Properties

Label 2736.2.s.q.1873.1
Level $2736$
Weight $2$
Character 2736.1873
Analytic conductor $21.847$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

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

Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1873.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2736.1873
Dual form 2736.2.s.q.577.1

$q$-expansion

\(f(q)\) \(=\) \(q+(1.50000 - 2.59808i) q^{5} +O(q^{10})\) \(q+(1.50000 - 2.59808i) q^{5} -4.00000 q^{11} +(2.50000 + 4.33013i) q^{13} +(-2.50000 + 4.33013i) q^{17} +(-4.00000 + 1.73205i) q^{19} +(0.500000 + 0.866025i) q^{23} +(-2.00000 - 3.46410i) q^{25} +(1.50000 + 2.59808i) q^{29} -4.00000 q^{31} +2.00000 q^{37} +(-2.50000 + 4.33013i) q^{41} +(-5.50000 + 9.52628i) q^{43} +(2.50000 + 4.33013i) q^{47} -7.00000 q^{49} +(-4.50000 - 7.79423i) q^{53} +(-6.00000 + 10.3923i) q^{55} +(-6.50000 + 11.2583i) q^{59} +(0.500000 + 0.866025i) q^{61} +15.0000 q^{65} +(-2.50000 - 4.33013i) q^{67} +(-0.500000 + 0.866025i) q^{71} +(4.50000 - 7.79423i) q^{73} +(8.50000 - 14.7224i) q^{79} +16.0000 q^{83} +(7.50000 + 12.9904i) q^{85} +(1.50000 + 2.59808i) q^{89} +(-1.50000 + 12.9904i) q^{95} +(6.50000 - 11.2583i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{5} + O(q^{10}) \) \( 2q + 3q^{5} - 8q^{11} + 5q^{13} - 5q^{17} - 8q^{19} + q^{23} - 4q^{25} + 3q^{29} - 8q^{31} + 4q^{37} - 5q^{41} - 11q^{43} + 5q^{47} - 14q^{49} - 9q^{53} - 12q^{55} - 13q^{59} + q^{61} + 30q^{65} - 5q^{67} - q^{71} + 9q^{73} + 17q^{79} + 32q^{83} + 15q^{85} + 3q^{89} - 3q^{95} + 13q^{97} + O(q^{100}) \)

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
\(5\) 1.50000 2.59808i 0.670820 1.16190i −0.306851 0.951757i \(-0.599275\pi\)
0.977672 0.210138i \(-0.0673912\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 2.50000 + 4.33013i 0.693375 + 1.20096i 0.970725 + 0.240192i \(0.0772105\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.50000 + 4.33013i −0.606339 + 1.05021i 0.385499 + 0.922708i \(0.374029\pi\)
−0.991838 + 0.127502i \(0.959304\pi\)
\(18\) 0 0
\(19\) −4.00000 + 1.73205i −0.917663 + 0.397360i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.500000 + 0.866025i 0.104257 + 0.180579i 0.913434 0.406986i \(-0.133420\pi\)
−0.809177 + 0.587565i \(0.800087\pi\)
\(24\) 0 0
\(25\) −2.00000 3.46410i −0.400000 0.692820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.50000 + 2.59808i 0.278543 + 0.482451i 0.971023 0.238987i \(-0.0768152\pi\)
−0.692480 + 0.721437i \(0.743482\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.50000 + 4.33013i −0.390434 + 0.676252i −0.992507 0.122189i \(-0.961009\pi\)
0.602072 + 0.798441i \(0.294342\pi\)
\(42\) 0 0
\(43\) −5.50000 + 9.52628i −0.838742 + 1.45274i 0.0522047 + 0.998636i \(0.483375\pi\)
−0.890947 + 0.454108i \(0.849958\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.50000 + 4.33013i 0.364662 + 0.631614i 0.988722 0.149763i \(-0.0478510\pi\)
−0.624059 + 0.781377i \(0.714518\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 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
\(55\) −6.00000 + 10.3923i −0.809040 + 1.40130i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.50000 + 11.2583i −0.846228 + 1.46571i 0.0383226 + 0.999265i \(0.487799\pi\)
−0.884551 + 0.466444i \(0.845535\pi\)
\(60\) 0 0
\(61\) 0.500000 + 0.866025i 0.0640184 + 0.110883i 0.896258 0.443533i \(-0.146275\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 15.0000 1.86052
\(66\) 0 0
\(67\) −2.50000 4.33013i −0.305424 0.529009i 0.671932 0.740613i \(-0.265465\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.500000 + 0.866025i −0.0593391 + 0.102778i −0.894169 0.447730i \(-0.852233\pi\)
0.834830 + 0.550508i \(0.185566\pi\)
\(72\) 0 0
\(73\) 4.50000 7.79423i 0.526685 0.912245i −0.472831 0.881153i \(-0.656768\pi\)
0.999517 0.0310925i \(-0.00989865\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.50000 14.7224i 0.956325 1.65640i 0.225018 0.974355i \(-0.427756\pi\)
0.731307 0.682048i \(-0.238911\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 16.0000 1.75623 0.878114 0.478451i \(-0.158802\pi\)
0.878114 + 0.478451i \(0.158802\pi\)
\(84\) 0 0
\(85\) 7.50000 + 12.9904i 0.813489 + 1.40900i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.50000 + 2.59808i 0.159000 + 0.275396i 0.934508 0.355942i \(-0.115840\pi\)
−0.775509 + 0.631337i \(0.782506\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.50000 + 12.9904i −0.153897 + 1.33278i
\(96\) 0 0
\(97\) 6.50000 11.2583i 0.659975 1.14311i −0.320647 0.947199i \(-0.603900\pi\)
0.980622 0.195911i \(-0.0627665\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 9.50000 + 16.4545i 0.945285 + 1.63728i 0.755179 + 0.655519i \(0.227550\pi\)
0.190106 + 0.981763i \(0.439117\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −3.50000 + 6.06218i −0.335239 + 0.580651i −0.983531 0.180741i \(-0.942150\pi\)
0.648292 + 0.761392i \(0.275484\pi\)
\(110\) 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
\(115\) 3.00000 0.279751
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) 7.50000 + 12.9904i 0.665517 + 1.15271i 0.979145 + 0.203164i \(0.0651224\pi\)
−0.313627 + 0.949546i \(0.601544\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.50000 12.9904i 0.655278 1.13497i −0.326546 0.945181i \(-0.605885\pi\)
0.981824 0.189794i \(-0.0607819\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.50000 4.33013i −0.213589 0.369948i 0.739246 0.673436i \(-0.235182\pi\)
−0.952835 + 0.303488i \(0.901849\pi\)
\(138\) 0 0
\(139\) 7.50000 + 12.9904i 0.636142 + 1.10183i 0.986272 + 0.165129i \(0.0528040\pi\)
−0.350130 + 0.936701i \(0.613863\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −10.0000 17.3205i −0.836242 1.44841i
\(144\) 0 0
\(145\) 9.00000 0.747409
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −8.50000 + 14.7224i −0.696347 + 1.20611i 0.273377 + 0.961907i \(0.411859\pi\)
−0.969724 + 0.244202i \(0.921474\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −6.00000 + 10.3923i −0.481932 + 0.834730i
\(156\) 0 0
\(157\) 6.50000 11.2583i 0.518756 0.898513i −0.481006 0.876717i \(-0.659728\pi\)
0.999762 0.0217953i \(-0.00693820\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.50000 + 4.33013i 0.193456 + 0.335075i 0.946393 0.323017i \(-0.104697\pi\)
−0.752937 + 0.658092i \(0.771364\pi\)
\(168\) 0 0
\(169\) −6.00000 + 10.3923i −0.461538 + 0.799408i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2.50000 + 4.33013i −0.190071 + 0.329213i −0.945274 0.326278i \(-0.894205\pi\)
0.755202 + 0.655492i \(0.227539\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −3.50000 6.06218i −0.260153 0.450598i 0.706129 0.708083i \(-0.250440\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.00000 5.19615i 0.220564 0.382029i
\(186\) 0 0
\(187\) 10.0000 17.3205i 0.731272 1.26660i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) −7.50000 + 12.9904i −0.539862 + 0.935068i 0.459049 + 0.888411i \(0.348190\pi\)
−0.998911 + 0.0466572i \(0.985143\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 0 0
\(199\) 1.50000 + 2.59808i 0.106332 + 0.184173i 0.914282 0.405079i \(-0.132756\pi\)
−0.807950 + 0.589252i \(0.799423\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 7.50000 + 12.9904i 0.523823 + 0.907288i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 16.0000 6.92820i 1.10674 0.479234i
\(210\) 0 0
\(211\) 4.50000 7.79423i 0.309793 0.536577i −0.668524 0.743690i \(-0.733074\pi\)
0.978317 + 0.207114i \(0.0664070\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 16.5000 + 28.5788i 1.12529 + 1.94906i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −25.0000 −1.68168
\(222\) 0 0
\(223\) −5.50000 + 9.52628i −0.368307 + 0.637927i −0.989301 0.145889i \(-0.953396\pi\)
0.620994 + 0.783815i \(0.286729\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.50000 9.52628i 0.360317 0.624087i −0.627696 0.778459i \(-0.716002\pi\)
0.988013 + 0.154371i \(0.0493352\pi\)
\(234\) 0 0
\(235\) 15.0000 0.978492
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 10.5000 + 18.1865i 0.676364 + 1.17150i 0.976068 + 0.217465i \(0.0697789\pi\)
−0.299704 + 0.954032i \(0.596888\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −10.5000 + 18.1865i −0.670820 + 1.16190i
\(246\) 0 0
\(247\) −17.5000 12.9904i −1.11350 0.826558i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 4.50000 + 7.79423i 0.284037 + 0.491967i 0.972375 0.233423i \(-0.0749927\pi\)
−0.688338 + 0.725390i \(0.741659\pi\)
\(252\) 0 0
\(253\) −2.00000 3.46410i −0.125739 0.217786i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −12.5000 21.6506i −0.779729 1.35053i −0.932098 0.362206i \(-0.882024\pi\)
0.152370 0.988324i \(-0.451310\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 9.50000 16.4545i 0.585795 1.01463i −0.408981 0.912543i \(-0.634116\pi\)
0.994776 0.102084i \(-0.0325510\pi\)
\(264\) 0 0
\(265\) −27.0000 −1.65860
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −8.50000 + 14.7224i −0.518254 + 0.897643i 0.481521 + 0.876435i \(0.340085\pi\)
−0.999775 + 0.0212079i \(0.993249\pi\)
\(270\) 0 0
\(271\) 14.5000 25.1147i 0.880812 1.52561i 0.0303728 0.999539i \(-0.490331\pi\)
0.850439 0.526073i \(-0.176336\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 8.00000 + 13.8564i 0.482418 + 0.835573i
\(276\) 0 0
\(277\) −30.0000 −1.80253 −0.901263 0.433273i \(-0.857359\pi\)
−0.901263 + 0.433273i \(0.857359\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 7.50000 + 12.9904i 0.447412 + 0.774941i 0.998217 0.0596933i \(-0.0190123\pi\)
−0.550804 + 0.834634i \(0.685679\pi\)
\(282\) 0 0
\(283\) −7.50000 + 12.9904i −0.445829 + 0.772198i −0.998110 0.0614601i \(-0.980424\pi\)
0.552281 + 0.833658i \(0.313758\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4.00000 6.92820i −0.235294 0.407541i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −10.0000 −0.584206 −0.292103 0.956387i \(-0.594355\pi\)
−0.292103 + 0.956387i \(0.594355\pi\)
\(294\) 0 0
\(295\) 19.5000 + 33.7750i 1.13533 + 1.96646i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.50000 + 4.33013i −0.144579 + 0.250418i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 3.00000 0.171780
\(306\) 0 0
\(307\) −1.50000 + 2.59808i −0.0856095 + 0.148280i −0.905651 0.424024i \(-0.860617\pi\)
0.820041 + 0.572304i \(0.193950\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) 2.50000 + 4.33013i 0.141308 + 0.244753i 0.927990 0.372606i \(-0.121536\pi\)
−0.786681 + 0.617359i \(0.788202\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 13.5000 + 23.3827i 0.758236 + 1.31330i 0.943750 + 0.330661i \(0.107272\pi\)
−0.185514 + 0.982642i \(0.559395\pi\)
\(318\) 0 0
\(319\) −6.00000 10.3923i −0.335936 0.581857i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.50000 21.6506i 0.139104 1.20467i
\(324\) 0 0
\(325\) 10.0000 17.3205i 0.554700 0.960769i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −15.0000 −0.819538
\(336\) 0 0
\(337\) −1.50000 + 2.59808i −0.0817102 + 0.141526i −0.903985 0.427565i \(-0.859372\pi\)
0.822274 + 0.569091i \(0.192705\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2.50000 + 4.33013i −0.134207 + 0.232453i −0.925294 0.379250i \(-0.876182\pi\)
0.791087 + 0.611703i \(0.209515\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10.0000 0.532246 0.266123 0.963939i \(-0.414257\pi\)
0.266123 + 0.963939i \(0.414257\pi\)
\(354\) 0 0
\(355\) 1.50000 + 2.59808i 0.0796117 + 0.137892i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −2.50000 + 4.33013i −0.131945 + 0.228535i −0.924426 0.381361i \(-0.875456\pi\)
0.792481 + 0.609896i \(0.208789\pi\)
\(360\) 0 0
\(361\) 13.0000 13.8564i 0.684211 0.729285i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −13.5000 23.3827i −0.706622 1.22391i
\(366\) 0 0
\(367\) −2.50000 4.33013i −0.130499 0.226031i 0.793370 0.608740i \(-0.208325\pi\)
−0.923869 + 0.382709i \(0.874991\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −7.50000 + 12.9904i −0.386270 + 0.669039i
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 7.50000 12.9904i 0.383232 0.663777i −0.608290 0.793715i \(-0.708144\pi\)
0.991522 + 0.129937i \(0.0414776\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −8.50000 14.7224i −0.430967 0.746457i 0.565990 0.824412i \(-0.308494\pi\)
−0.996957 + 0.0779554i \(0.975161\pi\)
\(390\) 0 0
\(391\) −5.00000 −0.252861
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −25.5000 44.1673i −1.28304 2.22230i
\(396\) 0 0
\(397\) −17.5000 + 30.3109i −0.878300 + 1.52126i −0.0250943 + 0.999685i \(0.507989\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 5.50000 9.52628i 0.274657 0.475720i −0.695392 0.718631i \(-0.744769\pi\)
0.970049 + 0.242911i \(0.0781024\pi\)
\(402\) 0 0
\(403\) −10.0000 17.3205i −0.498135 0.862796i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) −3.50000 6.06218i −0.173064 0.299755i 0.766426 0.642333i \(-0.222033\pi\)
−0.939490 + 0.342578i \(0.888700\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 24.0000 41.5692i 1.17811 2.04055i
\(416\) 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
\(421\) 14.5000 25.1147i 0.706687 1.22402i −0.259393 0.965772i \(-0.583522\pi\)
0.966079 0.258245i \(-0.0831443\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 20.0000 0.970143
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4.50000 + 7.79423i 0.216757 + 0.375435i 0.953815 0.300395i \(-0.0971186\pi\)
−0.737057 + 0.675830i \(0.763785\pi\)
\(432\) 0 0
\(433\) 12.5000 + 21.6506i 0.600712 + 1.04046i 0.992713 + 0.120499i \(0.0384494\pi\)
−0.392002 + 0.919964i \(0.628217\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.50000 2.59808i −0.167428 0.124283i
\(438\) 0 0
\(439\) −3.50000 + 6.06218i −0.167046 + 0.289332i −0.937380 0.348309i \(-0.886756\pi\)
0.770334 + 0.637641i \(0.220089\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −7.50000 12.9904i −0.356336 0.617192i 0.631010 0.775775i \(-0.282641\pi\)
−0.987346 + 0.158583i \(0.949307\pi\)
\(444\) 0 0
\(445\) 9.00000 0.426641
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 0 0
\(451\) 10.0000 17.3205i 0.470882 0.815591i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −0.500000 + 0.866025i −0.0232873 + 0.0403348i −0.877434 0.479697i \(-0.840747\pi\)
0.854147 + 0.520032i \(0.174080\pi\)
\(462\) 0 0
\(463\) 20.0000 0.929479 0.464739 0.885448i \(-0.346148\pi\)
0.464739 + 0.885448i \(0.346148\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 22.0000 38.1051i 1.01156 1.75208i
\(474\) 0 0
\(475\) 14.0000 + 10.3923i 0.642364 + 0.476832i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −13.5000 23.3827i −0.616831 1.06838i −0.990060 0.140643i \(-0.955083\pi\)
0.373230 0.927739i \(-0.378250\pi\)
\(480\) 0 0
\(481\) 5.00000 + 8.66025i 0.227980 + 0.394874i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −19.5000 33.7750i −0.885449 1.53364i
\(486\) 0 0
\(487\) 40.0000 1.81257 0.906287 0.422664i \(-0.138905\pi\)
0.906287 + 0.422664i \(0.138905\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −4.50000 + 7.79423i −0.203082 + 0.351749i −0.949520 0.313707i \(-0.898429\pi\)
0.746438 + 0.665455i \(0.231763\pi\)
\(492\) 0 0
\(493\) −15.0000 −0.675566
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −17.5000 + 30.3109i −0.783408 + 1.35690i 0.146538 + 0.989205i \(0.453187\pi\)
−0.929946 + 0.367697i \(0.880146\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −7.50000 12.9904i −0.334408 0.579212i 0.648963 0.760820i \(-0.275203\pi\)
−0.983371 + 0.181608i \(0.941870\pi\)
\(504\) 0 0
\(505\) 57.0000 2.53647
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −2.50000 4.33013i −0.110811 0.191930i 0.805287 0.592886i \(-0.202011\pi\)
−0.916097 + 0.400956i \(0.868678\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −10.0000 17.3205i −0.439799 0.761755i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) −14.5000 25.1147i −0.634041 1.09819i −0.986718 0.162446i \(-0.948062\pi\)
0.352677 0.935745i \(-0.385272\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 10.0000 17.3205i 0.435607 0.754493i
\(528\) 0 0
\(529\) 11.0000 19.0526i 0.478261 0.828372i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −25.0000 −1.08287
\(534\) 0 0
\(535\) −18.0000 + 31.1769i −0.778208 + 1.34790i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 28.0000 1.20605
\(540\) 0 0
\(541\) −5.50000 9.52628i −0.236463 0.409567i 0.723234 0.690604i \(-0.242655\pi\)
−0.959697 + 0.281037i \(0.909322\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 10.5000 + 18.1865i 0.449771 + 0.779026i
\(546\) 0 0
\(547\) 3.50000 + 6.06218i 0.149649 + 0.259200i 0.931098 0.364770i \(-0.118852\pi\)
−0.781449 + 0.623970i \(0.785519\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −10.5000 7.79423i −0.447315 0.332045i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −16.5000 28.5788i −0.699127 1.21092i −0.968769 0.247964i \(-0.920239\pi\)
0.269642 0.962961i \(-0.413095\pi\)
\(558\) 0 0
\(559\) −55.0000 −2.32625
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) −9.00000 + 15.5885i −0.378633 + 0.655811i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2.00000 3.46410i 0.0834058 0.144463i
\(576\) 0 0
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 18.0000 + 31.1769i 0.745484 + 1.29122i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −12.5000 + 21.6506i −0.515930 + 0.893617i 0.483899 + 0.875124i \(0.339220\pi\)
−0.999829 + 0.0184934i \(0.994113\pi\)
\(588\) 0 0
\(589\) 16.0000 6.92820i 0.659269 0.285472i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 17.5000 + 30.3109i 0.718639 + 1.24472i 0.961539 + 0.274668i \(0.0885679\pi\)
−0.242900 + 0.970051i \(0.578099\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 12.5000 + 21.6506i 0.510736 + 0.884621i 0.999923 + 0.0124417i \(0.00396043\pi\)
−0.489186 + 0.872179i \(0.662706\pi\)
\(600\) 0 0
\(601\) −34.0000 −1.38689 −0.693444 0.720510i \(-0.743908\pi\)
−0.693444 + 0.720510i \(0.743908\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 7.50000 12.9904i 0.304918 0.528134i
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −12.5000 + 21.6506i −0.505696 + 0.875891i
\(612\) 0 0
\(613\) 0.500000 0.866025i 0.0201948 0.0349784i −0.855751 0.517387i \(-0.826905\pi\)
0.875946 + 0.482409i \(0.160238\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.50000 + 2.59808i 0.0603877 + 0.104595i 0.894639 0.446790i \(-0.147433\pi\)
−0.834251 + 0.551385i \(0.814100\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 14.5000 25.1147i 0.580000 1.00459i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −5.00000 + 8.66025i −0.199363 + 0.345307i
\(630\) 0 0
\(631\) −14.5000 25.1147i −0.577236 0.999802i −0.995795 0.0916122i \(-0.970798\pi\)
0.418559 0.908190i \(-0.362535\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 45.0000 1.78577
\(636\) 0 0
\(637\) −17.5000 30.3109i −0.693375 1.20096i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −4.50000 + 7.79423i −0.177739 + 0.307854i −0.941106 0.338112i \(-0.890212\pi\)
0.763367 + 0.645966i \(0.223545\pi\)
\(642\) 0 0
\(643\) 2.50000 4.33013i 0.0985904 0.170764i −0.812511 0.582946i \(-0.801900\pi\)
0.911101 + 0.412182i \(0.135233\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 26.0000 45.0333i 1.02059 1.76771i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 0 0
\(655\) −22.5000 38.9711i −0.879148 1.52273i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −9.50000 16.4545i −0.370067 0.640976i 0.619508 0.784990i \(-0.287332\pi\)
−0.989576 + 0.144015i \(0.953999\pi\)
\(660\) 0 0
\(661\) 14.5000 + 25.1147i 0.563985 + 0.976850i 0.997143 + 0.0755324i \(0.0240656\pi\)
−0.433159 + 0.901318i \(0.642601\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1.50000 + 2.59808i −0.0580802 + 0.100598i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −2.00000 3.46410i −0.0772091 0.133730i
\(672\) 0 0
\(673\) 30.0000 1.15642 0.578208 0.815890i \(-0.303752\pi\)
0.578208 + 0.815890i \(0.303752\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 30.0000 1.15299 0.576497 0.817099i \(-0.304419\pi\)
0.576497 + 0.817099i \(0.304419\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) −15.0000 −0.573121
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 22.5000 38.9711i 0.857182 1.48468i
\(690\) 0 0
\(691\) −36.0000 −1.36950 −0.684752 0.728776i \(-0.740090\pi\)
−0.684752 + 0.728776i \(0.740090\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 45.0000 1.70695
\(696\) 0 0
\(697\) −12.5000 21.6506i −0.473471 0.820076i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 11.5000 19.9186i 0.434349 0.752315i −0.562893 0.826530i \(-0.690312\pi\)
0.997242 + 0.0742151i \(0.0236451\pi\)
\(702\) 0 0
\(703\) −8.00000 + 3.46410i −0.301726 + 0.130651i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 8.50000 + 14.7224i 0.319224 + 0.552913i 0.980326 0.197383i \(-0.0632444\pi\)
−0.661102 + 0.750296i \(0.729911\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −2.00000 3.46410i −0.0749006 0.129732i
\(714\) 0 0
\(715\) −60.0000 −2.24387
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 13.5000 23.3827i 0.503465 0.872027i −0.496527 0.868021i \(-0.665392\pi\)
0.999992 0.00400572i \(-0.00127506\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 6.00000 10.3923i 0.222834 0.385961i
\(726\) 0 0
\(727\) 6.50000 11.2583i 0.241072 0.417548i −0.719948 0.694028i \(-0.755834\pi\)
0.961020 + 0.276479i \(0.0891678\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −27.5000 47.6314i −1.01712 1.76171i
\(732\) 0 0
\(733\) −6.00000 −0.221615 −0.110808 0.993842i \(-0.535344\pi\)
−0.110808 + 0.993842i \(0.535344\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 10.0000 + 17.3205i 0.368355 + 0.638009i
\(738\) 0 0
\(739\) 18.5000 32.0429i 0.680534 1.17872i −0.294285 0.955718i \(-0.595081\pi\)
0.974818 0.223001i \(-0.0715853\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −14.5000 + 25.1147i −0.531953 + 0.921370i 0.467351 + 0.884072i \(0.345209\pi\)
−0.999304 + 0.0372984i \(0.988125\pi\)
\(744\) 0 0
\(745\) 25.5000 + 44.1673i 0.934248 + 1.61816i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 15.5000 + 26.8468i 0.565603 + 0.979653i 0.996993 + 0.0774878i \(0.0246899\pi\)
−0.431390 + 0.902165i \(0.641977\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −24.0000 + 41.5692i −0.873449 + 1.51286i
\(756\) 0 0
\(757\) −17.5000 + 30.3109i −0.636048 + 1.10167i 0.350244 + 0.936659i \(0.386099\pi\)
−0.986292 + 0.165009i \(0.947235\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −38.0000 −1.37750 −0.688749 0.724999i \(-0.741840\pi\)
−0.688749 + 0.724999i \(0.741840\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −65.0000 −2.34701
\(768\) 0 0
\(769\) −11.5000 19.9186i −0.414701 0.718283i 0.580696 0.814120i \(-0.302780\pi\)
−0.995397 + 0.0958377i \(0.969447\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −12.5000 21.6506i −0.449594 0.778719i 0.548766 0.835976i \(-0.315098\pi\)
−0.998359 + 0.0572570i \(0.981765\pi\)
\(774\) 0 0
\(775\) 8.00000 + 13.8564i 0.287368 + 0.497737i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2.50000 21.6506i 0.0895718 0.775715i
\(780\) 0 0
\(781\) 2.00000 3.46410i 0.0715656 0.123955i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −19.5000 33.7750i −0.695985 1.20548i
\(786\) 0 0
\(787\) −20.0000 −0.712923 −0.356462 0.934310i \(-0.616017\pi\)
−0.356462 + 0.934310i \(0.616017\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −2.50000 + 4.33013i −0.0887776 + 0.153767i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) −25.0000 −0.884436
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −18.0000 + 31.1769i −0.635206 + 1.10021i
\(804\) 0 0
\(805\) 0