Properties

Label 2736.2.s.v.1873.1
Level $2736$
Weight $2$
Character 2736.1873
Analytic conductor $21.847$
Analytic rank $0$
Dimension $4$
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] = [2736,2,Mod(577,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.577");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.s (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: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1873.1
Root \(-1.32288 + 2.29129i\) of defining polynomial
Character \(\chi\) \(=\) 2736.1873
Dual form 2736.2.s.v.577.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.822876 + 1.42526i) q^{5} -3.64575 q^{7} +O(q^{10})\) \(q+(-0.822876 + 1.42526i) q^{5} -3.64575 q^{7} -4.64575 q^{11} +(-1.00000 - 1.73205i) q^{13} +(-1.67712 + 4.02334i) q^{19} +(0.822876 + 1.42526i) q^{23} +(1.14575 + 1.98450i) q^{25} +(0.822876 + 1.42526i) q^{29} +5.64575 q^{31} +(3.00000 - 5.19615i) q^{35} +0.354249 q^{37} +(-0.145751 + 0.252449i) q^{41} +(5.64575 - 9.77873i) q^{43} +(-2.17712 - 3.77089i) q^{47} +6.29150 q^{49} +(-6.29150 - 10.8972i) q^{53} +(3.82288 - 6.62141i) q^{55} +(3.96863 - 6.87386i) q^{59} +(-0.468627 - 0.811686i) q^{61} +3.29150 q^{65} +(0.322876 + 0.559237i) q^{67} +(1.35425 - 2.34563i) q^{71} +(-0.854249 + 1.47960i) q^{73} +16.9373 q^{77} +(-2.00000 + 3.46410i) q^{79} +7.93725 q^{83} +(3.64575 + 6.31463i) q^{91} +(-4.35425 - 5.70105i) q^{95} +(1.85425 - 3.21165i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{5} - 4 q^{7} - 8 q^{11} - 4 q^{13} - 12 q^{19} - 2 q^{23} - 6 q^{25} - 2 q^{29} + 12 q^{31} + 12 q^{35} + 12 q^{37} + 10 q^{41} + 12 q^{43} - 14 q^{47} + 4 q^{49} - 4 q^{53} + 10 q^{55} + 14 q^{61} - 8 q^{65} - 4 q^{67} + 16 q^{71} - 14 q^{73} + 36 q^{77} - 8 q^{79} + 4 q^{91} - 28 q^{95} + 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −0.822876 + 1.42526i −0.368001 + 0.637397i −0.989253 0.146214i \(-0.953291\pi\)
0.621252 + 0.783611i \(0.286624\pi\)
\(6\) 0 0
\(7\) −3.64575 −1.37796 −0.688982 0.724778i \(-0.741942\pi\)
−0.688982 + 0.724778i \(0.741942\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.64575 −1.40075 −0.700373 0.713777i \(-0.746983\pi\)
−0.700373 + 0.713777i \(0.746983\pi\)
\(12\) 0 0
\(13\) −1.00000 1.73205i −0.277350 0.480384i 0.693375 0.720577i \(-0.256123\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) −1.67712 + 4.02334i −0.384759 + 0.923017i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.822876 + 1.42526i 0.171581 + 0.297188i 0.938973 0.343991i \(-0.111779\pi\)
−0.767391 + 0.641179i \(0.778446\pi\)
\(24\) 0 0
\(25\) 1.14575 + 1.98450i 0.229150 + 0.396900i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.822876 + 1.42526i 0.152804 + 0.264665i 0.932257 0.361796i \(-0.117836\pi\)
−0.779453 + 0.626461i \(0.784503\pi\)
\(30\) 0 0
\(31\) 5.64575 1.01401 0.507003 0.861944i \(-0.330753\pi\)
0.507003 + 0.861944i \(0.330753\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.00000 5.19615i 0.507093 0.878310i
\(36\) 0 0
\(37\) 0.354249 0.0582381 0.0291191 0.999576i \(-0.490730\pi\)
0.0291191 + 0.999576i \(0.490730\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.145751 + 0.252449i −0.0227625 + 0.0394259i −0.877182 0.480158i \(-0.840579\pi\)
0.854420 + 0.519583i \(0.173913\pi\)
\(42\) 0 0
\(43\) 5.64575 9.77873i 0.860969 1.49124i −0.0100257 0.999950i \(-0.503191\pi\)
0.870995 0.491292i \(-0.163475\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.17712 3.77089i −0.317566 0.550041i 0.662413 0.749138i \(-0.269532\pi\)
−0.979980 + 0.199098i \(0.936199\pi\)
\(48\) 0 0
\(49\) 6.29150 0.898786
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.29150 10.8972i −0.864204 1.49685i −0.867835 0.496852i \(-0.834489\pi\)
0.00363070 0.999993i \(-0.498844\pi\)
\(54\) 0 0
\(55\) 3.82288 6.62141i 0.515476 0.892831i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.96863 6.87386i 0.516671 0.894901i −0.483141 0.875542i \(-0.660504\pi\)
0.999813 0.0193585i \(-0.00616237\pi\)
\(60\) 0 0
\(61\) −0.468627 0.811686i −0.0600015 0.103926i 0.834464 0.551062i \(-0.185777\pi\)
−0.894466 + 0.447136i \(0.852444\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.29150 0.408261
\(66\) 0 0
\(67\) 0.322876 + 0.559237i 0.0394455 + 0.0683217i 0.885074 0.465450i \(-0.154108\pi\)
−0.845629 + 0.533772i \(0.820774\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.35425 2.34563i 0.160720 0.278375i −0.774407 0.632687i \(-0.781952\pi\)
0.935127 + 0.354313i \(0.115285\pi\)
\(72\) 0 0
\(73\) −0.854249 + 1.47960i −0.0999822 + 0.173174i −0.911677 0.410907i \(-0.865212\pi\)
0.811695 + 0.584082i \(0.198545\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 16.9373 1.93018
\(78\) 0 0
\(79\) −2.00000 + 3.46410i −0.225018 + 0.389742i −0.956325 0.292306i \(-0.905577\pi\)
0.731307 + 0.682048i \(0.238911\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.93725 0.871227 0.435613 0.900134i \(-0.356531\pi\)
0.435613 + 0.900134i \(0.356531\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) 3.64575 + 6.31463i 0.382179 + 0.661953i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.35425 5.70105i −0.446736 0.584915i
\(96\) 0 0
\(97\) 1.85425 3.21165i 0.188270 0.326094i −0.756403 0.654106i \(-0.773045\pi\)
0.944674 + 0.328012i \(0.106379\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.82288 11.8176i −0.678902 1.17589i −0.975312 0.220831i \(-0.929123\pi\)
0.296411 0.955061i \(-0.404210\pi\)
\(102\) 0 0
\(103\) 13.2915 1.30965 0.654825 0.755780i \(-0.272742\pi\)
0.654825 + 0.755780i \(0.272742\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 15.2915 1.47829 0.739143 0.673549i \(-0.235231\pi\)
0.739143 + 0.673549i \(0.235231\pi\)
\(108\) 0 0
\(109\) −7.29150 + 12.6293i −0.698399 + 1.20966i 0.270622 + 0.962686i \(0.412771\pi\)
−0.969021 + 0.246977i \(0.920563\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 15.5830 1.46593 0.732963 0.680269i \(-0.238137\pi\)
0.732963 + 0.680269i \(0.238137\pi\)
\(114\) 0 0
\(115\) −2.70850 −0.252569
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 10.5830 0.962091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −6.64575 11.5108i −0.589715 1.02142i −0.994270 0.106903i \(-0.965907\pi\)
0.404554 0.914514i \(-0.367427\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −0.968627 + 1.67771i −0.0846293 + 0.146582i −0.905233 0.424915i \(-0.860304\pi\)
0.820604 + 0.571497i \(0.193637\pi\)
\(132\) 0 0
\(133\) 6.11438 14.6681i 0.530184 1.27188i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 7.79150 + 13.4953i 0.665673 + 1.15298i 0.979102 + 0.203368i \(0.0651888\pi\)
−0.313429 + 0.949612i \(0.601478\pi\)
\(138\) 0 0
\(139\) 9.32288 + 16.1477i 0.790756 + 1.36963i 0.925499 + 0.378749i \(0.123646\pi\)
−0.134743 + 0.990881i \(0.543021\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.64575 + 8.04668i 0.388497 + 0.672897i
\(144\) 0 0
\(145\) −2.70850 −0.224928
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 5.46863 9.47194i 0.448007 0.775972i −0.550249 0.835001i \(-0.685467\pi\)
0.998256 + 0.0590292i \(0.0188005\pi\)
\(150\) 0 0
\(151\) −12.9373 −1.05282 −0.526409 0.850231i \(-0.676462\pi\)
−0.526409 + 0.850231i \(0.676462\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.64575 + 8.04668i −0.373156 + 0.646325i
\(156\) 0 0
\(157\) 5.29150 9.16515i 0.422308 0.731459i −0.573857 0.818956i \(-0.694553\pi\)
0.996165 + 0.0874969i \(0.0278868\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3.00000 5.19615i −0.236433 0.409514i
\(162\) 0 0
\(163\) −3.93725 −0.308390 −0.154195 0.988040i \(-0.549278\pi\)
−0.154195 + 0.988040i \(0.549278\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.00000 + 10.3923i 0.464294 + 0.804181i 0.999169 0.0407502i \(-0.0129748\pi\)
−0.534875 + 0.844931i \(0.679641\pi\)
\(168\) 0 0
\(169\) 4.50000 7.79423i 0.346154 0.599556i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −3.00000 + 5.19615i −0.228086 + 0.395056i −0.957241 0.289292i \(-0.906580\pi\)
0.729155 + 0.684349i \(0.239913\pi\)
\(174\) 0 0
\(175\) −4.17712 7.23499i −0.315761 0.546914i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.06275 0.303664 0.151832 0.988406i \(-0.451483\pi\)
0.151832 + 0.988406i \(0.451483\pi\)
\(180\) 0 0
\(181\) −11.1144 19.2507i −0.826125 1.43089i −0.901056 0.433702i \(-0.857207\pi\)
0.0749311 0.997189i \(-0.476126\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.291503 + 0.504897i −0.0214317 + 0.0371208i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6.58301 0.476330 0.238165 0.971225i \(-0.423454\pi\)
0.238165 + 0.971225i \(0.423454\pi\)
\(192\) 0 0
\(193\) −7.29150 + 12.6293i −0.524854 + 0.909074i 0.474727 + 0.880133i \(0.342547\pi\)
−0.999581 + 0.0289406i \(0.990787\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.64575 0.544737 0.272369 0.962193i \(-0.412193\pi\)
0.272369 + 0.962193i \(0.412193\pi\)
\(198\) 0 0
\(199\) −9.93725 17.2118i −0.704433 1.22011i −0.966896 0.255172i \(-0.917868\pi\)
0.262462 0.964942i \(-0.415465\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −3.00000 5.19615i −0.210559 0.364698i
\(204\) 0 0
\(205\) −0.239870 0.415468i −0.0167533 0.0290175i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 7.79150 18.6914i 0.538950 1.29291i
\(210\) 0 0
\(211\) −6.64575 + 11.5108i −0.457512 + 0.792435i −0.998829 0.0483843i \(-0.984593\pi\)
0.541316 + 0.840819i \(0.317926\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 9.29150 + 16.0934i 0.633675 + 1.09756i
\(216\) 0 0
\(217\) −20.5830 −1.39727
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −9.40588 + 16.2915i −0.629864 + 1.09096i 0.357714 + 0.933831i \(0.383556\pi\)
−0.987579 + 0.157126i \(0.949777\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −7.35425 −0.488119 −0.244059 0.969760i \(-0.578479\pi\)
−0.244059 + 0.969760i \(0.578479\pi\)
\(228\) 0 0
\(229\) 20.0000 1.32164 0.660819 0.750546i \(-0.270209\pi\)
0.660819 + 0.750546i \(0.270209\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.43725 16.3458i 0.618255 1.07085i −0.371549 0.928413i \(-0.621173\pi\)
0.989804 0.142436i \(-0.0454935\pi\)
\(234\) 0 0
\(235\) 7.16601 0.467459
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 3.79150 + 6.56708i 0.244232 + 0.423022i 0.961915 0.273347i \(-0.0881308\pi\)
−0.717683 + 0.696370i \(0.754797\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −5.17712 + 8.96704i −0.330754 + 0.572883i
\(246\) 0 0
\(247\) 8.64575 1.11847i 0.550116 0.0711668i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −14.6144 25.3128i −0.922451 1.59773i −0.795610 0.605810i \(-0.792849\pi\)
−0.126842 0.991923i \(-0.540484\pi\)
\(252\) 0 0
\(253\) −3.82288 6.62141i −0.240342 0.416285i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.14575 + 10.6448i 0.383361 + 0.664001i 0.991540 0.129798i \(-0.0414330\pi\)
−0.608179 + 0.793800i \(0.708100\pi\)
\(258\) 0 0
\(259\) −1.29150 −0.0802501
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 5.46863 9.47194i 0.337210 0.584065i −0.646697 0.762747i \(-0.723850\pi\)
0.983907 + 0.178682i \(0.0571834\pi\)
\(264\) 0 0
\(265\) 20.7085 1.27211
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(270\) 0 0
\(271\) 6.17712 10.6991i 0.375234 0.649924i −0.615128 0.788427i \(-0.710896\pi\)
0.990362 + 0.138503i \(0.0442292\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −5.32288 9.21949i −0.320981 0.555956i
\(276\) 0 0
\(277\) −27.5203 −1.65353 −0.826766 0.562546i \(-0.809822\pi\)
−0.826766 + 0.562546i \(0.809822\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 12.7288 + 22.0469i 0.759334 + 1.31520i 0.943191 + 0.332252i \(0.107808\pi\)
−0.183857 + 0.982953i \(0.558858\pi\)
\(282\) 0 0
\(283\) 15.3229 26.5400i 0.910850 1.57764i 0.0979848 0.995188i \(-0.468760\pi\)
0.812866 0.582451i \(-0.197906\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.531373 0.920365i 0.0313660 0.0543274i
\(288\) 0 0
\(289\) 8.50000 + 14.7224i 0.500000 + 0.866025i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −28.9373 −1.69053 −0.845266 0.534345i \(-0.820558\pi\)
−0.845266 + 0.534345i \(0.820558\pi\)
\(294\) 0 0
\(295\) 6.53137 + 11.3127i 0.380271 + 0.658649i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.64575 2.85052i 0.0951763 0.164850i
\(300\) 0 0
\(301\) −20.5830 + 35.6508i −1.18638 + 2.05488i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.54249 0.0883225
\(306\) 0 0
\(307\) 0.322876 0.559237i 0.0184275 0.0319173i −0.856665 0.515874i \(-0.827467\pi\)
0.875092 + 0.483956i \(0.160801\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 13.6458 0.773780 0.386890 0.922126i \(-0.373549\pi\)
0.386890 + 0.922126i \(0.373549\pi\)
\(312\) 0 0
\(313\) −4.43725 7.68555i −0.250808 0.434413i 0.712940 0.701225i \(-0.247363\pi\)
−0.963749 + 0.266812i \(0.914030\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3.00000 5.19615i −0.168497 0.291845i 0.769395 0.638774i \(-0.220558\pi\)
−0.937892 + 0.346929i \(0.887225\pi\)
\(318\) 0 0
\(319\) −3.82288 6.62141i −0.214040 0.370728i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 2.29150 3.96900i 0.127110 0.220160i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 7.93725 + 13.7477i 0.437595 + 0.757937i
\(330\) 0 0
\(331\) −19.8118 −1.08895 −0.544476 0.838776i \(-0.683272\pi\)
−0.544476 + 0.838776i \(0.683272\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.06275 −0.0580640
\(336\) 0 0
\(337\) 4.85425 8.40781i 0.264428 0.458002i −0.702986 0.711204i \(-0.748150\pi\)
0.967414 + 0.253202i \(0.0814836\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −26.2288 −1.42037
\(342\) 0 0
\(343\) 2.58301 0.139469
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 11.6144 20.1167i 0.623492 1.07992i −0.365338 0.930875i \(-0.619047\pi\)
0.988830 0.149046i \(-0.0476201\pi\)
\(348\) 0 0
\(349\) 21.1660 1.13299 0.566495 0.824065i \(-0.308299\pi\)
0.566495 + 0.824065i \(0.308299\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −12.8745 −0.685241 −0.342620 0.939474i \(-0.611314\pi\)
−0.342620 + 0.939474i \(0.611314\pi\)
\(354\) 0 0
\(355\) 2.22876 + 3.86032i 0.118290 + 0.204884i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2.46863 4.27579i 0.130289 0.225667i −0.793499 0.608572i \(-0.791743\pi\)
0.923788 + 0.382904i \(0.125076\pi\)
\(360\) 0 0
\(361\) −13.3745 13.4953i −0.703921 0.710278i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1.40588 2.43506i −0.0735872 0.127457i
\(366\) 0 0
\(367\) 8.11438 + 14.0545i 0.423567 + 0.733640i 0.996285 0.0861125i \(-0.0274444\pi\)
−0.572718 + 0.819752i \(0.694111\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 22.9373 + 39.7285i 1.19084 + 2.06260i
\(372\) 0 0
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.64575 2.85052i 0.0847605 0.146810i
\(378\) 0 0
\(379\) −10.7085 −0.550059 −0.275029 0.961436i \(-0.588688\pi\)
−0.275029 + 0.961436i \(0.588688\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −2.76013 + 4.78068i −0.141036 + 0.244282i −0.927887 0.372861i \(-0.878377\pi\)
0.786851 + 0.617143i \(0.211710\pi\)
\(384\) 0 0
\(385\) −13.9373 + 24.1400i −0.710308 + 1.23029i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 6.00000 + 10.3923i 0.304212 + 0.526911i 0.977086 0.212847i \(-0.0682735\pi\)
−0.672874 + 0.739758i \(0.734940\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −3.29150 5.70105i −0.165613 0.286851i
\(396\) 0 0
\(397\) −10.5314 + 18.2409i −0.528554 + 0.915483i 0.470891 + 0.882191i \(0.343932\pi\)
−0.999446 + 0.0332919i \(0.989401\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 13.7915 23.8876i 0.688715 1.19289i −0.283539 0.958961i \(-0.591509\pi\)
0.972254 0.233928i \(-0.0751581\pi\)
\(402\) 0 0
\(403\) −5.64575 9.77873i −0.281235 0.487113i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.64575 −0.0815769
\(408\) 0 0
\(409\) 3.79150 + 6.56708i 0.187478 + 0.324721i 0.944409 0.328774i \(-0.106635\pi\)
−0.756931 + 0.653495i \(0.773302\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −14.4686 + 25.0604i −0.711955 + 1.23314i
\(414\) 0 0
\(415\) −6.53137 + 11.3127i −0.320612 + 0.555317i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −31.7490 −1.55104 −0.775520 0.631322i \(-0.782512\pi\)
−0.775520 + 0.631322i \(0.782512\pi\)
\(420\) 0 0
\(421\) 12.4059 21.4876i 0.604626 1.04724i −0.387485 0.921876i \(-0.626656\pi\)
0.992111 0.125366i \(-0.0400106\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.70850 + 2.95920i 0.0826800 + 0.143206i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −13.9373 24.1400i −0.671334 1.16278i −0.977526 0.210815i \(-0.932388\pi\)
0.306192 0.951970i \(-0.400945\pi\)
\(432\) 0 0
\(433\) −8.93725 15.4798i −0.429497 0.743911i 0.567332 0.823489i \(-0.307976\pi\)
−0.996829 + 0.0795788i \(0.974642\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −7.11438 + 0.920365i −0.340327 + 0.0440270i
\(438\) 0 0
\(439\) 5.40588 9.36326i 0.258009 0.446884i −0.707700 0.706513i \(-0.750267\pi\)
0.965708 + 0.259629i \(0.0836004\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −5.32288 9.21949i −0.252897 0.438031i 0.711425 0.702762i \(-0.248050\pi\)
−0.964322 + 0.264731i \(0.914717\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 24.2915 1.14639 0.573193 0.819420i \(-0.305704\pi\)
0.573193 + 0.819420i \(0.305704\pi\)
\(450\) 0 0
\(451\) 0.677124 1.17281i 0.0318845 0.0552256i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −12.0000 −0.562569
\(456\) 0 0
\(457\) 32.8745 1.53780 0.768902 0.639366i \(-0.220803\pi\)
0.768902 + 0.639366i \(0.220803\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9.58301 16.5983i 0.446325 0.773058i −0.551818 0.833964i \(-0.686066\pi\)
0.998143 + 0.0609066i \(0.0193992\pi\)
\(462\) 0 0
\(463\) 38.4575 1.78727 0.893636 0.448792i \(-0.148146\pi\)
0.893636 + 0.448792i \(0.148146\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −19.3542 −0.895608 −0.447804 0.894132i \(-0.647794\pi\)
−0.447804 + 0.894132i \(0.647794\pi\)
\(468\) 0 0
\(469\) −1.17712 2.03884i −0.0543546 0.0941448i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −26.2288 + 45.4295i −1.20600 + 2.08885i
\(474\) 0 0
\(475\) −9.90588 + 1.28149i −0.454513 + 0.0587989i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −3.29150 5.70105i −0.150393 0.260488i 0.780979 0.624557i \(-0.214720\pi\)
−0.931372 + 0.364069i \(0.881387\pi\)
\(480\) 0 0
\(481\) −0.354249 0.613577i −0.0161523 0.0279767i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.05163 + 5.28558i 0.138567 + 0.240006i
\(486\) 0 0
\(487\) −4.22876 −0.191623 −0.0958116 0.995399i \(-0.530545\pi\)
−0.0958116 + 0.995399i \(0.530545\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −19.6458 + 34.0274i −0.886600 + 1.53564i −0.0427320 + 0.999087i \(0.513606\pi\)
−0.843868 + 0.536550i \(0.819727\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −4.93725 + 8.55157i −0.221466 + 0.383591i
\(498\) 0 0
\(499\) −2.38562 + 4.13202i −0.106795 + 0.184975i −0.914470 0.404653i \(-0.867392\pi\)
0.807675 + 0.589628i \(0.200726\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 20.4686 + 35.4527i 0.912651 + 1.58076i 0.810305 + 0.586009i \(0.199302\pi\)
0.102346 + 0.994749i \(0.467365\pi\)
\(504\) 0 0
\(505\) 22.4575 0.999346
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −15.8745 27.4955i −0.703625 1.21871i −0.967185 0.254072i \(-0.918230\pi\)
0.263560 0.964643i \(-0.415103\pi\)
\(510\) 0 0
\(511\) 3.11438 5.39426i 0.137772 0.238628i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −10.9373 + 18.9439i −0.481953 + 0.834767i
\(516\) 0 0
\(517\) 10.1144 + 17.5186i 0.444830 + 0.770468i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 11.7085 0.512959 0.256479 0.966550i \(-0.417437\pi\)
0.256479 + 0.966550i \(0.417437\pi\)
\(522\) 0 0
\(523\) −0.937254 1.62337i −0.0409833 0.0709851i 0.844806 0.535072i \(-0.179716\pi\)
−0.885789 + 0.464087i \(0.846382\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 10.1458 17.5730i 0.441120 0.764042i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0.583005 0.0252528
\(534\) 0 0
\(535\) −12.5830 + 21.7944i −0.544011 + 0.942254i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −29.2288 −1.25897
\(540\) 0 0
\(541\) −4.00000 6.92820i −0.171973 0.297867i 0.767136 0.641484i \(-0.221681\pi\)
−0.939110 + 0.343617i \(0.888348\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −12.0000 20.7846i −0.514024 0.890315i
\(546\) 0 0
\(547\) 5.64575 + 9.77873i 0.241395 + 0.418108i 0.961112 0.276159i \(-0.0890618\pi\)
−0.719717 + 0.694268i \(0.755728\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −7.11438 + 0.920365i −0.303083 + 0.0392089i
\(552\) 0 0
\(553\) 7.29150 12.6293i 0.310066 0.537050i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −2.70850 4.69126i −0.114763 0.198775i 0.802922 0.596084i \(-0.203277\pi\)
−0.917685 + 0.397309i \(0.869944\pi\)
\(558\) 0 0
\(559\) −22.5830 −0.955159
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 10.0627 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(564\) 0 0
\(565\) −12.8229 + 22.2099i −0.539462 + 0.934376i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −6.58301 −0.275974 −0.137987 0.990434i \(-0.544063\pi\)
−0.137987 + 0.990434i \(0.544063\pi\)
\(570\) 0 0
\(571\) −7.81176 −0.326912 −0.163456 0.986551i \(-0.552264\pi\)
−0.163456 + 0.986551i \(0.552264\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.88562 + 3.26599i −0.0786359 + 0.136201i
\(576\) 0 0
\(577\) 11.0000 0.457936 0.228968 0.973434i \(-0.426465\pi\)
0.228968 + 0.973434i \(0.426465\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −28.9373 −1.20052
\(582\) 0 0
\(583\) 29.2288 + 50.6257i 1.21053 + 2.09670i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 7.06275 12.2330i 0.291511 0.504911i −0.682656 0.730739i \(-0.739175\pi\)
0.974167 + 0.225828i \(0.0725088\pi\)
\(588\) 0 0
\(589\) −9.46863 + 22.7148i −0.390148 + 0.935946i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 14.8542 + 25.7283i 0.609991 + 1.05654i 0.991241 + 0.132063i \(0.0421602\pi\)
−0.381250 + 0.924472i \(0.624506\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 8.46863 + 14.6681i 0.346019 + 0.599322i 0.985538 0.169453i \(-0.0542000\pi\)
−0.639520 + 0.768775i \(0.720867\pi\)
\(600\) 0 0
\(601\) 10.4170 0.424918 0.212459 0.977170i \(-0.431853\pi\)
0.212459 + 0.977170i \(0.431853\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −8.70850 + 15.0836i −0.354051 + 0.613234i
\(606\) 0 0
\(607\) −6.93725 −0.281574 −0.140787 0.990040i \(-0.544963\pi\)
−0.140787 + 0.990040i \(0.544963\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −4.35425 + 7.54178i −0.176154 + 0.305108i
\(612\) 0 0
\(613\) −3.70850 + 6.42331i −0.149785 + 0.259435i −0.931148 0.364642i \(-0.881191\pi\)
0.781363 + 0.624077i \(0.214525\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −15.4373 26.7381i −0.621480 1.07644i −0.989210 0.146503i \(-0.953198\pi\)
0.367730 0.929933i \(-0.380135\pi\)
\(618\) 0 0
\(619\) 8.45751 0.339936 0.169968 0.985450i \(-0.445634\pi\)
0.169968 + 0.985450i \(0.445634\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 4.14575 7.18065i 0.165830 0.287226i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −12.4059 21.4876i −0.493870 0.855408i 0.506105 0.862472i \(-0.331085\pi\)
−0.999975 + 0.00706354i \(0.997752\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 21.8745 0.868063
\(636\) 0 0
\(637\) −6.29150 10.8972i −0.249278 0.431763i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 6.43725 11.1497i 0.254256 0.440385i −0.710437 0.703761i \(-0.751503\pi\)
0.964693 + 0.263376i \(0.0848360\pi\)
\(642\) 0 0
\(643\) −3.26013 + 5.64671i −0.128567 + 0.222685i −0.923122 0.384508i \(-0.874371\pi\)
0.794555 + 0.607193i \(0.207704\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −22.4575 −0.882896 −0.441448 0.897287i \(-0.645535\pi\)
−0.441448 + 0.897287i \(0.645535\pi\)
\(648\) 0 0
\(649\) −18.4373 + 31.9343i −0.723726 + 1.25353i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −24.0000 −0.939193 −0.469596 0.882881i \(-0.655601\pi\)
−0.469596 + 0.882881i \(0.655601\pi\)
\(654\) 0 0
\(655\) −1.59412 2.76110i −0.0622874 0.107885i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 9.29150 + 16.0934i 0.361946 + 0.626908i 0.988281 0.152646i \(-0.0487793\pi\)
−0.626335 + 0.779554i \(0.715446\pi\)
\(660\) 0 0
\(661\) −8.11438 14.0545i −0.315613 0.546657i 0.663955 0.747773i \(-0.268877\pi\)
−0.979568 + 0.201115i \(0.935543\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 15.8745 + 20.7846i 0.615587 + 0.805993i
\(666\) 0 0
\(667\) −1.35425 + 2.34563i −0.0524367 + 0.0908231i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 2.17712 + 3.77089i 0.0840470 + 0.145574i
\(672\) 0 0
\(673\) −13.8745 −0.534823 −0.267411 0.963582i \(-0.586168\pi\)
−0.267411 + 0.963582i \(0.586168\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 11.4170 0.438791 0.219395 0.975636i \(-0.429592\pi\)
0.219395 + 0.975636i \(0.429592\pi\)
\(678\) 0 0
\(679\) −6.76013 + 11.7089i −0.259430 + 0.449346i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −5.41699 −0.207276 −0.103638 0.994615i \(-0.533048\pi\)
−0.103638 + 0.994615i \(0.533048\pi\)
\(684\) 0 0
\(685\) −25.6458 −0.979874
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −12.5830 + 21.7944i −0.479374 + 0.830301i
\(690\) 0 0
\(691\) −2.58301 −0.0982622 −0.0491311 0.998792i \(-0.515645\pi\)
−0.0491311 + 0.998792i \(0.515645\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −30.6863 −1.16400
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 5.17712 8.96704i 0.195537 0.338681i −0.751539 0.659688i \(-0.770688\pi\)
0.947077 + 0.321008i \(0.104022\pi\)
\(702\) 0 0
\(703\) −0.594119 + 1.42526i −0.0224076 + 0.0537548i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 24.8745 + 43.0839i 0.935502 + 1.62034i
\(708\) 0 0
\(709\) −1.82288 3.15731i −0.0684595 0.118575i 0.829764 0.558115i \(-0.188475\pi\)
−0.898223 + 0.439539i \(0.855142\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 4.64575 + 8.04668i 0.173985 + 0.301350i
\(714\) 0 0
\(715\) −15.2915 −0.571870
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −1.35425 + 2.34563i −0.0505050 + 0.0874771i −0.890173 0.455623i \(-0.849416\pi\)
0.839668 + 0.543100i \(0.182750\pi\)
\(720\) 0 0
\(721\) −48.4575 −1.80465
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.88562 + 3.26599i −0.0700302 + 0.121296i
\(726\) 0 0
\(727\) 0.708497 1.22715i 0.0262767 0.0455126i −0.852588 0.522584i \(-0.824968\pi\)
0.878865 + 0.477071i \(0.158302\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −16.1033 −0.594788 −0.297394 0.954755i \(-0.596117\pi\)
−0.297394 + 0.954755i \(0.596117\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.50000 2.59808i −0.0552532 0.0957014i
\(738\) 0 0
\(739\) −16.9059 + 29.2818i −0.621893 + 1.07715i 0.367240 + 0.930126i \(0.380303\pi\)
−0.989133 + 0.147024i \(0.953031\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 23.7601 41.1538i 0.871675 1.50978i 0.0114112 0.999935i \(-0.496368\pi\)
0.860263 0.509850i \(-0.170299\pi\)
\(744\) 0 0
\(745\) 9.00000 + 15.5885i 0.329734 + 0.571117i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −55.7490 −2.03702
\(750\) 0 0
\(751\) −3.93725 6.81952i −0.143672 0.248848i 0.785204 0.619237i \(-0.212558\pi\)
−0.928877 + 0.370389i \(0.879225\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 10.6458 18.4390i 0.387439 0.671063i
\(756\) 0 0
\(757\) 2.29150 3.96900i 0.0832861 0.144256i −0.821374 0.570391i \(-0.806792\pi\)
0.904660 + 0.426135i \(0.140125\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −42.8745 −1.55420 −0.777100 0.629377i \(-0.783310\pi\)
−0.777100 + 0.629377i \(0.783310\pi\)
\(762\) 0 0
\(763\) 26.5830 46.0431i 0.962369 1.66687i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −15.8745 −0.573195
\(768\) 0 0
\(769\) −17.6458 30.5633i −0.636322 1.10214i −0.986233 0.165359i \(-0.947122\pi\)
0.349911 0.936783i \(-0.386212\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −2.46863 4.27579i −0.0887903 0.153789i 0.818210 0.574920i \(-0.194967\pi\)
−0.907000 + 0.421130i \(0.861633\pi\)
\(774\) 0 0
\(775\) 6.46863 + 11.2040i 0.232360 + 0.402459i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −0.771243 1.00979i −0.0276327 0.0361796i
\(780\) 0 0
\(781\) −6.29150 + 10.8972i −0.225128 + 0.389933i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 8.70850 + 15.0836i 0.310820 + 0.538355i
\(786\) 0 0
\(787\) 42.5203 1.51568 0.757842 0.652438i \(-0.226254\pi\)
0.757842 + 0.652438i \(0.226254\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −56.8118 −2.01999
\(792\) 0 0
\(793\) −0.937254 + 1.62337i −0.0332829 + 0.0576476i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 44.8118 1.58731 0.793657 0.608365i \(-0.208174\pi\)
0.793657 + 0.608365i \(0.208174\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 3.96863 6.87386i 0.140050 0.242573i
\(804\) 0 0
\(805\) 9.87451 0.348031
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 9.00000 0.316423 0.158212 0.987405i \(-0.449427\pi\)
0.158212 + 0.987405i \(0.449427\pi\)
\(810\) 0 0
\(811\) −15.6458 27.0992i −0.549397 0.951583i −0.998316 0.0580106i \(-0.981524\pi\)
0.448919 0.893572i \(-0.351809\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 3.23987 5.61162i 0.113488 0.196566i
\(816\) 0 0
\(817\) 29.8745 + 39.1149i 1.04518 + 1.36846i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −3.00000 5.19615i −0.104701 0.181347i 0.808915 0.587925i \(-0.200055\pi\)
−0.913616 + 0.406578i \(0.866722\pi\)
\(822\) 0 0
\(823\) −15.9373 27.6041i −0.555538 0.962220i −0.997861 0.0653641i \(-0.979179\pi\)
0.442324 0.896855i \(-0.354154\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 26.3229 + 45.5926i 0.915336 + 1.58541i 0.806408 + 0.591359i \(0.201408\pi\)
0.108928 + 0.994050i \(0.465258\pi\)
\(828\) 0 0
\(829\) −17.1660 −0.596200 −0.298100 0.954535i \(-0.596353\pi\)
−0.298100 + 0.954535i \(0.596353\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −19.7490 −0.683443
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 20.7601 35.9576i 0.716719 1.24139i −0.245573 0.969378i \(-0.578976\pi\)
0.962293 0.272016i \(-0.0876904\pi\)
\(840\) 0 0
\(841\) 13.1458 22.7691i 0.453302 0.785142i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 7.40588 + 12.8274i 0.254770 + 0.441275i
\(846\) 0 0
\(847\) −38.5830 −1.32573
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0.291503 + 0.504897i 0.00999258 + 0.0173077i
\(852\) 0 0
\(853\) −4.29150 + 7.43310i −0.146938 + 0.254505i −0.930094 0.367321i \(-0.880275\pi\)
0.783156 + 0.621825i \(0.213609\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 10.5000 18.1865i 0.358673 0.621240i −0.629066 0.777352i \(-0.716563\pi\)
0.987739 + 0.156112i \(0.0498959\pi\)
\(858\) 0 0
\(859\) 6.61438 + 11.4564i 0.225680 + 0.390889i 0.956523 0.291656i \(-0.0942064\pi\)
−0.730843 + 0.682545i \(0.760873\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −31.0627 −1.05739 −0.528694 0.848812i \(-0.677318\pi\)
−0.528694 + 0.848812i \(0.677318\pi\)
\(864\) 0 0
\(865\) −4.93725 8.55157i −0.167872 0.290762i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 9.29150 16.0934i 0.315193 0.545930i
\(870\) 0 0
\(871\) 0.645751 1.11847i 0.0218804 0.0378980i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 43.7490 1.47899
\(876\) 0 0
\(877\) 20.8229 36.0663i 0.703139 1.21787i −0.264221 0.964462i \(-0.585115\pi\)
0.967359 0.253409i \(-0.0815520\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 36.8745 1.24233 0.621167 0.783678i \(-0.286659\pi\)
0.621167 + 0.783678i \(0.286659\pi\)
\(882\) 0 0
\(883\) −14.1974 24.5906i −0.477780 0.827539i 0.521896 0.853009i \(-0.325225\pi\)
−0.999676 + 0.0254701i \(0.991892\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 8.70850 + 15.0836i 0.292403 + 0.506456i 0.974377 0.224919i \(-0.0722119\pi\)
−0.681975 + 0.731376i \(0.738879\pi\)
\(888\) 0 0
\(889\) 24.2288 + 41.9654i 0.812606 + 1.40748i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 18.8229 2.43506i 0.629884 0.0814861i
\(894\) 0 0
\(895\) −3.34313 + 5.79048i −0.111749 + 0.193554i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 4.64575 + 8.04668i 0.154944 + 0.268372i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 36.5830 1.21606
\(906\) 0 0
\(907\) 19.9686 34.5867i 0.663047 1.14843i −0.316763 0.948505i \(-0.602596\pi\)
0.979811 0.199927i \(-0.0640706\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 16.9373 0.561156 0.280578 0.959831i \(-0.409474\pi\)
0.280578 + 0.959831i \(0.409474\pi\)
\(912\) 0 0
\(913\) −36.8745 −1.22037
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 3.53137 6.11652i 0.116616 0.201985i
\(918\) 0 0
\(919\) 19.8745 0.655600 0.327800 0.944747i \(-0.393693\pi\)
0.327800 + 0.944747i \(0.393693\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −5.41699 −0.178303
\(924\) 0 0
\(925\) 0.405881 + 0.703006i 0.0133453 + 0.0231147i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −4.79150 + 8.29913i −0.157204 + 0.272285i −0.933859 0.357640i \(-0.883581\pi\)
0.776655 + 0.629926i \(0.216915\pi\)
\(930\) 0 0
\(931\) −10.5516 + 25.3128i −0.345816 + 0.829595i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −3.56275 6.17086i −0.116390 0.201593i 0.801945 0.597398i \(-0.203799\pi\)
−0.918334 + 0.395805i \(0.870466\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 8.41699 + 14.5787i 0.274386 + 0.475251i 0.969980 0.243184i \(-0.0781920\pi\)
−0.695594 + 0.718435i \(0.744859\pi\)
\(942\) 0 0
\(943\) −0.479741 −0.0156225
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 3.87451 6.71084i 0.125905 0.218073i −0.796182 0.605058i \(-0.793150\pi\)
0.922086 + 0.386985i \(0.126483\pi\)
\(948\) 0 0
\(949\) 3.41699 0.110920
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −6.72876 + 11.6545i −0.217966 + 0.377528i −0.954186 0.299214i \(-0.903275\pi\)
0.736220 + 0.676742i \(0.236609\pi\)
\(954\) 0 0
\(955\) −5.41699 + 9.38251i −0.175290 + 0.303611i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −28.4059 49.2004i −0.917274 1.58876i
\(960\) 0 0
\(961\) 0.874508 0.0282099
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −12.0000 20.7846i −0.386294 0.669080i
\(966\) 0 0
\(967\) −6.64575 + 11.5108i −0.213713 + 0.370162i −0.952874 0.303367i \(-0.901889\pi\)
0.739161 + 0.673529i \(0.235222\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 27.1974 47.1073i 0.872806 1.51174i 0.0137234 0.999906i \(-0.495632\pi\)
0.859082 0.511838i \(-0.171035\pi\)
\(972\) 0 0
\(973\) −33.9889 58.8705i −1.08963 1.88730i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 7.45751 0.238587 0.119293 0.992859i \(-0.461937\pi\)
0.119293 + 0.992859i \(0.461937\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −15.8745 + 27.4955i −0.506318 + 0.876969i 0.493655 + 0.869658i \(0.335661\pi\)
−0.999973 + 0.00731102i \(0.997673\pi\)
\(984\) 0 0
\(985\) −6.29150 + 10.8972i −0.200464 + 0.347214i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 18.5830 0.590905
\(990\) 0 0
\(991\) −1.41699 + 2.45431i −0.0450123 + 0.0779636i −0.887654 0.460511i \(-0.847666\pi\)
0.842641 + 0.538475i \(0.180999\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 32.7085 1.03693
\(996\) 0 0
\(997\) −8.11438 14.0545i −0.256985 0.445111i 0.708448 0.705763i \(-0.249396\pi\)
−0.965433 + 0.260652i \(0.916062\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2736.2.s.v.1873.1 4
3.2 odd 2 304.2.i.e.49.2 4
4.3 odd 2 342.2.g.f.163.1 4
12.11 even 2 38.2.c.b.11.1 yes 4
19.7 even 3 inner 2736.2.s.v.577.1 4
24.5 odd 2 1216.2.i.k.961.1 4
24.11 even 2 1216.2.i.l.961.2 4
57.8 even 6 5776.2.a.z.1.2 2
57.11 odd 6 5776.2.a.ba.1.1 2
57.26 odd 6 304.2.i.e.273.2 4
60.23 odd 4 950.2.j.g.49.3 8
60.47 odd 4 950.2.j.g.49.2 8
60.59 even 2 950.2.e.k.201.2 4
76.7 odd 6 342.2.g.f.235.1 4
76.11 odd 6 6498.2.a.ba.1.2 2
76.27 even 6 6498.2.a.bg.1.2 2
228.11 even 6 722.2.a.j.1.2 2
228.23 even 18 722.2.e.n.423.2 12
228.35 even 18 722.2.e.n.99.2 12
228.47 even 18 722.2.e.n.245.2 12
228.59 odd 18 722.2.e.o.595.2 12
228.71 odd 18 722.2.e.o.389.1 12
228.83 even 6 38.2.c.b.7.1 4
228.107 odd 6 722.2.c.j.653.2 4
228.119 even 18 722.2.e.n.389.2 12
228.131 even 18 722.2.e.n.595.1 12
228.143 odd 18 722.2.e.o.245.1 12
228.155 odd 18 722.2.e.o.99.1 12
228.167 odd 18 722.2.e.o.423.1 12
228.179 odd 6 722.2.a.g.1.1 2
228.203 odd 18 722.2.e.o.415.2 12
228.215 even 18 722.2.e.n.415.1 12
228.227 odd 2 722.2.c.j.429.2 4
456.83 even 6 1216.2.i.l.577.2 4
456.197 odd 6 1216.2.i.k.577.1 4
1140.83 odd 12 950.2.j.g.349.2 8
1140.539 even 6 950.2.e.k.501.2 4
1140.767 odd 12 950.2.j.g.349.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.2.c.b.7.1 4 228.83 even 6
38.2.c.b.11.1 yes 4 12.11 even 2
304.2.i.e.49.2 4 3.2 odd 2
304.2.i.e.273.2 4 57.26 odd 6
342.2.g.f.163.1 4 4.3 odd 2
342.2.g.f.235.1 4 76.7 odd 6
722.2.a.g.1.1 2 228.179 odd 6
722.2.a.j.1.2 2 228.11 even 6
722.2.c.j.429.2 4 228.227 odd 2
722.2.c.j.653.2 4 228.107 odd 6
722.2.e.n.99.2 12 228.35 even 18
722.2.e.n.245.2 12 228.47 even 18
722.2.e.n.389.2 12 228.119 even 18
722.2.e.n.415.1 12 228.215 even 18
722.2.e.n.423.2 12 228.23 even 18
722.2.e.n.595.1 12 228.131 even 18
722.2.e.o.99.1 12 228.155 odd 18
722.2.e.o.245.1 12 228.143 odd 18
722.2.e.o.389.1 12 228.71 odd 18
722.2.e.o.415.2 12 228.203 odd 18
722.2.e.o.423.1 12 228.167 odd 18
722.2.e.o.595.2 12 228.59 odd 18
950.2.e.k.201.2 4 60.59 even 2
950.2.e.k.501.2 4 1140.539 even 6
950.2.j.g.49.2 8 60.47 odd 4
950.2.j.g.49.3 8 60.23 odd 4
950.2.j.g.349.2 8 1140.83 odd 12
950.2.j.g.349.3 8 1140.767 odd 12
1216.2.i.k.577.1 4 456.197 odd 6
1216.2.i.k.961.1 4 24.5 odd 2
1216.2.i.l.577.2 4 456.83 even 6
1216.2.i.l.961.2 4 24.11 even 2
2736.2.s.v.577.1 4 19.7 even 3 inner
2736.2.s.v.1873.1 4 1.1 even 1 trivial
5776.2.a.z.1.2 2 57.8 even 6
5776.2.a.ba.1.1 2 57.11 odd 6
6498.2.a.ba.1.2 2 76.11 odd 6
6498.2.a.bg.1.2 2 76.27 even 6