Properties

Label 21.2.g.a.5.1
Level $21$
Weight $2$
Character 21.5
Analytic conductor $0.168$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 21.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.167685844245\)
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 5.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 21.5
Dual form 21.2.g.a.17.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.50000 + 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(0.500000 + 2.59808i) q^{7} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\) \(q+(-1.50000 + 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(0.500000 + 2.59808i) q^{7} +(1.50000 - 2.59808i) q^{9} +(3.00000 + 1.73205i) q^{12} -1.73205i q^{13} +(-2.00000 + 3.46410i) q^{16} +(-4.50000 - 2.59808i) q^{19} +(-3.00000 - 3.46410i) q^{21} +(2.50000 + 4.33013i) q^{25} +5.19615i q^{27} +(4.00000 - 3.46410i) q^{28} +(7.50000 - 4.33013i) q^{31} -6.00000 q^{36} +(-0.500000 + 0.866025i) q^{37} +(1.50000 + 2.59808i) q^{39} -5.00000 q^{43} -6.92820i q^{48} +(-6.50000 + 2.59808i) q^{49} +(-3.00000 + 1.73205i) q^{52} +9.00000 q^{57} +(6.00000 + 3.46410i) q^{61} +(7.50000 + 2.59808i) q^{63} +8.00000 q^{64} +(-5.50000 - 9.52628i) q^{67} +(-13.5000 + 7.79423i) q^{73} +(-7.50000 - 4.33013i) q^{75} +10.3923i q^{76} +(6.50000 - 11.2583i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(-3.00000 + 8.66025i) q^{84} +(4.50000 - 0.866025i) q^{91} +(-7.50000 + 12.9904i) q^{93} -13.8564i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{3} - 2q^{4} + q^{7} + 3q^{9} + O(q^{10}) \) \( 2q - 3q^{3} - 2q^{4} + q^{7} + 3q^{9} + 6q^{12} - 4q^{16} - 9q^{19} - 6q^{21} + 5q^{25} + 8q^{28} + 15q^{31} - 12q^{36} - q^{37} + 3q^{39} - 10q^{43} - 13q^{49} - 6q^{52} + 18q^{57} + 12q^{61} + 15q^{63} + 16q^{64} - 11q^{67} - 27q^{73} - 15q^{75} + 13q^{79} - 9q^{81} - 6q^{84} + 9q^{91} - 15q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) −1.50000 + 0.866025i −0.866025 + 0.500000i
\(4\) −1.00000 1.73205i −0.500000 0.866025i
\(5\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 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 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 3.00000 + 1.73205i 0.866025 + 0.500000i
\(13\) 1.73205i 0.480384i −0.970725 0.240192i \(-0.922790\pi\)
0.970725 0.240192i \(-0.0772105\pi\)
\(14\) 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
\(19\) −4.50000 2.59808i −1.03237 0.596040i −0.114708 0.993399i \(-0.536593\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) −3.00000 3.46410i −0.654654 0.755929i
\(22\) 0 0
\(23\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 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
\(31\) 7.50000 4.33013i 1.34704 0.777714i 0.359211 0.933257i \(-0.383046\pi\)
0.987829 + 0.155543i \(0.0497126\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −6.00000 −1.00000
\(37\) −0.500000 + 0.866025i −0.0821995 + 0.142374i −0.904194 0.427121i \(-0.859528\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 0 0
\(39\) 1.50000 + 2.59808i 0.240192 + 0.416025i
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 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 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 6.92820i 1.00000i
\(49\) −6.50000 + 2.59808i −0.928571 + 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) −3.00000 + 1.73205i −0.416025 + 0.240192i
\(53\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 9.00000 1.19208
\(58\) 0 0
\(59\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) 6.00000 + 3.46410i 0.768221 + 0.443533i 0.832240 0.554416i \(-0.187058\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) 7.50000 + 2.59808i 0.944911 + 0.327327i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −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 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −13.5000 + 7.79423i −1.58006 + 0.912245i −0.585206 + 0.810885i \(0.698986\pi\)
−0.994850 + 0.101361i \(0.967680\pi\)
\(74\) 0 0
\(75\) −7.50000 4.33013i −0.866025 0.500000i
\(76\) 10.3923i 1.19208i
\(77\) 0 0
\(78\) 0 0
\(79\) 6.50000 11.2583i 0.731307 1.26666i −0.225018 0.974355i \(-0.572244\pi\)
0.956325 0.292306i \(-0.0944227\pi\)
\(80\) 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 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) 0 0
\(91\) 4.50000 0.866025i 0.471728 0.0907841i
\(92\) 0 0
\(93\) −7.50000 + 12.9904i −0.777714 + 1.34704i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.8564i 1.40690i −0.710742 0.703452i \(-0.751641\pi\)
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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(102\) 0 0
\(103\) 16.5000 + 9.52628i 1.62579 + 0.938652i 0.985329 + 0.170664i \(0.0545913\pi\)
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
\(109\) 8.50000 + 14.7224i 0.814152 + 1.41015i 0.909935 + 0.414751i \(0.136131\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 1.73205i 0.164399i
\(112\) −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
\(117\) −4.50000 2.59808i −0.416025 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
\(124\) −15.0000 8.66025i −1.34704 0.777714i
\(125\) 0 0
\(126\) 0 0
\(127\) −19.0000 −1.68598 −0.842989 0.537931i \(-0.819206\pi\)
−0.842989 + 0.537931i \(0.819206\pi\)
\(128\) 0 0
\(129\) 7.50000 4.33013i 0.660338 0.381246i
\(130\) 0 0
\(131\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(132\) 0 0
\(133\) 4.50000 12.9904i 0.390199 1.12641i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(138\) 0 0
\(139\) 22.5167i 1.90984i 0.296866 + 0.954919i \(0.404058\pi\)
−0.296866 + 0.954919i \(0.595942\pi\)
\(140\) 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
\(148\) 2.00000 0.164399
\(149\) 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 0.773099 0.634285i \(-0.218706\pi\)
−0.935857 + 0.352381i \(0.885372\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 3.00000 5.19615i 0.240192 0.416025i
\(157\) 18.0000 10.3923i 1.43656 0.829396i 0.438948 0.898513i \(-0.355351\pi\)
0.997609 + 0.0691164i \(0.0220180\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4.00000 + 6.92820i −0.313304 + 0.542659i −0.979076 0.203497i \(-0.934769\pi\)
0.665771 + 0.746156i \(0.268103\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 10.0000 0.769231
\(170\) 0 0
\(171\) −13.5000 + 7.79423i −1.03237 + 0.596040i
\(172\) 5.00000 + 8.66025i 0.381246 + 0.660338i
\(173\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(174\) 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 \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(180\) 0 0
\(181\) 25.9808i 1.93113i −0.260153 0.965567i \(-0.583773\pi\)
0.260153 0.965567i \(-0.416227\pi\)
\(182\) 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 \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) −12.0000 + 6.92820i −0.866025 + 0.500000i
\(193\) −12.5000 21.6506i −0.899770 1.55845i −0.827788 0.561041i \(-0.810401\pi\)
−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 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −3.00000 + 1.73205i −0.212664 + 0.122782i −0.602549 0.798082i \(-0.705848\pi\)
0.389885 + 0.920864i \(0.372515\pi\)
\(200\) 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
\(208\) 6.00000 + 3.46410i 0.416025 + 0.240192i
\(209\) 0 0
\(210\) 0 0
\(211\) 16.0000 1.10149 0.550743 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 15.0000 + 17.3205i 1.01827 + 1.17579i
\(218\) 0 0
\(219\) 13.5000 23.3827i 0.912245 1.58006i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 10.3923i 0.695920i 0.937509 + 0.347960i \(0.113126\pi\)
−0.937509 + 0.347960i \(0.886874\pi\)
\(224\) 0 0
\(225\) 15.0000 1.00000
\(226\) 0 0
\(227\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(228\) −9.00000 15.5885i −0.596040 1.03237i
\(229\) −25.5000 14.7224i −1.68509 0.972886i −0.958187 0.286143i \(-0.907627\pi\)
−0.726900 0.686743i \(-0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 22.5167i 1.46261i
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −24.0000 + 13.8564i −1.54598 + 0.892570i −0.547533 + 0.836784i \(0.684433\pi\)
−0.998443 + 0.0557856i \(0.982234\pi\)
\(242\) 0 0
\(243\) 13.5000 + 7.79423i 0.866025 + 0.500000i
\(244\) 13.8564i 0.887066i
\(245\) 0 0
\(246\) 0 0
\(247\) −4.50000 + 7.79423i −0.286328 + 0.495935i
\(248\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(258\) 0 0
\(259\) −2.50000 0.866025i −0.155342 0.0538122i
\(260\) 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 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(270\) 0 0
\(271\) −15.0000 8.66025i −0.911185 0.526073i −0.0303728 0.999539i \(-0.509669\pi\)
−0.880812 + 0.473466i \(0.843003\pi\)
\(272\) 0 0
\(273\) −6.00000 + 5.19615i −0.363137 + 0.314485i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 15.5000 + 26.8468i 0.931305 + 1.61307i 0.781094 + 0.624413i \(0.214662\pi\)
0.150210 + 0.988654i \(0.452005\pi\)
\(278\) 0 0
\(279\) 25.9808i 1.55543i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 28.5000 16.4545i 1.69415 0.978117i 0.743048 0.669238i \(-0.233379\pi\)
0.951101 0.308879i \(-0.0999539\pi\)
\(284\) 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
\(291\) 12.0000 + 20.7846i 0.703452 + 1.21842i
\(292\) 27.0000 + 15.5885i 1.58006 + 0.912245i
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 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
\(304\) 18.0000 10.3923i 1.03237 0.596040i
\(305\) 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
\(309\) −33.0000 −1.87730
\(310\) 0 0
\(311\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) −4.50000 2.59808i −0.254355 0.146852i 0.367402 0.930062i \(-0.380247\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −26.0000 −1.46261
\(317\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(318\) 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
\(325\) 7.50000 4.33013i 0.416025 0.240192i
\(326\) 0 0
\(327\) −25.5000 14.7224i −1.41015 0.814152i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −0.500000 + 0.866025i −0.0274825 + 0.0476011i −0.879440 0.476011i \(-0.842082\pi\)
0.851957 + 0.523612i \(0.175416\pi\)
\(332\) 0 0
\(333\) 1.50000 + 2.59808i 0.0821995 + 0.142374i
\(334\) 0 0
\(335\) 0 0
\(336\) 18.0000 3.46410i 0.981981 0.188982i
\(337\) −5.00000 −0.272367 −0.136184 0.990684i \(-0.543484\pi\)
−0.136184 + 0.990684i \(0.543484\pi\)
\(338\) 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
\(349\) 34.6410i 1.85429i 0.374701 + 0.927146i \(0.377745\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 9.00000 0.480384
\(352\) 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 \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(360\) 0 0
\(361\) 4.00000 + 6.92820i 0.210526 + 0.364642i
\(362\) 0 0
\(363\) 19.0526i 1.00000i
\(364\) −6.00000 6.92820i −0.314485 0.363137i
\(365\) 0 0
\(366\) 0 0
\(367\) −13.5000 + 7.79423i −0.704694 + 0.406855i −0.809093 0.587680i \(-0.800041\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) 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
\(379\) 37.0000 1.90056 0.950281 0.311393i \(-0.100796\pi\)
0.950281 + 0.311393i \(0.100796\pi\)
\(380\) 0 0
\(381\) 28.5000 16.4545i 1.46010 0.842989i
\(382\) 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
\(387\) −7.50000 + 12.9904i −0.381246 + 0.660338i
\(388\) −24.0000 + 13.8564i −1.21842 + 0.703452i
\(389\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 16.5000 + 9.52628i 0.828111 + 0.478110i 0.853206 0.521575i \(-0.174655\pi\)
−0.0250943 + 0.999685i \(0.507989\pi\)
\(398\) 0 0
\(399\) 4.50000 + 23.3827i 0.225282 + 1.17060i
\(400\) −20.0000 −1.00000
\(401\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0 0
\(403\) −7.50000 12.9904i −0.373602 0.647097i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −34.5000 + 19.9186i −1.70592 + 0.984911i −0.766426 + 0.642333i \(0.777967\pi\)
−0.939490 + 0.342578i \(0.888700\pi\)
\(410\) 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
\(421\) −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
\(427\) −6.00000 + 17.3205i −0.290360 + 0.838198i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(432\) −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
\(436\) 17.0000 29.4449i 0.814152 1.41015i
\(437\) 0 0
\(438\) 0 0
\(439\) 27.0000 + 15.5885i 1.28864 + 0.743996i 0.978412 0.206666i \(-0.0662612\pi\)
0.310228 + 0.950662i \(0.399595\pi\)
\(440\) 0 0
\(441\) −3.00000 + 20.7846i −0.142857 + 0.989743i
\(442\) 0 0
\(443\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(444\) −3.00000 + 1.73205i −0.142374 + 0.0821995i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 4.00000 + 20.7846i 0.188982 + 0.981981i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 6.00000 + 3.46410i 0.281905 + 0.162758i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 20.5000 35.5070i 0.958950 1.66095i 0.233890 0.972263i \(-0.424854\pi\)
0.725059 0.688686i \(-0.241812\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 23.0000 1.06890 0.534450 0.845200i \(-0.320519\pi\)
0.534450 + 0.845200i \(0.320519\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(468\) 10.3923i 0.480384i
\(469\) 22.0000 19.0526i 1.01587 0.879765i
\(470\) 0 0
\(471\) −18.0000 + 31.1769i −0.829396 + 1.43656i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 25.9808i 1.19208i
\(476\) 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
\(481\) 1.50000 + 0.866025i 0.0683941 + 0.0394874i
\(482\) 0 0
\(483\) 0 0
\(484\) 22.0000 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −12.5000 21.6506i −0.566429 0.981084i −0.996915 0.0784867i \(-0.974991\pi\)
0.430486 0.902597i \(-0.358342\pi\)
\(488\) 0 0
\(489\) 13.8564i 0.626608i
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 34.6410i 1.55543i
\(497\) 0 0
\(498\) 0 0
\(499\) −21.5000 + 37.2391i −0.962472 + 1.66705i −0.246214 + 0.969216i \(0.579187\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 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
\(507\) −15.0000 + 8.66025i −0.666173 + 0.384615i
\(508\) 19.0000 + 32.9090i 0.842989 + 1.46010i
\(509\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(510\) 0 0
\(511\) −27.0000 31.1769i −1.19441 1.37919i
\(512\) 0 0
\(513\) 13.5000 23.3827i 0.596040 1.03237i
\(514\) 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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(522\) 0 0
\(523\) −25.5000 14.7224i −1.11504 0.643767i −0.174908 0.984585i \(-0.555963\pi\)
−0.940129 + 0.340818i \(0.889296\pi\)
\(524\) 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
\(532\) −27.0000 + 5.19615i −1.17060 + 0.225282i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −14.5000 + 25.1147i −0.623404 + 1.07977i 0.365444 + 0.930834i \(0.380917\pi\)
−0.988847 + 0.148933i \(0.952416\pi\)
\(542\) 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 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(558\) 0 0
\(559\) 8.66025i 0.366290i
\(560\) 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
\(567\) 18.0000 15.5885i 0.755929 0.654654i
\(568\) 0 0
\(569\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(570\) 0 0
\(571\) 15.5000 + 26.8468i 0.648655 + 1.12350i 0.983444 + 0.181210i \(0.0580014\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 12.0000 20.7846i 0.500000 0.866025i
\(577\) 28.5000 16.4545i 1.18647 0.685009i 0.228968 0.973434i \(-0.426465\pi\)
0.957503 + 0.288425i \(0.0931316\pi\)
\(578\) 0 0
\(579\) 37.5000 + 21.6506i 1.55845 + 0.899770i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) −24.0000 3.46410i −0.989743 0.142857i
\(589\) −45.0000 −1.85419
\(590\) 0 0
\(591\) 0 0
\(592\) −2.00000 3.46410i −0.0821995 0.142374i
\(593\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(594\) 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 \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(600\) 0 0
\(601\) 1.73205i 0.0706518i −0.999376 0.0353259i \(-0.988753\pi\)
0.999376 0.0353259i \(-0.0112469\pi\)
\(602\) 0 0
\(603\) −33.0000 −1.34386
\(604\) −4.00000 + 6.92820i −0.162758 + 0.281905i
\(605\) 0 0
\(606\) 0 0
\(607\) −4.50000 2.59808i −0.182649 0.105453i 0.405887 0.913923i \(-0.366962\pi\)
−0.588537 + 0.808470i \(0.700296\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 5.00000 + 8.66025i 0.201948 + 0.349784i 0.949156 0.314806i \(-0.101939\pi\)
−0.747208 + 0.664590i \(0.768606\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 7.50000 4.33013i 0.301450 0.174042i −0.341644 0.939829i \(-0.610984\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −12.0000 −0.480384
\(625\) −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
\(631\) 44.0000 1.75161 0.875806 0.482663i \(-0.160330\pi\)
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
\(637\) 4.50000 + 11.2583i 0.178296 + 0.446071i
\(638\) 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
\(643\) 50.2295i 1.98086i −0.138027 0.990429i \(-0.544076\pi\)
0.138027 0.990429i \(-0.455924\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −37.5000 12.9904i −1.46974 0.509133i
\(652\) 16.0000 0.626608
\(653\) 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
\(657\) 46.7654i 1.82449i
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 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
\(669\) −9.00000 15.5885i −0.347960 0.602685i
\(670\) 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
\(676\) −10.0000 17.3205i −0.384615 0.666173i
\(677\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(678\) 0 0
\(679\) 36.0000 6.92820i 1.38155 0.265880i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(684\) 27.0000 + 15.5885i 1.03237 + 0.596040i
\(685\) 0 0
\(686\) 0 0
\(687\) 51.0000 1.94577
\(688\) 10.0000 17.3205i 0.381246 0.660338i
\(689\) 0 0
\(690\) 0 0
\(691\) 16.5000 + 9.52628i 0.627690 + 0.362397i 0.779857 0.625958i \(-0.215292\pi\)
−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
\(700\) 25.0000 + 8.66025i 0.944911 + 0.327327i
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 4.50000 2.59808i 0.169721 0.0979883i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −11.0000 + 19.0526i −0.413114 + 0.715534i −0.995228 0.0975728i \(-0.968892\pi\)
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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(720\) 0 0
\(721\) −16.5000 + 47.6314i −0.614492 + 1.77389i
\(722\) 0 0
\(723\) 24.0000 41.5692i 0.892570 1.54598i
\(724\) −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
\(733\) −46.5000 26.8468i −1.71752 0.991609i −0.923396 0.383849i \(-0.874598\pi\)
−0.794121 0.607760i \(-0.792068\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −26.5000 45.8993i −0.974818 1.68843i −0.680534 0.732717i \(-0.738252\pi\)
−0.294285 0.955718i \(-0.595081\pi\)
\(740\) 0 0
\(741\) 15.5885i 0.572656i
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 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 −0.200698 0.979653i \(-0.564321\pi\)
0.948753 0.316017i \(-0.102346\pi\)
\(752\) 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 \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) 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
\(763\) −34.0000 + 29.4449i −1.23088 + 1.06598i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 24.0000 + 13.8564i 0.866025 + 0.500000i
\(769\) 25.9808i 0.936890i −0.883493 0.468445i \(-0.844814\pi\)
0.883493 0.468445i \(-0.155186\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −25.0000 + 43.3013i −0.899770 + 1.55845i
\(773\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(774\) 0 0
\(775\) 37.5000 + 21.6506i 1.34704 + 0.777714i
\(776\) 0 0
\(777\) 4.50000 0.866025i 0.161437 0.0310685i
\(778\) 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
\(787\) −3.00000 + 1.73205i −0.106938 + 0.0617409i −0.552515 0.833503i \(-0.686332\pi\)
0.445577 + 0.895244i \(0.352999\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 6.00000 10.3923i 0.213066 0.369042i
\(794\) 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
\(804\) 38.1051i