Properties

Label 1008.2.cs.e.271.1
Level $1008$
Weight $2$
Character 1008.271
Analytic conductor $8.049$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.cs (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 271.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1008.271
Dual form 1008.2.cs.e.703.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.50000 - 0.866025i) q^{5} +(0.500000 - 2.59808i) q^{7} +O(q^{10})\) \(q+(-1.50000 - 0.866025i) q^{5} +(0.500000 - 2.59808i) q^{7} +(-1.50000 + 0.866025i) q^{11} +(-3.00000 + 1.73205i) q^{17} +(1.00000 - 1.73205i) q^{19} +(-1.00000 - 1.73205i) q^{25} -9.00000 q^{29} +(-2.50000 - 4.33013i) q^{31} +(-3.00000 + 3.46410i) q^{35} +(-5.00000 + 8.66025i) q^{37} +10.3923i q^{41} +3.46410i q^{43} +(6.00000 - 10.3923i) q^{47} +(-6.50000 - 2.59808i) q^{49} +(-4.50000 - 7.79423i) q^{53} +3.00000 q^{55} +(-4.50000 - 7.79423i) q^{59} +(-12.0000 + 6.92820i) q^{67} -13.8564i q^{71} +(-6.00000 + 3.46410i) q^{73} +(1.50000 + 4.33013i) q^{77} +(4.50000 + 2.59808i) q^{79} -3.00000 q^{83} +6.00000 q^{85} +(3.00000 + 1.73205i) q^{89} +(-3.00000 + 1.73205i) q^{95} -19.0526i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{5} + q^{7} + O(q^{10}) \) \( 2q - 3q^{5} + q^{7} - 3q^{11} - 6q^{17} + 2q^{19} - 2q^{25} - 18q^{29} - 5q^{31} - 6q^{35} - 10q^{37} + 12q^{47} - 13q^{49} - 9q^{53} + 6q^{55} - 9q^{59} - 24q^{67} - 12q^{73} + 3q^{77} + 9q^{79} - 6q^{83} + 12q^{85} + 6q^{89} - 6q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

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
\(5\) −1.50000 0.866025i −0.670820 0.387298i 0.125567 0.992085i \(-0.459925\pi\)
−0.796387 + 0.604787i \(0.793258\pi\)
\(6\) 0 0
\(7\) 0.500000 2.59808i 0.188982 0.981981i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.50000 + 0.866025i −0.452267 + 0.261116i −0.708787 0.705422i \(-0.750757\pi\)
0.256520 + 0.966539i \(0.417424\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.00000 + 1.73205i −0.727607 + 0.420084i −0.817546 0.575863i \(-0.804666\pi\)
0.0899392 + 0.995947i \(0.471333\pi\)
\(18\) 0 0
\(19\) 1.00000 1.73205i 0.229416 0.397360i −0.728219 0.685344i \(-0.759652\pi\)
0.957635 + 0.287984i \(0.0929851\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) −1.00000 1.73205i −0.200000 0.346410i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) −2.50000 4.33013i −0.449013 0.777714i 0.549309 0.835619i \(-0.314891\pi\)
−0.998322 + 0.0579057i \(0.981558\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.00000 + 3.46410i −0.507093 + 0.585540i
\(36\) 0 0
\(37\) −5.00000 + 8.66025i −0.821995 + 1.42374i 0.0821995 + 0.996616i \(0.473806\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.3923i 1.62301i 0.584349 + 0.811503i \(0.301350\pi\)
−0.584349 + 0.811503i \(0.698650\pi\)
\(42\) 0 0
\(43\) 3.46410i 0.528271i 0.964486 + 0.264135i \(0.0850865\pi\)
−0.964486 + 0.264135i \(0.914913\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000 10.3923i 0.875190 1.51587i 0.0186297 0.999826i \(-0.494070\pi\)
0.856560 0.516047i \(-0.172597\pi\)
\(48\) 0 0
\(49\) −6.50000 2.59808i −0.928571 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.50000 7.79423i −0.618123 1.07062i −0.989828 0.142269i \(-0.954560\pi\)
0.371706 0.928351i \(-0.378773\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.50000 7.79423i −0.585850 1.01472i −0.994769 0.102151i \(-0.967427\pi\)
0.408919 0.912571i \(-0.365906\pi\)
\(60\) 0 0
\(61\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −12.0000 + 6.92820i −1.46603 + 0.846415i −0.999279 0.0379722i \(-0.987910\pi\)
−0.466755 + 0.884387i \(0.654577\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.8564i 1.64445i −0.569160 0.822226i \(-0.692732\pi\)
0.569160 0.822226i \(-0.307268\pi\)
\(72\) 0 0
\(73\) −6.00000 + 3.46410i −0.702247 + 0.405442i −0.808184 0.588930i \(-0.799549\pi\)
0.105937 + 0.994373i \(0.466216\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.50000 + 4.33013i 0.170941 + 0.493464i
\(78\) 0 0
\(79\) 4.50000 + 2.59808i 0.506290 + 0.292306i 0.731307 0.682048i \(-0.238911\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.00000 −0.329293 −0.164646 0.986353i \(-0.552648\pi\)
−0.164646 + 0.986353i \(0.552648\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.00000 + 1.73205i 0.317999 + 0.183597i 0.650500 0.759506i \(-0.274559\pi\)
−0.332501 + 0.943103i \(0.607893\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.00000 + 1.73205i −0.307794 + 0.177705i
\(96\) 0 0
\(97\) 19.0526i 1.93449i −0.253837 0.967247i \(-0.581693\pi\)
0.253837 0.967247i \(-0.418307\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 12.0000 6.92820i 1.19404 0.689382i 0.234823 0.972038i \(-0.424549\pi\)
0.959221 + 0.282656i \(0.0912155\pi\)
\(102\) 0 0
\(103\) 2.00000 3.46410i 0.197066 0.341328i −0.750510 0.660859i \(-0.770192\pi\)
0.947576 + 0.319531i \(0.103525\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 10.5000 + 6.06218i 1.01507 + 0.586053i 0.912673 0.408690i \(-0.134014\pi\)
0.102400 + 0.994743i \(0.467348\pi\)
\(108\) 0 0
\(109\) 2.00000 + 3.46410i 0.191565 + 0.331801i 0.945769 0.324840i \(-0.105310\pi\)
−0.754204 + 0.656640i \(0.771977\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.00000 + 8.66025i 0.275010 + 0.793884i
\(120\) 0 0
\(121\) −4.00000 + 6.92820i −0.363636 + 0.629837i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 12.1244i 1.08444i
\(126\) 0 0
\(127\) 5.19615i 0.461084i 0.973062 + 0.230542i \(0.0740499\pi\)
−0.973062 + 0.230542i \(0.925950\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −4.50000 + 7.79423i −0.393167 + 0.680985i −0.992865 0.119241i \(-0.961954\pi\)
0.599699 + 0.800226i \(0.295287\pi\)
\(132\) 0 0
\(133\) −4.00000 3.46410i −0.346844 0.300376i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.00000 10.3923i −0.512615 0.887875i −0.999893 0.0146279i \(-0.995344\pi\)
0.487278 0.873247i \(-0.337990\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 13.5000 + 7.79423i 1.12111 + 0.647275i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.00000 5.19615i 0.245770 0.425685i −0.716578 0.697507i \(-0.754293\pi\)
0.962348 + 0.271821i \(0.0876260\pi\)
\(150\) 0 0
\(151\) 4.50000 2.59808i 0.366205 0.211428i −0.305594 0.952162i \(-0.598855\pi\)
0.671799 + 0.740733i \(0.265522\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.66025i 0.695608i
\(156\) 0 0
\(157\) 6.00000 3.46410i 0.478852 0.276465i −0.241086 0.970504i \(-0.577504\pi\)
0.719938 + 0.694038i \(0.244170\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −12.0000 6.92820i −0.939913 0.542659i −0.0499796 0.998750i \(-0.515916\pi\)
−0.889933 + 0.456091i \(0.849249\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(174\) 0 0
\(175\) −5.00000 + 1.73205i −0.377964 + 0.130931i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −21.0000 + 12.1244i −1.56961 + 0.906217i −0.573400 + 0.819275i \(0.694376\pi\)
−0.996213 + 0.0869415i \(0.972291\pi\)
\(180\) 0 0
\(181\) 10.3923i 0.772454i 0.922404 + 0.386227i \(0.126222\pi\)
−0.922404 + 0.386227i \(0.873778\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 15.0000 8.66025i 1.10282 0.636715i
\(186\) 0 0
\(187\) 3.00000 5.19615i 0.219382 0.379980i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.0000 + 6.92820i 0.868290 + 0.501307i 0.866779 0.498692i \(-0.166186\pi\)
0.00151007 + 0.999999i \(0.499519\pi\)
\(192\) 0 0
\(193\) 2.50000 + 4.33013i 0.179954 + 0.311689i 0.941865 0.335993i \(-0.109072\pi\)
−0.761911 + 0.647682i \(0.775738\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −8.00000 13.8564i −0.567105 0.982255i −0.996850 0.0793045i \(-0.974730\pi\)
0.429745 0.902950i \(-0.358603\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −4.50000 + 23.3827i −0.315838 + 1.64114i
\(204\) 0 0
\(205\) 9.00000 15.5885i 0.628587 1.08875i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.46410i 0.239617i
\(210\) 0 0
\(211\) 6.92820i 0.476957i −0.971148 0.238479i \(-0.923351\pi\)
0.971148 0.238479i \(-0.0766487\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.00000 5.19615i 0.204598 0.354375i
\(216\) 0 0
\(217\) −12.5000 + 4.33013i −0.848555 + 0.293948i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 19.0000 1.27233 0.636167 0.771551i \(-0.280519\pi\)
0.636167 + 0.771551i \(0.280519\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −10.5000 18.1865i −0.696909 1.20708i −0.969533 0.244962i \(-0.921225\pi\)
0.272623 0.962121i \(-0.412109\pi\)
\(228\) 0 0
\(229\) −3.00000 1.73205i −0.198246 0.114457i 0.397591 0.917563i \(-0.369846\pi\)
−0.595837 + 0.803105i \(0.703180\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.00000 15.5885i 0.589610 1.02123i −0.404674 0.914461i \(-0.632615\pi\)
0.994283 0.106773i \(-0.0340517\pi\)
\(234\) 0 0
\(235\) −18.0000 + 10.3923i −1.17419 + 0.677919i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 6.92820i 0.448148i −0.974572 0.224074i \(-0.928064\pi\)
0.974572 0.224074i \(-0.0719358\pi\)
\(240\) 0 0
\(241\) 19.5000 11.2583i 1.25611 0.725213i 0.283790 0.958886i \(-0.408408\pi\)
0.972315 + 0.233674i \(0.0750747\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 7.50000 + 9.52628i 0.479157 + 0.608612i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 15.0000 0.946792 0.473396 0.880850i \(-0.343028\pi\)
0.473396 + 0.880850i \(0.343028\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.00000 + 5.19615i 0.561405 + 0.324127i 0.753709 0.657208i \(-0.228263\pi\)
−0.192304 + 0.981335i \(0.561596\pi\)
\(258\) 0 0
\(259\) 20.0000 + 17.3205i 1.24274 + 1.07624i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −15.0000 + 8.66025i −0.924940 + 0.534014i −0.885208 0.465196i \(-0.845984\pi\)
−0.0397320 + 0.999210i \(0.512650\pi\)
\(264\) 0 0
\(265\) 15.5885i 0.957591i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 13.5000 7.79423i 0.823110 0.475223i −0.0283781 0.999597i \(-0.509034\pi\)
0.851488 + 0.524375i \(0.175701\pi\)
\(270\) 0 0
\(271\) −5.50000 + 9.52628i −0.334101 + 0.578680i −0.983312 0.181928i \(-0.941766\pi\)
0.649211 + 0.760609i \(0.275099\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.00000 + 1.73205i 0.180907 + 0.104447i
\(276\) 0 0
\(277\) −14.0000 24.2487i −0.841178 1.45696i −0.888899 0.458103i \(-0.848529\pi\)
0.0477206 0.998861i \(-0.484804\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 24.0000 1.43172 0.715860 0.698244i \(-0.246035\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(282\) 0 0
\(283\) 7.00000 + 12.1244i 0.416107 + 0.720718i 0.995544 0.0942988i \(-0.0300609\pi\)
−0.579437 + 0.815017i \(0.696728\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 27.0000 + 5.19615i 1.59376 + 0.306719i
\(288\) 0 0
\(289\) −2.50000 + 4.33013i −0.147059 + 0.254713i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 5.19615i 0.303562i −0.988414 0.151781i \(-0.951499\pi\)
0.988414 0.151781i \(-0.0485009\pi\)
\(294\) 0 0
\(295\) 15.5885i 0.907595i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 9.00000 + 1.73205i 0.518751 + 0.0998337i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −9.00000 15.5885i −0.510343 0.883940i −0.999928 0.0119847i \(-0.996185\pi\)
0.489585 0.871956i \(-0.337148\pi\)
\(312\) 0 0
\(313\) −4.50000 2.59808i −0.254355 0.146852i 0.367402 0.930062i \(-0.380247\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −10.5000 + 18.1865i −0.589739 + 1.02146i 0.404528 + 0.914526i \(0.367436\pi\)
−0.994266 + 0.106932i \(0.965897\pi\)
\(318\) 0 0
\(319\) 13.5000 7.79423i 0.755855 0.436393i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 6.92820i 0.385496i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −24.0000 20.7846i −1.32316 1.14589i
\(330\) 0 0
\(331\) 12.0000 + 6.92820i 0.659580 + 0.380808i 0.792117 0.610370i \(-0.208979\pi\)
−0.132537 + 0.991178i \(0.542312\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 24.0000 1.31126
\(336\) 0 0
\(337\) −1.00000 −0.0544735 −0.0272367 0.999629i \(-0.508671\pi\)
−0.0272367 + 0.999629i \(0.508671\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 7.50000 + 4.33013i 0.406148 + 0.234490i
\(342\) 0 0
\(343\) −10.0000 + 15.5885i −0.539949 + 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 15.0000 8.66025i 0.805242 0.464907i −0.0400587 0.999197i \(-0.512754\pi\)
0.845301 + 0.534291i \(0.179421\pi\)
\(348\) 0 0
\(349\) 6.92820i 0.370858i −0.982658 0.185429i \(-0.940632\pi\)
0.982658 0.185429i \(-0.0593675\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −6.00000 + 3.46410i −0.319348 + 0.184376i −0.651102 0.758990i \(-0.725693\pi\)
0.331754 + 0.943366i \(0.392360\pi\)
\(354\) 0 0
\(355\) −12.0000 + 20.7846i −0.636894 + 1.10313i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −24.0000 13.8564i −1.26667 0.731313i −0.292315 0.956322i \(-0.594426\pi\)
−0.974357 + 0.225009i \(0.927759\pi\)
\(360\) 0 0
\(361\) 7.50000 + 12.9904i 0.394737 + 0.683704i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 12.0000 0.628109
\(366\) 0 0
\(367\) −9.50000 16.4545i −0.495896 0.858917i 0.504093 0.863649i \(-0.331827\pi\)
−0.999989 + 0.00473247i \(0.998494\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −22.5000 + 7.79423i −1.16814 + 0.404656i
\(372\) 0 0
\(373\) −1.00000 + 1.73205i −0.0517780 + 0.0896822i −0.890753 0.454488i \(-0.849822\pi\)
0.838975 + 0.544170i \(0.183156\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 24.2487i 1.24557i −0.782392 0.622786i \(-0.786001\pi\)
0.782392 0.622786i \(-0.213999\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 9.00000 15.5885i 0.459879 0.796533i −0.539076 0.842257i \(-0.681226\pi\)
0.998954 + 0.0457244i \(0.0145596\pi\)
\(384\) 0 0
\(385\) 1.50000 7.79423i 0.0764471 0.397231i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −3.00000 5.19615i −0.152106 0.263455i 0.779895 0.625910i \(-0.215272\pi\)
−0.932002 + 0.362454i \(0.881939\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4.50000 7.79423i −0.226420 0.392170i
\(396\) 0 0
\(397\) −18.0000 10.3923i −0.903394 0.521575i −0.0250943 0.999685i \(-0.507989\pi\)
−0.878300 + 0.478110i \(0.841322\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.00000 + 10.3923i −0.299626 + 0.518967i −0.976050 0.217545i \(-0.930195\pi\)
0.676425 + 0.736512i \(0.263528\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 17.3205i 0.858546i
\(408\) 0 0
\(409\) −19.5000 + 11.2583i −0.964213 + 0.556689i −0.897467 0.441081i \(-0.854595\pi\)
−0.0667458 + 0.997770i \(0.521262\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −22.5000 + 7.79423i −1.10715 + 0.383529i
\(414\) 0 0
\(415\) 4.50000 + 2.59808i 0.220896 + 0.127535i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −20.0000 −0.974740 −0.487370 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 6.00000 + 3.46410i 0.291043 + 0.168034i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −33.0000 + 19.0526i −1.58955 + 0.917729i −0.596174 + 0.802855i \(0.703313\pi\)
−0.993380 + 0.114874i \(0.963353\pi\)
\(432\) 0 0
\(433\) 34.6410i 1.66474i 0.554220 + 0.832370i \(0.313017\pi\)
−0.554220 + 0.832370i \(0.686983\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 14.5000 25.1147i 0.692047 1.19866i −0.279119 0.960257i \(-0.590042\pi\)
0.971166 0.238404i \(-0.0766244\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10.5000 6.06218i −0.498870 0.288023i 0.229377 0.973338i \(-0.426331\pi\)
−0.728247 + 0.685315i \(0.759665\pi\)
\(444\) 0 0
\(445\) −3.00000 5.19615i −0.142214 0.246321i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) −9.00000 15.5885i −0.423793 0.734032i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 15.5000 26.8468i 0.725059 1.25584i −0.233890 0.972263i \(-0.575146\pi\)
0.958950 0.283577i \(-0.0915211\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6.92820i 0.322679i 0.986899 + 0.161339i \(0.0515813\pi\)
−0.986899 + 0.161339i \(0.948419\pi\)
\(462\) 0 0
\(463\) 38.1051i 1.77090i −0.464739 0.885448i \(-0.653852\pi\)
0.464739 0.885448i \(-0.346148\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6.00000 10.3923i 0.277647 0.480899i −0.693153 0.720791i \(-0.743779\pi\)
0.970799 + 0.239892i \(0.0771121\pi\)
\(468\) 0 0
\(469\) 12.0000 + 34.6410i 0.554109 + 1.59957i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −3.00000 5.19615i −0.137940 0.238919i
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3.00000 + 5.19615i 0.137073 + 0.237418i 0.926388 0.376571i \(-0.122897\pi\)
−0.789314 + 0.613990i \(0.789564\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −16.5000 + 28.5788i −0.749226 + 1.29770i
\(486\) 0 0
\(487\) −7.50000 + 4.33013i −0.339857 + 0.196217i −0.660209 0.751082i \(-0.729532\pi\)
0.320352 + 0.947299i \(0.396199\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 25.9808i 1.17250i 0.810132 + 0.586248i \(0.199395\pi\)
−0.810132 + 0.586248i \(0.800605\pi\)
\(492\) 0 0
\(493\) 27.0000 15.5885i 1.21602 0.702069i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −36.0000 6.92820i −1.61482 0.310772i
\(498\) 0 0
\(499\) 3.00000 + 1.73205i 0.134298 + 0.0775372i 0.565644 0.824650i \(-0.308628\pi\)
−0.431346 + 0.902187i \(0.641961\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 0 0
\(505\) −24.0000 −1.06799
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −28.5000 16.4545i −1.26324 0.729332i −0.289540 0.957166i \(-0.593502\pi\)
−0.973700 + 0.227834i \(0.926836\pi\)
\(510\) 0 0
\(511\) 6.00000 + 17.3205i 0.265424 + 0.766214i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −6.00000 + 3.46410i −0.264392 + 0.152647i
\(516\) 0 0
\(517\) 20.7846i 0.914106i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −33.0000 + 19.0526i −1.44576 + 0.834708i −0.998225 0.0595604i \(-0.981030\pi\)
−0.447532 + 0.894268i \(0.647697\pi\)
\(522\) 0 0
\(523\) 14.0000 24.2487i 0.612177 1.06032i −0.378695 0.925521i \(-0.623627\pi\)
0.990873 0.134801i \(-0.0430394\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 15.0000 + 8.66025i 0.653410 + 0.377247i
\(528\) 0 0
\(529\) −11.5000 19.9186i −0.500000 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −10.5000 18.1865i −0.453955 0.786272i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 12.0000 1.73205i 0.516877 0.0746047i
\(540\) 0 0
\(541\) −11.0000 + 19.0526i −0.472927 + 0.819133i −0.999520 0.0309841i \(-0.990136\pi\)
0.526593 + 0.850118i \(0.323469\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 6.92820i 0.296772i
\(546\) 0 0
\(547\) 10.3923i 0.444343i −0.975008 0.222171i \(-0.928686\pi\)
0.975008 0.222171i \(-0.0713145\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −9.00000 + 15.5885i −0.383413 + 0.664091i
\(552\) 0 0
\(553\) 9.00000 10.3923i 0.382719 0.441926i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −13.5000 23.3827i −0.572013 0.990756i −0.996359 0.0852559i \(-0.972829\pi\)
0.424346 0.905500i \(-0.360504\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −22.5000 38.9711i −0.948262 1.64244i −0.749085 0.662474i \(-0.769506\pi\)
−0.199177 0.979963i \(-0.563827\pi\)
\(564\) 0 0
\(565\) 9.00000 + 5.19615i 0.378633 + 0.218604i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −3.00000 + 5.19615i −0.125767 + 0.217834i −0.922032 0.387113i \(-0.873472\pi\)
0.796266 + 0.604947i \(0.206806\pi\)
\(570\) 0 0
\(571\) −18.0000 + 10.3923i −0.753277 + 0.434904i −0.826877 0.562383i \(-0.809885\pi\)
0.0736000 + 0.997288i \(0.476551\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1.50000 + 0.866025i −0.0624458 + 0.0360531i −0.530898 0.847436i \(-0.678145\pi\)
0.468452 + 0.883489i \(0.344812\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1.50000 + 7.79423i −0.0622305 + 0.323359i
\(582\) 0 0
\(583\) 13.5000 + 7.79423i 0.559113 + 0.322804i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 33.0000 1.36206 0.681028 0.732257i \(-0.261533\pi\)
0.681028 + 0.732257i \(0.261533\pi\)
\(588\) 0 0
\(589\) −10.0000 −0.412043
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 24.0000 + 13.8564i 0.985562 + 0.569014i 0.903945 0.427649i \(-0.140658\pi\)
0.0816172 + 0.996664i \(0.473992\pi\)
\(594\) 0 0
\(595\) 3.00000 15.5885i 0.122988 0.639064i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −24.0000 + 13.8564i −0.980613 + 0.566157i −0.902455 0.430784i \(-0.858237\pi\)
−0.0781581 + 0.996941i \(0.524904\pi\)
\(600\) 0 0
\(601\) 15.5885i 0.635866i −0.948113 0.317933i \(-0.897011\pi\)
0.948113 0.317933i \(-0.102989\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 12.0000 6.92820i 0.487869 0.281672i
\(606\) 0 0
\(607\) −9.50000 + 16.4545i −0.385593 + 0.667867i −0.991851 0.127401i \(-0.959336\pi\)
0.606258 + 0.795268i \(0.292670\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 2.00000 + 3.46410i 0.0807792 + 0.139914i 0.903585 0.428409i \(-0.140926\pi\)
−0.822806 + 0.568323i \(0.807592\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −12.0000 −0.483102 −0.241551 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) 0 0
\(619\) 14.0000 + 24.2487i 0.562708 + 0.974638i 0.997259 + 0.0739910i \(0.0235736\pi\)
−0.434551 + 0.900647i \(0.643093\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 6.00000 6.92820i 0.240385 0.277573i
\(624\) 0 0
\(625\) 5.50000 9.52628i 0.220000 0.381051i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 34.6410i 1.38123i
\(630\) 0 0
\(631\) 32.9090i 1.31009i 0.755592 + 0.655043i \(0.227349\pi\)
−0.755592 + 0.655043i \(0.772651\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 4.50000 7.79423i 0.178577 0.309305i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 18.0000 + 31.1769i 0.710957 + 1.23141i 0.964498 + 0.264089i \(0.0850714\pi\)
−0.253541 + 0.967325i \(0.581595\pi\)
\(642\) 0 0
\(643\) 16.0000 0.630978 0.315489 0.948929i \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(648\) 0 0
\(649\) 13.5000 + 7.79423i 0.529921 + 0.305950i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −13.5000 + 23.3827i −0.528296 + 0.915035i 0.471160 + 0.882048i \(0.343835\pi\)
−0.999456 + 0.0329874i \(0.989498\pi\)
\(654\) 0 0
\(655\) 13.5000 7.79423i 0.527489 0.304546i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 10.3923i 0.404827i −0.979300 0.202413i \(-0.935122\pi\)
0.979300 0.202413i \(-0.0648785\pi\)
\(660\) 0 0
\(661\) 3.00000 1.73205i 0.116686 0.0673690i −0.440521 0.897742i \(-0.645206\pi\)
0.557207 + 0.830373i \(0.311873\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 3.00000 + 8.66025i 0.116335 + 0.335830i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −35.0000 −1.34915 −0.674575 0.738206i \(-0.735673\pi\)
−0.674575 + 0.738206i \(0.735673\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −25.5000 14.7224i −0.980045 0.565829i −0.0777610 0.996972i \(-0.524777\pi\)
−0.902284 + 0.431143i \(0.858110\pi\)
\(678\) 0 0
\(679\) −49.5000 9.52628i −1.89964 0.365585i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 22.5000 12.9904i 0.860939 0.497063i −0.00338791 0.999994i \(-0.501078\pi\)
0.864326 + 0.502931i \(0.167745\pi\)
\(684\) 0 0
\(685\) 20.7846i 0.794139i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 11.0000 19.0526i 0.418460 0.724793i −0.577325 0.816514i \(-0.695903\pi\)
0.995785 + 0.0917209i \(0.0292368\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −21.0000 12.1244i −0.796575 0.459903i
\(696\) 0 0
\(697\) −18.0000 31.1769i −0.681799 1.18091i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 45.0000 1.69963 0.849813 0.527084i \(-0.176715\pi\)
0.849813 + 0.527084i \(0.176715\pi\)
\(702\) 0 0
\(703\) 10.0000 + 17.3205i 0.377157 + 0.653255i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −12.0000 34.6410i −0.451306 1.30281i
\(708\) 0 0
\(709\) 5.00000 8.66025i 0.187779 0.325243i −0.756730 0.653727i \(-0.773204\pi\)
0.944509 + 0.328484i \(0.106538\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 9.00000 15.5885i 0.335643 0.581351i −0.647965 0.761670i \(-0.724380\pi\)
0.983608 + 0.180319i \(0.0577130\pi\)
\(720\) 0 0
\(721\) −8.00000 6.92820i −0.297936 0.258020i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 9.00000 + 15.5885i 0.334252 + 0.578941i
\(726\) 0 0
\(727\) −37.0000 −1.37225 −0.686127 0.727482i \(-0.740691\pi\)
−0.686127 + 0.727482i \(0.740691\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −6.00000 10.3923i −0.221918 0.384373i
\(732\) 0 0
\(733\) −9.00000 5.19615i −0.332423 0.191924i 0.324494 0.945888i \(-0.394806\pi\)
−0.656916 + 0.753964i \(0.728139\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 12.0000 20.7846i 0.442026 0.765611i
\(738\) 0 0
\(739\) 9.00000 5.19615i 0.331070 0.191144i −0.325246 0.945629i \(-0.605447\pi\)
0.656316 + 0.754486i \(0.272114\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 13.8564i 0.508342i 0.967159 + 0.254171i \(0.0818026\pi\)
−0.967159 + 0.254171i \(0.918197\pi\)
\(744\) 0 0
\(745\) −9.00000 + 5.19615i −0.329734 + 0.190372i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 21.0000 24.2487i 0.767323 0.886029i
\(750\) 0 0
\(751\) 4.50000 + 2.59808i 0.164207 + 0.0948051i 0.579852 0.814722i \(-0.303111\pi\)
−0.415644 + 0.909527i \(0.636444\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −9.00000 −0.327544
\(756\) 0 0
\(757\) −4.00000 −0.145382 −0.0726912 0.997354i \(-0.523159\pi\)
−0.0726912 + 0.997354i \(0.523159\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 42.0000 + 24.2487i 1.52250 + 0.879015i 0.999646 + 0.0265919i \(0.00846546\pi\)
0.522852 + 0.852423i \(0.324868\pi\)
\(762\) 0 0
\(763\) 10.0000 3.46410i 0.362024 0.125409i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 19.0526i 0.687053i 0.939143 + 0.343526i \(0.111621\pi\)
−0.939143 + 0.343526i \(0.888379\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(774\) 0 0
\(775\) −5.00000 + 8.66025i −0.179605 + 0.311086i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 18.0000 + 10.3923i 0.644917 + 0.372343i
\(780\) 0 0
\(781\) 12.0000 + 20.7846i 0.429394 + 0.743732i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −12.0000 −0.428298
\(786\) 0 0
\(787\) 14.0000 + 24.2487i 0.499046 + 0.864373i 0.999999 0.00110111i \(-0.000350496\pi\)
−0.500953 + 0.865474i \(0.667017\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.00000 + 15.5885i −0.106668 + 0.554262i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 5.19615i 0.184057i −0.995756 0.0920286i \(-0.970665\pi\)
0.995756 0.0920286i \(-0.0293351\pi\)
\(798\) 0 0
\(799\) 41.5692i 1.47061i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 6.00000 10.3923i 0.211735 0.366736i
\(804\) 0