.989490i \(0.546190\pi\)
\(744\) 0 0
\(745\) −1.88316 −0.0689935
\(746\) −15.4891 −0.567097
\(747\) 0 0
\(748\) 7.11684 0.260218
\(749\) −28.6277 −1.04603
\(750\) 0 0
\(751\) 16.2337 0.592376 0.296188 0.955130i \(-0.404285\pi\)
0.296188 + 0.955130i \(0.404285\pi\)
\(752\) 7.37228 0.268839
\(753\) 0 0
\(754\) 9.25544 0.337063
\(755\) −10.0000 −0.363937
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 17.8614 0.648756
\(759\) 0 0
\(760\) −2.37228 −0.0860517
\(761\) 51.0951 1.85220 0.926098 0.377283i \(-0.123142\pi\)
0.926098 + 0.377283i \(0.123142\pi\)
\(762\) 0 0
\(763\) 51.8397 1.87672
\(764\) −17.4891 −0.632734
\(765\) 0 0
\(766\) 22.9783 0.830238
\(767\) 29.4891 1.06479
\(768\) 0 0
\(769\) −46.8614 −1.68987 −0.844933 0.534873i \(-0.820360\pi\)
−0.844933 + 0.534873i \(0.820360\pi\)
\(770\) 14.7446 0.531357
\(771\) 0 0
\(772\) 21.1168 0.760012
\(773\) −3.25544 −0.117090 −0.0585450 0.998285i \(-0.518646\pi\)
−0.0585450 + 0.998285i \(0.518646\pi\)
\(774\) 0 0
\(775\) 4.74456 0.170430
\(776\) −8.37228 −0.300547
\(777\) 0 0
\(778\) 4.62772 0.165912
\(779\) −7.11684 −0.254987
\(780\) 0 0
\(781\) 26.2337 0.938715
\(782\) −2.23369 −0.0798765
\(783\) 0 0
\(784\) 4.37228 0.156153
\(785\) −6.74456 −0.240724
\(786\) 0 0
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) 5.48913 0.195542
\(789\) 0 0
\(790\) 2.00000 0.0711568
\(791\) −10.9783 −0.390342
\(792\) 0 0
\(793\) 54.7446 1.94404
\(794\) 22.7446 0.807174
\(795\) 0 0
\(796\) 13.4891 0.478109
\(797\) −14.7446 −0.522279 −0.261140 0.965301i \(-0.584098\pi\)
−0.261140 + 0.965301i \(0.584098\pi\)
\(798\) 0 0
\(799\) 12.0000 0.424529
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 1.11684 0.0394371
\(803\) 13.6277 0.480912
\(804\) 0 0
\(805\) −4.62772 −0.163106
\(806\) −32.0000 −1.12715
\(807\) 0 0
\(808\) −2.74456 −0.0965534
\(809\) −39.3505 −1.38349 −0.691746 0.722141i \(-0.743158\pi\)
−0.691746 + 0.722141i \(0.743158\pi\)
\(810\) 0 0
\(811\) 3.62772 0.127386 0.0636932 0.997970i \(-0.479712\pi\)
0.0636932 + 0.997970i \(0.479712\pi\)
\(812\) −4.62772 −0.162401
\(813\) 0 0
\(814\) −17.4891 −0.612994
\(815\) −21.4891 −0.752731
\(816\) 0 0
\(817\) 13.3505 0.467076
\(818\) 17.8614 0.624509
\(819\) 0 0
\(820\) 3.00000 0.104765
\(821\) 23.8397 0.832010 0.416005 0.909362i \(-0.363430\pi\)
0.416005 + 0.909362i \(0.363430\pi\)
\(822\) 0 0
\(823\) −25.0951 −0.874760 −0.437380 0.899277i \(-0.644094\pi\)
−0.437380 + 0.899277i \(0.644094\pi\)
\(824\) −16.0000 −0.557386
\(825\) 0 0
\(826\) −14.7446 −0.513029
\(827\) −36.8614 −1.28180 −0.640898 0.767626i \(-0.721438\pi\)
−0.640898 + 0.767626i \(0.721438\pi\)
\(828\) 0 0
\(829\) −50.1168 −1.74063 −0.870315 0.492496i \(-0.836085\pi\)
−0.870315 + 0.492496i \(0.836085\pi\)
\(830\) −7.37228 −0.255896
\(831\) 0 0
\(832\) −6.74456 −0.233826
\(833\) 7.11684 0.246584
\(834\) 0 0
\(835\) −13.3723 −0.462767
\(836\) −10.3723 −0.358733
\(837\) 0 0
\(838\) −25.7228 −0.888580
\(839\) −9.76631 −0.337171 −0.168585 0.985687i \(-0.553920\pi\)
−0.168585 + 0.985687i \(0.553920\pi\)
\(840\) 0 0
\(841\) −27.1168 −0.935064
\(842\) 30.4674 1.04998
\(843\) 0 0
\(844\) −18.7446 −0.645214
\(845\) 32.4891 1.11766
\(846\) 0 0
\(847\) 27.3723 0.940523
\(848\) 11.4891 0.394538
\(849\) 0 0
\(850\) 1.62772 0.0558303
\(851\) 5.48913 0.188165
\(852\) 0 0
\(853\) −36.2337 −1.24062 −0.620309 0.784358i \(-0.712993\pi\)
−0.620309 + 0.784358i \(0.712993\pi\)
\(854\) −27.3723 −0.936660
\(855\) 0 0
\(856\) −8.48913 −0.290152
\(857\) −0.510875 −0.0174511 −0.00872557 0.999962i \(-0.502777\pi\)
−0.00872557 + 0.999962i \(0.502777\pi\)
\(858\) 0 0
\(859\) −17.1168 −0.584019 −0.292010 0.956415i \(-0.594324\pi\)
−0.292010 + 0.956415i \(0.594324\pi\)
\(860\) −5.62772 −0.191904
\(861\) 0 0
\(862\) −8.23369 −0.280441
\(863\) 2.39403 0.0814938 0.0407469 0.999170i \(-0.487026\pi\)
0.0407469 + 0.999170i \(0.487026\pi\)
\(864\) 0 0
\(865\) −20.7446 −0.705336
\(866\) 6.37228 0.216539
\(867\) 0 0
\(868\) 16.0000 0.543075
\(869\) 8.74456 0.296639
\(870\) 0 0
\(871\) 47.2119 1.59972
\(872\) 15.3723 0.520571
\(873\) 0 0
\(874\) 3.25544 0.110117
\(875\) 3.37228 0.114004
\(876\) 0 0
\(877\) 35.9565 1.21416 0.607082 0.794639i \(-0.292340\pi\)
0.607082 + 0.794639i \(0.292340\pi\)
\(878\) −18.2337 −0.615357
\(879\) 0 0
\(880\) 4.37228 0.147390
\(881\) 24.3505 0.820390 0.410195 0.911998i \(-0.365461\pi\)
0.410195 + 0.911998i \(0.365461\pi\)
\(882\) 0 0
\(883\) −44.7228 −1.50504 −0.752521 0.658568i \(-0.771163\pi\)
−0.752521 + 0.658568i \(0.771163\pi\)
\(884\) −10.9783 −0.369239
\(885\) 0 0
\(886\) 3.51087 0.117950
\(887\) −19.7228 −0.662227 −0.331114 0.943591i \(-0.607424\pi\)
−0.331114 + 0.943591i \(0.607424\pi\)
\(888\) 0 0
\(889\) −27.3723 −0.918037
\(890\) −16.1168 −0.540238
\(891\) 0 0
\(892\) 6.62772 0.221912
\(893\) −17.4891 −0.585251
\(894\) 0 0
\(895\) 14.7446 0.492856
\(896\) 3.37228 0.112660
\(897\) 0 0
\(898\) −9.86141 −0.329079
\(899\) −6.51087 −0.217150
\(900\) 0 0
\(901\) 18.7011 0.623023
\(902\) 13.1168 0.436743
\(903\) 0 0
\(904\) −3.25544 −0.108274
\(905\) 20.8614 0.693457
\(906\) 0 0
\(907\) −7.00000 −0.232431 −0.116216 0.993224i \(-0.537076\pi\)
−0.116216 + 0.993224i \(0.537076\pi\)
\(908\) −19.1168 −0.634415
\(909\) 0 0
\(910\) −22.7446 −0.753975
\(911\) 42.0000 1.39152 0.695761 0.718273i \(-0.255067\pi\)
0.695761 + 0.718273i \(0.255067\pi\)
\(912\) 0 0
\(913\) −32.2337 −1.06678
\(914\) 12.8832 0.426137
\(915\) 0 0
\(916\) 12.6277 0.417232
\(917\) 9.25544 0.305641
\(918\) 0 0
\(919\) −26.4674 −0.873078 −0.436539 0.899685i \(-0.643796\pi\)
−0.436539 + 0.899685i \(0.643796\pi\)
\(920\) −1.37228 −0.0452428
\(921\) 0 0
\(922\) −1.88316 −0.0620184
\(923\) −40.4674 −1.33200
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 20.0000 0.657241
\(927\) 0 0
\(928\) −1.37228 −0.0450473
\(929\) −51.9565 −1.70464 −0.852319 0.523023i \(-0.824804\pi\)
−0.852319 + 0.523023i \(0.824804\pi\)
\(930\) 0 0
\(931\) −10.3723 −0.339938
\(932\) −7.11684 −0.233120
\(933\) 0 0
\(934\) −43.1168 −1.41083
\(935\) 7.11684 0.232746
\(936\) 0 0
\(937\) 39.7228 1.29769 0.648844 0.760922i \(-0.275253\pi\)
0.648844 + 0.760922i \(0.275253\pi\)
\(938\) −23.6060 −0.770762
\(939\) 0 0
\(940\) 7.37228 0.240457
\(941\) −43.3723 −1.41390 −0.706948 0.707266i \(-0.749929\pi\)
−0.706948 + 0.707266i \(0.749929\pi\)
\(942\) 0 0
\(943\) −4.11684 −0.134063
\(944\) −4.37228 −0.142306
\(945\) 0 0
\(946\) −24.6060 −0.800009
\(947\) 34.7228 1.12834 0.564170 0.825659i \(-0.309196\pi\)
0.564170 + 0.825659i \(0.309196\pi\)
\(948\) 0 0
\(949\) −21.0217 −0.682395
\(950\) −2.37228 −0.0769670
\(951\) 0 0
\(952\) 5.48913 0.177904
\(953\) −2.13859 −0.0692758 −0.0346379 0.999400i \(-0.511028\pi\)
−0.0346379 + 0.999400i \(0.511028\pi\)
\(954\) 0 0
\(955\) −17.4891 −0.565935
\(956\) 3.25544 0.105288
\(957\) 0 0
\(958\) 0.510875 0.0165056
\(959\) 64.4674 2.08176
\(960\) 0 0
\(961\) −8.48913 −0.273843
\(962\) 26.9783 0.869814
\(963\) 0 0
\(964\) −12.4891 −0.402248
\(965\) 21.1168 0.679775
\(966\) 0 0
\(967\) 36.1168 1.16144 0.580720 0.814104i \(-0.302771\pi\)
0.580720 + 0.814104i \(0.302771\pi\)
\(968\) 8.11684 0.260885
\(969\) 0 0
\(970\) −8.37228 −0.268818
\(971\) 22.9783 0.737407 0.368704 0.929547i \(-0.379802\pi\)
0.368704 + 0.929547i \(0.379802\pi\)
\(972\) 0 0
\(973\) 2.97825 0.0954783
\(974\) −12.7446 −0.408362
\(975\) 0 0
\(976\) −8.11684 −0.259814
\(977\) 22.8832 0.732097 0.366049 0.930596i \(-0.380710\pi\)
0.366049 + 0.930596i \(0.380710\pi\)
\(978\) 0 0
\(979\) −70.4674 −2.25215
\(980\) 4.37228 0.139667
\(981\) 0 0
\(982\) 36.6060 1.16814
\(983\) 12.8614 0.410215 0.205108 0.978739i \(-0.434246\pi\)
0.205108 + 0.978739i \(0.434246\pi\)
\(984\) 0 0
\(985\) 5.48913 0.174898
\(986\) −2.23369 −0.0711351
\(987\) 0 0
\(988\) 16.0000 0.509028
\(989\) 7.72281 0.245571
\(990\) 0 0
\(991\) 16.2337 0.515680 0.257840 0.966188i \(-0.416989\pi\)
0.257840 + 0.966188i \(0.416989\pi\)
\(992\) 4.74456 0.150640
\(993\) 0 0
\(994\) 20.2337 0.641774
\(995\) 13.4891 0.427634
\(996\) 0 0
\(997\) 14.0000 0.443384 0.221692 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) 15.1168 0.478515
\(999\) 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 810.2.a.k.1.2 2
3.2 odd 2 810.2.a.i.1.2 2
4.3 odd 2 6480.2.a.bn.1.1 2
5.2 odd 4 4050.2.c.ba.649.4 4
5.3 odd 4 4050.2.c.ba.649.1 4
5.4 even 2 4050.2.a.bo.1.1 2
9.2 odd 6 90.2.e.c.31.2 4
9.4 even 3 270.2.e.c.181.1 4
9.5 odd 6 90.2.e.c.61.1 yes 4
9.7 even 3 270.2.e.c.91.1 4
12.11 even 2 6480.2.a.be.1.1 2
15.2 even 4 4050.2.c.v.649.2 4
15.8 even 4 4050.2.c.v.649.3 4
15.14 odd 2 4050.2.a.bw.1.1 2
36.7 odd 6 2160.2.q.f.1441.2 4
36.11 even 6 720.2.q.f.481.1 4
36.23 even 6 720.2.q.f.241.2 4
36.31 odd 6 2160.2.q.f.721.2 4
45.2 even 12 450.2.j.g.49.2 8
45.4 even 6 1350.2.e.l.451.2 4
45.7 odd 12 1350.2.j.f.199.4 8
45.13 odd 12 1350.2.j.f.1099.4 8
45.14 odd 6 450.2.e.j.151.2 4
45.22 odd 12 1350.2.j.f.1099.1 8
45.23 even 12 450.2.j.g.349.2 8
45.29 odd 6 450.2.e.j.301.1 4
45.32 even 12 450.2.j.g.349.3 8
45.34 even 6 1350.2.e.l.901.2 4
45.38 even 12 450.2.j.g.49.3 8
45.43 odd 12 1350.2.j.f.199.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.2.e.c.31.2 4 9.2 odd 6
90.2.e.c.61.1 yes 4 9.5 odd 6
270.2.e.c.91.1 4 9.7 even 3
270.2.e.c.181.1 4 9.4 even 3
450.2.e.j.151.2 4 45.14 odd 6
450.2.e.j.301.1 4 45.29 odd 6
450.2.j.g.49.2 8 45.2 even 12
450.2.j.g.49.3 8 45.38 even 12
450.2.j.g.349.2 8 45.23 even 12
450.2.j.g.349.3 8 45.32 even 12
720.2.q.f.241.2 4 36.23 even 6
720.2.q.f.481.1 4 36.11 even 6
810.2.a.i.1.2 2 3.2 odd 2
810.2.a.k.1.2 2 1.1 even 1 trivial
1350.2.e.l.451.2 4 45.4 even 6
1350.2.e.l.901.2 4 45.34 even 6
1350.2.j.f.199.1 8 45.43 odd 12
1350.2.j.f.199.4 8 45.7 odd 12
1350.2.j.f.1099.1 8 45.22 odd 12
1350.2.j.f.1099.4 8 45.13 odd 12
2160.2.q.f.721.2 4 36.31 odd 6
2160.2.q.f.1441.2 4 36.7 odd 6
4050.2.a.bo.1.1 2 5.4 even 2
4050.2.a.bw.1.1 2 15.14 odd 2
4050.2.c.v.649.2 4 15.2 even 4
4050.2.c.v.649.3 4 15.8 even 4
4050.2.c.ba.649.1 4 5.3 odd 4
4050.2.c.ba.649.4 4 5.2 odd 4
6480.2.a.be.1.1 2 12.11 even 2
6480.2.a.bn.1.1 2 4.3 odd 2