Properties

Label 273.2.ba.b.38.1
Level $273$
Weight $2$
Character 273.38
Analytic conductor $2.180$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.ba (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 38.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 273.38
Dual form 273.2.ba.b.194.1

$q$-expansion

\(f(q)\) \(=\) \(q+(1.50000 + 0.866025i) q^{3} +(1.00000 - 1.73205i) q^{4} +(0.500000 - 2.59808i) q^{7} +(1.50000 + 2.59808i) q^{9} +O(q^{10})\) \(q+(1.50000 + 0.866025i) q^{3} +(1.00000 - 1.73205i) q^{4} +(0.500000 - 2.59808i) q^{7} +(1.50000 + 2.59808i) q^{9} +(3.00000 - 1.73205i) q^{12} +(-3.50000 - 0.866025i) q^{13} +(-2.00000 - 3.46410i) q^{16} +(3.50000 + 6.06218i) q^{19} +(3.00000 - 3.46410i) q^{21} +(-2.50000 + 4.33013i) q^{25} +5.19615i q^{27} +(-4.00000 - 3.46410i) q^{28} +(-3.50000 + 6.06218i) q^{31} +6.00000 q^{36} +(10.5000 - 6.06218i) q^{37} +(-4.50000 - 4.33013i) q^{39} -5.00000 q^{43} -6.92820i q^{48} +(-6.50000 - 2.59808i) q^{49} +(-5.00000 + 5.19615i) q^{52} +12.1244i q^{57} +(-6.00000 + 3.46410i) q^{61} +(7.50000 - 2.59808i) q^{63} -8.00000 q^{64} +(-10.5000 - 6.06218i) q^{67} +(-3.50000 + 6.06218i) q^{73} +(-7.50000 + 4.33013i) q^{75} +14.0000 q^{76} +(-6.50000 - 11.2583i) q^{79} +(-4.50000 + 7.79423i) q^{81} +(-3.00000 - 8.66025i) q^{84} +(-4.00000 + 8.66025i) q^{91} +(-10.5000 + 6.06218i) q^{93} +14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{3} + 2q^{4} + q^{7} + 3q^{9} + O(q^{10}) \) \( 2q + 3q^{3} + 2q^{4} + q^{7} + 3q^{9} + 6q^{12} - 7q^{13} - 4q^{16} + 7q^{19} + 6q^{21} - 5q^{25} - 8q^{28} - 7q^{31} + 12q^{36} + 21q^{37} - 9q^{39} - 10q^{43} - 13q^{49} - 10q^{52} - 12q^{61} + 15q^{63} - 16q^{64} - 21q^{67} - 7q^{73} - 15q^{75} + 28q^{76} - 13q^{79} - 9q^{81} - 6q^{84} - 8q^{91} - 21q^{93} + 28q^{97} + O(q^{100}) \)

Character values

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

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

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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(3\) 1.50000 + 0.866025i 0.866025 + 0.500000i
\(4\) 1.00000 1.73205i 0.500000 0.866025i
\(5\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) 0 0
\(7\) 0.500000 2.59808i 0.188982 0.981981i
\(8\) 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 \(0.166667\pi\)
\(12\) 3.00000 1.73205i 0.866025 0.500000i
\(13\) −3.50000 0.866025i −0.970725 0.240192i
\(14\) 0 0
\(15\) 0 0
\(16\) −2.00000 3.46410i −0.500000 0.866025i
\(17\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) 3.50000 + 6.06218i 0.802955 + 1.39076i 0.917663 + 0.397360i \(0.130073\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 3.00000 3.46410i 0.654654 0.755929i
\(22\) 0 0
\(23\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) 0 0
\(25\) −2.50000 + 4.33013i −0.500000 + 0.866025i
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) −4.00000 3.46410i −0.755929 0.654654i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −3.50000 + 6.06218i −0.628619 + 1.08880i 0.359211 + 0.933257i \(0.383046\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) 10.5000 6.06218i 1.72619 0.996616i 0.821995 0.569495i \(-0.192861\pi\)
0.904194 0.427121i \(-0.140472\pi\)
\(38\) 0 0
\(39\) −4.50000 4.33013i −0.720577 0.693375i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −5.00000 −0.762493 −0.381246 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(48\) 6.92820i 1.00000i
\(49\) −6.50000 2.59808i −0.928571 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) −5.00000 + 5.19615i −0.693375 + 0.720577i
\(53\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 12.1244i 1.60591i
\(58\) 0 0
\(59\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) −6.00000 + 3.46410i −0.768221 + 0.443533i −0.832240 0.554416i \(-0.812942\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 0 0
\(63\) 7.50000 2.59808i 0.944911 0.327327i
\(64\) −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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −3.50000 + 6.06218i −0.409644 + 0.709524i −0.994850 0.101361i \(-0.967680\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) −7.50000 + 4.33013i −0.866025 + 0.500000i
\(76\) 14.0000 1.60591
\(77\) 0 0
\(78\) 0 0
\(79\) −6.50000 11.2583i −0.731307 1.26666i −0.956325 0.292306i \(-0.905577\pi\)
0.225018 0.974355i \(-0.427756\pi\)
\(80\) 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 \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(90\) 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
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 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
\(103\) 16.5000 9.52628i 1.62579 0.938652i 0.640464 0.767988i \(-0.278742\pi\)
0.985329 0.170664i \(-0.0545913\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(108\) 9.00000 + 5.19615i 0.866025 + 0.500000i
\(109\) −10.5000 6.06218i −1.00572 0.580651i −0.0957826 0.995402i \(-0.530535\pi\)
−0.909935 + 0.414751i \(0.863869\pi\)
\(110\) 0 0
\(111\) 21.0000 1.99323
\(112\) −10.0000 + 3.46410i −0.944911 + 0.327327i
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −3.00000 10.3923i −0.277350 0.960769i
\(118\) 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
\(127\) 19.0000 1.68598 0.842989 0.537931i \(-0.180794\pi\)
0.842989 + 0.537931i \(0.180794\pi\)
\(128\) 0 0
\(129\) −7.50000 4.33013i −0.660338 0.381246i
\(130\) 0 0
\(131\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(132\) 0 0
\(133\) 17.5000 6.06218i 1.51744 0.525657i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(138\) 0 0
\(139\) 22.5167i 1.90984i −0.296866 0.954919i \(-0.595942\pi\)
0.296866 0.954919i \(-0.404058\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 6.00000 10.3923i 0.500000 0.866025i
\(145\) 0 0
\(146\) 0 0
\(147\) −7.50000 9.52628i −0.618590 0.785714i
\(148\) 24.2487i 1.99323i
\(149\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(150\) 0 0
\(151\) −21.0000 12.1244i −1.70896 0.986666i −0.935857 0.352381i \(-0.885372\pi\)
−0.773099 0.634285i \(-0.781294\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −12.0000 + 3.46410i −0.960769 + 0.277350i
\(157\) 18.0000 + 10.3923i 1.43656 + 0.829396i 0.997609 0.0691164i \(-0.0220180\pi\)
0.438948 + 0.898513i \(0.355351\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 21.0000 12.1244i 1.64485 0.949653i 0.665771 0.746156i \(-0.268103\pi\)
0.979076 0.203497i \(-0.0652307\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 11.5000 + 6.06218i 0.884615 + 0.466321i
\(170\) 0 0
\(171\) −10.5000 + 18.1865i −0.802955 + 1.39076i
\(172\) −5.00000 + 8.66025i −0.381246 + 0.660338i
\(173\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(174\) 0 0
\(175\) 10.0000 + 8.66025i 0.755929 + 0.654654i
\(176\) 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
\(181\) 25.9808i 1.93113i 0.260153 + 0.965567i \(0.416227\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 0 0
\(183\) −12.0000 −0.887066
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 13.5000 + 2.59808i 0.981981 + 0.188982i
\(190\) 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
\(193\) 10.5000 + 6.06218i 0.755807 + 0.436365i 0.827788 0.561041i \(-0.189599\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −11.0000 + 8.66025i −0.785714 + 0.618590i
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 3.00000 + 1.73205i 0.212664 + 0.122782i 0.602549 0.798082i \(-0.294152\pi\)
−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
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 14.0000 + 12.1244i 0.950382 + 0.823055i
\(218\) 0 0
\(219\) −10.5000 + 6.06218i −0.709524 + 0.409644i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −28.0000 −1.87502 −0.937509 0.347960i \(-0.886874\pi\)
−0.937509 + 0.347960i \(0.886874\pi\)
\(224\) 0 0
\(225\) −15.0000 −1.00000
\(226\) 0 0
\(227\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(228\) 21.0000 + 12.1244i 1.39076 + 0.802955i
\(229\) −3.50000 6.06218i −0.231287 0.400600i 0.726900 0.686743i \(-0.240960\pi\)
−0.958187 + 0.286143i \(0.907627\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 22.5167i 1.46261i
\(238\) 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 −0.547533 0.836784i \(-0.684433\pi\)
0.998443 + 0.0557856i \(0.0177663\pi\)
\(242\) 0 0
\(243\) −13.5000 + 7.79423i −0.866025 + 0.500000i
\(244\) 13.8564i 0.887066i
\(245\) 0 0
\(246\) 0 0
\(247\) −7.00000 24.2487i −0.445399 1.54291i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(258\) 0 0
\(259\) −10.5000 30.3109i −0.652438 1.88343i
\(260\) 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
\(268\) −21.0000 + 12.1244i −1.28278 + 0.740613i
\(269\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(270\) 0 0
\(271\) 14.0000 + 24.2487i 0.850439 + 1.47300i 0.880812 + 0.473466i \(0.156997\pi\)
−0.0303728 + 0.999539i \(0.509669\pi\)
\(272\) 0 0
\(273\) −13.5000 + 9.52628i −0.817057 + 0.576557i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 15.5000 26.8468i 0.931305 1.61307i 0.150210 0.988654i \(-0.452005\pi\)
0.781094 0.624413i \(-0.214662\pi\)
\(278\) 0 0
\(279\) −21.0000 −1.25724
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −28.5000 16.4545i −1.69415 0.978117i −0.951101 0.308879i \(-0.900046\pi\)
−0.743048 0.669238i \(-0.766621\pi\)
\(284\) 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
\(291\) 21.0000 + 12.1244i 1.23104 + 0.710742i
\(292\) 7.00000 + 12.1244i 0.409644 + 0.709524i
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 17.3205i 1.00000i
\(301\) −2.50000 + 12.9904i −0.144098 + 0.748753i
\(302\) 0 0
\(303\) 0 0
\(304\) 14.0000 24.2487i 0.802955 1.39076i
\(305\) 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
\(309\) 33.0000 1.87730
\(310\) 0 0
\(311\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(312\) 0 0
\(313\) 4.50000 2.59808i 0.254355 0.146852i −0.367402 0.930062i \(-0.619753\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −26.0000 −1.46261
\(317\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 9.00000 + 15.5885i 0.500000 + 0.866025i
\(325\) 12.5000 12.9904i 0.693375 0.720577i
\(326\) 0 0
\(327\) −10.5000 18.1865i −0.580651 1.00572i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 31.5000 18.1865i 1.73140 0.999622i 0.851957 0.523612i \(-0.175416\pi\)
0.879440 0.476011i \(-0.157918\pi\)
\(332\) 0 0
\(333\) 31.5000 + 18.1865i 1.72619 + 0.996616i
\(334\) 0 0
\(335\) 0 0
\(336\) −18.0000 3.46410i −0.981981 0.188982i
\(337\) 5.00000 0.272367 0.136184 0.990684i \(-0.456516\pi\)
0.136184 + 0.990684i \(0.456516\pi\)
\(338\) 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 \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 4.50000 18.1865i 0.240192 0.970725i
\(352\) 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 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(360\) 0 0
\(361\) −15.0000 + 25.9808i −0.789474 + 1.36741i
\(362\) 0 0
\(363\) 19.0526i 1.00000i
\(364\) 11.0000 + 15.5885i 0.576557 + 0.817057i
\(365\) 0 0
\(366\) 0 0
\(367\) −13.5000 7.79423i −0.704694 0.406855i 0.104399 0.994535i \(-0.466708\pi\)
−0.809093 + 0.587680i \(0.800041\pi\)
\(368\) 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
\(379\) 12.1244i 0.622786i 0.950281 + 0.311393i \(0.100796\pi\)
−0.950281 + 0.311393i \(0.899204\pi\)
\(380\) 0 0
\(381\) 28.5000 + 16.4545i 1.46010 + 0.842989i
\(382\) 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
\(387\) −7.50000 12.9904i −0.381246 0.660338i
\(388\) 14.0000 24.2487i 0.710742 1.23104i
\(389\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −17.5000 30.3109i −0.878300 1.52126i −0.853206 0.521575i \(-0.825345\pi\)
−0.0250943 0.999685i \(-0.507989\pi\)
\(398\) 0 0
\(399\) 31.5000 + 6.06218i 1.57697 + 0.303488i
\(400\) 20.0000 1.00000
\(401\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(402\) 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
\(409\) 3.50000 6.06218i 0.173064 0.299755i −0.766426 0.642333i \(-0.777967\pi\)
0.939490 + 0.342578i \(0.111300\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 38.1051i 1.87730i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 19.5000 33.7750i 0.954919 1.65397i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 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
\(427\) 6.00000 + 17.3205i 0.290360 + 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
\(433\) 22.5167i 1.08208i 0.840996 + 0.541041i \(0.181970\pi\)
−0.840996 + 0.541041i \(0.818030\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −21.0000 + 12.1244i −1.00572 + 0.580651i
\(437\) 0 0
\(438\) 0 0
\(439\) 27.0000 15.5885i 1.28864 0.743996i 0.310228 0.950662i \(-0.399595\pi\)
0.978412 + 0.206666i \(0.0662612\pi\)
\(440\) 0 0
\(441\) −3.00000 20.7846i −0.142857 0.989743i
\(442\) 0 0
\(443\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(444\) 21.0000 36.3731i 0.996616 1.72619i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −4.00000 + 20.7846i −0.188982 + 0.981981i
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 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 \(-0.575146\pi\)
0.725059 + 0.688686i \(0.241812\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 36.3731i 1.69040i −0.534450 0.845200i \(-0.679481\pi\)
0.534450 0.845200i \(-0.320519\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(468\) −21.0000 5.19615i −0.970725 0.240192i
\(469\) −21.0000 + 24.2487i −0.969690 + 1.11970i
\(470\) 0 0
\(471\) 18.0000 + 31.1769i 0.829396 + 1.43656i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −35.0000 −1.60591
\(476\) 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
\(481\) −42.0000 + 12.1244i −1.91504 + 0.552823i
\(482\) 0 0
\(483\) 0 0
\(484\) 22.0000 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 31.5000 + 18.1865i 1.42740 + 0.824110i 0.996915 0.0784867i \(-0.0250088\pi\)
0.430486 + 0.902597i \(0.358342\pi\)
\(488\) 0 0
\(489\) 42.0000 1.89931
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 28.0000 1.25724
\(497\) 0 0
\(498\) 0 0
\(499\) −10.5000 + 6.06218i −0.470045 + 0.271380i −0.716258 0.697835i \(-0.754147\pi\)
0.246214 + 0.969216i \(0.420813\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 12.0000 + 19.0526i 0.532939 + 0.846154i
\(508\) 19.0000 32.9090i 0.842989 1.46010i
\(509\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(510\) 0 0
\(511\) 14.0000 + 12.1244i 0.619324 + 0.536350i
\(512\) 0 0
\(513\) −31.5000 + 18.1865i −1.39076 + 0.802955i
\(514\) 0 0
\(515\) 0 0
\(516\) −15.0000 + 8.66025i −0.660338 + 0.381246i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(522\) 0 0
\(523\) −25.5000 + 14.7224i −1.11504 + 0.643767i −0.940129 0.340818i \(-0.889296\pi\)
−0.174908 + 0.984585i \(0.555963\pi\)
\(524\) 0 0
\(525\) 7.50000 + 21.6506i 0.327327 + 0.944911i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −11.5000 + 19.9186i −0.500000 + 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 7.00000 36.3731i 0.303488 1.57697i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 31.5000 18.1865i 1.35429 0.781900i 0.365444 0.930834i \(-0.380917\pi\)
0.988847 + 0.148933i \(0.0475840\pi\)
\(542\) 0 0
\(543\) −22.5000 + 38.9711i −0.965567 + 1.67241i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −40.0000 −1.71028 −0.855138 0.518400i \(-0.826528\pi\)
−0.855138 + 0.518400i \(0.826528\pi\)
\(548\) 0 0
\(549\) −18.0000 10.3923i −0.768221 0.443533i
\(550\) 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
\(559\) 17.5000 + 4.33013i 0.740171 + 0.183145i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 18.0000 + 15.5885i 0.755929 + 0.654654i
\(568\) 0 0
\(569\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(570\) 0 0
\(571\) −15.5000 + 26.8468i −0.648655 + 1.12350i 0.334790 + 0.942293i \(0.391335\pi\)
−0.983444 + 0.181210i \(0.941999\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −12.0000 20.7846i −0.500000 0.866025i
\(577\) −17.5000 + 30.3109i −0.728535 + 1.26186i 0.228968 + 0.973434i \(0.426465\pi\)
−0.957503 + 0.288425i \(0.906868\pi\)
\(578\) 0 0
\(579\) 10.5000 + 18.1865i 0.436365 + 0.755807i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) −24.0000 + 3.46410i −0.989743 + 0.142857i
\(589\) −49.0000 −2.01901
\(590\) 0 0
\(591\) 0 0
\(592\) −42.0000 24.2487i −1.72619 0.996616i
\(593\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 3.00000 + 5.19615i 0.122782 + 0.212664i
\(598\) 0 0
\(599\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(600\) 0 0
\(601\) 1.73205i 0.0706518i −0.999376 0.0353259i \(-0.988753\pi\)
0.999376 0.0353259i \(-0.0112469\pi\)
\(602\) 0 0
\(603\) 36.3731i 1.48123i
\(604\) −42.0000 + 24.2487i −1.70896 + 0.986666i
\(605\) 0 0
\(606\) 0 0
\(607\) −4.50000 + 2.59808i −0.182649 + 0.105453i −0.588537 0.808470i \(-0.700296\pi\)
0.405887 + 0.913923i \(0.366962\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 42.0000 + 24.2487i 1.69636 + 0.979396i 0.949156 + 0.314806i \(0.101939\pi\)
0.747208 + 0.664590i \(0.231394\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 24.5000 42.4352i 0.984738 1.70562i 0.341644 0.939829i \(-0.389016\pi\)
0.643094 0.765787i \(-0.277650\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −6.00000 + 24.2487i −0.240192 + 0.970725i
\(625\) −12.5000 21.6506i −0.500000 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 36.0000 20.7846i 1.43656 0.829396i
\(629\) 0 0
\(630\) 0 0
\(631\) 24.2487i 0.965326i −0.875806 0.482663i \(-0.839670\pi\)
0.875806 0.482663i \(-0.160330\pi\)
\(632\) 0 0
\(633\) −24.0000 13.8564i −0.953914 0.550743i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 20.5000 + 14.7224i 0.812240 + 0.583324i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(642\) 0 0
\(643\) 7.00000 0.276053 0.138027 0.990429i \(-0.455924\pi\)
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
\(651\) 10.5000 + 30.3109i 0.411527 + 1.18798i
\(652\) 48.4974i 1.89931i
\(653\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −21.0000 −0.819288
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 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
\(673\) 37.0000 1.42625 0.713123 0.701039i \(-0.247280\pi\)
0.713123 + 0.701039i \(0.247280\pi\)
\(674\) 0 0
\(675\) −22.5000 12.9904i −0.866025 0.500000i
\(676\) 22.0000 13.8564i 0.846154 0.532939i
\(677\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(678\) 0 0
\(679\) 7.00000 36.3731i 0.268635 1.39587i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(684\) 21.0000 + 36.3731i 0.802955 + 1.39076i
\(685\) 0 0
\(686\) 0 0
\(687\) 12.1244i 0.462573i
\(688\) 10.0000 + 17.3205i 0.381246 + 0.660338i
\(689\) 0 0
\(690\) 0 0
\(691\) 24.5000 + 42.4352i 0.932024 + 1.61431i 0.779857 + 0.625958i \(0.215292\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 25.0000 8.66025i 0.944911 0.327327i
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 73.5000 + 42.4352i 2.77211 + 1.60048i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −42.0000 + 24.2487i −1.57734 + 0.910679i −0.582115 + 0.813107i \(0.697775\pi\)
−0.995228 + 0.0975728i \(0.968892\pi\)
\(710\) 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
\(721\) −16.5000 47.6314i −0.614492 1.77389i
\(722\) 0 0
\(723\) 21.0000 12.1244i 0.780998 0.450910i
\(724\) 45.0000 + 25.9808i 1.67241 + 0.965567i
\(725\) 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
\(732\) −12.0000 + 20.7846i −0.443533 + 0.768221i
\(733\) −3.50000 6.06218i −0.129275 0.223912i 0.794121 0.607760i \(-0.207932\pi\)
−0.923396 + 0.383849i \(0.874598\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −10.5000 6.06218i −0.386249 0.223001i 0.294285 0.955718i \(-0.404919\pi\)
−0.680534 + 0.732717i \(0.738252\pi\)
\(740\) 0 0
\(741\) 10.5000 42.4352i 0.385727 1.55890i
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 20.5000 + 35.5070i 0.748056 + 1.29567i 0.948753 + 0.316017i \(0.102346\pi\)
−0.200698 + 0.979653i \(0.564321\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 18.0000 20.7846i 0.654654 0.755929i
\(757\) 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
\(761\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(762\) 0 0
\(763\) −21.0000 + 24.2487i −0.760251 + 0.877862i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −24.0000 + 13.8564i −0.866025 + 0.500000i
\(769\) 49.0000 1.76699 0.883493 0.468445i \(-0.155186\pi\)
0.883493 + 0.468445i \(0.155186\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 21.0000 12.1244i 0.755807 0.436365i
\(773\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(774\) 0 0
\(775\) −17.5000 30.3109i −0.628619 1.08880i
\(776\) 0 0
\(777\) 10.5000 54.5596i 0.376685 1.95731i
\(778\) 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
\(787\) −28.0000 + 48.4974i −0.998092 + 1.72875i −0.445577 + 0.895244i \(0.647001\pi\)
−0.552515 + 0.833503i \(0.686332\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 24.0000 6.92820i 0.852265 0.246028i
\(794\) 0 0
\(795\) 0 0
\(796\) 6.00000 3.46410i 0.212664 0.122782i
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) −42.0000 −1.48123