Properties

Label 5292.2.j.d.1765.2
Level $5292$
Weight $2$
Character 5292.1765
Analytic conductor $42.257$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5292,2,Mod(1765,5292)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5292, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5292.1765");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5292.j (of order \(3\), degree \(2\), not minimal)

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: no
Analytic conductor: \(42.2568327497\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1765.2
Root \(0.500000 - 1.41036i\) of defining polynomial
Character \(\chi\) \(=\) 5292.1765
Dual form 5292.2.j.d.3529.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.119562 + 0.207087i) q^{5} +O(q^{10})\) \(q+(-0.119562 + 0.207087i) q^{5} +(-2.56238 - 4.43818i) q^{11} +(-2.44282 + 4.23109i) q^{13} +3.70370 q^{17} -3.66019 q^{19} +(3.71053 - 6.42683i) q^{23} +(2.47141 + 4.28061i) q^{25} +(1.73229 + 3.00041i) q^{29} +(-0.358685 + 0.621261i) q^{31} +4.60301 q^{37} +(-2.80150 + 4.85235i) q^{41} +(-6.24433 - 10.8155i) q^{43} +(-2.16991 - 3.75839i) q^{47} +0.942820 q^{53} +1.22545 q^{55} +(-3.78947 + 6.56355i) q^{59} +(2.75404 + 4.77014i) q^{61} +(-0.584135 - 1.01175i) q^{65} +(0.330095 - 0.571741i) q^{67} -13.7414 q^{71} +3.66019 q^{73} +(3.11273 + 5.39140i) q^{79} +(-4.85185 - 8.40365i) q^{83} +(-0.442820 + 0.766987i) q^{85} -7.48865 q^{89} +(0.437618 - 0.757977i) q^{95} +(-8.57442 - 14.8513i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{5} + 2 q^{11} + 3 q^{13} + 4 q^{17} - 6 q^{19} + 14 q^{23} + 6 q^{25} + q^{29} - 3 q^{31} - 6 q^{37} - 3 q^{43} - 21 q^{47} - 12 q^{53} - 12 q^{55} - 31 q^{59} + 6 q^{61} + 15 q^{65} - 6 q^{67} - 34 q^{71} + 6 q^{73} + 9 q^{79} - 20 q^{83} + 15 q^{85} + 24 q^{89} + 20 q^{95} - 9 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(2647\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −0.119562 + 0.207087i −0.0534696 + 0.0926120i −0.891521 0.452979i \(-0.850361\pi\)
0.838052 + 0.545591i \(0.183695\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.56238 4.43818i −0.772587 1.33816i −0.936141 0.351626i \(-0.885629\pi\)
0.163554 0.986534i \(-0.447704\pi\)
\(12\) 0 0
\(13\) −2.44282 + 4.23109i −0.677516 + 1.17349i 0.298210 + 0.954500i \(0.403610\pi\)
−0.975727 + 0.218993i \(0.929723\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.70370 0.898278 0.449139 0.893462i \(-0.351731\pi\)
0.449139 + 0.893462i \(0.351731\pi\)
\(18\) 0 0
\(19\) −3.66019 −0.839705 −0.419853 0.907592i \(-0.637918\pi\)
−0.419853 + 0.907592i \(0.637918\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.71053 6.42683i 0.773700 1.34009i −0.161823 0.986820i \(-0.551737\pi\)
0.935522 0.353267i \(-0.114929\pi\)
\(24\) 0 0
\(25\) 2.47141 + 4.28061i 0.494282 + 0.856122i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.73229 + 3.00041i 0.321678 + 0.557162i 0.980834 0.194844i \(-0.0624200\pi\)
−0.659157 + 0.752006i \(0.729087\pi\)
\(30\) 0 0
\(31\) −0.358685 + 0.621261i −0.0644217 + 0.111582i −0.896437 0.443171i \(-0.853854\pi\)
0.832016 + 0.554752i \(0.187187\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.60301 0.756730 0.378365 0.925656i \(-0.376486\pi\)
0.378365 + 0.925656i \(0.376486\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.80150 + 4.85235i −0.437522 + 0.757810i −0.997498 0.0706992i \(-0.977477\pi\)
0.559976 + 0.828509i \(0.310810\pi\)
\(42\) 0 0
\(43\) −6.24433 10.8155i −0.952251 1.64935i −0.740538 0.672015i \(-0.765429\pi\)
−0.211713 0.977332i \(-0.567904\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.16991 3.75839i −0.316513 0.548217i 0.663245 0.748403i \(-0.269179\pi\)
−0.979758 + 0.200186i \(0.935845\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.942820 0.129506 0.0647531 0.997901i \(-0.479374\pi\)
0.0647531 + 0.997901i \(0.479374\pi\)
\(54\) 0 0
\(55\) 1.22545 0.165240
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.78947 + 6.56355i −0.493347 + 0.854501i −0.999971 0.00766579i \(-0.997560\pi\)
0.506624 + 0.862167i \(0.330893\pi\)
\(60\) 0 0
\(61\) 2.75404 + 4.77014i 0.352619 + 0.610754i 0.986707 0.162507i \(-0.0519579\pi\)
−0.634089 + 0.773260i \(0.718625\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.584135 1.01175i −0.0724530 0.125492i
\(66\) 0 0
\(67\) 0.330095 0.571741i 0.0403275 0.0698493i −0.845157 0.534518i \(-0.820493\pi\)
0.885485 + 0.464669i \(0.153827\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −13.7414 −1.63081 −0.815405 0.578891i \(-0.803486\pi\)
−0.815405 + 0.578891i \(0.803486\pi\)
\(72\) 0 0
\(73\) 3.66019 0.428393 0.214196 0.976791i \(-0.431287\pi\)
0.214196 + 0.976791i \(0.431287\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 3.11273 + 5.39140i 0.350209 + 0.606580i 0.986286 0.165046i \(-0.0527772\pi\)
−0.636077 + 0.771626i \(0.719444\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −4.85185 8.40365i −0.532560 0.922420i −0.999277 0.0380138i \(-0.987897\pi\)
0.466718 0.884406i \(-0.345436\pi\)
\(84\) 0 0
\(85\) −0.442820 + 0.766987i −0.0480306 + 0.0831914i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.48865 −0.793795 −0.396898 0.917863i \(-0.629913\pi\)
−0.396898 + 0.917863i \(0.629913\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.437618 0.757977i 0.0448987 0.0777668i
\(96\) 0 0
\(97\) −8.57442 14.8513i −0.870600 1.50792i −0.861377 0.507967i \(-0.830397\pi\)
−0.00922376 0.999957i \(-0.502936\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.59097 6.21975i −0.357315 0.618888i 0.630196 0.776436i \(-0.282974\pi\)
−0.987511 + 0.157548i \(0.949641\pi\)
\(102\) 0 0
\(103\) −6.41423 + 11.1098i −0.632013 + 1.09468i 0.355127 + 0.934818i \(0.384438\pi\)
−0.987140 + 0.159860i \(0.948896\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.56526 −0.731361 −0.365681 0.930740i \(-0.619164\pi\)
−0.365681 + 0.930740i \(0.619164\pi\)
\(108\) 0 0
\(109\) −6.98057 −0.668617 −0.334309 0.942464i \(-0.608503\pi\)
−0.334309 + 0.942464i \(0.608503\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −9.78495 + 16.9480i −0.920491 + 1.59434i −0.121834 + 0.992550i \(0.538878\pi\)
−0.798657 + 0.601787i \(0.794456\pi\)
\(114\) 0 0
\(115\) 0.887275 + 1.53681i 0.0827388 + 0.143308i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −7.63160 + 13.2183i −0.693782 + 1.20167i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −2.37756 −0.212655
\(126\) 0 0
\(127\) −16.8090 −1.49156 −0.745780 0.666192i \(-0.767923\pi\)
−0.745780 + 0.666192i \(0.767923\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.44966 + 4.24293i −0.214027 + 0.370706i −0.952971 0.303061i \(-0.901992\pi\)
0.738944 + 0.673767i \(0.235325\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.72257 + 4.71563i 0.232605 + 0.402884i 0.958574 0.284844i \(-0.0919417\pi\)
−0.725969 + 0.687727i \(0.758608\pi\)
\(138\) 0 0
\(139\) 2.83009 4.90187i 0.240046 0.415771i −0.720681 0.693266i \(-0.756171\pi\)
0.960727 + 0.277495i \(0.0895043\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 25.0377 2.09376
\(144\) 0 0
\(145\) −0.828460 −0.0687999
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.14132 1.97682i 0.0935002 0.161947i −0.815481 0.578783i \(-0.803528\pi\)
0.908982 + 0.416836i \(0.136861\pi\)
\(150\) 0 0
\(151\) −5.63160 9.75422i −0.458293 0.793787i 0.540578 0.841294i \(-0.318206\pi\)
−0.998871 + 0.0475071i \(0.984872\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.0857699 0.148558i −0.00688921 0.0119325i
\(156\) 0 0
\(157\) 2.77292 4.80283i 0.221303 0.383308i −0.733901 0.679256i \(-0.762302\pi\)
0.955204 + 0.295949i \(0.0956358\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 6.66019 0.521666 0.260833 0.965384i \(-0.416003\pi\)
0.260833 + 0.965384i \(0.416003\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.20370 + 3.81691i −0.170527 + 0.295362i −0.938604 0.344996i \(-0.887880\pi\)
0.768077 + 0.640357i \(0.221214\pi\)
\(168\) 0 0
\(169\) −5.43474 9.41325i −0.418057 0.724096i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −12.6654 21.9371i −0.962932 1.66785i −0.715072 0.699051i \(-0.753606\pi\)
−0.247860 0.968796i \(-0.579727\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −9.54583 −0.713489 −0.356744 0.934202i \(-0.616113\pi\)
−0.356744 + 0.934202i \(0.616113\pi\)
\(180\) 0 0
\(181\) −12.3743 −0.919774 −0.459887 0.887978i \(-0.652110\pi\)
−0.459887 + 0.887978i \(0.652110\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.550343 + 0.953223i −0.0404621 + 0.0700823i
\(186\) 0 0
\(187\) −9.49028 16.4377i −0.693998 1.20204i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6.58414 + 11.4041i 0.476411 + 0.825169i 0.999635 0.0270270i \(-0.00860400\pi\)
−0.523223 + 0.852196i \(0.675271\pi\)
\(192\) 0 0
\(193\) −5.57442 + 9.65518i −0.401256 + 0.694995i −0.993878 0.110486i \(-0.964759\pi\)
0.592622 + 0.805481i \(0.298093\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0.144194 0.0102734 0.00513669 0.999987i \(-0.498365\pi\)
0.00513669 + 0.999987i \(0.498365\pi\)
\(198\) 0 0
\(199\) 19.4692 1.38014 0.690068 0.723744i \(-0.257581\pi\)
0.690068 + 0.723744i \(0.257581\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −0.669905 1.16031i −0.0467882 0.0810395i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 9.37880 + 16.2446i 0.648745 + 1.12366i
\(210\) 0 0
\(211\) 1.61436 2.79615i 0.111137 0.192495i −0.805092 0.593150i \(-0.797884\pi\)
0.916229 + 0.400655i \(0.131217\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2.98633 0.203666
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −9.04746 + 15.6707i −0.608598 + 1.05412i
\(222\) 0 0
\(223\) 10.3856 + 17.9885i 0.695474 + 1.20460i 0.970021 + 0.243022i \(0.0781389\pi\)
−0.274547 + 0.961574i \(0.588528\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −10.9714 19.0030i −0.728198 1.26128i −0.957644 0.287955i \(-0.907025\pi\)
0.229446 0.973321i \(-0.426309\pi\)
\(228\) 0 0
\(229\) 11.3856 19.7205i 0.752384 1.30317i −0.194280 0.980946i \(-0.562237\pi\)
0.946664 0.322222i \(-0.104430\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 25.7817 1.68901 0.844507 0.535544i \(-0.179894\pi\)
0.844507 + 0.535544i \(0.179894\pi\)
\(234\) 0 0
\(235\) 1.03775 0.0676953
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −13.6488 + 23.6405i −0.882870 + 1.52918i −0.0347345 + 0.999397i \(0.511059\pi\)
−0.848136 + 0.529779i \(0.822275\pi\)
\(240\) 0 0
\(241\) 5.01724 + 8.69011i 0.323189 + 0.559779i 0.981144 0.193277i \(-0.0619118\pi\)
−0.657955 + 0.753057i \(0.728578\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 8.94119 15.4866i 0.568914 0.985388i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −28.3171 −1.78736 −0.893680 0.448705i \(-0.851885\pi\)
−0.893680 + 0.448705i \(0.851885\pi\)
\(252\) 0 0
\(253\) −38.0312 −2.39100
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −14.4315 + 24.9960i −0.900210 + 1.55921i −0.0729899 + 0.997333i \(0.523254\pi\)
−0.827221 + 0.561877i \(0.810079\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −0.604645 1.04728i −0.0372840 0.0645778i 0.846781 0.531941i \(-0.178537\pi\)
−0.884065 + 0.467363i \(0.845204\pi\)
\(264\) 0 0
\(265\) −0.112725 + 0.195246i −0.00692465 + 0.0119938i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 9.01367 0.549573 0.274787 0.961505i \(-0.411393\pi\)
0.274787 + 0.961505i \(0.411393\pi\)
\(270\) 0 0
\(271\) −17.6030 −1.06931 −0.534653 0.845071i \(-0.679558\pi\)
−0.534653 + 0.845071i \(0.679558\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 12.6654 21.9371i 0.763752 1.32286i
\(276\) 0 0
\(277\) −0.727085 1.25935i −0.0436863 0.0756669i 0.843355 0.537356i \(-0.180577\pi\)
−0.887042 + 0.461689i \(0.847244\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −10.1482 17.5771i −0.605388 1.04856i −0.991990 0.126316i \(-0.959685\pi\)
0.386602 0.922247i \(-0.373649\pi\)
\(282\) 0 0
\(283\) −2.30150 + 3.98632i −0.136810 + 0.236962i −0.926288 0.376817i \(-0.877018\pi\)
0.789477 + 0.613780i \(0.210352\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −3.28263 −0.193096
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 3.53667 6.12569i 0.206614 0.357867i −0.744031 0.668145i \(-0.767089\pi\)
0.950646 + 0.310278i \(0.100422\pi\)
\(294\) 0 0
\(295\) −0.906150 1.56950i −0.0527581 0.0913797i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 18.1283 + 31.3992i 1.04839 + 1.81586i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.31711 −0.0754175
\(306\) 0 0
\(307\) −15.7518 −0.899006 −0.449503 0.893279i \(-0.648399\pi\)
−0.449503 + 0.893279i \(0.648399\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.81191 16.9947i 0.556382 0.963682i −0.441412 0.897304i \(-0.645522\pi\)
0.997795 0.0663780i \(-0.0211443\pi\)
\(312\) 0 0
\(313\) −12.7427 22.0710i −0.720259 1.24753i −0.960896 0.276911i \(-0.910689\pi\)
0.240636 0.970615i \(-0.422644\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4.14132 + 7.17297i 0.232599 + 0.402874i 0.958572 0.284849i \(-0.0919436\pi\)
−0.725973 + 0.687723i \(0.758610\pi\)
\(318\) 0 0
\(319\) 8.87756 15.3764i 0.497048 0.860912i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −13.5562 −0.754289
\(324\) 0 0
\(325\) −24.1488 −1.33954
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 5.99028 + 10.3755i 0.329256 + 0.570288i 0.982364 0.186976i \(-0.0598688\pi\)
−0.653109 + 0.757264i \(0.726535\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0.0789334 + 0.136717i 0.00431259 + 0.00746963i
\(336\) 0 0
\(337\) 6.46006 11.1892i 0.351902 0.609512i −0.634681 0.772774i \(-0.718868\pi\)
0.986583 + 0.163262i \(0.0522017\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3.67635 0.199086
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 8.09329 14.0180i 0.434471 0.752526i −0.562781 0.826606i \(-0.690269\pi\)
0.997252 + 0.0740802i \(0.0236021\pi\)
\(348\) 0 0
\(349\) −9.05718 15.6875i −0.484820 0.839732i 0.515028 0.857173i \(-0.327781\pi\)
−0.999848 + 0.0174409i \(0.994448\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 5.84897 + 10.1307i 0.311309 + 0.539203i 0.978646 0.205553i \(-0.0658992\pi\)
−0.667337 + 0.744756i \(0.732566\pi\)
\(354\) 0 0
\(355\) 1.64295 2.84567i 0.0871987 0.151033i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −35.7245 −1.88547 −0.942734 0.333547i \(-0.891755\pi\)
−0.942734 + 0.333547i \(0.891755\pi\)
\(360\) 0 0
\(361\) −5.60301 −0.294895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −0.437618 + 0.757977i −0.0229060 + 0.0396743i
\(366\) 0 0
\(367\) 8.52696 + 14.7691i 0.445103 + 0.770942i 0.998059 0.0622687i \(-0.0198336\pi\)
−0.552956 + 0.833210i \(0.686500\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 12.9617 22.4503i 0.671131 1.16243i −0.306453 0.951886i \(-0.599142\pi\)
0.977584 0.210547i \(-0.0675246\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −16.9267 −0.871767
\(378\) 0 0
\(379\) 26.8446 1.37892 0.689458 0.724326i \(-0.257849\pi\)
0.689458 + 0.724326i \(0.257849\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −12.4263 + 21.5229i −0.634953 + 1.09977i 0.351573 + 0.936161i \(0.385647\pi\)
−0.986525 + 0.163610i \(0.947686\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −9.12640 15.8074i −0.462727 0.801466i 0.536369 0.843984i \(-0.319796\pi\)
−0.999096 + 0.0425174i \(0.986462\pi\)
\(390\) 0 0
\(391\) 13.7427 23.8030i 0.694998 1.20377i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −1.48865 −0.0749021
\(396\) 0 0
\(397\) −12.3743 −0.621048 −0.310524 0.950566i \(-0.600504\pi\)
−0.310524 + 0.950566i \(0.600504\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −5.48113 + 9.49359i −0.273714 + 0.474087i −0.969810 0.243862i \(-0.921586\pi\)
0.696096 + 0.717949i \(0.254919\pi\)
\(402\) 0 0
\(403\) −1.75241 3.03526i −0.0872935 0.151197i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −11.7947 20.4290i −0.584640 1.01263i
\(408\) 0 0
\(409\) −6.66019 + 11.5358i −0.329325 + 0.570408i −0.982378 0.186904i \(-0.940155\pi\)
0.653053 + 0.757312i \(0.273488\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 2.32038 0.113903
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2.35705 + 4.08253i −0.115149 + 0.199445i −0.917839 0.396952i \(-0.870068\pi\)
0.802690 + 0.596396i \(0.203401\pi\)
\(420\) 0 0
\(421\) −9.65856 16.7291i −0.470729 0.815327i 0.528710 0.848802i \(-0.322676\pi\)
−0.999440 + 0.0334755i \(0.989342\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 9.15335 + 15.8541i 0.444003 + 0.769036i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 30.2794 1.45851 0.729253 0.684244i \(-0.239868\pi\)
0.729253 + 0.684244i \(0.239868\pi\)
\(432\) 0 0
\(433\) −34.2060 −1.64384 −0.821918 0.569606i \(-0.807096\pi\)
−0.821918 + 0.569606i \(0.807096\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −13.5813 + 23.5234i −0.649680 + 1.12528i
\(438\) 0 0
\(439\) −0.311220 0.539049i −0.0148537 0.0257274i 0.858503 0.512809i \(-0.171395\pi\)
−0.873357 + 0.487081i \(0.838062\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2.58934 4.48486i −0.123023 0.213082i 0.797935 0.602743i \(-0.205926\pi\)
−0.920958 + 0.389661i \(0.872592\pi\)
\(444\) 0 0
\(445\) 0.895355 1.55080i 0.0424439 0.0735150i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 10.4977 0.495416 0.247708 0.968835i \(-0.420323\pi\)
0.247708 + 0.968835i \(0.420323\pi\)
\(450\) 0 0
\(451\) 28.7141 1.35209
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.60464 + 2.77933i 0.0750621 + 0.130011i 0.901113 0.433584i \(-0.142751\pi\)
−0.826051 + 0.563595i \(0.809418\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 18.1150 + 31.3762i 0.843702 + 1.46133i 0.886744 + 0.462261i \(0.152962\pi\)
−0.0430418 + 0.999073i \(0.513705\pi\)
\(462\) 0 0
\(463\) −14.5253 + 25.1586i −0.675049 + 1.16922i 0.301406 + 0.953496i \(0.402544\pi\)
−0.976455 + 0.215723i \(0.930789\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −37.4933 −1.73498 −0.867491 0.497452i \(-0.834269\pi\)
−0.867491 + 0.497452i \(0.834269\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −32.0007 + 55.4268i −1.47139 + 2.54853i
\(474\) 0 0
\(475\) −9.04583 15.6678i −0.415051 0.718890i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −14.9549 25.9026i −0.683305 1.18352i −0.973966 0.226693i \(-0.927209\pi\)
0.290661 0.956826i \(-0.406125\pi\)
\(480\) 0 0
\(481\) −11.2443 + 19.4757i −0.512697 + 0.888017i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4.10069 0.186203
\(486\) 0 0
\(487\) 21.2632 0.963528 0.481764 0.876301i \(-0.339996\pi\)
0.481764 + 0.876301i \(0.339996\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −10.6985 + 18.5303i −0.482816 + 0.836262i −0.999805 0.0197296i \(-0.993719\pi\)
0.516989 + 0.855992i \(0.327053\pi\)
\(492\) 0 0
\(493\) 6.41586 + 11.1126i 0.288956 + 0.500487i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −7.28263 + 12.6139i −0.326015 + 0.564675i −0.981717 0.190345i \(-0.939039\pi\)
0.655702 + 0.755020i \(0.272373\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 2.92339 0.130348 0.0651738 0.997874i \(-0.479240\pi\)
0.0651738 + 0.997874i \(0.479240\pi\)
\(504\) 0 0
\(505\) 1.71737 0.0764220
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 9.62025 16.6628i 0.426410 0.738564i −0.570141 0.821547i \(-0.693111\pi\)
0.996551 + 0.0829830i \(0.0264447\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.53379 2.65661i −0.0675869 0.117064i
\(516\) 0 0
\(517\) −11.1202 + 19.2608i −0.489068 + 0.847091i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 27.7486 1.21569 0.607844 0.794057i \(-0.292035\pi\)
0.607844 + 0.794057i \(0.292035\pi\)
\(522\) 0 0
\(523\) 2.73680 0.119672 0.0598360 0.998208i \(-0.480942\pi\)
0.0598360 + 0.998208i \(0.480942\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.32846 + 2.30096i −0.0578686 + 0.100231i
\(528\) 0 0
\(529\) −16.0361 27.7754i −0.697222 1.20762i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −13.6871 23.7068i −0.592856 1.02686i
\(534\) 0 0
\(535\) 0.904515 1.56667i 0.0391056 0.0677329i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −11.9773 −0.514944 −0.257472 0.966286i \(-0.582890\pi\)
−0.257472 + 0.966286i \(0.582890\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0.834608 1.44558i 0.0357507 0.0619220i
\(546\) 0 0
\(547\) 10.7346 + 18.5929i 0.458979 + 0.794975i 0.998907 0.0467363i \(-0.0148821\pi\)
−0.539928 + 0.841711i \(0.681549\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −6.34050 10.9821i −0.270114 0.467852i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 31.8493 1.34950 0.674748 0.738048i \(-0.264252\pi\)
0.674748 + 0.738048i \(0.264252\pi\)
\(558\) 0 0
\(559\) 61.0150 2.58066
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −17.7742 + 30.7857i −0.749091 + 1.29746i 0.199167 + 0.979966i \(0.436176\pi\)
−0.948259 + 0.317499i \(0.897157\pi\)
\(564\) 0 0
\(565\) −2.33981 4.05267i −0.0984366 0.170497i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 10.8743 + 18.8348i 0.455874 + 0.789597i 0.998738 0.0502237i \(-0.0159934\pi\)
−0.542864 + 0.839821i \(0.682660\pi\)
\(570\) 0 0
\(571\) 4.79987 8.31362i 0.200868 0.347914i −0.747940 0.663766i \(-0.768957\pi\)
0.948808 + 0.315852i \(0.102290\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 36.6810 1.52970
\(576\) 0 0
\(577\) −13.0183 −0.541960 −0.270980 0.962585i \(-0.587348\pi\)
−0.270980 + 0.962585i \(0.587348\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −2.41586 4.18440i −0.100055 0.173300i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −8.64364 14.9712i −0.356761 0.617928i 0.630657 0.776062i \(-0.282786\pi\)
−0.987418 + 0.158134i \(0.949452\pi\)
\(588\) 0 0
\(589\) 1.31285 2.27393i 0.0540952 0.0936957i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 12.4153 0.509836 0.254918 0.966963i \(-0.417952\pi\)
0.254918 + 0.966963i \(0.417952\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −3.94282 + 6.82916i −0.161099 + 0.279032i −0.935263 0.353953i \(-0.884837\pi\)
0.774164 + 0.632985i \(0.218171\pi\)
\(600\) 0 0
\(601\) 11.1413 + 19.2973i 0.454464 + 0.787154i 0.998657 0.0518055i \(-0.0164976\pi\)
−0.544193 + 0.838960i \(0.683164\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.82489 3.16081i −0.0741925 0.128505i
\(606\) 0 0
\(607\) 11.0458 19.1319i 0.448336 0.776541i −0.549942 0.835203i \(-0.685350\pi\)
0.998278 + 0.0586617i \(0.0186833\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 21.2028 0.857771
\(612\) 0 0
\(613\) −29.5264 −1.19256 −0.596280 0.802777i \(-0.703355\pi\)
−0.596280 + 0.802777i \(0.703355\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −5.01655 + 8.68892i −0.201959 + 0.349803i −0.949159 0.314796i \(-0.898064\pi\)
0.747201 + 0.664598i \(0.231397\pi\)
\(618\) 0 0
\(619\) −19.1283 33.1312i −0.768833 1.33166i −0.938196 0.346103i \(-0.887505\pi\)
0.169364 0.985554i \(-0.445829\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −12.0728 + 20.9107i −0.482911 + 0.836427i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 17.0482 0.679754
\(630\) 0 0
\(631\) −23.0377 −0.917118 −0.458559 0.888664i \(-0.651634\pi\)
−0.458559 + 0.888664i \(0.651634\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 2.00972 3.48093i 0.0797531 0.138136i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −8.68646 15.0454i −0.343094 0.594257i 0.641911 0.766779i \(-0.278142\pi\)
−0.985006 + 0.172522i \(0.944808\pi\)
\(642\) 0 0
\(643\) −9.47949 + 16.4190i −0.373835 + 0.647501i −0.990152 0.139997i \(-0.955291\pi\)
0.616317 + 0.787498i \(0.288624\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 19.0194 0.747731 0.373865 0.927483i \(-0.378032\pi\)
0.373865 + 0.927483i \(0.378032\pi\)
\(648\) 0 0
\(649\) 38.8402 1.52461
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −3.59329 + 6.22377i −0.140616 + 0.243555i −0.927729 0.373255i \(-0.878242\pi\)
0.787112 + 0.616810i \(0.211575\pi\)
\(654\) 0 0
\(655\) −0.585770 1.01458i −0.0228879 0.0396430i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 12.7261 + 22.0423i 0.495740 + 0.858647i 0.999988 0.00491209i \(-0.00156357\pi\)
−0.504248 + 0.863559i \(0.668230\pi\)
\(660\) 0 0
\(661\) 4.14295 7.17580i 0.161142 0.279106i −0.774136 0.633019i \(-0.781816\pi\)
0.935279 + 0.353912i \(0.115149\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 25.7108 0.995527
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 14.1138 24.4458i 0.544857 0.943721i
\(672\) 0 0
\(673\) 5.91586 + 10.2466i 0.228040 + 0.394977i 0.957227 0.289338i \(-0.0934350\pi\)
−0.729187 + 0.684314i \(0.760102\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −6.80314 11.7834i −0.261466 0.452872i 0.705166 0.709042i \(-0.250873\pi\)
−0.966632 + 0.256170i \(0.917539\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −3.58142 −0.137039 −0.0685196 0.997650i \(-0.521828\pi\)
−0.0685196 + 0.997650i \(0.521828\pi\)
\(684\) 0 0
\(685\) −1.30206 −0.0497492
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −2.30314 + 3.98916i −0.0877426 + 0.151975i
\(690\) 0 0
\(691\) 5.85868 + 10.1475i 0.222875 + 0.386031i 0.955680 0.294408i \(-0.0951225\pi\)
−0.732805 + 0.680439i \(0.761789\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0.676742 + 1.17215i 0.0256703 + 0.0444622i
\(696\) 0 0
\(697\) −10.3759 + 17.9716i −0.393016 + 0.680724i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 10.5926 0.400077 0.200039 0.979788i \(-0.435893\pi\)
0.200039 + 0.979788i \(0.435893\pi\)
\(702\) 0 0
\(703\) −16.8479 −0.635430
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −19.1488 33.1668i −0.719150 1.24560i −0.961337 0.275374i \(-0.911198\pi\)
0.242187 0.970230i \(-0.422135\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.66182 + 4.61042i 0.0996861 + 0.172661i
\(714\) 0 0
\(715\) −2.99355 + 5.18499i −0.111953 + 0.193908i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 41.6752 1.55422 0.777112 0.629362i \(-0.216684\pi\)
0.777112 + 0.629362i \(0.216684\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −8.56238 + 14.8305i −0.317999 + 0.550790i
\(726\) 0 0
\(727\) −16.4126 28.4274i −0.608709 1.05432i −0.991453 0.130461i \(-0.958354\pi\)
0.382744 0.923854i \(-0.374979\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −23.1271 40.0573i −0.855386 1.48157i
\(732\) 0 0
\(733\) 4.64884 8.05203i 0.171709 0.297408i −0.767309 0.641278i \(-0.778405\pi\)
0.939017 + 0.343870i \(0.111738\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −3.38332 −0.124626
\(738\) 0 0
\(739\) 11.3776 0.418530 0.209265 0.977859i \(-0.432893\pi\)
0.209265 + 0.977859i \(0.432893\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.16182 + 2.01234i −0.0426232 + 0.0738256i −0.886550 0.462633i \(-0.846905\pi\)
0.843927 + 0.536458i \(0.180238\pi\)
\(744\) 0 0
\(745\) 0.272915 + 0.472703i 0.00999883 + 0.0173185i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 5.56690 9.64215i 0.203139 0.351847i −0.746399 0.665498i \(-0.768219\pi\)
0.949538 + 0.313652i \(0.101552\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 2.69329 0.0980190
\(756\) 0 0
\(757\) 52.1639 1.89593 0.947964 0.318376i \(-0.103138\pi\)
0.947964 + 0.318376i \(0.103138\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 23.5127 40.7252i 0.852336 1.47629i −0.0267592 0.999642i \(-0.508519\pi\)
0.879095 0.476647i \(-0.158148\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −18.5140 32.0671i −0.668501 1.15788i
\(768\) 0 0
\(769\) −3.30314 + 5.72121i −0.119114 + 0.206312i −0.919417 0.393284i \(-0.871339\pi\)
0.800303 + 0.599596i \(0.204672\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 19.0870 0.686512 0.343256 0.939242i \(-0.388470\pi\)
0.343256 + 0.939242i \(0.388470\pi\)
\(774\) 0 0
\(775\) −3.54583 −0.127370
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 10.2540 17.7605i 0.367389 0.636337i
\(780\) 0 0
\(781\) 35.2108 + 60.9869i 1.25994 + 2.18228i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0.663069 + 1.14847i 0.0236659 + 0.0409906i
\(786\) 0 0
\(787\) 25.4503 44.0813i 0.907207 1.57133i 0.0892796 0.996007i \(-0.471544\pi\)
0.817927 0.575322i \(-0.195123\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −26.9105 −0.955620
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −4.38727 + 7.59898i −0.155405 + 0.269170i −0.933207 0.359341i \(-0.883002\pi\)
0.777801 + 0.628510i \(0.216335\pi\)
\(798\) 0 0
\(799\) −8.03667 13.9199i −0.284317 0.492451i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −9.37880 16.2446i −0.330971 0.573258i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 9.51384 0.334489 0.167244 0.985915i \(-0.446513\pi\)
0.167244 + 0.985915i \(0.446513\pi\)
\(810\) 0 0
\(811\) 25.0118 0.878282 0.439141 0.898418i \(-0.355283\pi\)
0.439141 + 0.898418i \(0.355283\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −0.796303 + 1.37924i −0.0278933 + 0.0483126i
\(816\) 0 0
\(817\) 22.8554 + 39.5867i 0.799610 + 1.38496i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −17.7970 30.8253i −0.621119 1.07581i −0.989278 0.146047i \(-0.953345\pi\)
0.368158 0.929763i \(-0.379988\pi\)
\(822\) 0 0
\(823\) −8.00000 + 13.8564i −0.278862 + 0.483004i −0.971102 0.238664i \(-0.923291\pi\)
0.692240 + 0.721668i \(0.256624\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 25.4531 0.885090 0.442545 0.896746i \(-0.354076\pi\)
0.442545 + 0.896746i \(0.354076\pi\)
\(828\) 0 0
\(829\) 17.5458 0.609392 0.304696 0.952450i \(-0.401445\pi\)
0.304696 + 0.952450i \(0.401445\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −0.526955 0.912713i −0.0182360 0.0315857i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −12.0562 20.8820i −0.416227 0.720927i 0.579329 0.815094i \(-0.303315\pi\)
−0.995556 + 0.0941668i \(0.969981\pi\)
\(840\) 0 0
\(841\) 8.49837 14.7196i 0.293047 0.507572i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2.59915 0.0894133
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 17.0796 29.5828i 0.585482 1.01408i
\(852\) 0 0
\(853\) 16.2616 + 28.1659i 0.556785 + 0.964381i 0.997762 + 0.0668621i \(0.0212988\pi\)
−0.440977 + 0.897518i \(0.645368\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0.299870 + 0.519390i 0.0102434 + 0.0177420i 0.871102 0.491103i \(-0.163406\pi\)
−0.860858 + 0.508845i \(0.830073\pi\)
\(858\) 0 0
\(859\) −13.2174 + 22.8932i −0.450971 + 0.781104i −0.998447 0.0557171i \(-0.982255\pi\)
0.547476 + 0.836822i \(0.315589\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −19.8454 −0.675545 −0.337773 0.941228i \(-0.609674\pi\)
−0.337773 + 0.941228i \(0.609674\pi\)
\(864\) 0 0
\(865\) 6.05718 0.205950
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 15.9520 27.6296i 0.541134 0.937271i
\(870\) 0 0
\(871\) 1.61273 + 2.79332i 0.0546451 + 0.0946481i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −10.2352 + 17.7278i −0.345617 + 0.598626i −0.985466 0.169875i \(-0.945664\pi\)
0.639849 + 0.768501i \(0.278997\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −31.1683 −1.05009 −0.525043 0.851076i \(-0.675951\pi\)
−0.525043 + 0.851076i \(0.675951\pi\)
\(882\) 0 0
\(883\) −2.64187 −0.0889060 −0.0444530 0.999011i \(-0.514154\pi\)
−0.0444530 + 0.999011i \(0.514154\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 11.5825 20.0615i 0.388902 0.673599i −0.603400 0.797439i \(-0.706188\pi\)
0.992302 + 0.123840i \(0.0395210\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 7.94226 + 13.7564i 0.265778 + 0.460341i
\(894\) 0 0
\(895\) 1.14132 1.97682i 0.0381500 0.0660777i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −2.48538 −0.0828921
\(900\) 0 0
\(901\) 3.49192 0.116333
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.47949 2.56255i 0.0491799 0.0851821i
\(906\) 0 0
\(907\) 25.0264 + 43.3470i 0.830988 + 1.43931i 0.897256 + 0.441511i \(0.145558\pi\)
−0.0662676 + 0.997802i \(0.521109\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −5.42231 9.39172i −0.179649 0.311161i 0.762111 0.647446i \(-0.224163\pi\)
−0.941760 + 0.336285i \(0.890830\pi\)
\(912\) 0 0
\(913\) −24.8646 + 43.0667i −0.822897 + 1.42530i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 11.1910 0.369156 0.184578 0.982818i \(-0.440908\pi\)
0.184578 + 0.982818i \(0.440908\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 33.5679 58.1413i 1.10490 1.91374i
\(924\) 0 0
\(925\) 11.3759 + 19.7037i 0.374038 + 0.647853i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −20.6478 35.7630i −0.677431 1.17335i −0.975752 0.218879i \(-0.929760\pi\)
0.298321 0.954466i \(-0.403573\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 4.53870 0.148431
\(936\) 0 0
\(937\) 33.5620 1.09642 0.548211 0.836340i \(-0.315309\pi\)
0.548211 + 0.836340i \(0.315309\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −25.2112 + 43.6671i −0.821862 + 1.42351i 0.0824315 + 0.996597i \(0.473731\pi\)
−0.904294 + 0.426911i \(0.859602\pi\)
\(942\) 0 0
\(943\) 20.7902 + 36.0096i 0.677021 + 1.17263i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 10.1212 + 17.5304i 0.328895 + 0.569662i 0.982293 0.187352i \(-0.0599905\pi\)
−0.653398 + 0.757014i \(0.726657\pi\)
\(948\) 0 0
\(949\) −8.94119 + 15.4866i −0.290243 + 0.502716i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 29.3685 0.951340 0.475670 0.879624i \(-0.342206\pi\)
0.475670 + 0.879624i \(0.342206\pi\)
\(954\) 0 0
\(955\) −3.14884 −0.101894
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 15.2427 + 26.4011i 0.491700 + 0.851649i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1.33297 2.30878i −0.0429099 0.0743222i
\(966\) 0 0
\(967\) 15.2157 26.3544i 0.489305 0.847501i −0.510619 0.859807i \(-0.670584\pi\)
0.999924 + 0.0123057i \(0.00391714\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 7.18659 0.230629 0.115314 0.993329i \(-0.463212\pi\)
0.115314 + 0.993329i \(0.463212\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 14.2713 24.7186i 0.456579 0.790818i −0.542199 0.840250i \(-0.682408\pi\)
0.998777 + 0.0494328i \(0.0157414\pi\)
\(978\) 0 0
\(979\) 19.1888 + 33.2359i 0.613276 + 1.06223i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −2.20821 3.82473i −0.0704310 0.121990i 0.828659 0.559753i \(-0.189104\pi\)
−0.899090 + 0.437763i \(0.855771\pi\)
\(984\) 0 0
\(985\) −0.0172400 + 0.0298606i −0.000549313 + 0.000951438i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −92.6791 −2.94702
\(990\) 0 0
\(991\) −5.81341 −0.184669 −0.0923345 0.995728i \(-0.529433\pi\)
−0.0923345 + 0.995728i \(0.529433\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −2.32777 + 4.03182i −0.0737953 + 0.127817i
\(996\) 0 0
\(997\) −26.3204 45.5882i −0.833575 1.44379i −0.895186 0.445694i \(-0.852957\pi\)
0.0616108 0.998100i \(-0.480376\pi\)
\(998\) 0 0
\(999\) 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 5292.2.j.d.1765.2 6
3.2 odd 2 1764.2.j.e.589.3 6
7.2 even 3 5292.2.l.f.361.2 6
7.3 odd 6 5292.2.i.f.1549.2 6
7.4 even 3 5292.2.i.e.1549.2 6
7.5 odd 6 5292.2.l.e.361.2 6
7.6 odd 2 756.2.j.b.253.2 6
9.2 odd 6 1764.2.j.e.1177.3 6
9.7 even 3 inner 5292.2.j.d.3529.2 6
21.2 odd 6 1764.2.l.f.949.1 6
21.5 even 6 1764.2.l.e.949.3 6
21.11 odd 6 1764.2.i.d.373.2 6
21.17 even 6 1764.2.i.g.373.2 6
21.20 even 2 252.2.j.a.85.1 6
28.27 even 2 3024.2.r.j.1009.2 6
63.2 odd 6 1764.2.i.d.1537.2 6
63.11 odd 6 1764.2.l.f.961.1 6
63.13 odd 6 2268.2.a.h.1.2 3
63.16 even 3 5292.2.i.e.2125.2 6
63.20 even 6 252.2.j.a.169.1 yes 6
63.25 even 3 5292.2.l.f.3313.2 6
63.34 odd 6 756.2.j.b.505.2 6
63.38 even 6 1764.2.l.e.961.3 6
63.41 even 6 2268.2.a.i.1.2 3
63.47 even 6 1764.2.i.g.1537.2 6
63.52 odd 6 5292.2.l.e.3313.2 6
63.61 odd 6 5292.2.i.f.2125.2 6
84.83 odd 2 1008.2.r.j.337.3 6
252.83 odd 6 1008.2.r.j.673.3 6
252.139 even 6 9072.2.a.bv.1.2 3
252.167 odd 6 9072.2.a.by.1.2 3
252.223 even 6 3024.2.r.j.2017.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.j.a.85.1 6 21.20 even 2
252.2.j.a.169.1 yes 6 63.20 even 6
756.2.j.b.253.2 6 7.6 odd 2
756.2.j.b.505.2 6 63.34 odd 6
1008.2.r.j.337.3 6 84.83 odd 2
1008.2.r.j.673.3 6 252.83 odd 6
1764.2.i.d.373.2 6 21.11 odd 6
1764.2.i.d.1537.2 6 63.2 odd 6
1764.2.i.g.373.2 6 21.17 even 6
1764.2.i.g.1537.2 6 63.47 even 6
1764.2.j.e.589.3 6 3.2 odd 2
1764.2.j.e.1177.3 6 9.2 odd 6
1764.2.l.e.949.3 6 21.5 even 6
1764.2.l.e.961.3 6 63.38 even 6
1764.2.l.f.949.1 6 21.2 odd 6
1764.2.l.f.961.1 6 63.11 odd 6
2268.2.a.h.1.2 3 63.13 odd 6
2268.2.a.i.1.2 3 63.41 even 6
3024.2.r.j.1009.2 6 28.27 even 2
3024.2.r.j.2017.2 6 252.223 even 6
5292.2.i.e.1549.2 6 7.4 even 3
5292.2.i.e.2125.2 6 63.16 even 3
5292.2.i.f.1549.2 6 7.3 odd 6
5292.2.i.f.2125.2 6 63.61 odd 6
5292.2.j.d.1765.2 6 1.1 even 1 trivial
5292.2.j.d.3529.2 6 9.7 even 3 inner
5292.2.l.e.361.2 6 7.5 odd 6
5292.2.l.e.3313.2 6 63.52 odd 6
5292.2.l.f.361.2 6 7.2 even 3
5292.2.l.f.3313.2 6 63.25 even 3
9072.2.a.bv.1.2 3 252.139 even 6
9072.2.a.by.1.2 3 252.167 odd 6