Properties

Label 273.2.u.b.251.1
Level $273$
Weight $2$
Character 273.251
Analytic conductor $2.180$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.u (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 251.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 273.251
Dual form 273.2.u.b.62.1

$q$-expansion

\(f(q)\) \(=\) \(q+(1.50000 + 0.866025i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-2.50000 + 0.866025i) q^{7} +(1.50000 + 2.59808i) q^{9} +O(q^{10})\) \(q+(1.50000 + 0.866025i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-2.50000 + 0.866025i) q^{7} +(1.50000 + 2.59808i) q^{9} +3.46410i q^{12} +(2.50000 - 2.59808i) q^{13} +(-2.00000 + 3.46410i) q^{16} +(-4.00000 - 6.92820i) q^{19} +(-4.50000 - 0.866025i) q^{21} +5.00000 q^{25} +5.19615i q^{27} +(-4.00000 - 3.46410i) q^{28} +7.00000 q^{31} +(-3.00000 + 5.19615i) q^{36} +(6.00000 + 3.46410i) q^{37} +(6.00000 - 1.73205i) q^{39} +(-6.50000 - 11.2583i) q^{43} +(-6.00000 + 3.46410i) q^{48} +(5.50000 - 4.33013i) q^{49} +(7.00000 + 1.73205i) q^{52} -13.8564i q^{57} +(-7.50000 + 4.33013i) q^{61} +(-6.00000 - 5.19615i) q^{63} -8.00000 q^{64} +(-10.5000 - 6.06218i) q^{67} -17.0000 q^{73} +(7.50000 + 4.33013i) q^{75} +(8.00000 - 13.8564i) q^{76} +13.0000 q^{79} +(-4.50000 + 7.79423i) q^{81} +(-3.00000 - 8.66025i) q^{84} +(-4.00000 + 8.66025i) q^{91} +(10.5000 + 6.06218i) q^{93} +(-2.50000 - 4.33013i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{3} + 2q^{4} - 5q^{7} + 3q^{9} + O(q^{10}) \) \( 2q + 3q^{3} + 2q^{4} - 5q^{7} + 3q^{9} + 5q^{13} - 4q^{16} - 8q^{19} - 9q^{21} + 10q^{25} - 8q^{28} + 14q^{31} - 6q^{36} + 12q^{37} + 12q^{39} - 13q^{43} - 12q^{48} + 11q^{49} + 14q^{52} - 15q^{61} - 12q^{63} - 16q^{64} - 21q^{67} - 34q^{73} + 15q^{75} + 16q^{76} + 26q^{79} - 9q^{81} - 6q^{84} - 8q^{91} + 21q^{93} - 5q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(-1\)

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 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(3\) 1.50000 + 0.866025i 0.866025 + 0.500000i
\(4\) 1.00000 + 1.73205i 0.500000 + 0.866025i
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −2.50000 + 0.866025i −0.944911 + 0.327327i
\(8\) 0 0
\(9\) 1.50000 + 2.59808i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(12\) 3.46410i 1.00000i
\(13\) 2.50000 2.59808i 0.693375 0.720577i
\(14\) 0 0
\(15\) 0 0
\(16\) −2.00000 + 3.46410i −0.500000 + 0.866025i
\(17\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) −4.00000 6.92820i −0.917663 1.58944i −0.802955 0.596040i \(-0.796740\pi\)
−0.114708 0.993399i \(-0.536593\pi\)
\(20\) 0 0
\(21\) −4.50000 0.866025i −0.981981 0.188982i
\(22\) 0 0
\(23\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) −4.00000 3.46410i −0.755929 0.654654i
\(29\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −3.00000 + 5.19615i −0.500000 + 0.866025i
\(37\) 6.00000 + 3.46410i 0.986394 + 0.569495i 0.904194 0.427121i \(-0.140472\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) 0 0
\(39\) 6.00000 1.73205i 0.960769 0.277350i
\(40\) 0 0
\(41\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(42\) 0 0
\(43\) −6.50000 11.2583i −0.991241 1.71688i −0.609994 0.792406i \(-0.708828\pi\)
−0.381246 0.924473i \(-0.624505\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −6.00000 + 3.46410i −0.866025 + 0.500000i
\(49\) 5.50000 4.33013i 0.785714 0.618590i
\(50\) 0 0
\(51\) 0 0
\(52\) 7.00000 + 1.73205i 0.970725 + 0.240192i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 13.8564i 1.83533i
\(58\) 0 0
\(59\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) −7.50000 + 4.33013i −0.960277 + 0.554416i −0.896258 0.443533i \(-0.853725\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) −6.00000 5.19615i −0.755929 0.654654i
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −10.5000 6.06218i −1.28278 0.740613i −0.305424 0.952217i \(-0.598798\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(72\) 0 0
\(73\) −17.0000 −1.98970 −0.994850 0.101361i \(-0.967680\pi\)
−0.994850 + 0.101361i \(0.967680\pi\)
\(74\) 0 0
\(75\) 7.50000 + 4.33013i 0.866025 + 0.500000i
\(76\) 8.00000 13.8564i 0.917663 1.58944i
\(77\) 0 0
\(78\) 0 0
\(79\) 13.0000 1.46261 0.731307 0.682048i \(-0.238911\pi\)
0.731307 + 0.682048i \(0.238911\pi\)
\(80\) 0 0
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −3.00000 8.66025i −0.327327 0.944911i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(90\) 0 0
\(91\) −4.00000 + 8.66025i −0.419314 + 0.907841i
\(92\) 0 0
\(93\) 10.5000 + 6.06218i 1.08880 + 0.628619i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.50000 4.33013i −0.253837 0.439658i 0.710742 0.703452i \(-0.248359\pi\)
−0.964579 + 0.263795i \(0.915026\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 5.00000 + 8.66025i 0.500000 + 0.866025i
\(101\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(102\) 0 0
\(103\) 15.5885i 1.53598i −0.640464 0.767988i \(-0.721258\pi\)
0.640464 0.767988i \(-0.278742\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(108\) −9.00000 + 5.19615i −0.866025 + 0.500000i
\(109\) 8.66025i 0.829502i 0.909935 + 0.414751i \(0.136131\pi\)
−0.909935 + 0.414751i \(0.863869\pi\)
\(110\) 0 0
\(111\) 6.00000 + 10.3923i 0.569495 + 0.986394i
\(112\) 2.00000 10.3923i 0.188982 0.981981i
\(113\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 10.5000 + 2.59808i 0.970725 + 0.240192i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.50000 + 9.52628i 0.500000 + 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 7.00000 + 12.1244i 0.628619 + 1.08880i
\(125\) 0 0
\(126\) 0 0
\(127\) −0.500000 + 0.866025i −0.0443678 + 0.0768473i −0.887357 0.461084i \(-0.847461\pi\)
0.842989 + 0.537931i \(0.180794\pi\)
\(128\) 0 0
\(129\) 22.5167i 1.98248i
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 16.0000 + 13.8564i 1.38738 + 1.20150i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(138\) 0 0
\(139\) −19.5000 + 11.2583i −1.65397 + 0.954919i −0.678551 + 0.734553i \(0.737392\pi\)
−0.975417 + 0.220366i \(0.929275\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −12.0000 −1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) 12.0000 1.73205i 0.989743 0.142857i
\(148\) 13.8564i 1.13899i
\(149\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(150\) 0 0
\(151\) 24.2487i 1.97333i 0.162758 + 0.986666i \(0.447961\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 9.00000 + 8.66025i 0.720577 + 0.693375i
\(157\) 22.5167i 1.79703i 0.438948 + 0.898513i \(0.355351\pi\)
−0.438948 + 0.898513i \(0.644649\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4.50000 + 2.59808i −0.352467 + 0.203497i −0.665771 0.746156i \(-0.731897\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(168\) 0 0
\(169\) −0.500000 12.9904i −0.0384615 0.999260i
\(170\) 0 0
\(171\) 12.0000 20.7846i 0.917663 1.58944i
\(172\) 13.0000 22.5167i 0.991241 1.71688i
\(173\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(174\) 0 0
\(175\) −12.5000 + 4.33013i −0.944911 + 0.327327i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(180\) 0 0
\(181\) 6.92820i 0.514969i −0.966282 0.257485i \(-0.917106\pi\)
0.966282 0.257485i \(-0.0828937\pi\)
\(182\) 0 0
\(183\) −15.0000 −1.10883
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −4.50000 12.9904i −0.327327 0.944911i
\(190\) 0 0
\(191\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(192\) −12.0000 6.92820i −0.866025 0.500000i
\(193\) 13.5000 + 7.79423i 0.971751 + 0.561041i 0.899770 0.436365i \(-0.143734\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 13.0000 + 5.19615i 0.928571 + 0.371154i
\(197\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(198\) 0 0
\(199\) −19.5000 + 11.2583i −1.38232 + 0.798082i −0.992434 0.122782i \(-0.960818\pi\)
−0.389885 + 0.920864i \(0.627485\pi\)
\(200\) 0 0
\(201\) −10.5000 18.1865i −0.740613 1.28278i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 4.00000 + 13.8564i 0.277350 + 0.960769i
\(209\) 0 0
\(210\) 0 0
\(211\) 6.50000 11.2583i 0.447478 0.775055i −0.550743 0.834675i \(-0.685655\pi\)
0.998221 + 0.0596196i \(0.0189888\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −17.5000 + 6.06218i −1.18798 + 0.411527i
\(218\) 0 0
\(219\) −25.5000 14.7224i −1.72313 0.994850i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 14.0000 24.2487i 0.937509 1.62381i 0.167412 0.985887i \(-0.446459\pi\)
0.770097 0.637927i \(-0.220208\pi\)
\(224\) 0 0
\(225\) 7.50000 + 12.9904i 0.500000 + 0.866025i
\(226\) 0 0
\(227\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(228\) 24.0000 13.8564i 1.58944 0.917663i
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 19.5000 + 11.2583i 1.26666 + 0.731307i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 7.00000 + 12.1244i 0.450910 + 0.780998i 0.998443 0.0557856i \(-0.0177663\pi\)
−0.547533 + 0.836784i \(0.684433\pi\)
\(242\) 0 0
\(243\) −13.5000 + 7.79423i −0.866025 + 0.500000i
\(244\) −15.0000 8.66025i −0.960277 0.554416i
\(245\) 0 0
\(246\) 0 0
\(247\) −28.0000 6.92820i −1.78160 0.440831i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(252\) 3.00000 15.5885i 0.188982 0.981981i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −8.00000 13.8564i −0.500000 0.866025i
\(257\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(258\) 0 0
\(259\) −18.0000 3.46410i −1.11847 0.215249i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 24.2487i 1.48123i
\(269\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(270\) 0 0
\(271\) −14.5000 + 25.1147i −0.880812 + 1.52561i −0.0303728 + 0.999539i \(0.509669\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) 0 0
\(273\) −13.5000 + 9.52628i −0.817057 + 0.576557i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −13.0000 22.5167i −0.781094 1.35290i −0.931305 0.364241i \(-0.881328\pi\)
0.150210 0.988654i \(-0.452005\pi\)
\(278\) 0 0
\(279\) 10.5000 + 18.1865i 0.628619 + 1.08880i
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 19.5000 + 11.2583i 1.15915 + 0.669238i 0.951101 0.308879i \(-0.0999539\pi\)
0.208053 + 0.978117i \(0.433287\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.50000 14.7224i 0.500000 0.866025i
\(290\) 0 0
\(291\) 8.66025i 0.507673i
\(292\) −17.0000 29.4449i −0.994850 1.72313i
\(293\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 17.3205i 1.00000i
\(301\) 26.0000 + 22.5167i 1.49862 + 1.29784i
\(302\) 0 0
\(303\) 0 0
\(304\) 32.0000 1.83533
\(305\) 0 0
\(306\) 0 0
\(307\) −35.0000 −1.99756 −0.998778 0.0494267i \(-0.984261\pi\)
−0.998778 + 0.0494267i \(0.984261\pi\)
\(308\) 0 0
\(309\) 13.5000 23.3827i 0.767988 1.33019i
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 32.9090i 1.86012i −0.367402 0.930062i \(-0.619753\pi\)
0.367402 0.930062i \(-0.380247\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 13.0000 + 22.5167i 0.731307 + 1.26666i
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −18.0000 −1.00000
\(325\) 12.5000 12.9904i 0.693375 0.720577i
\(326\) 0 0
\(327\) −7.50000 + 12.9904i −0.414751 + 0.718370i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −16.5000 + 9.52628i −0.906922 + 0.523612i −0.879440 0.476011i \(-0.842082\pi\)
−0.0274825 + 0.999622i \(0.508749\pi\)
\(332\) 0 0
\(333\) 20.7846i 1.13899i
\(334\) 0 0
\(335\) 0 0
\(336\) 12.0000 13.8564i 0.654654 0.755929i
\(337\) 29.0000 1.57973 0.789865 0.613280i \(-0.210150\pi\)
0.789865 + 0.613280i \(0.210150\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −10.0000 + 15.5885i −0.539949 + 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(348\) 0 0
\(349\) 11.5000 19.9186i 0.615581 1.06622i −0.374701 0.927146i \(-0.622255\pi\)
0.990282 0.139072i \(-0.0444119\pi\)
\(350\) 0 0
\(351\) 13.5000 + 12.9904i 0.720577 + 0.693375i
\(352\) 0 0
\(353\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −22.5000 + 38.9711i −1.18421 + 2.05111i
\(362\) 0 0
\(363\) 19.0526i 1.00000i
\(364\) −19.0000 + 1.73205i −0.995871 + 0.0907841i
\(365\) 0 0
\(366\) 0 0
\(367\) −19.5000 11.2583i −1.01789 0.587680i −0.104399 0.994535i \(-0.533292\pi\)
−0.913493 + 0.406855i \(0.866625\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 24.2487i 1.25724i
\(373\) −6.50000 11.2583i −0.336557 0.582934i 0.647225 0.762299i \(-0.275929\pi\)
−0.983783 + 0.179364i \(0.942596\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −10.5000 6.06218i −0.539349 0.311393i 0.205466 0.978664i \(-0.434129\pi\)
−0.744815 + 0.667271i \(0.767462\pi\)
\(380\) 0 0
\(381\) −1.50000 + 0.866025i −0.0768473 + 0.0443678i
\(382\) 0 0
\(383\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 19.5000 33.7750i 0.991241 1.71688i
\(388\) 5.00000 8.66025i 0.253837 0.439658i
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0.500000 + 0.866025i 0.0250943 + 0.0434646i 0.878300 0.478110i \(-0.158678\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 0 0
\(399\) 12.0000 + 34.6410i 0.600751 + 1.73422i
\(400\) −10.0000 + 17.3205i −0.500000 + 0.866025i
\(401\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(402\) 0 0
\(403\) 17.5000 18.1865i 0.871737 0.905936i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 15.5000 + 26.8468i 0.766426 + 1.32749i 0.939490 + 0.342578i \(0.111300\pi\)
−0.173064 + 0.984911i \(0.555367\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 27.0000 15.5885i 1.33019 0.767988i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −39.0000 −1.90984
\(418\) 0 0
\(419\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(420\) 0 0
\(421\) 36.3731i 1.77271i 0.463002 + 0.886357i \(0.346772\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 15.0000 17.3205i 0.725901 0.838198i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(432\) −18.0000 10.3923i −0.866025 0.500000i
\(433\) 19.5000 11.2583i 0.937110 0.541041i 0.0480569 0.998845i \(-0.484697\pi\)
0.889053 + 0.457804i \(0.151364\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −15.0000 + 8.66025i −0.718370 + 0.414751i
\(437\) 0 0
\(438\) 0 0
\(439\) 34.5000 + 19.9186i 1.64660 + 0.950662i 0.978412 + 0.206666i \(0.0662612\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) 19.5000 + 7.79423i 0.928571 + 0.371154i
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) −12.0000 + 20.7846i −0.569495 + 0.986394i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 20.0000 6.92820i 0.944911 0.327327i
\(449\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −21.0000 + 36.3731i −0.986666 + 1.70896i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −10.5000 6.06218i −0.491169 0.283577i 0.233890 0.972263i \(-0.424854\pi\)
−0.725059 + 0.688686i \(0.758188\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(462\) 0 0
\(463\) 1.73205i 0.0804952i −0.999190 0.0402476i \(-0.987185\pi\)
0.999190 0.0402476i \(-0.0128147\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 6.00000 + 20.7846i 0.277350 + 0.960769i
\(469\) 31.5000 + 6.06218i 1.45453 + 0.279925i
\(470\) 0 0
\(471\) −19.5000 + 33.7750i −0.898513 + 1.55627i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −20.0000 34.6410i −0.917663 1.58944i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(480\) 0 0
\(481\) 24.0000 6.92820i 1.09431 0.315899i
\(482\) 0 0
\(483\) 0 0
\(484\) −11.0000 + 19.0526i −0.500000 + 0.866025i
\(485\) 0 0
\(486\) 0 0
\(487\) −3.00000 + 1.73205i −0.135943 + 0.0784867i −0.566429 0.824110i \(-0.691675\pi\)
0.430486 + 0.902597i \(0.358342\pi\)
\(488\) 0 0
\(489\) −9.00000 −0.406994
\(490\) 0 0
\(491\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −14.0000 + 24.2487i −0.628619 + 1.08880i
\(497\) 0 0
\(498\) 0 0
\(499\) 31.1769i 1.39567i −0.716258 0.697835i \(-0.754147\pi\)
0.716258 0.697835i \(-0.245853\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 10.5000 19.9186i 0.466321 0.884615i
\(508\) −2.00000 −0.0887357
\(509\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(510\) 0 0
\(511\) 42.5000 14.7224i 1.88009 0.651282i
\(512\) 0 0
\(513\) 36.0000 20.7846i 1.58944 0.917663i
\(514\) 0 0
\(515\) 0 0
\(516\) 39.0000 22.5167i 1.71688 0.991241i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) −39.0000 22.5167i −1.70535 0.984585i −0.940129 0.340818i \(-0.889296\pi\)
−0.765222 0.643767i \(-0.777371\pi\)
\(524\) 0 0
\(525\) −22.5000 4.33013i −0.981981 0.188982i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −11.5000 19.9186i −0.500000 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) −8.00000 + 41.5692i −0.346844 + 1.80225i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 43.3013i 1.86167i −0.365444 0.930834i \(-0.619083\pi\)
0.365444 0.930834i \(-0.380917\pi\)
\(542\) 0 0
\(543\) 6.00000 10.3923i 0.257485 0.445976i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 41.0000 1.75303 0.876517 0.481371i \(-0.159861\pi\)
0.876517 + 0.481371i \(0.159861\pi\)
\(548\) 0 0
\(549\) −22.5000 12.9904i −0.960277 0.554416i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −32.5000 + 11.2583i −1.38204 + 0.478753i
\(554\) 0 0
\(555\) 0 0
\(556\) −39.0000 22.5167i −1.65397 0.954919i
\(557\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(558\) 0 0
\(559\) −45.5000 11.2583i −1.92444 0.476177i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 4.50000 23.3827i 0.188982 0.981981i
\(568\) 0 0
\(569\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(570\) 0 0
\(571\) 16.0000 0.669579 0.334790 0.942293i \(-0.391335\pi\)
0.334790 + 0.942293i \(0.391335\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −12.0000 20.7846i −0.500000 0.866025i
\(577\) −46.0000 −1.91501 −0.957503 0.288425i \(-0.906868\pi\)
−0.957503 + 0.288425i \(0.906868\pi\)
\(578\) 0 0
\(579\) 13.5000 + 23.3827i 0.561041 + 0.971751i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(588\) 15.0000 + 19.0526i 0.618590 + 0.785714i
\(589\) −28.0000 48.4974i −1.15372 1.99830i
\(590\) 0 0
\(591\) 0 0
\(592\) −24.0000 + 13.8564i −0.986394 + 0.569495i
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −39.0000 −1.59616
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 36.0000 + 20.7846i 1.46847 + 0.847822i 0.999376 0.0353259i \(-0.0112469\pi\)
0.469095 + 0.883148i \(0.344580\pi\)
\(602\) 0 0
\(603\) 36.3731i 1.48123i
\(604\) −42.0000 + 24.2487i −1.70896 + 0.986666i
\(605\) 0 0
\(606\) 0 0
\(607\) 39.0000 22.5167i 1.58296 0.913923i 0.588537 0.808470i \(-0.299704\pi\)
0.994424 0.105453i \(-0.0336291\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −28.5000 16.4545i −1.15110 0.664590i −0.201948 0.979396i \(-0.564727\pi\)
−0.949156 + 0.314806i \(0.898061\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(618\) 0 0
\(619\) 17.0000 0.683288 0.341644 0.939829i \(-0.389016\pi\)
0.341644 + 0.939829i \(0.389016\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −6.00000 + 24.2487i −0.240192 + 0.970725i
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −39.0000 + 22.5167i −1.55627 + 0.898513i
\(629\) 0 0
\(630\) 0 0
\(631\) 43.5000 25.1147i 1.73171 0.999802i 0.855901 0.517139i \(-0.173003\pi\)
0.875806 0.482663i \(-0.160330\pi\)
\(632\) 0 0
\(633\) 19.5000 11.2583i 0.775055 0.447478i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2.50000 25.1147i 0.0990536 0.995082i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(642\) 0 0
\(643\) 23.5000 + 40.7032i 0.926750 + 1.60518i 0.788723 + 0.614749i \(0.210743\pi\)
0.138027 + 0.990429i \(0.455924\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −31.5000 6.06218i −1.23458 0.237595i
\(652\) −9.00000 5.19615i −0.352467 0.203497i
\(653\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −25.5000 44.1673i −0.994850 1.72313i
\(658\) 0 0
\(659\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(660\) 0 0
\(661\) −24.5000 + 42.4352i −0.952940 + 1.65054i −0.213925 + 0.976850i \(0.568625\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 42.0000 24.2487i 1.62381 0.937509i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −6.50000 + 11.2583i −0.250557 + 0.433977i −0.963679 0.267063i \(-0.913947\pi\)
0.713123 + 0.701039i \(0.247280\pi\)
\(674\) 0 0
\(675\) 25.9808i 1.00000i
\(676\) 22.0000 13.8564i 0.846154 0.532939i
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 10.0000 + 8.66025i 0.383765 + 0.332350i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(684\) 48.0000 1.83533
\(685\) 0 0
\(686\) 0 0
\(687\) 33.0000 + 19.0526i 1.25903 + 0.726900i
\(688\) 52.0000 1.98248
\(689\) 0 0
\(690\) 0 0
\(691\) −20.5000 + 35.5070i −0.779857 + 1.35075i 0.152167 + 0.988355i \(0.451375\pi\)
−0.932024 + 0.362397i \(0.881959\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −20.0000 17.3205i −0.755929 0.654654i
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 55.4256i 2.09042i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 4.50000 2.59808i 0.169001 0.0975728i −0.413114 0.910679i \(-0.635559\pi\)
0.582115 + 0.813107i \(0.302225\pi\)
\(710\) 0 0
\(711\) 19.5000 + 33.7750i 0.731307 + 1.26666i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(720\) 0 0
\(721\) 13.5000 + 38.9711i 0.502766 + 1.45136i
\(722\) 0 0
\(723\) 24.2487i 0.901819i
\(724\) 12.0000 6.92820i 0.445976 0.257485i
\(725\) 0 0
\(726\) 0 0
\(727\) 22.5167i 0.835097i 0.908655 + 0.417548i \(0.137111\pi\)
−0.908655 + 0.417548i \(0.862889\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) −15.0000 25.9808i −0.554416 0.960277i
\(733\) 7.00000 0.258551 0.129275 0.991609i \(-0.458735\pi\)
0.129275 + 0.991609i \(0.458735\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 45.0000 + 25.9808i 1.65535 + 0.955718i 0.974818 + 0.223001i \(0.0715853\pi\)
0.680534 + 0.732717i \(0.261748\pi\)
\(740\) 0 0
\(741\) −36.0000 34.6410i −1.32249 1.27257i
\(742\) 0 0
\(743\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −26.0000 + 45.0333i −0.948753 + 1.64329i −0.200698 + 0.979653i \(0.564321\pi\)
−0.748056 + 0.663636i \(0.769012\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 18.0000 20.7846i 0.654654 0.755929i
\(757\) −13.0000 + 22.5167i −0.472493 + 0.818382i −0.999505 0.0314762i \(-0.989979\pi\)
0.527011 + 0.849858i \(0.323312\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(762\) 0 0
\(763\) −7.50000 21.6506i −0.271518 0.783806i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 27.7128i 1.00000i
\(769\) 1.00000 1.73205i 0.0360609 0.0624593i −0.847432 0.530904i \(-0.821852\pi\)
0.883493 + 0.468445i \(0.155186\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 31.1769i 1.12208i
\(773\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(774\) 0 0
\(775\) 35.0000 1.25724
\(776\) 0 0
\(777\) −24.0000 20.7846i −0.860995 0.745644i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 4.00000 + 27.7128i 0.142857 + 0.989743i
\(785\) 0 0
\(786\) 0 0
\(787\) 12.5000 + 21.6506i 0.445577 + 0.771762i 0.998092 0.0617409i \(-0.0196653\pi\)
−0.552515 + 0.833503i \(0.686332\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −7.50000 + 30.3109i −0.266333 + 1.07637i
\(794\) 0 0
\(795\) 0 0
\(796\) −39.0000 22.5167i −1.38232 0.798082i
\(797\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 21.0000 36.3731i 0.740613 1.28278i