Properties

Label 273.2.r.a.269.1
Level $273$
Weight $2$
Character 273.269
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.r (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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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 269.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 273.269
Dual form 273.2.r.a.68.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.50000 + 0.866025i) q^{3} +2.00000 q^{4} +(2.00000 - 1.73205i) q^{7} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\) \(q+(-1.50000 + 0.866025i) q^{3} +2.00000 q^{4} +(2.00000 - 1.73205i) q^{7} +(1.50000 - 2.59808i) q^{9} +(-3.00000 + 1.73205i) q^{12} +(-1.00000 + 3.46410i) q^{13} +4.00000 q^{16} +(7.50000 + 4.33013i) q^{19} +(-1.50000 + 4.33013i) q^{21} +(2.50000 - 4.33013i) q^{25} +5.19615i q^{27} +(4.00000 - 3.46410i) q^{28} +(-7.50000 - 4.33013i) q^{31} +(3.00000 - 5.19615i) q^{36} -11.0000 q^{37} +(-1.50000 - 6.06218i) q^{39} +(4.00000 + 6.92820i) q^{43} +(-6.00000 + 3.46410i) q^{48} +(1.00000 - 6.92820i) q^{49} +(-2.00000 + 6.92820i) q^{52} -15.0000 q^{57} +(-13.5000 - 7.79423i) q^{61} +(-1.50000 - 7.79423i) q^{63} +8.00000 q^{64} +(-5.50000 - 9.52628i) q^{67} +(-12.0000 - 6.92820i) q^{73} +8.66025i q^{75} +(15.0000 + 8.66025i) q^{76} +(6.50000 + 11.2583i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(-3.00000 + 8.66025i) q^{84} +(4.00000 + 8.66025i) q^{91} +15.0000 q^{93} +(-4.50000 + 2.59808i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} + 4 q^{4} + 4 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} + 4 q^{4} + 4 q^{7} + 3 q^{9} - 6 q^{12} - 2 q^{13} + 8 q^{16} + 15 q^{19} - 3 q^{21} + 5 q^{25} + 8 q^{28} - 15 q^{31} + 6 q^{36} - 22 q^{37} - 3 q^{39} + 8 q^{43} - 12 q^{48} + 2 q^{49} - 4 q^{52} - 30 q^{57} - 27 q^{61} - 3 q^{63} + 16 q^{64} - 11 q^{67} - 24 q^{73} + 30 q^{76} + 13 q^{79} - 9 q^{81} - 6 q^{84} + 8 q^{91} + 30 q^{93} - 9 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) −1.50000 + 0.866025i −0.866025 + 0.500000i
\(4\) 2.00000 1.00000
\(5\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 0 0
\(7\) 2.00000 1.73205i 0.755929 0.654654i
\(8\) 0 0
\(9\) 1.50000 2.59808i 0.500000 0.866025i
\(10\) 0 0
\(11\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) −3.00000 + 1.73205i −0.866025 + 0.500000i
\(13\) −1.00000 + 3.46410i −0.277350 + 0.960769i
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 7.50000 + 4.33013i 1.72062 + 0.993399i 0.917663 + 0.397360i \(0.130073\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 0 0
\(21\) −1.50000 + 4.33013i −0.327327 + 0.944911i
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 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 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) −7.50000 4.33013i −1.34704 0.777714i −0.359211 0.933257i \(-0.616954\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 3.00000 5.19615i 0.500000 0.866025i
\(37\) −11.0000 −1.80839 −0.904194 0.427121i \(-0.859528\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 0 0
\(39\) −1.50000 6.06218i −0.240192 0.970725i
\(40\) 0 0
\(41\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(42\) 0 0
\(43\) 4.00000 + 6.92820i 0.609994 + 1.05654i 0.991241 + 0.132068i \(0.0421616\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) −6.00000 + 3.46410i −0.866025 + 0.500000i
\(49\) 1.00000 6.92820i 0.142857 0.989743i
\(50\) 0 0
\(51\) 0 0
\(52\) −2.00000 + 6.92820i −0.277350 + 0.960769i
\(53\) 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
\(57\) −15.0000 −1.98680
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −13.5000 7.79423i −1.72850 0.997949i −0.896258 0.443533i \(-0.853725\pi\)
−0.832240 0.554416i \(-0.812942\pi\)
\(62\) 0 0
\(63\) −1.50000 7.79423i −0.188982 0.981981i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −5.50000 9.52628i −0.671932 1.16382i −0.977356 0.211604i \(-0.932131\pi\)
0.305424 0.952217i \(-0.401202\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(72\) 0 0
\(73\) −12.0000 6.92820i −1.40449 0.810885i −0.409644 0.912245i \(-0.634347\pi\)
−0.994850 + 0.101361i \(0.967680\pi\)
\(74\) 0 0
\(75\) 8.66025i 1.00000i
\(76\) 15.0000 + 8.66025i 1.72062 + 0.993399i
\(77\) 0 0
\(78\) 0 0
\(79\) 6.50000 + 11.2583i 0.731307 + 1.26666i 0.956325 + 0.292306i \(0.0944227\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) −4.50000 7.79423i −0.500000 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −3.00000 + 8.66025i −0.327327 + 0.944911i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 4.00000 + 8.66025i 0.419314 + 0.907841i
\(92\) 0 0
\(93\) 15.0000 1.55543
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.50000 + 2.59808i −0.456906 + 0.263795i −0.710742 0.703452i \(-0.751641\pi\)
0.253837 + 0.967247i \(0.418307\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 5.00000 8.66025i 0.500000 0.866025i
\(101\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(102\) 0 0
\(103\) 3.00000 1.73205i 0.295599 0.170664i −0.344865 0.938652i \(-0.612075\pi\)
0.640464 + 0.767988i \(0.278742\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 10.3923i 1.00000i
\(109\) 1.00000 1.73205i 0.0957826 0.165900i −0.814152 0.580651i \(-0.802798\pi\)
0.909935 + 0.414751i \(0.136131\pi\)
\(110\) 0 0
\(111\) 16.5000 9.52628i 1.56611 0.904194i
\(112\) 8.00000 6.92820i 0.755929 0.654654i
\(113\) 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
\(117\) 7.50000 + 7.79423i 0.693375 + 0.720577i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −5.50000 + 9.52628i −0.500000 + 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) −15.0000 8.66025i −1.34704 0.777714i
\(125\) 0 0
\(126\) 0 0
\(127\) −10.0000 + 17.3205i −0.887357 + 1.53695i −0.0443678 + 0.999015i \(0.514127\pi\)
−0.842989 + 0.537931i \(0.819206\pi\)
\(128\) 0 0
\(129\) −12.0000 6.92820i −1.05654 0.609994i
\(130\) 0 0
\(131\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(132\) 0 0
\(133\) 22.5000 4.33013i 1.95100 0.375470i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 19.5000 11.2583i 1.65397 0.954919i 0.678551 0.734553i \(-0.262608\pi\)
0.975417 0.220366i \(-0.0707252\pi\)
\(140\) 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
\(147\) 4.50000 + 11.2583i 0.371154 + 0.928571i
\(148\) −22.0000 −1.80839
\(149\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(150\) 0 0
\(151\) −2.00000 + 3.46410i −0.162758 + 0.281905i −0.935857 0.352381i \(-0.885372\pi\)
0.773099 + 0.634285i \(0.218706\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −3.00000 12.1244i −0.240192 0.970725i
\(157\) −1.50000 0.866025i −0.119713 0.0691164i 0.438948 0.898513i \(-0.355351\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −8.50000 + 14.7224i −0.665771 + 1.15315i 0.313304 + 0.949653i \(0.398564\pi\)
−0.979076 + 0.203497i \(0.934769\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(168\) 0 0
\(169\) −11.0000 6.92820i −0.846154 0.532939i
\(170\) 0 0
\(171\) 22.5000 12.9904i 1.72062 0.993399i
\(172\) 8.00000 + 13.8564i 0.609994 + 1.05654i
\(173\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(174\) 0 0
\(175\) −2.50000 12.9904i −0.188982 0.981981i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(180\) 0 0
\(181\) 19.0526i 1.41617i 0.706129 + 0.708083i \(0.250440\pi\)
−0.706129 + 0.708083i \(0.749560\pi\)
\(182\) 0 0
\(183\) 27.0000 1.99590
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 9.00000 + 10.3923i 0.654654 + 0.755929i
\(190\) 0 0
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) −12.0000 + 6.92820i −0.866025 + 0.500000i
\(193\) 1.00000 + 1.73205i 0.0719816 + 0.124676i 0.899770 0.436365i \(-0.143734\pi\)
−0.827788 + 0.561041i \(0.810401\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 2.00000 13.8564i 0.142857 0.989743i
\(197\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(198\) 0 0
\(199\) 25.9808i 1.84173i 0.389885 + 0.920864i \(0.372515\pi\)
−0.389885 + 0.920864i \(0.627485\pi\)
\(200\) 0 0
\(201\) 16.5000 + 9.52628i 1.16382 + 0.671932i
\(202\) 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
\(211\) 14.5000 25.1147i 0.998221 1.72897i 0.447478 0.894295i \(-0.352322\pi\)
0.550743 0.834675i \(-0.314345\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −22.5000 + 4.33013i −1.52740 + 0.293948i
\(218\) 0 0
\(219\) 24.0000 1.62177
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −9.00000 5.19615i −0.602685 0.347960i 0.167412 0.985887i \(-0.446459\pi\)
−0.770097 + 0.637927i \(0.779792\pi\)
\(224\) 0 0
\(225\) −7.50000 12.9904i −0.500000 0.866025i
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) −30.0000 −1.98680
\(229\) −7.50000 + 4.33013i −0.495614 + 0.286143i −0.726900 0.686743i \(-0.759040\pi\)
0.231287 + 0.972886i \(0.425707\pi\)
\(230\) 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
\(237\) −19.5000 11.2583i −1.26666 0.731307i
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 27.7128i 1.78514i −0.450910 0.892570i \(-0.648900\pi\)
0.450910 0.892570i \(-0.351100\pi\)
\(242\) 0 0
\(243\) 13.5000 + 7.79423i 0.866025 + 0.500000i
\(244\) −27.0000 15.5885i −1.72850 0.997949i
\(245\) 0 0
\(246\) 0 0
\(247\) −22.5000 + 21.6506i −1.43164 + 1.37760i
\(248\) 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
\(256\) 16.0000 1.00000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) −22.0000 + 19.0526i −1.36701 + 1.18387i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −11.0000 19.0526i −0.671932 1.16382i
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 32.9090i 1.99908i −0.0303728 0.999539i \(-0.509669\pi\)
0.0303728 0.999539i \(-0.490331\pi\)
\(272\) 0 0
\(273\) −13.5000 9.52628i −0.817057 0.576557i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5.00000 0.300421 0.150210 0.988654i \(-0.452005\pi\)
0.150210 + 0.988654i \(0.452005\pi\)
\(278\) 0 0
\(279\) −22.5000 + 12.9904i −1.34704 + 0.777714i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −9.00000 + 5.19615i −0.534994 + 0.308879i −0.743048 0.669238i \(-0.766621\pi\)
0.208053 + 0.978117i \(0.433287\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 4.50000 7.79423i 0.263795 0.456906i
\(292\) −24.0000 13.8564i −1.40449 0.810885i
\(293\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 17.3205i 1.00000i
\(301\) 20.0000 + 6.92820i 1.15278 + 0.399335i
\(302\) 0 0
\(303\) 0 0
\(304\) 30.0000 + 17.3205i 1.72062 + 0.993399i
\(305\) 0 0
\(306\) 0 0
\(307\) 1.73205i 0.0988534i −0.998778 0.0494267i \(-0.984261\pi\)
0.998778 0.0494267i \(-0.0157394\pi\)
\(308\) 0 0
\(309\) −3.00000 + 5.19615i −0.170664 + 0.295599i
\(310\) 0 0
\(311\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(312\) 0 0
\(313\) 24.0000 13.8564i 1.35656 0.783210i 0.367402 0.930062i \(-0.380247\pi\)
0.989158 + 0.146852i \(0.0469141\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 13.0000 + 22.5167i 0.731307 + 1.26666i
\(317\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(318\) 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
\(327\) 3.46410i 0.191565i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 16.0000 27.7128i 0.879440 1.52323i 0.0274825 0.999622i \(-0.491251\pi\)
0.851957 0.523612i \(-0.175416\pi\)
\(332\) 0 0
\(333\) −16.5000 + 28.5788i −0.904194 + 1.56611i
\(334\) 0 0
\(335\) 0 0
\(336\) −6.00000 + 17.3205i −0.327327 + 0.944911i
\(337\) 34.0000 1.85210 0.926049 0.377403i \(-0.123183\pi\)
0.926049 + 0.377403i \(0.123183\pi\)
\(338\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 4.50000 + 2.59808i 0.240879 + 0.139072i 0.615581 0.788074i \(-0.288921\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) −18.0000 5.19615i −0.960769 0.277350i
\(352\) 0 0
\(353\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(360\) 0 0
\(361\) 28.0000 + 48.4974i 1.47368 + 2.55250i
\(362\) 0 0
\(363\) 19.0526i 1.00000i
\(364\) 8.00000 + 17.3205i 0.419314 + 0.907841i
\(365\) 0 0
\(366\) 0 0
\(367\) 33.0000 19.0526i 1.72259 0.994535i 0.809093 0.587680i \(-0.199959\pi\)
0.913493 0.406855i \(-0.133375\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 30.0000 1.55543
\(373\) 6.50000 11.2583i 0.336557 0.582934i −0.647225 0.762299i \(-0.724071\pi\)
0.983783 + 0.179364i \(0.0574041\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −18.5000 + 32.0429i −0.950281 + 1.64594i −0.205466 + 0.978664i \(0.565871\pi\)
−0.744815 + 0.667271i \(0.767462\pi\)
\(380\) 0 0
\(381\) 34.6410i 1.77471i
\(382\) 0 0
\(383\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 24.0000 1.21999
\(388\) −9.00000 + 5.19615i −0.456906 + 0.263795i
\(389\) 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
\(397\) 18.0000 + 10.3923i 0.903394 + 0.521575i 0.878300 0.478110i \(-0.158678\pi\)
0.0250943 + 0.999685i \(0.492011\pi\)
\(398\) 0 0
\(399\) −30.0000 + 25.9808i −1.50188 + 1.30066i
\(400\) 10.0000 17.3205i 0.500000 0.866025i
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 22.5000 21.6506i 1.12080 1.07849i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 13.8564i 0.685155i 0.939490 + 0.342578i \(0.111300\pi\)
−0.939490 + 0.342578i \(0.888700\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 6.00000 3.46410i 0.295599 0.170664i
\(413\) 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 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(420\) 0 0
\(421\) −19.0000 −0.926003 −0.463002 0.886357i \(-0.653228\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −40.5000 + 7.79423i −1.95993 + 0.377189i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(432\) 20.7846i 1.00000i
\(433\) 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
\(436\) 2.00000 3.46410i 0.0957826 0.165900i
\(437\) 0 0
\(438\) 0 0
\(439\) 8.66025i 0.413331i −0.978412 0.206666i \(-0.933739\pi\)
0.978412 0.206666i \(-0.0662612\pi\)
\(440\) 0 0
\(441\) −16.5000 12.9904i −0.785714 0.618590i
\(442\) 0 0
\(443\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(444\) 33.0000 19.0526i 1.56611 0.904194i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 16.0000 13.8564i 0.755929 0.654654i
\(449\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 6.92820i 0.325515i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −41.0000 −1.91790 −0.958950 0.283577i \(-0.908479\pi\)
−0.958950 + 0.283577i \(0.908479\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(462\) 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
\(467\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(468\) 15.0000 + 15.5885i 0.693375 + 0.720577i
\(469\) −27.5000 9.52628i −1.26983 0.439883i
\(470\) 0 0
\(471\) 3.00000 0.138233
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 37.5000 21.6506i 1.72062 0.993399i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(480\) 0 0
\(481\) 11.0000 38.1051i 0.501557 1.73744i
\(482\) 0 0
\(483\) 0 0
\(484\) −11.0000 + 19.0526i −0.500000 + 0.866025i
\(485\) 0 0
\(486\) 0 0
\(487\) 19.0000 0.860972 0.430486 0.902597i \(-0.358342\pi\)
0.430486 + 0.902597i \(0.358342\pi\)
\(488\) 0 0
\(489\) 29.4449i 1.33154i
\(490\) 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
\(496\) −30.0000 17.3205i −1.34704 0.777714i
\(497\) 0 0
\(498\) 0 0
\(499\) 5.50000 + 9.52628i 0.246214 + 0.426455i 0.962472 0.271380i \(-0.0874801\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 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
\(507\) 22.5000 + 0.866025i 0.999260 + 0.0384615i
\(508\) −20.0000 + 34.6410i −0.887357 + 1.53695i
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −36.0000 + 6.92820i −1.59255 + 0.306486i
\(512\) 0 0
\(513\) −22.5000 + 38.9711i −0.993399 + 1.72062i
\(514\) 0 0
\(515\) 0 0
\(516\) −24.0000 13.8564i −1.05654 0.609994i
\(517\) 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
\(523\) 15.5885i 0.681636i 0.940129 + 0.340818i \(0.110704\pi\)
−0.940129 + 0.340818i \(0.889296\pi\)
\(524\) 0 0
\(525\) 15.0000 + 17.3205i 0.654654 + 0.755929i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 45.0000 8.66025i 1.95100 0.375470i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 23.0000 + 39.8372i 0.988847 + 1.71273i 0.623404 + 0.781900i \(0.285749\pi\)
0.365444 + 0.930834i \(0.380917\pi\)
\(542\) 0 0
\(543\) −16.5000 28.5788i −0.708083 1.22644i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.00000 −0.0427569 −0.0213785 0.999771i \(-0.506805\pi\)
−0.0213785 + 0.999771i \(0.506805\pi\)
\(548\) 0 0
\(549\) −40.5000 + 23.3827i −1.72850 + 0.997949i
\(550\) 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 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(558\) 0 0
\(559\) −28.0000 + 6.92820i −1.18427 + 0.293032i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −22.5000 7.79423i −0.944911 0.327327i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −23.5000 + 40.7032i −0.983444 + 1.70338i −0.334790 + 0.942293i \(0.608665\pi\)
−0.648655 + 0.761083i \(0.724668\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 12.0000 20.7846i 0.500000 0.866025i
\(577\) 40.5000 + 23.3827i 1.68604 + 0.973434i 0.957503 + 0.288425i \(0.0931316\pi\)
0.728535 + 0.685009i \(0.240202\pi\)
\(578\) 0 0
\(579\) −3.00000 1.73205i −0.124676 0.0719816i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(588\) 9.00000 + 22.5167i 0.371154 + 0.928571i
\(589\) −37.5000 64.9519i −1.54516 2.67630i
\(590\) 0 0
\(591\) 0 0
\(592\) −44.0000 −1.80839
\(593\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −22.5000 38.9711i −0.920864 1.59498i
\(598\) 0 0
\(599\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(600\) 0 0
\(601\) −37.5000 21.6506i −1.52966 0.883148i −0.999376 0.0353259i \(-0.988753\pi\)
−0.530281 0.847822i \(-0.677914\pi\)
\(602\) 0 0
\(603\) −33.0000 −1.34386
\(604\) −4.00000 + 6.92820i −0.162758 + 0.281905i
\(605\) 0 0
\(606\) 0 0
\(607\) −34.5000 19.9186i −1.40031 0.808470i −0.405887 0.913923i \(-0.633038\pi\)
−0.994424 + 0.105453i \(0.966371\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −23.5000 40.7032i −0.949156 1.64399i −0.747208 0.664590i \(-0.768606\pi\)
−0.201948 0.979396i \(-0.564727\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(618\) 0 0
\(619\) −33.0000 19.0526i −1.32638 0.765787i −0.341644 0.939829i \(-0.610984\pi\)
−0.984738 + 0.174042i \(0.944317\pi\)
\(620\) 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
\(628\) −3.00000 1.73205i −0.119713 0.0691164i
\(629\) 0 0
\(630\) 0 0
\(631\) 21.5000 + 37.2391i 0.855901 + 1.48246i 0.875806 + 0.482663i \(0.160330\pi\)
−0.0199047 + 0.999802i \(0.506336\pi\)
\(632\) 0 0
\(633\) 50.2295i 1.99644i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 23.0000 + 10.3923i 0.911293 + 0.411758i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 27.0000 15.5885i 1.06478 0.614749i 0.138027 0.990429i \(-0.455924\pi\)
0.926750 + 0.375680i \(0.122591\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 30.0000 25.9808i 1.17579 1.01827i
\(652\) −17.0000 + 29.4449i −0.665771 + 1.15315i
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −36.0000 + 20.7846i −1.40449 + 0.810885i
\(658\) 0 0
\(659\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(660\) 0 0
\(661\) −13.5000 + 7.79423i −0.525089 + 0.303160i −0.739014 0.673690i \(-0.764708\pi\)
0.213925 + 0.976850i \(0.431375\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 18.0000 0.695920
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 25.0000 43.3013i 0.963679 1.66914i 0.250557 0.968102i \(-0.419386\pi\)
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
\(679\) −4.50000 + 12.9904i −0.172694 + 0.498525i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 45.0000 25.9808i 1.72062 0.993399i
\(685\) 0 0
\(686\) 0 0
\(687\) 7.50000 12.9904i 0.286143 0.495614i
\(688\) 16.0000 + 27.7128i 0.609994 + 1.05654i
\(689\) 0 0
\(690\) 0 0
\(691\) 51.9615i 1.97671i 0.152167 + 0.988355i \(0.451375\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −5.00000 25.9808i −0.188982 0.981981i
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) −82.5000 47.6314i −3.11155 1.79645i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −15.5000 + 26.8468i −0.582115 + 1.00825i 0.413114 + 0.910679i \(0.364441\pi\)
−0.995228 + 0.0975728i \(0.968892\pi\)
\(710\) 0 0
\(711\) 39.0000 1.46261
\(712\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(720\) 0 0
\(721\) 3.00000 8.66025i 0.111726 0.322525i
\(722\) 0 0
\(723\) 24.0000 + 41.5692i 0.892570 + 1.54598i
\(724\) 38.1051i 1.41617i
\(725\) 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
\(732\) 54.0000 1.99590
\(733\) 46.5000 26.8468i 1.71752 0.991609i 0.794121 0.607760i \(-0.207932\pi\)
0.923396 0.383849i \(-0.125402\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 18.5000 + 32.0429i 0.680534 + 1.17872i 0.974818 + 0.223001i \(0.0715853\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) 15.0000 51.9615i 0.551039 1.90885i
\(742\) 0 0
\(743\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −11.0000 −0.401396 −0.200698 0.979653i \(-0.564321\pi\)
−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
\(757\) 13.0000 22.5167i 0.472493 0.818382i −0.527011 0.849858i \(-0.676688\pi\)
0.999505 + 0.0314762i \(0.0100208\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(762\) 0 0
\(763\) −1.00000 5.19615i −0.0362024 0.188113i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −24.0000 + 13.8564i −0.866025 + 0.500000i
\(769\) 25.5000 + 14.7224i 0.919554 + 0.530904i 0.883493 0.468445i \(-0.155186\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2.00000 + 3.46410i 0.0719816 + 0.124676i
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) −37.5000 + 21.6506i −1.34704 + 0.777714i
\(776\) 0 0
\(777\) 16.5000 47.6314i 0.591934 1.70877i
\(778\) 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
\(787\) 46.7654i 1.66701i −0.552515 0.833503i \(-0.686332\pi\)
0.552515 0.833503i \(-0.313668\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 40.5000 38.9711i 1.43820 1.38391i
\(794\) 0 0
\(795\) 0 0
\(796\) 51.9615i 1.84173i
\(797\) 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
\(802\)