Properties

Label 1764.2.bm.b.1685.3
Level $1764$
Weight $2$
Character 1764.1685
Analytic conductor $14.086$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,2,Mod(1685,1764)]
 
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, 1, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.1685");
 
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.bm (of order \(6\), 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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3x^{14} - 9x^{12} - 9x^{10} + 225x^{8} - 81x^{6} - 729x^{4} - 2187x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1685.3
Root \(-0.744857 - 1.56371i\) of defining polynomial
Character \(\chi\) \(=\) 1764.1685
Dual form 1764.2.bm.b.1697.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.744857 - 1.56371i) q^{3} -0.553827 q^{5} +(-1.89038 + 2.32948i) q^{9} +O(q^{10})\) \(q+(-0.744857 - 1.56371i) q^{3} -0.553827 q^{5} +(-1.89038 + 2.32948i) q^{9} -4.65896i q^{11} +(3.58265 - 2.06844i) q^{13} +(0.412522 + 0.866025i) q^{15} +(3.62264 + 6.27459i) q^{17} +(5.81722 + 3.35857i) q^{19} -5.60371i q^{23} -4.69328 q^{25} +(5.05069 + 1.22087i) q^{27} +(1.16599 + 0.673187i) q^{29} +(0.830741 + 0.479629i) q^{31} +(-7.28526 + 3.47026i) q^{33} +(3.53478 - 6.12241i) q^{37} +(-5.90301 - 4.06153i) q^{39} +(-2.39152 - 4.14224i) q^{41} +(-1.02846 + 1.78135i) q^{43} +(1.04694 - 1.29013i) q^{45} +(-4.90301 - 8.49226i) q^{47} +(7.11329 - 10.3384i) q^{51} +(-7.30235 + 4.21601i) q^{53} +2.58026i q^{55} +(0.918838 - 11.5981i) q^{57} +(3.89955 - 6.75422i) q^{59} +(5.37336 - 3.10231i) q^{61} +(-1.98417 + 1.14556i) q^{65} +(1.68814 - 2.92394i) q^{67} +(-8.76257 + 4.17396i) q^{69} -0.407556i q^{71} +(-7.47870 + 4.31783i) q^{73} +(3.49582 + 7.33892i) q^{75} +(-0.318176 - 0.551097i) q^{79} +(-1.85295 - 8.80719i) q^{81} +(2.78840 - 4.82965i) q^{83} +(-2.00632 - 3.47504i) q^{85} +(0.184170 - 2.32470i) q^{87} +(3.46568 - 6.00274i) q^{89} +(0.131217 - 1.65629i) q^{93} +(-3.22174 - 1.86007i) q^{95} +(7.48798 + 4.32318i) q^{97} +(10.8530 + 8.80719i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 6 q^{9} - 12 q^{15} + 16 q^{25} - 12 q^{29} - 2 q^{37} + 18 q^{39} + 4 q^{43} + 6 q^{51} - 36 q^{53} - 42 q^{57} + 24 q^{65} + 14 q^{67} + 20 q^{79} + 54 q^{81} + 6 q^{85} + 30 q^{93} + 60 q^{95} + 90 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) −0.744857 1.56371i −0.430043 0.902808i
\(4\) 0 0
\(5\) −0.553827 −0.247679 −0.123840 0.992302i \(-0.539521\pi\)
−0.123840 + 0.992302i \(0.539521\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −1.89038 + 2.32948i −0.630125 + 0.776493i
\(10\) 0 0
\(11\) 4.65896i 1.40473i −0.711817 0.702365i \(-0.752128\pi\)
0.711817 0.702365i \(-0.247872\pi\)
\(12\) 0 0
\(13\) 3.58265 2.06844i 0.993648 0.573683i 0.0872856 0.996183i \(-0.472181\pi\)
0.906363 + 0.422500i \(0.138847\pi\)
\(14\) 0 0
\(15\) 0.412522 + 0.866025i 0.106513 + 0.223607i
\(16\) 0 0
\(17\) 3.62264 + 6.27459i 0.878619 + 1.52181i 0.852857 + 0.522144i \(0.174868\pi\)
0.0257612 + 0.999668i \(0.491799\pi\)
\(18\) 0 0
\(19\) 5.81722 + 3.35857i 1.33456 + 0.770510i 0.985995 0.166774i \(-0.0533351\pi\)
0.348567 + 0.937284i \(0.386668\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.60371i 1.16845i −0.811590 0.584227i \(-0.801398\pi\)
0.811590 0.584227i \(-0.198602\pi\)
\(24\) 0 0
\(25\) −4.69328 −0.938655
\(26\) 0 0
\(27\) 5.05069 + 1.22087i 0.972006 + 0.234957i
\(28\) 0 0
\(29\) 1.16599 + 0.673187i 0.216520 + 0.125008i 0.604338 0.796728i \(-0.293438\pi\)
−0.387818 + 0.921736i \(0.626771\pi\)
\(30\) 0 0
\(31\) 0.830741 + 0.479629i 0.149206 + 0.0861438i 0.572744 0.819734i \(-0.305879\pi\)
−0.423539 + 0.905878i \(0.639212\pi\)
\(32\) 0 0
\(33\) −7.28526 + 3.47026i −1.26820 + 0.604094i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.53478 6.12241i 0.581114 1.00652i −0.414234 0.910170i \(-0.635950\pi\)
0.995348 0.0963482i \(-0.0307162\pi\)
\(38\) 0 0
\(39\) −5.90301 4.06153i −0.945238 0.650365i
\(40\) 0 0
\(41\) −2.39152 4.14224i −0.373493 0.646909i 0.616607 0.787271i \(-0.288507\pi\)
−0.990100 + 0.140362i \(0.955173\pi\)
\(42\) 0 0
\(43\) −1.02846 + 1.78135i −0.156839 + 0.271653i −0.933727 0.357986i \(-0.883464\pi\)
0.776888 + 0.629639i \(0.216797\pi\)
\(44\) 0 0
\(45\) 1.04694 1.29013i 0.156069 0.192321i
\(46\) 0 0
\(47\) −4.90301 8.49226i −0.715177 1.23872i −0.962891 0.269890i \(-0.913013\pi\)
0.247714 0.968833i \(-0.420321\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 7.11329 10.3384i 0.996060 1.44767i
\(52\) 0 0
\(53\) −7.30235 + 4.21601i −1.00305 + 0.579114i −0.909151 0.416468i \(-0.863268\pi\)
−0.0939038 + 0.995581i \(0.529935\pi\)
\(54\) 0 0
\(55\) 2.58026i 0.347922i
\(56\) 0 0
\(57\) 0.918838 11.5981i 0.121703 1.53621i
\(58\) 0 0
\(59\) 3.89955 6.75422i 0.507678 0.879325i −0.492282 0.870436i \(-0.663837\pi\)
0.999960 0.00888893i \(-0.00282947\pi\)
\(60\) 0 0
\(61\) 5.37336 3.10231i 0.687989 0.397211i −0.114869 0.993381i \(-0.536645\pi\)
0.802858 + 0.596170i \(0.203312\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.98417 + 1.14556i −0.246106 + 0.142089i
\(66\) 0 0
\(67\) 1.68814 2.92394i 0.206239 0.357217i −0.744288 0.667859i \(-0.767211\pi\)
0.950527 + 0.310642i \(0.100544\pi\)
\(68\) 0 0
\(69\) −8.76257 + 4.17396i −1.05489 + 0.502486i
\(70\) 0 0
\(71\) 0.407556i 0.0483680i −0.999708 0.0241840i \(-0.992301\pi\)
0.999708 0.0241840i \(-0.00769875\pi\)
\(72\) 0 0
\(73\) −7.47870 + 4.31783i −0.875316 + 0.505364i −0.869111 0.494617i \(-0.835308\pi\)
−0.00620487 + 0.999981i \(0.501975\pi\)
\(74\) 0 0
\(75\) 3.49582 + 7.33892i 0.403662 + 0.847426i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −0.318176 0.551097i −0.0357976 0.0620032i 0.847572 0.530681i \(-0.178064\pi\)
−0.883369 + 0.468678i \(0.844730\pi\)
\(80\) 0 0
\(81\) −1.85295 8.80719i −0.205884 0.978576i
\(82\) 0 0
\(83\) 2.78840 4.82965i 0.306066 0.530123i −0.671432 0.741066i \(-0.734320\pi\)
0.977498 + 0.210944i \(0.0676537\pi\)
\(84\) 0 0
\(85\) −2.00632 3.47504i −0.217616 0.376921i
\(86\) 0 0
\(87\) 0.184170 2.32470i 0.0197451 0.249235i
\(88\) 0 0
\(89\) 3.46568 6.00274i 0.367362 0.636289i −0.621790 0.783184i \(-0.713594\pi\)
0.989152 + 0.146895i \(0.0469278\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0.131217 1.65629i 0.0136065 0.171750i
\(94\) 0 0
\(95\) −3.22174 1.86007i −0.330543 0.190839i
\(96\) 0 0
\(97\) 7.48798 + 4.32318i 0.760289 + 0.438953i 0.829399 0.558656i \(-0.188683\pi\)
−0.0691107 + 0.997609i \(0.522016\pi\)
\(98\) 0 0
\(99\) 10.8530 + 8.80719i 1.09076 + 0.885156i
\(100\) 0 0
\(101\) −4.68453 −0.466128 −0.233064 0.972461i \(-0.574875\pi\)
−0.233064 + 0.972461i \(0.574875\pi\)
\(102\) 0 0
\(103\) 7.39937i 0.729082i −0.931187 0.364541i \(-0.881226\pi\)
0.931187 0.364541i \(-0.118774\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.38260 + 2.53029i 0.423682 + 0.244613i 0.696651 0.717410i \(-0.254673\pi\)
−0.272970 + 0.962023i \(0.588006\pi\)
\(108\) 0 0
\(109\) −5.88142 10.1869i −0.563337 0.975729i −0.997202 0.0747510i \(-0.976184\pi\)
0.433865 0.900978i \(-0.357150\pi\)
\(110\) 0 0
\(111\) −12.2066 0.967044i −1.15860 0.0917877i
\(112\) 0 0
\(113\) −1.51895 + 0.876965i −0.142891 + 0.0824979i −0.569741 0.821824i \(-0.692956\pi\)
0.426850 + 0.904322i \(0.359623\pi\)
\(114\) 0 0
\(115\) 3.10349i 0.289402i
\(116\) 0 0
\(117\) −1.95416 + 12.2559i −0.180662 + 1.13305i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.7059 −0.973264
\(122\) 0 0
\(123\) −4.69592 + 6.82503i −0.423417 + 0.615392i
\(124\) 0 0
\(125\) 5.36840 0.480164
\(126\) 0 0
\(127\) 11.0822 0.983385 0.491693 0.870769i \(-0.336378\pi\)
0.491693 + 0.870769i \(0.336378\pi\)
\(128\) 0 0
\(129\) 3.55157 + 0.281366i 0.312698 + 0.0247729i
\(130\) 0 0
\(131\) −18.4729 −1.61398 −0.806990 0.590565i \(-0.798905\pi\)
−0.806990 + 0.590565i \(0.798905\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −2.79721 0.676152i −0.240746 0.0581939i
\(136\) 0 0
\(137\) 22.0013i 1.87970i −0.341585 0.939851i \(-0.610964\pi\)
0.341585 0.939851i \(-0.389036\pi\)
\(138\) 0 0
\(139\) −8.55986 + 4.94204i −0.726038 + 0.419178i −0.816971 0.576679i \(-0.804348\pi\)
0.0909332 + 0.995857i \(0.471015\pi\)
\(140\) 0 0
\(141\) −9.62739 + 13.9924i −0.810773 + 1.17837i
\(142\) 0 0
\(143\) −9.63680 16.6914i −0.805870 1.39581i
\(144\) 0 0
\(145\) −0.645760 0.372830i −0.0536274 0.0309618i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.35171i 0.602275i −0.953581 0.301138i \(-0.902634\pi\)
0.953581 0.301138i \(-0.0973664\pi\)
\(150\) 0 0
\(151\) −4.33199 −0.352532 −0.176266 0.984343i \(-0.556402\pi\)
−0.176266 + 0.984343i \(0.556402\pi\)
\(152\) 0 0
\(153\) −21.4647 3.42248i −1.73532 0.276691i
\(154\) 0 0
\(155\) −0.460087 0.265632i −0.0369551 0.0213360i
\(156\) 0 0
\(157\) 1.54152 + 0.889998i 0.123027 + 0.0710296i 0.560251 0.828323i \(-0.310705\pi\)
−0.437224 + 0.899353i \(0.644038\pi\)
\(158\) 0 0
\(159\) 12.0318 + 8.27842i 0.954185 + 0.656522i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.81818 4.88122i 0.220737 0.382327i −0.734295 0.678830i \(-0.762487\pi\)
0.955032 + 0.296503i \(0.0958206\pi\)
\(164\) 0 0
\(165\) 4.03478 1.92192i 0.314107 0.149622i
\(166\) 0 0
\(167\) −2.38803 4.13618i −0.184791 0.320067i 0.758715 0.651423i \(-0.225827\pi\)
−0.943506 + 0.331355i \(0.892494\pi\)
\(168\) 0 0
\(169\) 2.05692 3.56270i 0.158225 0.274054i
\(170\) 0 0
\(171\) −18.8205 + 7.20213i −1.43924 + 0.550761i
\(172\) 0 0
\(173\) −2.83766 4.91497i −0.215743 0.373678i 0.737759 0.675064i \(-0.235884\pi\)
−0.953502 + 0.301386i \(0.902551\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −13.4662 1.06684i −1.01219 0.0801885i
\(178\) 0 0
\(179\) −10.9383 + 6.31525i −0.817570 + 0.472024i −0.849578 0.527463i \(-0.823143\pi\)
0.0320079 + 0.999488i \(0.489810\pi\)
\(180\) 0 0
\(181\) 22.7424i 1.69043i 0.534426 + 0.845215i \(0.320528\pi\)
−0.534426 + 0.845215i \(0.679472\pi\)
\(182\) 0 0
\(183\) −8.85351 6.09160i −0.654470 0.450304i
\(184\) 0 0
\(185\) −1.95766 + 3.39076i −0.143930 + 0.249294i
\(186\) 0 0
\(187\) 29.2331 16.8777i 2.13773 1.23422i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.0537 5.80452i 0.727462 0.420000i −0.0900309 0.995939i \(-0.528697\pi\)
0.817493 + 0.575939i \(0.195363\pi\)
\(192\) 0 0
\(193\) −3.16599 + 5.48366i −0.227893 + 0.394723i −0.957184 0.289482i \(-0.906517\pi\)
0.729290 + 0.684204i \(0.239850\pi\)
\(194\) 0 0
\(195\) 3.26925 + 2.24939i 0.234116 + 0.161082i
\(196\) 0 0
\(197\) 3.10030i 0.220888i −0.993882 0.110444i \(-0.964773\pi\)
0.993882 0.110444i \(-0.0352272\pi\)
\(198\) 0 0
\(199\) 13.6198 7.86341i 0.965484 0.557422i 0.0676272 0.997711i \(-0.478457\pi\)
0.897856 + 0.440288i \(0.145124\pi\)
\(200\) 0 0
\(201\) −5.82962 0.461841i −0.411190 0.0325758i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 1.32449 + 2.29409i 0.0925065 + 0.160226i
\(206\) 0 0
\(207\) 13.0537 + 10.5931i 0.907297 + 0.736273i
\(208\) 0 0
\(209\) 15.6475 27.1022i 1.08236 1.87470i
\(210\) 0 0
\(211\) 3.51263 + 6.08406i 0.241820 + 0.418844i 0.961233 0.275739i \(-0.0889225\pi\)
−0.719413 + 0.694582i \(0.755589\pi\)
\(212\) 0 0
\(213\) −0.637299 + 0.303571i −0.0436670 + 0.0208003i
\(214\) 0 0
\(215\) 0.569590 0.986559i 0.0388457 0.0672828i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 12.3224 + 8.47836i 0.832671 + 0.572914i
\(220\) 0 0
\(221\) 25.9573 + 14.9864i 1.74608 + 1.00810i
\(222\) 0 0
\(223\) 21.3477 + 12.3251i 1.42955 + 0.825350i 0.997085 0.0763008i \(-0.0243109\pi\)
0.432464 + 0.901651i \(0.357644\pi\)
\(224\) 0 0
\(225\) 8.87206 10.9329i 0.591470 0.728859i
\(226\) 0 0
\(227\) 20.0743 1.33238 0.666190 0.745782i \(-0.267924\pi\)
0.666190 + 0.745782i \(0.267924\pi\)
\(228\) 0 0
\(229\) 7.80422i 0.515717i 0.966183 + 0.257859i \(0.0830169\pi\)
−0.966183 + 0.257859i \(0.916983\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.89843 3.98281i −0.451931 0.260922i 0.256714 0.966487i \(-0.417360\pi\)
−0.708645 + 0.705565i \(0.750693\pi\)
\(234\) 0 0
\(235\) 2.71542 + 4.70325i 0.177135 + 0.306806i
\(236\) 0 0
\(237\) −0.624760 + 0.908023i −0.0405825 + 0.0589824i
\(238\) 0 0
\(239\) 23.2059 13.3979i 1.50107 0.866640i 0.501066 0.865409i \(-0.332941\pi\)
0.999999 0.00123146i \(-0.000391987\pi\)
\(240\) 0 0
\(241\) 0.874407i 0.0563255i −0.999603 0.0281628i \(-0.991034\pi\)
0.999603 0.0281628i \(-0.00896567\pi\)
\(242\) 0 0
\(243\) −12.3917 + 9.45758i −0.794928 + 0.606704i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 27.7881 1.76811
\(248\) 0 0
\(249\) −9.62912 0.762849i −0.610221 0.0483436i
\(250\) 0 0
\(251\) −2.32082 −0.146489 −0.0732445 0.997314i \(-0.523335\pi\)
−0.0732445 + 0.997314i \(0.523335\pi\)
\(252\) 0 0
\(253\) −26.1075 −1.64136
\(254\) 0 0
\(255\) −3.93954 + 5.72570i −0.246703 + 0.358558i
\(256\) 0 0
\(257\) −24.6035 −1.53472 −0.767361 0.641215i \(-0.778431\pi\)
−0.767361 + 0.641215i \(0.778431\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −3.77234 + 1.44358i −0.233502 + 0.0893556i
\(262\) 0 0
\(263\) 23.7024i 1.46155i 0.682619 + 0.730775i \(0.260841\pi\)
−0.682619 + 0.730775i \(0.739159\pi\)
\(264\) 0 0
\(265\) 4.04424 2.33494i 0.248436 0.143434i
\(266\) 0 0
\(267\) −11.9680 0.948141i −0.732429 0.0580253i
\(268\) 0 0
\(269\) 10.3106 + 17.8585i 0.628648 + 1.08885i 0.987823 + 0.155580i \(0.0497245\pi\)
−0.359176 + 0.933270i \(0.616942\pi\)
\(270\) 0 0
\(271\) 12.0771 + 6.97270i 0.733630 + 0.423562i 0.819749 0.572723i \(-0.194113\pi\)
−0.0861186 + 0.996285i \(0.527446\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 21.8658i 1.31856i
\(276\) 0 0
\(277\) 20.7881 1.24904 0.624518 0.781011i \(-0.285296\pi\)
0.624518 + 0.781011i \(0.285296\pi\)
\(278\) 0 0
\(279\) −2.68770 + 1.02852i −0.160908 + 0.0615757i
\(280\) 0 0
\(281\) −20.3371 11.7416i −1.21321 0.700448i −0.249754 0.968309i \(-0.580350\pi\)
−0.963457 + 0.267862i \(0.913683\pi\)
\(282\) 0 0
\(283\) 11.1906 + 6.46089i 0.665211 + 0.384060i 0.794260 0.607578i \(-0.207859\pi\)
−0.129048 + 0.991638i \(0.541192\pi\)
\(284\) 0 0
\(285\) −0.508878 + 6.42335i −0.0301433 + 0.380486i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −17.7470 + 30.7387i −1.04394 + 1.80816i
\(290\) 0 0
\(291\) 1.18274 14.9292i 0.0693332 0.875164i
\(292\) 0 0
\(293\) −1.99115 3.44878i −0.116324 0.201480i 0.801984 0.597346i \(-0.203778\pi\)
−0.918308 + 0.395866i \(0.870445\pi\)
\(294\) 0 0
\(295\) −2.15968 + 3.74067i −0.125741 + 0.217790i
\(296\) 0 0
\(297\) 5.68799 23.5310i 0.330051 1.36541i
\(298\) 0 0
\(299\) −11.5910 20.0761i −0.670322 1.16103i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 3.48931 + 7.32525i 0.200455 + 0.420824i
\(304\) 0 0
\(305\) −2.97592 + 1.71815i −0.170401 + 0.0983808i
\(306\) 0 0
\(307\) 22.3162i 1.27365i −0.771008 0.636825i \(-0.780247\pi\)
0.771008 0.636825i \(-0.219753\pi\)
\(308\) 0 0
\(309\) −11.5705 + 5.51147i −0.658221 + 0.313537i
\(310\) 0 0
\(311\) −8.18371 + 14.1746i −0.464056 + 0.803768i −0.999158 0.0410190i \(-0.986940\pi\)
0.535103 + 0.844787i \(0.320273\pi\)
\(312\) 0 0
\(313\) 16.5547 9.55785i 0.935726 0.540242i 0.0471079 0.998890i \(-0.485000\pi\)
0.888618 + 0.458648i \(0.151666\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −20.8798 + 12.0550i −1.17273 + 0.677074i −0.954321 0.298784i \(-0.903419\pi\)
−0.218405 + 0.975858i \(0.570086\pi\)
\(318\) 0 0
\(319\) 3.13635 5.43232i 0.175602 0.304152i
\(320\) 0 0
\(321\) 0.692237 8.73781i 0.0386369 0.487697i
\(322\) 0 0
\(323\) 48.6676i 2.70794i
\(324\) 0 0
\(325\) −16.8144 + 9.70778i −0.932693 + 0.538491i
\(326\) 0 0
\(327\) −11.5486 + 16.7846i −0.638637 + 0.928191i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −5.70077 9.87403i −0.313343 0.542726i 0.665741 0.746183i \(-0.268116\pi\)
−0.979084 + 0.203457i \(0.934782\pi\)
\(332\) 0 0
\(333\) 7.57998 + 19.8079i 0.415380 + 1.08546i
\(334\) 0 0
\(335\) −0.934938 + 1.61936i −0.0510811 + 0.0884751i
\(336\) 0 0
\(337\) 2.16599 + 3.75161i 0.117989 + 0.204363i 0.918971 0.394326i \(-0.129022\pi\)
−0.800981 + 0.598689i \(0.795689\pi\)
\(338\) 0 0
\(339\) 2.50272 + 1.72198i 0.135929 + 0.0935251i
\(340\) 0 0
\(341\) 2.23457 3.87039i 0.121009 0.209593i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 4.85295 2.31165i 0.261274 0.124455i
\(346\) 0 0
\(347\) 4.62254 + 2.66882i 0.248151 + 0.143270i 0.618917 0.785456i \(-0.287572\pi\)
−0.370766 + 0.928726i \(0.620905\pi\)
\(348\) 0 0
\(349\) 8.78031 + 5.06931i 0.469999 + 0.271354i 0.716239 0.697855i \(-0.245862\pi\)
−0.246240 + 0.969209i \(0.579195\pi\)
\(350\) 0 0
\(351\) 20.6202 6.07312i 1.10062 0.324159i
\(352\) 0 0
\(353\) 15.3513 0.817067 0.408533 0.912743i \(-0.366040\pi\)
0.408533 + 0.912743i \(0.366040\pi\)
\(354\) 0 0
\(355\) 0.225715i 0.0119797i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.51381 0.874000i −0.0798960 0.0461280i 0.459520 0.888168i \(-0.348022\pi\)
−0.539416 + 0.842040i \(0.681355\pi\)
\(360\) 0 0
\(361\) 13.0600 + 22.6207i 0.687371 + 1.19056i
\(362\) 0 0
\(363\) 7.97437 + 16.7409i 0.418546 + 0.878671i
\(364\) 0 0
\(365\) 4.14191 2.39133i 0.216798 0.125168i
\(366\) 0 0
\(367\) 12.9004i 0.673393i 0.941613 + 0.336697i \(0.109310\pi\)
−0.941613 + 0.336697i \(0.890690\pi\)
\(368\) 0 0
\(369\) 14.1702 + 2.25939i 0.737669 + 0.117619i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −10.6956 −0.553799 −0.276900 0.960899i \(-0.589307\pi\)
−0.276900 + 0.960899i \(0.589307\pi\)
\(374\) 0 0
\(375\) −3.99869 8.39462i −0.206492 0.433496i
\(376\) 0 0
\(377\) 5.56980 0.286859
\(378\) 0 0
\(379\) 0.566797 0.0291144 0.0145572 0.999894i \(-0.495366\pi\)
0.0145572 + 0.999894i \(0.495366\pi\)
\(380\) 0 0
\(381\) −8.25464 17.3293i −0.422898 0.887808i
\(382\) 0 0
\(383\) 18.8612 0.963761 0.481880 0.876237i \(-0.339954\pi\)
0.481880 + 0.876237i \(0.339954\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2.20543 5.76320i −0.112109 0.292960i
\(388\) 0 0
\(389\) 16.5332i 0.838267i −0.907925 0.419133i \(-0.862334\pi\)
0.907925 0.419133i \(-0.137666\pi\)
\(390\) 0 0
\(391\) 35.1610 20.3002i 1.77817 1.02663i
\(392\) 0 0
\(393\) 13.7596 + 28.8862i 0.694082 + 1.45712i
\(394\) 0 0
\(395\) 0.176215 + 0.305213i 0.00886632 + 0.0153569i
\(396\) 0 0
\(397\) 29.5384 + 17.0540i 1.48249 + 0.855917i 0.999802 0.0198756i \(-0.00632700\pi\)
0.482688 + 0.875792i \(0.339660\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 21.5230i 1.07481i 0.843325 + 0.537404i \(0.180595\pi\)
−0.843325 + 0.537404i \(0.819405\pi\)
\(402\) 0 0
\(403\) 3.96834 0.197677
\(404\) 0 0
\(405\) 1.02622 + 4.87766i 0.0509931 + 0.242373i
\(406\) 0 0
\(407\) −28.5241 16.4684i −1.41389 0.816308i
\(408\) 0 0
\(409\) −19.2516 11.1149i −0.951933 0.549599i −0.0582520 0.998302i \(-0.518553\pi\)
−0.893681 + 0.448703i \(0.851886\pi\)
\(410\) 0 0
\(411\) −34.4037 + 16.3879i −1.69701 + 0.808353i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −1.54429 + 2.67479i −0.0758063 + 0.131300i
\(416\) 0 0
\(417\) 14.1038 + 9.70402i 0.690665 + 0.475208i
\(418\) 0 0
\(419\) −6.81490 11.8038i −0.332930 0.576651i 0.650155 0.759802i \(-0.274704\pi\)
−0.983085 + 0.183150i \(0.941371\pi\)
\(420\) 0 0
\(421\) 13.7071 23.7414i 0.668043 1.15708i −0.310408 0.950603i \(-0.600466\pi\)
0.978451 0.206480i \(-0.0662009\pi\)
\(422\) 0 0
\(423\) 29.0511 + 4.63211i 1.41251 + 0.225221i
\(424\) 0 0
\(425\) −17.0020 29.4484i −0.824720 1.42846i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −18.9225 + 27.5019i −0.913587 + 1.32780i
\(430\) 0 0
\(431\) 30.1663 17.4165i 1.45306 0.838925i 0.454406 0.890795i \(-0.349851\pi\)
0.998654 + 0.0518699i \(0.0165181\pi\)
\(432\) 0 0
\(433\) 19.5648i 0.940223i 0.882607 + 0.470112i \(0.155786\pi\)
−0.882607 + 0.470112i \(0.844214\pi\)
\(434\) 0 0
\(435\) −0.101999 + 1.28749i −0.00489046 + 0.0617302i
\(436\) 0 0
\(437\) 18.8205 32.5980i 0.900305 1.55937i
\(438\) 0 0
\(439\) −24.7361 + 14.2814i −1.18059 + 0.681614i −0.956151 0.292874i \(-0.905388\pi\)
−0.224439 + 0.974488i \(0.572055\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 21.8612 12.6216i 1.03866 0.599669i 0.119205 0.992870i \(-0.461966\pi\)
0.919453 + 0.393201i \(0.128632\pi\)
\(444\) 0 0
\(445\) −1.91939 + 3.32448i −0.0909878 + 0.157596i
\(446\) 0 0
\(447\) −11.4959 + 5.47597i −0.543739 + 0.259005i
\(448\) 0 0
\(449\) 15.1042i 0.712811i 0.934331 + 0.356405i \(0.115998\pi\)
−0.934331 + 0.356405i \(0.884002\pi\)
\(450\) 0 0
\(451\) −19.2985 + 11.1420i −0.908733 + 0.524657i
\(452\) 0 0
\(453\) 3.22671 + 6.77397i 0.151604 + 0.318269i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.75138 3.03348i −0.0819261 0.141900i 0.822151 0.569269i \(-0.192774\pi\)
−0.904077 + 0.427369i \(0.859440\pi\)
\(458\) 0 0
\(459\) 10.6364 + 36.1138i 0.496462 + 1.68565i
\(460\) 0 0
\(461\) 1.98765 3.44272i 0.0925743 0.160343i −0.816019 0.578025i \(-0.803824\pi\)
0.908594 + 0.417681i \(0.137157\pi\)
\(462\) 0 0
\(463\) −5.18494 8.98058i −0.240965 0.417363i 0.720025 0.693948i \(-0.244130\pi\)
−0.960989 + 0.276585i \(0.910797\pi\)
\(464\) 0 0
\(465\) −0.0726714 + 0.917300i −0.00337006 + 0.0425388i
\(466\) 0 0
\(467\) 9.80952 16.9906i 0.453930 0.786230i −0.544696 0.838634i \(-0.683355\pi\)
0.998626 + 0.0524035i \(0.0166882\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0.243486 3.07342i 0.0112192 0.141615i
\(472\) 0 0
\(473\) 8.29923 + 4.79156i 0.381599 + 0.220316i
\(474\) 0 0
\(475\) −27.3018 15.7627i −1.25269 0.723243i
\(476\) 0 0
\(477\) 3.98307 24.9805i 0.182372 1.14378i
\(478\) 0 0
\(479\) −31.6474 −1.44600 −0.723002 0.690846i \(-0.757238\pi\)
−0.723002 + 0.690846i \(0.757238\pi\)
\(480\) 0 0
\(481\) 29.2460i 1.33350i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4.14705 2.39430i −0.188308 0.108719i
\(486\) 0 0
\(487\) 9.72923 + 16.8515i 0.440874 + 0.763616i 0.997755 0.0669768i \(-0.0213353\pi\)
−0.556881 + 0.830592i \(0.688002\pi\)
\(488\) 0 0
\(489\) −9.73196 0.770996i −0.440094 0.0348656i
\(490\) 0 0
\(491\) 23.9525 13.8290i 1.08096 0.624094i 0.149806 0.988715i \(-0.452135\pi\)
0.931156 + 0.364622i \(0.118802\pi\)
\(492\) 0 0
\(493\) 9.75485i 0.439336i
\(494\) 0 0
\(495\) −6.01066 4.87766i −0.270159 0.219235i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −9.01263 −0.403461 −0.201730 0.979441i \(-0.564656\pi\)
−0.201730 + 0.979441i \(0.564656\pi\)
\(500\) 0 0
\(501\) −4.68905 + 6.81504i −0.209491 + 0.304474i
\(502\) 0 0
\(503\) −27.1572 −1.21088 −0.605440 0.795891i \(-0.707003\pi\)
−0.605440 + 0.795891i \(0.707003\pi\)
\(504\) 0 0
\(505\) 2.59442 0.115450
\(506\) 0 0
\(507\) −7.10314 0.562733i −0.315461 0.0249918i
\(508\) 0 0
\(509\) 13.1360 0.582242 0.291121 0.956686i \(-0.405972\pi\)
0.291121 + 0.956686i \(0.405972\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 25.2806 + 24.0652i 1.11617 + 1.06250i
\(514\) 0 0
\(515\) 4.09797i 0.180578i
\(516\) 0 0
\(517\) −39.5651 + 22.8429i −1.74007 + 1.00463i
\(518\) 0 0
\(519\) −5.57193 + 8.09822i −0.244581 + 0.355472i
\(520\) 0 0
\(521\) −2.01609 3.49198i −0.0883266 0.152986i 0.818477 0.574539i \(-0.194819\pi\)
−0.906804 + 0.421553i \(0.861485\pi\)
\(522\) 0 0
\(523\) 0.516117 + 0.297980i 0.0225682 + 0.0130298i 0.511242 0.859437i \(-0.329186\pi\)
−0.488673 + 0.872467i \(0.662519\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.95008i 0.302750i
\(528\) 0 0
\(529\) −8.40154 −0.365285
\(530\) 0 0
\(531\) 8.36220 + 21.8519i 0.362889 + 0.948294i
\(532\) 0 0
\(533\) −17.1360 9.89347i −0.742242 0.428534i
\(534\) 0 0
\(535\) −2.42720 1.40135i −0.104937 0.0605855i
\(536\) 0 0
\(537\) 18.0227 + 12.4004i 0.777738 + 0.535118i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 7.05061 12.2120i 0.303129 0.525035i −0.673714 0.738992i \(-0.735302\pi\)
0.976843 + 0.213957i \(0.0686353\pi\)
\(542\) 0 0
\(543\) 35.5625 16.9398i 1.52613 0.726958i
\(544\) 0 0
\(545\) 3.25729 + 5.64179i 0.139527 + 0.241668i
\(546\) 0 0
\(547\) −18.9921 + 32.8952i −0.812042 + 1.40650i 0.0993905 + 0.995049i \(0.468311\pi\)
−0.911433 + 0.411450i \(0.865023\pi\)
\(548\) 0 0
\(549\) −2.93091 + 18.3817i −0.125088 + 0.784511i
\(550\) 0 0
\(551\) 4.52190 + 7.83216i 0.192639 + 0.333661i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 6.76034 + 0.535575i 0.286960 + 0.0227339i
\(556\) 0 0
\(557\) −36.1181 + 20.8528i −1.53037 + 0.883562i −0.531031 + 0.847353i \(0.678195\pi\)
−0.999344 + 0.0362098i \(0.988472\pi\)
\(558\) 0 0
\(559\) 8.50926i 0.359903i
\(560\) 0 0
\(561\) −48.1663 33.1406i −2.03358 1.39920i
\(562\) 0 0
\(563\) −0.938436 + 1.62542i −0.0395504 + 0.0685033i −0.885123 0.465357i \(-0.845926\pi\)
0.845573 + 0.533860i \(0.179259\pi\)
\(564\) 0 0
\(565\) 0.841235 0.485687i 0.0353910 0.0204330i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 20.6391 11.9160i 0.865237 0.499545i −0.000525844 1.00000i \(-0.500167\pi\)
0.865762 + 0.500455i \(0.166834\pi\)
\(570\) 0 0
\(571\) −14.8719 + 25.7589i −0.622370 + 1.07798i 0.366674 + 0.930350i \(0.380497\pi\)
−0.989043 + 0.147626i \(0.952837\pi\)
\(572\) 0 0
\(573\) −16.5652 11.3976i −0.692020 0.476140i
\(574\) 0 0
\(575\) 26.2997i 1.09678i
\(576\) 0 0
\(577\) 12.6950 7.32947i 0.528501 0.305130i −0.211905 0.977290i \(-0.567967\pi\)
0.740406 + 0.672160i \(0.234633\pi\)
\(578\) 0 0
\(579\) 10.9331 + 0.866152i 0.454363 + 0.0359961i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 19.6422 + 34.0213i 0.813498 + 1.40902i
\(584\) 0 0
\(585\) 1.08227 6.78763i 0.0447462 0.280634i
\(586\) 0 0
\(587\) −22.2189 + 38.4843i −0.917074 + 1.58842i −0.113238 + 0.993568i \(0.536122\pi\)
−0.803836 + 0.594851i \(0.797211\pi\)
\(588\) 0 0
\(589\) 3.22174 + 5.58021i 0.132749 + 0.229929i
\(590\) 0 0
\(591\) −4.84798 + 2.30928i −0.199419 + 0.0949912i
\(592\) 0 0
\(593\) −18.5593 + 32.1457i −0.762140 + 1.32006i 0.179606 + 0.983739i \(0.442518\pi\)
−0.941746 + 0.336326i \(0.890816\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −22.4409 15.4403i −0.918445 0.631931i
\(598\) 0 0
\(599\) 12.0534 + 6.95901i 0.492487 + 0.284338i 0.725606 0.688111i \(-0.241560\pi\)
−0.233119 + 0.972448i \(0.574893\pi\)
\(600\) 0 0
\(601\) 0.377613 + 0.218015i 0.0154032 + 0.00889301i 0.507682 0.861545i \(-0.330503\pi\)
−0.492279 + 0.870438i \(0.663836\pi\)
\(602\) 0 0
\(603\) 3.62005 + 9.45984i 0.147420 + 0.385235i
\(604\) 0 0
\(605\) 5.92923 0.241057
\(606\) 0 0
\(607\) 2.21837i 0.0900410i 0.998986 + 0.0450205i \(0.0143353\pi\)
−0.998986 + 0.0450205i \(0.985665\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −35.1315 20.2832i −1.42127 0.820570i
\(612\) 0 0
\(613\) −19.1011 33.0841i −0.771488 1.33626i −0.936748 0.350006i \(-0.886180\pi\)
0.165260 0.986250i \(-0.447154\pi\)
\(614\) 0 0
\(615\) 2.60073 3.77989i 0.104872 0.152420i
\(616\) 0 0
\(617\) 21.1043 12.1846i 0.849628 0.490533i −0.0108970 0.999941i \(-0.503469\pi\)
0.860525 + 0.509407i \(0.170135\pi\)
\(618\) 0 0
\(619\) 29.3845i 1.18106i −0.807015 0.590531i \(-0.798918\pi\)
0.807015 0.590531i \(-0.201082\pi\)
\(620\) 0 0
\(621\) 6.84140 28.3026i 0.274536 1.13574i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 20.4932 0.819728
\(626\) 0 0
\(627\) −54.0351 4.28083i −2.15795 0.170960i
\(628\) 0 0
\(629\) 51.2209 2.04231
\(630\) 0 0
\(631\) 5.17077 0.205845 0.102923 0.994689i \(-0.467181\pi\)
0.102923 + 0.994689i \(0.467181\pi\)
\(632\) 0 0
\(633\) 6.89729 10.0245i 0.274143 0.398438i
\(634\) 0 0
\(635\) −6.13762 −0.243564
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0.949393 + 0.770434i 0.0375574 + 0.0304779i
\(640\) 0 0
\(641\) 6.14769i 0.242819i 0.992603 + 0.121410i \(0.0387414\pi\)
−0.992603 + 0.121410i \(0.961259\pi\)
\(642\) 0 0
\(643\) 23.9599 13.8333i 0.944886 0.545530i 0.0533976 0.998573i \(-0.482995\pi\)
0.891489 + 0.453043i \(0.149662\pi\)
\(644\) 0 0
\(645\) −1.96696 0.155828i −0.0774488 0.00613574i
\(646\) 0 0
\(647\) −2.40729 4.16954i −0.0946402 0.163922i 0.814818 0.579717i \(-0.196837\pi\)
−0.909458 + 0.415795i \(0.863503\pi\)
\(648\) 0 0
\(649\) −31.4676 18.1679i −1.23521 0.713151i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 24.4518i 0.956874i 0.878122 + 0.478437i \(0.158796\pi\)
−0.878122 + 0.478437i \(0.841204\pi\)
\(654\) 0 0
\(655\) 10.2308 0.399749
\(656\) 0 0
\(657\) 4.07926 25.5838i 0.159147 0.998120i
\(658\) 0 0
\(659\) −20.7514 11.9808i −0.808359 0.466706i 0.0380267 0.999277i \(-0.487893\pi\)
−0.846386 + 0.532570i \(0.821226\pi\)
\(660\) 0 0
\(661\) 6.73275 + 3.88715i 0.261874 + 0.151193i 0.625189 0.780473i \(-0.285022\pi\)
−0.363315 + 0.931666i \(0.618355\pi\)
\(662\) 0 0
\(663\) 4.09999 51.7524i 0.159230 2.00990i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 3.77234 6.53389i 0.146066 0.252993i
\(668\) 0 0
\(669\) 3.37190 42.5621i 0.130365 1.64554i
\(670\) 0 0
\(671\) −14.4536 25.0343i −0.557973 0.966438i
\(672\) 0 0
\(673\) −14.7281 + 25.5097i −0.567725 + 0.983328i 0.429066 + 0.903273i \(0.358843\pi\)
−0.996790 + 0.0800548i \(0.974490\pi\)
\(674\) 0 0
\(675\) −23.7043 5.72988i −0.912378 0.220543i
\(676\) 0 0
\(677\) 17.8293 + 30.8812i 0.685235 + 1.18686i 0.973363 + 0.229270i \(0.0736337\pi\)
−0.288128 + 0.957592i \(0.593033\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −14.9525 31.3904i −0.572981 1.20288i
\(682\) 0 0
\(683\) −11.3423 + 6.54846i −0.433999 + 0.250570i −0.701049 0.713113i \(-0.747285\pi\)
0.267050 + 0.963683i \(0.413951\pi\)
\(684\) 0 0
\(685\) 12.1850i 0.465563i
\(686\) 0 0
\(687\) 12.2035 5.81302i 0.465594 0.221781i
\(688\) 0 0
\(689\) −17.4412 + 30.2090i −0.664456 + 1.15087i
\(690\) 0 0
\(691\) −7.81992 + 4.51483i −0.297484 + 0.171752i −0.641312 0.767280i \(-0.721610\pi\)
0.343828 + 0.939033i \(0.388276\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4.74068 2.73704i 0.179824 0.103822i
\(696\) 0 0
\(697\) 17.3273 30.0117i 0.656316 1.13677i
\(698\) 0 0
\(699\) −1.08962 + 13.7538i −0.0412131 + 0.520215i
\(700\) 0 0
\(701\) 0.259274i 0.00979264i 0.999988 + 0.00489632i \(0.00155855\pi\)
−0.999988 + 0.00489632i \(0.998441\pi\)
\(702\) 0 0
\(703\) 41.1252 23.7436i 1.55107 0.895508i
\(704\) 0 0
\(705\) 5.33191 7.74938i 0.200811 0.291858i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −3.08574 5.34467i −0.115888 0.200723i 0.802247 0.596993i \(-0.203638\pi\)
−0.918134 + 0.396270i \(0.870305\pi\)
\(710\) 0 0
\(711\) 1.88524 + 0.300596i 0.0707021 + 0.0112732i
\(712\) 0 0
\(713\) 2.68770 4.65523i 0.100655 0.174340i
\(714\) 0 0
\(715\) 5.33712 + 9.24417i 0.199597 + 0.345712i
\(716\) 0 0
\(717\) −38.2356 26.3078i −1.42793 0.982481i
\(718\) 0 0
\(719\) −9.76375 + 16.9113i −0.364127 + 0.630686i −0.988636 0.150332i \(-0.951966\pi\)
0.624509 + 0.781018i \(0.285299\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −1.36732 + 0.651308i −0.0508511 + 0.0242224i
\(724\) 0 0
\(725\) −5.47233 3.15945i −0.203237 0.117339i
\(726\) 0 0
\(727\) 0.425312 + 0.245554i 0.0157740 + 0.00910710i 0.507866 0.861436i \(-0.330434\pi\)
−0.492092 + 0.870543i \(0.663768\pi\)
\(728\) 0 0
\(729\) 24.0189 + 12.3325i 0.889591 + 0.456759i
\(730\) 0 0
\(731\) −14.9030 −0.551206
\(732\) 0 0
\(733\) 7.71829i 0.285082i 0.989789 + 0.142541i \(0.0455272\pi\)
−0.989789 + 0.142541i \(0.954473\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −13.6225 7.86498i −0.501793 0.289710i
\(738\) 0 0
\(739\) −5.45417 9.44690i −0.200635 0.347510i 0.748098 0.663588i \(-0.230967\pi\)
−0.948733 + 0.316078i \(0.897634\pi\)
\(740\) 0 0
\(741\) −20.6982 43.4525i −0.760366 1.59627i
\(742\) 0 0
\(743\) 27.0051 15.5914i 0.990722 0.571994i 0.0852322 0.996361i \(-0.472837\pi\)
0.905490 + 0.424367i \(0.139503\pi\)
\(744\) 0 0
\(745\) 4.07158i 0.149171i
\(746\) 0 0
\(747\) 5.97944 + 15.6254i 0.218776 + 0.571702i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −46.0734 −1.68124 −0.840622 0.541623i \(-0.817810\pi\)
−0.840622 + 0.541623i \(0.817810\pi\)
\(752\) 0 0
\(753\) 1.72868 + 3.62909i 0.0629966 + 0.132252i
\(754\) 0 0
\(755\) 2.39917 0.0873149
\(756\) 0 0
\(757\) 40.6419 1.47715 0.738577 0.674169i \(-0.235498\pi\)
0.738577 + 0.674169i \(0.235498\pi\)
\(758\) 0 0
\(759\) 19.4463 + 40.8245i 0.705856 + 1.48183i
\(760\) 0 0
\(761\) −7.28380 −0.264037 −0.132019 0.991247i \(-0.542146\pi\)
−0.132019 + 0.991247i \(0.542146\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 11.8877 + 1.89546i 0.429802 + 0.0685306i
\(766\) 0 0
\(767\) 32.2640i 1.16499i
\(768\) 0 0
\(769\) −35.8261 + 20.6842i −1.29192 + 0.745892i −0.978995 0.203886i \(-0.934643\pi\)
−0.312927 + 0.949777i \(0.601310\pi\)
\(770\) 0 0
\(771\) 18.3261 + 38.4727i 0.659997 + 1.38556i
\(772\) 0 0
\(773\) −8.55914 14.8249i −0.307851 0.533213i 0.670041 0.742324i \(-0.266276\pi\)
−0.977892 + 0.209111i \(0.932943\pi\)
\(774\) 0 0
\(775\) −3.89890 2.25103i −0.140053 0.0808594i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 32.1284i 1.15112i
\(780\) 0 0
\(781\) −1.89879 −0.0679439
\(782\) 0 0
\(783\) 5.06720 + 4.82359i 0.181087 + 0.172381i
\(784\) 0 0
\(785\) −0.853737 0.492906i −0.0304712 0.0175926i
\(786\) 0 0
\(787\) 25.7426 + 14.8625i 0.917623 + 0.529790i 0.882876 0.469606i \(-0.155604\pi\)
0.0347472 + 0.999396i \(0.488937\pi\)
\(788\) 0 0
\(789\) 37.0636 17.6549i 1.31950 0.628530i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 12.8339 22.2290i 0.455746 0.789375i
\(794\) 0 0
\(795\) −6.66355 4.58482i −0.236332 0.162607i
\(796\) 0 0
\(797\) 20.6019 + 35.6836i 0.729757 + 1.26398i 0.956986 + 0.290135i \(0.0937002\pi\)
−0.227229 + 0.973841i \(0.572966\pi\)
\(798\) 0 0
\(799\) 35.5236 61.5288i 1.25674 2.17673i
\(800\) 0 0
\(801\) 7.43182 + 19.4207i 0.262590 + 0.686196i
\(802\) 0 0
\(803\) 20.1166 + 34.8430i 0.709900 + 1.22958i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 20.2455 29.4248i 0.712677 1.03580i
\(808\) 0 0
\(809\) −10.7589 + 6.21165i −0.378262 + 0.218390i −0.677062 0.735926i \(-0.736747\pi\)
0.298800 + 0.954316i \(0.403414\pi\)
\(810\) 0 0
\(811\) 32.8713i 1.15427i −0.816650 0.577133i \(-0.804171\pi\)
0.816650 0.577133i \(-0.195829\pi\)
\(812\) 0 0
\(813\) 1.90759 24.0787i 0.0669021 0.844477i
\(814\) 0 0
\(815\) −1.56078 + 2.70336i −0.0546719 + 0.0946944i
\(816\) 0 0
\(817\) −11.9656 + 6.90833i −0.418623 + 0.241692i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 11.7411 6.77873i 0.409768 0.236579i −0.280922 0.959731i \(-0.590640\pi\)
0.690690 + 0.723151i \(0.257307\pi\)
\(822\) 0 0
\(823\) 12.2565 21.2289i 0.427235 0.739993i −0.569391 0.822067i \(-0.692821\pi\)
0.996626 + 0.0820737i \(0.0261543\pi\)
\(824\) 0 0
\(825\) 34.1917 16.2869i 1.19040 0.567036i
\(826\) 0 0
\(827\) 14.7323i 0.512292i 0.966638 + 0.256146i \(0.0824527\pi\)
−0.966638 + 0.256146i \(0.917547\pi\)
\(828\) 0 0
\(829\) 11.7079 6.75957i 0.406633 0.234770i −0.282709 0.959206i \(-0.591233\pi\)
0.689342 + 0.724436i \(0.257900\pi\)
\(830\) 0 0
\(831\) −15.4842 32.5065i −0.537139 1.12764i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 1.32255 + 2.29073i 0.0457689 + 0.0792740i
\(836\) 0 0
\(837\) 3.61025 + 3.43668i 0.124789 + 0.118789i
\(838\) 0 0
\(839\) −0.511154 + 0.885345i −0.0176470 + 0.0305655i −0.874714 0.484639i \(-0.838951\pi\)
0.857067 + 0.515205i \(0.172284\pi\)
\(840\) 0 0
\(841\) −13.5936 23.5449i −0.468746 0.811892i
\(842\) 0 0
\(843\) −3.21228 + 40.5472i −0.110637 + 1.39652i
\(844\) 0 0
\(845\) −1.13918 + 1.97312i −0.0391890 + 0.0678773i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 1.76757 22.3113i 0.0606628 0.765721i
\(850\) 0 0
\(851\) −34.3082 19.8079i −1.17607 0.679005i
\(852\) 0 0
\(853\) 8.70682 + 5.02689i 0.298116 + 0.172117i 0.641596 0.767043i \(-0.278273\pi\)
−0.343480 + 0.939160i \(0.611606\pi\)
\(854\) 0 0
\(855\) 10.4233 3.98874i 0.356469 0.136412i
\(856\) 0 0
\(857\) 28.2543 0.965150 0.482575 0.875855i \(-0.339702\pi\)
0.482575 + 0.875855i \(0.339702\pi\)
\(858\) 0 0
\(859\) 31.1538i 1.06295i 0.847073 + 0.531476i \(0.178363\pi\)
−0.847073 + 0.531476i \(0.821637\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 49.4703 + 28.5617i 1.68399 + 0.972252i 0.958963 + 0.283530i \(0.0915055\pi\)
0.725026 + 0.688722i \(0.241828\pi\)
\(864\) 0 0
\(865\) 1.57157 + 2.72204i 0.0534351 + 0.0925522i
\(866\) 0 0
\(867\) 61.2854 + 4.85522i 2.08136 + 0.164892i
\(868\) 0 0
\(869\) −2.56754 + 1.48237i −0.0870978 + 0.0502859i
\(870\) 0 0
\(871\) 13.9673i 0.473264i
\(872\) 0 0
\(873\) −24.2259 + 9.27064i −0.819921 + 0.313764i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −31.3382 −1.05822 −0.529108 0.848554i \(-0.677473\pi\)
−0.529108 + 0.848554i \(0.677473\pi\)
\(878\) 0 0
\(879\) −3.90976 + 5.68243i −0.131873 + 0.191664i
\(880\) 0 0
\(881\) 12.2822 0.413799 0.206900 0.978362i \(-0.433663\pi\)
0.206900 + 0.978362i \(0.433663\pi\)
\(882\) 0 0
\(883\) 3.61496 0.121653 0.0608266 0.998148i \(-0.480626\pi\)
0.0608266 + 0.998148i \(0.480626\pi\)
\(884\) 0 0
\(885\) 7.45798 + 0.590844i 0.250697 + 0.0198610i
\(886\) 0 0
\(887\) 18.8853 0.634106 0.317053 0.948408i \(-0.397307\pi\)
0.317053 + 0.948408i \(0.397307\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −41.0323 + 8.63284i −1.37463 + 0.289211i
\(892\) 0 0
\(893\) 65.8685i 2.20420i
\(894\) 0 0
\(895\) 6.05795 3.49756i 0.202495 0.116911i
\(896\) 0 0
\(897\) −22.7596 + 33.0787i −0.759922 + 1.10447i
\(898\) 0 0
\(899\) 0.645760 + 1.11849i 0.0215373 + 0.0373037i
\(900\) 0 0
\(901\) −52.9075 30.5462i −1.76260 1.01764i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 12.5954i 0.418684i
\(906\) 0 0
\(907\) −57.7603 −1.91790 −0.958950 0.283576i \(-0.908479\pi\)
−0.958950 + 0.283576i \(0.908479\pi\)
\(908\) 0 0
\(909\) 8.85553 10.9125i 0.293719 0.361945i
\(910\) 0 0
\(911\) −33.7810 19.5034i −1.11921 0.646178i −0.178013 0.984028i \(-0.556967\pi\)
−0.941200 + 0.337850i \(0.890300\pi\)
\(912\) 0 0
\(913\) −22.5011 12.9910i −0.744679 0.429940i
\(914\) 0 0
\(915\) 4.90331 + 3.37370i 0.162099 + 0.111531i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 15.9047 27.5478i 0.524649 0.908719i −0.474939 0.880019i \(-0.657530\pi\)
0.999588 0.0287001i \(-0.00913677\pi\)
\(920\) 0 0
\(921\) −34.8960 + 16.6223i −1.14986 + 0.547725i
\(922\) 0 0
\(923\) −0.843006 1.46013i −0.0277479 0.0480607i
\(924\) 0 0
\(925\) −16.5897 + 28.7342i −0.545465 + 0.944774i
\(926\) 0 0
\(927\) 17.2367 + 13.9876i 0.566127 + 0.459413i
\(928\) 0 0
\(929\) 19.0424 + 32.9825i 0.624762 + 1.08212i 0.988587 + 0.150653i \(0.0481375\pi\)
−0.363824 + 0.931468i \(0.618529\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 28.2607 + 2.23890i 0.925212 + 0.0732982i
\(934\) 0 0
\(935\) −16.1901 + 9.34735i −0.529472 + 0.305691i
\(936\) 0 0
\(937\) 37.6261i 1.22919i −0.788842 0.614596i \(-0.789319\pi\)
0.788842 0.614596i \(-0.210681\pi\)
\(938\) 0 0
\(939\) −27.2766 18.7675i −0.890137 0.612454i
\(940\) 0 0
\(941\) −0.837737 + 1.45100i −0.0273094 + 0.0473014i −0.879357 0.476163i \(-0.842027\pi\)
0.852048 + 0.523464i \(0.175361\pi\)
\(942\) 0 0
\(943\) −23.2119 + 13.4014i −0.755884 + 0.436410i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −19.2331 + 11.1042i −0.624992 + 0.360839i −0.778810 0.627260i \(-0.784176\pi\)
0.153818 + 0.988099i \(0.450843\pi\)
\(948\) 0 0
\(949\) −17.8624 + 30.9386i −0.579838 + 1.00431i
\(950\) 0 0
\(951\) 34.4029 + 23.6707i 1.11559 + 0.767576i
\(952\) 0 0
\(953\) 27.1505i 0.879491i 0.898122 + 0.439746i \(0.144931\pi\)
−0.898122 + 0.439746i \(0.855069\pi\)
\(954\) 0 0
\(955\) −5.56803 + 3.21470i −0.180177 + 0.104025i
\(956\) 0 0
\(957\) −10.8307 0.858043i −0.350107 0.0277366i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0399 26.0499i −0.485158 0.840319i
\(962\) 0 0
\(963\) −14.1790 + 5.42596i −0.456913 + 0.174849i
\(964\) 0 0
\(965\) 1.75341 3.03700i 0.0564444 0.0977646i
\(966\) 0 0
\(967\) −16.7553 29.0211i −0.538815 0.933255i −0.998968 0.0454157i \(-0.985539\pi\)
0.460153 0.887840i \(-0.347795\pi\)
\(968\) 0 0
\(969\) 76.1020 36.2504i 2.44475 1.16453i
\(970\) 0 0
\(971\) 17.8733 30.9574i 0.573580 0.993470i −0.422614 0.906310i \(-0.638887\pi\)
0.996194 0.0871606i \(-0.0277793\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 27.7044 + 19.0619i 0.887252 + 0.610469i
\(976\) 0 0
\(977\) 39.5696 + 22.8455i 1.26594 + 0.730892i 0.974218 0.225610i \(-0.0724376\pi\)
0.291725 + 0.956502i \(0.405771\pi\)
\(978\) 0 0
\(979\) −27.9665 16.1465i −0.893814 0.516044i
\(980\) 0 0
\(981\) 34.8483 + 5.55646i 1.11262 + 0.177404i
\(982\) 0 0
\(983\) −18.6743 −0.595616 −0.297808 0.954626i \(-0.596256\pi\)
−0.297808 + 0.954626i \(0.596256\pi\)
\(984\) 0 0
\(985\) 1.71703i 0.0547092i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 9.98215 + 5.76320i 0.317414 + 0.183259i
\(990\) 0 0
\(991\) −24.2806 42.0552i −0.771299 1.33593i −0.936851 0.349727i \(-0.886274\pi\)
0.165553 0.986201i \(-0.447059\pi\)
\(992\) 0 0
\(993\) −11.1939 + 16.2691i −0.355226 + 0.516284i
\(994\) 0 0
\(995\) −7.54303 + 4.35497i −0.239130 + 0.138062i
\(996\) 0 0
\(997\) 52.1903i 1.65288i 0.563022 + 0.826442i \(0.309639\pi\)
−0.563022 + 0.826442i \(0.690361\pi\)
\(998\) 0 0
\(999\) 25.3277 26.6069i 0.801334 0.841805i
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.bm.b.1685.3 16
3.2 odd 2 5292.2.bm.b.4625.6 16
7.2 even 3 252.2.x.a.209.8 yes 16
7.3 odd 6 1764.2.w.a.1109.6 16
7.4 even 3 1764.2.w.a.1109.3 16
7.5 odd 6 252.2.x.a.209.1 yes 16
7.6 odd 2 inner 1764.2.bm.b.1685.6 16
9.4 even 3 5292.2.w.a.1097.6 16
9.5 odd 6 1764.2.w.a.509.6 16
21.2 odd 6 756.2.x.a.629.3 16
21.5 even 6 756.2.x.a.629.6 16
21.11 odd 6 5292.2.w.a.521.3 16
21.17 even 6 5292.2.w.a.521.6 16
21.20 even 2 5292.2.bm.b.4625.3 16
28.19 even 6 1008.2.cc.c.209.8 16
28.23 odd 6 1008.2.cc.c.209.1 16
63.2 odd 6 2268.2.f.b.1133.12 16
63.4 even 3 5292.2.bm.b.2285.3 16
63.5 even 6 252.2.x.a.41.8 yes 16
63.13 odd 6 5292.2.w.a.1097.3 16
63.16 even 3 2268.2.f.b.1133.6 16
63.23 odd 6 252.2.x.a.41.1 16
63.31 odd 6 5292.2.bm.b.2285.6 16
63.32 odd 6 inner 1764.2.bm.b.1697.6 16
63.40 odd 6 756.2.x.a.125.3 16
63.41 even 6 1764.2.w.a.509.3 16
63.47 even 6 2268.2.f.b.1133.5 16
63.58 even 3 756.2.x.a.125.6 16
63.59 even 6 inner 1764.2.bm.b.1697.3 16
63.61 odd 6 2268.2.f.b.1133.11 16
84.23 even 6 3024.2.cc.c.2897.3 16
84.47 odd 6 3024.2.cc.c.2897.6 16
252.23 even 6 1008.2.cc.c.545.8 16
252.103 even 6 3024.2.cc.c.881.3 16
252.131 odd 6 1008.2.cc.c.545.1 16
252.247 odd 6 3024.2.cc.c.881.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.x.a.41.1 16 63.23 odd 6
252.2.x.a.41.8 yes 16 63.5 even 6
252.2.x.a.209.1 yes 16 7.5 odd 6
252.2.x.a.209.8 yes 16 7.2 even 3
756.2.x.a.125.3 16 63.40 odd 6
756.2.x.a.125.6 16 63.58 even 3
756.2.x.a.629.3 16 21.2 odd 6
756.2.x.a.629.6 16 21.5 even 6
1008.2.cc.c.209.1 16 28.23 odd 6
1008.2.cc.c.209.8 16 28.19 even 6
1008.2.cc.c.545.1 16 252.131 odd 6
1008.2.cc.c.545.8 16 252.23 even 6
1764.2.w.a.509.3 16 63.41 even 6
1764.2.w.a.509.6 16 9.5 odd 6
1764.2.w.a.1109.3 16 7.4 even 3
1764.2.w.a.1109.6 16 7.3 odd 6
1764.2.bm.b.1685.3 16 1.1 even 1 trivial
1764.2.bm.b.1685.6 16 7.6 odd 2 inner
1764.2.bm.b.1697.3 16 63.59 even 6 inner
1764.2.bm.b.1697.6 16 63.32 odd 6 inner
2268.2.f.b.1133.5 16 63.47 even 6
2268.2.f.b.1133.6 16 63.16 even 3
2268.2.f.b.1133.11 16 63.61 odd 6
2268.2.f.b.1133.12 16 63.2 odd 6
3024.2.cc.c.881.3 16 252.103 even 6
3024.2.cc.c.881.6 16 252.247 odd 6
3024.2.cc.c.2897.3 16 84.23 even 6
3024.2.cc.c.2897.6 16 84.47 odd 6
5292.2.w.a.521.3 16 21.11 odd 6
5292.2.w.a.521.6 16 21.17 even 6
5292.2.w.a.1097.3 16 63.13 odd 6
5292.2.w.a.1097.6 16 9.4 even 3
5292.2.bm.b.2285.3 16 63.4 even 3
5292.2.bm.b.2285.6 16 63.31 odd 6
5292.2.bm.b.4625.3 16 21.20 even 2
5292.2.bm.b.4625.6 16 3.2 odd 2