Properties

Label 5292.2.l.f.3313.2
Level $5292$
Weight $2$
Character 5292.3313
Analytic conductor $42.257$
Analytic rank $0$
Dimension $6$
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] = [5292,2,Mod(361,5292)]
 
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, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5292.361");
 
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.l (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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 3313.2
Root \(0.500000 + 1.41036i\) of defining polynomial
Character \(\chi\) \(=\) 5292.3313
Dual form 5292.2.l.f.361.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.239123 q^{5} +O(q^{10})\) \(q+0.239123 q^{5} +5.12476 q^{11} +(-2.44282 - 4.23109i) q^{13} +(-1.85185 - 3.20750i) q^{17} +(1.83009 - 3.16982i) q^{19} -7.42107 q^{23} -4.94282 q^{25} +(1.73229 - 3.00041i) q^{29} +(-0.358685 + 0.621261i) q^{31} +(-2.30150 + 3.98632i) q^{37} +(-2.80150 - 4.85235i) q^{41} +(-6.24433 + 10.8155i) q^{43} +(-2.16991 - 3.75839i) q^{47} +(-0.471410 - 0.816506i) q^{53} +1.22545 q^{55} +(-3.78947 + 6.56355i) q^{59} +(2.75404 + 4.77014i) q^{61} +(-0.584135 - 1.01175i) q^{65} +(0.330095 - 0.571741i) q^{67} -13.7414 q^{71} +(-1.83009 - 3.16982i) q^{73} +(3.11273 + 5.39140i) q^{79} +(-4.85185 + 8.40365i) q^{83} +(-0.442820 - 0.766987i) q^{85} +(3.74433 - 6.48536i) q^{89} +(0.437618 - 0.757977i) q^{95} +(-8.57442 + 14.8513i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{5} - 4 q^{11} + 3 q^{13} - 2 q^{17} + 3 q^{19} - 28 q^{23} - 12 q^{25} + q^{29} - 3 q^{31} + 3 q^{37} - 3 q^{43} - 21 q^{47} + 6 q^{53} - 12 q^{55} - 31 q^{59} + 6 q^{61} + 15 q^{65} - 6 q^{67} - 34 q^{71} - 3 q^{73} + 9 q^{79} - 20 q^{83} + 15 q^{85} - 12 q^{89} + 20 q^{95} - 9 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(1081\) \(2647\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\)

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
\(5\) 0.239123 0.106939 0.0534696 0.998569i \(-0.482972\pi\)
0.0534696 + 0.998569i \(0.482972\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.12476 1.54517 0.772587 0.634909i \(-0.218962\pi\)
0.772587 + 0.634909i \(0.218962\pi\)
\(12\) 0 0
\(13\) −2.44282 4.23109i −0.677516 1.17349i −0.975727 0.218993i \(-0.929723\pi\)
0.298210 0.954500i \(-0.403610\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.85185 3.20750i −0.449139 0.777932i 0.549191 0.835697i \(-0.314936\pi\)
−0.998330 + 0.0577649i \(0.981603\pi\)
\(18\) 0 0
\(19\) 1.83009 3.16982i 0.419853 0.727206i −0.576072 0.817399i \(-0.695415\pi\)
0.995924 + 0.0901932i \(0.0287484\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.42107 −1.54740 −0.773700 0.633553i \(-0.781596\pi\)
−0.773700 + 0.633553i \(0.781596\pi\)
\(24\) 0 0
\(25\) −4.94282 −0.988564
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.73229 3.00041i 0.321678 0.557162i −0.659157 0.752006i \(-0.729087\pi\)
0.980834 + 0.194844i \(0.0624200\pi\)
\(30\) 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
\(37\) −2.30150 + 3.98632i −0.378365 + 0.655348i −0.990825 0.135154i \(-0.956847\pi\)
0.612459 + 0.790502i \(0.290180\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.80150 4.85235i −0.437522 0.757810i 0.559976 0.828509i \(-0.310810\pi\)
−0.997498 + 0.0706992i \(0.977477\pi\)
\(42\) 0 0
\(43\) −6.24433 + 10.8155i −0.952251 + 1.64935i −0.211713 + 0.977332i \(0.567904\pi\)
−0.740538 + 0.672015i \(0.765429\pi\)
\(44\) 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
\(53\) −0.471410 0.816506i −0.0647531 0.112156i 0.831831 0.555029i \(-0.187293\pi\)
−0.896584 + 0.442873i \(0.853959\pi\)
\(54\) 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
\(73\) −1.83009 3.16982i −0.214196 0.370999i 0.738827 0.673895i \(-0.235380\pi\)
−0.953024 + 0.302896i \(0.902047\pi\)
\(74\) 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 0.466718 + 0.884406i \(0.345436\pi\)
−0.999277 + 0.0380138i \(0.987897\pi\)
\(84\) 0 0
\(85\) −0.442820 0.766987i −0.0480306 0.0831914i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.74433 6.48536i 0.396898 0.687447i −0.596444 0.802655i \(-0.703420\pi\)
0.993341 + 0.115208i \(0.0367535\pi\)
\(90\) 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 −0.00922376 + 0.999957i \(0.502936\pi\)
−0.861377 + 0.507967i \(0.830397\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.18194 0.714630 0.357315 0.933984i \(-0.383692\pi\)
0.357315 + 0.933984i \(0.383692\pi\)
\(102\) 0 0
\(103\) 12.8285 1.26403 0.632013 0.774958i \(-0.282229\pi\)
0.632013 + 0.774958i \(0.282229\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.78263 6.55171i 0.365681 0.633377i −0.623204 0.782059i \(-0.714170\pi\)
0.988885 + 0.148681i \(0.0475029\pi\)
\(108\) 0 0
\(109\) 3.49028 + 6.04535i 0.334309 + 0.579040i 0.983352 0.181712i \(-0.0581639\pi\)
−0.649043 + 0.760752i \(0.724831\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −9.78495 16.9480i −0.920491 1.59434i −0.798657 0.601787i \(-0.794456\pi\)
−0.121834 0.992550i \(-0.538878\pi\)
\(114\) 0 0
\(115\) −1.77455 −0.165478
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 15.2632 1.38756
\(122\) 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
\(131\) 4.89931 0.428055 0.214027 0.976828i \(-0.431342\pi\)
0.214027 + 0.976828i \(0.431342\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −5.44514 −0.465210 −0.232605 0.972571i \(-0.574725\pi\)
−0.232605 + 0.972571i \(0.574725\pi\)
\(138\) 0 0
\(139\) 2.83009 + 4.90187i 0.240046 + 0.415771i 0.960727 0.277495i \(-0.0895043\pi\)
−0.720681 + 0.693266i \(0.756171\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −12.5189 21.6833i −1.04688 1.81325i
\(144\) 0 0
\(145\) 0.414230 0.717468i 0.0343999 0.0595824i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2.28263 −0.187000 −0.0935002 0.995619i \(-0.529806\pi\)
−0.0935002 + 0.995619i \(0.529806\pi\)
\(150\) 0 0
\(151\) 11.2632 0.916586 0.458293 0.888801i \(-0.348461\pi\)
0.458293 + 0.888801i \(0.348461\pi\)
\(152\) 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
\(163\) −3.33009 + 5.76789i −0.260833 + 0.451776i −0.966464 0.256804i \(-0.917331\pi\)
0.705630 + 0.708580i \(0.250664\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.20370 3.81691i −0.170527 0.295362i 0.768077 0.640357i \(-0.221214\pi\)
−0.938604 + 0.344996i \(0.887880\pi\)
\(168\) 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
\(179\) 4.77292 + 8.26693i 0.356744 + 0.617899i 0.987415 0.158151i \(-0.0505534\pi\)
−0.630670 + 0.776051i \(0.717220\pi\)
\(180\) 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
\(199\) −9.73461 16.8608i −0.690068 1.19523i −0.971815 0.235744i \(-0.924247\pi\)
0.281747 0.959489i \(-0.409086\pi\)
\(200\) 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 0.916229 0.400655i \(-0.131217\pi\)
−0.805092 + 0.593150i \(0.797884\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1.49316 + 2.58623i −0.101833 + 0.176380i
\(216\) 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 −0.274547 0.961574i \(-0.588528\pi\)
0.970021 0.243022i \(-0.0781389\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 21.9428 1.45640 0.728198 0.685367i \(-0.240358\pi\)
0.728198 + 0.685367i \(0.240358\pi\)
\(228\) 0 0
\(229\) −22.7713 −1.50477 −0.752384 0.658724i \(-0.771096\pi\)
−0.752384 + 0.658724i \(0.771096\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −12.8908 + 22.3276i −0.844507 + 1.46273i 0.0415414 + 0.999137i \(0.486773\pi\)
−0.886049 + 0.463592i \(0.846560\pi\)
\(234\) 0 0
\(235\) −0.518875 0.898718i −0.0338477 0.0586259i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −13.6488 23.6405i −0.882870 1.52918i −0.848136 0.529779i \(-0.822275\pi\)
−0.0347345 0.999397i \(-0.511059\pi\)
\(240\) 0 0
\(241\) −10.0345 −0.646378 −0.323189 0.946334i \(-0.604755\pi\)
−0.323189 + 0.946334i \(0.604755\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −17.8824 −1.13783
\(248\) 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
\(257\) 28.8629 1.80042 0.900210 0.435455i \(-0.143413\pi\)
0.900210 + 0.435455i \(0.143413\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.20929 0.0745680 0.0372840 0.999305i \(-0.488129\pi\)
0.0372840 + 0.999305i \(0.488129\pi\)
\(264\) 0 0
\(265\) −0.112725 0.195246i −0.00692465 0.0119938i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −4.50684 7.80607i −0.274787 0.475944i 0.695295 0.718725i \(-0.255274\pi\)
−0.970081 + 0.242780i \(0.921941\pi\)
\(270\) 0 0
\(271\) 8.80150 15.2447i 0.534653 0.926047i −0.464527 0.885559i \(-0.653776\pi\)
0.999180 0.0404876i \(-0.0128911\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −25.3308 −1.52750
\(276\) 0 0
\(277\) 1.45417 0.0873726 0.0436863 0.999045i \(-0.486090\pi\)
0.0436863 + 0.999045i \(0.486090\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −10.1482 + 17.5771i −0.605388 + 1.04856i 0.386602 + 0.922247i \(0.373649\pi\)
−0.991990 + 0.126316i \(0.959685\pi\)
\(282\) 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
\(289\) 1.64132 2.84284i 0.0965479 0.167226i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 3.53667 + 6.12569i 0.206614 + 0.357867i 0.950646 0.310278i \(-0.100422\pi\)
−0.744031 + 0.668145i \(0.767089\pi\)
\(294\) 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
\(305\) 0.658555 + 1.14065i 0.0377088 + 0.0653135i
\(306\) 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
\(325\) 12.0744 + 20.9135i 0.669768 + 1.16007i
\(326\) 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 0.986583 0.163262i \(-0.0522017\pi\)
−0.634681 + 0.772774i \(0.718868\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.83818 + 3.18381i −0.0995428 + 0.172413i
\(342\) 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 −0.999848 0.0174409i \(-0.994448\pi\)
0.515028 + 0.857173i \(0.327781\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −11.6979 −0.622618 −0.311309 0.950309i \(-0.600767\pi\)
−0.311309 + 0.950309i \(0.600767\pi\)
\(354\) 0 0
\(355\) −3.28590 −0.174397
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 17.8623 30.9383i 0.942734 1.63286i 0.182507 0.983205i \(-0.441579\pi\)
0.760227 0.649658i \(-0.225088\pi\)
\(360\) 0 0
\(361\) 2.80150 + 4.85235i 0.147448 + 0.255387i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −0.437618 0.757977i −0.0229060 0.0396743i
\(366\) 0 0
\(367\) −17.0539 −0.890207 −0.445103 0.895479i \(-0.646833\pi\)
−0.445103 + 0.895479i \(0.646833\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −25.9234 −1.34226 −0.671131 0.741339i \(-0.734191\pi\)
−0.671131 + 0.741339i \(0.734191\pi\)
\(374\) 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
\(383\) 24.8525 1.26991 0.634953 0.772551i \(-0.281020\pi\)
0.634953 + 0.772551i \(0.281020\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 18.2528 0.925454 0.462727 0.886501i \(-0.346871\pi\)
0.462727 + 0.886501i \(0.346871\pi\)
\(390\) 0 0
\(391\) 13.7427 + 23.8030i 0.694998 + 1.20377i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0.744325 + 1.28921i 0.0374511 + 0.0648671i
\(396\) 0 0
\(397\) 6.18715 10.7164i 0.310524 0.537843i −0.667952 0.744204i \(-0.732829\pi\)
0.978476 + 0.206361i \(0.0661622\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 10.9623 0.547429 0.273714 0.961811i \(-0.411748\pi\)
0.273714 + 0.961811i \(0.411748\pi\)
\(402\) 0 0
\(403\) 3.50481 0.174587
\(404\) 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
\(415\) −1.16019 + 2.00951i −0.0569515 + 0.0986429i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2.35705 4.08253i −0.115149 0.199445i 0.802690 0.596396i \(-0.203401\pi\)
−0.917839 + 0.396952i \(0.870068\pi\)
\(420\) 0 0
\(421\) −9.65856 + 16.7291i −0.470729 + 0.815327i −0.999440 0.0334755i \(-0.989342\pi\)
0.528710 + 0.848802i \(0.322676\pi\)
\(422\) 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
\(431\) −15.1397 26.2227i −0.729253 1.26310i −0.957199 0.289429i \(-0.906535\pi\)
0.227947 0.973674i \(-0.426799\pi\)
\(432\) 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
\(451\) −14.3571 24.8671i −0.676047 1.17095i
\(452\) 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 −0.0430418 0.999073i \(-0.513705\pi\)
0.886744 0.462261i \(-0.152962\pi\)
\(462\) 0 0
\(463\) −14.5253 25.1586i −0.675049 1.16922i −0.976455 0.215723i \(-0.930789\pi\)
0.301406 0.953496i \(-0.402544\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 18.7466 32.4701i 0.867491 1.50254i 0.00293952 0.999996i \(-0.499064\pi\)
0.864552 0.502544i \(-0.167602\pi\)
\(468\) 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
\(479\) 29.9097 1.36661 0.683305 0.730133i \(-0.260542\pi\)
0.683305 + 0.730133i \(0.260542\pi\)
\(480\) 0 0
\(481\) 22.4887 1.02539
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.05034 + 3.55130i −0.0931013 + 0.161256i
\(486\) 0 0
\(487\) −10.6316 18.4145i −0.481764 0.834439i 0.518017 0.855370i \(-0.326670\pi\)
−0.999781 + 0.0209309i \(0.993337\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −10.6985 18.5303i −0.482816 0.836262i 0.516989 0.855992i \(-0.327053\pi\)
−0.999805 + 0.0197296i \(0.993719\pi\)
\(492\) 0 0
\(493\) −12.8317 −0.577912
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 14.5653 0.652031 0.326015 0.945364i \(-0.394294\pi\)
0.326015 + 0.945364i \(0.394294\pi\)
\(500\) 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
\(509\) −19.2405 −0.852820 −0.426410 0.904530i \(-0.640222\pi\)
−0.426410 + 0.904530i \(0.640222\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3.06758 0.135174
\(516\) 0 0
\(517\) −11.1202 19.2608i −0.489068 0.847091i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −13.8743 24.0310i −0.607844 1.05282i −0.991595 0.129380i \(-0.958701\pi\)
0.383751 0.923436i \(-0.374632\pi\)
\(522\) 0 0
\(523\) −1.36840 + 2.37014i −0.0598360 + 0.103639i −0.894392 0.447285i \(-0.852391\pi\)
0.834556 + 0.550924i \(0.185724\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.65692 0.115737
\(528\) 0 0
\(529\) 32.0722 1.39444
\(530\) 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
\(541\) 5.98865 10.3726i 0.257472 0.445955i −0.708092 0.706120i \(-0.750444\pi\)
0.965564 + 0.260165i \(0.0837771\pi\)
\(542\) 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 −0.539928 0.841711i \(-0.681549\pi\)
0.998907 + 0.0467363i \(0.0148821\pi\)
\(548\) 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
\(557\) −15.9246 27.5823i −0.674748 1.16870i −0.976542 0.215325i \(-0.930919\pi\)
0.301794 0.953373i \(-0.402415\pi\)
\(558\) 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
\(577\) 6.50916 + 11.2742i 0.270980 + 0.469351i 0.969113 0.246617i \(-0.0793190\pi\)
−0.698133 + 0.715968i \(0.745986\pi\)
\(578\) 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 −0.987418 0.158134i \(-0.949452\pi\)
0.630657 + 0.776062i \(0.282786\pi\)
\(588\) 0 0
\(589\) 1.31285 + 2.27393i 0.0540952 + 0.0936957i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −6.20765 + 10.7520i −0.254918 + 0.441531i −0.964873 0.262716i \(-0.915382\pi\)
0.709955 + 0.704247i \(0.248715\pi\)
\(594\) 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 −0.544193 0.838960i \(-0.683164\pi\)
0.998657 + 0.0518055i \(0.0164976\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 3.64979 0.148385
\(606\) 0 0
\(607\) −22.0917 −0.896673 −0.448336 0.893865i \(-0.647983\pi\)
−0.448336 + 0.893865i \(0.647983\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −10.6014 + 18.3621i −0.428886 + 0.742852i
\(612\) 0 0
\(613\) 14.7632 + 25.5706i 0.596280 + 1.03279i 0.993365 + 0.115005i \(0.0366885\pi\)
−0.397085 + 0.917782i \(0.629978\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −5.01655 8.68892i −0.201959 0.349803i 0.747201 0.664598i \(-0.231397\pi\)
−0.949159 + 0.314796i \(0.898064\pi\)
\(618\) 0 0
\(619\) 38.2567 1.53767 0.768833 0.639450i \(-0.220838\pi\)
0.768833 + 0.639450i \(0.220838\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 24.1456 0.965823
\(626\) 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
\(635\) −4.01943 −0.159506
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 17.3729 0.686189 0.343094 0.939301i \(-0.388525\pi\)
0.343094 + 0.939301i \(0.388525\pi\)
\(642\) 0 0
\(643\) −9.47949 16.4190i −0.373835 0.647501i 0.616317 0.787498i \(-0.288624\pi\)
−0.990152 + 0.139997i \(0.955291\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −9.50972 16.4713i −0.373865 0.647554i 0.616291 0.787518i \(-0.288634\pi\)
−0.990157 + 0.139964i \(0.955301\pi\)
\(648\) 0 0
\(649\) −19.4201 + 33.6366i −0.762306 + 1.32035i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 7.18659 0.281233 0.140616 0.990064i \(-0.455092\pi\)
0.140616 + 0.990064i \(0.455092\pi\)
\(654\) 0 0
\(655\) 1.17154 0.0457758
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 12.7261 22.0423i 0.495740 0.858647i −0.504248 0.863559i \(-0.668230\pi\)
0.999988 + 0.00491209i \(0.00156357\pi\)
\(660\) 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
\(667\) −12.8554 + 22.2662i −0.497764 + 0.862152i
\(668\) 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 −0.729187 0.684314i \(-0.760102\pi\)
0.957227 + 0.289338i \(0.0934350\pi\)
\(674\) 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
\(683\) 1.79071 + 3.10160i 0.0685196 + 0.118679i 0.898250 0.439485i \(-0.144839\pi\)
−0.829730 + 0.558165i \(0.811506\pi\)
\(684\) 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
\(703\) 8.42395 + 14.5907i 0.317715 + 0.550299i
\(704\) 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
\(719\) −20.8376 + 36.0918i −0.777112 + 1.34600i 0.156488 + 0.987680i \(0.449983\pi\)
−0.933600 + 0.358318i \(0.883350\pi\)
\(720\) 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 0.382744 + 0.923854i \(0.374979\pi\)
−0.991453 + 0.130461i \(0.958354\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 46.2542 1.71077
\(732\) 0 0
\(733\) −9.29768 −0.343418 −0.171709 0.985148i \(-0.554929\pi\)
−0.171709 + 0.985148i \(0.554929\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.69166 2.93004i 0.0623130 0.107929i
\(738\) 0 0
\(739\) −5.68878 9.85326i −0.209265 0.362458i 0.742218 0.670158i \(-0.233774\pi\)
−0.951483 + 0.307701i \(0.900441\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.16182 2.01234i −0.0426232 0.0738256i 0.843927 0.536458i \(-0.180238\pi\)
−0.886550 + 0.462633i \(0.846905\pi\)
\(744\) 0 0
\(745\) −0.545830 −0.0199977
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −11.1338 −0.406278 −0.203139 0.979150i \(-0.565114\pi\)
−0.203139 + 0.979150i \(0.565114\pi\)
\(752\) 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
\(761\) −47.0255 −1.70467 −0.852336 0.522995i \(-0.824815\pi\)
−0.852336 + 0.522995i \(0.824815\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 37.0279 1.33700
\(768\) 0 0
\(769\) −3.30314 5.72121i −0.119114 0.206312i 0.800303 0.599596i \(-0.204672\pi\)
−0.919417 + 0.393284i \(0.871339\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −9.54351 16.5298i −0.343256 0.594537i 0.641779 0.766889i \(-0.278197\pi\)
−0.985035 + 0.172352i \(0.944863\pi\)
\(774\) 0 0
\(775\) 1.77292 3.07078i 0.0636850 0.110306i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −20.5081 −0.734778
\(780\) 0 0
\(781\) −70.4217 −2.51989
\(782\) 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
\(793\) 13.4552 23.3052i 0.477810 0.827591i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −4.38727 7.59898i −0.155405 0.269170i 0.777801 0.628510i \(-0.216335\pi\)
−0.933207 + 0.359341i \(0.883002\pi\)
\(798\) 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
\(809\) −4.75692 8.23923i −0.167244 0.289676i 0.770206 0.637796i \(-0.220154\pi\)
−0.937450 + 0.348120i \(0.886820\pi\)
\(810\) 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
\(829\) −8.77292 15.1951i −0.304696 0.527749i 0.672498 0.740099i \(-0.265222\pi\)
−0.977194 + 0.212350i \(0.931888\pi\)
\(830\) 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 −0.995556 0.0941668i \(-0.969981\pi\)
0.579329 + 0.815094i \(0.303315\pi\)
\(840\) 0 0
\(841\) 8.49837 + 14.7196i 0.293047 + 0.507572i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1.29957 + 2.25093i −0.0447067 + 0.0774342i
\(846\) 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 −0.440977 0.897518i \(-0.645368\pi\)
0.997762 0.0668621i \(-0.0212988\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −0.599740 −0.0204867 −0.0102434 0.999948i \(-0.503261\pi\)
−0.0102434 + 0.999948i \(0.503261\pi\)
\(858\) 0 0
\(859\) 26.4347 0.901942 0.450971 0.892539i \(-0.351078\pi\)
0.450971 + 0.892539i \(0.351078\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 9.92270 17.1866i 0.337773 0.585039i −0.646241 0.763134i \(-0.723660\pi\)
0.984013 + 0.178094i \(0.0569932\pi\)
\(864\) 0 0
\(865\) −3.02859 5.24567i −0.102975 0.178358i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 15.9520 + 27.6296i 0.541134 + 0.937271i
\(870\) 0 0
\(871\) −3.22545 −0.109290
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 20.4703 0.691234 0.345617 0.938376i \(-0.387670\pi\)
0.345617 + 0.938376i \(0.387670\pi\)
\(878\) 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
\(887\) −23.1650 −0.777805 −0.388902 0.921279i \(-0.627146\pi\)
−0.388902 + 0.921279i \(0.627146\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −15.8845 −0.531555
\(894\) 0 0
\(895\) 1.14132 + 1.97682i 0.0381500 + 0.0660777i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1.24269 + 2.15240i 0.0414460 + 0.0717866i
\(900\) 0 0
\(901\) −1.74596 + 3.02409i −0.0581664 + 0.100747i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2.95898 −0.0983599
\(906\) 0 0
\(907\) −50.0528 −1.66198 −0.830988 0.556290i \(-0.812224\pi\)
−0.830988 + 0.556290i \(0.812224\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −5.42231 + 9.39172i −0.179649 + 0.311161i −0.941760 0.336285i \(-0.890830\pi\)
0.762111 + 0.647446i \(0.224163\pi\)
\(912\) 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
\(919\) −5.59549 + 9.69166i −0.184578 + 0.319699i −0.943434 0.331560i \(-0.892425\pi\)
0.758856 + 0.651258i \(0.225759\pi\)
\(920\) 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
\(935\) −2.26935 3.93063i −0.0742156 0.128545i
\(936\) 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
\(955\) 1.57442 + 2.72698i 0.0509470 + 0.0882429i
\(956\) 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 0.999924 0.0123057i \(-0.00391714\pi\)
−0.510619 + 0.859807i \(0.670584\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −3.59329 + 6.22377i −0.115314 + 0.199730i −0.917905 0.396799i \(-0.870121\pi\)
0.802591 + 0.596530i \(0.203454\pi\)
\(972\) 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
\(983\) 4.41642 0.140862 0.0704310 0.997517i \(-0.477563\pi\)
0.0704310 + 0.997517i \(0.477563\pi\)
\(984\) 0 0
\(985\) 0.0344801 0.00109863
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 46.3396 80.2625i 1.47351 2.55220i
\(990\) 0 0
\(991\) 2.90671 + 5.03456i 0.0923345 + 0.159928i 0.908493 0.417900i \(-0.137234\pi\)
−0.816159 + 0.577828i \(0.803900\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −2.32777 4.03182i −0.0737953 0.127817i
\(996\) 0 0
\(997\) 52.6408 1.66715 0.833575 0.552407i \(-0.186290\pi\)
0.833575 + 0.552407i \(0.186290\pi\)
\(998\) 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.l.f.3313.2 6
3.2 odd 2 1764.2.l.f.961.1 6
7.2 even 3 5292.2.j.d.3529.2 6
7.3 odd 6 5292.2.i.f.2125.2 6
7.4 even 3 5292.2.i.e.2125.2 6
7.5 odd 6 756.2.j.b.505.2 6
7.6 odd 2 5292.2.l.e.3313.2 6
9.4 even 3 5292.2.i.e.1549.2 6
9.5 odd 6 1764.2.i.d.373.2 6
21.2 odd 6 1764.2.j.e.1177.3 6
21.5 even 6 252.2.j.a.169.1 yes 6
21.11 odd 6 1764.2.i.d.1537.2 6
21.17 even 6 1764.2.i.g.1537.2 6
21.20 even 2 1764.2.l.e.961.3 6
28.19 even 6 3024.2.r.j.2017.2 6
63.4 even 3 inner 5292.2.l.f.361.2 6
63.5 even 6 252.2.j.a.85.1 6
63.13 odd 6 5292.2.i.f.1549.2 6
63.23 odd 6 1764.2.j.e.589.3 6
63.31 odd 6 5292.2.l.e.361.2 6
63.32 odd 6 1764.2.l.f.949.1 6
63.40 odd 6 756.2.j.b.253.2 6
63.41 even 6 1764.2.i.g.373.2 6
63.47 even 6 2268.2.a.i.1.2 3
63.58 even 3 5292.2.j.d.1765.2 6
63.59 even 6 1764.2.l.e.949.3 6
63.61 odd 6 2268.2.a.h.1.2 3
84.47 odd 6 1008.2.r.j.673.3 6
252.47 odd 6 9072.2.a.by.1.2 3
252.103 even 6 3024.2.r.j.1009.2 6
252.131 odd 6 1008.2.r.j.337.3 6
252.187 even 6 9072.2.a.bv.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.j.a.85.1 6 63.5 even 6
252.2.j.a.169.1 yes 6 21.5 even 6
756.2.j.b.253.2 6 63.40 odd 6
756.2.j.b.505.2 6 7.5 odd 6
1008.2.r.j.337.3 6 252.131 odd 6
1008.2.r.j.673.3 6 84.47 odd 6
1764.2.i.d.373.2 6 9.5 odd 6
1764.2.i.d.1537.2 6 21.11 odd 6
1764.2.i.g.373.2 6 63.41 even 6
1764.2.i.g.1537.2 6 21.17 even 6
1764.2.j.e.589.3 6 63.23 odd 6
1764.2.j.e.1177.3 6 21.2 odd 6
1764.2.l.e.949.3 6 63.59 even 6
1764.2.l.e.961.3 6 21.20 even 2
1764.2.l.f.949.1 6 63.32 odd 6
1764.2.l.f.961.1 6 3.2 odd 2
2268.2.a.h.1.2 3 63.61 odd 6
2268.2.a.i.1.2 3 63.47 even 6
3024.2.r.j.1009.2 6 252.103 even 6
3024.2.r.j.2017.2 6 28.19 even 6
5292.2.i.e.1549.2 6 9.4 even 3
5292.2.i.e.2125.2 6 7.4 even 3
5292.2.i.f.1549.2 6 63.13 odd 6
5292.2.i.f.2125.2 6 7.3 odd 6
5292.2.j.d.1765.2 6 63.58 even 3
5292.2.j.d.3529.2 6 7.2 even 3
5292.2.l.e.361.2 6 63.31 odd 6
5292.2.l.e.3313.2 6 7.6 odd 2
5292.2.l.f.361.2 6 63.4 even 3 inner
5292.2.l.f.3313.2 6 1.1 even 1 trivial
9072.2.a.bv.1.2 3 252.187 even 6
9072.2.a.by.1.2 3 252.47 odd 6