Properties

Label 2736.2.s.k.577.1
Level $2736$
Weight $2$
Character 2736.577
Analytic conductor $21.847$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 577.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2736.577
Dual form 2736.2.s.k.1873.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{7} +O(q^{10})\) \(q-1.00000 q^{7} -2.00000 q^{11} +(1.50000 - 2.59808i) q^{13} +(2.00000 + 3.46410i) q^{17} +(-4.00000 + 1.73205i) q^{19} +(-2.00000 + 3.46410i) q^{23} +(2.50000 - 4.33013i) q^{25} +3.00000 q^{31} -5.00000 q^{37} +(2.00000 + 3.46410i) q^{41} +(-4.50000 - 7.79423i) q^{43} +(-5.00000 + 8.66025i) q^{47} -6.00000 q^{49} +(-2.00000 + 3.46410i) q^{53} +(7.00000 + 12.1244i) q^{59} +(-5.50000 + 9.52628i) q^{61} +(1.50000 - 2.59808i) q^{67} +(-7.00000 - 12.1244i) q^{71} +(5.50000 + 9.52628i) q^{73} +2.00000 q^{77} +(0.500000 + 0.866025i) q^{79} +8.00000 q^{83} +(-7.00000 + 12.1244i) q^{89} +(-1.50000 + 2.59808i) q^{91} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{7} - 4 q^{11} + 3 q^{13} + 4 q^{17} - 8 q^{19} - 4 q^{23} + 5 q^{25} + 6 q^{31} - 10 q^{37} + 4 q^{41} - 9 q^{43} - 10 q^{47} - 12 q^{49} - 4 q^{53} + 14 q^{59} - 11 q^{61} + 3 q^{67} - 14 q^{71} + 11 q^{73} + 4 q^{77} + q^{79} + 16 q^{83} - 14 q^{89} - 3 q^{91} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 1.50000 2.59808i 0.416025 0.720577i −0.579510 0.814965i \(-0.696756\pi\)
0.995535 + 0.0943882i \(0.0300895\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000 + 3.46410i 0.485071 + 0.840168i 0.999853 0.0171533i \(-0.00546033\pi\)
−0.514782 + 0.857321i \(0.672127\pi\)
\(18\) 0 0
\(19\) −4.00000 + 1.73205i −0.917663 + 0.397360i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.00000 + 3.46410i −0.417029 + 0.722315i −0.995639 0.0932891i \(-0.970262\pi\)
0.578610 + 0.815604i \(0.303595\pi\)
\(24\) 0 0
\(25\) 2.50000 4.33013i 0.500000 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.00000 −0.821995 −0.410997 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 + 3.46410i 0.312348 + 0.541002i 0.978870 0.204483i \(-0.0655513\pi\)
−0.666523 + 0.745485i \(0.732218\pi\)
\(42\) 0 0
\(43\) −4.50000 7.79423i −0.686244 1.18861i −0.973044 0.230618i \(-0.925925\pi\)
0.286801 0.957990i \(-0.407408\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.00000 + 8.66025i −0.729325 + 1.26323i 0.227844 + 0.973698i \(0.426832\pi\)
−0.957169 + 0.289530i \(0.906501\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.00000 + 3.46410i −0.274721 + 0.475831i −0.970065 0.242846i \(-0.921919\pi\)
0.695344 + 0.718677i \(0.255252\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.00000 + 12.1244i 0.911322 + 1.57846i 0.812198 + 0.583382i \(0.198271\pi\)
0.0991242 + 0.995075i \(0.468396\pi\)
\(60\) 0 0
\(61\) −5.50000 + 9.52628i −0.704203 + 1.21972i 0.262776 + 0.964857i \(0.415362\pi\)
−0.966978 + 0.254858i \(0.917971\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.50000 2.59808i 0.183254 0.317406i −0.759733 0.650236i \(-0.774670\pi\)
0.942987 + 0.332830i \(0.108004\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −7.00000 12.1244i −0.830747 1.43890i −0.897447 0.441123i \(-0.854580\pi\)
0.0666994 0.997773i \(-0.478753\pi\)
\(72\) 0 0
\(73\) 5.50000 + 9.52628i 0.643726 + 1.11497i 0.984594 + 0.174855i \(0.0559458\pi\)
−0.340868 + 0.940111i \(0.610721\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) 0.500000 + 0.866025i 0.0562544 + 0.0974355i 0.892781 0.450490i \(-0.148751\pi\)
−0.836527 + 0.547926i \(0.815418\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.00000 + 12.1244i −0.741999 + 1.28518i 0.209585 + 0.977790i \(0.432789\pi\)
−0.951584 + 0.307389i \(0.900545\pi\)
\(90\) 0 0
\(91\) −1.50000 + 2.59808i −0.157243 + 0.272352i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.00000 1.73205i −0.101535 0.175863i 0.810782 0.585348i \(-0.199042\pi\)
−0.912317 + 0.409484i \(0.865709\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −5.00000 + 8.66025i −0.497519 + 0.861727i −0.999996 0.00286291i \(-0.999089\pi\)
0.502477 + 0.864590i \(0.332422\pi\)
\(102\) 0 0
\(103\) 3.00000 0.295599 0.147799 0.989017i \(-0.452781\pi\)
0.147799 + 0.989017i \(0.452781\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −10.0000 −0.966736 −0.483368 0.875417i \(-0.660587\pi\)
−0.483368 + 0.875417i \(0.660587\pi\)
\(108\) 0 0
\(109\) −7.00000 12.1244i −0.670478 1.16130i −0.977769 0.209687i \(-0.932756\pi\)
0.307290 0.951616i \(-0.400578\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.00000 3.46410i −0.183340 0.317554i
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −4.00000 + 6.92820i −0.354943 + 0.614779i −0.987108 0.160055i \(-0.948833\pi\)
0.632166 + 0.774833i \(0.282166\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.00000 + 5.19615i 0.262111 + 0.453990i 0.966803 0.255524i \(-0.0822479\pi\)
−0.704692 + 0.709514i \(0.748915\pi\)
\(132\) 0 0
\(133\) 4.00000 1.73205i 0.346844 0.150188i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −3.00000 + 5.19615i −0.256307 + 0.443937i −0.965250 0.261329i \(-0.915839\pi\)
0.708942 + 0.705266i \(0.249173\pi\)
\(138\) 0 0
\(139\) −9.50000 + 16.4545i −0.805779 + 1.39565i 0.109984 + 0.993933i \(0.464920\pi\)
−0.915764 + 0.401718i \(0.868413\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −3.00000 + 5.19615i −0.250873 + 0.434524i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −11.0000 19.0526i −0.901155 1.56085i −0.825997 0.563675i \(-0.809387\pi\)
−0.0751583 0.997172i \(-0.523946\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 10.5000 + 18.1865i 0.837991 + 1.45144i 0.891572 + 0.452880i \(0.149603\pi\)
−0.0535803 + 0.998564i \(0.517063\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.00000 3.46410i 0.157622 0.273009i
\(162\) 0 0
\(163\) −11.0000 −0.861586 −0.430793 0.902451i \(-0.641766\pi\)
−0.430793 + 0.902451i \(0.641766\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3.00000 + 5.19615i −0.232147 + 0.402090i −0.958440 0.285295i \(-0.907908\pi\)
0.726293 + 0.687386i \(0.241242\pi\)
\(168\) 0 0
\(169\) 2.00000 + 3.46410i 0.153846 + 0.266469i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −12.0000 20.7846i −0.912343 1.58022i −0.810745 0.585399i \(-0.800938\pi\)
−0.101598 0.994826i \(-0.532395\pi\)
\(174\) 0 0
\(175\) −2.50000 + 4.33013i −0.188982 + 0.327327i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 9.00000 15.5885i 0.668965 1.15868i −0.309229 0.950988i \(-0.600071\pi\)
0.978194 0.207693i \(-0.0665956\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −4.00000 6.92820i −0.292509 0.506640i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) 6.50000 + 11.2583i 0.467880 + 0.810392i 0.999326 0.0366998i \(-0.0116845\pi\)
−0.531446 + 0.847092i \(0.678351\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) 2.50000 4.33013i 0.177220 0.306955i −0.763707 0.645563i \(-0.776623\pi\)
0.940927 + 0.338608i \(0.109956\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8.00000 3.46410i 0.553372 0.239617i
\(210\) 0 0
\(211\) 12.5000 + 21.6506i 0.860535 + 1.49049i 0.871413 + 0.490550i \(0.163204\pi\)
−0.0108774 + 0.999941i \(0.503462\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −3.00000 −0.203653
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) 4.50000 + 7.79423i 0.301342 + 0.521940i 0.976440 0.215788i \(-0.0692320\pi\)
−0.675098 + 0.737728i \(0.735899\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 8.00000 0.530979 0.265489 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) 0 0
\(229\) −11.0000 −0.726900 −0.363450 0.931614i \(-0.618401\pi\)
−0.363450 + 0.931614i \(0.618401\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.00000 + 15.5885i 0.589610 + 1.02123i 0.994283 + 0.106773i \(0.0340517\pi\)
−0.404674 + 0.914461i \(0.632615\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 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
\(241\) −12.5000 + 21.6506i −0.805196 + 1.39464i 0.110963 + 0.993825i \(0.464606\pi\)
−0.916159 + 0.400815i \(0.868727\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1.50000 + 12.9904i −0.0954427 + 0.826558i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 9.00000 15.5885i 0.568075 0.983935i −0.428681 0.903456i \(-0.641022\pi\)
0.996756 0.0804789i \(-0.0256450\pi\)
\(252\) 0 0
\(253\) 4.00000 6.92820i 0.251478 0.435572i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −6.00000 + 10.3923i −0.374270 + 0.648254i −0.990217 0.139533i \(-0.955440\pi\)
0.615948 + 0.787787i \(0.288773\pi\)
\(258\) 0 0
\(259\) 5.00000 0.310685
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −9.00000 15.5885i −0.554964 0.961225i −0.997906 0.0646755i \(-0.979399\pi\)
0.442943 0.896550i \(-0.353935\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1.00000 + 1.73205i 0.0609711 + 0.105605i 0.894900 0.446267i \(-0.147247\pi\)
−0.833929 + 0.551872i \(0.813914\pi\)
\(270\) 0 0
\(271\) −8.00000 13.8564i −0.485965 0.841717i 0.513905 0.857847i \(-0.328199\pi\)
−0.999870 + 0.0161307i \(0.994865\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −5.00000 + 8.66025i −0.301511 + 0.522233i
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −4.00000 + 6.92820i −0.238620 + 0.413302i −0.960319 0.278906i \(-0.910028\pi\)
0.721699 + 0.692207i \(0.243362\pi\)
\(282\) 0 0
\(283\) −10.0000 17.3205i −0.594438 1.02960i −0.993626 0.112728i \(-0.964041\pi\)
0.399188 0.916869i \(-0.369292\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.00000 3.46410i −0.118056 0.204479i
\(288\) 0 0
\(289\) 0.500000 0.866025i 0.0294118 0.0509427i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.00000 + 10.3923i 0.346989 + 0.601003i
\(300\) 0 0
\(301\) 4.50000 + 7.79423i 0.259376 + 0.449252i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −6.00000 10.3923i −0.342438 0.593120i 0.642447 0.766330i \(-0.277919\pi\)
−0.984885 + 0.173210i \(0.944586\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 10.0000 0.567048 0.283524 0.958965i \(-0.408496\pi\)
0.283524 + 0.958965i \(0.408496\pi\)
\(312\) 0 0
\(313\) −3.00000 + 5.19615i −0.169570 + 0.293704i −0.938269 0.345907i \(-0.887571\pi\)
0.768699 + 0.639611i \(0.220905\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.00000 10.3923i 0.336994 0.583690i −0.646872 0.762598i \(-0.723923\pi\)
0.983866 + 0.178908i \(0.0572566\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −14.0000 10.3923i −0.778981 0.578243i
\(324\) 0 0
\(325\) −7.50000 12.9904i −0.416025 0.720577i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 5.00000 8.66025i 0.275659 0.477455i
\(330\) 0 0
\(331\) −15.0000 −0.824475 −0.412237 0.911077i \(-0.635253\pi\)
−0.412237 + 0.911077i \(0.635253\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 9.50000 + 16.4545i 0.517498 + 0.896333i 0.999793 + 0.0203242i \(0.00646983\pi\)
−0.482295 + 0.876009i \(0.660197\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −8.00000 13.8564i −0.429463 0.743851i 0.567363 0.823468i \(-0.307964\pi\)
−0.996826 + 0.0796169i \(0.974630\pi\)
\(348\) 0 0
\(349\) −29.0000 −1.55233 −0.776167 0.630527i \(-0.782839\pi\)
−0.776167 + 0.630527i \(0.782839\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 12.0000 0.638696 0.319348 0.947638i \(-0.396536\pi\)
0.319348 + 0.947638i \(0.396536\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 18.0000 + 31.1769i 0.950004 + 1.64545i 0.745409 + 0.666608i \(0.232254\pi\)
0.204595 + 0.978847i \(0.434412\pi\)
\(360\) 0 0
\(361\) 13.0000 13.8564i 0.684211 0.729285i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 11.5000 19.9186i 0.600295 1.03974i −0.392481 0.919760i \(-0.628383\pi\)
0.992776 0.119982i \(-0.0382835\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.00000 3.46410i 0.103835 0.179847i
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 13.0000 0.667765 0.333883 0.942615i \(-0.391641\pi\)
0.333883 + 0.942615i \(0.391641\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −7.00000 12.1244i −0.357683 0.619526i 0.629890 0.776684i \(-0.283100\pi\)
−0.987573 + 0.157159i \(0.949767\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −8.00000 + 13.8564i −0.405616 + 0.702548i −0.994393 0.105748i \(-0.966276\pi\)
0.588777 + 0.808296i \(0.299610\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −7.50000 12.9904i −0.376414 0.651969i 0.614123 0.789210i \(-0.289510\pi\)
−0.990538 + 0.137241i \(0.956176\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 9.00000 + 15.5885i 0.449439 + 0.778450i 0.998350 0.0574304i \(-0.0182907\pi\)
−0.548911 + 0.835881i \(0.684957\pi\)
\(402\) 0 0
\(403\) 4.50000 7.79423i 0.224161 0.388258i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 10.0000 0.495682
\(408\) 0 0
\(409\) 15.0000 25.9808i 0.741702 1.28467i −0.210017 0.977698i \(-0.567352\pi\)
0.951720 0.306968i \(-0.0993146\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −7.00000 12.1244i −0.344447 0.596601i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 18.0000 0.879358 0.439679 0.898155i \(-0.355092\pi\)
0.439679 + 0.898155i \(0.355092\pi\)
\(420\) 0 0
\(421\) −5.00000 8.66025i −0.243685 0.422075i 0.718076 0.695965i \(-0.245023\pi\)
−0.961761 + 0.273890i \(0.911690\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 20.0000 0.970143
\(426\) 0 0
\(427\) 5.50000 9.52628i 0.266164 0.461009i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5.00000 8.66025i 0.240842 0.417150i −0.720113 0.693857i \(-0.755910\pi\)
0.960954 + 0.276707i \(0.0892433\pi\)
\(432\) 0 0
\(433\) 4.50000 7.79423i 0.216256 0.374567i −0.737404 0.675452i \(-0.763949\pi\)
0.953660 + 0.300885i \(0.0972820\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.00000 17.3205i 0.0956730 0.828552i
\(438\) 0 0
\(439\) 9.50000 + 16.4545i 0.453410 + 0.785330i 0.998595 0.0529862i \(-0.0168739\pi\)
−0.545185 + 0.838316i \(0.683541\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −12.0000 + 20.7846i −0.570137 + 0.987507i 0.426414 + 0.904528i \(0.359777\pi\)
−0.996551 + 0.0829786i \(0.973557\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 36.0000 1.69895 0.849473 0.527633i \(-0.176920\pi\)
0.849473 + 0.527633i \(0.176920\pi\)
\(450\) 0 0
\(451\) −4.00000 6.92820i −0.188353 0.326236i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.00000 0.233890 0.116945 0.993138i \(-0.462690\pi\)
0.116945 + 0.993138i \(0.462690\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.00000 + 5.19615i 0.139724 + 0.242009i 0.927392 0.374091i \(-0.122045\pi\)
−0.787668 + 0.616100i \(0.788712\pi\)
\(462\) 0 0
\(463\) 9.00000 0.418265 0.209133 0.977887i \(-0.432936\pi\)
0.209133 + 0.977887i \(0.432936\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) −1.50000 + 2.59808i −0.0692636 + 0.119968i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 9.00000 + 15.5885i 0.413820 + 0.716758i
\(474\) 0 0
\(475\) −2.50000 + 21.6506i −0.114708 + 0.993399i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −3.00000 + 5.19615i −0.137073 + 0.237418i −0.926388 0.376571i \(-0.877103\pi\)
0.789314 + 0.613990i \(0.210436\pi\)
\(480\) 0 0
\(481\) −7.50000 + 12.9904i −0.341971 + 0.592310i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −32.0000 −1.45006 −0.725029 0.688718i \(-0.758174\pi\)
−0.725029 + 0.688718i \(0.758174\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −15.0000 25.9808i −0.676941 1.17250i −0.975898 0.218229i \(-0.929972\pi\)
0.298957 0.954267i \(-0.403361\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7.00000 + 12.1244i 0.313993 + 0.543852i
\(498\) 0 0
\(499\) 2.50000 + 4.33013i 0.111915 + 0.193843i 0.916542 0.399937i \(-0.130968\pi\)
−0.804627 + 0.593780i \(0.797635\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −8.00000 + 13.8564i −0.356702 + 0.617827i −0.987408 0.158196i \(-0.949432\pi\)
0.630705 + 0.776022i \(0.282766\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 13.0000 22.5167i 0.576215 0.998033i −0.419694 0.907666i \(-0.637862\pi\)
0.995908 0.0903676i \(-0.0288042\pi\)
\(510\) 0 0
\(511\) −5.50000 9.52628i −0.243306 0.421418i
\(512\) 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
\(521\) −34.0000 −1.48957 −0.744784 0.667306i \(-0.767447\pi\)
−0.744784 + 0.667306i \(0.767447\pi\)
\(522\) 0 0
\(523\) 18.5000 32.0429i 0.808949 1.40114i −0.104644 0.994510i \(-0.533370\pi\)
0.913593 0.406630i \(-0.133296\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.00000 + 10.3923i 0.261364 + 0.452696i
\(528\) 0 0
\(529\) 3.50000 + 6.06218i 0.152174 + 0.263573i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 12.0000 0.516877
\(540\) 0 0
\(541\) 15.5000 26.8468i 0.666397 1.15423i −0.312507 0.949915i \(-0.601169\pi\)
0.978905 0.204318i \(-0.0654977\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 5.50000 9.52628i 0.235163 0.407314i −0.724157 0.689635i \(-0.757771\pi\)
0.959320 + 0.282321i \(0.0911043\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −0.500000 0.866025i −0.0212622 0.0368271i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 17.0000 29.4449i 0.720313 1.24762i −0.240561 0.970634i \(-0.577331\pi\)
0.960874 0.276985i \(-0.0893352\pi\)
\(558\) 0 0
\(559\) −27.0000 −1.14198
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −6.00000 −0.252870 −0.126435 0.991975i \(-0.540353\pi\)
−0.126435 + 0.991975i \(0.540353\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) 7.00000 0.292941 0.146470 0.989215i \(-0.453209\pi\)
0.146470 + 0.989215i \(0.453209\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 10.0000 + 17.3205i 0.417029 + 0.722315i
\(576\) 0 0
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) 0 0
\(583\) 4.00000 6.92820i 0.165663 0.286937i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.00000 + 1.73205i 0.0412744 + 0.0714894i 0.885925 0.463829i \(-0.153525\pi\)
−0.844650 + 0.535319i \(0.820192\pi\)
\(588\) 0 0
\(589\) −12.0000 + 5.19615i −0.494451 + 0.214104i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 17.0000 29.4449i 0.698106 1.20916i −0.271016 0.962575i \(-0.587360\pi\)
0.969122 0.246581i \(-0.0793071\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 23.0000 39.8372i 0.939755 1.62770i 0.173826 0.984776i \(-0.444387\pi\)
0.765928 0.642926i \(-0.222280\pi\)
\(600\) 0 0
\(601\) 27.0000 1.10135 0.550676 0.834719i \(-0.314370\pi\)
0.550676 + 0.834719i \(0.314370\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −5.00000 −0.202944 −0.101472 0.994838i \(-0.532355\pi\)
−0.101472 + 0.994838i \(0.532355\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 15.0000 + 25.9808i 0.606835 + 1.05107i
\(612\) 0 0
\(613\) −1.00000 1.73205i −0.0403896 0.0699569i 0.845124 0.534570i \(-0.179527\pi\)
−0.885514 + 0.464614i \(0.846193\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −21.0000 + 36.3731i −0.845428 + 1.46432i 0.0398207 + 0.999207i \(0.487321\pi\)
−0.885249 + 0.465118i \(0.846012\pi\)
\(618\) 0 0
\(619\) −1.00000 −0.0401934 −0.0200967 0.999798i \(-0.506397\pi\)
−0.0200967 + 0.999798i \(0.506397\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 7.00000 12.1244i 0.280449 0.485752i
\(624\) 0 0
\(625\) −12.5000 21.6506i −0.500000 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −10.0000 17.3205i −0.398726 0.690614i
\(630\) 0 0
\(631\) 8.50000 14.7224i 0.338380 0.586091i −0.645748 0.763550i \(-0.723455\pi\)
0.984128 + 0.177459i \(0.0567879\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −9.00000 + 15.5885i −0.356593 + 0.617637i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −7.00000 12.1244i −0.276483 0.478883i 0.694025 0.719951i \(-0.255836\pi\)
−0.970508 + 0.241068i \(0.922502\pi\)
\(642\) 0 0
\(643\) 18.5000 + 32.0429i 0.729569 + 1.26365i 0.957066 + 0.289871i \(0.0936125\pi\)
−0.227497 + 0.973779i \(0.573054\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 48.0000 1.88707 0.943537 0.331266i \(-0.107476\pi\)
0.943537 + 0.331266i \(0.107476\pi\)
\(648\) 0 0
\(649\) −14.0000 24.2487i −0.549548 0.951845i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 42.0000 1.64359 0.821794 0.569785i \(-0.192974\pi\)
0.821794 + 0.569785i \(0.192974\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 7.00000 12.1244i 0.272681 0.472298i −0.696866 0.717201i \(-0.745423\pi\)
0.969548 + 0.244903i \(0.0787562\pi\)
\(660\) 0 0
\(661\) −5.00000 + 8.66025i −0.194477 + 0.336845i −0.946729 0.322031i \(-0.895634\pi\)
0.752252 + 0.658876i \(0.228968\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 11.0000 19.0526i 0.424650 0.735516i
\(672\) 0 0
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) 1.00000 + 1.73205i 0.0383765 + 0.0664700i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 6.00000 + 10.3923i 0.228582 + 0.395915i
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −8.00000 + 13.8564i −0.303022 + 0.524849i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −22.0000 38.1051i −0.830929 1.43921i −0.897303 0.441416i \(-0.854476\pi\)
0.0663742 0.997795i \(-0.478857\pi\)
\(702\) 0 0
\(703\) 20.0000 8.66025i 0.754314 0.326628i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 5.00000 8.66025i 0.188044 0.325702i
\(708\) 0 0
\(709\) 15.5000 26.8468i 0.582115 1.00825i −0.413114 0.910679i \(-0.635559\pi\)
0.995228 0.0975728i \(-0.0311079\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −6.00000 + 10.3923i −0.224702 + 0.389195i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −20.0000 34.6410i −0.745874 1.29189i −0.949785 0.312903i \(-0.898699\pi\)
0.203911 0.978989i \(-0.434635\pi\)
\(720\) 0 0
\(721\) −3.00000 −0.111726
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 8.50000 + 14.7224i 0.315248 + 0.546025i 0.979490 0.201492i \(-0.0645791\pi\)
−0.664243 + 0.747517i \(0.731246\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 18.0000 31.1769i 0.665754 1.15312i
\(732\) 0 0
\(733\) −2.00000 −0.0738717 −0.0369358 0.999318i \(-0.511760\pi\)
−0.0369358 + 0.999318i \(0.511760\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −3.00000 + 5.19615i −0.110506 + 0.191403i
\(738\) 0 0
\(739\) −2.50000 4.33013i −0.0919640 0.159286i 0.816373 0.577524i \(-0.195981\pi\)
−0.908337 + 0.418238i \(0.862648\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 9.00000 + 15.5885i 0.330178 + 0.571885i 0.982547 0.186017i \(-0.0595579\pi\)
−0.652369 + 0.757902i \(0.726225\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 10.0000 0.365392
\(750\) 0 0
\(751\) 6.50000 11.2583i 0.237188 0.410822i −0.722718 0.691143i \(-0.757107\pi\)
0.959906 + 0.280321i \(0.0904408\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −2.50000 4.33013i −0.0908640 0.157381i 0.817011 0.576622i \(-0.195630\pi\)
−0.907875 + 0.419241i \(0.862296\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 34.0000 1.23250 0.616250 0.787551i \(-0.288651\pi\)
0.616250 + 0.787551i \(0.288651\pi\)
\(762\) 0 0
\(763\) 7.00000 + 12.1244i 0.253417 + 0.438931i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 42.0000 1.51653
\(768\) 0 0
\(769\) −13.5000 + 23.3827i −0.486822 + 0.843201i −0.999885 0.0151499i \(-0.995177\pi\)
0.513063 + 0.858351i \(0.328511\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −9.00000 + 15.5885i −0.323708 + 0.560678i −0.981250 0.192740i \(-0.938263\pi\)
0.657542 + 0.753418i \(0.271596\pi\)
\(774\) 0 0
\(775\) 7.50000 12.9904i 0.269408 0.466628i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −14.0000 10.3923i −0.501602 0.372343i
\(780\) 0 0
\(781\) 14.0000 + 24.2487i 0.500959 + 0.867687i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 47.0000 1.67537 0.837685 0.546154i \(-0.183909\pi\)
0.837685 + 0.546154i \(0.183909\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 10.0000 0.355559
\(792\) 0 0
\(793\) 16.5000 + 28.5788i 0.585932 + 1.01486i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −26.0000 −0.920967 −0.460484 0.887668i \(-0.652324\pi\)
−0.460484 + 0.887668i \(0.652324\pi\)
\(798\) 0 0
\(799\) −40.0000 −1.41510
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −11.0000 19.0526i −0.388182 0.672350i
\(804\) 0 0