Properties

Label 512.2.a.e.1.1
Level $512$
Weight $2$
Character 512.1
Self dual yes
Analytic conductor $4.088$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [512,2,Mod(1,512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(512, 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("512.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 512 = 2^{9} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 512.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: \(4.08834058349\)
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: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 512.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421 q^{3} +2.00000 q^{5} -2.82843 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.41421 q^{3} +2.00000 q^{5} -2.82843 q^{7} -1.00000 q^{9} +4.24264 q^{11} +6.00000 q^{13} -2.82843 q^{15} -4.24264 q^{19} +4.00000 q^{21} +8.48528 q^{23} -1.00000 q^{25} +5.65685 q^{27} +2.00000 q^{29} +5.65685 q^{31} -6.00000 q^{33} -5.65685 q^{35} +6.00000 q^{37} -8.48528 q^{39} +6.00000 q^{41} -4.24264 q^{43} -2.00000 q^{45} +1.00000 q^{49} -2.00000 q^{53} +8.48528 q^{55} +6.00000 q^{57} +1.41421 q^{59} +6.00000 q^{61} +2.82843 q^{63} +12.0000 q^{65} +12.7279 q^{67} -12.0000 q^{69} -8.48528 q^{71} -12.0000 q^{73} +1.41421 q^{75} -12.0000 q^{77} -5.65685 q^{79} -5.00000 q^{81} +4.24264 q^{83} -2.82843 q^{87} -12.0000 q^{89} -16.9706 q^{91} -8.00000 q^{93} -8.48528 q^{95} -8.00000 q^{97} -4.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{5} - 2 q^{9} + 12 q^{13} + 8 q^{21} - 2 q^{25} + 4 q^{29} - 12 q^{33} + 12 q^{37} + 12 q^{41} - 4 q^{45} + 2 q^{49} - 4 q^{53} + 12 q^{57} + 12 q^{61} + 24 q^{65} - 24 q^{69} - 24 q^{73} - 24 q^{77} - 10 q^{81} - 24 q^{89} - 16 q^{93} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.41421 −0.816497 −0.408248 0.912871i \(-0.633860\pi\)
−0.408248 + 0.912871i \(0.633860\pi\)
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) −2.82843 −1.06904 −0.534522 0.845154i \(-0.679509\pi\)
−0.534522 + 0.845154i \(0.679509\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.24264 1.27920 0.639602 0.768706i \(-0.279099\pi\)
0.639602 + 0.768706i \(0.279099\pi\)
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) −2.82843 −0.730297
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −4.24264 −0.973329 −0.486664 0.873589i \(-0.661786\pi\)
−0.486664 + 0.873589i \(0.661786\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) 8.48528 1.76930 0.884652 0.466252i \(-0.154396\pi\)
0.884652 + 0.466252i \(0.154396\pi\)
\(24\) 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 \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 5.65685 1.01600 0.508001 0.861357i \(-0.330385\pi\)
0.508001 + 0.861357i \(0.330385\pi\)
\(32\) 0 0
\(33\) −6.00000 −1.04447
\(34\) 0 0
\(35\) −5.65685 −0.956183
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) −8.48528 −1.35873
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −4.24264 −0.646997 −0.323498 0.946229i \(-0.604859\pi\)
−0.323498 + 0.946229i \(0.604859\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 8.48528 1.14416
\(56\) 0 0
\(57\) 6.00000 0.794719
\(58\) 0 0
\(59\) 1.41421 0.184115 0.0920575 0.995754i \(-0.470656\pi\)
0.0920575 + 0.995754i \(0.470656\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 2.82843 0.356348
\(64\) 0 0
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) 12.7279 1.55496 0.777482 0.628906i \(-0.216497\pi\)
0.777482 + 0.628906i \(0.216497\pi\)
\(68\) 0 0
\(69\) −12.0000 −1.44463
\(70\) 0 0
\(71\) −8.48528 −1.00702 −0.503509 0.863990i \(-0.667958\pi\)
−0.503509 + 0.863990i \(0.667958\pi\)
\(72\) 0 0
\(73\) −12.0000 −1.40449 −0.702247 0.711934i \(-0.747820\pi\)
−0.702247 + 0.711934i \(0.747820\pi\)
\(74\) 0 0
\(75\) 1.41421 0.163299
\(76\) 0 0
\(77\) −12.0000 −1.36753
\(78\) 0 0
\(79\) −5.65685 −0.636446 −0.318223 0.948016i \(-0.603086\pi\)
−0.318223 + 0.948016i \(0.603086\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) 4.24264 0.465690 0.232845 0.972514i \(-0.425196\pi\)
0.232845 + 0.972514i \(0.425196\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −2.82843 −0.303239
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) −16.9706 −1.77900
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) −8.48528 −0.870572
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 0 0
\(99\) −4.24264 −0.426401
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) 2.82843 0.278693 0.139347 0.990244i \(-0.455500\pi\)
0.139347 + 0.990244i \(0.455500\pi\)
\(104\) 0 0
\(105\) 8.00000 0.780720
\(106\) 0 0
\(107\) −7.07107 −0.683586 −0.341793 0.939775i \(-0.611034\pi\)
−0.341793 + 0.939775i \(0.611034\pi\)
\(108\) 0 0
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) −8.48528 −0.805387
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 16.9706 1.58251
\(116\) 0 0
\(117\) −6.00000 −0.554700
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) −8.48528 −0.765092
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 5.65685 0.501965 0.250982 0.967992i \(-0.419246\pi\)
0.250982 + 0.967992i \(0.419246\pi\)
\(128\) 0 0
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) 1.41421 0.123560 0.0617802 0.998090i \(-0.480322\pi\)
0.0617802 + 0.998090i \(0.480322\pi\)
\(132\) 0 0
\(133\) 12.0000 1.04053
\(134\) 0 0
\(135\) 11.3137 0.973729
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −4.24264 −0.359856 −0.179928 0.983680i \(-0.557586\pi\)
−0.179928 + 0.983680i \(0.557586\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 25.4558 2.12872
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) 0 0
\(147\) −1.41421 −0.116642
\(148\) 0 0
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 0 0
\(151\) −14.1421 −1.15087 −0.575435 0.817847i \(-0.695167\pi\)
−0.575435 + 0.817847i \(0.695167\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 11.3137 0.908739
\(156\) 0 0
\(157\) 6.00000 0.478852 0.239426 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) 0 0
\(159\) 2.82843 0.224309
\(160\) 0 0
\(161\) −24.0000 −1.89146
\(162\) 0 0
\(163\) 21.2132 1.66155 0.830773 0.556611i \(-0.187899\pi\)
0.830773 + 0.556611i \(0.187899\pi\)
\(164\) 0 0
\(165\) −12.0000 −0.934199
\(166\) 0 0
\(167\) 8.48528 0.656611 0.328305 0.944572i \(-0.393522\pi\)
0.328305 + 0.944572i \(0.393522\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 4.24264 0.324443
\(172\) 0 0
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) 2.82843 0.213809
\(176\) 0 0
\(177\) −2.00000 −0.150329
\(178\) 0 0
\(179\) −18.3848 −1.37414 −0.687071 0.726590i \(-0.741104\pi\)
−0.687071 + 0.726590i \(0.741104\pi\)
\(180\) 0 0
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 0 0
\(183\) −8.48528 −0.627250
\(184\) 0 0
\(185\) 12.0000 0.882258
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −16.0000 −1.16383
\(190\) 0 0
\(191\) −16.9706 −1.22795 −0.613973 0.789327i \(-0.710430\pi\)
−0.613973 + 0.789327i \(0.710430\pi\)
\(192\) 0 0
\(193\) 24.0000 1.72756 0.863779 0.503871i \(-0.168091\pi\)
0.863779 + 0.503871i \(0.168091\pi\)
\(194\) 0 0
\(195\) −16.9706 −1.21529
\(196\) 0 0
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) 0 0
\(199\) 19.7990 1.40351 0.701757 0.712417i \(-0.252399\pi\)
0.701757 + 0.712417i \(0.252399\pi\)
\(200\) 0 0
\(201\) −18.0000 −1.26962
\(202\) 0 0
\(203\) −5.65685 −0.397033
\(204\) 0 0
\(205\) 12.0000 0.838116
\(206\) 0 0
\(207\) −8.48528 −0.589768
\(208\) 0 0
\(209\) −18.0000 −1.24509
\(210\) 0 0
\(211\) 4.24264 0.292075 0.146038 0.989279i \(-0.453348\pi\)
0.146038 + 0.989279i \(0.453348\pi\)
\(212\) 0 0
\(213\) 12.0000 0.822226
\(214\) 0 0
\(215\) −8.48528 −0.578691
\(216\) 0 0
\(217\) −16.0000 −1.08615
\(218\) 0 0
\(219\) 16.9706 1.14676
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −5.65685 −0.378811 −0.189405 0.981899i \(-0.560656\pi\)
−0.189405 + 0.981899i \(0.560656\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −12.7279 −0.844782 −0.422391 0.906414i \(-0.638809\pi\)
−0.422391 + 0.906414i \(0.638809\pi\)
\(228\) 0 0
\(229\) 18.0000 1.18947 0.594737 0.803921i \(-0.297256\pi\)
0.594737 + 0.803921i \(0.297256\pi\)
\(230\) 0 0
\(231\) 16.9706 1.11658
\(232\) 0 0
\(233\) −12.0000 −0.786146 −0.393073 0.919507i \(-0.628588\pi\)
−0.393073 + 0.919507i \(0.628588\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) 16.9706 1.09773 0.548867 0.835910i \(-0.315059\pi\)
0.548867 + 0.835910i \(0.315059\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) −9.89949 −0.635053
\(244\) 0 0
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) −25.4558 −1.61972
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 12.7279 0.803379 0.401690 0.915776i \(-0.368423\pi\)
0.401690 + 0.915776i \(0.368423\pi\)
\(252\) 0 0
\(253\) 36.0000 2.26330
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) −16.9706 −1.05450
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) −8.48528 −0.523225 −0.261612 0.965173i \(-0.584254\pi\)
−0.261612 + 0.965173i \(0.584254\pi\)
\(264\) 0 0
\(265\) −4.00000 −0.245718
\(266\) 0 0
\(267\) 16.9706 1.03858
\(268\) 0 0
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) −22.6274 −1.37452 −0.687259 0.726413i \(-0.741186\pi\)
−0.687259 + 0.726413i \(0.741186\pi\)
\(272\) 0 0
\(273\) 24.0000 1.45255
\(274\) 0 0
\(275\) −4.24264 −0.255841
\(276\) 0 0
\(277\) 6.00000 0.360505 0.180253 0.983620i \(-0.442309\pi\)
0.180253 + 0.983620i \(0.442309\pi\)
\(278\) 0 0
\(279\) −5.65685 −0.338667
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 0 0
\(283\) −12.7279 −0.756596 −0.378298 0.925684i \(-0.623491\pi\)
−0.378298 + 0.925684i \(0.623491\pi\)
\(284\) 0 0
\(285\) 12.0000 0.710819
\(286\) 0 0
\(287\) −16.9706 −1.00174
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 11.3137 0.663221
\(292\) 0 0
\(293\) 10.0000 0.584206 0.292103 0.956387i \(-0.405645\pi\)
0.292103 + 0.956387i \(0.405645\pi\)
\(294\) 0 0
\(295\) 2.82843 0.164677
\(296\) 0 0
\(297\) 24.0000 1.39262
\(298\) 0 0
\(299\) 50.9117 2.94430
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 0 0
\(303\) 14.1421 0.812444
\(304\) 0 0
\(305\) 12.0000 0.687118
\(306\) 0 0
\(307\) −4.24264 −0.242140 −0.121070 0.992644i \(-0.538633\pi\)
−0.121070 + 0.992644i \(0.538633\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −8.48528 −0.481156 −0.240578 0.970630i \(-0.577337\pi\)
−0.240578 + 0.970630i \(0.577337\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 0 0
\(315\) 5.65685 0.318728
\(316\) 0 0
\(317\) 10.0000 0.561656 0.280828 0.959758i \(-0.409391\pi\)
0.280828 + 0.959758i \(0.409391\pi\)
\(318\) 0 0
\(319\) 8.48528 0.475085
\(320\) 0 0
\(321\) 10.0000 0.558146
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −6.00000 −0.332820
\(326\) 0 0
\(327\) 8.48528 0.469237
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −29.6985 −1.63238 −0.816188 0.577786i \(-0.803917\pi\)
−0.816188 + 0.577786i \(0.803917\pi\)
\(332\) 0 0
\(333\) −6.00000 −0.328798
\(334\) 0 0
\(335\) 25.4558 1.39080
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 0 0
\(339\) −8.48528 −0.460857
\(340\) 0 0
\(341\) 24.0000 1.29967
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) −24.0000 −1.29212
\(346\) 0 0
\(347\) −21.2132 −1.13878 −0.569392 0.822066i \(-0.692821\pi\)
−0.569392 + 0.822066i \(0.692821\pi\)
\(348\) 0 0
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 0 0
\(351\) 33.9411 1.81164
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) −16.9706 −0.900704
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −8.48528 −0.447836 −0.223918 0.974608i \(-0.571885\pi\)
−0.223918 + 0.974608i \(0.571885\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) −9.89949 −0.519589
\(364\) 0 0
\(365\) −24.0000 −1.25622
\(366\) 0 0
\(367\) −11.3137 −0.590571 −0.295285 0.955409i \(-0.595415\pi\)
−0.295285 + 0.955409i \(0.595415\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 5.65685 0.293689
\(372\) 0 0
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 0 0
\(375\) 16.9706 0.876356
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 12.7279 0.653789 0.326895 0.945061i \(-0.393998\pi\)
0.326895 + 0.945061i \(0.393998\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) −33.9411 −1.73431 −0.867155 0.498038i \(-0.834054\pi\)
−0.867155 + 0.498038i \(0.834054\pi\)
\(384\) 0 0
\(385\) −24.0000 −1.22315
\(386\) 0 0
\(387\) 4.24264 0.215666
\(388\) 0 0
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −2.00000 −0.100887
\(394\) 0 0
\(395\) −11.3137 −0.569254
\(396\) 0 0
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 0 0
\(399\) −16.9706 −0.849591
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 33.9411 1.69073
\(404\) 0 0
\(405\) −10.0000 −0.496904
\(406\) 0 0
\(407\) 25.4558 1.26180
\(408\) 0 0
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) 8.48528 0.418548
\(412\) 0 0
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) 8.48528 0.416526
\(416\) 0 0
\(417\) 6.00000 0.293821
\(418\) 0 0
\(419\) 12.7279 0.621800 0.310900 0.950443i \(-0.399370\pi\)
0.310900 + 0.950443i \(0.399370\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −16.9706 −0.821263
\(428\) 0 0
\(429\) −36.0000 −1.73810
\(430\) 0 0
\(431\) −16.9706 −0.817443 −0.408722 0.912659i \(-0.634025\pi\)
−0.408722 + 0.912659i \(0.634025\pi\)
\(432\) 0 0
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 0 0
\(435\) −5.65685 −0.271225
\(436\) 0 0
\(437\) −36.0000 −1.72211
\(438\) 0 0
\(439\) 2.82843 0.134993 0.0674967 0.997719i \(-0.478499\pi\)
0.0674967 + 0.997719i \(0.478499\pi\)
\(440\) 0 0
\(441\) −1.00000 −0.0476190
\(442\) 0 0
\(443\) −12.7279 −0.604722 −0.302361 0.953194i \(-0.597775\pi\)
−0.302361 + 0.953194i \(0.597775\pi\)
\(444\) 0 0
\(445\) −24.0000 −1.13771
\(446\) 0 0
\(447\) 31.1127 1.47158
\(448\) 0 0
\(449\) 24.0000 1.13263 0.566315 0.824189i \(-0.308369\pi\)
0.566315 + 0.824189i \(0.308369\pi\)
\(450\) 0 0
\(451\) 25.4558 1.19867
\(452\) 0 0
\(453\) 20.0000 0.939682
\(454\) 0 0
\(455\) −33.9411 −1.59118
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 26.0000 1.21094 0.605470 0.795868i \(-0.292985\pi\)
0.605470 + 0.795868i \(0.292985\pi\)
\(462\) 0 0
\(463\) 5.65685 0.262896 0.131448 0.991323i \(-0.458037\pi\)
0.131448 + 0.991323i \(0.458037\pi\)
\(464\) 0 0
\(465\) −16.0000 −0.741982
\(466\) 0 0
\(467\) −4.24264 −0.196326 −0.0981630 0.995170i \(-0.531297\pi\)
−0.0981630 + 0.995170i \(0.531297\pi\)
\(468\) 0 0
\(469\) −36.0000 −1.66233
\(470\) 0 0
\(471\) −8.48528 −0.390981
\(472\) 0 0
\(473\) −18.0000 −0.827641
\(474\) 0 0
\(475\) 4.24264 0.194666
\(476\) 0 0
\(477\) 2.00000 0.0915737
\(478\) 0 0
\(479\) −16.9706 −0.775405 −0.387702 0.921785i \(-0.626731\pi\)
−0.387702 + 0.921785i \(0.626731\pi\)
\(480\) 0 0
\(481\) 36.0000 1.64146
\(482\) 0 0
\(483\) 33.9411 1.54437
\(484\) 0 0
\(485\) −16.0000 −0.726523
\(486\) 0 0
\(487\) −36.7696 −1.66619 −0.833094 0.553132i \(-0.813433\pi\)
−0.833094 + 0.553132i \(0.813433\pi\)
\(488\) 0 0
\(489\) −30.0000 −1.35665
\(490\) 0 0
\(491\) −26.8701 −1.21263 −0.606314 0.795225i \(-0.707353\pi\)
−0.606314 + 0.795225i \(0.707353\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −8.48528 −0.381385
\(496\) 0 0
\(497\) 24.0000 1.07655
\(498\) 0 0
\(499\) 4.24264 0.189927 0.0949633 0.995481i \(-0.469727\pi\)
0.0949633 + 0.995481i \(0.469727\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 0 0
\(503\) 42.4264 1.89170 0.945850 0.324604i \(-0.105231\pi\)
0.945850 + 0.324604i \(0.105231\pi\)
\(504\) 0 0
\(505\) −20.0000 −0.889988
\(506\) 0 0
\(507\) −32.5269 −1.44457
\(508\) 0 0
\(509\) −34.0000 −1.50702 −0.753512 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 0 0
\(511\) 33.9411 1.50147
\(512\) 0 0
\(513\) −24.0000 −1.05963
\(514\) 0 0
\(515\) 5.65685 0.249271
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −19.7990 −0.869079
\(520\) 0 0
\(521\) 42.0000 1.84005 0.920027 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 0 0
\(523\) 4.24264 0.185518 0.0927589 0.995689i \(-0.470431\pi\)
0.0927589 + 0.995689i \(0.470431\pi\)
\(524\) 0 0
\(525\) −4.00000 −0.174574
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 49.0000 2.13043
\(530\) 0 0
\(531\) −1.41421 −0.0613716
\(532\) 0 0
\(533\) 36.0000 1.55933
\(534\) 0 0
\(535\) −14.1421 −0.611418
\(536\) 0 0
\(537\) 26.0000 1.12198
\(538\) 0 0
\(539\) 4.24264 0.182743
\(540\) 0 0
\(541\) −42.0000 −1.80572 −0.902861 0.429934i \(-0.858537\pi\)
−0.902861 + 0.429934i \(0.858537\pi\)
\(542\) 0 0
\(543\) −25.4558 −1.09241
\(544\) 0 0
\(545\) −12.0000 −0.514024
\(546\) 0 0
\(547\) −21.2132 −0.907011 −0.453506 0.891253i \(-0.649827\pi\)
−0.453506 + 0.891253i \(0.649827\pi\)
\(548\) 0 0
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) −8.48528 −0.361485
\(552\) 0 0
\(553\) 16.0000 0.680389
\(554\) 0 0
\(555\) −16.9706 −0.720360
\(556\) 0 0
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 0 0
\(559\) −25.4558 −1.07667
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 29.6985 1.25164 0.625821 0.779967i \(-0.284764\pi\)
0.625821 + 0.779967i \(0.284764\pi\)
\(564\) 0 0
\(565\) 12.0000 0.504844
\(566\) 0 0
\(567\) 14.1421 0.593914
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −12.7279 −0.532647 −0.266323 0.963884i \(-0.585809\pi\)
−0.266323 + 0.963884i \(0.585809\pi\)
\(572\) 0 0
\(573\) 24.0000 1.00261
\(574\) 0 0
\(575\) −8.48528 −0.353861
\(576\) 0 0
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) 0 0
\(579\) −33.9411 −1.41055
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) −8.48528 −0.351424
\(584\) 0 0
\(585\) −12.0000 −0.496139
\(586\) 0 0
\(587\) 15.5563 0.642079 0.321040 0.947066i \(-0.395968\pi\)
0.321040 + 0.947066i \(0.395968\pi\)
\(588\) 0 0
\(589\) −24.0000 −0.988903
\(590\) 0 0
\(591\) −14.1421 −0.581730
\(592\) 0 0
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −28.0000 −1.14596
\(598\) 0 0
\(599\) 25.4558 1.04010 0.520049 0.854137i \(-0.325914\pi\)
0.520049 + 0.854137i \(0.325914\pi\)
\(600\) 0 0
\(601\) −12.0000 −0.489490 −0.244745 0.969587i \(-0.578704\pi\)
−0.244745 + 0.969587i \(0.578704\pi\)
\(602\) 0 0
\(603\) −12.7279 −0.518321
\(604\) 0 0
\(605\) 14.0000 0.569181
\(606\) 0 0
\(607\) 22.6274 0.918419 0.459209 0.888328i \(-0.348133\pi\)
0.459209 + 0.888328i \(0.348133\pi\)
\(608\) 0 0
\(609\) 8.00000 0.324176
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −6.00000 −0.242338 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) 0 0
\(615\) −16.9706 −0.684319
\(616\) 0 0
\(617\) −36.0000 −1.44931 −0.724653 0.689114i \(-0.758000\pi\)
−0.724653 + 0.689114i \(0.758000\pi\)
\(618\) 0 0
\(619\) 38.1838 1.53474 0.767368 0.641207i \(-0.221566\pi\)
0.767368 + 0.641207i \(0.221566\pi\)
\(620\) 0 0
\(621\) 48.0000 1.92617
\(622\) 0 0
\(623\) 33.9411 1.35982
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 25.4558 1.01661
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 31.1127 1.23858 0.619288 0.785164i \(-0.287421\pi\)
0.619288 + 0.785164i \(0.287421\pi\)
\(632\) 0 0
\(633\) −6.00000 −0.238479
\(634\) 0 0
\(635\) 11.3137 0.448971
\(636\) 0 0
\(637\) 6.00000 0.237729
\(638\) 0 0
\(639\) 8.48528 0.335673
\(640\) 0 0
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 0 0
\(643\) 12.7279 0.501940 0.250970 0.967995i \(-0.419250\pi\)
0.250970 + 0.967995i \(0.419250\pi\)
\(644\) 0 0
\(645\) 12.0000 0.472500
\(646\) 0 0
\(647\) 25.4558 1.00077 0.500386 0.865802i \(-0.333191\pi\)
0.500386 + 0.865802i \(0.333191\pi\)
\(648\) 0 0
\(649\) 6.00000 0.235521
\(650\) 0 0
\(651\) 22.6274 0.886838
\(652\) 0 0
\(653\) 2.00000 0.0782660 0.0391330 0.999234i \(-0.487540\pi\)
0.0391330 + 0.999234i \(0.487540\pi\)
\(654\) 0 0
\(655\) 2.82843 0.110516
\(656\) 0 0
\(657\) 12.0000 0.468165
\(658\) 0 0
\(659\) −41.0122 −1.59761 −0.798804 0.601591i \(-0.794534\pi\)
−0.798804 + 0.601591i \(0.794534\pi\)
\(660\) 0 0
\(661\) 18.0000 0.700119 0.350059 0.936727i \(-0.386161\pi\)
0.350059 + 0.936727i \(0.386161\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 24.0000 0.930680
\(666\) 0 0
\(667\) 16.9706 0.657103
\(668\) 0 0
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) 25.4558 0.982712
\(672\) 0 0
\(673\) 8.00000 0.308377 0.154189 0.988041i \(-0.450724\pi\)
0.154189 + 0.988041i \(0.450724\pi\)
\(674\) 0 0
\(675\) −5.65685 −0.217732
\(676\) 0 0
\(677\) 38.0000 1.46046 0.730229 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) 0 0
\(679\) 22.6274 0.868361
\(680\) 0 0
\(681\) 18.0000 0.689761
\(682\) 0 0
\(683\) −38.1838 −1.46106 −0.730531 0.682880i \(-0.760727\pi\)
−0.730531 + 0.682880i \(0.760727\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) 0 0
\(687\) −25.4558 −0.971201
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −29.6985 −1.12978 −0.564892 0.825165i \(-0.691082\pi\)
−0.564892 + 0.825165i \(0.691082\pi\)
\(692\) 0 0
\(693\) 12.0000 0.455842
\(694\) 0 0
\(695\) −8.48528 −0.321865
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 16.9706 0.641886
\(700\) 0 0
\(701\) 46.0000 1.73740 0.868698 0.495342i \(-0.164957\pi\)
0.868698 + 0.495342i \(0.164957\pi\)
\(702\) 0 0
\(703\) −25.4558 −0.960085
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 28.2843 1.06374
\(708\) 0 0
\(709\) −18.0000 −0.676004 −0.338002 0.941145i \(-0.609751\pi\)
−0.338002 + 0.941145i \(0.609751\pi\)
\(710\) 0 0
\(711\) 5.65685 0.212149
\(712\) 0 0
\(713\) 48.0000 1.79761
\(714\) 0 0
\(715\) 50.9117 1.90399
\(716\) 0 0
\(717\) −24.0000 −0.896296
\(718\) 0 0
\(719\) 16.9706 0.632895 0.316448 0.948610i \(-0.397510\pi\)
0.316448 + 0.948610i \(0.397510\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) 2.82843 0.104901 0.0524503 0.998624i \(-0.483297\pi\)
0.0524503 + 0.998624i \(0.483297\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 42.0000 1.55131 0.775653 0.631160i \(-0.217421\pi\)
0.775653 + 0.631160i \(0.217421\pi\)
\(734\) 0 0
\(735\) −2.82843 −0.104328
\(736\) 0 0
\(737\) 54.0000 1.98912
\(738\) 0 0
\(739\) −46.6690 −1.71675 −0.858374 0.513024i \(-0.828525\pi\)
−0.858374 + 0.513024i \(0.828525\pi\)
\(740\) 0 0
\(741\) 36.0000 1.32249
\(742\) 0 0
\(743\) −25.4558 −0.933884 −0.466942 0.884288i \(-0.654644\pi\)
−0.466942 + 0.884288i \(0.654644\pi\)
\(744\) 0 0
\(745\) −44.0000 −1.61204
\(746\) 0 0
\(747\) −4.24264 −0.155230
\(748\) 0 0
\(749\) 20.0000 0.730784
\(750\) 0 0
\(751\) −5.65685 −0.206422 −0.103211 0.994660i \(-0.532912\pi\)
−0.103211 + 0.994660i \(0.532912\pi\)
\(752\) 0 0
\(753\) −18.0000 −0.655956
\(754\) 0 0
\(755\) −28.2843 −1.02937
\(756\) 0 0
\(757\) 42.0000 1.52652 0.763258 0.646094i \(-0.223599\pi\)
0.763258 + 0.646094i \(0.223599\pi\)
\(758\) 0 0
\(759\) −50.9117 −1.84798
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 0 0
\(763\) 16.9706 0.614376
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.48528 0.306386
\(768\) 0 0
\(769\) −24.0000 −0.865462 −0.432731 0.901523i \(-0.642450\pi\)
−0.432731 + 0.901523i \(0.642450\pi\)
\(770\) 0 0
\(771\) −8.48528 −0.305590
\(772\) 0 0
\(773\) 14.0000 0.503545 0.251773 0.967786i \(-0.418987\pi\)
0.251773 + 0.967786i \(0.418987\pi\)
\(774\) 0 0
\(775\) −5.65685 −0.203200
\(776\) 0 0
\(777\) 24.0000 0.860995
\(778\) 0 0
\(779\) −25.4558 −0.912050
\(780\) 0 0
\(781\) −36.0000 −1.28818
\(782\) 0 0
\(783\) 11.3137 0.404319
\(784\) 0 0
\(785\) 12.0000 0.428298
\(786\) 0 0
\(787\) −29.6985 −1.05864 −0.529318 0.848423i \(-0.677552\pi\)
−0.529318 + 0.848423i \(0.677552\pi\)
\(788\) 0 0
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) −16.9706 −0.603404
\(792\) 0 0
\(793\) 36.0000 1.27840
\(794\) 0 0
\(795\) 5.65685 0.200628
\(796\) 0 0
\(797\) −2.00000 −0.0708436 −0.0354218 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 12.0000 0.423999
\(802\) 0 0
\(803\) −50.9117 −1.79663
\(804\) 0 0
\(805\) −48.0000 −1.69178
\(806\) 0 0
\(807\) 19.7990 0.696957
\(808\) 0 0
\(809\) −42.0000 −1.47664 −0.738321 0.674450i \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) 0 0
\(811\) 4.24264 0.148979 0.0744896 0.997222i \(-0.476267\pi\)
0.0744896 + 0.997222i \(0.476267\pi\)
\(812\) 0 0
\(813\) 32.0000 1.12229
\(814\) 0 0
\(815\) 42.4264 1.48613
\(816\) 0 0
\(817\) 18.0000 0.629740
\(818\) 0 0
\(819\) 16.9706 0.592999
\(820\) 0 0
\(821\) 38.0000 1.32621 0.663105 0.748527i \(-0.269238\pi\)
0.663105 + 0.748527i \(0.269238\pi\)
\(822\) 0 0
\(823\) −19.7990 −0.690149 −0.345075 0.938575i \(-0.612146\pi\)
−0.345075 + 0.938575i \(0.612146\pi\)
\(824\) 0 0
\(825\) 6.00000 0.208893
\(826\) 0 0
\(827\) 43.8406 1.52449 0.762244 0.647290i \(-0.224098\pi\)
0.762244 + 0.647290i \(0.224098\pi\)
\(828\) 0 0
\(829\) −18.0000 −0.625166 −0.312583 0.949890i \(-0.601194\pi\)
−0.312583 + 0.949890i \(0.601194\pi\)
\(830\) 0 0
\(831\) −8.48528 −0.294351
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 16.9706 0.587291
\(836\) 0 0
\(837\) 32.0000 1.10608
\(838\) 0 0
\(839\) 8.48528 0.292944 0.146472 0.989215i \(-0.453208\pi\)
0.146472 + 0.989215i \(0.453208\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) −16.9706 −0.584497
\(844\) 0 0
\(845\) 46.0000 1.58245
\(846\) 0 0
\(847\) −19.7990 −0.680301
\(848\) 0 0
\(849\) 18.0000 0.617758
\(850\) 0 0
\(851\) 50.9117 1.74523
\(852\) 0 0
\(853\) −18.0000 −0.616308 −0.308154 0.951336i \(-0.599711\pi\)
−0.308154 + 0.951336i \(0.599711\pi\)
\(854\) 0 0
\(855\) 8.48528 0.290191
\(856\) 0 0
\(857\) −54.0000 −1.84460 −0.922302 0.386469i \(-0.873695\pi\)
−0.922302 + 0.386469i \(0.873695\pi\)
\(858\) 0 0
\(859\) 46.6690 1.59233 0.796164 0.605081i \(-0.206859\pi\)
0.796164 + 0.605081i \(0.206859\pi\)
\(860\) 0 0
\(861\) 24.0000 0.817918
\(862\) 0 0
\(863\) −50.9117 −1.73305 −0.866527 0.499130i \(-0.833653\pi\)
−0.866527 + 0.499130i \(0.833653\pi\)
\(864\) 0 0
\(865\) 28.0000 0.952029
\(866\) 0 0
\(867\) 24.0416 0.816497
\(868\) 0 0
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) 76.3675 2.58762
\(872\) 0 0
\(873\) 8.00000 0.270759
\(874\) 0 0
\(875\) 33.9411 1.14742
\(876\) 0 0
\(877\) −6.00000 −0.202606 −0.101303 0.994856i \(-0.532301\pi\)
−0.101303 + 0.994856i \(0.532301\pi\)
\(878\) 0 0
\(879\) −14.1421 −0.477002
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) −38.1838 −1.28499 −0.642493 0.766292i \(-0.722100\pi\)
−0.642493 + 0.766292i \(0.722100\pi\)
\(884\) 0 0
\(885\) −4.00000 −0.134459
\(886\) 0 0
\(887\) −42.4264 −1.42454 −0.712270 0.701906i \(-0.752333\pi\)
−0.712270 + 0.701906i \(0.752333\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) −21.2132 −0.710669
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −36.7696 −1.22907
\(896\) 0 0
\(897\) −72.0000 −2.40401
\(898\) 0 0
\(899\) 11.3137 0.377333
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −16.9706 −0.564745
\(904\) 0 0
\(905\) 36.0000 1.19668
\(906\) 0 0
\(907\) −4.24264 −0.140875 −0.0704373 0.997516i \(-0.522439\pi\)
−0.0704373 + 0.997516i \(0.522439\pi\)
\(908\) 0 0
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) −33.9411 −1.12452 −0.562260 0.826961i \(-0.690068\pi\)
−0.562260 + 0.826961i \(0.690068\pi\)
\(912\) 0 0
\(913\) 18.0000 0.595713
\(914\) 0 0
\(915\) −16.9706 −0.561029
\(916\) 0 0
\(917\) −4.00000 −0.132092
\(918\) 0 0
\(919\) −2.82843 −0.0933012 −0.0466506 0.998911i \(-0.514855\pi\)
−0.0466506 + 0.998911i \(0.514855\pi\)
\(920\) 0 0
\(921\) 6.00000 0.197707
\(922\) 0 0
\(923\) −50.9117 −1.67578
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) 0 0
\(927\) −2.82843 −0.0928977
\(928\) 0 0
\(929\) −24.0000 −0.787414 −0.393707 0.919236i \(-0.628808\pi\)
−0.393707 + 0.919236i \(0.628808\pi\)
\(930\) 0 0
\(931\) −4.24264 −0.139047
\(932\) 0 0
\(933\) 12.0000 0.392862
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 20.0000 0.653372 0.326686 0.945133i \(-0.394068\pi\)
0.326686 + 0.945133i \(0.394068\pi\)
\(938\) 0 0
\(939\) −8.48528 −0.276907
\(940\) 0 0
\(941\) −22.0000 −0.717180 −0.358590 0.933495i \(-0.616742\pi\)
−0.358590 + 0.933495i \(0.616742\pi\)
\(942\) 0 0
\(943\) 50.9117 1.65791
\(944\) 0 0
\(945\) −32.0000 −1.04096
\(946\) 0 0
\(947\) 41.0122 1.33272 0.666359 0.745631i \(-0.267852\pi\)
0.666359 + 0.745631i \(0.267852\pi\)
\(948\) 0 0
\(949\) −72.0000 −2.33722
\(950\) 0 0
\(951\) −14.1421 −0.458590
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) −33.9411 −1.09831
\(956\) 0 0
\(957\) −12.0000 −0.387905
\(958\) 0 0
\(959\) 16.9706 0.548008
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 7.07107 0.227862
\(964\) 0 0
\(965\) 48.0000 1.54517
\(966\) 0 0
\(967\) 31.1127 1.00052 0.500258 0.865876i \(-0.333238\pi\)
0.500258 + 0.865876i \(0.333238\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 29.6985 0.953070 0.476535 0.879156i \(-0.341893\pi\)
0.476535 + 0.879156i \(0.341893\pi\)
\(972\) 0 0
\(973\) 12.0000 0.384702
\(974\) 0 0
\(975\) 8.48528 0.271746
\(976\) 0 0
\(977\) −48.0000 −1.53566 −0.767828 0.640656i \(-0.778662\pi\)
−0.767828 + 0.640656i \(0.778662\pi\)
\(978\) 0 0
\(979\) −50.9117 −1.62714
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) 0 0
\(983\) −42.4264 −1.35319 −0.676596 0.736354i \(-0.736546\pi\)
−0.676596 + 0.736354i \(0.736546\pi\)
\(984\) 0 0
\(985\) 20.0000 0.637253
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −36.0000 −1.14473
\(990\) 0 0
\(991\) 45.2548 1.43757 0.718784 0.695234i \(-0.244699\pi\)
0.718784 + 0.695234i \(0.244699\pi\)
\(992\) 0 0
\(993\) 42.0000 1.33283
\(994\) 0 0
\(995\) 39.5980 1.25534
\(996\) 0 0
\(997\) −18.0000 −0.570066 −0.285033 0.958518i \(-0.592005\pi\)
−0.285033 + 0.958518i \(0.592005\pi\)
\(998\) 0 0
\(999\) 33.9411 1.07385
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 512.2.a.e.1.1 yes 2
3.2 odd 2 4608.2.a.c.1.1 2
4.3 odd 2 inner 512.2.a.e.1.2 yes 2
8.3 odd 2 512.2.a.b.1.1 2
8.5 even 2 512.2.a.b.1.2 yes 2
12.11 even 2 4608.2.a.c.1.2 2
16.3 odd 4 512.2.b.e.257.3 4
16.5 even 4 512.2.b.e.257.4 4
16.11 odd 4 512.2.b.e.257.2 4
16.13 even 4 512.2.b.e.257.1 4
24.5 odd 2 4608.2.a.p.1.1 2
24.11 even 2 4608.2.a.p.1.2 2
32.3 odd 8 1024.2.e.n.257.1 4
32.5 even 8 1024.2.e.n.769.2 4
32.11 odd 8 1024.2.e.n.769.1 4
32.13 even 8 1024.2.e.n.257.2 4
32.19 odd 8 1024.2.e.h.257.2 4
32.21 even 8 1024.2.e.h.769.1 4
32.27 odd 8 1024.2.e.h.769.2 4
32.29 even 8 1024.2.e.h.257.1 4
48.5 odd 4 4608.2.d.j.2305.2 4
48.11 even 4 4608.2.d.j.2305.1 4
48.29 odd 4 4608.2.d.j.2305.4 4
48.35 even 4 4608.2.d.j.2305.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
512.2.a.b.1.1 2 8.3 odd 2
512.2.a.b.1.2 yes 2 8.5 even 2
512.2.a.e.1.1 yes 2 1.1 even 1 trivial
512.2.a.e.1.2 yes 2 4.3 odd 2 inner
512.2.b.e.257.1 4 16.13 even 4
512.2.b.e.257.2 4 16.11 odd 4
512.2.b.e.257.3 4 16.3 odd 4
512.2.b.e.257.4 4 16.5 even 4
1024.2.e.h.257.1 4 32.29 even 8
1024.2.e.h.257.2 4 32.19 odd 8
1024.2.e.h.769.1 4 32.21 even 8
1024.2.e.h.769.2 4 32.27 odd 8
1024.2.e.n.257.1 4 32.3 odd 8
1024.2.e.n.257.2 4 32.13 even 8
1024.2.e.n.769.1 4 32.11 odd 8
1024.2.e.n.769.2 4 32.5 even 8
4608.2.a.c.1.1 2 3.2 odd 2
4608.2.a.c.1.2 2 12.11 even 2
4608.2.a.p.1.1 2 24.5 odd 2
4608.2.a.p.1.2 2 24.11 even 2
4608.2.d.j.2305.1 4 48.11 even 4
4608.2.d.j.2305.2 4 48.5 odd 4
4608.2.d.j.2305.3 4 48.35 even 4
4608.2.d.j.2305.4 4 48.29 odd 4