Properties

Label 441.4.e.a.361.1
Level $441$
Weight $4$
Character 441.361
Analytic conductor $26.020$
Analytic rank $0$
Dimension $2$
CM discriminant -7
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.0198423125\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 49)
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label 361.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 441.361
Dual form 441.4.e.a.226.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-2.50000 - 4.33013i) q^{2} +(-8.50000 + 14.7224i) q^{4} +45.0000 q^{8} +O(q^{10})\) \(q+(-2.50000 - 4.33013i) q^{2} +(-8.50000 + 14.7224i) q^{4} +45.0000 q^{8} +(-34.0000 + 58.8897i) q^{11} +(-44.5000 - 77.0763i) q^{16} +340.000 q^{22} +(-20.0000 - 34.6410i) q^{23} +(62.5000 - 108.253i) q^{25} +166.000 q^{29} +(-42.5000 + 73.6122i) q^{32} +(-225.000 - 389.711i) q^{37} -180.000 q^{43} +(-578.000 - 1001.13i) q^{44} +(-100.000 + 173.205i) q^{46} -625.000 q^{50} +(295.000 - 510.955i) q^{53} +(-415.000 - 718.801i) q^{58} -287.000 q^{64} +(370.000 - 640.859i) q^{67} -688.000 q^{71} +(-1125.00 + 1948.56i) q^{74} +(692.000 + 1198.58i) q^{79} +(450.000 + 779.423i) q^{86} +(-1530.00 + 2650.04i) q^{88} +680.000 q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 5q^{2} - 17q^{4} + 90q^{8} + O(q^{10}) \) \( 2q - 5q^{2} - 17q^{4} + 90q^{8} - 68q^{11} - 89q^{16} + 680q^{22} - 40q^{23} + 125q^{25} + 332q^{29} - 85q^{32} - 450q^{37} - 360q^{43} - 1156q^{44} - 200q^{46} - 1250q^{50} + 590q^{53} - 830q^{58} - 574q^{64} + 740q^{67} - 1376q^{71} - 2250q^{74} + 1384q^{79} + 900q^{86} - 3060q^{88} + 1360q^{92} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.50000 4.33013i −0.883883 1.53093i −0.846988 0.531612i \(-0.821586\pi\)
−0.0368954 0.999319i \(-0.511747\pi\)
\(3\) 0 0
\(4\) −8.50000 + 14.7224i −1.06250 + 1.84030i
\(5\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 45.0000 1.98874
\(9\) 0 0
\(10\) 0 0
\(11\) −34.0000 + 58.8897i −0.931944 + 1.61417i −0.151950 + 0.988388i \(0.548555\pi\)
−0.779994 + 0.625786i \(0.784778\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −44.5000 77.0763i −0.695312 1.20432i
\(17\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 340.000 3.29492
\(23\) −20.0000 34.6410i −0.181317 0.314050i 0.761012 0.648737i \(-0.224703\pi\)
−0.942329 + 0.334687i \(0.891369\pi\)
\(24\) 0 0
\(25\) 62.5000 108.253i 0.500000 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 166.000 1.06295 0.531473 0.847075i \(-0.321639\pi\)
0.531473 + 0.847075i \(0.321639\pi\)
\(30\) 0 0
\(31\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(32\) −42.5000 + 73.6122i −0.234782 + 0.406654i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −225.000 389.711i −0.999724 1.73157i −0.520223 0.854030i \(-0.674151\pi\)
−0.479500 0.877542i \(-0.659182\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −180.000 −0.638366 −0.319183 0.947693i \(-0.603408\pi\)
−0.319183 + 0.947693i \(0.603408\pi\)
\(44\) −578.000 1001.13i −1.98038 3.43012i
\(45\) 0 0
\(46\) −100.000 + 173.205i −0.320526 + 0.555167i
\(47\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −625.000 −1.76777
\(51\) 0 0
\(52\) 0 0
\(53\) 295.000 510.955i 0.764554 1.32425i −0.175928 0.984403i \(-0.556293\pi\)
0.940482 0.339843i \(-0.110374\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −415.000 718.801i −0.939520 1.62730i
\(59\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −287.000 −0.560547
\(65\) 0 0
\(66\) 0 0
\(67\) 370.000 640.859i 0.674667 1.16856i −0.301899 0.953340i \(-0.597621\pi\)
0.976566 0.215218i \(-0.0690461\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −688.000 −1.15001 −0.575004 0.818151i \(-0.695000\pi\)
−0.575004 + 0.818151i \(0.695000\pi\)
\(72\) 0 0
\(73\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(74\) −1125.00 + 1948.56i −1.76728 + 3.06102i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 692.000 + 1198.58i 0.985520 + 1.70697i 0.639602 + 0.768706i \(0.279099\pi\)
0.345918 + 0.938265i \(0.387568\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 450.000 + 779.423i 0.564241 + 0.977295i
\(87\) 0 0
\(88\) −1530.00 + 2650.04i −1.85339 + 3.21017i
\(89\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 680.000 0.770597
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1062.50 + 1840.30i 1.06250 + 1.84030i
\(101\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(102\) 0 0
\(103\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2950.00 −2.70311
\(107\) −790.000 1368.32i −0.713759 1.23627i −0.963436 0.267937i \(-0.913658\pi\)
0.249678 0.968329i \(-0.419675\pi\)
\(108\) 0 0
\(109\) 27.0000 46.7654i 0.0237260 0.0410946i −0.853919 0.520407i \(-0.825780\pi\)
0.877645 + 0.479312i \(0.159114\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 670.000 0.557773 0.278886 0.960324i \(-0.410035\pi\)
0.278886 + 0.960324i \(0.410035\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1411.00 + 2443.92i −1.12938 + 1.95614i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1646.50 2851.82i −1.23704 2.14262i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −2000.00 −1.39741 −0.698706 0.715409i \(-0.746240\pi\)
−0.698706 + 0.715409i \(0.746240\pi\)
\(128\) 1057.50 + 1831.64i 0.730240 + 1.26481i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −3700.00 −2.38531
\(135\) 0 0
\(136\) 0 0
\(137\) 1555.00 2693.34i 0.969727 1.67962i 0.273388 0.961904i \(-0.411856\pi\)
0.696339 0.717713i \(-0.254811\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1720.00 + 2979.13i 1.01647 + 1.76058i
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 7650.00 4.24883
\(149\) 407.000 + 704.945i 0.223777 + 0.387593i 0.955952 0.293524i \(-0.0948280\pi\)
−0.732175 + 0.681117i \(0.761495\pi\)
\(150\) 0 0
\(151\) 1476.00 2556.51i 0.795465 1.37779i −0.127079 0.991893i \(-0.540560\pi\)
0.922544 0.385893i \(-0.126107\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(158\) 3460.00 5992.90i 1.74217 3.01753i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −890.000 1541.53i −0.427670 0.740746i 0.568996 0.822340i \(-0.307332\pi\)
−0.996666 + 0.0815946i \(0.973999\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −2197.00 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 1530.00 2650.04i 0.678264 1.17479i
\(173\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 6052.00 2.59197
\(177\) 0 0
\(178\) 0 0
\(179\) −1042.00 + 1804.80i −0.435099 + 0.753614i −0.997304 0.0733844i \(-0.976620\pi\)
0.562205 + 0.826998i \(0.309953\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −900.000 1558.85i −0.360592 0.624563i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2036.00 3526.46i −0.771308 1.33594i −0.936846 0.349741i \(-0.886270\pi\)
0.165539 0.986203i \(-0.447064\pi\)
\(192\) 0 0
\(193\) 2295.00 3975.06i 0.855947 1.48254i −0.0198172 0.999804i \(-0.506308\pi\)
0.875764 0.482740i \(-0.160358\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2210.00 0.799269 0.399634 0.916675i \(-0.369137\pi\)
0.399634 + 0.916675i \(0.369137\pi\)
\(198\) 0 0
\(199\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(200\) 2812.50 4871.39i 0.994369 1.72230i
\(201\) 0 0
\(202\) 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
\(211\) 5868.00 1.91455 0.957274 0.289181i \(-0.0933830\pi\)
0.957274 + 0.289181i \(0.0933830\pi\)
\(212\) 5015.00 + 8686.23i 1.62468 + 2.81402i
\(213\) 0 0
\(214\) −3950.00 + 6841.60i −1.26176 + 2.18543i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −270.000 −0.0838840
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −1675.00 2901.19i −0.493006 0.853911i
\(227\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(228\) 0 0
\(229\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 7470.00 2.11392
\(233\) −2365.00 4096.30i −0.664963 1.15175i −0.979295 0.202436i \(-0.935114\pi\)
0.314333 0.949313i \(-0.398219\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 7376.00 1.99629 0.998146 0.0608655i \(-0.0193861\pi\)
0.998146 + 0.0608655i \(0.0193861\pi\)
\(240\) 0 0
\(241\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(242\) −8232.50 + 14259.1i −2.18680 + 3.78765i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 2720.00 0.675909
\(254\) 5000.00 + 8660.25i 1.23515 + 2.13934i
\(255\) 0 0
\(256\) 4139.50 7169.82i 1.01062 1.75045i
\(257\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 3760.00 6512.51i 0.881565 1.52691i 0.0319637 0.999489i \(-0.489824\pi\)
0.849601 0.527426i \(-0.176843\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 6290.00 + 10894.6i 1.43367 + 2.48319i
\(269\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(270\) 0 0
\(271\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −15550.0 −3.42850
\(275\) 4250.00 + 7361.22i 0.931944 + 1.61417i
\(276\) 0 0
\(277\) −3655.00 + 6330.65i −0.792807 + 1.37318i 0.131415 + 0.991327i \(0.458048\pi\)
−0.924222 + 0.381855i \(0.875285\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −4342.00 −0.921786 −0.460893 0.887456i \(-0.652471\pi\)
−0.460893 + 0.887456i \(0.652471\pi\)
\(282\) 0 0
\(283\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(284\) 5848.00 10129.0i 1.22188 2.11636i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2456.50 + 4254.78i 0.500000 + 0.866025i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −10125.0 17537.0i −1.98819 3.44364i
\(297\) 0 0
\(298\) 2035.00 3524.72i 0.395585 0.685174i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) −14760.0 −2.81239
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(312\) 0 0
\(313\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −23528.0 −4.18846
\(317\) −3485.00 6036.20i −0.617467 1.06948i −0.989946 0.141444i \(-0.954825\pi\)
0.372479 0.928041i \(-0.378508\pi\)
\(318\) 0 0
\(319\) −5644.00 + 9775.69i −0.990606 + 1.71578i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) −4450.00 + 7707.63i −0.756021 + 1.30947i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −5454.00 9446.61i −0.905677 1.56868i −0.820006 0.572354i \(-0.806030\pi\)
−0.0856702 0.996324i \(-0.527303\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −3330.00 −0.538269 −0.269135 0.963103i \(-0.586738\pi\)
−0.269135 + 0.963103i \(0.586738\pi\)
\(338\) 5492.50 + 9513.29i 0.883883 + 1.53093i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) −8100.00 −1.26954
\(345\) 0 0
\(346\) 0 0
\(347\) −2050.00 + 3550.70i −0.317146 + 0.549314i −0.979891 0.199532i \(-0.936058\pi\)
0.662745 + 0.748845i \(0.269391\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2890.00 5005.63i −0.437607 0.757957i
\(353\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 10420.0 1.53831
\(359\) −4052.00 7018.27i −0.595700 1.03178i −0.993448 0.114288i \(-0.963541\pi\)
0.397747 0.917495i \(-0.369792\pi\)
\(360\) 0 0
\(361\) 3429.50 5940.07i 0.500000 0.866025i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(368\) −1780.00 + 3083.05i −0.252144 + 0.436726i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 6985.00 + 12098.4i 0.969624 + 1.67944i 0.696643 + 0.717418i \(0.254676\pi\)
0.272980 + 0.962020i \(0.411991\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 11916.0 1.61500 0.807498 0.589870i \(-0.200821\pi\)
0.807498 + 0.589870i \(0.200821\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −10180.0 + 17632.3i −1.36349 + 2.36164i
\(383\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −22950.0 −3.02623
\(387\) 0 0
\(388\) 0 0
\(389\) −5263.00 + 9115.78i −0.685976 + 1.18815i 0.287153 + 0.957885i \(0.407291\pi\)
−0.973129 + 0.230261i \(0.926042\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) −5525.00 9569.58i −0.706461 1.22363i
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −11125.0 −1.39062
\(401\) 799.000 + 1383.91i 0.0995016 + 0.172342i 0.911478 0.411348i \(-0.134942\pi\)
−0.811977 + 0.583690i \(0.801608\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 30600.0 3.72675
\(408\) 0 0
\(409\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 15262.0 1.76680 0.883402 0.468616i \(-0.155247\pi\)
0.883402 + 0.468616i \(0.155247\pi\)
\(422\) −14670.0 25409.2i −1.69224 2.93104i
\(423\) 0 0
\(424\) 13275.0 22993.0i 1.52050 2.63358i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 26860.0 3.03347
\(429\) 0 0
\(430\) 0 0
\(431\) −4304.00 + 7454.75i −0.481012 + 0.833138i −0.999763 0.0217878i \(-0.993064\pi\)
0.518750 + 0.854926i \(0.326398\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 459.000 + 795.011i 0.0504177 + 0.0873260i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 9290.00 + 16090.8i 0.996346 + 1.72572i 0.572140 + 0.820156i \(0.306113\pi\)
0.424205 + 0.905566i \(0.360553\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 2686.00 0.282317 0.141158 0.989987i \(-0.454917\pi\)
0.141158 + 0.989987i \(0.454917\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −5695.00 + 9864.03i −0.592633 + 1.02647i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −4005.00 6936.86i −0.409947 0.710050i 0.584936 0.811079i \(-0.301120\pi\)
−0.994883 + 0.101030i \(0.967786\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −8440.00 −0.847171 −0.423585 0.905856i \(-0.639229\pi\)
−0.423585 + 0.905856i \(0.639229\pi\)
\(464\) −7387.00 12794.7i −0.739079 1.28012i
\(465\) 0 0
\(466\) −11825.0 + 20481.5i −1.17550 + 2.03602i
\(467\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 6120.00 10600.2i 0.594922 1.03043i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −18440.0 31939.0i −1.76449 3.05619i
\(479\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 55981.0 5.25742
\(485\) 0 0
\(486\) 0 0
\(487\) −10620.0 + 18394.4i −0.988169 + 1.71156i −0.361261 + 0.932465i \(0.617654\pi\)
−0.626908 + 0.779094i \(0.715680\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −20372.0 −1.87246 −0.936228 0.351394i \(-0.885708\pi\)
−0.936228 + 0.351394i \(0.885708\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 3618.00 + 6266.56i 0.324577 + 0.562184i 0.981427 0.191838i \(-0.0614447\pi\)
−0.656850 + 0.754022i \(0.728111\pi\)
\(500\) 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
\(506\) −6800.00 11777.9i −0.597425 1.03477i
\(507\) 0 0
\(508\) 17000.0 29444.9i 1.48475 2.57166i
\(509\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −24475.0 −2.11260
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 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
\(523\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −37600.0 −3.11680
\(527\) 0 0
\(528\) 0 0
\(529\) 5283.50 9151.29i 0.434248 0.752140i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 16650.0 28838.6i 1.34174 2.32395i
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −7939.00 13750.8i −0.630914 1.09277i −0.987365 0.158461i \(-0.949347\pi\)
0.356452 0.934314i \(-0.383986\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 12980.0 1.01460 0.507299 0.861770i \(-0.330644\pi\)
0.507299 + 0.861770i \(0.330644\pi\)
\(548\) 26435.0 + 45786.8i 2.06067 + 3.56919i
\(549\) 0 0
\(550\) 21250.0 36806.1i 1.64746 2.85348i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 36550.0 2.80300
\(555\) 0 0
\(556\) 0 0
\(557\) 10235.0 17727.5i 0.778583 1.34855i −0.154175 0.988044i \(-0.549272\pi\)
0.932758 0.360502i \(-0.117395\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 10855.0 + 18801.4i 0.814752 + 1.41119i
\(563\) 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
\(567\) 0 0
\(568\) −30960.0 −2.28706
\(569\) −13453.0 23301.3i −0.991176 1.71677i −0.610382 0.792107i \(-0.708984\pi\)
−0.380794 0.924660i \(-0.624349\pi\)
\(570\) 0 0
\(571\) 3394.00 5878.58i 0.248747 0.430842i −0.714431 0.699705i \(-0.753315\pi\)
0.963178 + 0.268863i \(0.0866479\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −5000.00 −0.362634
\(576\) 0 0
\(577\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(578\) 12282.5 21273.9i 0.883883 1.53093i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 20060.0 + 34744.9i 1.42504 + 2.46825i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −20025.0 + 34684.3i −1.39024 + 2.40797i
\(593\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −13838.0 −0.951051
\(597\) 0 0
\(598\) 0 0
\(599\) −12368.0 + 21422.0i −0.843644 + 1.46123i 0.0431495 + 0.999069i \(0.486261\pi\)
−0.886794 + 0.462166i \(0.847073\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 25092.0 + 43460.6i 1.69036 + 2.92779i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −7505.00 + 12999.0i −0.494493 + 0.856487i −0.999980 0.00634752i \(-0.997980\pi\)
0.505487 + 0.862834i \(0.331313\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −30550.0 −1.99335 −0.996675 0.0814823i \(-0.974035\pi\)
−0.996675 + 0.0814823i \(0.974035\pi\)
\(618\) 0 0
\(619\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −7812.50 13531.6i −0.500000 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −26192.0 −1.65244 −0.826218 0.563351i \(-0.809512\pi\)
−0.826218 + 0.563351i \(0.809512\pi\)
\(632\) 31140.0 + 53936.1i 1.95994 + 3.39472i
\(633\) 0 0
\(634\) −17425.0 + 30181.0i −1.09154 + 1.89060i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 56440.0 3.50232
\(639\) 0 0
\(640\) 0 0
\(641\) 4439.00 7688.57i 0.273526 0.473760i −0.696236 0.717813i \(-0.745143\pi\)
0.969762 + 0.244052i \(0.0784768\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 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
\(651\) 0 0
\(652\) 30260.0 1.81760
\(653\) 13525.0 + 23426.0i 0.810527 + 1.40387i 0.912496 + 0.409086i \(0.134153\pi\)
−0.101969 + 0.994788i \(0.532514\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1804.00 0.106637 0.0533186 0.998578i \(-0.483020\pi\)
0.0533186 + 0.998578i \(0.483020\pi\)
\(660\) 0 0
\(661\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(662\) −27270.0 + 47233.0i −1.60103 + 2.77306i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −3320.00 5750.41i −0.192730 0.333818i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −33570.0 −1.92278 −0.961388 0.275196i \(-0.911257\pi\)
−0.961388 + 0.275196i \(0.911257\pi\)
\(674\) 8325.00 + 14419.3i 0.475767 + 0.824053i
\(675\) 0 0
\(676\) 18674.5 32345.2i 1.06250 1.84030i
\(677\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −17030.0 + 29496.8i −0.954077 + 1.65251i −0.217612 + 0.976035i \(0.569827\pi\)
−0.736466 + 0.676475i \(0.763507\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 8010.00 + 13873.7i 0.443864 + 0.768795i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 20500.0 1.12128
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 4198.00 0.226186 0.113093 0.993584i \(-0.463924\pi\)
0.113093 + 0.993584i \(0.463924\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 9758.00 16901.4i 0.522398 0.904821i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −6273.00 10865.2i −0.332281 0.575528i 0.650677 0.759354i \(-0.274485\pi\)
−0.982959 + 0.183826i \(0.941152\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −17714.0 30681.5i −0.924586 1.60143i
\(717\) 0 0
\(718\) −20260.0 + 35091.3i −1.05306 + 1.82395i
\(719\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −34295.0 −1.76777
\(723\) 0 0
\(724\) 0 0
\(725\) 10375.0 17970.0i 0.531473 0.920538i
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 3400.00 0.170279
\(737\) 25160.0 + 43578.4i 1.25750 + 2.17806i
\(738\) 0 0
\(739\) 12662.0 21931.2i 0.630283 1.09168i −0.357211 0.934024i \(-0.616272\pi\)
0.987494 0.157658i \(-0.0503945\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −25160.0 −1.24230 −0.621151 0.783691i \(-0.713335\pi\)
−0.621151 + 0.783691i \(0.713335\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 34925.0 60491.9i 1.71407 2.96885i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 1224.00 + 2120.03i 0.0594732 + 0.103011i 0.894229 0.447610i \(-0.147725\pi\)
−0.834756 + 0.550620i \(0.814391\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −34830.0 −1.67228 −0.836141 0.548514i \(-0.815194\pi\)
−0.836141 + 0.548514i \(0.815194\pi\)
\(758\) −29790.0 51597.8i −1.42747 2.47245i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 69224.0 3.27806
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 39015.0 + 67576.0i 1.81889 + 3.15040i
\(773\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 52630.0 2.42529
\(779\) 0 0
\(780\) 0 0
\(781\) 23392.0 40516.1i 1.07174 1.85631i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(788\) −18785.0 + 32536.6i −0.849223 + 1.47090i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 5312.50 + 9201.52i 0.234782 + 0.406654i
\(801\) 0 0
\(802\) 3995.00 6919.54i 0.175896 0.304660i
\(803\) 0