Properties

Label 1764.2.w.a.1109.6
Level $1764$
Weight $2$
Character 1764.1109
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(509,1764)]
 
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, 5, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.509");
 
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.w (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 1109.6
Root \(-0.744857 + 1.56371i\) of defining polynomial
Character \(\chi\) \(=\) 1764.1109
Dual form 1764.2.w.a.509.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.981784 - 1.42692i) q^{3} +(-0.276914 + 0.479629i) q^{5} +(-1.07220 - 2.80185i) q^{9} +O(q^{10})\) \(q+(0.981784 - 1.42692i) q^{3} +(-0.276914 + 0.479629i) q^{5} +(-1.07220 - 2.80185i) q^{9} +(-4.03478 + 2.32948i) q^{11} +(-3.58265 + 2.06844i) q^{13} +(0.412522 + 0.866025i) q^{15} +(-3.62264 + 6.27459i) q^{17} +(5.81722 - 3.35857i) q^{19} +(4.85295 + 2.80185i) q^{23} +(2.34664 + 4.06450i) q^{25} +(-5.05069 - 1.22087i) q^{27} +(1.16599 + 0.673187i) q^{29} +0.959257i q^{31} +(-0.637299 + 8.04435i) q^{33} +(3.53478 + 6.12241i) q^{37} +(-0.565885 + 7.14292i) q^{39} +(2.39152 + 4.14224i) q^{41} +(-1.02846 + 1.78135i) q^{43} +(1.64076 + 0.261614i) q^{45} -9.80602 q^{47} +(5.39669 + 11.3295i) q^{51} +(7.30235 + 4.21601i) q^{53} -2.58026i q^{55} +(0.918838 - 11.5981i) q^{57} +7.79910 q^{59} -6.20463i q^{61} -2.29112i q^{65} -3.37628 q^{67} +(8.76257 - 4.17396i) q^{69} -0.407556i q^{71} +(-7.47870 - 4.31783i) q^{73} +(8.10360 + 0.641993i) q^{75} +0.636352 q^{79} +(-6.70077 + 6.00830i) q^{81} +(-2.78840 + 4.82965i) q^{83} +(-2.00632 - 3.47504i) q^{85} +(2.10534 - 1.00286i) q^{87} +(-3.46568 - 6.00274i) q^{89} +(1.36878 + 0.941783i) q^{93} +3.72014i q^{95} +(-7.48798 - 4.32318i) q^{97} +(10.8530 + 8.80719i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{9} - 6 q^{11} - 12 q^{15} - 6 q^{23} - 8 q^{25} - 12 q^{29} - 2 q^{37} - 36 q^{39} + 4 q^{43} + 12 q^{51} + 36 q^{53} - 42 q^{57} - 28 q^{67} - 40 q^{79} - 18 q^{81} + 6 q^{85} - 6 q^{93} + 90 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 0.981784 1.42692i 0.566833 0.823833i
\(4\) 0 0
\(5\) −0.276914 + 0.479629i −0.123840 + 0.214496i −0.921279 0.388903i \(-0.872854\pi\)
0.797439 + 0.603399i \(0.206187\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −1.07220 2.80185i −0.357400 0.933951i
\(10\) 0 0
\(11\) −4.03478 + 2.32948i −1.21653 + 0.702365i −0.964174 0.265270i \(-0.914539\pi\)
−0.252357 + 0.967634i \(0.581206\pi\)
\(12\) 0 0
\(13\) −3.58265 + 2.06844i −0.993648 + 0.573683i −0.906363 0.422500i \(-0.861153\pi\)
−0.0872856 + 0.996183i \(0.527819\pi\)
\(14\) 0 0
\(15\) 0.412522 + 0.866025i 0.106513 + 0.223607i
\(16\) 0 0
\(17\) −3.62264 + 6.27459i −0.878619 + 1.52181i −0.0257612 + 0.999668i \(0.508201\pi\)
−0.852857 + 0.522144i \(0.825132\pi\)
\(18\) 0 0
\(19\) 5.81722 3.35857i 1.33456 0.770510i 0.348567 0.937284i \(-0.386668\pi\)
0.985995 + 0.166774i \(0.0533351\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.85295 + 2.80185i 1.01191 + 0.584227i 0.911751 0.410744i \(-0.134731\pi\)
0.100160 + 0.994971i \(0.468064\pi\)
\(24\) 0 0
\(25\) 2.34664 + 4.06450i 0.469328 + 0.812899i
\(26\) 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
\(31\) 0.959257i 0.172288i 0.996283 + 0.0861438i \(0.0274545\pi\)
−0.996283 + 0.0861438i \(0.972546\pi\)
\(32\) 0 0
\(33\) −0.637299 + 8.04435i −0.110939 + 1.40034i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.53478 + 6.12241i 0.581114 + 1.00652i 0.995348 + 0.0963482i \(0.0307162\pi\)
−0.414234 + 0.910170i \(0.635950\pi\)
\(38\) 0 0
\(39\) −0.565885 + 7.14292i −0.0906141 + 1.14378i
\(40\) 0 0
\(41\) 2.39152 + 4.14224i 0.373493 + 0.646909i 0.990100 0.140362i \(-0.0448265\pi\)
−0.616607 + 0.787271i \(0.711493\pi\)
\(42\) 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
\(45\) 1.64076 + 0.261614i 0.244590 + 0.0389991i
\(46\) 0 0
\(47\) −9.80602 −1.43035 −0.715177 0.698943i \(-0.753654\pi\)
−0.715177 + 0.698943i \(0.753654\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 5.39669 + 11.3295i 0.755688 + 1.58645i
\(52\) 0 0
\(53\) 7.30235 + 4.21601i 1.00305 + 0.579114i 0.909151 0.416468i \(-0.136732\pi\)
0.0939038 + 0.995581i \(0.470065\pi\)
\(54\) 0 0
\(55\) 2.58026i 0.347922i
\(56\) 0 0
\(57\) 0.918838 11.5981i 0.121703 1.53621i
\(58\) 0 0
\(59\) 7.79910 1.01536 0.507678 0.861547i \(-0.330504\pi\)
0.507678 + 0.861547i \(0.330504\pi\)
\(60\) 0 0
\(61\) 6.20463i 0.794421i −0.917728 0.397211i \(-0.869978\pi\)
0.917728 0.397211i \(-0.130022\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.29112i 0.284179i
\(66\) 0 0
\(67\) −3.37628 −0.412478 −0.206239 0.978502i \(-0.566122\pi\)
−0.206239 + 0.978502i \(0.566122\pi\)
\(68\) 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 −0.00620487 0.999981i \(-0.501975\pi\)
−0.869111 + 0.494617i \(0.835308\pi\)
\(74\) 0 0
\(75\) 8.10360 + 0.641993i 0.935723 + 0.0741309i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0.636352 0.0715952 0.0357976 0.999359i \(-0.488603\pi\)
0.0357976 + 0.999359i \(0.488603\pi\)
\(80\) 0 0
\(81\) −6.70077 + 6.00830i −0.744530 + 0.667589i
\(82\) 0 0
\(83\) −2.78840 + 4.82965i −0.306066 + 0.530123i −0.977498 0.210944i \(-0.932346\pi\)
0.671432 + 0.741066i \(0.265680\pi\)
\(84\) 0 0
\(85\) −2.00632 3.47504i −0.217616 0.376921i
\(86\) 0 0
\(87\) 2.10534 1.00286i 0.225716 0.107517i
\(88\) 0 0
\(89\) −3.46568 6.00274i −0.367362 0.636289i 0.621790 0.783184i \(-0.286406\pi\)
−0.989152 + 0.146895i \(0.953072\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1.36878 + 0.941783i 0.141936 + 0.0976584i
\(94\) 0 0
\(95\) 3.72014i 0.381678i
\(96\) 0 0
\(97\) −7.48798 4.32318i −0.760289 0.438953i 0.0691107 0.997609i \(-0.477984\pi\)
−0.829399 + 0.558656i \(0.811317\pi\)
\(98\) 0 0
\(99\) 10.8530 + 8.80719i 1.09076 + 0.885156i
\(100\) 0 0
\(101\) −2.34227 4.05692i −0.233064 0.403679i 0.725644 0.688070i \(-0.241542\pi\)
−0.958708 + 0.284391i \(0.908209\pi\)
\(102\) 0 0
\(103\) −6.40804 3.69969i −0.631403 0.364541i 0.149892 0.988702i \(-0.452107\pi\)
−0.781295 + 0.624162i \(0.785441\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.38260 + 2.53029i −0.423682 + 0.244613i −0.696651 0.717410i \(-0.745327\pi\)
0.272970 + 0.962023i \(0.411994\pi\)
\(108\) 0 0
\(109\) −5.88142 + 10.1869i −0.563337 + 0.975729i 0.433865 + 0.900978i \(0.357150\pi\)
−0.997202 + 0.0747510i \(0.976184\pi\)
\(110\) 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
\(115\) −2.68770 + 1.55174i −0.250629 + 0.144701i
\(116\) 0 0
\(117\) 9.63680 + 7.82028i 0.890922 + 0.722985i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.35295 9.27159i 0.486632 0.842872i
\(122\) 0 0
\(123\) 8.25861 + 0.654273i 0.744654 + 0.0589938i
\(124\) 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
\(129\) 1.53211 + 3.21643i 0.134895 + 0.283191i
\(130\) 0 0
\(131\) −9.23643 + 15.9980i −0.806990 + 1.39775i 0.107949 + 0.994156i \(0.465572\pi\)
−0.914939 + 0.403592i \(0.867762\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 1.98417 2.08438i 0.170770 0.179395i
\(136\) 0 0
\(137\) −19.0537 + 11.0007i −1.62787 + 0.939851i −0.643142 + 0.765747i \(0.722370\pi\)
−0.984727 + 0.174104i \(0.944297\pi\)
\(138\) 0 0
\(139\) 8.55986 4.94204i 0.726038 0.419178i −0.0909332 0.995857i \(-0.528985\pi\)
0.816971 + 0.576679i \(0.195652\pi\)
\(140\) 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
\(149\) 6.36677 + 3.67585i 0.521586 + 0.301138i 0.737583 0.675256i \(-0.235967\pi\)
−0.215997 + 0.976394i \(0.569300\pi\)
\(150\) 0 0
\(151\) 2.16599 + 3.75161i 0.176266 + 0.305302i 0.940599 0.339520i \(-0.110265\pi\)
−0.764333 + 0.644822i \(0.776931\pi\)
\(152\) 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
\(157\) 1.78000i 0.142059i 0.997474 + 0.0710296i \(0.0226285\pi\)
−0.997474 + 0.0710296i \(0.977372\pi\)
\(158\) 0 0
\(159\) 13.1852 6.28065i 1.04566 0.498088i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.81818 + 4.88122i 0.220737 + 0.382327i 0.955032 0.296503i \(-0.0958206\pi\)
−0.734295 + 0.678830i \(0.762487\pi\)
\(164\) 0 0
\(165\) −3.68182 2.53326i −0.286630 0.197214i
\(166\) 0 0
\(167\) 2.38803 + 4.13618i 0.184791 + 0.320067i 0.943506 0.331355i \(-0.107506\pi\)
−0.758715 + 0.651423i \(0.774173\pi\)
\(168\) 0 0
\(169\) 2.05692 3.56270i 0.158225 0.274054i
\(170\) 0 0
\(171\) −15.6475 12.6979i −1.19659 0.971036i
\(172\) 0 0
\(173\) −5.67531 −0.431486 −0.215743 0.976450i \(-0.569217\pi\)
−0.215743 + 0.976450i \(0.569217\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 7.65703 11.1287i 0.575538 0.836484i
\(178\) 0 0
\(179\) 10.9383 + 6.31525i 0.817570 + 0.472024i 0.849578 0.527463i \(-0.176857\pi\)
−0.0320079 + 0.999488i \(0.510190\pi\)
\(180\) 0 0
\(181\) 22.7424i 1.69043i −0.534426 0.845215i \(-0.679472\pi\)
0.534426 0.845215i \(-0.320528\pi\)
\(182\) 0 0
\(183\) −8.85351 6.09160i −0.654470 0.450304i
\(184\) 0 0
\(185\) −3.91531 −0.287860
\(186\) 0 0
\(187\) 33.7554i 2.46844i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 11.6090i 0.840001i 0.907524 + 0.420000i \(0.137970\pi\)
−0.907524 + 0.420000i \(0.862030\pi\)
\(192\) 0 0
\(193\) 6.33199 0.455787 0.227893 0.973686i \(-0.426816\pi\)
0.227893 + 0.973686i \(0.426816\pi\)
\(194\) 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 0.897856 0.440288i \(-0.145124\pi\)
0.0676272 + 0.997711i \(0.478457\pi\)
\(200\) 0 0
\(201\) −3.31478 + 4.81768i −0.233806 + 0.339813i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −2.64898 −0.185013
\(206\) 0 0
\(207\) 2.64705 16.6014i 0.183982 1.15388i
\(208\) 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
\(213\) −0.581549 0.400132i −0.0398471 0.0274166i
\(214\) 0 0
\(215\) −0.569590 0.986559i −0.0388457 0.0672828i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −13.5037 + 6.43233i −0.912493 + 0.434657i
\(220\) 0 0
\(221\) 29.9729i 2.01619i
\(222\) 0 0
\(223\) −21.3477 12.3251i −1.42955 0.825350i −0.432464 0.901651i \(-0.642356\pi\)
−0.997085 + 0.0763008i \(0.975689\pi\)
\(224\) 0 0
\(225\) 8.87206 10.9329i 0.591470 0.728859i
\(226\) 0 0
\(227\) 10.0372 + 17.3849i 0.666190 + 1.15388i 0.978961 + 0.204047i \(0.0654095\pi\)
−0.312771 + 0.949829i \(0.601257\pi\)
\(228\) 0 0
\(229\) 6.75865 + 3.90211i 0.446624 + 0.257859i 0.706403 0.707809i \(-0.250316\pi\)
−0.259779 + 0.965668i \(0.583650\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.89843 3.98281i 0.451931 0.260922i −0.256714 0.966487i \(-0.582640\pi\)
0.708645 + 0.705565i \(0.249307\pi\)
\(234\) 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
\(241\) 0.757259 0.437203i 0.0487793 0.0281628i −0.475412 0.879763i \(-0.657701\pi\)
0.524191 + 0.851601i \(0.324368\pi\)
\(242\) 0 0
\(243\) 1.99465 + 15.4603i 0.127957 + 0.991780i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −13.8940 + 24.0652i −0.884057 + 1.53123i
\(248\) 0 0
\(249\) 4.15391 + 8.72049i 0.263244 + 0.552639i
\(250\) 0 0
\(251\) 2.32082 0.146489 0.0732445 0.997314i \(-0.476665\pi\)
0.0732445 + 0.997314i \(0.476665\pi\)
\(252\) 0 0
\(253\) −26.1075 −1.64136
\(254\) 0 0
\(255\) −6.92837 0.548888i −0.433872 0.0343727i
\(256\) 0 0
\(257\) −12.3017 + 21.3072i −0.767361 + 1.32911i 0.171628 + 0.985162i \(0.445097\pi\)
−0.938989 + 0.343947i \(0.888236\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0.635992 3.98874i 0.0393669 0.246897i
\(262\) 0 0
\(263\) 20.5268 11.8512i 1.26574 0.730775i 0.291560 0.956552i \(-0.405826\pi\)
0.974179 + 0.225778i \(0.0724922\pi\)
\(264\) 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 0.359176 + 0.933270i \(0.383058\pi\)
−0.987823 + 0.155580i \(0.950275\pi\)
\(270\) 0 0
\(271\) 12.0771 6.97270i 0.733630 0.423562i −0.0861186 0.996285i \(-0.527446\pi\)
0.819749 + 0.572723i \(0.194113\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −18.9363 10.9329i −1.14190 0.659278i
\(276\) 0 0
\(277\) −10.3940 18.0030i −0.624518 1.08170i −0.988634 0.150343i \(-0.951962\pi\)
0.364116 0.931354i \(-0.381371\pi\)
\(278\) 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
\(283\) 12.9218i 0.768120i 0.923308 + 0.384060i \(0.125474\pi\)
−0.923308 + 0.384060i \(0.874526\pi\)
\(284\) 0 0
\(285\) 5.30834 + 3.65237i 0.314439 + 0.216348i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −17.7470 30.7387i −1.04394 1.80816i
\(290\) 0 0
\(291\) −13.5204 + 6.44031i −0.792581 + 0.377538i
\(292\) 0 0
\(293\) 1.99115 + 3.44878i 0.116324 + 0.201480i 0.918308 0.395866i \(-0.129555\pi\)
−0.801984 + 0.597346i \(0.796222\pi\)
\(294\) 0 0
\(295\) −2.15968 + 3.74067i −0.125741 + 0.217790i
\(296\) 0 0
\(297\) 23.2224 6.83954i 1.34750 0.396870i
\(298\) 0 0
\(299\) −23.1819 −1.34064
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −8.08850 0.640797i −0.464672 0.0368128i
\(304\) 0 0
\(305\) 2.97592 + 1.71815i 0.170401 + 0.0983808i
\(306\) 0 0
\(307\) 22.3162i 1.27365i 0.771008 + 0.636825i \(0.219753\pi\)
−0.771008 + 0.636825i \(0.780247\pi\)
\(308\) 0 0
\(309\) −11.5705 + 5.51147i −0.658221 + 0.313537i
\(310\) 0 0
\(311\) −16.3674 −0.928111 −0.464056 0.885806i \(-0.653606\pi\)
−0.464056 + 0.885806i \(0.653606\pi\)
\(312\) 0 0
\(313\) 19.1157i 1.08048i −0.841510 0.540242i \(-0.818333\pi\)
0.841510 0.540242i \(-0.181667\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 24.1099i 1.35415i −0.735915 0.677074i \(-0.763248\pi\)
0.735915 0.677074i \(-0.236752\pi\)
\(318\) 0 0
\(319\) −6.27270 −0.351204
\(320\) 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
\(327\) 8.76163 + 18.3937i 0.484519 + 1.01717i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 11.4015 0.626686 0.313343 0.949640i \(-0.398551\pi\)
0.313343 + 0.949640i \(0.398551\pi\)
\(332\) 0 0
\(333\) 13.3641 16.4684i 0.732349 0.902462i
\(334\) 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
\(339\) −0.239920 + 3.02841i −0.0130307 + 0.164480i
\(340\) 0 0
\(341\) −2.23457 3.87039i −0.121009 0.209593i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −0.424526 + 5.35861i −0.0228557 + 0.288498i
\(346\) 0 0
\(347\) 5.33765i 0.286540i −0.989684 0.143270i \(-0.954238\pi\)
0.989684 0.143270i \(-0.0457617\pi\)
\(348\) 0 0
\(349\) −8.78031 5.06931i −0.469999 0.271354i 0.246240 0.969209i \(-0.420805\pi\)
−0.716239 + 0.697855i \(0.754138\pi\)
\(350\) 0 0
\(351\) 20.6202 6.07312i 1.10062 0.324159i
\(352\) 0 0
\(353\) 7.67564 + 13.2946i 0.408533 + 0.707600i 0.994726 0.102571i \(-0.0327070\pi\)
−0.586192 + 0.810172i \(0.699374\pi\)
\(354\) 0 0
\(355\) 0.195475 + 0.112858i 0.0103748 + 0.00598987i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1.51381 0.874000i 0.0798960 0.0461280i −0.459520 0.888168i \(-0.651978\pi\)
0.539416 + 0.842040i \(0.318645\pi\)
\(360\) 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
\(367\) −11.1720 + 6.45018i −0.583176 + 0.336697i −0.762394 0.647113i \(-0.775976\pi\)
0.179219 + 0.983809i \(0.442643\pi\)
\(368\) 0 0
\(369\) 9.04176 11.1420i 0.470695 0.580030i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 5.34782 9.26269i 0.276900 0.479604i −0.693713 0.720251i \(-0.744026\pi\)
0.970613 + 0.240647i \(0.0773597\pi\)
\(374\) 0 0
\(375\) −5.27061 + 7.66028i −0.272173 + 0.395575i
\(376\) 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
\(381\) 10.8803 15.8134i 0.557415 0.810145i
\(382\) 0 0
\(383\) 9.43059 16.3343i 0.481880 0.834641i −0.517903 0.855439i \(-0.673287\pi\)
0.999784 + 0.0207978i \(0.00662061\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 6.09379 + 0.971637i 0.309765 + 0.0493911i
\(388\) 0 0
\(389\) −14.3182 + 8.26660i −0.725960 + 0.419133i −0.816943 0.576719i \(-0.804333\pi\)
0.0909822 + 0.995853i \(0.470999\pi\)
\(390\) 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 0.482688 0.875792i \(-0.339660\pi\)
0.999802 + 0.0198756i \(0.00632700\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −18.6395 10.7615i −0.930811 0.537404i −0.0437428 0.999043i \(-0.513928\pi\)
−0.887068 + 0.461639i \(0.847262\pi\)
\(402\) 0 0
\(403\) −1.98417 3.43668i −0.0988386 0.171193i
\(404\) 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
\(409\) 22.2299i 1.09920i −0.835429 0.549599i \(-0.814781\pi\)
0.835429 0.549599i \(-0.185219\pi\)
\(410\) 0 0
\(411\) −3.00956 + 37.9884i −0.148451 + 1.87383i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −1.54429 2.67479i −0.0758063 0.131300i
\(416\) 0 0
\(417\) 1.35204 17.0662i 0.0662098 0.835738i
\(418\) 0 0
\(419\) 6.81490 + 11.8038i 0.332930 + 0.576651i 0.983085 0.183150i \(-0.0586294\pi\)
−0.650155 + 0.759802i \(0.725296\pi\)
\(420\) 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
\(423\) 10.5140 + 27.4750i 0.511209 + 1.33588i
\(424\) 0 0
\(425\) −34.0041 −1.64944
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −14.3561 30.1383i −0.693118 1.45509i
\(430\) 0 0
\(431\) −30.1663 17.4165i −1.45306 0.838925i −0.454406 0.890795i \(-0.650149\pi\)
−0.998654 + 0.0518699i \(0.983482\pi\)
\(432\) 0 0
\(433\) 19.5648i 0.940223i −0.882607 0.470112i \(-0.844214\pi\)
0.882607 0.470112i \(-0.155786\pi\)
\(434\) 0 0
\(435\) −0.101999 + 1.28749i −0.00489046 + 0.0617302i
\(436\) 0 0
\(437\) 37.6409 1.80061
\(438\) 0 0
\(439\) 28.5628i 1.36323i 0.731712 + 0.681614i \(0.238722\pi\)
−0.731712 + 0.681614i \(0.761278\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 25.2431i 1.19934i 0.800248 + 0.599669i \(0.204701\pi\)
−0.800248 + 0.599669i \(0.795299\pi\)
\(444\) 0 0
\(445\) 3.83878 0.181976
\(446\) 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
\(453\) 7.47979 + 0.592572i 0.351431 + 0.0278415i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3.50276 0.163852 0.0819261 0.996638i \(-0.473893\pi\)
0.0819261 + 0.996638i \(0.473893\pi\)
\(458\) 0 0
\(459\) 25.9573 27.2682i 1.21158 1.27277i
\(460\) 0 0
\(461\) −1.98765 + 3.44272i −0.0925743 + 0.160343i −0.908594 0.417681i \(-0.862843\pi\)
0.816019 + 0.578025i \(0.196176\pi\)
\(462\) 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
\(465\) −0.830741 + 0.395715i −0.0385247 + 0.0183508i
\(466\) 0 0
\(467\) −9.80952 16.9906i −0.453930 0.786230i 0.544696 0.838634i \(-0.316645\pi\)
−0.998626 + 0.0524035i \(0.983312\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 2.53991 + 1.74757i 0.117033 + 0.0805239i
\(472\) 0 0
\(473\) 9.58312i 0.440632i
\(474\) 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
\(479\) −15.8237 27.4074i −0.723002 1.25228i −0.959791 0.280715i \(-0.909428\pi\)
0.236789 0.971561i \(-0.423905\pi\)
\(480\) 0 0
\(481\) −25.3277 14.6230i −1.15485 0.666751i
\(482\) 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 −0.556881 0.830592i \(-0.688002\pi\)
0.997755 + 0.0669768i \(0.0213353\pi\)
\(488\) 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
\(493\) −8.44795 + 4.87743i −0.380476 + 0.219668i
\(494\) 0 0
\(495\) −7.22951 + 2.76656i −0.324942 + 0.124347i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 4.50632 7.80517i 0.201730 0.349407i −0.747356 0.664424i \(-0.768677\pi\)
0.949086 + 0.315017i \(0.102010\pi\)
\(500\) 0 0
\(501\) 8.24653 + 0.653316i 0.368428 + 0.0291880i
\(502\) 0 0
\(503\) 27.1572 1.21088 0.605440 0.795891i \(-0.292997\pi\)
0.605440 + 0.795891i \(0.292997\pi\)
\(504\) 0 0
\(505\) 2.59442 0.115450
\(506\) 0 0
\(507\) −3.06423 6.43286i −0.136087 0.285693i
\(508\) 0 0
\(509\) 6.56799 11.3761i 0.291121 0.504236i −0.682954 0.730461i \(-0.739305\pi\)
0.974075 + 0.226225i \(0.0726384\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −33.4814 + 9.86104i −1.47824 + 0.435376i
\(514\) 0 0
\(515\) 3.54895 2.04899i 0.156385 0.0902892i
\(516\) 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 \(-0.805181\pi\)
0.906804 + 0.421553i \(0.138515\pi\)
\(522\) 0 0
\(523\) 0.516117 0.297980i 0.0225682 0.0130298i −0.488673 0.872467i \(-0.662519\pi\)
0.511242 + 0.859437i \(0.329186\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −6.01895 3.47504i −0.262189 0.151375i
\(528\) 0 0
\(529\) 4.20077 + 7.27595i 0.182642 + 0.316346i
\(530\) 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
\(535\) 2.80269i 0.121171i
\(536\) 0 0
\(537\) 19.7504 9.40792i 0.852295 0.405982i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 7.05061 + 12.2120i 0.303129 + 0.525035i 0.976843 0.213957i \(-0.0686353\pi\)
−0.673714 + 0.738992i \(0.735302\pi\)
\(542\) 0 0
\(543\) −32.4516 22.3281i −1.39263 0.958192i
\(544\) 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
\(549\) −17.3845 + 6.65260i −0.741951 + 0.283926i
\(550\) 0 0
\(551\) 9.04380 0.385279
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −3.84399 + 5.58684i −0.163168 + 0.237148i
\(556\) 0 0
\(557\) 36.1181 + 20.8528i 1.53037 + 0.883562i 0.999344 + 0.0362098i \(0.0115285\pi\)
0.531031 + 0.847353i \(0.321805\pi\)
\(558\) 0 0
\(559\) 8.50926i 0.359903i
\(560\) 0 0
\(561\) −48.1663 33.1406i −2.03358 1.39920i
\(562\) 0 0
\(563\) −1.87687 −0.0791007 −0.0395504 0.999218i \(-0.512593\pi\)
−0.0395504 + 0.999218i \(0.512593\pi\)
\(564\) 0 0
\(565\) 0.971375i 0.0408660i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 23.8320i 0.999089i 0.866288 + 0.499545i \(0.166499\pi\)
−0.866288 + 0.499545i \(0.833501\pi\)
\(570\) 0 0
\(571\) 29.7438 1.24474 0.622370 0.782723i \(-0.286170\pi\)
0.622370 + 0.782723i \(0.286170\pi\)
\(572\) 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 0.740406 0.672160i \(-0.234633\pi\)
−0.211905 + 0.977290i \(0.567967\pi\)
\(578\) 0 0
\(579\) 6.21664 9.03524i 0.258355 0.375492i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −39.2845 −1.62700
\(584\) 0 0
\(585\) −6.41939 + 2.45654i −0.265409 + 0.101566i
\(586\) 0 0
\(587\) 22.2189 38.4843i 0.917074 1.58842i 0.113238 0.993568i \(-0.463878\pi\)
0.803836 0.594851i \(-0.202789\pi\)
\(588\) 0 0
\(589\) 3.22174 + 5.58021i 0.132749 + 0.229929i
\(590\) 0 0
\(591\) −4.42389 3.04383i −0.181974 0.125206i
\(592\) 0 0
\(593\) 18.5593 + 32.1457i 0.762140 + 1.32006i 0.941746 + 0.336326i \(0.109184\pi\)
−0.179606 + 0.983739i \(0.557482\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 24.5922 11.7142i 1.00649 0.479431i
\(598\) 0 0
\(599\) 13.9180i 0.568675i −0.958724 0.284338i \(-0.908226\pi\)
0.958724 0.284338i \(-0.0917737\pi\)
\(600\) 0 0
\(601\) −0.377613 0.218015i −0.0154032 0.00889301i 0.492279 0.870438i \(-0.336164\pi\)
−0.507682 + 0.861545i \(0.669497\pi\)
\(602\) 0 0
\(603\) 3.62005 + 9.45984i 0.147420 + 0.385235i
\(604\) 0 0
\(605\) 2.96461 + 5.13486i 0.120529 + 0.208762i
\(606\) 0 0
\(607\) 1.92117 + 1.10919i 0.0779778 + 0.0450205i 0.538482 0.842637i \(-0.318998\pi\)
−0.460504 + 0.887658i \(0.652331\pi\)
\(608\) 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 0.165260 + 0.986250i \(0.447154\pi\)
−0.936748 + 0.350006i \(0.886180\pi\)
\(614\) 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
\(619\) 25.4477 14.6922i 1.02283 0.590531i 0.107907 0.994161i \(-0.465585\pi\)
0.914922 + 0.403630i \(0.132252\pi\)
\(620\) 0 0
\(621\) −21.0901 20.0761i −0.846315 0.805627i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −10.2466 + 17.7476i −0.409864 + 0.709906i
\(626\) 0 0
\(627\) 23.3102 + 48.9362i 0.930921 + 1.95432i
\(628\) 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
\(633\) 12.1301 + 0.960985i 0.482128 + 0.0381957i
\(634\) 0 0
\(635\) −3.06881 + 5.31533i −0.121782 + 0.210933i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −1.14191 + 0.436981i −0.0451733 + 0.0172867i
\(640\) 0 0
\(641\) 5.32405 3.07384i 0.210287 0.121410i −0.391158 0.920324i \(-0.627925\pi\)
0.601445 + 0.798914i \(0.294592\pi\)
\(642\) 0 0
\(643\) −23.9599 + 13.8333i −0.944886 + 0.545530i −0.891489 0.453043i \(-0.850338\pi\)
−0.0533976 + 0.998573i \(0.517005\pi\)
\(644\) 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 \(-0.803163\pi\)
0.909458 + 0.415795i \(0.136497\pi\)
\(648\) 0 0
\(649\) −31.4676 + 18.1679i −1.23521 + 0.713151i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −21.1759 12.2259i −0.828677 0.478437i 0.0247223 0.999694i \(-0.492130\pi\)
−0.853400 + 0.521257i \(0.825463\pi\)
\(654\) 0 0
\(655\) −5.11539 8.86011i −0.199875 0.346193i
\(656\) 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
\(661\) 7.77431i 0.302386i 0.988504 + 0.151193i \(0.0483114\pi\)
−0.988504 + 0.151193i \(0.951689\pi\)
\(662\) 0 0
\(663\) −42.7689 29.4269i −1.66101 1.14285i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 3.77234 + 6.53389i 0.146066 + 0.252993i
\(668\) 0 0
\(669\) −38.5458 + 18.3609i −1.49027 + 0.709873i
\(670\) 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
\(675\) −6.88992 23.3935i −0.265193 0.900414i
\(676\) 0 0
\(677\) 35.6586 1.37047 0.685235 0.728322i \(-0.259700\pi\)
0.685235 + 0.728322i \(0.259700\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 34.6612 + 2.74597i 1.32822 + 0.105226i
\(682\) 0 0
\(683\) 11.3423 + 6.54846i 0.433999 + 0.250570i 0.701049 0.713113i \(-0.252715\pi\)
−0.267050 + 0.963683i \(0.586049\pi\)
\(684\) 0 0
\(685\) 12.1850i 0.465563i
\(686\) 0 0
\(687\) 12.2035 5.81302i 0.465594 0.221781i
\(688\) 0 0
\(689\) −34.8823 −1.32891
\(690\) 0 0
\(691\) 9.02967i 0.343505i 0.985140 + 0.171752i \(0.0549429\pi\)
−0.985140 + 0.171752i \(0.945057\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5.47407i 0.207643i
\(696\) 0 0
\(697\) −34.6545 −1.31263
\(698\) 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
\(705\) −4.04520 8.49226i −0.152351 0.319837i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 6.17149 0.231775 0.115888 0.993262i \(-0.463029\pi\)
0.115888 + 0.993262i \(0.463029\pi\)
\(710\) 0 0
\(711\) −0.682297 1.78296i −0.0255881 0.0668664i
\(712\) 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
\(717\) 3.66541 46.2668i 0.136887 1.72787i
\(718\) 0 0
\(719\) 9.76375 + 16.9113i 0.364127 + 0.630686i 0.988636 0.150332i \(-0.0480342\pi\)
−0.624509 + 0.781018i \(0.714701\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0.119610 1.50979i 0.00444835 0.0561496i
\(724\) 0 0
\(725\) 6.31890i 0.234678i
\(726\) 0 0
\(727\) −0.425312 0.245554i −0.0157740 0.00910710i 0.492092 0.870543i \(-0.336232\pi\)
−0.507866 + 0.861436i \(0.669566\pi\)
\(728\) 0 0
\(729\) 24.0189 + 12.3325i 0.889591 + 0.456759i
\(730\) 0 0
\(731\) −7.45149 12.9064i −0.275603 0.477359i
\(732\) 0 0
\(733\) 6.68424 + 3.85915i 0.246888 + 0.142541i 0.618338 0.785912i \(-0.287806\pi\)
−0.371450 + 0.928453i \(0.621139\pi\)
\(734\) 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 −0.948733 0.316078i \(-0.897634\pi\)
0.748098 + 0.663588i \(0.230967\pi\)
\(740\) 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
\(745\) −3.52609 + 2.03579i −0.129186 + 0.0745855i
\(746\) 0 0
\(747\) 16.5217 + 2.63433i 0.604497 + 0.0963852i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 23.0367 39.9008i 0.840622 1.45600i −0.0487482 0.998811i \(-0.515523\pi\)
0.889370 0.457188i \(-0.151143\pi\)
\(752\) 0 0
\(753\) 2.27855 3.31163i 0.0830349 0.120682i
\(754\) 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
\(759\) −25.6319 + 37.2532i −0.930378 + 1.35221i
\(760\) 0 0
\(761\) −3.64190 + 6.30795i −0.132019 + 0.228663i −0.924455 0.381292i \(-0.875479\pi\)
0.792436 + 0.609955i \(0.208813\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −7.58538 + 9.34735i −0.274250 + 0.337954i
\(766\) 0 0
\(767\) −27.9415 + 16.1320i −1.00891 + 0.582493i
\(768\) 0 0
\(769\) 35.8261 20.6842i 1.29192 0.745892i 0.312927 0.949777i \(-0.398690\pi\)
0.978995 + 0.203886i \(0.0653570\pi\)
\(770\) 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 \(-0.733724\pi\)
0.977892 + 0.209111i \(0.0670569\pi\)
\(774\) 0 0
\(775\) −3.89890 + 2.25103i −0.140053 + 0.0808594i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 27.8241 + 16.0642i 0.996900 + 0.575561i
\(780\) 0 0
\(781\) 0.949393 + 1.64440i 0.0339719 + 0.0588411i
\(782\) 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
\(787\) 29.7250i 1.05958i 0.848129 + 0.529790i \(0.177729\pi\)
−0.848129 + 0.529790i \(0.822271\pi\)
\(788\) 0 0
\(789\) 3.24224 40.9255i 0.115427 1.45698i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 12.8339 + 22.2290i 0.455746 + 0.789375i
\(794\) 0 0
\(795\) −0.638793 + 8.06322i −0.0226557 + 0.285973i
\(796\) 0 0
\(797\) −20.6019 35.6836i −0.729757 1.26398i −0.956986 0.290135i \(-0.906300\pi\)
0.227229 0.973841i \(-0.427034\pi\)
\(798\) 0 0
\(799\) 35.5236 61.5288i 1.25674 2.17673i
\(800\) 0 0
\(801\) −13.1029 + 16.1465i −0.462968 + 0.570508i
\(802\) 0 0
\(803\) 40.2332 1.41980
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 15.3598 + 32.2455i 0.540691 + 1.13510i
\(808\) 0 0
\(809\) 10.7589 + 6.21165i 0.378262 + 0.218390i 0.677062 0.735926i \(-0.263253\pi\)
−0.298800 + 0.954316i \(0.596586\pi\)
\(810\) 0 0
\(811\) 32.8713i 1.15427i 0.816650 + 0.577133i \(0.195829\pi\)
−0.816650 + 0.577133i \(0.804171\pi\)
\(812\) 0 0
\(813\) 1.90759 24.0787i 0.0669021 0.844477i
\(814\) 0 0
\(815\) −3.12157 −0.109344
\(816\) 0 0
\(817\) 13.8167i 0.483384i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 13.5575i 0.473159i 0.971612 + 0.236579i \(0.0760263\pi\)
−0.971612 + 0.236579i \(0.923974\pi\)
\(822\) 0 0
\(823\) −24.5130 −0.854470 −0.427235 0.904141i \(-0.640512\pi\)
−0.427235 + 0.904141i \(0.640512\pi\)
\(824\) 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 0.689342 0.724436i \(-0.257900\pi\)
−0.282709 + 0.959206i \(0.591233\pi\)
\(830\) 0 0
\(831\) −35.8936 2.84360i −1.24513 0.0986434i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −2.64511 −0.0915378
\(836\) 0 0
\(837\) 1.17113 4.84491i 0.0404802 0.167465i
\(838\) 0 0
\(839\) 0.511154 0.885345i 0.0176470 0.0305655i −0.857067 0.515205i \(-0.827716\pi\)
0.874714 + 0.484639i \(0.161049\pi\)
\(840\) 0 0
\(841\) −13.5936 23.5449i −0.468746 0.811892i
\(842\) 0 0
\(843\) −36.7210 + 17.4917i −1.26474 + 0.602446i
\(844\) 0 0
\(845\) 1.13918 + 1.97312i 0.0391890 + 0.0678773i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 18.4383 + 12.6864i 0.632802 + 0.435396i
\(850\) 0 0
\(851\) 39.6157i 1.35801i
\(852\) 0 0
\(853\) −8.70682 5.02689i −0.298116 0.172117i 0.343480 0.939160i \(-0.388394\pi\)
−0.641596 + 0.767043i \(0.721727\pi\)
\(854\) 0 0
\(855\) 10.4233 3.98874i 0.356469 0.136412i
\(856\) 0 0
\(857\) 14.1272 + 24.4690i 0.482575 + 0.835844i 0.999800 0.0200052i \(-0.00636826\pi\)
−0.517225 + 0.855850i \(0.673035\pi\)
\(858\) 0 0
\(859\) 26.9800 + 15.5769i 0.920544 + 0.531476i 0.883809 0.467849i \(-0.154971\pi\)
0.0367356 + 0.999325i \(0.488304\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −49.4703 + 28.5617i −1.68399 + 0.972252i −0.725026 + 0.688722i \(0.758172\pi\)
−0.958963 + 0.283530i \(0.908495\pi\)
\(864\) 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
\(871\) 12.0960 6.98365i 0.409858 0.236632i
\(872\) 0 0
\(873\) −4.08432 + 25.6155i −0.138233 + 0.866955i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 15.6691 27.1397i 0.529108 0.916443i −0.470315 0.882498i \(-0.655860\pi\)
0.999424 0.0339441i \(-0.0108068\pi\)
\(878\) 0 0
\(879\) 6.87601 + 0.544739i 0.231922 + 0.0183736i
\(880\) 0 0
\(881\) −12.2822 −0.413799 −0.206900 0.978362i \(-0.566337\pi\)
−0.206900 + 0.978362i \(0.566337\pi\)
\(882\) 0 0
\(883\) 3.61496 0.121653 0.0608266 0.998148i \(-0.480626\pi\)
0.0608266 + 0.998148i \(0.480626\pi\)
\(884\) 0 0
\(885\) 3.21730 + 6.75422i 0.108148 + 0.227041i
\(886\) 0 0
\(887\) 9.44264 16.3551i 0.317053 0.549152i −0.662819 0.748780i \(-0.730640\pi\)
0.979872 + 0.199628i \(0.0639734\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 13.0399 39.8515i 0.436854 1.33507i
\(892\) 0 0
\(893\) −57.0438 + 32.9342i −1.90890 + 1.10210i
\(894\) 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
\(905\) 10.9079 + 6.29769i 0.362591 + 0.209342i
\(906\) 0 0
\(907\) 28.8802 + 50.0219i 0.958950 + 1.66095i 0.725059 + 0.688687i \(0.241813\pi\)
0.233891 + 0.972263i \(0.424854\pi\)
\(908\) 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
\(913\) 25.9821i 0.859881i
\(914\) 0 0
\(915\) 5.37336 2.55955i 0.177638 0.0846160i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 15.9047 + 27.5478i 0.524649 + 0.908719i 0.999588 + 0.0287001i \(0.00913677\pi\)
−0.474939 + 0.880019i \(0.657530\pi\)
\(920\) 0 0
\(921\) 31.8434 + 21.9096i 1.04927 + 0.721947i
\(922\) 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
\(927\) −3.49527 + 21.9212i −0.114800 + 0.719987i
\(928\) 0 0
\(929\) 38.0849 1.24952 0.624762 0.780815i \(-0.285196\pi\)
0.624762 + 0.780815i \(0.285196\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −16.0693 + 23.3550i −0.526084 + 0.764608i
\(934\) 0 0
\(935\) 16.1901 + 9.34735i 0.529472 + 0.305691i
\(936\) 0 0
\(937\) 37.6261i 1.22919i 0.788842 + 0.614596i \(0.210681\pi\)
−0.788842 + 0.614596i \(0.789319\pi\)
\(938\) 0 0
\(939\) −27.2766 18.7675i −0.890137 0.612454i
\(940\) 0 0
\(941\) −1.67547 −0.0546189 −0.0273094 0.999627i \(-0.508694\pi\)
−0.0273094 + 0.999627i \(0.508694\pi\)
\(942\) 0 0
\(943\) 26.8028i 0.872820i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 22.2085i 0.721679i −0.932628 0.360839i \(-0.882490\pi\)
0.932628 0.360839i \(-0.117510\pi\)
\(948\) 0 0
\(949\) 35.7248 1.15968
\(950\) 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
\(957\) −6.15844 + 8.95065i −0.199074 + 0.289333i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 30.0798 0.970317
\(962\) 0 0
\(963\) 11.7885 + 9.56641i 0.379880 + 0.308273i
\(964\) 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
\(969\) 69.4447 + 47.7811i 2.23089 + 1.53495i
\(970\) 0 0
\(971\) −17.8733 30.9574i −0.573580 0.993470i −0.996194 0.0871606i \(-0.972221\pi\)
0.422614 0.906310i \(-0.361113\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −30.3603 + 14.4618i −0.972308 + 0.463149i
\(976\) 0 0
\(977\) 45.6910i 1.46178i −0.682493 0.730892i \(-0.739104\pi\)
0.682493 0.730892i \(-0.260896\pi\)
\(978\) 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
\(983\) −9.33713 16.1724i −0.297808 0.515819i 0.677826 0.735222i \(-0.262922\pi\)
−0.975634 + 0.219404i \(0.929589\pi\)
\(984\) 0 0
\(985\) 1.48699 + 0.858517i 0.0473796 + 0.0273546i
\(986\) 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 0.165553 + 0.986201i \(0.447059\pi\)
−0.936851 + 0.349727i \(0.886274\pi\)
\(992\) 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
\(997\) −45.1982 + 26.0952i −1.43144 + 0.826442i −0.997231 0.0743700i \(-0.976305\pi\)
−0.434209 + 0.900812i \(0.642972\pi\)
\(998\) 0 0
\(999\) −10.3784 35.2379i −0.328358 1.11488i
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.w.a.1109.6 16
3.2 odd 2 5292.2.w.a.521.6 16
7.2 even 3 1764.2.bm.b.1685.6 16
7.3 odd 6 252.2.x.a.209.8 yes 16
7.4 even 3 252.2.x.a.209.1 yes 16
7.5 odd 6 1764.2.bm.b.1685.3 16
7.6 odd 2 inner 1764.2.w.a.1109.3 16
9.4 even 3 5292.2.bm.b.2285.6 16
9.5 odd 6 1764.2.bm.b.1697.3 16
21.2 odd 6 5292.2.bm.b.4625.3 16
21.5 even 6 5292.2.bm.b.4625.6 16
21.11 odd 6 756.2.x.a.629.6 16
21.17 even 6 756.2.x.a.629.3 16
21.20 even 2 5292.2.w.a.521.3 16
28.3 even 6 1008.2.cc.c.209.1 16
28.11 odd 6 1008.2.cc.c.209.8 16
63.4 even 3 756.2.x.a.125.3 16
63.5 even 6 inner 1764.2.w.a.509.6 16
63.11 odd 6 2268.2.f.b.1133.5 16
63.13 odd 6 5292.2.bm.b.2285.3 16
63.23 odd 6 inner 1764.2.w.a.509.3 16
63.25 even 3 2268.2.f.b.1133.11 16
63.31 odd 6 756.2.x.a.125.6 16
63.32 odd 6 252.2.x.a.41.8 yes 16
63.38 even 6 2268.2.f.b.1133.12 16
63.40 odd 6 5292.2.w.a.1097.6 16
63.41 even 6 1764.2.bm.b.1697.6 16
63.52 odd 6 2268.2.f.b.1133.6 16
63.58 even 3 5292.2.w.a.1097.3 16
63.59 even 6 252.2.x.a.41.1 16
84.11 even 6 3024.2.cc.c.2897.6 16
84.59 odd 6 3024.2.cc.c.2897.3 16
252.31 even 6 3024.2.cc.c.881.6 16
252.59 odd 6 1008.2.cc.c.545.8 16
252.67 odd 6 3024.2.cc.c.881.3 16
252.95 even 6 1008.2.cc.c.545.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.x.a.41.1 16 63.59 even 6
252.2.x.a.41.8 yes 16 63.32 odd 6
252.2.x.a.209.1 yes 16 7.4 even 3
252.2.x.a.209.8 yes 16 7.3 odd 6
756.2.x.a.125.3 16 63.4 even 3
756.2.x.a.125.6 16 63.31 odd 6
756.2.x.a.629.3 16 21.17 even 6
756.2.x.a.629.6 16 21.11 odd 6
1008.2.cc.c.209.1 16 28.3 even 6
1008.2.cc.c.209.8 16 28.11 odd 6
1008.2.cc.c.545.1 16 252.95 even 6
1008.2.cc.c.545.8 16 252.59 odd 6
1764.2.w.a.509.3 16 63.23 odd 6 inner
1764.2.w.a.509.6 16 63.5 even 6 inner
1764.2.w.a.1109.3 16 7.6 odd 2 inner
1764.2.w.a.1109.6 16 1.1 even 1 trivial
1764.2.bm.b.1685.3 16 7.5 odd 6
1764.2.bm.b.1685.6 16 7.2 even 3
1764.2.bm.b.1697.3 16 9.5 odd 6
1764.2.bm.b.1697.6 16 63.41 even 6
2268.2.f.b.1133.5 16 63.11 odd 6
2268.2.f.b.1133.6 16 63.52 odd 6
2268.2.f.b.1133.11 16 63.25 even 3
2268.2.f.b.1133.12 16 63.38 even 6
3024.2.cc.c.881.3 16 252.67 odd 6
3024.2.cc.c.881.6 16 252.31 even 6
3024.2.cc.c.2897.3 16 84.59 odd 6
3024.2.cc.c.2897.6 16 84.11 even 6
5292.2.w.a.521.3 16 21.20 even 2
5292.2.w.a.521.6 16 3.2 odd 2
5292.2.w.a.1097.3 16 63.58 even 3
5292.2.w.a.1097.6 16 63.40 odd 6
5292.2.bm.b.2285.3 16 63.13 odd 6
5292.2.bm.b.2285.6 16 9.4 even 3
5292.2.bm.b.4625.3 16 21.2 odd 6
5292.2.bm.b.4625.6 16 21.5 even 6