Properties

Label 1764.2.l.f.961.1
Level $1764$
Weight $2$
Character 1764.961
Analytic conductor $14.086$
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] = [1764,2,Mod(949,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, 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("1764.949");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.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: \(14.0856109166\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 961.1
Root \(0.500000 - 1.41036i\) of defining polynomial
Character \(\chi\) \(=\) 1764.961
Dual form 1764.2.l.f.949.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.09097 - 1.34528i) q^{3} -0.239123 q^{5} +(-0.619562 + 2.93533i) q^{9} +O(q^{10})\) \(q+(-1.09097 - 1.34528i) q^{3} -0.239123 q^{5} +(-0.619562 + 2.93533i) q^{9} -5.12476 q^{11} +(-2.44282 - 4.23109i) q^{13} +(0.260877 + 0.321688i) q^{15} +(1.85185 + 3.20750i) q^{17} +(1.83009 - 3.16982i) q^{19} +7.42107 q^{23} -4.94282 q^{25} +(4.62476 - 2.36887i) q^{27} +(-1.73229 + 3.00041i) q^{29} +(-0.358685 + 0.621261i) q^{31} +(5.59097 + 6.89425i) q^{33} +(-2.30150 + 3.98632i) q^{37} +(-3.02696 + 7.90228i) q^{39} +(2.80150 + 4.85235i) q^{41} +(-6.24433 + 10.8155i) q^{43} +(0.148152 - 0.701905i) q^{45} +(2.16991 + 3.75839i) q^{47} +(2.29467 - 5.99054i) q^{51} +(0.471410 + 0.816506i) q^{53} +1.22545 q^{55} +(-6.26088 + 0.996189i) q^{57} +(3.78947 - 6.56355i) q^{59} +(2.75404 + 4.77014i) q^{61} +(0.584135 + 1.01175i) q^{65} +(0.330095 - 0.571741i) q^{67} +(-8.09617 - 9.98342i) q^{69} +13.7414 q^{71} +(-1.83009 - 3.16982i) q^{73} +(5.39248 + 6.64948i) q^{75} +(3.11273 + 5.39140i) q^{79} +(-8.23229 - 3.63723i) q^{81} +(4.85185 - 8.40365i) q^{83} +(-0.442820 - 0.766987i) q^{85} +(5.92627 - 0.942948i) q^{87} +(-3.74433 + 6.48536i) q^{89} +(1.22708 - 0.195246i) q^{93} +(-0.437618 + 0.757977i) q^{95} +(-8.57442 + 14.8513i) q^{97} +(3.17511 - 15.0429i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{3} - 2 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{3} - 2 q^{5} - 4 q^{9} + 4 q^{11} + 3 q^{13} + q^{15} + 2 q^{17} + 3 q^{19} + 28 q^{23} - 12 q^{25} - 7 q^{27} - q^{29} - 3 q^{31} + 25 q^{33} + 3 q^{37} + 18 q^{39} - 3 q^{43} + 10 q^{45} + 21 q^{47} - 13 q^{51} - 6 q^{53} - 12 q^{55} - 37 q^{57} + 31 q^{59} + 6 q^{61} - 15 q^{65} - 6 q^{67} - 5 q^{69} + 34 q^{71} - 3 q^{73} + 7 q^{75} + 9 q^{79} - 40 q^{81} + 20 q^{83} + 15 q^{85} - 16 q^{87} + 12 q^{89} + 33 q^{93} - 20 q^{95} - 9 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) −1.09097 1.34528i −0.629873 0.776698i
\(4\) 0 0
\(5\) −0.239123 −0.106939 −0.0534696 0.998569i \(-0.517028\pi\)
−0.0534696 + 0.998569i \(0.517028\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.619562 + 2.93533i −0.206521 + 0.978442i
\(10\) 0 0
\(11\) −5.12476 −1.54517 −0.772587 0.634909i \(-0.781038\pi\)
−0.772587 + 0.634909i \(0.781038\pi\)
\(12\) 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
\(15\) 0.260877 + 0.321688i 0.0673581 + 0.0830595i
\(16\) 0 0
\(17\) 1.85185 + 3.20750i 0.449139 + 0.777932i 0.998330 0.0577649i \(-0.0183974\pi\)
−0.549191 + 0.835697i \(0.685064\pi\)
\(18\) 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 \(-0.218404\pi\)
0.773700 + 0.633553i \(0.218404\pi\)
\(24\) 0 0
\(25\) −4.94282 −0.988564
\(26\) 0 0
\(27\) 4.62476 2.36887i 0.890036 0.455890i
\(28\) 0 0
\(29\) −1.73229 + 3.00041i −0.321678 + 0.557162i −0.980834 0.194844i \(-0.937580\pi\)
0.659157 + 0.752006i \(0.270913\pi\)
\(30\) 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
\(33\) 5.59097 + 6.89425i 0.973263 + 1.20013i
\(34\) 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
\(39\) −3.02696 + 7.90228i −0.484701 + 1.26538i
\(40\) 0 0
\(41\) 2.80150 + 4.85235i 0.437522 + 0.757810i 0.997498 0.0706992i \(-0.0225230\pi\)
−0.559976 + 0.828509i \(0.689190\pi\)
\(42\) 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
\(45\) 0.148152 0.701905i 0.0220851 0.104634i
\(46\) 0 0
\(47\) 2.16991 + 3.75839i 0.316513 + 0.548217i 0.979758 0.200186i \(-0.0641545\pi\)
−0.663245 + 0.748403i \(0.730821\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 2.29467 5.99054i 0.321318 0.838844i
\(52\) 0 0
\(53\) 0.471410 + 0.816506i 0.0647531 + 0.112156i 0.896584 0.442873i \(-0.146041\pi\)
−0.831831 + 0.555029i \(0.812707\pi\)
\(54\) 0 0
\(55\) 1.22545 0.165240
\(56\) 0 0
\(57\) −6.26088 + 0.996189i −0.829273 + 0.131948i
\(58\) 0 0
\(59\) 3.78947 6.56355i 0.493347 0.854501i −0.506624 0.862167i \(-0.669107\pi\)
0.999971 + 0.00766579i \(0.00244012\pi\)
\(60\) 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
\(69\) −8.09617 9.98342i −0.974665 1.20186i
\(70\) 0 0
\(71\) 13.7414 1.63081 0.815405 0.578891i \(-0.196514\pi\)
0.815405 + 0.578891i \(0.196514\pi\)
\(72\) 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
\(75\) 5.39248 + 6.64948i 0.622670 + 0.767816i
\(76\) 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
\(81\) −8.23229 3.63723i −0.914699 0.404137i
\(82\) 0 0
\(83\) 4.85185 8.40365i 0.532560 0.922420i −0.466718 0.884406i \(-0.654564\pi\)
0.999277 0.0380138i \(-0.0121031\pi\)
\(84\) 0 0
\(85\) −0.442820 0.766987i −0.0480306 0.0831914i
\(86\) 0 0
\(87\) 5.92627 0.942948i 0.635363 0.101095i
\(88\) 0 0
\(89\) −3.74433 + 6.48536i −0.396898 + 0.687447i −0.993341 0.115208i \(-0.963247\pi\)
0.596444 + 0.802655i \(0.296580\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1.22708 0.195246i 0.127243 0.0202460i
\(94\) 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
\(99\) 3.17511 15.0429i 0.319110 1.51186i
\(100\) 0 0
\(101\) −7.18194 −0.714630 −0.357315 0.933984i \(-0.616308\pi\)
−0.357315 + 0.933984i \(0.616308\pi\)
\(102\) 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 −0.988885 0.148681i \(-0.952497\pi\)
0.623204 + 0.782059i \(0.285830\pi\)
\(108\) 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
\(111\) 7.87360 1.25280i 0.747329 0.118910i
\(112\) 0 0
\(113\) 9.78495 + 16.9480i 0.920491 + 1.59434i 0.798657 + 0.601787i \(0.205544\pi\)
0.121834 + 0.992550i \(0.461122\pi\)
\(114\) 0 0
\(115\) −1.77455 −0.165478
\(116\) 0 0
\(117\) 13.9331 4.54906i 1.28812 0.420560i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 15.2632 1.38756
\(122\) 0 0
\(123\) 3.47141 9.06259i 0.313007 0.817146i
\(124\) 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
\(129\) 21.3623 3.39902i 1.88084 0.299267i
\(130\) 0 0
\(131\) −4.89931 −0.428055 −0.214027 0.976828i \(-0.568658\pi\)
−0.214027 + 0.976828i \(0.568658\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −1.10589 + 0.566453i −0.0951797 + 0.0487525i
\(136\) 0 0
\(137\) 5.44514 0.465210 0.232605 0.972571i \(-0.425275\pi\)
0.232605 + 0.972571i \(0.425275\pi\)
\(138\) 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
\(141\) 2.68878 7.01942i 0.226436 0.591142i
\(142\) 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 \(-0.470194\pi\)
0.0935002 + 0.995619i \(0.470194\pi\)
\(150\) 0 0
\(151\) 11.2632 0.916586 0.458293 0.888801i \(-0.348461\pi\)
0.458293 + 0.888801i \(0.348461\pi\)
\(152\) 0 0
\(153\) −10.5624 + 3.44854i −0.853918 + 0.278798i
\(154\) 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
\(159\) 0.584135 1.52496i 0.0463249 0.120938i
\(160\) 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
\(165\) −1.33693 1.64857i −0.104080 0.128341i
\(166\) 0 0
\(167\) 2.20370 + 3.81691i 0.170527 + 0.295362i 0.938604 0.344996i \(-0.112120\pi\)
−0.768077 + 0.640357i \(0.778786\pi\)
\(168\) 0 0
\(169\) −5.43474 + 9.41325i −0.418057 + 0.724096i
\(170\) 0 0
\(171\) 8.17059 + 7.33582i 0.624821 + 0.560984i
\(172\) 0 0
\(173\) 12.6654 + 21.9371i 0.962932 + 1.66785i 0.715072 + 0.699051i \(0.246394\pi\)
0.247860 + 0.968796i \(0.420273\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −12.9640 + 2.06275i −0.974435 + 0.155046i
\(178\) 0 0
\(179\) −4.77292 8.26693i −0.356744 0.617899i 0.630670 0.776051i \(-0.282780\pi\)
−0.987415 + 0.158151i \(0.949447\pi\)
\(180\) 0 0
\(181\) −12.3743 −0.919774 −0.459887 0.887978i \(-0.652110\pi\)
−0.459887 + 0.887978i \(0.652110\pi\)
\(182\) 0 0
\(183\) 3.41260 8.90904i 0.252266 0.658575i
\(184\) 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 0.523223 0.852196i \(-0.324729\pi\)
−0.999635 + 0.0270270i \(0.991396\pi\)
\(192\) 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
\(195\) 0.723815 1.88962i 0.0518335 0.135318i
\(196\) 0 0
\(197\) −0.144194 −0.0102734 −0.00513669 0.999987i \(-0.501635\pi\)
−0.00513669 + 0.999987i \(0.501635\pi\)
\(198\) 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
\(201\) −1.12928 + 0.179683i −0.0796530 + 0.0126739i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −0.669905 1.16031i −0.0467882 0.0810395i
\(206\) 0 0
\(207\) −4.59781 + 21.7833i −0.319570 + 1.51404i
\(208\) 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
\(213\) −14.9915 18.4861i −1.02720 1.26665i
\(214\) 0 0
\(215\) 1.49316 2.58623i 0.101833 0.176380i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −2.26771 + 5.92017i −0.153238 + 0.400048i
\(220\) 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
\(225\) 3.06238 14.5088i 0.204159 0.967253i
\(226\) 0 0
\(227\) −21.9428 −1.45640 −0.728198 0.685367i \(-0.759642\pi\)
−0.728198 + 0.685367i \(0.759642\pi\)
\(228\) 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 \(-0.513227\pi\)
0.886049 0.463592i \(-0.153440\pi\)
\(234\) 0 0
\(235\) −0.518875 0.898718i −0.0338477 0.0586259i
\(236\) 0 0
\(237\) 3.85705 10.0694i 0.250542 0.654075i
\(238\) 0 0
\(239\) 13.6488 + 23.6405i 0.882870 + 1.52918i 0.848136 + 0.529779i \(0.177725\pi\)
0.0347345 + 0.999397i \(0.488941\pi\)
\(240\) 0 0
\(241\) −10.0345 −0.646378 −0.323189 0.946334i \(-0.604755\pi\)
−0.323189 + 0.946334i \(0.604755\pi\)
\(242\) 0 0
\(243\) 4.08809 + 15.0429i 0.262251 + 0.965000i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −17.8824 −1.13783
\(248\) 0 0
\(249\) −16.5985 + 2.64104i −1.05189 + 0.167369i
\(250\) 0 0
\(251\) 28.3171 1.78736 0.893680 0.448705i \(-0.148115\pi\)
0.893680 + 0.448705i \(0.148115\pi\)
\(252\) 0 0
\(253\) −38.0312 −2.39100
\(254\) 0 0
\(255\) −0.548709 + 1.43248i −0.0343615 + 0.0897053i
\(256\) 0 0
\(257\) −28.8629 −1.80042 −0.900210 0.435455i \(-0.856587\pi\)
−0.900210 + 0.435455i \(0.856587\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −7.73392 6.94377i −0.478718 0.429808i
\(262\) 0 0
\(263\) −1.20929 −0.0745680 −0.0372840 0.999305i \(-0.511871\pi\)
−0.0372840 + 0.999305i \(0.511871\pi\)
\(264\) 0 0
\(265\) −0.112725 0.195246i −0.00692465 0.0119938i
\(266\) 0 0
\(267\) 12.8096 2.03818i 0.783934 0.124734i
\(268\) 0 0
\(269\) 4.50684 + 7.80607i 0.274787 + 0.475944i 0.970081 0.242780i \(-0.0780594\pi\)
−0.695295 + 0.718725i \(0.744726\pi\)
\(270\) 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
\(279\) −1.60138 1.43777i −0.0958718 0.0860768i
\(280\) 0 0
\(281\) 10.1482 17.5771i 0.605388 1.04856i −0.386602 0.922247i \(-0.626351\pi\)
0.991990 0.126316i \(-0.0403154\pi\)
\(282\) 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
\(285\) 1.49712 0.238212i 0.0886818 0.0141105i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.64132 2.84284i 0.0965479 0.167226i
\(290\) 0 0
\(291\) 29.3337 4.66738i 1.71957 0.273607i
\(292\) 0 0
\(293\) −3.53667 6.12569i −0.206614 0.357867i 0.744031 0.668145i \(-0.232911\pi\)
−0.950646 + 0.310278i \(0.899578\pi\)
\(294\) 0 0
\(295\) −0.906150 + 1.56950i −0.0527581 + 0.0913797i
\(296\) 0 0
\(297\) −23.7008 + 12.1399i −1.37526 + 0.704430i
\(298\) 0 0
\(299\) −18.1283 31.3992i −1.04839 1.81586i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 7.83530 + 9.66173i 0.450126 + 0.555052i
\(304\) 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
\(309\) −13.9955 17.2579i −0.796175 0.981767i
\(310\) 0 0
\(311\) −9.81191 + 16.9947i −0.556382 + 0.963682i 0.441412 + 0.897304i \(0.354478\pi\)
−0.997795 + 0.0663780i \(0.978856\pi\)
\(312\) 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 0.725973 0.687723i \(-0.241390\pi\)
−0.958572 + 0.284849i \(0.908056\pi\)
\(318\) 0 0
\(319\) 8.87756 15.3764i 0.497048 0.860912i
\(320\) 0 0
\(321\) 12.9406 2.05903i 0.722276 0.114924i
\(322\) 0 0
\(323\) 13.5562 0.754289
\(324\) 0 0
\(325\) 12.0744 + 20.9135i 0.669768 + 1.16007i
\(326\) 0 0
\(327\) 4.32489 11.2907i 0.239167 0.624378i
\(328\) 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
\(333\) −10.2752 9.22544i −0.563080 0.505551i
\(334\) 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
\(339\) 12.1248 31.6533i 0.658527 1.71917i
\(340\) 0 0
\(341\) 1.83818 3.18381i 0.0995428 0.172413i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 1.93598 + 2.38727i 0.104230 + 0.128526i
\(346\) 0 0
\(347\) −8.09329 + 14.0180i −0.434471 + 0.752526i −0.997252 0.0740802i \(-0.976398\pi\)
0.562781 + 0.826606i \(0.309731\pi\)
\(348\) 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
\(351\) −21.3204 13.7811i −1.13800 0.735578i
\(352\) 0 0
\(353\) 11.6979 0.622618 0.311309 0.950309i \(-0.399233\pi\)
0.311309 + 0.950309i \(0.399233\pi\)
\(354\) 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 \(0.558421\pi\)
−0.760227 + 0.649658i \(0.774912\pi\)
\(360\) 0 0
\(361\) 2.80150 + 4.85235i 0.147448 + 0.255387i
\(362\) 0 0
\(363\) −16.6517 20.5333i −0.873989 1.07772i
\(364\) 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
\(369\) −15.9789 + 5.21700i −0.831830 + 0.271586i
\(370\) 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
\(375\) −2.59385 3.19849i −0.133946 0.165169i
\(376\) 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
\(381\) 18.3382 + 22.6129i 0.939493 + 1.15849i
\(382\) 0 0
\(383\) −24.8525 −1.26991 −0.634953 0.772551i \(-0.718980\pi\)
−0.634953 + 0.772551i \(0.718980\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −27.8782 25.0300i −1.41713 1.27235i
\(388\) 0 0
\(389\) −18.2528 −0.925454 −0.462727 0.886501i \(-0.653129\pi\)
−0.462727 + 0.886501i \(0.653129\pi\)
\(390\) 0 0
\(391\) 13.7427 + 23.8030i 0.694998 + 1.20377i
\(392\) 0 0
\(393\) 5.34501 + 6.59095i 0.269620 + 0.332470i
\(394\) 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 \(-0.588252\pi\)
−0.273714 + 0.961811i \(0.588252\pi\)
\(402\) 0 0
\(403\) 3.50481 0.174587
\(404\) 0 0
\(405\) 1.96853 + 0.869747i 0.0978171 + 0.0432181i
\(406\) 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
\(411\) −5.94050 7.32525i −0.293023 0.361328i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −1.16019 + 2.00951i −0.0569515 + 0.0986429i
\(416\) 0 0
\(417\) 3.50684 9.15507i 0.171731 0.448326i
\(418\) 0 0
\(419\) 2.35705 + 4.08253i 0.115149 + 0.199445i 0.917839 0.396952i \(-0.129932\pi\)
−0.802690 + 0.596396i \(0.796599\pi\)
\(420\) 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
\(423\) −12.3765 + 4.04083i −0.601765 + 0.196472i
\(424\) 0 0
\(425\) −9.15335 15.8541i −0.444003 0.769036i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 15.5124 40.4973i 0.748947 1.95523i
\(430\) 0 0
\(431\) 15.1397 + 26.2227i 0.729253 + 1.26310i 0.957199 + 0.289429i \(0.0934655\pi\)
−0.227947 + 0.973674i \(0.573201\pi\)
\(432\) 0 0
\(433\) −34.2060 −1.64384 −0.821918 0.569606i \(-0.807096\pi\)
−0.821918 + 0.569606i \(0.807096\pi\)
\(434\) 0 0
\(435\) −1.41711 + 0.225481i −0.0679452 + 0.0108110i
\(436\) 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 0.920958 0.389661i \(-0.127408\pi\)
−0.797935 + 0.602743i \(0.794074\pi\)
\(444\) 0 0
\(445\) 0.895355 1.55080i 0.0424439 0.0735150i
\(446\) 0 0
\(447\) −2.49028 3.07078i −0.117786 0.145243i
\(448\) 0 0
\(449\) −10.4977 −0.495416 −0.247708 0.968835i \(-0.579677\pi\)
−0.247708 + 0.968835i \(0.579677\pi\)
\(450\) 0 0
\(451\) −14.3571 24.8671i −0.676047 1.17095i
\(452\) 0 0
\(453\) −12.2878 15.1522i −0.577333 0.711911i
\(454\) 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
\(459\) 16.1625 + 10.4471i 0.754402 + 0.487629i
\(460\) 0 0
\(461\) −18.1150 + 31.3762i −0.843702 + 1.46133i 0.0430418 + 0.999073i \(0.486295\pi\)
−0.886744 + 0.462261i \(0.847038\pi\)
\(462\) 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
\(465\) −0.293425 + 0.0466878i −0.0136072 + 0.00216509i
\(466\) 0 0
\(467\) −18.7466 + 32.4701i −0.867491 + 1.50254i −0.00293952 + 0.999996i \(0.500936\pi\)
−0.864552 + 0.502544i \(0.832398\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −9.48633 + 1.50940i −0.437107 + 0.0695495i
\(472\) 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
\(477\) −2.68878 + 0.877867i −0.123111 + 0.0401948i
\(478\) 0 0
\(479\) −29.9097 −1.36661 −0.683305 0.730133i \(-0.739458\pi\)
−0.683305 + 0.730133i \(0.739458\pi\)
\(480\) 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
\(489\) 11.3925 1.81270i 0.515186 0.0819729i
\(490\) 0 0
\(491\) 10.6985 + 18.5303i 0.482816 + 0.836262i 0.999805 0.0197296i \(-0.00628054\pi\)
−0.516989 + 0.855992i \(0.672947\pi\)
\(492\) 0 0
\(493\) −12.8317 −0.577912
\(494\) 0 0
\(495\) −0.759242 + 3.59710i −0.0341254 + 0.161677i
\(496\) 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
\(501\) 2.73065 7.12874i 0.121997 0.318488i
\(502\) 0 0
\(503\) −2.92339 −0.130348 −0.0651738 0.997874i \(-0.520760\pi\)
−0.0651738 + 0.997874i \(0.520760\pi\)
\(504\) 0 0
\(505\) 1.71737 0.0764220
\(506\) 0 0
\(507\) 18.5926 2.95833i 0.825727 0.131384i
\(508\) 0 0
\(509\) 19.2405 0.852820 0.426410 0.904530i \(-0.359778\pi\)
0.426410 + 0.904530i \(0.359778\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0.954858 18.9949i 0.0421580 0.838646i
\(514\) 0 0
\(515\) −3.06758 −0.135174
\(516\) 0 0
\(517\) −11.1202 19.2608i −0.489068 0.847091i
\(518\) 0 0
\(519\) 15.6940 40.9713i 0.688889 1.79844i
\(520\) 0 0
\(521\) 13.8743 + 24.0310i 0.607844 + 1.05282i 0.991595 + 0.129380i \(0.0412986\pi\)
−0.383751 + 0.923436i \(0.625368\pi\)
\(522\) 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
\(531\) 16.9184 + 15.1898i 0.734194 + 0.659183i
\(532\) 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
\(537\) −5.91423 + 15.4399i −0.255218 + 0.666281i
\(538\) 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
\(543\) 13.5000 + 16.6469i 0.579340 + 0.714387i
\(544\) 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
\(549\) −15.7082 + 5.12861i −0.670410 + 0.218884i
\(550\) 0 0
\(551\) 6.34050 + 10.9821i 0.270114 + 0.467852i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −1.88276 + 0.299573i −0.0799188 + 0.0127161i
\(556\) 0 0
\(557\) 15.9246 + 27.5823i 0.674748 + 1.16870i 0.976542 + 0.215325i \(0.0690812\pi\)
−0.301794 + 0.953373i \(0.597585\pi\)
\(558\) 0 0
\(559\) 61.0150 2.58066
\(560\) 0 0
\(561\) −11.7596 + 30.7001i −0.496492 + 1.29616i
\(562\) 0 0
\(563\) 17.7742 30.7857i 0.749091 1.29746i −0.199167 0.979966i \(-0.563824\pi\)
0.948259 0.317499i \(-0.102843\pi\)
\(564\) 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 0.542864 0.839821i \(-0.317340\pi\)
−0.998738 + 0.0502237i \(0.984007\pi\)
\(570\) 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
\(573\) −8.15856 + 21.2990i −0.340829 + 0.889779i
\(574\) 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
\(579\) 19.0705 3.03437i 0.792541 0.126104i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −2.41586 4.18440i −0.100055 0.173300i
\(584\) 0 0
\(585\) −3.33173 + 1.08778i −0.137750 + 0.0449744i
\(586\) 0 0
\(587\) 8.64364 14.9712i 0.356761 0.617928i −0.630657 0.776062i \(-0.717214\pi\)
0.987418 + 0.158134i \(0.0505477\pi\)
\(588\) 0 0
\(589\) 1.31285 + 2.27393i 0.0540952 + 0.0936957i
\(590\) 0 0
\(591\) 0.157311 + 0.193981i 0.00647092 + 0.00797931i
\(592\) 0 0
\(593\) 6.20765 10.7520i 0.254918 0.441531i −0.709955 0.704247i \(-0.751285\pi\)
0.964873 + 0.262716i \(0.0846182\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −12.0624 + 31.4905i −0.493680 + 1.28882i
\(598\) 0 0
\(599\) 3.94282 6.82916i 0.161099 0.279032i −0.774164 0.632985i \(-0.781829\pi\)
0.935263 + 0.353953i \(0.115163\pi\)
\(600\) 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
\(603\) 1.47373 + 1.32317i 0.0600151 + 0.0538835i
\(604\) 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
\(615\) −0.830095 + 2.16708i −0.0334727 + 0.0873849i
\(616\) 0 0
\(617\) 5.01655 + 8.68892i 0.201959 + 0.349803i 0.949159 0.314796i \(-0.101936\pi\)
−0.747201 + 0.664598i \(0.768603\pi\)
\(618\) 0 0
\(619\) 38.2567 1.53767 0.768833 0.639450i \(-0.220838\pi\)
0.768833 + 0.639450i \(0.220838\pi\)
\(620\) 0 0
\(621\) 34.3207 17.5796i 1.37724 0.705444i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 24.1456 0.965823
\(626\) 0 0
\(627\) 32.0855 5.10523i 1.28137 0.203883i
\(628\) 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
\(633\) 2.00039 5.22229i 0.0795084 0.207567i
\(634\) 0 0
\(635\) 4.01943 0.159506
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −8.51367 + 40.3356i −0.336796 + 1.59565i
\(640\) 0 0
\(641\) −17.3729 −0.686189 −0.343094 0.939301i \(-0.611475\pi\)
−0.343094 + 0.939301i \(0.611475\pi\)
\(642\) 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
\(645\) −5.10821 + 0.812785i −0.201136 + 0.0320034i
\(646\) 0 0
\(647\) 9.50972 + 16.4713i 0.373865 + 0.647554i 0.990157 0.139964i \(-0.0446988\pi\)
−0.616291 + 0.787518i \(0.711366\pi\)
\(648\) 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 \(-0.544908\pi\)
−0.140616 + 0.990064i \(0.544908\pi\)
\(654\) 0 0
\(655\) 1.17154 0.0457758
\(656\) 0 0
\(657\) 10.4383 3.40803i 0.407237 0.132960i
\(658\) 0 0
\(659\) −12.7261 + 22.0423i −0.495740 + 0.858647i −0.999988 0.00491209i \(-0.998436\pi\)
0.504248 + 0.863559i \(0.331770\pi\)
\(660\) 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
\(663\) −30.9520 + 4.92487i −1.20208 + 0.191266i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −12.8554 + 22.2662i −0.497764 + 0.862152i
\(668\) 0 0
\(669\) −35.5300 + 5.65329i −1.37367 + 0.218569i
\(670\) 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
\(675\) −22.8594 + 11.7089i −0.879858 + 0.450676i
\(676\) 0 0
\(677\) 6.80314 + 11.7834i 0.261466 + 0.452872i 0.966632 0.256170i \(-0.0824608\pi\)
−0.705166 + 0.709042i \(0.749127\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 23.9390 + 29.5193i 0.917344 + 1.13118i
\(682\) 0 0
\(683\) −1.79071 3.10160i −0.0685196 0.118679i 0.829730 0.558165i \(-0.188494\pi\)
−0.898250 + 0.439485i \(0.855161\pi\)
\(684\) 0 0
\(685\) −1.30206 −0.0497492
\(686\) 0 0
\(687\) 24.8428 + 30.6338i 0.947813 + 1.16875i
\(688\) 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
\(699\) −44.1004 + 7.01697i −1.66803 + 0.265406i
\(700\) 0 0
\(701\) −10.5926 −0.400077 −0.200039 0.979788i \(-0.564107\pi\)
−0.200039 + 0.979788i \(0.564107\pi\)
\(702\) 0 0
\(703\) 8.42395 + 14.5907i 0.317715 + 0.550299i
\(704\) 0 0
\(705\) −0.642950 + 1.67851i −0.0242149 + 0.0632163i
\(706\) 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
\(711\) −17.7540 + 5.79656i −0.665829 + 0.217388i
\(712\) 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
\(717\) 16.9126 44.1526i 0.631612 1.64891i
\(718\) 0 0
\(719\) 20.8376 36.0918i 0.777112 1.34600i −0.156488 0.987680i \(-0.550017\pi\)
0.933600 0.358318i \(-0.116650\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 10.9473 + 13.4992i 0.407136 + 0.502040i
\(724\) 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
\(729\) 15.7769 21.9110i 0.584329 0.811517i
\(730\) 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
\(741\) 19.5092 + 24.0568i 0.716687 + 0.883749i
\(742\) 0 0
\(743\) 1.16182 + 2.01234i 0.0426232 + 0.0738256i 0.886550 0.462633i \(-0.153095\pi\)
−0.843927 + 0.536458i \(0.819762\pi\)
\(744\) 0 0
\(745\) −0.545830 −0.0199977
\(746\) 0 0
\(747\) 21.6614 + 19.4483i 0.792550 + 0.711577i
\(748\) 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
\(753\) −30.8932 38.0945i −1.12581 1.38824i
\(754\) 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
\(759\) 41.4910 + 51.1627i 1.50603 + 1.85709i
\(760\) 0 0
\(761\) 47.0255 1.70467 0.852336 0.522995i \(-0.175185\pi\)
0.852336 + 0.522995i \(0.175185\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 2.52571 0.824626i 0.0913173 0.0298144i
\(766\) 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
\(771\) 31.4887 + 38.8288i 1.13404 + 1.39838i
\(772\) 0 0
\(773\) 9.54351 + 16.5298i 0.343256 + 0.594537i 0.985035 0.172352i \(-0.0551368\pi\)
−0.641779 + 0.766889i \(0.721803\pi\)
\(774\) 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
\(783\) −0.903827 + 17.9797i −0.0323001 + 0.642544i
\(784\) 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
\(789\) 1.31930 + 1.62683i 0.0469683 + 0.0579168i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 13.4552 23.3052i 0.477810 0.827591i
\(794\) 0 0
\(795\) −0.139680 + 0.364654i −0.00495395 + 0.0129330i
\(796\) 0 0
\(797\) 4.38727 + 7.59898i 0.155405 + 0.269170i 0.933207 0.359341i \(-0.116998\pi\)
−0.777801 + 0.628510i \(0.783665\pi\)
\(798\) 0 0
\(799\) −8.03667 + 13.9199i −0.284317 + 0.492451i
\(800\) 0 0
\(801\) −16.7168 15.0089i −0.590660 0.530313i
\(802\) 0 0
\(803\) 9.37880 + 16.2446i 0.330971 + 0.573258i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 5.58453 14.5792i 0.196585 0.513211i
\(808\) 0 0
\(809\) 4.75692 + 8.23923i 0.167244 + 0.289676i 0.937450 0.348120i \(-0.113180\pi\)
−0.770206 + 0.637796i \(0.779846\pi\)
\(810\) 0 0
\(811\) 25.0118 0.878282 0.439141 0.898418i \(-0.355283\pi\)
0.439141 + 0.898418i \(0.355283\pi\)
\(812\) 0 0
\(813\) −30.1105 + 4.79099i −1.05602 + 0.168027i
\(814\) 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 \(0.0466551\pi\)
−0.368158 + 0.929763i \(0.620012\pi\)
\(822\) 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
\(825\) −27.6352 34.0770i −0.962133 1.18641i
\(826\) 0 0
\(827\) −25.4531 −0.885090 −0.442545 0.896746i \(-0.645924\pi\)
−0.442545 + 0.896746i \(0.645924\pi\)
\(828\) 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
\(831\) −1.58646 1.95627i −0.0550336 0.0678622i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −0.526955 0.912713i −0.0182360 0.0315857i
\(836\) 0 0
\(837\) −0.187145 + 3.72286i −0.00646868 + 0.128681i
\(838\) 0 0
\(839\) 12.0562 20.8820i 0.416227 0.720927i −0.579329 0.815094i \(-0.696685\pi\)
0.995556 + 0.0941668i \(0.0300187\pi\)
\(840\) 0 0
\(841\) 8.49837 + 14.7196i 0.293047 + 0.507572i
\(842\) 0 0
\(843\) −34.7175 + 5.52402i −1.19573 + 0.190257i
\(844\) 0 0
\(845\) 1.29957 2.25093i 0.0447067 0.0774342i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 7.87360 1.25280i 0.270221 0.0429958i
\(850\) 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
\(855\) −1.95378 1.75417i −0.0668178 0.0599912i
\(856\) 0 0
\(857\) 0.599740 0.0204867 0.0102434 0.999948i \(-0.496739\pi\)
0.0102434 + 0.999948i \(0.496739\pi\)
\(858\) 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 −0.984013 0.178094i \(-0.943007\pi\)
0.646241 + 0.763134i \(0.276340\pi\)
\(864\) 0 0
\(865\) −3.02859 5.24567i −0.102975 0.178358i
\(866\) 0 0
\(867\) −5.61505 + 0.893429i −0.190697 + 0.0303424i
\(868\) 0 0
\(869\) −15.9520 27.6296i −0.541134 0.937271i
\(870\) 0 0
\(871\) −3.22545 −0.109290
\(872\) 0 0
\(873\) −38.2811 34.3700i −1.29562 1.16325i
\(874\) 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
\(879\) −4.38237 + 11.4408i −0.147814 + 0.385888i
\(880\) 0 0
\(881\) 31.1683 1.05009 0.525043 0.851076i \(-0.324049\pi\)
0.525043 + 0.851076i \(0.324049\pi\)
\(882\) 0 0
\(883\) −2.64187 −0.0889060 −0.0444530 0.999011i \(-0.514154\pi\)
−0.0444530 + 0.999011i \(0.514154\pi\)
\(884\) 0 0
\(885\) 3.10000 0.493251i 0.104205 0.0165805i
\(886\) 0 0
\(887\) 23.1650 0.777805 0.388902 0.921279i \(-0.372854\pi\)
0.388902 + 0.921279i \(0.372854\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 42.1885 + 18.6400i 1.41337 + 0.624462i
\(892\) 0 0
\(893\) 15.8845 0.531555
\(894\) 0 0
\(895\) 1.14132 + 1.97682i 0.0381500 + 0.0660777i
\(896\) 0 0
\(897\) −22.4632 + 58.6433i −0.750026 + 1.95804i
\(898\) 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
\(909\) 4.44966 21.0814i 0.147586 0.699224i
\(910\) 0 0
\(911\) 5.42231 9.39172i 0.179649 0.311161i −0.762111 0.647446i \(-0.775837\pi\)
0.941760 + 0.336285i \(0.109170\pi\)
\(912\) 0 0
\(913\) −24.8646 + 43.0667i −0.822897 + 1.42530i
\(914\) 0 0
\(915\) −0.816031 + 2.13036i −0.0269772 + 0.0704275i
\(916\) 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
\(921\) 17.1848 + 21.1907i 0.566259 + 0.698256i
\(922\) 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
\(927\) −7.94802 + 37.6557i −0.261047 + 1.23678i
\(928\) 0 0
\(929\) 20.6478 + 35.7630i 0.677431 + 1.17335i 0.975752 + 0.218879i \(0.0702401\pi\)
−0.298321 + 0.954466i \(0.596427\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 33.5672 5.34099i 1.09894 0.174856i
\(934\) 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
\(939\) −15.7898 + 41.2213i −0.515279 + 1.34521i
\(940\) 0 0
\(941\) 25.2112 43.6671i 0.821862 1.42351i −0.0824315 0.996597i \(-0.526269\pi\)
0.904294 0.426911i \(-0.140398\pi\)
\(942\) 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 0.653398 0.757014i \(-0.273343\pi\)
−0.982293 + 0.187352i \(0.940009\pi\)
\(948\) 0 0
\(949\) −8.94119 + 15.4866i −0.290243 + 0.502716i
\(950\) 0 0
\(951\) −5.13160 + 13.3967i −0.166404 + 0.434419i
\(952\) 0 0
\(953\) −29.3685 −0.951340 −0.475670 0.879624i \(-0.657794\pi\)
−0.475670 + 0.879624i \(0.657794\pi\)
\(954\) 0 0
\(955\) 1.57442 + 2.72698i 0.0509470 + 0.0882429i
\(956\) 0 0
\(957\) −30.3707 + 4.83239i −0.981746 + 0.156209i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 15.2427 + 26.4011i 0.491700 + 0.851649i
\(962\) 0 0
\(963\) −16.8878 15.1624i −0.544203 0.488603i
\(964\) 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
\(969\) −14.7895 18.2369i −0.475106 0.585855i
\(970\) 0 0
\(971\) 3.59329 6.22377i 0.115314 0.199730i −0.802591 0.596530i \(-0.796546\pi\)
0.917905 + 0.396799i \(0.129879\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 14.9617 39.0595i 0.479158 1.25091i
\(976\) 0 0
\(977\) −14.2713 + 24.7186i −0.456579 + 0.790818i −0.998777 0.0494328i \(-0.984259\pi\)
0.542199 + 0.840250i \(0.317592\pi\)
\(978\) 0 0
\(979\) 19.1888 33.2359i 0.613276 1.06223i
\(980\) 0 0
\(981\) −19.9075 + 6.49966i −0.635598 + 0.207518i
\(982\) 0 0
\(983\) −4.41642 −0.140862 −0.0704310 0.997517i \(-0.522437\pi\)
−0.0704310 + 0.997517i \(0.522437\pi\)
\(984\) 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
\(993\) 7.42270 19.3780i 0.235552 0.614941i
\(994\) 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
\(999\) −1.20082 + 23.8878i −0.0379922 + 0.755776i
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 1764.2.l.f.961.1 6
3.2 odd 2 5292.2.l.f.3313.2 6
7.2 even 3 1764.2.j.e.1177.3 6
7.3 odd 6 1764.2.i.g.1537.2 6
7.4 even 3 1764.2.i.d.1537.2 6
7.5 odd 6 252.2.j.a.169.1 yes 6
7.6 odd 2 1764.2.l.e.961.3 6
9.4 even 3 1764.2.i.d.373.2 6
9.5 odd 6 5292.2.i.e.1549.2 6
21.2 odd 6 5292.2.j.d.3529.2 6
21.5 even 6 756.2.j.b.505.2 6
21.11 odd 6 5292.2.i.e.2125.2 6
21.17 even 6 5292.2.i.f.2125.2 6
21.20 even 2 5292.2.l.e.3313.2 6
28.19 even 6 1008.2.r.j.673.3 6
63.4 even 3 inner 1764.2.l.f.949.1 6
63.5 even 6 756.2.j.b.253.2 6
63.13 odd 6 1764.2.i.g.373.2 6
63.23 odd 6 5292.2.j.d.1765.2 6
63.31 odd 6 1764.2.l.e.949.3 6
63.32 odd 6 5292.2.l.f.361.2 6
63.40 odd 6 252.2.j.a.85.1 6
63.41 even 6 5292.2.i.f.1549.2 6
63.47 even 6 2268.2.a.h.1.2 3
63.58 even 3 1764.2.j.e.589.3 6
63.59 even 6 5292.2.l.e.361.2 6
63.61 odd 6 2268.2.a.i.1.2 3
84.47 odd 6 3024.2.r.j.2017.2 6
252.47 odd 6 9072.2.a.bv.1.2 3
252.103 even 6 1008.2.r.j.337.3 6
252.131 odd 6 3024.2.r.j.1009.2 6
252.187 even 6 9072.2.a.by.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.j.a.85.1 6 63.40 odd 6
252.2.j.a.169.1 yes 6 7.5 odd 6
756.2.j.b.253.2 6 63.5 even 6
756.2.j.b.505.2 6 21.5 even 6
1008.2.r.j.337.3 6 252.103 even 6
1008.2.r.j.673.3 6 28.19 even 6
1764.2.i.d.373.2 6 9.4 even 3
1764.2.i.d.1537.2 6 7.4 even 3
1764.2.i.g.373.2 6 63.13 odd 6
1764.2.i.g.1537.2 6 7.3 odd 6
1764.2.j.e.589.3 6 63.58 even 3
1764.2.j.e.1177.3 6 7.2 even 3
1764.2.l.e.949.3 6 63.31 odd 6
1764.2.l.e.961.3 6 7.6 odd 2
1764.2.l.f.949.1 6 63.4 even 3 inner
1764.2.l.f.961.1 6 1.1 even 1 trivial
2268.2.a.h.1.2 3 63.47 even 6
2268.2.a.i.1.2 3 63.61 odd 6
3024.2.r.j.1009.2 6 252.131 odd 6
3024.2.r.j.2017.2 6 84.47 odd 6
5292.2.i.e.1549.2 6 9.5 odd 6
5292.2.i.e.2125.2 6 21.11 odd 6
5292.2.i.f.1549.2 6 63.41 even 6
5292.2.i.f.2125.2 6 21.17 even 6
5292.2.j.d.1765.2 6 63.23 odd 6
5292.2.j.d.3529.2 6 21.2 odd 6
5292.2.l.e.361.2 6 63.59 even 6
5292.2.l.e.3313.2 6 21.20 even 2
5292.2.l.f.361.2 6 63.32 odd 6
5292.2.l.f.3313.2 6 3.2 odd 2
9072.2.a.bv.1.2 3 252.47 odd 6
9072.2.a.by.1.2 3 252.187 even 6