Properties

Label 507.2.k.a.488.1
Level $507$
Weight $2$
Character 507.488
Analytic conductor $4.048$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.k (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

Embedding invariants

Embedding label 488.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 507.488
Dual form 507.2.k.a.80.1

$q$-expansion

\(f(q)\) \(=\) \(q+(0.866025 - 1.50000i) q^{3} +(1.73205 - 1.00000i) q^{4} +(-2.86603 - 0.767949i) q^{7} +(-1.50000 - 2.59808i) q^{9} +O(q^{10})\) \(q+(0.866025 - 1.50000i) q^{3} +(1.73205 - 1.00000i) q^{4} +(-2.86603 - 0.767949i) q^{7} +(-1.50000 - 2.59808i) q^{9} -3.46410i q^{12} +(2.00000 - 3.46410i) q^{16} +(2.09808 - 7.83013i) q^{19} +(-3.63397 + 3.63397i) q^{21} +5.00000i q^{25} -5.19615 q^{27} +(-5.73205 + 1.53590i) q^{28} +(7.83013 + 7.83013i) q^{31} +(-5.19615 - 3.00000i) q^{36} +(0.562178 + 2.09808i) q^{37} +(1.50000 - 0.866025i) q^{43} +(-3.46410 - 6.00000i) q^{48} +(1.56218 + 0.901924i) q^{49} +(-9.92820 - 9.92820i) q^{57} +(4.33013 + 7.50000i) q^{61} +(2.30385 + 8.59808i) q^{63} -8.00000i q^{64} +(0.767949 - 0.205771i) q^{67} +(9.36603 - 9.36603i) q^{73} +(7.50000 + 4.33013i) q^{75} +(-4.19615 - 15.6603i) q^{76} +12.1244 q^{79} +(-4.50000 + 7.79423i) q^{81} +(-2.66025 + 9.92820i) q^{84} +(18.5263 - 4.96410i) q^{93} +(-4.40192 + 16.4282i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{7} - 6q^{9} + O(q^{10}) \) \( 4q - 8q^{7} - 6q^{9} + 8q^{16} - 2q^{19} - 18q^{21} - 16q^{28} + 14q^{31} - 22q^{37} + 6q^{43} - 18q^{49} - 12q^{57} + 30q^{63} + 10q^{67} + 34q^{73} + 30q^{75} + 4q^{76} - 18q^{81} + 24q^{84} + 36q^{93} - 28q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(-1\) \(e\left(\frac{11}{12}\right)\)

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 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(3\) 0.866025 1.50000i 0.500000 0.866025i
\(4\) 1.73205 1.00000i 0.866025 0.500000i
\(5\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(6\) 0 0
\(7\) −2.86603 0.767949i −1.08326 0.290258i −0.327327 0.944911i \(-0.606148\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) −1.50000 2.59808i −0.500000 0.866025i
\(10\) 0 0
\(11\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(12\) 3.46410i 1.00000i
\(13\) 0 0
\(14\) 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
\(19\) 2.09808 7.83013i 0.481332 1.79635i −0.114708 0.993399i \(-0.536593\pi\)
0.596040 0.802955i \(-0.296740\pi\)
\(20\) 0 0
\(21\) −3.63397 + 3.63397i −0.792998 + 0.792998i
\(22\) 0 0
\(23\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) 5.00000i 1.00000i
\(26\) 0 0
\(27\) −5.19615 −1.00000
\(28\) −5.73205 + 1.53590i −1.08326 + 0.290258i
\(29\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) 7.83013 + 7.83013i 1.40633 + 1.40633i 0.777714 + 0.628619i \(0.216379\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −5.19615 3.00000i −0.866025 0.500000i
\(37\) 0.562178 + 2.09808i 0.0924215 + 0.344922i 0.996616 0.0821995i \(-0.0261945\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(42\) 0 0
\(43\) 1.50000 0.866025i 0.228748 0.132068i −0.381246 0.924473i \(-0.624505\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) −3.46410 6.00000i −0.500000 0.866025i
\(49\) 1.56218 + 0.901924i 0.223168 + 0.128846i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −9.92820 9.92820i −1.31502 1.31502i
\(58\) 0 0
\(59\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(60\) 0 0
\(61\) 4.33013 + 7.50000i 0.554416 + 0.960277i 0.997949 + 0.0640184i \(0.0203916\pi\)
−0.443533 + 0.896258i \(0.646275\pi\)
\(62\) 0 0
\(63\) 2.30385 + 8.59808i 0.290258 + 1.08326i
\(64\) 8.00000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 0.767949 0.205771i 0.0938199 0.0251390i −0.211604 0.977356i \(-0.567869\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(72\) 0 0
\(73\) 9.36603 9.36603i 1.09621 1.09621i 0.101361 0.994850i \(-0.467680\pi\)
0.994850 0.101361i \(-0.0323196\pi\)
\(74\) 0 0
\(75\) 7.50000 + 4.33013i 0.866025 + 0.500000i
\(76\) −4.19615 15.6603i −0.481332 1.79635i
\(77\) 0 0
\(78\) 0 0
\(79\) 12.1244 1.36410 0.682048 0.731307i \(-0.261089\pi\)
0.682048 + 0.731307i \(0.261089\pi\)
\(80\) 0 0
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) −2.66025 + 9.92820i −0.290258 + 1.08326i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 18.5263 4.96410i 1.92109 0.514753i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.40192 + 16.4282i −0.446948 + 1.66803i 0.263795 + 0.964579i \(0.415026\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 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 \(0.278742\pi\)
−0.640464 + 0.767988i \(0.721258\pi\)
\(104\) 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
\(109\) −5.16987 5.16987i −0.495184 0.495184i 0.414751 0.909935i \(-0.363869\pi\)
−0.909935 + 0.414751i \(0.863869\pi\)
\(110\) 0 0
\(111\) 3.63397 + 0.973721i 0.344922 + 0.0924215i
\(112\) −8.39230 + 8.39230i −0.792998 + 0.792998i
\(113\) 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
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −9.52628 + 5.50000i −0.866025 + 0.500000i
\(122\) 0 0
\(123\) 0 0
\(124\) 21.3923 + 5.73205i 1.92109 + 0.514753i
\(125\) 0 0
\(126\) 0 0
\(127\) 0.866025 + 0.500000i 0.0768473 + 0.0443678i 0.537931 0.842989i \(-0.319206\pi\)
−0.461084 + 0.887357i \(0.652539\pi\)
\(128\) 0 0
\(129\) 3.00000i 0.264135i
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) −12.0263 + 20.8301i −1.04281 + 1.80620i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(138\) 0 0
\(139\) 3.50000 + 6.06218i 0.296866 + 0.514187i 0.975417 0.220366i \(-0.0707252\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −12.0000 −1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) 2.70577 1.56218i 0.223168 0.128846i
\(148\) 3.07180 + 3.07180i 0.252500 + 0.252500i
\(149\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(150\) 0 0
\(151\) 14.1244 14.1244i 1.14942 1.14942i 0.162758 0.986666i \(-0.447961\pi\)
0.986666 0.162758i \(-0.0520389\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −11.0000 −0.877896 −0.438948 0.898513i \(-0.644649\pi\)
−0.438948 + 0.898513i \(0.644649\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −20.6244 5.52628i −1.61542 0.432852i −0.665771 0.746156i \(-0.731897\pi\)
−0.949653 + 0.313304i \(0.898564\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) −23.4904 + 6.29423i −1.79635 + 0.481332i
\(172\) 1.73205 3.00000i 0.132068 0.228748i
\(173\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(174\) 0 0
\(175\) 3.83975 14.3301i 0.290258 1.08326i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(180\) 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
\(189\) 14.8923 + 3.99038i 1.08326 + 0.290258i
\(190\) 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
\(193\) 7.06218 + 26.3564i 0.508347 + 1.89718i 0.436365 + 0.899770i \(0.356266\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 3.60770 0.257693
\(197\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(198\) 0 0
\(199\) 14.7224 8.50000i 1.04365 0.602549i 0.122782 0.992434i \(-0.460818\pi\)
0.920864 + 0.389885i \(0.127485\pi\)
\(200\) 0 0
\(201\) 0.356406 1.33013i 0.0251390 0.0938199i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 12.9904 22.5000i 0.894295 1.54896i 0.0596196 0.998221i \(-0.481011\pi\)
0.834675 0.550743i \(-0.185655\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −16.4282 28.4545i −1.11522 1.93162i
\(218\) 0 0
\(219\) −5.93782 22.1603i −0.401241 1.49745i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 12.0263 3.22243i 0.805339 0.215790i 0.167412 0.985887i \(-0.446459\pi\)
0.637927 + 0.770097i \(0.279792\pi\)
\(224\) 0 0
\(225\) 12.9904 7.50000i 0.866025 0.500000i
\(226\) 0 0
\(227\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(228\) −27.1244 7.26795i −1.79635 0.481332i
\(229\) −21.3923 + 21.3923i −1.41364 + 1.41364i −0.686743 + 0.726900i \(0.740960\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 10.5000 18.1865i 0.682048 1.18134i
\(238\) 0 0
\(239\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(240\) 0 0
\(241\) −9.36603 2.50962i −0.603319 0.161659i −0.0557856 0.998443i \(-0.517766\pi\)
−0.547533 + 0.836784i \(0.684433\pi\)
\(242\) 0 0
\(243\) 7.79423 + 13.5000i 0.500000 + 0.866025i
\(244\) 15.0000 + 8.66025i 0.960277 + 0.554416i
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(252\) 12.5885 + 12.5885i 0.792998 + 0.792998i
\(253\) 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
\(259\) 6.44486i 0.400464i
\(260\) 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
\(268\) 1.12436 1.12436i 0.0686810 0.0686810i
\(269\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(270\) 0 0
\(271\) −2.45448 9.16025i −0.149099 0.556446i −0.999539 0.0303728i \(-0.990331\pi\)
0.850439 0.526073i \(-0.176336\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −18.0000 + 10.3923i −1.08152 + 0.624413i −0.931305 0.364241i \(-0.881328\pi\)
−0.150210 + 0.988654i \(0.547995\pi\)
\(278\) 0 0
\(279\) 8.59808 32.0885i 0.514753 1.92109i
\(280\) 0 0
\(281\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(282\) 0 0
\(283\) −21.6506 12.5000i −1.28700 0.743048i −0.308879 0.951101i \(-0.599954\pi\)
−0.978117 + 0.208053i \(0.933287\pi\)
\(284\) 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
\(291\) 20.8301 + 20.8301i 1.22108 + 1.22108i
\(292\) 6.85641 25.5885i 0.401241 1.49745i
\(293\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 17.3205 1.00000
\(301\) −4.96410 + 1.33013i −0.286126 + 0.0766672i
\(302\) 0 0
\(303\) 0 0
\(304\) −22.9282 22.9282i −1.31502 1.31502i
\(305\) 0 0
\(306\) 0 0
\(307\) −16.6340 + 16.6340i −0.949351 + 0.949351i −0.998778 0.0494267i \(-0.984261\pi\)
0.0494267 + 0.998778i \(0.484261\pi\)
\(308\) 0 0
\(309\) 23.3827 + 13.5000i 1.33019 + 0.767988i
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −32.9090 −1.86012 −0.930062 0.367402i \(-0.880247\pi\)
−0.930062 + 0.367402i \(0.880247\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 21.0000 12.1244i 1.18134 0.682048i
\(317\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 18.0000i 1.00000i
\(325\) 0 0
\(326\) 0 0
\(327\) −12.2321 + 3.27757i −0.676434 + 0.181250i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −9.16025 + 34.1865i −0.503493 + 1.87906i −0.0274825 + 0.999622i \(0.508749\pi\)
−0.476011 + 0.879440i \(0.657918\pi\)
\(332\) 0 0
\(333\) 4.60770 4.60770i 0.252500 0.252500i
\(334\) 0 0
\(335\) 0 0
\(336\) 5.32051 + 19.8564i 0.290258 + 1.08326i
\(337\) 29.0000i 1.57973i −0.613280 0.789865i \(-0.710150\pi\)
0.613280 0.789865i \(-0.289850\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 10.9019 + 10.9019i 0.588649 + 0.588649i
\(344\) 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
\(349\) −1.17949 4.40192i −0.0631368 0.235630i 0.927146 0.374701i \(-0.122255\pi\)
−0.990282 + 0.139072i \(0.955588\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(360\) 0 0
\(361\) −40.4545 23.3564i −2.12918 1.22928i
\(362\) 0 0
\(363\) 19.0526i 1.00000i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −15.5000 + 26.8468i −0.809093 + 1.40139i 0.104399 + 0.994535i \(0.466708\pi\)
−0.913493 + 0.406855i \(0.866625\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 27.1244 27.1244i 1.40633 1.40633i
\(373\) −18.1865 31.5000i −0.941663 1.63101i −0.762299 0.647225i \(-0.775929\pi\)
−0.179364 0.983783i \(-0.557404\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −16.9904 + 4.55256i −0.872737 + 0.233849i −0.667271 0.744815i \(-0.732538\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) 1.50000 0.866025i 0.0768473 0.0443678i
\(382\) 0 0
\(383\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.50000 2.59808i −0.228748 0.132068i
\(388\) 8.80385 + 32.8564i 0.446948 + 1.66803i
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 27.8923 + 7.47372i 1.39987 + 0.375095i 0.878300 0.478110i \(-0.158678\pi\)
0.521575 + 0.853206i \(0.325345\pi\)
\(398\) 0 0
\(399\) 20.8301 + 36.0788i 1.04281 + 1.80620i
\(400\) 17.3205 + 10.0000i 0.866025 + 0.500000i
\(401\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −0.918584 + 3.42820i −0.0454211 + 0.169514i −0.984911 0.173064i \(-0.944633\pi\)
0.939490 + 0.342578i \(0.111300\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 15.5885 + 27.0000i 0.767988 + 1.33019i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 12.1244 0.593732
\(418\) 0 0
\(419\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(420\) 0 0
\(421\) −27.6865 27.6865i −1.34936 1.34936i −0.886357 0.463002i \(-0.846772\pi\)
−0.463002 0.886357i \(-0.653228\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −6.65064 24.8205i −0.321847 1.20115i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(432\) −10.3923 + 18.0000i −0.500000 + 0.866025i
\(433\) −30.3109 + 17.5000i −1.45665 + 0.840996i −0.998845 0.0480569i \(-0.984697\pi\)
−0.457804 + 0.889053i \(0.651364\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −14.1244 3.78461i −0.676434 0.181250i
\(437\) 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
\(441\) 5.41154i 0.257693i
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 7.26795 1.94744i 0.344922 0.0924215i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −6.14359 + 22.9282i −0.290258 + 1.08326i
\(449\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −8.95448 33.4186i −0.420718 1.57014i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 36.2846 9.72243i 1.69732 0.454796i 0.725059 0.688686i \(-0.241812\pi\)
0.972263 + 0.233890i \(0.0751456\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(462\) 0 0
\(463\) 22.3660 22.3660i 1.03944 1.03944i 0.0402476 0.999190i \(-0.487185\pi\)
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 \(0.5\pi\)
\(468\) 0 0
\(469\) −2.35898 −0.108928
\(470\) 0 0
\(471\) −9.52628 + 16.5000i −0.438948 + 0.760280i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 39.1506 + 10.4904i 1.79635 + 0.481332i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −11.0000 + 19.0526i −0.500000 + 0.866025i
\(485\) 0 0
\(486\) 0 0
\(487\) −7.41858 + 27.6865i −0.336168 + 1.25460i 0.566429 + 0.824110i \(0.308325\pi\)
−0.902597 + 0.430486i \(0.858342\pi\)
\(488\) 0 0
\(489\) −26.1506 + 26.1506i −1.18257 + 1.18257i
\(490\) 0 0
\(491\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 42.7846 11.4641i 1.92109 0.514753i
\(497\) 0 0
\(498\) 0 0
\(499\) −0.411543 0.411543i −0.0184232 0.0184232i 0.697835 0.716258i \(-0.254147\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 2.00000 0.0887357
\(509\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(510\) 0 0
\(511\) −34.0359 + 19.6506i −1.50566 + 0.869293i
\(512\) 0 0
\(513\) −10.9019 + 40.6865i −0.481332 + 1.79635i
\(514\) 0 0
\(515\) 0 0
\(516\) −3.00000 5.19615i −0.132068 0.228748i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 4.00000 6.92820i 0.174908 0.302949i −0.765222 0.643767i \(-0.777371\pi\)
0.940129 + 0.340818i \(0.110704\pi\)
\(524\) 0 0
\(525\) −18.1699 18.1699i −0.792998 0.792998i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 11.5000 + 19.9186i 0.500000 + 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 48.1051i 2.08562i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −13.1506 + 13.1506i −0.565390 + 0.565390i −0.930834 0.365444i \(-0.880917\pi\)
0.365444 + 0.930834i \(0.380917\pi\)
\(542\) 0 0
\(543\) −10.3923 6.00000i −0.445976 0.257485i
\(544\) 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
\(549\) 12.9904 22.5000i 0.554416 0.960277i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −34.7487 9.31089i −1.47767 0.395939i
\(554\) 0 0
\(555\) 0 0
\(556\) 12.1244 + 7.00000i 0.514187 + 0.296866i
\(557\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 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
\(567\) 18.8827 18.8827i 0.792998 0.792998i
\(568\) 0 0
\(569\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(570\) 0 0
\(571\) 16.0000i 0.669579i −0.942293 0.334790i \(-0.891335\pi\)
0.942293 0.334790i \(-0.108665\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −20.7846 + 12.0000i −0.866025 + 0.500000i
\(577\) 16.0718 + 16.0718i 0.669078 + 0.669078i 0.957503 0.288425i \(-0.0931316\pi\)
−0.288425 + 0.957503i \(0.593132\pi\)
\(578\) 0 0
\(579\) 45.6506 + 12.2321i 1.89718 + 0.508347i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(588\) 3.12436 5.41154i 0.128846 0.223168i
\(589\) 77.7391 44.8827i 3.20318 1.84936i
\(590\) 0 0
\(591\) 0 0
\(592\) 8.39230 + 2.24871i 0.344922 + 0.0924215i
\(593\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 29.4449i 1.20510i
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −20.7846 + 36.0000i −0.847822 + 1.46847i 0.0353259 + 0.999376i \(0.488753\pi\)
−0.883148 + 0.469095i \(0.844580\pi\)
\(602\) 0 0
\(603\) −1.68653 1.68653i −0.0686810 0.0686810i
\(604\) 10.3397 38.5885i 0.420718 1.57014i
\(605\) 0 0
\(606\) 0 0
\(607\) 10.0000 + 17.3205i 0.405887 + 0.703018i 0.994424 0.105453i \(-0.0336291\pi\)
−0.588537 + 0.808470i \(0.700296\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −47.7487 + 12.7942i −1.92855 + 0.516754i −0.949156 + 0.314806i \(0.898061\pi\)
−0.979396 + 0.201948i \(0.935273\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(618\) 0 0
\(619\) 31.8827 31.8827i 1.28147 1.28147i 0.341644 0.939829i \(-0.389016\pi\)
0.939829 0.341644i \(-0.110984\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −25.0000 −1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −19.0526 + 11.0000i −0.760280 + 0.438948i
\(629\) 0 0
\(630\) 0 0
\(631\) −33.6244 9.00962i −1.33856 0.358667i −0.482663 0.875806i \(-0.660330\pi\)
−0.855901 + 0.517139i \(0.826997\pi\)
\(632\) 0 0
\(633\) −22.5000 38.9711i −0.894295 1.54896i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(642\) 0 0
\(643\) 5.11474 19.0885i 0.201706 0.752775i −0.788723 0.614749i \(-0.789257\pi\)
0.990429 0.138027i \(-0.0440759\pi\)
\(644\) 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
\(651\) −56.9090 −2.23044
\(652\) −41.2487 + 11.0526i −1.61542 + 0.432852i
\(653\) 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
\(657\) −38.3827 10.2846i −1.49745 0.401241i
\(658\) 0 0
\(659\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(660\) 0 0
\(661\) 11.8205 + 44.1147i 0.459764 + 1.71586i 0.673690 + 0.739014i \(0.264708\pi\)
−0.213925 + 0.976850i \(0.568625\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 5.58142 20.8301i 0.215790 0.805339i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −43.5000 25.1147i −1.67680 0.968102i −0.963679 0.267063i \(-0.913947\pi\)
−0.713123 0.701039i \(-0.752720\pi\)
\(674\) 0 0
\(675\) 25.9808i 1.00000i
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 25.2321 43.7032i 0.968317 1.67717i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(684\) −34.3923 + 34.3923i −1.31502 + 1.31502i
\(685\) 0 0
\(686\) 0 0
\(687\) 13.5622 + 50.6147i 0.517429 + 1.93107i
\(688\) 6.92820i 0.264135i
\(689\) 0 0
\(690\) 0 0
\(691\) 5.52628 1.48076i 0.210230 0.0563308i −0.152167 0.988355i \(-0.548625\pi\)
0.362397 + 0.932024i \(0.381959\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −7.67949 28.6603i −0.290258 1.08326i
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 17.6077 0.664087
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 32.6506 + 8.74871i 1.22622 + 0.328565i 0.813107 0.582115i \(-0.197775\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 0 0
\(711\) −18.1865 31.5000i −0.682048 1.18134i
\(712\) 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
\(721\) 11.9711 44.6769i 0.445829 1.66386i
\(722\) 0 0
\(723\) −11.8756 + 11.8756i −0.441660 + 0.441660i
\(724\) −6.92820 12.0000i −0.257485 0.445976i
\(725\) 0 0
\(726\) 0 0
\(727\) 49.0000i 1.81731i 0.417548 + 0.908655i \(0.362889\pi\)
−0.417548 + 0.908655i \(0.637111\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 25.9808 15.0000i 0.960277 0.554416i
\(733\) 30.3468 + 30.3468i 1.12088 + 1.12088i 0.991609 + 0.129275i \(0.0412651\pi\)
0.129275 + 0.991609i \(0.458735\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −12.4378 46.4186i −0.457533 1.70754i −0.680534 0.732717i \(-0.738252\pi\)
0.223001 0.974818i \(-0.428415\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 15.0000 + 8.66025i 0.547358 + 0.316017i 0.748056 0.663636i \(-0.230988\pi\)
−0.200698 + 0.979653i \(0.564321\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 29.7846 7.98076i 1.08326 0.290258i
\(757\) 24.2487 42.0000i 0.881334 1.52652i 0.0314762 0.999505i \(-0.489979\pi\)
0.849858 0.527011i \(-0.176688\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(762\) 0 0
\(763\) 10.8468 + 18.7872i 0.392680 + 0.680142i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −27.7128 −1.00000
\(769\) −36.4904 + 9.77757i −1.31588 + 0.352588i −0.847432 0.530904i \(-0.821852\pi\)
−0.468445 + 0.883493i \(0.655186\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 38.5885 + 38.5885i 1.38883 + 1.38883i
\(773\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(774\) 0 0
\(775\) −39.1506 + 39.1506i −1.40633 + 1.40633i
\(776\) 0 0
\(777\) −9.66730 5.58142i −0.346812 0.200232i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 6.24871 3.60770i 0.223168 0.128846i
\(785\) 0 0
\(786\) 0 0
\(787\) −51.3827 13.7679i −1.83159 0.490774i −0.833503 0.552515i \(-0.813668\pi\)
−0.998092 + 0.0617409i \(0.980335\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 17.0000 29.4449i 0.602549 1.04365i
\(797\) 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
\(804\) −0.712813 2.66025i −0.0251390 0.0938199i
\(805\) 0 0