Properties

Label 1008.2.cs.c.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 112)
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.c.703.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.50000 - 0.866025i) q^{5} +(-2.00000 + 1.73205i) q^{7} +O(q^{10})\) \(q+(-1.50000 - 0.866025i) q^{5} +(-2.00000 + 1.73205i) q^{7} +(-1.50000 + 0.866025i) q^{11} +(4.50000 - 2.59808i) q^{17} +(3.50000 - 6.06218i) q^{19} +(7.50000 + 4.33013i) q^{23} +(-1.00000 - 1.73205i) q^{25} +6.00000 q^{29} +(2.50000 + 4.33013i) q^{31} +(4.50000 - 0.866025i) q^{35} +(2.50000 - 4.33013i) q^{37} -6.92820i q^{41} +3.46410i q^{43} +(-1.50000 + 2.59808i) q^{47} +(1.00000 - 6.92820i) q^{49} +(-4.50000 - 7.79423i) q^{53} +3.00000 q^{55} +(-4.50000 - 7.79423i) q^{59} +(7.50000 + 4.33013i) q^{61} +(-4.50000 + 2.59808i) q^{67} +3.46410i q^{71} +(1.50000 - 0.866025i) q^{73} +(1.50000 - 4.33013i) q^{77} +(4.50000 + 2.59808i) q^{79} +12.0000 q^{83} -9.00000 q^{85} +(10.5000 + 6.06218i) q^{89} +(-10.5000 + 6.06218i) q^{95} +6.92820i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{5} - 4q^{7} + O(q^{10}) \) \( 2q - 3q^{5} - 4q^{7} - 3q^{11} + 9q^{17} + 7q^{19} + 15q^{23} - 2q^{25} + 12q^{29} + 5q^{31} + 9q^{35} + 5q^{37} - 3q^{47} + 2q^{49} - 9q^{53} + 6q^{55} - 9q^{59} + 15q^{61} - 9q^{67} + 3q^{73} + 3q^{77} + 9q^{79} + 24q^{83} - 18q^{85} + 21q^{89} - 21q^{95} + O(q^{100}) \)

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
\(7\) −2.00000 + 1.73205i −0.755929 + 0.654654i
\(8\) 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
\(17\) 4.50000 2.59808i 1.09141 0.630126i 0.157459 0.987526i \(-0.449670\pi\)
0.933952 + 0.357400i \(0.116337\pi\)
\(18\) 0 0
\(19\) 3.50000 6.06218i 0.802955 1.39076i −0.114708 0.993399i \(-0.536593\pi\)
0.917663 0.397360i \(-0.130073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.50000 + 4.33013i 1.56386 + 0.902894i 0.996861 + 0.0791743i \(0.0252284\pi\)
0.566997 + 0.823720i \(0.308105\pi\)
\(24\) 0 0
\(25\) −1.00000 1.73205i −0.200000 0.346410i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 2.50000 + 4.33013i 0.449013 + 0.777714i 0.998322 0.0579057i \(-0.0184423\pi\)
−0.549309 + 0.835619i \(0.685109\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.50000 0.866025i 0.760639 0.146385i
\(36\) 0 0
\(37\) 2.50000 4.33013i 0.410997 0.711868i −0.584002 0.811752i \(-0.698514\pi\)
0.994999 + 0.0998840i \(0.0318472\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.92820i 1.08200i −0.841021 0.541002i \(-0.818045\pi\)
0.841021 0.541002i \(-0.181955\pi\)
\(42\) 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
\(47\) −1.50000 + 2.59808i −0.218797 + 0.378968i −0.954441 0.298401i \(-0.903547\pi\)
0.735643 + 0.677369i \(0.236880\pi\)
\(48\) 0 0
\(49\) 1.00000 6.92820i 0.142857 0.989743i
\(50\) 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 \(1.63409\pi\)
\(60\) 0 0
\(61\) 7.50000 + 4.33013i 0.960277 + 0.554416i 0.896258 0.443533i \(-0.146275\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.50000 + 2.59808i −0.549762 + 0.317406i −0.749026 0.662540i \(-0.769478\pi\)
0.199264 + 0.979946i \(0.436145\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.46410i 0.411113i 0.978645 + 0.205557i \(0.0659005\pi\)
−0.978645 + 0.205557i \(0.934100\pi\)
\(72\) 0 0
\(73\) 1.50000 0.866025i 0.175562 0.101361i −0.409644 0.912245i \(-0.634347\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 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
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) −9.00000 −0.976187
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.5000 + 6.06218i 1.11300 + 0.642590i 0.939604 0.342263i \(-0.111193\pi\)
0.173394 + 0.984853i \(0.444527\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −10.5000 + 6.06218i −1.07728 + 0.621966i
\(96\) 0 0
\(97\) 6.92820i 0.703452i 0.936103 + 0.351726i \(0.114405\pi\)
−0.936103 + 0.351726i \(0.885595\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 4.50000 2.59808i 0.447767 0.258518i −0.259120 0.965845i \(-0.583432\pi\)
0.706887 + 0.707327i \(0.250099\pi\)
\(102\) 0 0
\(103\) −0.500000 + 0.866025i −0.0492665 + 0.0853320i −0.889607 0.456727i \(-0.849022\pi\)
0.840341 + 0.542059i \(0.182355\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.50000 2.59808i −0.435031 0.251166i 0.266456 0.963847i \(-0.414147\pi\)
−0.701488 + 0.712681i \(0.747481\pi\)
\(108\) 0 0
\(109\) −5.50000 9.52628i −0.526804 0.912452i −0.999512 0.0312328i \(-0.990057\pi\)
0.472708 0.881219i \(-0.343277\pi\)
\(110\) 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
\(115\) −7.50000 12.9904i −0.699379 1.21136i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −4.50000 + 12.9904i −0.412514 + 1.19083i
\(120\) 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
\(127\) 3.46410i 0.307389i −0.988118 0.153695i \(-0.950883\pi\)
0.988118 0.153695i \(-0.0491172\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 10.5000 18.1865i 0.917389 1.58896i 0.114024 0.993478i \(-0.463626\pi\)
0.803365 0.595487i \(-0.203041\pi\)
\(132\) 0 0
\(133\) 3.50000 + 18.1865i 0.303488 + 1.57697i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.50000 + 2.59808i 0.128154 + 0.221969i 0.922961 0.384893i \(-0.125762\pi\)
−0.794808 + 0.606861i \(0.792428\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −9.00000 5.19615i −0.747409 0.431517i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −4.50000 + 7.79423i −0.368654 + 0.638528i −0.989355 0.145519i \(-0.953515\pi\)
0.620701 + 0.784047i \(0.286848\pi\)
\(150\) 0 0
\(151\) −10.5000 + 6.06218i −0.854478 + 0.493333i −0.862159 0.506637i \(-0.830888\pi\)
0.00768132 + 0.999970i \(0.497555\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.66025i 0.695608i
\(156\) 0 0
\(157\) 13.5000 7.79423i 1.07742 0.622047i 0.147219 0.989104i \(-0.452968\pi\)
0.930199 + 0.367057i \(0.119635\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −22.5000 + 4.33013i −1.77325 + 0.341262i
\(162\) 0 0
\(163\) 10.5000 + 6.06218i 0.822423 + 0.474826i 0.851251 0.524758i \(-0.175844\pi\)
−0.0288280 + 0.999584i \(0.509178\pi\)
\(164\) 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
\(173\) −7.50000 4.33013i −0.570214 0.329213i 0.187021 0.982356i \(-0.440117\pi\)
−0.757235 + 0.653143i \(0.773450\pi\)
\(174\) 0 0
\(175\) 5.00000 + 1.73205i 0.377964 + 0.130931i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −13.5000 + 7.79423i −1.00904 + 0.582568i −0.910910 0.412606i \(-0.864619\pi\)
−0.0981277 + 0.995174i \(0.531285\pi\)
\(180\) 0 0
\(181\) 6.92820i 0.514969i −0.966282 0.257485i \(-0.917106\pi\)
0.966282 0.257485i \(-0.0828937\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −7.50000 + 4.33013i −0.551411 + 0.318357i
\(186\) 0 0
\(187\) −4.50000 + 7.79423i −0.329073 + 0.569970i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −10.5000 6.06218i −0.759753 0.438644i 0.0694538 0.997585i \(-0.477874\pi\)
−0.829207 + 0.558941i \(0.811208\pi\)
\(192\) 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
\(199\) −5.50000 9.52628i −0.389885 0.675300i 0.602549 0.798082i \(-0.294152\pi\)
−0.992434 + 0.122782i \(0.960818\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −12.0000 + 10.3923i −0.842235 + 0.729397i
\(204\) 0 0
\(205\) −6.00000 + 10.3923i −0.419058 + 0.725830i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 12.1244i 0.838659i
\(210\) 0 0
\(211\) 24.2487i 1.66935i −0.550743 0.834675i \(-0.685655\pi\)
0.550743 0.834675i \(-0.314345\pi\)
\(212\) 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
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 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 \(1.58789\pi\)
\(228\) 0 0
\(229\) −10.5000 6.06218i −0.693860 0.400600i 0.111197 0.993798i \(-0.464532\pi\)
−0.805056 + 0.593198i \(0.797865\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.50000 2.59808i 0.0982683 0.170206i −0.812700 0.582683i \(-0.802003\pi\)
0.910968 + 0.412477i \(0.135336\pi\)
\(234\) 0 0
\(235\) 4.50000 2.59808i 0.293548 0.169480i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 10.3923i 0.672222i 0.941822 + 0.336111i \(0.109112\pi\)
−0.941822 + 0.336111i \(0.890888\pi\)
\(240\) 0 0
\(241\) −10.5000 + 6.06218i −0.676364 + 0.390499i −0.798484 0.602016i \(-0.794364\pi\)
0.122119 + 0.992515i \(0.461031\pi\)
\(242\) 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
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −15.0000 −0.943042
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 16.5000 + 9.52628i 1.02924 + 0.594233i 0.916767 0.399422i \(-0.130789\pi\)
0.112474 + 0.993655i \(0.464122\pi\)
\(258\) 0 0
\(259\) 2.50000 + 12.9904i 0.155342 + 0.807183i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −7.50000 + 4.33013i −0.462470 + 0.267007i −0.713082 0.701080i \(-0.752701\pi\)
0.250612 + 0.968088i \(0.419368\pi\)
\(264\) 0 0
\(265\) 15.5885i 0.957591i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1.50000 + 0.866025i −0.0914566 + 0.0528025i −0.545031 0.838416i \(-0.683482\pi\)
0.453574 + 0.891219i \(0.350149\pi\)
\(270\) 0 0
\(271\) −0.500000 + 0.866025i −0.0303728 + 0.0526073i −0.880812 0.473466i \(-0.843003\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.00000 + 1.73205i 0.180907 + 0.104447i
\(276\) 0 0
\(277\) 8.50000 + 14.7224i 0.510716 + 0.884585i 0.999923 + 0.0124177i \(0.00395278\pi\)
−0.489207 + 0.872167i \(0.662714\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) −5.50000 9.52628i −0.326941 0.566279i 0.654962 0.755662i \(-0.272685\pi\)
−0.981903 + 0.189383i \(0.939351\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000 + 13.8564i 0.708338 + 0.817918i
\(288\) 0 0
\(289\) 5.00000 8.66025i 0.294118 0.509427i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 20.7846i 1.21425i 0.794606 + 0.607125i \(0.207677\pi\)
−0.794606 + 0.607125i \(0.792323\pi\)
\(294\) 0 0
\(295\) 15.5885i 0.907595i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −6.00000 6.92820i −0.345834 0.399335i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −7.50000 12.9904i −0.429449 0.743827i
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −16.5000 28.5788i −0.935629 1.62056i −0.773508 0.633786i \(-0.781500\pi\)
−0.162121 0.986771i \(1.44817\pi\)
\(312\) 0 0
\(313\) 25.5000 + 14.7224i 1.44135 + 0.832161i 0.997940 0.0641600i \(-0.0204368\pi\)
0.443406 + 0.896321i \(0.353770\pi\)
\(314\) 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
\(319\) −9.00000 + 5.19615i −0.503903 + 0.290929i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 36.3731i 2.02385i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −1.50000 7.79423i −0.0826977 0.429710i
\(330\) 0 0
\(331\) −25.5000 14.7224i −1.40161 0.809218i −0.407049 0.913406i \(-0.633442\pi\)
−0.994558 + 0.104188i \(0.966776\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 9.00000 0.491723
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 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
\(347\) −7.50000 + 4.33013i −0.402621 + 0.232453i −0.687614 0.726076i \(-0.741342\pi\)
0.284993 + 0.958530i \(0.408009\pi\)
\(348\) 0 0
\(349\) 27.7128i 1.48343i 0.670714 + 0.741716i \(0.265988\pi\)
−0.670714 + 0.741716i \(0.734012\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 16.5000 9.52628i 0.878206 0.507033i 0.00813978 0.999967i \(-0.497409\pi\)
0.870067 + 0.492934i \(0.164076\pi\)
\(354\) 0 0
\(355\) 3.00000 5.19615i 0.159223 0.275783i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 13.5000 + 7.79423i 0.712503 + 0.411364i 0.811987 0.583675i \(-0.198386\pi\)
−0.0994843 + 0.995039i \(0.531719\pi\)
\(360\) 0 0
\(361\) −15.0000 25.9808i −0.789474 1.36741i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3.00000 −0.157027
\(366\) 0 0
\(367\) 0.500000 + 0.866025i 0.0260998 + 0.0452062i 0.878780 0.477227i \(-0.158358\pi\)
−0.852680 + 0.522433i \(0.825025\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 22.5000 + 7.79423i 1.16814 + 0.404656i
\(372\) 0 0
\(373\) 6.50000 11.2583i 0.336557 0.582934i −0.647225 0.762299i \(-0.724071\pi\)
0.983783 + 0.179364i \(0.0574041\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 10.3923i 0.533817i 0.963722 + 0.266908i \(0.0860021\pi\)
−0.963722 + 0.266908i \(0.913998\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −13.5000 + 23.3827i −0.689818 + 1.19480i 0.282079 + 0.959391i \(0.408976\pi\)
−0.971897 + 0.235408i \(0.924357\pi\)
\(384\) 0 0
\(385\) −6.00000 + 5.19615i −0.305788 + 0.264820i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −10.5000 18.1865i −0.532371 0.922094i −0.999286 0.0377914i \(-0.987968\pi\)
0.466915 0.884302i \(-0.345366\pi\)
\(390\) 0 0
\(391\) 45.0000 2.27575
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4.50000 7.79423i −0.226420 0.392170i
\(396\) 0 0
\(397\) −10.5000 6.06218i −0.526980 0.304252i 0.212806 0.977095i \(-0.431740\pi\)
−0.739786 + 0.672843i \(0.765073\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.50000 2.59808i 0.0749064 0.129742i −0.826139 0.563466i \(-0.809468\pi\)
0.901046 + 0.433724i \(0.142801\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 8.66025i 0.429273i
\(408\) 0 0
\(409\) −4.50000 + 2.59808i −0.222511 + 0.128467i −0.607112 0.794616i \(-0.707672\pi\)
0.384602 + 0.923083i \(0.374339\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 22.5000 + 7.79423i 1.10715 + 0.383529i
\(414\) 0 0
\(415\) −18.0000 10.3923i −0.883585 0.510138i
\(416\) 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
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −9.00000 5.19615i −0.436564 0.252050i
\(426\) 0 0
\(427\) −22.5000 + 4.33013i −1.08885 + 0.209550i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −25.5000 + 14.7224i −1.22829 + 0.709155i −0.966672 0.256017i \(-0.917590\pi\)
−0.261619 + 0.965171i \(0.584257\pi\)
\(432\) 0 0
\(433\) 34.6410i 1.66474i −0.554220 0.832370i \(-0.686983\pi\)
0.554220 0.832370i \(-0.313017\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 52.5000 30.3109i 2.51142 1.44997i
\(438\) 0 0
\(439\) 9.50000 16.4545i 0.453410 0.785330i −0.545185 0.838316i \(-0.683541\pi\)
0.998595 + 0.0529862i \(0.0168739\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 19.5000 + 11.2583i 0.926473 + 0.534899i 0.885694 0.464269i \(-0.153683\pi\)
0.0407786 + 0.999168i \(0.487016\pi\)
\(444\) 0 0
\(445\) −10.5000 18.1865i −0.497748 0.862124i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 6.00000 + 10.3923i 0.282529 + 0.489355i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.500000 0.866025i 0.0233890 0.0405110i −0.854094 0.520119i \(-0.825888\pi\)
0.877483 + 0.479608i \(0.159221\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 27.7128i 1.29071i −0.763881 0.645357i \(-0.776709\pi\)
0.763881 0.645357i \(-0.223291\pi\)
\(462\) 0 0
\(463\) 3.46410i 0.160990i −0.996755 0.0804952i \(-0.974350\pi\)
0.996755 0.0804952i \(-0.0256502\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1.50000 + 2.59808i −0.0694117 + 0.120225i −0.898642 0.438682i \(-0.855446\pi\)
0.829231 + 0.558906i \(0.188779\pi\)
\(468\) 0 0
\(469\) 4.50000 12.9904i 0.207791 0.599840i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −3.00000 5.19615i −0.137940 0.238919i
\(474\) 0 0
\(475\) −14.0000 −0.642364
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −4.50000 7.79423i −0.205610 0.356127i 0.744717 0.667381i \(-0.232585\pi\)
−0.950327 + 0.311253i \(0.899251\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6.00000 10.3923i 0.272446 0.471890i
\(486\) 0 0
\(487\) 7.50000 4.33013i 0.339857 0.196217i −0.320352 0.947299i \(-0.603801\pi\)
0.660209 + 0.751082i \(0.270468\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 17.3205i 0.781664i 0.920462 + 0.390832i \(0.127813\pi\)
−0.920462 + 0.390832i \(0.872187\pi\)
\(492\) 0 0
\(493\) 27.0000 15.5885i 1.21602 0.702069i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −6.00000 6.92820i −0.269137 0.310772i
\(498\) 0 0
\(499\) 10.5000 + 6.06218i 0.470045 + 0.271380i 0.716258 0.697835i \(-0.245853\pi\)
−0.246214 + 0.969216i \(0.579187\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) −9.00000 −0.400495
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −13.5000 7.79423i −0.598377 0.345473i 0.170026 0.985440i \(-0.445615\pi\)
−0.768403 + 0.639966i \(0.778948\pi\)
\(510\) 0 0
\(511\) −1.50000 + 4.33013i −0.0663561 + 0.191554i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.50000 0.866025i 0.0660979 0.0381616i
\(516\) 0 0
\(517\) 5.19615i 0.228527i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −25.5000 + 14.7224i −1.11718 + 0.645001i −0.940678 0.339300i \(-0.889810\pi\)
−0.176497 + 0.984301i \(0.556477\pi\)
\(522\) 0 0
\(523\) 11.5000 19.9186i 0.502860 0.870979i −0.497135 0.867673i \(-0.665615\pi\)
0.999995 0.00330547i \(-0.00105217\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 22.5000 + 12.9904i 0.980115 + 0.565870i
\(528\) 0 0
\(529\) 26.0000 + 45.0333i 1.13043 + 1.95797i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 4.50000 + 7.79423i 0.194552 + 0.336974i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 4.50000 + 11.2583i 0.193829 + 0.484931i
\(540\) 0 0
\(541\) −3.50000 + 6.06218i −0.150477 + 0.260633i −0.931403 0.363990i \(-0.881414\pi\)
0.780926 + 0.624623i \(0.214748\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 19.0526i 0.816122i
\(546\) 0 0
\(547\) 24.2487i 1.03680i 0.855138 + 0.518400i \(0.173472\pi\)
−0.855138 + 0.518400i \(0.826528\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 21.0000 36.3731i 0.894630 1.54954i
\(552\) 0 0
\(553\) −13.5000 + 2.59808i −0.574078 + 0.110481i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.50000 + 2.59808i 0.0635570 + 0.110084i 0.896053 0.443947i \(-0.146422\pi\)
−0.832496 + 0.554031i \(0.813089\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 7.50000 + 12.9904i 0.316087 + 0.547479i 0.979668 0.200625i \(-0.0642974\pi\)
−0.663581 + 0.748105i \(0.730964\pi\)
\(564\) 0 0
\(565\) 9.00000 + 5.19615i 0.378633 + 0.218604i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 19.5000 33.7750i 0.817483 1.41592i −0.0900490 0.995937i \(-0.528702\pi\)
0.907532 0.419984i \(-0.137964\pi\)
\(570\) 0 0
\(571\) −10.5000 + 6.06218i −0.439411 + 0.253694i −0.703348 0.710846i \(-0.748312\pi\)
0.263937 + 0.964540i \(0.414979\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 17.3205i 0.722315i
\(576\) 0 0
\(577\) 13.5000 7.79423i 0.562012 0.324478i −0.191940 0.981407i \(-0.561478\pi\)
0.753953 + 0.656929i \(0.228145\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −24.0000 + 20.7846i −0.995688 + 0.862291i
\(582\) 0 0
\(583\) 13.5000 + 7.79423i 0.559113 + 0.322804i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) 35.0000 1.44215
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 16.5000 + 9.52628i 0.677574 + 0.391197i 0.798940 0.601410i \(-0.205394\pi\)
−0.121367 + 0.992608i \(0.538728\pi\)
\(594\) 0 0
\(595\) 18.0000 15.5885i 0.737928 0.639064i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 28.5000 16.4545i 1.16448 0.672312i 0.212105 0.977247i \(-0.431968\pi\)
0.952373 + 0.304935i \(0.0986347\pi\)
\(600\) 0 0
\(601\) 6.92820i 0.282607i −0.989966 0.141304i \(-0.954871\pi\)
0.989966 0.141304i \(-0.0451294\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 12.0000 6.92820i 0.487869 0.281672i
\(606\) 0 0
\(607\) −14.5000 + 25.1147i −0.588537 + 1.01938i 0.405887 + 0.913923i \(0.366962\pi\)
−0.994424 + 0.105453i \(0.966371\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −5.50000 9.52628i −0.222143 0.384763i 0.733316 0.679888i \(-0.237972\pi\)
−0.955458 + 0.295126i \(0.904638\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) −3.50000 6.06218i −0.140677 0.243659i 0.787075 0.616858i \(-0.211595\pi\)
−0.927752 + 0.373198i \(0.878261\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −31.5000 + 6.06218i −1.26202 + 0.242876i
\(624\) 0 0
\(625\) 5.50000 9.52628i 0.220000 0.381051i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 25.9808i 1.03592i
\(630\) 0 0
\(631\) 45.0333i 1.79275i −0.443298 0.896374i \(-0.646192\pi\)
0.443298 0.896374i \(-0.353808\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −3.00000 + 5.19615i −0.119051 + 0.206203i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −4.50000 7.79423i −0.177739 0.307854i 0.763367 0.645966i \(-0.223545\pi\)
−0.941106 + 0.338112i \(0.890212\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 7.50000 + 12.9904i 0.294855 + 0.510705i 0.974951 0.222419i \(-0.0713952\pi\)
−0.680096 + 0.733123i \(0.738062\pi\)
\(648\) 0 0
\(649\) 13.5000 + 7.79423i 0.529921 + 0.305950i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.50000 2.59808i 0.0586995 0.101671i −0.835182 0.549973i \(-0.814638\pi\)
0.893882 + 0.448303i \(0.147971\pi\)
\(654\) 0 0
\(655\) −31.5000 + 18.1865i −1.23081 + 0.710607i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 24.2487i 0.944596i 0.881439 + 0.472298i \(0.156575\pi\)
−0.881439 + 0.472298i \(0.843425\pi\)
\(660\) 0 0
\(661\) −34.5000 + 19.9186i −1.34189 + 0.774743i −0.987085 0.160196i \(-0.948788\pi\)
−0.354809 + 0.934939i \(0.615454\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 10.5000 30.3109i 0.407173 1.17541i
\(666\) 0 0
\(667\) 45.0000 + 25.9808i 1.74241 + 1.00598i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −15.0000 −0.579069
\(672\) 0 0
\(673\) −50.0000 −1.92736 −0.963679 0.267063i \(-0.913947\pi\)
−0.963679 + 0.267063i \(0.913947\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 34.5000 + 19.9186i 1.32594 + 0.765533i 0.984669 0.174431i \(-0.0558085\pi\)
0.341273 + 0.939964i \(0.389142\pi\)
\(678\) 0 0
\(679\) −12.0000 13.8564i −0.460518 0.531760i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −37.5000 + 21.6506i −1.43490 + 0.828439i −0.997489 0.0708242i \(-0.977437\pi\)
−0.437409 + 0.899263i \(0.644104\pi\)
\(684\) 0 0
\(685\) 5.19615i 0.198535i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −6.50000 + 11.2583i −0.247272 + 0.428287i −0.962768 0.270330i \(-0.912867\pi\)
0.715496 + 0.698617i \(0.246201\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.00000 3.46410i −0.227593 0.131401i
\(696\) 0 0
\(697\) −18.0000 31.1769i −0.681799 1.18091i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) −17.5000 30.3109i −0.660025 1.14320i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −4.50000 + 12.9904i −0.169240 + 0.488554i
\(708\) 0 0
\(709\) −17.5000 + 30.3109i −0.657226 + 1.13835i 0.324104 + 0.946021i \(0.394937\pi\)
−0.981331 + 0.192328i \(0.938396\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 43.3013i 1.62165i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 16.5000 28.5788i 0.615346 1.06581i −0.374978 0.927034i \(-0.622350\pi\)
0.990324 0.138777i \(-0.0443171\pi\)
\(720\) 0 0
\(721\) −0.500000 2.59808i −0.0186210 0.0967574i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −6.00000 10.3923i −0.222834 0.385961i
\(726\) 0 0
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 9.00000 + 15.5885i 0.332877 + 0.576560i
\(732\) 0 0
\(733\) −16.5000 9.52628i −0.609441 0.351861i 0.163305 0.986576i \(-0.447784\pi\)
−0.772747 + 0.634714i \(0.781118\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.50000 7.79423i 0.165760 0.287104i
\(738\) 0 0
\(739\) 1.50000 0.866025i 0.0551784 0.0318573i −0.472157 0.881514i \(-0.656524\pi\)
0.527335 + 0.849657i \(0.323191\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 38.1051i 1.39794i −0.715150 0.698971i \(-0.753642\pi\)
0.715150 0.698971i \(-0.246358\pi\)
\(744\) 0 0
\(745\) 13.5000 7.79423i 0.494602 0.285558i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 13.5000 2.59808i 0.493279 0.0949316i
\(750\) 0 0
\(751\) 34.5000 + 19.9186i 1.25892 + 0.726839i 0.972865 0.231373i \(-0.0743217\pi\)
0.286058 + 0.958212i \(0.407655\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 21.0000 0.764268
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4.50000 + 2.59808i 0.163125 + 0.0941802i 0.579340 0.815086i \(-0.303310\pi\)
−0.416215 + 0.909266i \(0.636644\pi\)
\(762\) 0 0
\(763\) 27.5000 + 9.52628i 0.995567 + 0.344874i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 6.92820i 0.249837i −0.992167 0.124919i \(-0.960133\pi\)
0.992167 0.124919i \(-0.0398670\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −7.50000 + 4.33013i −0.269756 + 0.155744i −0.628777 0.777586i \(-0.716444\pi\)
0.359021 + 0.933330i \(0.383111\pi\)
\(774\) 0 0
\(775\) 5.00000 8.66025i 0.179605 0.311086i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −42.0000 24.2487i −1.50481 0.868800i
\(780\) 0 0
\(781\) −3.00000 5.19615i −0.107348 0.185933i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −27.0000 −0.963671
\(786\) 0 0
\(787\) −3.50000 6.06218i −0.124762 0.216093i 0.796878 0.604140i \(-0.206483\pi\)
−0.921640 + 0.388047i \(0.873150\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 12.0000 10.3923i 0.426671 0.369508i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 13.8564i 0.490819i −0.969419 0.245410i \(-0.921078\pi\)
0.969419 0.245410i \(-0.0789224\pi\)
\(798\) 0 0
\(799\) 15.5885i 0.551480i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1.50000 +